a
Fermi National Accelerator
Laboratory FERMILAB-Conf-90/164-T August
QCD
and Collider
Physics
14, 1990
’
R. K. Ellis Fermi
National
P. 0. Box
Accelerator
500, Batavia,
Laboratory
Illinois
60510,
USA
and
W. J. Stirling Departments University
An introduction
of Physics of Durham,
to perturbative
and Mathematical Durham
Quantum
DH13LE,
Cbromodynamics
Sciences England
and collider
physics
is
giVCIl.
‘Bared on lecture Operafad
by Unlverritles
given at the CERN and CERN-JINX Aerearch
Association
Inc. under
contract
SchooL of Physics. with the United
States Department
of Energy
-l-
FERMILAB-Conf-90/164-T
Contents 1. Fundamentals
of Perturbative
1.1
Lagrangian
1.2
Feynman
1.3
The running
1.4
The beta function
2. QCD
of QCD
QCD
2 2
............................
rules ............................... coupling
4 ......................
constant
and the A parameter
5 ............
in QCD
8
16
in e+e- + Hadrons
2.2
Jet cross sections
.............................
23
2.3
Colour coherence
.............................
25
Inelastic
for e+e-
Scattering
Deep inelastic
3.2
Scaling
3.3
QCD
fits to deep inelastic
3.4
Small
z behaviour
QCD
Parton
4. The 4.1
The
4.2
Factorisation
4.3
Parton
5. Large
QCD
scattering
Two-jet
5.3
Comparison
5.4
Multijet
in Hadron-Hadron
cross sections
production
..........
model
.............
49
Collisions
51 51
....................
53
......................
........................... .......................
............................
36
....................
in Hadron-Hadron
experiment
31
45
55
............................
and jet definition.
with
equations
distributions
of the cross section
pi Jet Production
5.2
model .............
.....................
data
parton
luminosities
Kinematics
and the parton
of the parton
Model
31
Distributions
and the Altarelli-Parisi
improved
5.1
hadrons
and Parton
3.1
violations
+
16
The total
3. Deep
cross section
...............
2.1
Collisions
00 60 61 65 68
-2-
6.
7.
photon
production
FERMILAB-Conf-90/164-T
5.5
Direct
. . . . . . . . . . . . .
The
Production
6.1
The Drell-Yan
6.2
W and 2 production
6.3
W and 2 decay
properties
.....................
6.4
Lepton
distribution
in W and 2 decay
6.5
W and 2 transverse
momentum
6.6
Multijet
with
The
Production
7.1
The decays
of heavy
quarks
7.2
The theory
of heavy
quark
7.3
Higher
7.4
Results
7.5
The search
7.6
Heavy
of Vector
Bosons
mechanism
angular
production
corrections
on the production
quark
.
76
. .
79
. .
82
..........
.
86
..........
. .
87
. .
91
.
................
94
Quarks . . . . . . . . . . . . . . production
to heavy of charm
for the top quark in jets
distributions
72
76
.....................
W and 2
. .
Collisions
........................
of Heavy
order
in Hadronic
. . . . . .
.
quark
. . . .
production
and bottom . .
. . . . . . . . . .
.
.
. .
94
. .
. .
96
.
101
. .
. .
107
. .
. .
111
.
. .
117
. quarks
.
. .
.
.
. . . . . . .
-3-
1.
Fundamentals In this
tigate
of Perturbative
set of lectures
the behaviour
are applicable
we shall
of strong
because
calculated,
and how the results
QED,
describe
of the property
lecture.
Subsequent
treatment
QCD the use of perturbative
interactions
in the first
The
FERMILAB-Conf-90/164-T
distances.
of asymptotic
lectures
describe
how short
QCD
of perturbative
which
QED
will
distance
compare
is developed
to inves-
Perturbative
freedom
of these calculations
of perturb&m
and an understanding
at short
methods
is therefore
be described
cross sections
with
in analogy
methods
are
experiment. with
perturbative
a prerequisite
for this
colnse.
1.1
Lagrangian
We begin which
with
of QCD
a brief
description
can be derived
introduce
notation
tum
theory.
field
Introductions Just
from
an effective
the reader
Lagrangian
rules
and the Feynman
which
which
rules than
of quantexts
[1,2].
in refs.[3,4,5,6,7]. calculation
the interactions analysis
is given
more
structure
to the standard
the perturbative
describing
does little
to the elegant
is referred
for a perturbstive
density
guide
can be found
Electrodynamics,
rules required
Lagrangian
does not do justice
QCD
the use of Feynman
QCD
is a practical
For more details,
to perturbative
The Feynman
This
and certainly
as in Quantum
requires
it.
of the
of any process
of quarks
of QCD
and gluons.
can be derived
from
by
c G.(i~ - m),sqa +Ig.“se-f&jng +L&& (1.1) L=-iFGFiP +&*our, This
Lagrangian
massless
spin-l
density
describes
gluons.
F$
the interaction
is the field
strength
of spin-l/2 tensor
quarks
derived
of mass m and
from
the glum
field
AA-3 F$ and the indices
A,B,C
field.
third
It is the
distinguishes and ultimately
QCD
= &,d; [ run
over
the eight
‘non-Abelian’
from
QED,
to the property
aad,”
giving
term
-gfABCd:d; colour on the
rise to triplet
of asymptotic
1
degrees
(1.2)
of freedom
right-hand-side and quartic
freedom.
of the glum
of Eq.(1.2) glum
which
self-interactions
-4-
The
sum over the flavours
coupling
constant
quanta,
and fABC
group.
The
(a = 1,2,3) covariant
over the nf
determines
fields
qa are in the
derivative
flavours
of quarks,
of the interaction
(A, B, C = 1, . . . . 8) are the structure
quark
=
different
the strength
and D is the covariant
(D&,
where
which
runs
FERMILAB-Conf-90/164-T
triplet
Acting
between
constants
coloured
of the SU(3)
representation
derivative.
g is the
of the
on triplet
colour
colour
and octet
group,
fields
the
takes the form
8&b
+ ig
(&)AB
(tcd:)ab,
t and T are matrices
=
in the fundamental
arn6AB
+ iS(T’d:)AB,
and adjoint
(1.3)
representations
of SU(3)
respectively: [tA,tB] b in Eq.(l.l) have
been
with
metric
normalisation
= ifABCtC,
[TA,TB]
is a symbolic suppressed. given
notation
Otherwise
for r,,D’
matrices
this
choice
the notation
is chosen
We cannot gauge fixing making
matrices
obey
fABCfABD=
perturbation
term.
It is impossible The
theory
with
to define
theory ghost
the class of covariant such as QCD Lagrangian,
this
which
DreII
[l] the
(1.5)
the following
relations,
CA scD,
(1.6)
CA = N = 3.
the Lagrangian the propagator
of Eq.(l.l) for the gluon
(1.7) without
the
field without
choice,
C 8mn.e-firing = -A fixes
and
(N=3)
-
of gauge.
of Bjorken
of rP and q.
4,
perform
a choice
indices
(1.4)
;.
tA tA .b c bc TrTCTD
-ifABC.
to be,
= TR cYAB, TR=
the SU( N) colour
=
and set h. = c = 1. By convention
by 9-D = diag(l,-1,-1,-l)
of the SU(N)
(TA)BC
and the spinor
we follow
Tr tAtB With
= ifABCTC,
gauges
and X is the gauge
covariant is given
(1.8)
(-k2)‘,
gauge-fixing
term
parameter. must
In a non-Abelian
be supplemented
by a
by
c ghost = ad + (DZB~B).
(1.9)
-5-
Here
qA 1s ’ a complex
the form
scalar
of the ghost
field
Lagrangian
due to Fadeev
degrees
which
explanation
1.2
Feynman
Eqs.(l.l),
(1.8)
density. mally contains
perturbation
[9]. The ghost propagate
the action
to derive theory
operator
fields
the reader
derivation
of
formalism
[a]
cancel
in covariant
all the terms
in a covariant
unphysical
gauges.
is referred
bilinear
lagrangian
in the fields,
rules
gauge.
d4t rather into
For an
to ref. [lo].
than
which
The
should
Feynman
from
be rules
the Lagrangian
a free piece
and an interaction
&,
which
piece,
nor-
~21, which
all the rest:
‘PI = i
d%C:o(r),
The practical
recipe
derived
-@o, whereas
from
are treated
This
recipe
the following
to determine
a theory
contains
only
which
bilinear
are included
the Feynman
minus
approaches contains
terms,
only
A for the I$ field
approach
as the interaction perturbation
is given
A=;+
K is regarded
Lagrangian,
theory
by inserting
($)K’($)+
the inverse
from
@,r =
parts
sign) can be understood
a complex
propagator
is
of the theory
@I.
scalar
[ll]
by considering
of a theory.
For simplicity,
field
+ = 6 (K + K’) 4. In the first
A= In the second
rules is that
to the quantisation
in the free Lagrangian,
the propagator
(1.10)
rules for the interacting
are derived
the extra
two different
consider
the Feynman
as perturbations
(including
d’rC:r(z). I
J
K'
The integral
the Feynman
ip = i JL
the effective
@po= i
which
statistics.
by the path
by ghost fields,
are sufficient
We can separate contains
otherwise
role played
and (1.9)
from
Fermi
rules
used in weak coupling are defined
and Popov
would
of the physical
obeys
is best provided
and the procedures of freedom
which
FERMILAB-Conf-90/164-T
4 and an action
which
approach,
both
Using
the above
@o = @ (K + K') 4.
K and rule
by
-’
K+K" as the free Lagrangian, &K'qb.
the interaction (s)K’($)K’($)+...=
Now term
Qo = &K&
‘PI is included an infinite
and K'
to all orders
number KylK,
in
of times: (1.12)
-6-
Note
that
with
the choice
is the same in both Using obtain
quark
propagator
field
approaches,
and
gluon
in momentum
for an incoming depends
on a single
Similarly
momentum
have no inverse.
is added
= i&AB
to check that
1.3
The
In order
$B,
,p&P)
A(“,
dimensionless assumption masses.
because This
that
therefore
true
function
a-
ie prescription
in exactly
(1 -
-$+,
=
of the quark
field
for the
the same way as in
is found
to be
1
;)P,PO
(1.14)
term
this function
A is as given
would
in Table
1:
(=C, SY)(~) = s,“~;
+
(I
momenta,
-
(1.15)
1
#F
Eqs.(l.l),
(1.6)
(1.16)
and (1.9)
can
1.
constant of the running tl which
bigger
than
coupling,
consider
depends
on a single
all other
dimensionful
zero mass limit.)
large scale, R should
is not however
-
set the masses to zero.
R has a sensible
there is a single
result
observable
the scale Q is much
We shall
assumption
coupling
physical
fermion
the identification
1. The
propagator
the appropriate
the concept
the inverse
(1.13)
the gauge fixing
= s,C;
all the rules in Table
to introduce
in Eq.( 1.1) one can readily
to be
in Table
$‘%a0
for the gluon
running
of the recipe.
by making
of the gluon
result
with
consistency
example,
causality,
without
(BC,‘&)
be used to derive
given
$B, ,P)(P)
derivatives
of the 4 field
space the two point
to preserve
A@)
Replacing
propagator
= 46 ab (j - m),
propagator
The
for
p. It is found
the inverse
It is straightforward
given
Thus,
In momentum
of the propagator
pole of the propagator QED[l].
Lagrangian
space can be obtained
field.
is the inverse
the full
the internal
propagators.
rSJ(p) which
above
demonstrating
the free piece to of the QCD
the
-ip”
of signs described
FERMILAB-Conf-90/164-T
(This Naive
energy
scaling
quantum
scale Q.
parameters
step requires
have a constant
in a renormalisable
as an example
By
such as
the additional
would
suggest
value independent field
a
theory.
When
that of Q. we
-7-
FERMILAB-Conf-90/164-T
1
sa= -p+(1~.A)ppps i pa + i& p’ 1 A ----c---
P
B
6A= : pa + ie
a,i
P
b,j
6”b
+ ic
(i-Ltis)ji
P P A -‘La
B,P T c,7
-gfA=C
[F
(P -
9)’
+ g@-r (n -
(aU momenta
7)”
+ gyp (r - Pq
incoming)
B,P
-%a
-;g2fXACfXBD _igZfXADfXBC -igafxABfxcD
x CT-7
b47,s
- SPbSST)
hv776
-
!Av%S)
(g=).%6
-
SP6SS-r)
D76
‘%a B’
k /
lq ‘c
Table
gf ABcqa
1: Feynman
rules for QCD
in a covariant
gauge
-a-
calculate with
R as a perturbation
the fine structure
isation
to remove
introduces
on the choice
However of the
as R cannot depend
Therefore,
a choice
if we hold
on the choice
on the ratio
dependence
parameter.
though
depend
It follows
Q’,/p’
this
the
subtractions
which
- R depends
in general
point
coupling
as
p.
The Lagrangian
of QCD
of p is required the bare coupling
to define fixed,
makes no mention the
physical
theory
coupling
at the
quantities
R is dimensionless,
such
it can only
QS. Mathematically,
the p
by
acllaa, this equation
re-
on the ratio
the renormalised
+ p2~~
To rewrite
renormalprocedure
made for p. Since
of R may be quantified
in analogy
renormalisation
also that
and the renormalised
(defined
series requires
at which
are performed
constant.
,u is an arbitrary
level.
Because
made for the subtraction
scale p, even
quantum
divergences.
divergences
QS = g2/4n,
the perturbation
mass scale p - the point
and is not therefore
depends
of QED),
ultra-violet
move the ultra-violet Q/p
series in the coupling
constant
a second
FERMILAB-Conf-90/164-T
in a more compact
form
1
we introduce
R=
0.
(1.17)
the notations
= $2, t =l=(s), P(cQ-) and rewrite
Eq.(1.17)
as
1
(1.19)
R = 0.
This
first
function
order
partial
- the running
differential coupling t=
I
equation as(Q)
- as follows:
4Q)
dz
as
By differentiating
and hence that
is solved
by implicitly
defining
(1.20)
%(PL) = as.
PO’
a new
Eq.( 1.20) we can show that
R(l,
““;I”’
= p(crs(Q)),
as(Q))
is a solution
all of the scale dependence
in R enters
“-(;’
=
of Eq.(1.19). through
‘;$)‘.
(1.21)
The above
the running
analysis
of the coupling
shows that constant
-9-
as(Q).
It follows
that
perturbation
theory,
Eq.( 1.20).
In the next
ory. This large
1.4
allows
means that
Q, therefore,
The
equation.
us to predict
section, as(Q)
of the quantity
we shall
becomes
we can always
beta
The running
knowledge
In &CD,
the variation show that
smaller
constant
the p function
nf
is the number
sometimes
of active
using
in fixed
Q if we can solve
is an asymptotically
free the-
For sufficiently
perturbation
A parameter
theory.
in QCD
by the renormalisation
group
expansion
t b’as + O(a;))
light
b’ =
053 - 19nt) 2~(33 - 2nf)
flavours.
(1.22)
’
An alternative
notation
which
is
A (z)(n+l)
p,, = 4xb = 11 - ;nf, The p function
coefficients
p1 = 16r’bb’
can be extracted
to the bare vertices
of the theory,
of the non-Abelian
interactions
from
as in QED. in QCD.
= 102 - 23nf,
the higher
order
(loop)
Here we see for the first
In QED
(with
(1.23)
...
one fermion
time
corrections the effect
flavour)
the p
is 1 &ED(a)
and thus the b coefficients From
Eq.(1.21)
as(p)
in QED
=
%a2
and QCD
+
(1.24)
.. .
have the opposite
sign.
we may write, aas at
If both
order
used is
P(as) = --as *$
function
QCD
has the perturbative
b = (33 - 2nt 1, 12n where
of R with
a.s is determined
= -b&l
calculated
as the scale Q increases.
and the
P(w)
R(l,as),
solve Eq.(1.20)
function
of the coupling
FERMILAB-Conf-90/164-T
and as(Q)
series on the right-hand-side
=
-b&Q)
[1+
b’as(Q)
are in the perturbative
f C+:(Q))].
region
and solve the resulting
it makes sense to truncate differential
equation
the
for as(Q).
-lO-
For example,
neglecting
the b’ and higher
R-(Q) = This
gives
region. zero.
the relation
Evidently This
between
decreases
crucial.
the opposite
With
ffs(c1)
very
and
large,
a&),
power
gives the solution
(1.26)
if both
the running
freedom.
like an inverse
in Eq.(1.25)
t = I*($).
w(p)bt’
of asymptotic
since as only
coefficients
us(Q)
as t becomes
is the property
t
1
FERMILAB-Conf-90/164-T
are in the
coupling
as(Q)
The approach
of 1ogQ’.
sign of b the coupling
decreases
to zero is rather
Notice
would
perturbative
that
increase
to slow
the sign of b is
at large
Qa, as it
the next-to-leading
order
does in QED. It is relatively coefficient
b’ yields
practice,
-(Q)
1
w(Q) Note that
to show
+ “ln
4P)
( (1 + b’as(Q)
this is now an implicit given values
to any desired
that
including
the solution
1
--~
equation
for these parameters,
) - b’ln( for as(Q) as(Q)
(1+
as(pL) ) = bt.
(1.27)
b’as(p)
as a function can easily
oft
and as(‘).
be obtained
In
numerically
accuracy.
Returning which
straightforward
to the physical
the renormalisation
quantity
group
R, we can now demonstrate
resums.
Assume
that
the type
in perturbation
of terms
theory
R has
the expansion R = as + . . . where
. . represents
the special using
choice
terms
of order
of R given
(1.28)
cxi and higher.
by Eq.(1.28)
The
solution
R(l,crs(Q))
- can be re-expressed
in terms
- for of a&)
Eq.(1.26):
R(l,as(Q))
= as(p)
z
(-l)j(w(r)bt)j
= as(p)
[l-
as(p)bt
+ &p)(bt)’
+
.] (1.29)
Thus
order
by order
in perturbation
are automatically
resummed
R - represented
by the dots
logarithms be discussed
per power
by using
chapter.
there
the running
in Eq.(1.28)
of (1s. An explicit
in the next
theory
- when example
are logarithms
of Q*//L”
coupling.
Higher
expanded
give
of how this
order
terms
works
which terms
with
in practice
in
fewer will
FERMILAB-Conf-90/164-T
-ll-
Perturbative
QCD
the absolute
value
can choose
as ‘the’
tells
itself.
The
at a convenient
domain,
Ma for example.
this parameter
from
of the theory
scale which
is large enough
approach
for specifying
a dimensionful
varies
has to be obtained
An alternative standard
constant
parameter
reference
and is now the de facto
convention
latter
fundamental
constant
- is to introduce
us how the coupling
parameter
is called
A and is a constant
experiment.
Thus
we
of the coupling
to be in the perturbative
the strength into
the scale, not
the value
- which
directly
with
was adopted
historically
of the strong
interaction
the definition
of as(Q).
of integration
defined
By
by (1.30)
In effect,
A represents
the scale at which
arbitrariness
of the integration
Ax
in Eq.(1.30)
constant The
introduction
In leading perform
still
constant
(LO),
the integral
i.e.
is reflected
gives a solution
of A alkws
order
us to write
retaining
in Eq.(1.30)
only
that
for large
of A is extended
Q this
as(Q)
becomes
in the fact
that
to the differential the correct
The
replacing
equation
asymptotic
the b coefficient
strong.
A by
for as(Q).
solution
for as.
in the p function,
we can
to obtain
Q-(Q) = Note
the coupling
solution
1
agrees with
to next-to-leading
order
(1.31)
bln(Qa/A’)’ Eq.(1.26)
(NLO)
as it must.
by including
The
definition
also the 6’ coefficient
in
the integral: 1
“=dQ) ) = bin($). ( 1 + b’czs(Q)
-as(Q) + “l* Again,
this
ternatively, powers
allows
a numerical
we can obtain
determination
an approximate
as(Q) that
this
A to Eq.(1.32).
The
true expansion
contain
of order
a term
can be used
1-
of Eq.(1.32)
expression
corresponds of as(Q)
constant/log2.
to remove
El*l*(Q2/A2)
+ “’
b ln(Q*/Aa)
= bln(Q”/h*)
however,
constant
solution
for a given
value
in terms
of A.
Al-
of inverse
of log(Q’/A’): 1
Note,
of as(Q)
(1.32)
this
to a slightly
in inverse
However term.
1 ’
different
definition
of
powers
of log(Q’/A”)
would
the freedom
to multiply
A by a
Specifically,
if we call
Eqs.(1.32)
and
-12-
Table (1.33)
definitions
are related
2: as(Q)
for Q = 5 GeV and A = 200 MeV
1 and 2 respectively,
then
be clear from
fundamental unwary.
the above
parameter
First,
multiplying induced
of QCD
A can be defined
A by a constant in as(Q)
are one order
phenomenology
Eq.(1.33)
can be used to define Since in practice
important
when
comparing
A from
using different
conventions
flavours.
Values
the two A’s
of traps
or next-to-leading
an equally
acceptable
as which
- although
and in each case The
Either
between
small - can be comparable
alI preciEq.( 1.32) or
are used in the
experimentally,
the same equation
Differences
the
differences
Nowadays,
definitions
is measured
to check that
constant.
order
order.
case, and both
it is usually
can ensnare
theory.
at next-to-leading
A as the
which
definition.
in perturbation
A in this
A values
(1.34)
the use of the parameter
a number
to leading
higher
(nr = 5).
that
with
the above definitions
of A for different
in Table
and (1.33)
by the conditions both
a solution
From
Eq.(1.33)
that
2 where
respectively.
is that
numbers
of as at the scale p = m, where
is illustrated
Eqs(l.31)
of as(Q)
it is
has been used
the results
obtained
to present-day
mea-
errors.
the continuity This
presents
the coupling
A second difficulty active
discussion
is performed
literature.
to determine
m l.l48Ar,
gives
sion QCD
surement
for the same value
by Ar = (;)“A1
It will
FERMILAB-Conf-90/164-T
of flavours
for sll values
of the renormalisation
correct
are defined
couplings
matching
of the momenta group
on the number
the coupling
equation
quark
are calculated
prescription
of
by imposing
m is the mass of the heavy
the LO and NLO The
A depends
(121. using
is determined constant
and also a continuous
must
be
function.
for p > ms we have,
os(” For m, < p < ms, the coupling
5, = b(5) ln(~~/A(5)r) evolves
with
[l-...I.
four active
flavours,
and the correct
form
-13-
500
, I I -
I I I /
I
Comparison
of
matching
at
I I I
I I I I
Lambda
mb=5
FERMILAB-Conf-90/164-T
for
I I I I
4 and
I I
I I
5 flavours
GeV.
400
100
7
I I III 100
Figure
200
1: Comparison
300 A(4)
400
WV1
of A for 4 and 5 light
I I
quark
500
flavours,
with
600
matching
at ms = 5
GeV. to use is 1 4% where
the square
continuity
= ““l;“l(~i”“’
bracket
is the same as in Eq.(1.35).
the next-to-leading
order
A(4)
Fig.(l)
(1.36)
The
constant
is fixed
by the
condition, as(ms,
Using
+ constant
4)
illustrates
the relation
form
for as(Q)
x A(5)(%)’
between
(1.37)
4) = (LS(mt., 5). one can show then
that
[In($$)]r?
A(4)
and A(5)
(1.38)
graphically.
In summary,
it is
-14-
important
when
flavours
comparing
assumed
is illustrated
Consider
and also whether
in Table
The third
different
FERMILAB-Conf-90/164-T
A values
to establish
the LO or NLO
the number
expressions
of light
quark
have been used.
This
2.
troubling
property
two calculations
of A is that it depends
of the renormalised
on the renormalisation
coupling
constant
which
scheme.
start
from
the
bare parameters.
same
0; = .PaO, 0; = ZWaO, The
two
schemes
renormalisation theory.
start
from
constants
Therefore
the same bare
ZA and 2’
coupling
must
the two renormalised
(1.39) a:.
The
infinite
be the same in all orders
coupling
constants
must
parts
of the
of perturbation
be related
by a finite
renormalisation: 0; Note that
the first
transformation. Eq.(1.30)
= c&l
two coefficients
They
of the /3 function,
are therefore
we see that
independent
the two values
b and b’, are unchanged
of the renormalisation
of A are related
equality
follows
be true for all values always
determined
Nowadays,
most
in the modified violet
from
taking
of Q*. Therefore
the limit
by the one loop calculation
calculations minimal
loop divergences
in fixed
subtraction are regulated
Q -+ 00, because
relations
order
between which
different
fixes
QCD
perturbation
renormalisation
scheme.
by reducing
to n < 4:
scheme.
From
the relation
must
Cl = zb’
ba+(l+...) last
by such a
by,
dx
The
(1.40)
+ crcz; + . . .).
the number
definitions
of A are
cr:
theory In this
are performed approach,
of space-time
dimensions
d’-l’k (1.43)
(2x)4-” where
c = 2 - y. Note
the couplings
ultra-
that
and the fields.
the renormalisation Loop
integrals
scale p preserves of the form
d”k/[P
the dimensions t ma]’
then
of
lead to
-15-
poles at E = 0. The
minimal
subtraction
off these poles and to replace In practice
the poles
always
ditional
constant)
constants
ples of schemes Eqs(l.40)
appear
t
A leads
errors
measure in Fig.(2)
but analysis
coupling
a.+).
(1.44)
-YE,
minimal
off as well.
These
above,
subtraction
two schemes
scheme
these
are therefore
and it is straightforward
ad-
exam-
to show using
that
the expression
mathematically actually
ln(4n)
A and B introduced
to an error
is to subtract
by the renormalised
modified
Ab Lastly,
prescription
in the combination
and in the
are subtracted
and (1.42)
renormalisation
the bare coupling
i
(7~ is Euler’s
FERMILAB-Conf-90/164-T
which
= ,&e@=‘(4rhE)~
(1.45)
of the experimentally
measured
coupling
is both
magnified
and asymmetric.
correct
but
us.
A partial
exponentially
depressing
are too large
of jet data in e+e-
for an experimenter
compilation
of measurements
to conclude
that
annihilation
since
most
is shown
as has a logarithmic
demonstrates
(XS in terms
that
of
This
is
experiments
in Fig.(2). fall-off
The
with
as does decrease
p, with
scale (see later). Guided lectures
by Fig.(2),
we shall
for the phenomenalogical
corresponds
to about
0.12. Lack of knowledge of the size of QCD cross sections
made
in the following
assume 100 MeV
This
predictions
which
< A&5)
a 20% uncertainty of as directly
cross sections. begin
in order
< 250 MeV.
at the mass of the 2: 0.10 < as(Mx)
translates
Thus
(1.46)
into an uncertainty
we should
CYSof about
expect
20%.
errors
<
in the prediction in the prediction
of
-16-
FERMILAB-Conf-90/164-T
.25
.2
.15
.1
------A
t .05
I
I I ,,,,I
3
I
2: Measurements
I I ,,,,I
10
I 100
p
Figure
I
_I
of as compared
[Gev”]”
with
predictions
for various
values
of A(5).
-17-
2.
QCD Many
tion.
in e+e- -+ Hadrons
of the basic ideas and properties
considering
the process
e+e-
+
We show how the order
scheme dependence
enters
one of the most precise experimental
experiment.
at order
The property
of the total
final
QCD
can be illustrated
by discussing
are calculated,
the total
by
cross sec-
and how renormalisation
hadronic
of the strong
also predicts
total
e+e-
of colour
hadronic
cross section
coupling,
du dcos9
=
predictions
the high
also provides
and we quote
the latest
g
+ cos
compare
is R’+‘-,
production efe-
with
the ratio
cross section.
--t ff
is mediated
the centre-of-mass
cross section
+-4QfAeAds)
QCD
pair
the process
with
by either
scattering
f a light a virtual
angle of the
is:
cos’~)(Q; - 2Q/KVfxl(s)
(1+ 1
hadrons.
hadrons
of perturbative
Denoting
state
is also discussed.
2 -B 2 process
order
by 8, the differential
for the final
and how the predictions
to the muon
energy
f # e. In lowest
structure
for e+e- +
cross section
or a 2” in the s-channel. state pair
‘jet’
coherence
section
cleanest
by considering fermion,
a rich
can be defined,
cross
One of the theoretically
photon
We begin
czi. The total
measurements
QCD
The
charged
hadrons.
cxs corrections
We show how jet cross sections
We begin
of perturbative
results.
Perturbative
2.1
FERMILAB-Conf-90/164-T
t (A:
+ 8A.V.AfVfx,(s))
+ V:)(A;
+ v&(a))
1
(2.1)
where
Xl(S)
=
fi
s(s - M;) (s - A!fp + rl,lvr;
x2(s) = 2 K = (fi;-ayfy and (Vf, Af)
are the vector
in Eq.(6.11).
The ~2 term
and axial
couplings
of the fermions
(2.2) to the Z given
comes from the square of the Z-exchange
amplitude
explicitly and the
-1%
x1 term ,/i
from
the photon-2
far below
the 2 peak,
that
the weak effects
- are small
interference. the ratio
- manifest
Now
s/M;
at centre-of-mass
is small
in the terms
and can be neglected.
FERMILAB-Conf-90/164-T
Eq.(2.1)
and so 1 >> x1 >
involving then
scattering
the vector
reduces
x2.
energies
This
and axial
means
couplings
to
do -= dcosB
(2.3)
The Mandelstam
variables
setting
gives the total
Qr = -1
where
fi
When
are denoted
energy.
an electron
and r~ positron
annihilate
Although
the formation
theory
calculated
perturbative
using
to give an accurate
form
antiquark. l/Q,
a photon
cross section
methods.
can be understood
of the total
of virtuality
the quarks
principle
a later
time
strong
interactions.
modify
characterised
the outgoing
interactions
state,
one expect
hadronic
production
this fluctuation
occurs
but they
be predicted
themselves
which occur
where
into
can be theory
cross section? The into
The
electron
a quark
and
and an
in a space time volume by perturbation
hadrons.
quarks
too late
is not governed
perturbation
This
A is the typical
change
in the
of hadrons
in space-time. fluctuates
form
hadrons
state hadrons
which
by the scale l/A,
The
final
would
rate should
and gluons
can also produce
the event
over B and
+ p+p-:
for the production
Q = fi
and if Q is large the production
Subsequently
Why
by visualising
By the uncertainty
they
of the observed
the total
description
by S, t and u. Integrating
for e+e-
centre-of-mass
by perturbation
positron
cross section
is the total
final state.
answer
as usual
happens
at
mass scale of the
and gluons
to modify
theory.
into
hadrons
the probability
for an
event to happen. In leading obtained
order
by simply
perturbation summing
theory,
therefore,
the total
over all kinematically
hadronic
accessible
flavours
cross section
is
and colours
of
quarks: @PM
With
q = a, d, 8, C, b we obtain
value
is about
3.9.
Even
= mce+eu(e+e-
-+ nq) = 3xQ;. + p+p-) p
RQPM = 1113 = 3.67.
allowing
At 4
for the 2 contribution
= 34 GeV the measured (ARZ
N 0.05),
this
result
-19-
FERMILAB-Conf-90/164-T
b)
Figure
3: Feynman
section
in e+e-
diagrams
corrections
for the O(as)
to the total
hadronic
cross
annihilation
is some 5% higher
than
the lowest QCD
order
is due to higher
order
corrections,
and experiment
gives one of the most
prediction.
It turns
and in fact
the comparison
precise
determinations
out that
the difference between
of the strong
theory coupling
constant. The
O(as)
corrections
real and virtual shown
in Fig.(3b)
gluon
to the total
diagrams
it is convenient
diPs
shown
hadronic in Fig.(3).
to write
cross section
For the real gluon
the three-body
=
1 6p, d3p, d3k ---6’(q (2~)~ 2E1 2Ez 2&
=
&d&dcos
are calculated
B,d&adrldzz
-ply-
emission
from
the
diagrams
phase space integration
as
p2 - k)
(2.6)
-2o-
where
&,6’1,&a
are Euler
fractions
of the final
a matrix
element
cross section
angles,
state quark
which
that
and 2, = 2E,lJ;; and antiquark.
depends
only
the integration the integrals
a0 3 -&Q”
region
Integrating
space where
gluon
is soft, E, +
/dzldzl
2
the gluon
is collinear
0. Evidently
with
we require
are suitable.
One can give the gluon
manifest
of space-time
- before
off-mass-shell
by a small
amount.
procedure
is to use dimensional
cast in n dimensions,
now R > 4. With
of how the calculation
is that
the cross section q-
proceeds
of Eq.(2.7)
LIZ go 3 T”‘*
0,
we see regions
of
-P 0, or where
mass, or take
the
procedure
-
A variety
of
the final
state
In each case the singularities
mass. regular&&on,
the three-body
the soft and collinear
Details
from
come
can be completed.
a small
of the regulating
dimensions
quark,
the calculation
as logarithms
A more elegant
The
to the total
some sort of regularisation
methods
H(e)
angles gives
21)
singularities
either
finite
and antiquark
(1 -zl,‘,;f
at Zi = 1. These
the integrals
where
out the Euler
5 1, z1 + zz 1 1. Unfortunately,
to render
are then
are the energy
on z1 and zz and the contribution
is: 0 < +1,+,
are divergent
phase
quark
and z1 = 2Ez/J;j
is uqcg =
where
FERMILAB-Conf-90/164-T
the number
phase space integrals
singularities
can be found
with
appear
for example
now
as poles at n = 4. in ref.(4].
The result
becomes
H(e)
q 3a
[;
- z + y
+ O(e)],
= 1 + O(c).
virtual
gluon
contributions
lar fashion,
with
dimensional
divergences
in the loops.
shown
regularisation
The result
gnw = When
the two contributions
cancel
and the result
Ret’-
H(E)
Eqs.(2.8)
used to render
[-S
finite
in a simithe i&a-red
+ 5 - 8 + O(e)].
and (2.9) are added
in the limit =3
again
can be calculated
is
~03yQ;2
is finite
in Fig.(3a)
‘&Q”
together
the poles exactly
E -+ 0: {I+
:
+ O(a:)}.
(2.10)
-21-
Note that about
the next-to-leading
0.15, can accommodate
contrast,
diagrams
[13] and fined
hadronic
The
Lee,
O(ai)
will
Indeed
state,
divergences
are theorems
+ q$
with
= 34 GeV.
In
the real and virtual
- the Bloch,
- which
state
that
in the massless whereas
Nordsieck suitably
limit.
de-
The
total
the cross section
for
is not.
are also known.
to Ret’-
in the MS
on the renormalisation
there
for as of
gluon.
between
of such B quantity,
associated
renormalisation,
singularities
a value
at fi
for a scalar
be free of singularities
i.e. u(e+e-
and with
measurement
[14] theorems
is an example
corrections
is positive,
is negative
Nauenberg
quantities
qp final
ultra-violet After
is not accidental.
cross section
the exclusive
correction
of the soft and collinear
Kinoshita,
inclusive
correction
the experimental
the corresponding
The cancellation gluon
order
FERMILAB-Conf-90/164-T
At this
order
the renormalisation
scheme for example,
we encounter
of the strong
the O(czi)
the
coupling.
coefficient
depends
scale p:
R e+e- =
{I+++
3CQ;
[33;22nfIn$+?$
(2.11) and C(3) =
1.2021.
exactly
as specified
ln(p’/s)
is exactly
Specialising
that
by the
the p-dependence
renormalisation
blr, where
b is the
to the case of p = 4
Ret’-
What
Note
=
3 TQ’
{ 1+
order
an explicit
calculation
calculation
of the third
order coefficient
In general
the coefficients
made for the renormalisation exactly
compensates
predictions
terms
coefficient
is
the
coefficient
of
defined
in
Eq.(1.22).
in this perturbation
(2.12)
+ . . . }.
perturbation
all we can say is that
they
series?
will
Before
be of O(ai).
A
series has been performed
[16].
of any QCD perturb&iv= scale /.J. As p is varied,
of p. However
coefficient
becomes
in this
to be in error
order
i.e.
+ 1.411 (“s;fi))’
the change in the coupling
are independent
equation,
p function
“s’,fi)
performing
are now known
group
and nf = 5, Eq.(2.11)
can one say of the higher
[15], but the results
of the second
a.+)
expansion
depend
the change
in the coefficients
in such a way that
this p-independence
on the choice
breaks
the physical
down
whenever
-22-
Deviation A”’
QPM result
from
(two
FERMILAB-Conf-90/164-T
loop)
= 230
in QCD
MeV.
4-
3-“““““““““’ 0
40
20
80
60
100
P [GeVl Figure
4: The
denotes
quantity
the QCD
x =
prediction
[R(j)/RQPM - l] for Ret’-
truncated
the series is truncated.
One can show in fact
quantity
- which
such 88 Re+e-
as a function
of the s&e
/L, where
R(j)
at O(ai) that
changing
has been calculated
the scale in a physical
to O(cr”,)
- induces
changes
of
O( a;“). The
dependence
is shown definite
in Fig.4. prediction.
the ‘best’ orders
choice
predictions
particular.
of R’+‘-
on the scale p retaining
As expected,
the inclusion
In the absence of scale, equal.
In the fastest
defined
of higher
of higher order
the first
order
corrections,
as the scale which
In the literature, apparent
only
makes
two such choices
convergence
approach
terms
or second
terms
leads to a more
one can try the truncated
to guess and
aII-
have been advocated
in
[17], one chooses
the scale
-23-
FERMILAB-Conf-90/164-T
7.0 R 6.5 6.0
CESR.DORIS
5.5
*
PETRA
P
TRISTAN
5.0 4.5 4.0 3.5 3.0
20.0
10.0
30.0
40.0
50.0
60.0
v3 (Cd) Figure
5: Combined
QCD-electroweak
fit to Ii’+‘-,
from
reference[lg]
P = PFAC,wh=r= R(')(PFK) On the other
hand,
the principle
(2.13)
= R@$+~~). of minimal
sensitivity
[18] suggests
a scale choice
P = PPMS, wh=r=
P-$R%)I,,~,= 0. These there
two special
scales can be identified
are no theorems
say is that
the theoretical
the scale is simply Finally,
that
Fig.5
that
shows
error on a quantity
a recent
this
QCD
GeV)
to remember
are correct.
to O(a;)
All
is O(atfl).
that
one can Varying
uncertainty.
are displayed.
order
a~(34
calculated
fit [I91 to data on R’+“-
contributions
the second
It is important
any of these schemes
one way of quantifying
The weak and QCD scheme and using
prove
in Fig.4.
(2.14)
prediction,
The
over a broad
fitted
value
energy
range.
of as, in the MS
is
= 0.158 & 0.020
(2.15)
-24-
which
corresponds
to Am-
= 440 $320 -220
MS
2.2
Jet cross
The
expression
very concise, final
state.
(2.16)
MeV.
sections
given
for the total
hadronic
but it tells us nothing If the hadronic
momentum
FERMILAB-Conf-90/164-T
relative
about
fragments
to the
quark
cross section
the kinematic
distrihtion
of a fast moving momentum,
in the previous
quark
then
section
of hadrons
have limited
the lowest
be interpreted as the production ( e+e- -+ qq), can naively In this section we investigate how higher order perturbative
order
is
in the
transverse contribution,
of two back-to-back corrections
jets.
modify
this
picture. Consider previous
first
the
section,
next-to-leading
process
that
collinear This
the cross section
with
gluon tion
words
to (a) only the gluon
is required
to lowest
of the final
ferred
configurations
orders
from
state
or more
qqg.
From
Eq.(2.7)
in the
that
is maintained
‘two-jet-like’,
distinguishable
jets.
state
This
a smaller
A more
(a) the gluon
momentum
collinear
with
appear,
qualitative
probability
complete
that
two jet the pre-
fragmentation
holds
to a final
relative
the both
parton
(determined
discussion
- a configura-
since
result
leads
If the
is suppressed
order, (after
emission
1 respectively.
the quarks.
therefore,
is
goes to eero.
the quarks
the cross section
indistinguishable
order.
either
of the Ei approaching
to next-to-leading
Multigluon with
when
in phase space from
It would
theory.
large
to be soft and/or
of as.
at lowest
(2.17)
za)’
or (b) the gluon
- then
give a final
of perturbation
predominantly
quarks,
jet event’
by one power
nature
infinitely
one and (b) both
prefers
to a ‘three
order
to hadrons)
becomes
to be well-separated
corresponding
z: t z; (1 - r1)(1-
2as 3n
one of the outgoing
corresponds
In other
+
we have 1 dau --=Q dr,dxl
Recall
e+e-
in fact state
to all
which
is
by as) for three
can be found
in reference
PO1 To quantify procedure
this statement
for classifying
a final
we need to introduce state of hadrons
the concept
(experimentally)
of a jet measure, or quarks
i.e. a
and gluons
-25-
(theoretically)
according
to the number
be free of soft and collinear should
also be relatively
and gluons
into
Consider
invariant
widely
a @g final
centre-of-mass
state.
It is immediately
clear
that
of the matrix
is equivalent
to
pairs
+
4Lizi&)-G],
=
1-
R,, so that
state.
is defined
than
than
the O(crs)
correction
Note
that
makes
to multi-jet identify
Then
all parton/clusters of clusters
as one in which
some fixed
fraction
the
y of the
(2.16)
space avoids
the soft and collinear
of the energy
fractions,
Eq.(2.18)
(2.19)
21 + 22 > 1+ y.
singularities
only
ya then
‘cluster’.
mass’ algo-
= q,P,s.
In fact in terms
the result
generalisation state,
invariant
then
to O(aS)
we obtain
(2.20)
RI1 >
number
of quarks
R3
Clearly
until
and
Lig(y)=-[&In(*).
0.
final
should
&CD,
fragmentation
R2 and R3 to be the two and three jet fractions
y -+
a single
event
of phase
< 1 - y,
the soft and collinear
is greater
jet
i,j
region
element.
Note that
parton
measure
in perturbative
is the ‘minimum
are all larger
> Yst
this
0 < Zl,ZZ
R1
a jet
energy:
singularities
The
measures A three
(Pi + Pj)’
If we define
calculated
to the non-perturbative
used jet
masses of the parton
overall
when
To be useful,
hadrons.
One of the most rithm.
of jets.
singularities
insensitive
FERMILAB-Conf-90/164-T
the number
an n-parton
large
to RZ is perturbatively
with
is straightforward. the lowest
invariant
of jets is n. If not,
combine
for the (n - 1).parton/cluster
have a relative remaining
as large logarithms
sense for y values
fractions
the pair
repeat
reappear
is then final
invariant by definition
state
mass squared the number
can give any number
in the limit
enough
such
that
small. Starting
from
mass squared. the lowest final
state,
greater
of jets
If this pair into
and
than
of jets
an n-
ys.
so on The
in the final
between
7~ (all
-26-
partons
and 2 (f or example,
well-separated)
and collinear Since
or collinear
multiplicity
of jets,
in the total
cross section
gluon
emitted
the cancellation
Now in general
calculation
=
The
is applied
state
Monte
corrections
is shown
with
the string
effect
is a result
determine
by measuring Note
than
have shown
that
and therefore
in
experimentally However
studies
- at least
at high
the QCD data
exhibit,
a decrease
that
partons.
the experimental
effect
it is entirely
unremarkable
that
However,
that
such interference
the authors high
the pattern
energy,
[23,24].
pa&on-level An example
[Zl].
and
such interference
the colour
effects
of the radiation
In the language
it is interesting
of ref.(24]
of associated
the coherence
of constructive
theory.
which
At sufficiently
dependence
coherence
to the string
phenomenon
theory.
in Fig.(7).
leads
indicates
defined
One can therefore
rather
are small
compared
partons
evidence
the
was evident
all the energy
in Fig.(G).
Carlos
the hard
field
that
in perturbation
coupling
hadrons
annihilation
in quantum
change
and the jet fractions
as(&).
visible
For the case of three jet events in e+e-
cowse,
by soft
i 2 0,
of the strong
to final
can be reliably
Colour
QCD,
does not
singularities
take place,
in the coupling
shower/fragmentation
of such a comparison
2.3
line
y is dimensionless
effect is clearly
- the fragmentation
predictions
accompanied
~ij(y)(os~+))j,
the running
RJ as J;; increases.
parton
a quark
to all orders
criterion
is contained
in principle,
the algorithm
from
can still
(“sLG’)i
since the jet fraction
of the jet fractions
energy
quarks
we have
R;+2(VGY)
using
hard
of soft and collinear
this way are free of such singularities
at least
two
gluons).
a soft
Note that
FERMILAB-Conf-90/164-T
to note
survive
radiation.
interference. should
that
be observed experimental
the hadronisation
process,
duality.
of the hard final
Because
Of
the
call local parton-hadron structure
of perturbative
destructive effects
from
the distribution
state partons
will
of this radia-
a
-27-
30
,I((
-
,,/I
I,,,
Energy
,,/,
FERMILAB-Conf-90/164-T
,,,,
dependence
,,,(
of three
o x 4
;!I
jet
,,(,
,,,
production
JADE Mark II TASS0 TRISTAN OPAL
n
25
,,,,
i $1 i ________________________________________--------------
20 -
I? ------
15
20
Figure
30
lie between
the hard
altered
the jets
dependence
depend
90
100
of three jet production[ZZ]
by hadronisation
will
1
the observed
on the colour
pattern
of the partons
cf the hadrons participating
in
scatter.
We illustrate e+e-
g &eBvq70 a0
40
6: The energy
is not significantly
which
i
a,=const
1111’1111’1111’1111’111111111’1111’111111111 10
tion
1
--t qqg.
the directions (; = 1 - cosB;,
the Soft
derivation gluons
of the hard where
of the
angle
are emitted
only
partons
ordered inside
q, Q and g.
Bi is the angle
between
the soft gluon
Bij is the angle between
these variables
factor
which
certain
angular
We introduce
and (;j = 1 - cos Bij where the eikonal
approximation
describes
hard
regions
the angular and the hard
partons
the emission
in the process
i and j.
around variables parton
i,
In terms
of
of soft radiation
may
be written,
($${$+i-i})+(i++j)
(2.22)
-28-
100 8
OPAL E,,= , \ \
80
.-.
91 GeV
Z-jet
.I*’ P l
Jii 2
FERMILAB-Conf-90/164-T
mtA 2-.3-,4-, 5.jet data
60
z g
40
E ‘7 e
20
vi ‘% *\7 ‘L, t l
d
0 0.0
l\X \.X 5. . ..__ ..x+jet . ...7LA..i-“~ Ly==x-x
+&@*-, .
0.05
I
I
I
0.10
0.15
Ycut Figure
7: Jet fractions
fits with where
different
choices
lkj represents
nected.
The eikonal
photon
approximation
at [; = 0 but angle
the OPAL
collaboration
for the renormalisation
the energy factor
in Eq.(2.22)
at [j
The expression
lines
i and j are colour
parton
obtained
in braces contains when
averaged
i, it vanishes
QCD
con-
in the soft
the collinear
pole
over the azimuthal
outside
the ccme c; = [id.
[25,24], $
+
i
-
k
=
:O((ij
3 J
averaging
we mavI write.
Eq.(2.24) dynamically
-
(2.23)
(ii).
I I
{
each term with
respect
to azimuth
around
its direction
of singularity,
, M = &@(fij
has the same form imposed
An elegant scattering
The
is the same as the factor
of hard
$f
Hence,
[ZZ]. Perturbative
scale ti are shown
= 0. Furthermore,
the direction
at LEP
of the soft gluon.
in QED[l].
not that
& around
In fact
from
angular
way to examine
event is to compare
- Ci)+ &O(b f
as the incoherent constraint
radiation
emission
result
but
with
on the phase space.
the pattern e+e-
(2.24)
- cj).
annihilation
of soft radiation into
associated
three jets with
with
annihilation
a hard into
a
-29-
Figure two jets deduce
8: Particle
flow as a function
and a photon.
The
parton
that the soft radiation
by angular of this
ordering
argument
regarded
to lie between
as a qp system,
in the qqg event
is then
quark
and the gluon
occurs
predominantly
colour
of freedom
the quark
connected
expected
between
[27] are shown
and the third
jet is assumed
the
1 and 2, the distributions
region
between
and
connected
the gluon
Eq.(2.24)
quark
For
the
line.
antiquark
The soft radiation the gluon
for the qijy event
events
purposes
can be approximately
between
In the angular
we
constrained
to the outgoing
the antiquark.
of the qqg and qe
1 and 2, opposite
lines.
The jets are ordered
to be the gluon.
From
is dynamically
of the gluon
In contrast
quark
flow)
connected
to lie predominantly
in Fig.(S).
of jets
part
of the event
are nqg and qq7.
to the outgoing
and the antiquark.
collaboration
jets
states
the
degrees
with
part
of angle in the plane
(and hence the particle
the colour
line and the antiquark
final
FERMILAB-Conf-90/164-T
Data
and the
the radiation from
the TPC
in energy
Et > Es > Es
regions
near the cores
agree very
jet or the photon,
well.
the data
In the show
a
-3o-
depletion
in particle
A heuristic
explanation
using
a simple
which
decays into
and
By the uncertainty approximately
pair.
differs
(Ei + Ej + Eb) - (Ei+r,
=
@iG-z+
of momentum
The virtual
the
final
state consisting
state
containing
+ Ej) (2.25)
B;k this becomes,
AE
N Irc’l0:.
the virtual
electron
(2.26) state
lives
for a time
At
which
In this given
interval
N l/(koik) of time
At
the electron
wavelength
and positron
of the emitted separate
soft photon.
a transverse
distance
by
If &, > 8ij, the separation length
of the emitted
positron
pair
the other positron This
hand, pair,
charge photon
indicates
z 2.
emitted
neutral
property
can, to a good
object the
perceives
and no radiation cone described
wave-
the electronoccurs.
If, on
by the electron
is uninhibited.
is complicated
but the angular
is less than the transverse
soft photon
lies within
the reason for angular
to QCD
(2.28)
and positron
The
&k < Bij, the radiation
It is an interesting state
soft photon.
the emitted
example
charge,
of the electron
as an unresolved
of this argument
final
(2.27) &k
is the transverse
Ad = &eij
colour
is
by IklS,?,
XT N l/kT
an
AE,
At-=!-+?, where
[26]
photon
Ii1 - J(p7.
large P; and small
given
from
virtual
soft photon
pair.
by an energy
can be obtained
an incoming
An additional
=
principle
ordering
Consider
in energy
and a soft photon
of very
for angular
from the electron-positron
a positron
AE
to qQ^/.
argument.
a.n electron-positron
a positron
In the limit
principle
radiated
of an electron
in qqg compared
of the reason
uncertainty
k is subsequently
electron,
production
FERMILAB-Conf-90/164-T
ordering of the
approximation,
ordering
by the fact that result theory
in QED.
The generalisation
the gluons
themselves
carry
emission
of gluons
in the
persists. that
the
be represented
by a semi-classical
parton
-31-
‘branching’ gluons that
or ‘cascade’
etc.
This
the eikonal
approximately photon
vertex
picture,
property factor
obtained
the virtualities,
until
in terms
of QCD
cannot
made up of pions, be described
determined
by fitting
constructed. different
Different
models
the interference
The
produced
at the
(i.e.
are off mass shell)
have virtualities
kaons and other
perturbatively, to the data.
the partons
hadrons.
In this
then
the non-perturbative with
experimental
of the
can be described
The hadronisation
can be mod&d,
of the
reducing
of the order
‘hadronise’
way jet fragmentation
can be compared
takes place,
of the fragmentation
but instead
ways of performing
[28] which
quarks
partons
Finally,
it is shown can be
state
theory.
more
diagrams
branching
part
where
emit
of Feynman
have ‘virtuality’
This
in turn
in Eq.(2.24)
Parton
GeV)).
perturbation
which
energy.
all the final
mass scale (0(1
gluons
for example
annihilation
centre-of-mass
hadronic
states
from
emit
as a sum of probabilities.
after an e+e-
order of the total
i.e. the quarks
is evident
represented
FERMILAB-Conf-90/164-T
to give final of the partons
the parameters Monte hadronisation data.
being
Carlos
are
lead to
-32-
Deep
3.
Inelastic
The original, of Bjorken
theory
and still
scaling
structure
analyses
begin
by discussing
show
how
QCD
and discuss
asymptotic
cross sections
generalisation
and gluons.
3.1
Deep
Consider
of the parton
of the parton
quarks
inelastic
the scattering
the incoming
of a high
of the target
momentum
transfer
hadron
property
at small
for general
tests
model.
data,
and
+. Finally,
model
off a hadron
theory.
we describe
parton
four-momenta
model,
calculate
the
to be a proton) deep inelastic
the the
involving
target.
If we
by kp and k’g respectively,
hereafter
we
We then
of the parton
processes
lepton
for use
In this lecture
scattering
charged
of the
in hadrons
hard
and
(assumed
precise
in perturbation
experimental
by q’ = kp - k’p, th en the standard
the
by pi’ and the variables
are
by: Q'
where
scaling
with
lepton
deep inelastic
parton
can be calculated
energy
Nowadays,
collisions.
and the ‘naive’
Bjorken
scattering
is the breaking
of partons
hadron
distributions
picture
and outgoing
momentum
defined
scattering
predictions
QCD
some of the most
distributions
violations’
the theoretical behaviour
provide
the simple ‘scaling
scattering.
in high energy
deep inelastic
how these
We compare
only
Distributions
test of perturbative
the momentum
modifies
Parton
lepton-hadron
not
but also determine in predicting
and
the most powerful,
in deep inelastic
function
as input
label
Scattering
FERMILAB-Conf-90/164-T
the energy
or muon,
The as ‘seen’
variables
then the scattering
mq2,
pz = Ma
Q"
Q2
z =
2p=2M(E-E’)
’
cl.P k.p
=
= 1 - E’IE
refer to the target is mediated
structure functions Fi(t, by the virtual
=
photon
Q’)
,
rest frame.
by the exchange
- which
- are then
parametrise defined
If the lepton of a virtual
is an electron photon,
the structure
in terms
of the lepton
Fig.(S).
of the target scattering
-33-
FERMILAB-Conf-90/164-T
k’
--I k
Figure cross sections.
9: Deep inelastic
For charged
lepton
+(1 and for neutrino
scattering,
structure
Bjorken
limit
functions
dimensionless
is defined
as Q’,p
an approximate
is illustrated
IX,
- (M/2E)zyF;m
1 ,
(3.2)
vp -+ IX,
+ (-)
y( 1 - y/2)4*‘)
. q -+ 00 with scaling
law,
i.e.
+ fixed. they
1 In this depend
limit
only
the
on the
z: Fi(z,
This
scattering
=
obey
variable
lp +
scattering,
+yW,‘(“) The
lepton-proton
- 2rF;m)
- Y)(r
(antineutrino) #,y’W __ dzdy
charged
in Fig.(lO),
where
Q’)
-
E(z).
data on the electromagnetic
(3.4) structure
function
-34-
FERMILAB-Conf-90/164-T
.l i-I
+
0
-I
+*
.
n L I
I
- 0
.1
I
I
.2
I .4
.3
I
2. * . I I I cp .8 .6 .7
I .5
x Figure
10: The FZ structure
Fz, measured
with
of experiments,
from
measurements data
points
magnitude,
the BCDMS
is shown.
Note
since otherwise l/Q0
model’
we consider carries
a fraction
Eq.(3.2)
that
is moving
[30].
even though
the
Only Q’
[29] to the most
values
photon
scatters
structure
functions
would
where proton’s
two decades
a representative
the virtual
scattering
collaborations
data span nearly
vary
the data lie on a universal
is most
the photon momentum.
easily
scatters Setting
depend
orders
of
constituents,
on the ratio The
formulated
Q/Q,,, ‘parton
in a frame
frame.
quark
which
= 0, we can rewrite
as
1 .
Now
in
In this frame,
off a pointlike M”
of
curve.
off pointlike
momentum
recent
sample
by two
the size of the constituents.
very fast - the infinite
model
< of the
The
and BCDMS
measurements
collaboration
scale characterizing
of deep inelastic
a simple
SLAC-MIT
approximation
implies
some length
the proton
are displayed.
that
the dimensionless
picture
which
target,
from
scaling
from the SLAC-MIT
the original
to a good
Bjorken
with
a proton
function
the spin-averaged
matrix
element
squared
for massless
eq -t
eq scattering
(3.5) is
-3%
obtained
simply
by crossing
ered in the previous
FERMILAB-Conf-90/164-T
the corresponding
lecture,
cf.Eq.(2.3).
matrix
In terms
element
for e+e-
of the usual
+
qq consid-
Mandelstam
variables
i. i. fi we have ElMI” The notation Eq.(3.1)
Cdenotes
= 2eg
the average
we can substitute
(sum)
over initial
for the deep inelastic
i = Qz/xy.
The differential
Comparing
Eqs.(3.5)
cross section
and (3.7)
result
with
momentum
bution
The
l
than
of momentum above
ideas
variables:
g’Ives us the structure
the structure
a delta function,
1^= -Q’,
scattering
functions
and spins. G = i(y
process
in this
is therefore
simple
model: (34
Fs(r)
‘probes’
a quark
the measured
structure
function
suggesting
Using
- 1) and
- () = 2zil. function
z. Now clearly
colours
that
the quark
constituent is a distri-
constituents
carry
fractions. are incorporated
in what
is now
known
as the
‘naive
parton
[31]: q([)d<
represents
between l
that
fraction
in + rather
a range
model’
suggests
(final)
for the quark
kg = +eib(r This
; Y
the virtual
the
probability
that
a quark
q carries
momentum
fraction
< and t + d( photon
scatters
incoherently
off the quark
constituents
Thus
z=i6(* -0 J’dz)= C 4 dtn(E) = $:“p(x). and so for the scattering F;‘-(z)
of a charged = + ;u(x)
lepton
+ id(+)
off a proton + $(z)
(3.9)
target,
+ ;a(~)
1
+ ... .
(3.10)
-36-
For neutrino
scattering
distributions
weighted
- vp
This
inverted
= 2x d(x) 1
probe
measures
the
structure
=
2x[d + s - ii -E]
F.f
=
2++
ZF3”
=
2r[u+c--d-z]
F;”
=
x[;(u+u+c+z)+$(d+d+s+s)]
22F,
=
Fs.
evident
quark
picture
functions
follows
from
the spin-i
property
the above relations
of measured
structure
functions,
distribution
functions
themselves.
The
charge
sea of light
qij pairs.
flavours
with
Q. Thus
is given
below.
(3.12)
emerges.
the electric
(3.11)
c+d+a]
in Eq.(3.8)
number
1
+ E(X) f . . . .
encountered
ZF3y
m, <
W+
+ a(+)
2x[d+s+ti+E]
infinite
with
+ z(x)
=
to give the quark
carry
virtual
F;
sufficient
the following which
- the
list of the most commonly
last result With
IX
by the weak charge:
F,‘(x) A complete
+
FERMILAB-Conf-90/164-T
proton
and baryon When
consists quantum
probed
From
of the quarks.
such an analysis,
of three
valence
quarks
numbers
of the proton,
at scale Q, the sea contains
at a scale of O(1 GeV)
u(x)
=. w(x)
d(z)
=
&(+)+S(x)
ii(r)
=
J(c) = S(x).
can be
(uud) and an
all quark
we have
+ S(z)
(3.13)
the sum rules dzuv(r)
=
2,
dx dv(r)
= 1
4 dz z(q(z) The about
last
+ q(i))
of these is an experimental
50% of the proton’s
momentum.
N
result.
0.5. It indicates
that
The rest is attributed
the quarks to glvon
only
carry
constituents.
-37-
Figure
.4
0
.2
11: Quark
and gluon
the gluons
ing, their
presence
momentum
jet
are not directly
is evident
gluon
distribution
functions
measured
1
hard scattering
photon
production
distributions
at Q’ = 10 GeV’
in deep inelastic
in other
and prompt
and
.8
.6 x
Although
set of quark
FERMILAB-Conf-90/164-T
processes
from
hadron
Fig.(U)
fits
scatter-
such as large transverse
(see later).
extracted
lepton
shows
a typical
to deep inelastic
data,
at
p2 = 10 GeV’. Closer
examination
scaling:
the structure
opposite
hehsviour
violations
3.2
function at small
are understood
Scaling
In the ‘naive’ the asymptotic logarithms to the
of Fig.(lO)
decreases
in perturb&v=
model
(Bjorken)
a systematic with
limit:
eq scattering
functions
Q* -+ co, x fixed.
process
exact
at large
we discuss
Altarelli-Parisi
of Q. To see how this Q’ dependence
eq -+
section,
Q’
from
Bjorken
z and has the
how these scaling
QCD.
and the the structure
deviation
increasing
+. In the following
violations
parton
reveals
considered
scale, In QCD,
equations i.e. this
arises, consider in the
previous
F(z,
Q’)
scaling
--t F(r)
in
is broken
by
the O(as)
corrections
section.
An explicit
-3%
calculation
FERMILAB-Conf-90/164-T
gives
+,Q’)
= (3.15)
where
P, C are calculable
virtuality
2
= -pi)
functions
which
arises when
the gluon
not subject
to the theorems
lecture, pair
because
carrying
is introduced
is emitted
and choose to define
parallel
photon
the same overall integrate
(for
the
example,
collinear
to the incoming
quark.
of singularities
can resolve
a quark
divergence This
discussed and
the
which
divergence
the above
result
Q’-dependent
a collinear
quark-gluon
quark Q’)
with
the quark
distributions
distribution
function
can we interpret constant,
The collinear
singularities
scale’ ~0, which
the limit
by (3.16)
= ~;+q(=,Q), P
prediction
(3.17)
-+ O? Exactly
2
we can regard
into
is how the distribution of Eq.(3.17)
distribution varies with
at a ‘factorisation
scale. n(z, p).
p2. Thus
There What
is therefore the theory
function
- known
equation
describing
The above derivation and extends
as the
the result.
The full
does
we obtain
Altarelli-Parisi
the variation
is rather
no
if we define t = ln(p*/&)
$n(d = ffs(t) 271.% $&WY;). equation
of
bare distribution.
this bare distribution
role to the renormalisation
for the ‘renormalised’
and take the t-derivative
as for the renormalisation
n( z ) as an unmeasureable,
are absorbed
plays a similar
tell us, however,
This
q(t)
to O(as),
the coupling
absolute
is
in the second
n(z,~)=n(.)+~[~n(O{fi;)ln$+C(;)}+....
How
quark
momentum.
Fz(l, then we find
to control
for cancellation
the virtual
If we again
and n is a regulator
heuristic, prediction
equation of as(t)
with
- is the
analogue
of the p
t.
but a more complete of the theory
(3.18)
is most
treatment easily
confirms
cast in terms
-39-
of the moments
(Mellin
transforms)
FERMILAB-Conf-90/164-T
of the distributions:
q(i,t)
1 d+ zj-1
=
q(z,Q
(3.19)
j In terms given
of these
moments,
the t dependence
of the quark
distribution
function
is
by ddj, t) = -hq (i as(t)) dt
We next
define
P,, as the inverse
Mellin
(3.20)
q (i t) .
transform
of m,,
= & J 4 2-j-&,as), gf pw(Z,QS) where
the integration
contour
axis and to the right transform
in the complex
of all singularities
of Eq.(3.20),
we obtain
j plane
(3.21)
is parallel
of the integrand.
to the imaginary
Taking
the inverse
M&n
in I space,
dq(z,t)
dz 6(~ - b)P,,(z,
dt
as(t))q(L
t) (3.22)
P,, has a perturbative
expansion
in the running
coupling,
PdZ,W) = P;;)(z)+ SP(l)(,) + ... 2K ‘I’l Retaining with
only
the first
term
in this expansion
gives precisely
(3.23) the result
in Eq.(3.18),
P = Pi:). In fact the above derivations
distributions,
are strictly
q = q; - qj. In general,
only
correct
the Altarelli-Parisi
for di&wzces (AP)
between
equation
quark
is a matrix
equation,
The AP kernels finding
parton
of the parent
P&?“‘(z) have an attractive i in a parton
parton
of type j with
and a transverse
physical
interpretation
a fraction
momentum
as the probability
I of the longitudinal
much less than
momentum
p. The interpretation
of
-4o-
as probabilities satisfy
implies
the following
that
the AP
kernels
FERMILAB-Conf-90/164-T
are positive
definite
for z < 1.
They
relations:
4 dxPg’(x) =0 =0 dxEP$‘(x)+ P;,“‘(x)] % [ 4 dx 2 [znfP$(z) + P;;‘(z)] = These tion
equations
correspond
in the splittings The
kernels
coupling lution
as. kernels
kernels
to quark
of quarks
of the AP
Both
the lowest
equations order
are calculable terms
have been calculated.
and momentum
conserva-
as a power
[32] and the first
The lowest
order
series
correction
approximations
in the
strong
[33] to the evoto the evolution
are:
P;;‘(x)
=
P;;‘(x)
=
Po’p”‘(x)
=
‘plus
1-X
x +x(1-x)
prescription’
on the singular
4ffxf(~M~)l+ In terms
conservation
(3.25)
and gluons.
J’:;‘(x) =
The
number
0.
of moments
these four
parts
=
evolution
J
1
+6(1-r)
of the kernels
(11N
6
. (3.26)
is defined
dz (f(x) - f(l)) s(x).
kernels
- 4n,TA)
take the form
as (3.27)
-41-
FERMILAB-Conf-90/164-T
(3.28)
In general of quarks,
the AP equation
antiquarks
so the matrix
is a (2nt + 1) dimensional
and gluons.
equation
However
matrix
equation
not all of the evolution
can be considerably
simplified.
Because
kernels of charge
in the space are distinct conjugation
we have that, p,, At lowest
order
= pa,
we have in addition
the following
P$=O, The solution
of the AP equation
non-singlet with
(in flavour
the flavour
space)
singlet
P&lj=O
moments,
this
equation
drops
notation
combinations
out and we have,
which
are
the mixing
(V = q; - gj),
P*,(t) @V(z, t119 integral
of Eq.(3.22).
Taking
becomes
dt the lowest
by considering
for the convolution
dV(j, t) Inserting
(3.30)
such as Qi - qi or qi - Qj. In this combination
gluons
8 is a shorthand
relations, (i#j).
is simplified
$x,t) = f$ where
(3.29)
ppo = pm.
order
form
as(t) (0) ,
= Tjpq
(3) w,
for the running
coupling,
(3.32)
t), we find
the solution
(3.33)
It is straightforward This
in turn
implies
I and increases
to show that
at small
that
as p increases I.
Physically,
&(l)
= 0 and
the distribution this
that
d,,(j)
function
can be understood
< 0 for j decreases
as an increase
2
2.
at large in the
-42-
LO 0030 I ..*.aI
’
FERMILAB-Conf-90/164-T
““‘II
/
I
1
x i 003 EMC
O,LO 0,30
F:' .
.
x
0,30 .J
ix,021
...**
.
.
.
.
.
.
.
.
0.05
I
o,o*
i
0,125
. x
OLO’C 030 r
i
.
.
. x
l
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
l
l
l
.
0 LO 0 30 It
2 030 J
F,
.
.
.
.
l
.
.
.
0 20
.
15
. x
.
.
0 15 0 2s 1 00
.
x =0,175
l
l
025
z
0,35
. x
[
i
.
X*;
,‘,;
l
10
0 10 006 1 005 303
.
.
‘d
;~;j ,:,,,,,, l((,[;lt!; ,jl 23L6810 20 30 51 100 200 a2 IGd Figure
12: The structure
*
function
Fz measured
in high
.
energy
muon-proton
scattering
by the EMC phase space for gluon degradation
in momentum.
data on the structure We now turn all quark
emission
flavours
function
to the flavour to be given
by the quarks
The
trend
F;’
is clearly
measured
singlet
as p increases, visible
a corresponding
in the data.
by the EMC
combination
with
Fig.(l2)
[34].
of moments.
Define
the sum over
by C,
c = CC%+ a). I From Eq.(3.24),
which
holds
shows
for all flavours
of quarks,
(3.34)
we derive
the equation
for the
FERMILAB-Conf-90/164-T
-43-
flavour
singlet
combination
of parton
distributions,
(3.35) This
equation
is most easily
with
an input
distribution
We can illustrate Taking
by direct
obtained
from
some simple
the second
The eigenvectors
solved
numerical
of the distributions
(j = 2) moment
of Eq.(3.35)
and corresponding
eigenvalues
C(2) +g(2)
o-(2)
=
C(2) - Zs(2) F
Note that
the combination
the quarks
and gluons,
that
of this
system
is independent
t:
corresponds oft.
The eigenvector
d-(2)
asymptotically
we have
g(2)
fractions
the moments.
of equations
are
(gcF
(3.37)
momentum
O- vanishes
-( $F
=
+ y)
carried
by
at asymptotic
+ 7~)
(3.38)
2zrb
’
w -=
The momentum
: -
to the total
d-(2) +0,
So that
using
: 0
Eigenvalue
which
Of,
we find
Eigenvalue
=
in I space starting
data.
properties
o+(2)
integration
carried
nf -4cF
Nnf = 2(Na
by the quarks
(3.39)
- 1)’
and gluons
in the p +
00 limit
are therefore
qt=_ Note,
however,
and is therefore
that quite
= (40::,,) the approach slow.
3 g(2Jit=_ to the asymptotic
For a tabulation
(3.40)
= (4czf)~ limit
is controlled
of the eigenvectors
by t N lnFa
and eigenvalues
of
-44-
, , , ,,,,,
1000 -
, , ) ,1,,,
Gluon
, , , ,,,,,
, , , ,,,,,
FERMILAB-Conf-90/164-T
, , , ,,,,,
, , , ,,,/(
II
distribution
100 r
z * x x
10
x=10-’
IF
.1 ’ “l111f’ ’ “11111’ “11d ’ “1 1oa 10’ 10’ 10’
Ir* [Q
Figure
1
13: The
,
,,,,
1,,
,
scale dependence
,,,,,,,
Up valence
m
,,,,
,
,,,,,,,
of the glum
,
,,,1,,,
,
,,I
distribution
,,,,
,
,rrmp
distribution x=1o-s
.5 -
=
-*
2 -
Figure
14: The scale dependence
of the valence
up distribution
-45-
1
FERMILAB-Conf-90/164-T
k ’ “““‘1 ’ “““‘1 ’ “““‘I ’ ‘1”“11’ “““‘I ’ “““‘I ’ ‘1” Down
valence
distribution
3
x=1o-3 = -1
.l 3 * a x
=
-1
.03
.Ol
,003
.oo 1 LOI
$111111L I,,,,,, I I l,,,,, , , I, loa LOS 10’
Ir* [Ge Figure
15: The
scale dependence
of the valence
down
distribution
of the anti-up
quark
distribution
ti distribution 3-
Figure
16: The scale dependence
-46-
2 x’ 25 x x
.l
F
.03
7
,001
Figure
FERMILAB-Conf-90/164-T
d
u S
1 ~~~‘~W~’ lOI lo2
’ ‘111111’ ’ “111111 ’ “I lo3 10’
yL’ ‘Yi P’ [Ge 1
17: Momentum
fractions
carried
by the quarks
““k and gluons
as functions
of the
scale the moments
of Eq.(3.35)
the scale dependence
3.3
QCD
of the quark
fits
In the the previous
to deep section
of the structure
functions,
Quantitatively,
the variation
s&e
parameter
therefore,
A.
provide
determination
we refer the reader
Deep
and gluon
inelastic
we saw that rather
than
with inelastic
[S]. Figs.(l3)
data
perturbative
QCD
predicts
the Q’ evolution
the size and shape of the functions
scattering
- (17) show
distributions.
Q’ is controlled
one of the ‘precision’
of Am
to reference
data
tests of QCD
by as(Q) of the
and hence type
and, arguably,
shown
themselves. by the QCD in Fig.(l2),
the most accurate
-47-
Although structure This
the theoretical
function
is because
some form
and accurate
distributions
are then evolved the F&,Q1)
using
function
fit is performed extent
can also be quantified
they
evaluated
the ‘best’ value
and used to derive is not, however,
small
integrals.
to obtain
function
Note
that
Finally,
without
on the other
for
in this
z depends a global
values for the parameters,
a systematic
A
values
at a given
Eq.(3.24).
of A depends
I, and
These distributions
are measured.
at < > +, c.f.
of
Q0 and parametrise
equations,
Q’ of the structure
the measured
The above procedure
the moment value
in terms
from the data.
and very
QO) = Az”(l-z)~.
where
to determine
to which
to very large
the AltarelIi-Parisi
with
expressed
such moments
to construct
e.g. n(t,
regions
the rate of change
when
is to choose a reference
value,
in the kinematic
numerical A. The
do not extend
method
numerically,
simplest
to extract
is required
at that
on the structure
appear
it is very difficult
of ad hoc extrapolation
the parton
only
moments,
the measurements
more practical
approach
predictions
FERMILAB-Conf-90/164-T
including parameters
error.
problems.
The most serious
of these
are: l
In QCD,
the structure
functions
are much
more
to estimate
difficult F(z,Q’)
where
the
sion -
superscripts
spin)
analysis
=
have
on the
be performed
twist’
power
corrections,
which
_. ,
(3.41)
quantitatively:
F(‘)(z,Q’)
+
F0;;Q2)
right-hand-side
of the contributing
must
‘higher
operators.
at large
+
refer To avoid
Q’ where
to the these
the power
‘twist’
= (dimen-
complications,
suppressed
the
terms
are
negligible. . The structure
function
quark’
and ‘valence
tively.
Hence,
a p&vi
J’s can be decomposed quark’)
except
unknown
parts,
at large
gluon
which
z, the Q’
distribution
into
dominate
singlet at small
dependence
and there
and non-singlet and large
z respec-
of Fz is sensitive
is potentially
(‘sea
a strong
to the A-gluon
correlation. . Non-singlet
structure
(see Eq.(3.31)),
but
differences
between
additional
systematic
functions these
do not suffer from the gluon
are only
cross sections, and statistical
measurable e.g.
experimentally
c+‘p - a’“.
uncertainties.
correlation
This
problem
by constructing
inevitably
introduces
-4%
FERMILAB-Conf-90/164-T
-,
“0 x2 ,=a
me-
ro (x4) I
10 e-t.
l
--
W-c’.
.
*.
-
*
,
r=O
tr
14
r=O
Cll~W
(x3)
18
(12)
“=a 225 (x1,5,
~9-
---+-TH
.=a275 CL17)
10-
IILL.,~,__10
Figure
18:
Data
on the
structure
function
~I---L-.L~‘.
J
,... 1I?? :;’
Fz in muon-hydrogen
(&g’)
scattering,
from
BCDMS The
most
problems
recent
generation
by collecting
contemporary
high
data demands
in the fits.
Beyond
and in practice
leading
statistics that
almost
always
data
experiments
at large
a specific
the MS
order
renormalisation
scheme.
partially
+ and Q”.
the next-to-leading
order
this is usually
the literature
of deep inelastic
For this
Fig.(l8)
muon-hydrogen
scattering.
Q” values Q2 derivative are negative, Also shown
of several
QCD predictions scheme
reason
of logF2 consistent
the structure
The
are the predictions
function
measurements
GeV’.
as a function with
of
are used
must
be chosen,
the results
quoted
in
refer to Am
shows
hundred
these
In fact the precision
Some of the most precise recent data comes from the BCDMS As an example,
solve
a structure
Fig.(lS) of z.
Fz measured
Note
that which order
[30,35].
in deep inelastic
up to + values
sh ows the corresponding
function
of next-to-leading
extend
collaboration
of 0.75 and logarithmic
the derivatives
in this
decreases
increasing
with
QCD for three
different
region Q”. values
-49-
FERMILAB-Conf-90/164-T
bl
al
-0 3 0
Figure
19:
figure
I
,
0,2
04
06
Logarithmic
with
of Am
1
QCD
Q’
result
about
equations
collider
valid
The
widely
220
04
0~6
Knowing
A=,
a parton
+
15 rt
function
I,
J
0,a
in the previous
50 MeV
determinations
measure
phenomenology.
such parametrisations
0~2
of the Fz structure
with
experiments
each time
approximation,
=
is compared
Q = 15 GeV.
hadron
,
iit gives [35]
for Am
Deep inelastic
1
BCDMS
A$ This
,
1
0~8 0 x
derivative
fits, from
A detailed
1,
.-BCOMS Hz o--- BCIJMS C
quark
from
densities
these can then
Instead
to a sufficient
processes
over a broad to higher
integrating
is required,
accuracy
other
be evolved
of laboriously
distribution
(3.42)
it is useful
over a prescribed
range
in Fig.(2). in z up to
p and used for
the Altarelli-Parisi to have an analytic (2, p) range.
Several
are available.
used Duke
and Owens
parametrisations
1361, for example,
form q(z, Q)
=
Az”(l
A
=
Ao+A,s+
+ cz)(l Ad
- z)* etc.
are of the
-5o-
s with
the parameters
an accuracy cantly
Ao, Al,
constrain
the gluon
‘soft gluon’,
range, ation
experiments and ‘hard of parton
next-to-leading
Because
distributions order
data from prompt
- for example
QCD
photon
fits
range in + and p, and are ideal future
hadron-hadron
Small
3.4 From
x behaviour
Fig.(l3),
asymptotic
we see that
limit
where
of the distributions The z + anomalous
for making
from
of the parton
dimensions
r(j)
The
most
from
sets [38] - are obtained
from
data,
The distributions predictions
parton
by,
as
for present
and
distributions grows
the Altarelli-Parisi distributions
rapidly
at small
to determine
is controlled
near j = 1. Considering
for the moments
z.
In the
the behaviour
equations.
the gluon
by the behaviour only
we have
x Jff-. j-l of the gluon
s(i,t) = s(i to)exp(,$ and f is defined
as well
cover a wide
Eq.(3.28),
the solution
z
gener-
quantitative
distribution
$(j) In this limit
prompt
recent
$4i 4 = $4~~)(j)g(j, t) where
and a
colliders.
of the
the gluon
gluon’
in
in the medium
of deep inelastic
z -+ 0 and p --t 00 it is possible
directly
0 limit
variety
pair production.
and lepton-hadron
[37].
signifi-
- to include
fixed-target
particularly
out
to give
does not
a ‘hard
precision
the HMRS
to a wide
and lepton
high
are ruled
evolution
scattering
typically
the gluon,
parametrisations
order
- in the past
distributions,
are able to constrain gluon’
leading
deep inelastic
Nowadays,
(3.43)
0,
it was usual
of gluon
its own A value.
>
to an exact
distribution,
a choice
each with
ln(t~{$~~~)
. . . fitted
of a few per cent.
the parametrisations
photon
=
FERMILAB-Conf-90/164-T
distribution
l))f
is,
of the
-51-
To return
to + space we perform G(r,t)
the inverse
z zg(z,t)
=
FERMILAB-Conf-90/164-T
Mellin
&
transform
as given
by Eq.(3.21)
dj z-(j-l)g(j,t)
(3.48)
I = where
the exponent
in which
by expanding
(3.49)
f is, f(i)
In the limit
to) exp [f(j)] dj,to) J4s(j,
1 2xi
both
about
= [(j - 1)141/z) ln(l/z)
+ Tb(yy
and f tend
the saddle
point
(3.50)
l)].
to infinity
we can estimate
this integral
of the exponential:
f(j)=$G+ O(i -joI’,i0=1t z$+,
y = zln(l/r).
We therefore
find
for the asymptotic G(*,
which
expressed
s(z) Notice
-
that
limits
1 - exp +
at fixed
which
the growth
momentum
partons estimate
f/y
the initial
distribution.
G(r,
limit
l/p.
to happen
of hadron
Area
of parton
is beyond
the range
enters
via the j,,th
only
as an overall
moment factor.
[39] is the mechanism
In the infinite
momentum
of gluons of gluons
new effects
- /A
of the hadron.
does not exceed
(3.53)
investigation
is provided
Area
(33l-;f).
enters
If the number
the nucleon
is the radius
of G(z,t)
active
b=
distribution
t) gives the number
inside
this begins
N=3,
2’
information
of the gluon
(3.52)
=xp $6,
yields
lni
under
t) =
4
on the starting
to overlap
T N l/m=
saturation
In&$/As
than
of when
the value
2”
is presently
G(r, where
variables
size greater
start
t) = s(h
lnpa/Al
distribution
a transverse
solution
4N
the dependence
of g. Therefore A topic
in the original
(3.51)
will
frame
per unit grows
which
the gluon
of rapidity so large
come into
play.
with
that
A crude
by, - pa 25 GeV-‘,
At presently
3 or 4, so, if the above of the present
the
colliders.
attainable estimate
(3.54) values is correct,
of I the
-52-
4.
The
QCD
Parton
In this lecture involving
we shall
two hadrons
4.1
The
The high model.
QCD
energy
fractions
the application
in the initial
state.
improved
parton
of hadrons
the quarks
The incoming
varying
in Hadron-Hadron
a hard scattering
between
hadrons.
consider
interactions
In this model
interaction
Model
model
to processes
model
process
by the QCD
improved
two hadrons
is the result
between
which
provide
broad
of their
parent
of the momenta
Collisions
of the parton
are described
and gluons
hadrons
FERMILAB-Conf-90/164-T
are the constituents band
beams
hadrons,
parton of an
of the incoming
of partons
which
as described
possess
in the previous
lecture. The
cross
section
four-momenta
for
a hard
PI and PZ can be written u(f’~lp~)
= C dzldzl w J
The parton
model
for hard
the partons
which
participate
The
characteristic
The
energy,
functions
of type
the short
the running
distance as.
is given
the c(j) are functions
In the leading to the normal
The
as the cross section
in Fig.(20).
with
quark,
by Q.
quark
Since
the
or gluon
could
be, for
momentum distributions,
for the scattering
coupling
is small
as a perturbation
approximation
of
and pa = z2P2.
This
cross section
can be calculated
the nth order
The momenta
or the transverse
QCD
distance
by >;j.
(4.1)
are pl = z,P,
is denoted
usual
short
at high series in
to the short
distance
c%:(l+&Y%+
of the kinematic
approximation parton
hadrons
by k=
where
are the
cross section Therefore
interaction
or heavy
i and j is denoted
coupling
cross section
scale p.
by two
*~j(p~~p~t~s(/t),Q).
is depicted
scattering
boson
fi(c,p)
at factorisation
of partons
events
scale of the hard
of a jet.
initiated
as
in the hard
the mass of a weak
process
fi(zl,P)fj(+l,P)
scattering
example,
defined
scattering
scattering for a QED
(4.2) variables.
(n = 0) the short cross section process.
distance
calculated
In higher
orders,
cross section in exactly the short
the
is identical same
distance
way cross
-53-
Figure
20: Schematic
section
is derived
from
pieces and factoring section
involves
momentum
purely
high
all orders
in perturbation
heuristic
argument
a fundamental tool with
approximations,
distance
and is insensitive
to the physics
of low
cross section
or the type
does not depend
of the incoming
in perturbation
hadron. theory
of the cross section
For more details, of factorisation turns
see for example is given QCD
distinguishing
it from
It is a
because
of
can be proved
to
reference
in the next
into
on
a reliable the ‘naive’
[40]. A
section.
It is
calculational parton
model
[31].
scale p in Eq.(4.1)
the order
long
cross
distance
which
by removing
process
The remaining
property
of the theory
scattering
functions.
and is calculable
for the validity
controllable
of Feynman
the short
theory.
property
cross section
transfers
factorisation
of a hard
distribution
wave function
This
description
scattering
momentum
construct
freedom.
model
the parton
In particular,
short-distance
The
into
of the hadron
asymptotic
more
the parton
them
only
scales.
the details
of the parton
FERMILAB-Conf-90/164-T
of the hard
terms
is an arbitrary
scale Q which
are included
parameter.
characterises
in the perturbative
It should
be chosen
the parton-parton
expansion,
the weaker
to be of
interaction.
The
the dependence
on p. Finally,
it should
be emphasised
that
Eq.(4.1)
is not a description
of the bulk
of
-54-
the events which to describe
4.2
occur
the most
at a hadron-hadron interesting
Factorisation
The
property
inelastic
of factorisation
illustrates
section
why the factorisation
of a hard process a massive
This
observed ucts
respects
vector
boson
are observed
consequently
through
hadron
the influence
H, before
Soft gluons
theoretical
the hard
which
+
hard
colour
to describe
classical
vector
model
that
fail.
As an example
boson
V - in practice
(4.3)
involving
no colour
two hadrons,
and its leptonic
the easiest
to analyse
since the decay prod-
theoretically
and
attention.
fields,
occurs,
long before
model
a simple
issue in this process is whether
scattering
are created
parton
v +x.
process
theoretical
of their
interaction.
of two hadrons,
state carries
the most
QCD
of a massive
It is therefore
has received
a hard
and when it should
+ &(Ps)
in the final
directly.
A very important HI,
holds
the production
the simplest
involve
present
W or 2 - in the collision
is in many
which
see, it can be used
section
we shall
&(Pl)
but as we shall
us to use the
property
we consider
photon,
cross
allows
In this
collider,
classes of events
of the
processes.
FERMILAB-Conf-90/164-T
change
the partons
the distribution
thus spoiling
the collision
the simple
are potentially
in hadron
of partons parton
in
picture.
troublesome
in
this respect. We shall a simple current
argue
model density
that
soft
[41] from
classical
J is given
by [42]
A’(t,lc) where sider velocity
the delta a particle p. The
gluons
=
function with
J
dt’dZ’
non-zero
in fact
electrodynamics.
the
The
parton
vector
+ 12 - $1 - t),
the retarded
e travelling
components
spoil
qt
;;(y$)
provides charge
do not
behaviour
in the
positive
of the current
J’(t,i?)
=
e6(Z-
J=(t,q
=
q?6(z’-
required
potential
using
due to a
e = 1, by causality.
z direction
density
picture,
with
Con-
constant
are
F(t)) T(t)),
F(t) = pi,
(4.5)
FERMILAB-Conf-90/164-T
-55-
where
i is a unit
at time
t = 0.
coordinates
z,y
vector At
in the
z direction.
an observation
point
and z, the vector
moving
charge is obtained
density
of Eq.(4.5).
(the
potential
by performing
The result
The
charge
position
at time
passes through
of hadron
Hz)
the origin described
t due to the passage
the integrations
in Eq.(4.4)
by
of the fast
using
the current
is
A’(t,5’) = +a + Y2_“;ypt _ *)‘I
where
7’ = l/(1
A-(&5)
=
0
A’(t,S)
=
0
A’(6 5)
=
&’
- pa).
The observation
Hz which
is at rest near the origin,
non-zero
(Pt - .z) some components
of 7, suggesting the arrival pure
will
a covariant
the most efficient
method
so that
-y x a/m2.
be non-zero
formulation to handle
fields
energy.
to be the target
Note that tend
which
However
which
these large
terms
in the vector
are of order
and hence of order
The
of the quark,
the distribution
theory
implication
potential
is a of this
A will
have large fields
not
be
which
The leading ml/s
potential
have no effect
we compute
terms in 7 cancel and the field strengths
‘. For example,
the electric
field
along
the
is
the force experienced
breakdown
with
at large +y the potential
er(Pt - 2) [z’ + y2 + rypt
arrival
independent
are not in coincidence
since we will
E’(t.$) = p s i?g + !g = Thus
for large 7 and fixed
to a constant
uses the vector
this problem,
hadron
effect.
from Eq.(4.6).
z direction
can be taken
of the potential
the field strengths l/y’
_ z)‘] ’
does not lead to E or B fields.
have no physical
To show that
point
even at high
piece and hence
is that
ultimately
there
of the particle,
gauge
result
that
+ ys y;ypt
by a charge in the hadron
decreases
of quarks
as ml/s
H,,
but
at order
l/s2
is therefore
and has been demonstrated range
are residual
in hadron
of factorisation
due to the long
‘. There
nature
explicitly
of the vector
H,,
their
effects
at any fixed
vanish
Note
time before
In the realistic
which
at high
that
(4.7)
*
interactions
to be expected
in ref. [43]. field.
- .)‘I+
the
distort
energies.
A
in perturbation these effects
are
case of an incoming
-56-
colour that
neutral
hadron
there
the factorisation
theory. terms
suppressed
valid
description
4.3
Parton
centre
lecture
only
collision
nosities.
we will
in the full
consider
of s. The
fields.
vector
QCD
It is therefore
theory
boson
improved
than
in perturbation
production,
parton
possible
dropping
model
will
all
provide
a
process.
luminosities carry
of mass energy
hadron
colour
is even better
by powers of this
Since partons
are no long-range
property
In the next
FERMILAB-Conf-90/164-T
a fraction
parent
of a parton-parton
energy.
Consider
of their
collision
A convenient
a generic
hard
hadron’s
is less than
way to quantify
process
initiated
momentum
this
the available
the overall
is to define
by two hadrons
hadron-
pm-ton
lumi-
of momenta
P1
and P2 and s = (PI + Pz)‘,
O(8)= c w We may define
the parton 1
dLij YF=-
1+
If b depends
d+1 ‘d~z 4
J
only
luminosity
dxldzs
&j
fi(Zl,p)fj(zsvr)
(4.8)
as follows:
[(L~~~(xI,P)
on the product
~~j(~lP~,zaPZl,S(P)).
XJj(zg,P))
21~s the parton
+ (1 *
2)IS(T
cross section
- XI’S).
(4.9)
can be written
as, (4.10)
where
i =
z1zs8
object
in square
square brackets knowing
the sum now
brackets
We assume
The gluon jet cross section two powers
of as x 0.1.
over
all pairs
estimate
for the production that
4
= 2 TeV
can be calculated
of partons
of a cross section.
and is approximately
we can roughly
the cross section
= 40 TeV.
runs
has the dimensions
is dimensionless
the luminosities
can estimate at 4
and
by couplings.
cross sections.
and from
first
pi
in
Hence
As an example
jets with
Fig.(21)
to be approximately
The
The second object
determined
of two gluon
{ij}.
we
> 1 TeV
we find
10 pb after including
FERMILAB-Conf-90/164-T
-57-
---+A
x’s=6
TeV
Figure
21: Luminosity
I
plot
, II11h1
I “‘I
g E(q+$
v
< -
10’
-pp,
q/s=40
TeV
to’
-.- -‘pp.
u’s=17
TeV
10°
----+I,
u’s=6
TeV
lo-’ lo-’ lo-’
------pp
, ‘/s=1.8
---pb
, v’r=0.63 I I1111
I 10
30
\
“\
~1,
‘*.. I
: \
ssts :
TeV TeV I
\ ,I
:
‘\,
: :, : ;>
1000
100
Figure
luminosity
22: Luminosity
plot
3000
10000
-5%
10’
FERMILAB-Conf-90/164-T
uu luminosity
10’
g
:I:
c <
los
+ < -
10’
-pp,
qs=40
T&x
d’s=17
TeV \
“\\,
10’
-.-.-.-.pp,
loo
-----pi
, ds=tl
------pp
, v’.=1.8
---pp
, d/9=0.53
TeV
I I111111 30 100
f
lo-’ 1o-c 10-% 10
TeV
\
TeV
Figure
~Q“\\,, :.\,, \
%I : ‘!
‘> I
‘\
lllll~ 1000
23: Luminosity
3000
10000
plot
I ’ ” UU luminosity
”
10=
-pp.
u’s=40
TeV
10’
-.-‘-‘-pp.
d/9=17
TeV
, u’s=6
TeV
----pp
loo
-------pi,,
10-l
---pb
lo-’ 1o-a
TeV
, d/s=O.t?3 TeV
1 10
\/s=l.B
I ,111, 30
100
Figure
1000
24: Luminosity
plot
3000
10000
-59-
----pi,
ds=B TeV
----pp
, \/s=O.63
TeV
Figure
10’
:.‘
10’ ‘3;
106
A c 5
10’
?
10’
-
.:.
._ -..‘ic “‘~<~~~+
~
10’
-pp, -‘-‘----pp,
10°
----PC, ------pi
10-l
---pi,
10’ lo-a 10
25: Luminosity
plot
._,_ .;... \.
loa
FERMILAB-Conf-90/164-T
L. *..:.\+ . ‘\ .\.< X,‘.. *.., “1 ‘\ \, ***..< q..,, \ \
v’s=40
Ted
4.~17
TeV
i
‘i
Js=B TeV , ~‘~-1.8 ds=O.63
I I111111 30 100
Figure
TeV TeV I 1000
26: Luminosity
plot
3000
10000
-6O-
I
,
FERMILAB-Conf-90/164-T
I11111,
Uahinosity
-i- 10’lo10° lo-’
ds=6
------~6,
~‘~1.6
---pj,
10-l 10-
----pi,
, \la=0.63
TeV TeV TeV
10
Figure
27: Luminosity
plot
-61-
Large
5.
pT Jet
The scattering of QED.
The
processes
of large
QCD
discussion of direct
parton
photon
band
beams
terms of variables
and
in
After
is calculated
angular
distributions,
data.
simply
distribution boosted
We extend
the related
hadrons
beams
the
process
functions. with
longitudinal
boosts.
of of
to the centre
the final
state in
For this purpose
pr and the azimuthal
of momenta
two
The centre
respect
to classify
momentum
provides
have a spectrum
useful
the four components
of a particle
angle of mass
as
p” = (dpccosh(y),pT y is therefore
sin 4,pT cos 4, $wsinh(y))
defined
under
~ln(~),
the restrictive
the .z direction.
the rapidity
(5.1)
by
y=
along
under
of two
incoming
It is therefore
y, the transverse
of these variables,
and is additive
These
is normally
hadrons.
transform
the rapidity
In practice
collisions.
cross section
describe
scattering
by the parton
scattering
which
m may be written
to a boost
. . . can be studied
the experimental and finally
the
partons.
determined
of mass of the two incoming
The rapidity
the pi
tests
definition
lecture,
of incoming
mass of the parton-parton
4. In terms
with
cross sections,
previous
momenta
we introduce
We study
predictions
fundamental
jets in hadron-hadron
how the jet inclusive
and jet
in the
longitudinal
-+ 77, . . . provide
production.
Kinematics
As described
Collisions
qp -+ qq, qP -+ gg,
momentum
model.
multijet
e+e-
for &CD,
we show
the theoretical
to include
in Hadron-Hadron
-+ e+e-,
transverse
improved
and compare
broad
e+e-
some kinematics,
in the
5.1
processes
analog
the production defining
Production
FERMILAB-Conf-90/164-T
(5.2)
class of Lorentz
Rapidity
is normally
differences replaced
7 = -Intan(
transformations are boost
corresponding
invariant.
by the pseudorapidity
7,
(5.3)
-62-
which
coincides
with
experimentally, detector.
the rapidity
since the angle
It is also standard
momentum
for similar
by a jet.
There
in the m -+ 0 limit. 6’ from
to use the transverse
reasons.
analyses
is a cluster
of transverse
Many
methods
axis of the jet. the measured
5.2
jet cross-section
cross
two-jet
an incoming
events
which
partons
are produced
of-mass
frame.
result
of constant
depend
when
AR
used definition
describe
and of a jet
a circle
around
convenience,
the and
chosen.
equal
to produce
and opposite
one hadron
two high
momenta
partons
and balanced
from
transverse
conservation
are produced,
of the incoming
in azimuth
parton
From momentum
two partons
momentum
theoretical
where
on the value
an incoming
as jets.
with
If only
is meant
sections
are observed
be back-to-back
will
what
in the
+ (W)al.
from the other hadron
parton
partons
transverse
both
The cone size can be chosen at the experiment&t’s
Two-jet
In QCD,
lines
the transverse
than
be sure that
ET in a cone of size AR,
y, 4 plane,
rather
A commonly
AR = JNAY)’ In the two-dimensional
directly
energy
but one must
variable
is measured
can be used to define
use the same definition. energy
It is a more convenient
the beam direction
is no best definition,
experimental
FERMILAB-Conf-90/164-T
in the subprocess
is neglected,
then
momentum
off
momentum
the two final
and the relatively
in transverse
scatters
small
state centre-
intrinsic
the two jets
will
in the laboratory
frame. For a 2 -+ 2 parton
scattering
p**toni(pl)
described
process
+ P**tonj(Pz)
by a matrix
element
M,
-+
p**tonk(P3)
the parton
(5.5)
+ p&rton,(P&
cross section
is
(5.6) All
parton
grams
shown
Expressions
processes
which
in Fig.(28) for
the
contribute
by including
leading
order
in lowest other
matrix
order
diagrams elements
can be derived which
squared
are related CIMI’,
from
the dia-
by crossing. averaged
and
-63-
FERMILAB-Conf-90/164-T
lb)
Figure
28: Diagrams
summed
over initial
notation
i = (pl + pz)*, i = (~1 - p3)a and fi = (pa - P~)~.
The two-jet contribution
cross section
state
(k,Z)
partons.
and
colours
are given
as a sum of terms
due to a particular
Using
Eq.(5.6)
the result
in Table
3 in the
each representing
combination
the
of incoming
(;, j)
for the two jet inclusive
cross
is,
d%
fi(~~ll))(~j(~~))CIM(ij
dYsdYrdP$ where
spins
may be written
to the cross section
and outgoing section
and final
for jet production
the
u,c,d,&g laboratory
=ik+g
fi(z,,u) I...
etc.),
rapidities
3 represent evaluated
+
hl)j2 &
(5.7)
I the
number
distributions
at momentum
of the outgoing
partons.
for
partons
scale p, and ys and For massless
partons
of type
i (i
ya represent the
= the
rapidities
FERMILAB-Conf-90/164-T
-64-
Process
em=
CIW/d I
1
nl2
I 4 2 + 2 --p-9 3.26
0.22
nP-‘qP
2.59
99
Table
processes
99-99
;(3-g-;-i!)
over initial
the that
statistical
(aooSt)
of the two jets observed
30.4
squared
The colour
factor
necessary
and jet algorithm
jets may be identified
We now consider rapidity
9
CIiVI”
for two-to-two
and spin indices
may be used interchangeably.
the detector
of the outgoing
iti
elements
partons.
+
parton
are averaged
sub-
(summed)
states.
and pseudorapidities
assume
matrix
massless
(final)
introduces
9
invariant
72 + ia
4 P+tP
--
with
1.04
3P+51 1 i’ + 7? --~ -i; 8 i1 6
+clq
gq-+!-Jq
3: The
8 i’ + GiL)
32 i2 + 6’ z---Jg--iyr
qq+99
the kinematics
of the two-parton in the parton-parton
rapidities
with
for
The
identical
final
are 100% efficient, those of the outgoing
of the two produced system
Kronecker
and the equal
centre-of-mass
state
function
partons.
the rapidities
If we and pi
partons.
jets in detail. and opposite
system
delta
are given
The laboratory rapidities in terms
(&y’) of the
by: nooat = (YS + Y,)/2,
Y’ = (Ys - yr)/2.
(5.8)
-65-
For a massless
parton
the centre
FERMILAB-Conf-90/164-T
of mass scattering
angle
0’ is given
by,
cos8’ = g = fo;;;,, = tanh(Y3 ; y’), where
y’ = ys - yb,,,,,,.
the laboratory
frame
The
longitudinal
Eq.(5.7)
are given
determines
in terms
zr
= 2p~/Js.
fractions
of p~,ys
cosh(y’),
of the rapidity
the subprocess
momentum
z1 = zTe-*where
The measurement
of the
the invariant
of the two jets in
of mass scattering
incoming
and y, by momentum
ra = xre-**s’
Lastly,
centre
difference
partons
angle 0’.
zr
and
conservation: y,,oo,r = 1, 2 n 21 + , 1
cosh(y’),
za in
mass of the jet-jet
system
(5.10)
can be written
as, Mj, Given
a knowledge
periments, duction
Eq.(5.7)
of the parton
the parton
level
distributions
from
may be used to make leading
in hadron-hadron
collisions.
may be obtained
(5.11)
= i = 4~; cosh”(y’).
order
For example,
by integrating
deep inelastic QCD
predictions
the inclusive Eq.(5.6)
scattering
jet
ex-
for jet pro-
cross section
at
over the momentum
of one
+ i + Ii),
(5.12)
of the jets. Ed35 d=p where
t and c are fixed
d%f E dyd=pr
= $&~lIlq’6(i
by i and the centre
of mass scattering
angle,
i = --i (I -case*) ij
Again p&,,,,,
assuming
that
the single
the parton
2
the detector
jet inclusive
-;
functions:
E,d=& -=d%
1 167Gs i,j, 2x, =p,g xIM(ij
-+ kl)l’
(5.13)
COSP).
and jet algorithm
cross section
distribution
(l$
are 100% efficient,
is obtained
from
Eq.(5.12)
so that
p,:, =
by folding
in
d+ldzl fi(zl*P)fj(zlvP) Xl =a &6(2
+ i + 6).
(5.14)
FERMILAB-Conf-90/164-T
-66-
Note that tion
this result
corresponds
is made between
5.3
quark
Comparison
Although
with
has been studied
at different
and UA2
collaborations
at the CERN
pp collider
that
difficult
collaboration
only
at the FNAL
at these very
Two quantities
the jets from
useful obtained
Ed=0 -q-= the third
Fig.(29)
term
shows
CDF
collaboration.
order
(i.e.
from
reference
O(ai))
dependence
follows
The
[38]. The
about
The
from
width
next-to-leading
centre
matrix
elements.
scattering
energies
with
by
are approximately
calculated
contributions
massless.
= 1.8 TeV,
parton
precise
reduce
treatment especially
in the theoretical
the
distributions
considerably
is excellent,
from
in next-to-leading
the HMRSB
the
of effects considering
prediction.
quark-gluon
at the lower
The
(5.15)
prediction,
comes from
it is
experiment.
cross section
a more
and
in the event.
at fi
agreement
no free parameters
gluon-gluon
quantity
parton
The
theory
the jets
~1, and allow
of the jet.
At lower
in pp collisions
order
= 1.8 TeV).
dETdq
et al. [44] and using
half the cross section
at the high second
that
(&
(ru
2~8~
the
does the identification
the inclusive
is the QCD
on the scale parameter
coming
scattering
curve
collider
hadrons
1
the jet ET distribution
there are essentially
this energy
from
i.e. from
= 546 GeV and 630 GeV)
energies
for comparing
over a period
pp colliders,
(4
‘underlying’
if we assume
machines
unambiguous.
d=a -4-d=pdy
by S. D. Ellis
due to the finite
relatively
energy
Tevatron
collision
the other
are particularly
is the jet pr distribution,
where
high
of large pT jets become
to separate
no distinc-
experiment
the high
the CDF
and that
jets.
data are from
measurement
half
and gluons
the definitive
It appears
that
quarks
years,
and from
first
and gluon
large pi jet production
of many UAl
to massless
Note that
scattering,
at
the other
ET end, and quark-(anti)quark
ET end. of interest
of mass, the angular The differential
an angle 0’ to the beam direction
is the jet distribution
angular
distribution.
is sensitive
In the psrton-
to the form
of the 2 +
cross section
for a jet pair of mass MJJ
produced
in the jet-jet
centre
be obtained
of mass can readily
2 at
-67-
Inclusive
jet
10-
cross
FERMILAB-Conf-90/164-T
section
(AR=0.7)
uncertainty
I Normalisatmn
7
$
10-l
24 ij
lcip
$ $
lo+
1o-4 : \ I
1 I
I
I
I
100
I
I
I
I
I
I
200
I
I
300
I
I
I
400
ET [-VI
Figure
29:
Jet
next-to-leading from
Eq.(5.7)
ET
distribution
order using
QCD
from
prediction
the from
CDF
collaboration,
compared
with
a
[44]
the transformation dp$dysdyr
E ;drldzldcos
8’
(5.16)
to give ba dMj,d
CO8
dzld+l
8’ z.7
TJdlij(rJ,p)
-
9
dn
fi(zl,P)fj(z~,P) dk’j dcose”
~(ZIZZS
- #J)
dtf:e.
(5.17)
-68-
with
7~ = Mj,/s
and dg “I‘j
=
d COS 8’
32,hj,
Note that for each subprocess Thus,
FERMILAB-Conf-90/164-T
ClWG
the d+/dcos
--t kUl’&
6” is symmetrised
in t^ and 6 (unless
k E I).
for example,
d@
na; -2Mj,
=
dcos6” Numerically
at small
4 + (1 -
9
(1 -
(I +
characteristic
have the familiar
of the exchange
d+ d cos 8’
1
(5.19)
are gg + gg, gq --t gq and qq --t qp. For
subprocesses
the B’ distributions
angle,
cose*)~ cose*)l .
4 + (1 +
the most important
each of these,
cose*)~+ co8 e*)l
4
Rutherford
of a vector
scattering
boson
behaviour
in the t-channel:
1 N
(5.20)
sin’($)’
It is convenient
to plot the data in terms of the variable
ford singularity
[48],
x, which
removes
the Ruther-
case* l-case-’
I + x= In the small
Data
angle
limit
on the angular
with
the leading
that
these
data
For example,
(x + co) the cross section
distribution
order
QCD
a model
are very
of the colour
imation
the gg +
scatter
by exchanging
the angular
4/9
quark
a scalar
of the Feynman can be used
the angular
ratios
respectively. diagrams. as the
dependence
Note
mechanisms. gluon
would
angle. of the dominant
shows the cos 0’ dependence
and (4/g)’
in Fig.(30),
agreement.
scattering
dependences
to gg -+ gg. These
structure
i.e.
at small
are shown
is excellent
quarks
Fig.(31)
gg subprocess
in Eq.(5.17),
there other
that
in x is then
collaboration
certain
(sin-r(P/2))
values
differential
out
normalised
at the numerical
given
Again,
to note
similar.
in terms
result
prediction. rule
behaviour
qij -+ qcj subprocesses stant
the CDF
in which
It is also interesting processes
from
automatically
give a less singular
(5.21)
of the qg -t qg and
are evidently This Thus
‘universal’ effectively
sub-
rather
con-
can be understood to a good subprocess factors
approxin the
out leaving
-69-
I
I
I
1
I
I
10 -
I
FERMILAB-Conf-90/164-T
I
I
I
I
I
I
I
I
MU > 200
GeV
CDF
statistical
data,
errors
only
6-
6 -.
2-
I
0
I
I
I
I
2
I
I
I
I
4
I
I
6
I 6
I
I
I 10
X
Figure order
30: QCD
x distribution
of parton
appmcimation
[48].
Multijet
collaboration
compared
with
the leading
This
distributions.
is called
the single
eflectiue
subprocess
production
As long
as the jets
sections
can be calculated
in the final
the CDF
prediction
a convolution
5.4
from
state.
are required
from scattering
In this
jets which
satisfy,
In leading
order
‘tree-level’
Feynman
to be well
QCD,
processes
way one defines
say, p$ > pi@“,
separated
an n-jet
in phase
involving crms
/q’] < qmaa and ARij
these
cross sections
diagrams,
i.e. diagrams
are calculated without
space,
many section > A&i,,
multijet
quarks
and gluons
u” for producing
n
for i, j = 1, . . ..n.
at the parton
any internal
cross
loops.
level
from
The general
-7O-
.6
I
I
I
FERMILAB-Conf-90/164-T
I
I
1
I
I
I
5
(4/Q)
-___--------------------------------.4
.2 -
.l
qg
+
gg
-+ gg
qg
(4/Q)’
__________-_____________________
-
0
__ _ __ _ _
I
-? ss
gg
+
I
I .2
0
ss
.m
I .4 lcos
Figure that
31:
Quark-antiquark
and
I .6
I
I
I
I
.6
1
e-1
quark-glum
angular
distributions,
normalised
to
for 99 -+ 99
expression
is again q”
obtained
from
Eq.(4.8):
=
dzldz*
fi(zl,
p) cij-k-k=.
~)fj(~1,
(5.23)
iAh,...,z ~=P,P % The matrix Since
elements
each n-jet
geometrically Events tudes
with
cross section
with with
for all the 2 +
increasing
three jets
two incoming
the two-to-three
parton
2,3,4,5
is proportional
QCD
processes
to a;,
are known
exactly
the cross sections
fall
[45].
roughly
n.
at large partons scattering
transverse and three processes
energy outgoing
are described partons.
have been given
in QCD
Very elegant by Berends
by ampliresults et al.
for [46].
-71-
For a complete
description
it is sufficient
FERMILAB-Conf-90/164-T
to consider
the following
four
processes.
(A) qh) + d(n) + q(m)+ q’(pr)+ s(k) (W q(pd+ q(n) -+ q(m)+ n(n) + g(k) cc) ‘?(P.)+ hb) -+ dP1) + !7(P2) + dP3) cm The
momentum
elements
dP1)
assignments
for two-to-three
above four
+ !dPz)
dP3)
for the partons parton
+ dP4)
are given
amplitudes
(5.24)
+ dPS).
in brackets.
may be obtained
AU other
by crossing
matrix from
the
processes.
The matrix
elements
squared
the initial
(final)
colours
and
the quarks
equal
to zero.
With
element
+
[47] for process
~(@)I’
= ‘$
The kinematic
For compactness
for the processes spins
are given
the momentum
(A - D),
below.
averaged
We have
assignments
(summed)
set the
of Eq.(5.24)
over
masses
of
the matrix
(A) is, (“’
variables
+ “‘;+$
+ “‘s)
are defined
(2C+4]
341).
(5.25)
as follows,
9 = (Pl + Pl)“,
t = (PI - pay,
Q’ = (P3 + PJ,
t’ = (pa - p,)“,
of notation
+ [23]) + $2;
we have introduced
‘11= (PI - pa)?, (5.26)
11’ = (p2 - pg.
the eikonal
factor
[ij]
which
is defined
as, (5.27) We have also defined
the following
[12;34] Note
that
pendence
this
combination
on the SU(N)
sum of eikonal
terms,
= 21121 + 2[34] - [13] - [14] - [23] - [24]. is free from colour
group
collinear
is shown
singularities. explicitly,
In Eq.(5.25)
(CA = N = ~,CF
(5.28) the de= 4/3).
FERMILAB-Conf-90/164-T
-72-
In the same notation
the result
for process
(B)
with
four identical
quarks
[47] may
be written,
xlM(B)l’
g’cF N
=
2 + 8’1 + 211+ 21’1 2tv
( +
g*cF N
_
22121’
(
To write notation
the
written
2&([13]
+ [24]) + $[12;34] > 2cF(
4tvuu
[12] + [34]) + $12;
341
)(
results
for
for the dot product
the
remaining
the momentum
two
.
> (5.29)
processes
we introduce
a compact
of two momenta, {ij}
Using
>
I( - tt’ - d)
(8’ + 8”)(88’
N1
+ [23]) + -$12;34]
>(
2 + d2 + t2 + t’l
2g’cF
2C~([14]
assignments
Z pi ’
(5.30)
pj.
of EqJ5.24)
the
result
for process
(C)
may
be
as [46],
1) 3 {ai)(bi}({ai}~+ {bi}y ,= {al}{a2}{43}{al}{b2}{~3} (z; x lab] + N1 tab} - ~‘u1”a2~;;42”a1’)
~pq
= m-;
~4W~(~~lHb2~
)I.
+ {WbllJ {23)(311
The
sums run
final
state Using
over the three
cyclic
permutations
P of the momentum
labels
(5.31)
of the
gluons. the momentum
labels
of Eq.(5.24)
the result
for process
(D)
is [46],
(5.32) The sums run over the 120 permutations These matrix
elements
sion of soft and collinear
of the momentum
display
the typical
gluons
predominating.
bremsstrahlung This
labels. structure
is particularly
with
the emis-
clear
from
the
FERMILAB-Conf-90/164-T
form
of the result
from
the region
given
in which
can also show that production
centre
the eikonal
are large.
to specify
and two
respect
to the axis
(massless)
partons
is specified
defined
written
using
valid
by the colliding
such that
direction,
the three
then
particle
The
at
variables
the three final-state
last
system
variable
of the outgoing
with
is an overall
partons
scaled such
zs > +* > zs and Bi is the angle between
the subprocess
massless
Two
of the three-jet
partons.
[49].
configuration,
variables. between
one
for two-jet
production
parton
is shared
come
results
is relevant
for three-jet
the final-state
If zs, L,, and zs are the energies
i and the beam
which
serve to fix the orientation
+s + z, + 2s = 2 and ordered
parton
function
energy
contributions
From the tree graph
by five independent
how the available
variables
angle.
structure
good approximation
of mass energy,
partons,
azimuthal
factors
the same effective
final-state
are required
that
w h em the dominant
is also to a very
For three fixed
in Eqs.(5.25,5.29)
differential
cross section
can be
phase space of Eq.(2.6):
d’& cm Bldpb = (10;4,r)CIM12~
drsdx4d In EqJ5.33)
the variable
and the plane There
containing
is again
jet-l
excellent
the experimental zs measured
11,is the angle between
data.
Direct
High
transverse
jet
scattering
phase space alone.
5.5
production
view,
better energy
are two
of jets:
of direct
direct
closely photon
the energy than
scale are smaller. of photons
need for a jet algorithm
data
containing
by the incoming
between
the above theoretical
Fig.(32)
sh ows the distribution
The
solid
amplitudes, clearly
line
and jet-3
partons. predictions
and
in the variable
is the prediction
and the dashed
favour
jet-2
from
QCD,
line is the prediction
the former.
production
momentum
for photons
and energy
The
photon
the study
the study
agreement
collaboration.
based on the 2 -+ 3 parton from
and the axis defined
As an example,
by the CDF
the plane
photon
related
production phenomena.
production
resolution
has several
and systematic
transverse
momentum
an experimental advantages
with
calorimeter
point
of
respect
to
is generally
uncertainties
on the photon
since photons
do not fragment,
the direction
measured
in the calorimeter
is straightforwardly which
From
of the electromagnetic
it is for hadrons, Furthermore,
and high
is required
to reconstruct
a jet.
Only
without
the
the relatively
-74-
FERMILAB-Conf-90/164-T
700
/:+;t\,\! : : ?J\\ : $I :: : : A 100
600
500
400
300
200
1
:
‘, :
Ii,
I\
o”I’II”““““““‘~ .5
Figure
32: Distribution
measured QCD,
in the variable
by the CDF
collaboration.
.6
ii .7 3%
zs and +, in a sample The solid,
dashed
lines
.6
of three
: .Q
jet events,
are the predictions
as
from
phase space respectively
low rate for the production jet production quantitative
processes QCD
The leading the Compton squared
example,
subprocesses
process
on the values of 4
have limited
photons
and the non-negligible
the usefulness
of the direct
background photons
from
for making
tests.
order
are given
of direct
are (a) the annihilation
qg -t yq shown
in Table
4. Depending
and p~(s pg), either
in proton-proton
process
dominates
while
process
is more important.
in Fig.(33)
process
The
on the nature
invariant
in proton-antiproton
collisions collisions
matrix
of the colliding
of these two subprocesses
or proton-nucleus
qcj -+ rg and (b)
at medium at high
elements
hadrons
and
can dominate.
For
pr
the Compton
pi the annihilation
-75-
FERMILAB-Conf-90/164-T
b)
Figure
33: Diagrams
for direct
photon
or vector
boson
at large PT.
production
Process (N2 - 1) t’ + IL’ + 2s(s + t + n) nq+r*s
r
4:
spin
indices
Lowest
order
are averaged
(s + t + u) = 0.
1 81 + us + 2t(a + t + u)
--
gq+-r*‘I
Table
tu
N1 *u
N
processes (summed)
for
virtual over
photon initial
(final)
production. states.
The For
colour
a real
and
photon
-76-
FERMILAB-Conf-90/164-T
10 > s a a
a "0 ‘b D w
1
D ::
WA70
pp-XX
-----
HMRSiB) HMRSI E)
IO-'
pT (GeVlcl ;ure
34: Direct
pi distribution
photon
rves are next-to-leading All direct he most
photon
he latest
QCD
at J;;
the WA70
= 23 GeV.
based on the next-to-leading HMRS(E,B)
hese two sets are chosen
parton
by the WA70
calculations,
data show good agreement
precise data is from
zta on pp -+ yX :ctions,
order
measured
as described with
collaboration
Th e curves order
distributions
to fit the WA70
QCD
over a large energy
[50].
Fig(34)
of Aurenche
1381. In fact
The
in the text
shows
are the fully-corrected
calculation
data.
collaboration.
the gluon
QCD
range. WA70 cross
et al. [51], using distributions
in
-7-T-
The
6.
Production
In this lecture collisions.
we review
We begin
annihilation
into
standard
6.1
with
The
massive
photon
second lecture,
- the Drell-Yan
process.
we next
special
emphasis
is easily
obtained
because
of the averaging
section
for the production
colour
matches
singlet
final
components
from
hadronic this
with
state
du dM=
of the
quantity
process
QCD
review
of the
of W, 2 production effects.
to a lepton
pair via an intermediate
-+ 49 cross section
the e+e-
presented
in the
factor
qq --t e+e-
of l/N
place.
of the initial
- MS),
of the antiquark In the
can be written
us dzldr~S(rp~s = 3 %
quarks.
that
The
partons
~(21,0,0,21)
Pa =
~(22,0,0,-+d
centre-of-mass Using
energy Eq.(4.1),
factor
of l/N’
differential given
cross
by
only
when
the colour
can the annihilation
centre-of-mass
=
by 6 = zlzzs.
by a colour
k-0 = g.
is due to the fact
of the incoming
parton
is smaller
pair of mass M is therefore
= ??Q:S(j
the colour take
4naa 1 = TNQ:.
efe-)
of a lepton
of momenta
square
on perturbative
over the colours
Pl
The
a brief
the phenomenology
annihilation
process,
- d& dM=
quark
After
by quark-antiquark
Eq.(2.3):
the time-reversed
The overall
pairs
in hadron-hadron
mechanism
c(qcj + Note that
discuss
Collisions
boson production of lepton
for quark-antiquark
photon
of vector
in Hadronic
the production
model,
Drell-Yan
The cross section
Bosons
the physics
by discussing
a virtual
electroweak
in pp collisions,
of Vector
FERMILAB-Conf-90/164-T
frame
of the two
may be written
i is related the parton
into
of the a colour
hadrons
the
as
to the model
corresponding
cross section
for
as - Ma) [T
Q:(s7k(Q,f‘)&(Q,P)
+ 11 *
21)
(‘3.4)
-7%
Apart
from
pair
the mild
cross section
M3da -=dM
exhibits
8m+r
behaviour
scaling
functions,
the lepton
7 = Ml/s:
1
Q:(crd~~,~)%(=z,~) + [1 t+ 21) = F(r).
% the rapidity
in the distribution
in the variable
dwM(wr~)[-&
3N
From Eq.(6.3)
logarithmic
FERMILAB-Conf-90/164-T
of the produced
lepton
pair is found
(6.5) to be y = l/2 ln(zJzx),
and hence z1 = &e”, The double
differential
cross section
du
all = N,
dM=dy with
z1 and ~2 given
mass of the produced functions
rections graphs
lepton
there
Q:(qk(“l,/‘)&(%/‘)
pair
+ [l t-t 21)
By measuring
the
one can in principle
I
distribution
measure
in rapidity
the quark
and
distribution
hadrons.
exists
a systematic
to all orders. shown
(6.6)
is therefore
by Eq.(6.6).
of the incoming
In QCD
22 = &e-“.
The
procedure
next-to-leading
for calculating
order
corrections
the perturbative are obtained
cor-
from
the
>,
(‘5.8)
in Fig.(35):
do -=dM2
(rr, Ns 4
drld+2dzS(zlr~r
- 7)
‘?:(Pk(“~ti‘)s%k(%P) + [1 ++21)Ii s(l - 2) + Q:(g(%P)(Qk(=%P) where
the correction
f,(z)
= ;
terms
The overall
size of the center-of-mass
- 2) + ; - 52 + ;*a
are defined
O(as)
21)] ~$fAz)]
are [52,53]
(2 + (1 - z)l)ln(l
and the plus distributions
+ f?kk(%f‘)) + [I ++
correction
energy.
as in Eq.(3.27). depends
At fixed-target
on the lepton energies
(6.9)
1 )
pair
and masses
mass
and on the
the correction
is
-79-
FERMILAB-Conf-90/164-T
>+P-+Y-+k”-
Figure large
35: The leading and positive,
(negative)
overall
the lowest
order
Several
mass lepton
small
2: behaviour
is proportional
sea quark collaboration
from
the quark-gluon
by about
pieces
of the parton
in high distributions.
to the sea quark to deep inelastic
distributions
scattering
distribution,
data,
in the latest
[54], compared
with
term
the f, term
is quite
large small.
correction
r, the
However
is more important
the O(as)
can be obtained energy
hadron
from
collisions
In pp or pN collisions ~(z,P).
This
and in fact Drell-Yan MRS
of relatively
process
and the increases
25% - 30%.
of information
production
for the Drell-Yan
regime
For W and 2 production
cross section
pair
diagrams
In this
where T is much smaller,
is smaller.
important
Low
information
50% or more.
energies,
correction
order
of order
contribution
at pp collider
and next-to-leading
data.
is sensitive
to the
the cross section complementary
data is used to constrain
fits [38]. Fig.(36)
the next-to-leading
provides
Drell-Yan
shows data from
order
QCD
calculation
the
the E605 using
FERMILAB-Conf-90/164-T
-8O-
,()3
~~,~7~~~, .,. ‘FT’i~
E ‘\\
1
E 01 u z N-
/~~I~~ r-,-~,-r~r7-rrrn-mrim~~~--
x
\\ j_ hj j\
1'
\
>, w
‘~\. \~
f
‘1~
2 ::
1 \ ‘1
E605
pN---+k-X
1 ~, -----
HMRS(B1 HMRStE)
\
-x ‘F, "
o., -.l.
Figure
36:
theoretical
Drell-Yan
from
the E605
collaborstion
with
next-to-leading
order
predictions
the HMRS(E,B)
distributions.
Equally
in pions can be extracted
of quarks
A comprehensive Fig.(37)
data
review
ing the effects
from
of Drell-Yan
shows the predictions of the 2 pole.
important
is the fact
Drell-Yan
data
phenomenology
for lepton Fig.(37)
that
the distr!.butions
in =p and
can be found
pair production at collider the influence al so illustrates
TN
collisions.
in reference energies, of higher
[55].
includorder
corrections.
6.2
W
and
The discovery
2 production
in 1983 of the W and 2 weak bosons
of the Glashow-S&m-Weinberg we discuss
the physics
of W
electroweak and
model.
2 production
provided
dramatic
In the remainder in pp collisions,
confirmation of this lecture
starting
with
an
-81-
FERMILAB-Conf-90/164-T
~““““““““““““““~ a+a-
100
;
h
tt F lo :\‘\\ II ::i, ‘\\ ; 5& \\’\\ 8!3 : \\ \ l.J \\\\ .J/:” :
pairs
from
DY
-
\/S
=
1.8 TeV
with
----
\/S
=
1.8 TeV
without
----
\/S
= 0.83
TeV
and
Z
O(as)
with
O(a,) O(as)
1
%\ ’ :‘,1’ ..__’ ‘,\\’ \\ ‘\\\ ‘\ N %\ \.‘\ ‘1 ‘\ ‘. ‘.‘. -. -. ‘. ‘. \ --. L
E
Y G
.l
.Ol
i .oo 1
rl I I I I I I I I I I I I I I I*p.lI I I I Ii-50
Figure elementary
37: The
actions
predicted
introduction
The Lagrangian
150 M [GeV]
100
e+e-
pair
200
production
to the electroweak
250
cross section
J
300
in pp collisions
model.
for the Glashow-Weinberg-S&m
model
of the electroweak
inter-
is
&WS
=
-z
xqi-yP(l
- 2 capon
where
l?w is the Weinberg
raising
and lowering
- r”)(T+WL
x$ir’( ,
- e ~Q$&&d,, t (6.10)
vi - Ad)$~Zp,
angle
operators
+ T-W;)&
and gw
= e/sin
and the vector
Bw.
and axial
T+
and T-
couplings
are the isospin
of the Z are given
by vi = t&i)
- 2Qi sin”(Bw),
Ai = k&(i),
(6.11)
FERMILAB-Conf-90/164-T
-82-
where
ts~(i)
is the weak isospin
and Q; is the charge level
the Fermi
of the fermion
of the fermion
constant
in units
can be written
using
the Josephson
effect
GF Using
neutrino-nucleon predictions
total
of the coupling
At tree-graph
SW: (6.12)
constants lifetime
137.03604(11)
=
1.16637(Z)
are measured
to high
precision
respectively:
lo-’
x
angle derived
cross sections,
(6.13)
GeV-‘.
from charged
and neutral
sina Bw = 0.23 [56], we obtain
deep inelastic
the leading
order
for the masses: MW
Mz The most
The
coupling
=
the value for the Weinberg
charge.
=x
and the muon
a-’
for di and ei),
GF
0M& and Fermi
for ‘Al; and Vi, -i
of the positron
in terms
&
The electromagnetic
(+i
recent
=
=
-
measured
values
(6.14)
a~ 89 GeV.
cm ew
[57,58,59]
for the masses are
Mw
=
79.91 i
0.35(stat)
f 0.24(sys)
f O.lS(scale)
: CDF(ev)
Mw
=
79.90 f 0.53(stat)
f 0.32(sys)
f O.OE$scaIe)
: CDF(pv)
Mw
=
80.79 & 0.3l(stat)
f O.Zl(sys)
f 0.8l(scaIe)
: UAZ(ev)
Mz
=
91.49 5 0.35(stat)
+ O.lZ(sys)
& 0.92(scale)
: UAZ(e+e-)
Mz
=
differences
between
ments
are due to higher
taken
into
account
In analogy
: LEP
91.150 f 0.032
with
the predictions order
electroweak
[60], the agreement the Drell-Yan
in Eq.(6.14) perturbative
between
cross section
theory
+ SLC
and the experimental corrections. and experiment
in the previous
section,
(6.15) measure-
When
these are
is excellent. the subprocess
-83-
cross sections
for W and 2 production >d-.W
&.P+~ where
V,,
The O(crs)
section
is ‘flavour
(times
measurements
from
and statistical
errors
Mz
= 91.16
tion
GeV.
to allow
electroweak
leptonic
tutes
a non-trivial
calculation
they
scattering
data.
and couples
ratios
experiments
bosons
- see next
been
error
Note
due to the parton
and - most significantly Evidently
on the evolution
are being
evaluated
and
Z decay
At leading
order
in electroweak
I-(2’
where
N is a normalisation
latter,
the W+
to be Mw
The
= 80 GeV,
set, with
the
Note
known that
distributions,
higher
p values
theory
the partial
predic-
to higher
partially
than
this
scale
order O(ai) consti-
since in this the deep inelastic
properties perturbation
(in the standard
W’+
good.
of the parton
at much
in quadrature.
distributions,
is very
with
the systematic
on the theoretical
- to the only
the agreement
compared
that
HMRS(B) band
for the W and
section)
chosen
are the
a ilO%
in the previous
predictions
[61,62,63].
distributions
is the same
in the same way to the annihilat-
have
check
are given
of the same mass) discussed
of the vector
W
2 bosom
element.
have been combined
parton
(6.16)
to the W and 2 cross sections
branching
We have included
[64].
matrix
on the measurements
corrections,
corrections
- Mi),
sh ows the theoretical
for the uncertainties
QCD
6.3
blind’
Fig.(38)
The
p = Mw,z.
+ A:)&(;
correction
the pp collider
of the masses
choice
:fiGFMi(V,’
to be
IV,q~I” a(; - M$)
(for a photon
and antiquark.
2 cross sections
values
=
QCD
correction
- the gluon
ing quark
i4iiGFM&
calculated
Kobayashi-Maskawa
perturbative
as the Drell-Yan
are readily
=
is the appropriate
FERMILAB-Conf-90/164-T
+
model)
GF%$ 6Jzs
=
N
--+ ff)
=
N s(V;
which
of the W and
by
ff”)
factor
widths
(6.17)
+ A;),
is 1 for leptons
decay rate refers to the sum of the decays
and 3 for quarks. to a given
quark
For the of charge
-84-
3
,,ll,,,,,,l,i,,l,,,
FERMILAB-Conf-90/164-T
.3c, L
- $ UAl 2.5 -f
z
(1989)
UA2
(1990)
CDF
(1990)
- 5 UAl
(1989)
.25 - $ uA2
(1990)
,I’-
2
-
,,‘-
.2 1
T al
1.5
5 a 1 b’ I’ ,’ .5
:I / #’
0
I’1111’1111’1114’01 1000 1600 600 ds (GeV)
I
Figure
38:
Comparison
2000 (GeV)
ds
of W and
2 cross
section
measurements
with
theoretical
predictions i and all antiquarks
of charge
there is an additional Using modes. mr > mw
factor
these relations By counting
5, e.g. W+ --) ud + US + ~6. For any individual
from
the Kobayashi-Maskawa
we can calculate decay
modes
mixing
the branching
we obtain
for the
ratios
mode
matrix.
for the observed
W (if the top quark
decay
is heavy:
- ms), BR(W+ BR(W+ BR(W+
--te+fi,/~+~,r+o) --, ud + US + &) --. cd+
c.? + ca)
=
3+;+3
x
33.3%
x
33.3%.
z 11.1%
(6.18)
-85-
FERMILAB-Conf-90/164-T
For the 2 we obtain e+e-
w&
[l + (1 - 4 sin’ ~9,)~]
I [2]
Choosing
ufi I [l + (1 - t sin’ Bw)‘]
-+ e’e-,p+p-,r+r-)
Note the large
branching
Although
the hadronic
at hadronic
colliders
production.
A statistically
fraction
x
20.4%
BR(Z’
+ u&cz)
ss
11.8%
da&.$
EZ 15.2%.
observing
the top quark.
serious
significant
signal
quarks
quark
is reduced
from
b&) into
relative
background
t6 is of great interest
from
the partial given
quarks.
normal
modes,
QCD
by the UA2
two-jet
collabora-
since it offers the possibility
the mass of the top quark
the expression
and bottom to the leptonic
has been reported
equal to zero),
(6.20)
neutrinos
are enhanced
is B very
Taking
the mass of the bottom
+
of the 2 boson
[65]. The W decay mode into
3.4%
p#<)
decay modes
there
x
BR(Z’-+
BR(Z’
bottom
(6.19)
[l + (1 - i sin’ 0w)‘]
sin’ Bw = 0.23 gives BR(Z’
tion
dd
into
width
account,
(but
of the W into
for qiqj above.
The correct
of
setting top and result
is Iyw+ qw+ where
TW = m:/M&.
given leptonic
depending
channel,
tq
-+ e’u.) Counting
= 3(v,&l
such as e+v.,
is forbidden.
if rnt < Mz/2.
Including
- TW)(l
up all modes
- ?(1+
we see that
TW)), the branching
ratio
into
a
is
on the mass of the top quark.
the top quark ratios
+
A massive
The larger top
quark
value
holds
can also
the effect of the top quark
affect
in both
when the
the decay
to
2 branching
the matrix
element
and the phase space we find qz”
+ tq
l?(ZO 4
ZLTq
=,/~[1+(1-~sin’t?~)‘+2r~((l-~sin’Bw)’-2)]
(6.23)
-86-
where
TZ = m:/rn%.
Because on mt and
the total
widths
(for
2)
the production
the
(and hence the branching
on the
and leptonic
Nowadays
we know
and from
direct
rnt
FERMILAB-Conf-90/164-T
> 89 GeV
[66].
a light
still
contributing
with
collider
so we consider
the widths
instead input
W and 2 decays
an indirect
decays
on these quantities. at LEP
that
that
N, = 3,
p&i collider
the collider
It is not impossible, could
of
evade direct
that W and
for example, discovery
while
width.
of the W and 2 are hard method,
which
[67]. The idea is to express in terms
measurements
at the Tevatron to check
these results.
W decay
species,
information
collaboration
non-standard
of the W and 2 depend
of the 2 width
nevertheless
with
to the total
neutrino
measurements
by the CDF
are consistent
At a hadron
of light
rates can provide
It is important
top quark
of theoretical
decay
precision
searches
2 measurements that
from
number
ratios)
of the ratio
to measure
however
the ratio
of production
requires
directly,
a certain
R of the number cross sections
and
amount
of observed
and branching
fractions: R = Number
of decays
W +
TV
Number
of decays
2 +
ee
R The input
ratio
theoretical
distributions. predictions
theoretically,
z
with
In Fig.(39) are shown
each prediction
is indicative
[38]. The most
recent
The results
=
BR(W ’ BR(Z
--+ ev) +
ee)
= R,
RBR
qw + rv) r(z -+ d) B(W -+ Iv) BR= B(Z + z+z-) = r(w -+ d) r(z -+ i+k)’
12, is calculable
parton
ow
theory
as functions
of the theoretical
experimental
9.38+;:;‘:(stat)
R =
10.2 f O.B(stat) consistent
with
error
due to ignorance
is compared
with
of mt, for N, uncertainty
measurements
R =
are evidently
a certain
(6.24)
experiment.
= 3,4,5.
from
of the
parton
The
The band
on
distributions
for R are [62,68]:
f 0.25(sys)
:
UA2
f 0.4(sys)
:
CDF.
the N, = 3, mt > 90 GeV hypothesis.
(6.25)
-87-
FERMILAB-Conf-90/164-T
NV=5
12 -
a
40
60
Figure
6.4
a0
m, [GeVl
Lepton
100
120
39: Theoretical
angular
values
distribution
Another
important
test of the theory
current,
Eq(6.10).
For the process d(m)
where derived
the momentum from
ClM(da
labels
the electroweak
+
WI2
of the R ratio
in
concerns
+ a(~4
are shown
W
the V-A
--t e-(~4
compared
with
and
Z decay
structure
data.
of the weak charged
(6.26)
+ o(pv),
in brackets,
we obtain
(using
the couplings
Lagrangian),
= 64(G~)‘lVLd2
L((pu +,,:‘-$&
+ Mgr2l.
(‘5.27)
-88-
Likewise,
for the charge
CIWud--,
conjugate
e+vf
angle of emission
of the incident constituents
we have
= 64(G321V~12
where now p, is the momentum e+(e-)
process,
p(p),
FERMILAB-Conf-90/164-T
[(CPU +,,~:$;,
of the incoming
u quark
in the W rest frame,
and
of the proton
if we assume
that
(antiproton),
then
+ Mgrzl’ etc. If we define
measured
with
all incoming for both
respect
(6.28) 0’ to be the
to the direction
quarks
(antiqusrks)
of the above
matrix
are
elements
we have (P” . p.)l Thus
the cross section
the direction tum
of the incoming
argument
fermions
for this.
and positive
fore requires fermion
(quark),
hypothesis.
which
and UA2 Note,
the scattering replaced
proton
helicity
collaborations
however,
amplitude,
that
fermions
since there
the arguments
is more complicated
is a combination is an admixture
of left-
momen-
to negative
helicity
the direction
there-
of the incoming
proton.
Fig.(40)
shows 8’ distributions
Th e d a t a are consistent
with
are two W-fermion-fermion
are unchanged
Fig.(41)
The curve
shows the standard
W and
shows
if the
ever, part
of the total
momentum
bosons. The
the angular model
diagrams
from
prediction
with
momentum with
cross section
corresponds
relevant
pieces,
the relative
distribution
are produced
The
Because
and right-handed
2 transverse
W and 2 bosons
qg -a Vq.
for 2 decay.
of (1 f cos 8*)1 terms,
by sinBw.
Most
[61,69].
in
angular
conservation
of the incoming the data.
moves
the V-A vertices
in
(1 - 75) coupling
is
by (1 + 75).
The situation
6.5
visiblein
is a simple
momentum
to follow
the direction
(positron)
the W couples
Angular
(electron)
is clearly
electron
There
model,
antifermions.
is usually
(6.29)
the outgoing
(antiproton).
fermion
asymmetry
from the UAl
when
In the standard
the outgoing
The lepton
bution
is maximal
- (1 + CO80.)2.
relatively
mechanisms
are identical
the coupling the lepton amounts
the CDF
of the Z to angular
being
distri-
determined
collaboration
[70].
sin’ 0~ = 0.231.
distributions little
transverse
to the production are the 2 +
to those for large pi
momentum.
of large transverse
2 processes direct
How-
photon
44 -+ Vg and production,
-89-
FERMILAB-Conf-90/164-T
l-
*0 g ? 3
40
30
2o
t 1
(1+cc 1
I,
CT
-I
I
,R?
ll&
-.4
-.8
0
.a
.4
l e
Figure Fig.(33),
with
and the annihilation
similar
transverse ments
in Fig.(42).
l
0
.4
,
from
W-boson
distribution
of leptons
and Compton
matrix
elements
decay.
are (Table
= Tas&GFM:, ; t2+u2;2M’s
~I~rH”~2
= nas&q+M& i ” + u~-u2”M’,
results
for the
parton
on the pi The
[71], using
over the complete
2 obtained
distribution
by changing
is then
obtained
distributions
in the usual
distribution
of the W from
curve
is a next-to-leading
HMRS(B) pi
range,
parton although
.B
co.53
~I~+V#l’
momentum
with
Data
Reno
40: Angular
,‘,
-.4
-.a
the overall
4)
(6.30)
couplings.
by convoluting
The
these matrix
W ele-
way. the CDF
order
distributions. it is clearly
QCD The
collaboration prediction agreement
not possible
[72] are shown from is very
Arnold
and
reasonable
yet to use such data for
-9o-
FERMILAB-Conf-90/164-T
.12
CDF
.l
Preliminary
l QJ .08 .08
.06 .04
P
.02
1’
-
II II IiIi
0
1
11 11
-11
I
l
II II II II II II II II II II II II II
-.5
-1
1
.5
0
1
cd’
Figure a precision
41: Angular
hgfg measurement.
for cz.s to leading events
distribution
order
of leptons
The UA2
by comparing
from
Z-boson
collaboration
the relative
decay, from
have, however,
rstes
of W
CDF
derived
a value
+ 1 jet and W
+ 0 jet
1731:
as(MS,@ From
Fig.(2)
= Mw)
= 0.13 h
we see that
0.03 (stat)
the result
& 0.03 (expsys)
is consistent
with
f
0.02 (th.sys).
measurements
from
(6.31) other
pro-
cesses. At small This
transverse
is s reflection
momentum,
of the i&a-red
t = 0 and u = 0 in the expressions becomes
smaller,
the emission
the theoretical singularity given
of multiple
cross section
in the matrix
in Eq.(6.30)). soft gluons
in Fig.(42)
element
(i.e.
As the transverse becomes
important.
diverges. the poles at momentum The generic
-91-
FERMILAB-Conf-90/164-T
W+ + W- production d/s
I
1
.l
1
=
1.8 TeV,
I
I
I
at
CDF
I
pr
preliminary
I
I
I
50
0
large
I
I
I
I 150
100 PT [GaVl
Figure
42: W transverse
next-to-leading expression
order
QCD
in this limit
Q(P;) N Ar-+ogg PT
the A; are coefficients
distribution
Ma
of O(1).
w(P;) this corresponds
the large logarithms For more details regulates
the
collaboration,
logs $
+ ...,
with
is: 4(P$) AaT
+
The higher
w
order
Ml x
-
to pr values less than
in Eq.(6.32) see reference
cross section
the CDF
(6.32)
PT
when
In practice,
from
predictions
for the cross section 1 du -0 44
where
momentum
at small
The PT.
result The
are evidently
important (6.33)
1. about
crm be resummedto [40].
terms
10 - 15 GeV/c.
Fortunately,
all orders in perturbation is B ‘Sudakov’
small
pT QCD
form
factor
cross section
theory. which is most
-92-
naturally
expressed
parameter’
as a Fourier
vector
schematically,
transform.
g, which
is the Fourier
p
db b J&v)
dzl To the extent
that
pi
the exponent
distribution
there
are some difficulties
smearing
must
be included
introduces
QCD
measurement
transverse
of theory
with
energy
is singular
evidently
agree quite
One of the most
jets.
by the production
by the UA2
is given,
as a test
of QCD.
in Eq.(6.34)
converge
It is also difficult
cut-off
or
at large
b.
to make an accurate
sh ows an example
collaboration
on Am,
is of the same order as the
momentum
Fig.(43)
(6.34)
In practice,
some non-perturbative
the b integral
uncertainty.
q(z2, b-l).
of a comparison
[74]
[72]. The solid line is the resummed
line is the O(ai)
fixed
with
order
Eq.(6.32).
prediction.
Note
Experiment
that
and theory
well.
production important
with
standard
model
of vector
process
bosom
collaboration
in high
accompanying
(heavy
quarks, with
accurately.
mentioned
of the strong
2
processes
in association
these backgrounds
- we have already
and
W
of a W or 2 with
any new physics
to be able to estimate are possible
q(zlrb-‘)
be used
= 0, in accordance
is the production
Essentially
QCD
the cross section
d e p en d s on as and hence
- for example,
the CDF
at pi
Multijet
S in Eq.(6.34)
resolution.
and the dashed
the latter
collisions
‘impact
M))
- $)
when the transverse
data from
prediction
6.6
of p;,
exp( -S(b,
dz~6(zlza
to make
some theoretical
experimental missing
the two-dimensional
conjugate
can in principle
however,
This
Introducing
by
1 du -00 dp$
the small
FERMILAB-Conf-90/164-T
energy
hadron-hadron
hadron
(quark
SUSY,...) jets.
can be mimicked
It is therefore
In addition
important
quantitative
in an earlier
section
as from
the relative
coupling
or gluon)
tests of
the measurement rate of W + 1
jet and W + 0 jet production. As long as the jets are required
to be well-separated
beam, the cross sections
can be calculated
parton
V + kl . . . k,,, where
processes:
how the matrix
ij
+
element
calculations
from
from
the matrix
each other elements
V = W, Z and i, j, k
are performed,
together
with
and from
the
for the tree-level = q,g.
references
Details’of to earlier
-93-
‘,‘!
W’+W-
:
CDF
at
FERMILAB-Conf-90/164-T
ds=1.8
Tei
preliminary
data
: --I---c
I$$~ -
1.5
Resummed
I
1
.5
1 r
0
I
!’
I
I
I 5
0
1
I PT
Figure
43: W transverse
oration,
with
momentum
resummed
work,
can be found
model
predictions
QCD
I
I
I
[Gevl
pr,
from
the CDF
collab-
ref. 1741
[45]. As an illustration,
for the jet fractious
I
I
15
at small
from
I
10
distribution
predictions
in reference
I
f,, defined
Fig.(44)
shows
the standard
by
f = u(pp-+ W + n jets) n c%,(ti -+ w + w at J;; = 1.8 TeV, leptons
and jets.
[75] with
using
The predictions
the recent
the almost
a representative
exact
related
geometric
of the multijet
in this
simple
combine
V + 4 jet calculation relation
of cuts, the 0,. . . , 4 jet fractions complexity
set of pi,
way.
11 and AR
the V+O, from
between
1,2,3
reference the jet
are well-parametrised
calculations,
(6.35)
it is surprising
cuts for the final
jet calculations
of reference
[76]. It is interesting fractions,
i.e.
with
by f,, = fo(0.19)“. that
the final
state
to note this
choice
Given
predictions
the are
-94-
FERMILAB-Conf-90/164-T
2
3
I
.l
42
.Ol
,001
.OOOl 0
Figure
1
44: Predictions
4
n for the jet fractions
in W production
-95-
7.
The The
Production
production
for collider
predicting
the
to produce
quarks
rates
number with
sufficient
The
disadvantage
distinguished
from
a large
by using
7.1
The
decays.
Therefore
of hadrons fraction
containing
We shall start
mt
heavy
< mw.
on-shell
W boson
one power
width
is given
are never This
&s the bottom
to study
their
decays
to observe
CP violation
produced
b’sis
they
hadrons.
It is therefore
from
by
offer the potential
that
the background.
in in
have to be necessary This
to
is done
products.
quarks
is deduced
measures
makes
by observation
the cross section
assumptions
of their
for the production
concerning
the decays of a free top quark
the branching
of the Fermi
in the standard
as well as the experimentally
of a very
and a b quark.
This
constant.
massive
process
top
quark
has a semi-weak
In the limit
in which
model.
less favoured which decay
decays
case
into
an
rate involving
mt >> mry
the total
t
by,
the top quark
scale the top
quark.
mode.
the decay
I‘(t
When
quarks
the case mt > rnw
Consider
only
heavy which
by considering
consider
top
processes
such
interactions
necessary
events
decay
as the
quarks
containing
any experiment
to the observed
We shall
of their
such
objects,
hadronic
of other
of heavy
of hadrons
heavy
quarks
the bottom
properties
decays
The existence
objects,
b’s it may be possible
background
One of the motivations
of such production
of hadronically
way to isolate
the special
are large,
of bottom
the b system.
find an efficient
heavy
for the known
the cross sections
For example,
new
issue.
to test our understanding
production
the large
Quarks
is an important
is to discover
important
Because
detail.
of heavy
experiments
It is therefore
quark.
of Heavy
FERMILAB-Conf-90/164-T
quark
--t bW) = 2
is so heavy decays
that
IV&l’
x 170 MeV
the width
becomes
before
it hadronises.
with
the top quark
bigger
Hadrons
than
a typical
containing
hadronic
the top quark
formed. should
be compared
decay for mt < mw
- mb which
is a
-96-
10’
Total
10-s!/’
top
’ ’ ’ 50
’
quark
’ ’ ’ ’ 100
Figure scaled-up
version
dashed
shows
imation forbidden
In this
because
modes e&., PO,,, r&
modes
’ ’ 200
width
width
for general
derived ratio
colours +
’ ’ 250
’
’ ’
’ ’ J 300
of the top quark
from
into
values
Eq.(i’.l)
to leptons
e3 is given
by,
ratio
is given
=
1 3+3t3
in the simplest
the decay
is given
of uH and CS.
e+ti)
of the top mass.
The
and Eq.(7.2).
Assuming
mr > mw - mb, the branching
BR(W+
’
lKbla x 2.3 keV
for the W decay.
and three
’ ’
case the partial
of the top quark
cases the top branching
by counting
’ ’
m, [GeVl
--t befi) = -G-4 192+’
the width
mass
’ ’ ’ ’ ’ 160
lines show the asymptotes
In both
vs.
45: The total
of Jo decay.
r(t
Fig.(45)
width
FERMILAB-Conf-90/164-T
!% 11%
channel
by counting
approxto t6 is the decay
-97-
All direct into
searches
leptons.
especially
It is important if they
example,
for the top quark
alter
we consider
of a second
investigated
in ref.
[78] one must spontaneous
couple
all quarks
symmetry
neutral
particles.
quark
is not to a leptonic
into
the
Top
to avoid
quark
leptonic
decay
we are left
with
in this
one charged
but rather
to the charged
value.
As an extreme
it is found
[77] that
the semi-weak Eq.(7.2).
in this model. values
decay
quark
modes
This
fv,
exceeds
the
weak
is called
of the u and c quarks, strong
interaction
details
and references
7.2
The
the signature
to be equal
model,
the b decays,
of the top
(7-4) expectation
as determined of the light
the branching
in analogy
papers
of heavy
quark
However
must
can be appreciable
to the original
processes
7 and
of parameters,
If the vacuum
since the quarks
play no role in the decay. to which
Higgs
fraction
from
top quark expectation into
cz is
[77].
of the B-meson
the spectator
After
and rnb = 4.7 GeV
range
width
to ci and ?v,.
= 25 GeV
the decays
corrections
theory order
are taken
is 31% for m,
in the B-meson
The leading
for a large
of the q+ determine
doublet.
vacuum
= 25 GeV
that
greatly
fields
m,
currents
m;, m3
It is clear
The T+ decays predominantly
We may also treat muon.
Eq.(7.4)
of the two Higgs
64% and into
for mt = 30 GeV,
neutral
Higgs,
c ) an d v is the normal
r 2 0.4MeV.
width
The
example,
the
has been
decay mode
77%;- m; $2m~ms)Xf(m:,
= ((a-b-c)‘-4b
involves
physical
qt--+ bq+) >&$bJ: + InthisequationX(a,b,c)
As an
model
one Higgs
If mt > m, + rnb the dominant
mode,
mode.
which
changing
to only
ratio
of the top quark,
decay
model
strangeness charge
the branching
decays
of the standard
of a given
breaking
three
ratio
doublet.
In order
about
unconventional
extension
Higgs
[77].
Higgs
branching
a simple
introduction
make assumptions
to investigate
the
FERMILAB-Conf-90/164-T
with
the decay of a free
which
accompany
in this case the finite
be taken because
into
as(mb)
account. is large.
the b masses
In addition For further
see ref. [79].
for the production
production of a heavy
(a)
n(pd
+ u(n)
-+
Q(n)
(b)
g(n)
+ g(n)
+
Q(Ps) + Tj(,,)
quark
Q
of mass m are,
+ G(n) (7.5) ,
-98-
FERMILAB-Conf-90/164-T
b)
Figure
46: Lowest
where
the four momenta
which
contribute
invariant
matrix
elements
in Table
over initial
(final)
ratios
elements of scalar
5.
colours
matrix
form,
for heavy
are given squared
[80,81]
in brackets. in O(g’)
w h ‘C h result
elements
and spins,
in a compact
diagrams
elements
squared
The
squared
(as indicated
from
by C).
production
The Feynman
are shown
have
we have introduced
quark
diagrams
in Fig.(46).
The
the diagrams
in Fig. (46)
been averaged
(summed)
In order
the following
to express notation
the
for the
products,
2Pl .P3
7-l = -,
In leading
Feynman
of the partons
to the matrix
are given
matrix
order
order
h
the short
rz=-,
2Pl.P3 i
distance
p=---,
4ma 5
cross section
i=(PltP2)1.
is obtained
from
the invariant
-99-
Table
5: Lowest
matrix
element
initial
(final)
matrix
order squared.
element
d+ii first
The
in the normal
factor
and spin
indices
cjq’
is the invariant
are averaged
(summed)
+ pa -pa
- pr) CIMijl,.
factor
for massless
incoming
particles.
the phase space for two-to-two
theory
the momenta
cwer
distance
pl = zIPI,
[82]. Consider
cross section
pa = rsP2
is i = ~1~8,
for the two final
state
which
centre
heavy
The
other
is defined
production
cross section.
are moving
terms
of the total
in terms
by s where
parton particles.
of their
is described Let us denote
in the z direction,
is to be evaluated
the masses of the incoming
partons
quark
of mass energy
in Eq(4.1)
and hence the square
if we ignore
that
first the differential
hadrons,
of the total
(7.7)
scattering.
why it is plausible
of the incoming
and Pa and the square
Using
production.
(2r)‘b’(p1
(27r)32&
is the flux
by perturbation
momenta
quark
[l]:
d3p4
(2*)32.&
We shall now illustrate
short
colour
fashion
@P3
= ii
come from
The
for heavy
states.
1
The
processes
FERMILAB-Conf-90/164-T
s = (PI + 4)‘.
for parton centre
momenta
of mass energy
The rapidity
energies
by PI
variable
and longitudinal
as.
Eqs.(4.1)
and (7.7)
the result
for the invariant
cross section
may be written
as,
do dy,&d%
The energy
momentum
delta
function
in Eq.(7.7)
fixes
the values
of z1 and za if we
-lOO-
know
the value
of the pi
of mass system four momenta
and rapidity
of the incoming
of the outgoing
hadrons
we may write
heavy
quarks.
In the centre
the components
of the parton
as (E , p I, p II, p 7.)
Pl
=
~/2(%O,O,ZI)
pa
=
J;;l2(Z%O,
~3
=
(fm
energy
0, -4
coshy,,pTr
O,mT
(W-h?/,,
P4 = Applying
FERMILAB-Conf-90/164-T
and momentum
sinhys) (7.10)
-PnO,WSinhY,).
conservation
we obtain, (7.11)
The
transverse
mass
of the
heavy
Ay = ys - yd is the rapidity Using
Eqs.(7.9)
two massive
difference
and (7.11),
quarks
calculated
dc = 64?r%#
Expressed
in terms
and Ay
=
Zlfi(zl,P)
the matrix
+ p$)
and
for the production
of
quarks.
the CIOSB section perturbation
theory
z2fj(z2tP)
elements
J(mz
as, )J”Glz
’ t7’12)
for the two processes
in
5 are,
~IM,,1’ that,
section,
because
Eq.(7.12),
two heavy
quarks
contribution produced gluon
C ,,
by rn~
the two heavy
order
1 + cosh(Ay))2
~I-%d
Note
we may write in lowest
of m,mT
is denoted
between
1
dy&&w
Table
quarks
= F(,
(coahhd
= $(6;;;($A;;)(cosh(Ay) of the
specific
is strongly becomes
to the total by qq annihilation
fusion.
+ co;h(Ay))
form
damped
large.
+2$ of the matrix as the
rapidity
It is therefore
cross section are more
closely
elements separation
the region
correlated
(7.14)
- 2s).
to be expected
comes from
(7.13)
+ $)
Ay
those
squared, Ay that
the
cross
between
the
the dominant
5 1. Heavy produced
quarks
by gluon-
-lOl-
We now consider the above variables
the propagators they
that
= 2p1 .pr
contribution, QCD.
= -2pl.p3
= -n&l
+ &‘).
also that than
the quark
transverse
quark central,
which
would
incoming
transform
heavy
by powers ignored.
quark
The and
mass
because
cut-off
It is of
cut-off is less
give the dominant
not be calculable
in perturbative
is provided
by the mass
is controlled
by QS evaluated
from
of pr which
However
of the heavy
we expect
heavy
falling
mass [83].
parton
mechanism
quark
spectator
F or a sufficiently
The
might
scale
of the oris
Final
state inter-
will
not change this
the debris
partons heavy
theory
production
spoil
with
quark.
the large
momenta
hadrons
which
heavy with
to the cross
provides
fluxes.
of
momentum
of perturbation
the observed
of the produced
of the increase
of the mass of the heavy
which
are
cross section
contribution
close in rapidity.
into
these interactions
because
have transverse
are produced
quarks
differential
the methods quark
quarks
values
on the transverse
the dominant
of the order
A possible
quark
The
flux decreases
of the rapidly
the heavy
be the interaction
hadrons.
the lower mass, which
virtualities
Since all dependence
momenta
the size of the cross section. picture
the parton
mass combination,
production.
der of the heavy
quark
production
It is the mass of the heavy
quark
by the light
will
m’.
the production
diagrams
small
of order
from
by these
the lower
quark
quark
mass is also suppressed.
to Eq(7.11).
for a sufficiently
predominantly
quark
(7.15)
of least
to the cross section
and as mr increases,
to be applicable.
with
quark,
heavy
the contribution
comes from
in heavy
is provided
propagators
that
m the transverse
Thus
is produced
of a heavy
of
scale.
zr and zr according
actions
quark
of a light
plausible
quark
falls like l/m%
section
of a light
scale A. Since
greater
appears
the production
of the propagators
m. It is therefore
much
by a quantity
a light
In the production
Note
are all off-shell
the production
at the heavy
1 + co& Ay)
(Pa - ~3)~ -ma
When
the QCD
= 2m;(
+ .-Au)
a heavy
In terms
as,
= --n&l
distinguishes
than
in Fig.(46).
= -2pl.ps
this fact which
on the virtuality
shown
(PI - ~3)’ - ma
the denominators
quark.
in the diagrams
can be written
(?‘I + Pa )’
Note
FERMILAB-Conf-90/164-T
simple of the
are suppressed
quark
they
can be
-102-
The theoretical
arguments
summarized
quark
in all regions
of phase space is well described
perturbative
corrections.
Higher
The lowest
order
order
in the running
terms
heavy
above
the charmed
7.3
is sufficiently
quark
specify
perturbative
The
full
refer
subtractions p.
calculation
The
perform
to ref.
dependence
distance
partons.
processes
hadrons
(a) and (b) and their
quark
production
of a systematic
expansion
(7.16)
it)
cross-section
The
+4nas(P)[C$)(p)
involves
The
7;j,
for the production
where
dimensionless
[84].
both
real
In order
for renormalisation on p of the
+$~‘(p)ln($)]
functions
which
renormalisation
required
(p,
of the functions
of diagrams
both
F.Fij
the short
in Eq.(7.6).
of the types
the reader
we must
of charmed
the indices
of
i and j
functions
~;j
have
expansion,
= @J’(P)
p is defined
Examples
by only
are the beginning
=
of the annihilating
Ej(P,$) where
describes
ma)
of mass m in terms
the types
the following
the hadroproduction
the issue of whether
coupling,
completely
a heavy
do not address
to heavy
above
+ij(S,
Eq.(7.16)
that
corrections presented
FERMILAB-Conf-90/164-T
J$’
are completely
contribute
to 6:’
virtual
corrections.
and
to calculate
the
and factorisation and
non-leading
known
are shown
[84,85].
in Fig.(47).
For full
details
7
term
we
theory
of mass singularities.
factorisation order
(7.17)
+ O(g’)
The
are done
at mass
scale
is displayed
explicitly
in
Eq.(7.17). As discussed predictions turbation p will
should theory.
in previous be invariant
lectures under
If we have performed
lead to corrections
of O(o$),
p is an unphysical changes
parameter.
of n at the appropriate
a calculation
to O(as),
variations
The order
physical in per-
of the scale
-103-
FERMILAB-Conf-90/164-T
2
Real
emission
diagrams 2
Virtual
Figure In this
equation
Eq.(7.18) order
47: Examples
we find
perturbative
emission
of higher
order
o is the hadronic that
diagrams
corrections
cross section
the term
7”’
contributions
which
is tied
to heavy
as determined
controls
in terms
running
this
coupling,
result
we have
used
the
production
by Eq. (4.1).
the p dependence
of the lower
dzl F$)(:)PIG(z,)-J In obtaining
quark
order
of the higher result
dza S$‘($)P~j(~~)
renormabsation
group
Using
equation
7(O):
1
. (7.19)
for
the
Eq.( 1.22)
” b=
d ---a&J) d/S 33--?r2nf)
= -acc:(l+ b, =
b’as + . . .) 153 - 19nr
21r(33 - 2nf)
(7.20)
-104-
and the Altarelli-Parisi
This illustrates
equation,
an important
improved
perturbation
depends
on the choice
made
the physical
dependence
is formally
small
current
illustrate
interest.
duction
First,
of the higher respect
order
to changes
Fig.(49)
rections, of p.
leads
in p.
The
which
are subject
is one in which hadron.
state.
that
in calculating
treated
the
sensitive
In the conclusion. should
from
of a flavour
following When
the other excitation
excitation
prediction
with
different.
cross section
In
is shown.
of the higher
production
order
under
cor-
changes
at collider
energies
excitation
diagram
considering
the total
to reside hadron
gluon.
an analysis
wilI
and appears is shown
contain these
to the hadronic
on shell
in the
in Fig.(SOa).
Note
heavy
quark
transfer
between
a factor
l/q’
graphs
the
coming
appear
size scale.
is
to be
This
casts
to these processes. [82] which
cross section,
The net contribution
in the incoming
the incoming
Therefore
QCD
already
the momentum
cross section
of perturbative sketch
A flavour
contribution
scales all the way down
we shall
not be included.
The inclusion
is quite
quark
diagram
If we denote
of the exchanged
on the applicability
for the hadropro-
order.
quark
excitation.
is considered
as q, the parton
to momentum
found
in two cases of
of the prediction
for bottom
the ~1
A pronounced
found
by the inclusion
the stability
Thus
series.
of the theoretical
bottom
in such
does not assure
for all series.
and next-to-leading
for the bottom
of flavour
flavour
the flavour
the propagator
for p.
This
the p dependence
doubled
by a gluon
partons
small
in as.
correction
the result
made
perturbation
of the predicted
as it were on its mass shell.
two incoming
doubt
situation
group
uncertainty.
heavy
An example
order
to a stabilisation
to the question
It is excited
final
in leading
the predictions
to considerable
We now turn
of the choice
numerically
to improve
that
changes
we show the p dependence
is approximately do nothing
of renormalisation
of the perturbative
the n dependence
by showing
top quark
terms
It is apparent
from
point
the scale dependence
The cross section
coefficient
of an untrustworthy
in Fig.(48),
of a 120 GeV
The
feature
it is of higher
is actually
this
is a general
is independent because
on p is a signal
We shall
which
for p, but
result
the p dependence
dependence
point
series in QCD.
a way that
us that
FERMILAB-Conf-90/164-T
flavour
of these diagrams
leads
to an important
excitation is already
contributions included
as
FERMILAB-Conf-90/164-T
-105-
Top :
: :
cross-section
vs.
d/s =1800
GeV, m,
DFLM, &=
0.1’70
scale
/J,
= 120 GeV
GeV
40 z ti b
30
20 cl
(1’, ,,[,(,;yI:;;*;**;-;.,-, ,,,,1 100
50
200
150 P [GeVl
250
Figure 48: Scale dependence of the top quark cross section in second and third a higher order correction the observation
to the gluon-gluon
that the flavour
excitation
the first two diagrams shown in Fig.(SOb). accurately
represent
the results
which is the signature
order
fusion process. This analysis begins from graph is already present as a subgraph D oes the flavour excitation
of these diagrams?
of the presence of the flavour
In particular excitation
of
approximation is the l/q’
diagrams,
pole,
present in
these diagrams? We shall now indicate diagrams displayed
why the l/q”
in Fig.(SOb).
behaviour
is not present in the sum of sll three
Let us denote the ‘plus’ and ‘minus’
components
of
any vector q as follows: qf = qo + q3,
We choose the upper
incoming
q- = qo - q3,
parton
- pT . PT.
(7.22)
to be directed
along the ‘plus’
qs = q+q-
in Fig.(50b)
-106-
60
I
I1
I,
I
I
I
I,
Bottom 70 -
I
I
I
I,
I
cross-section
d/s =1800
I
vs.
I
II
I
scale
I
II
/I,
GeV, mb = 5 GeV
DFLM, &,s= 0.170
60 -
FERMILAB-Conf-90/164-T
GeV
50 -
-
L+NL
-----2 b
4030 20 -
*.------------
---_________
-------------___________________
10 I
0
I
I
I
5
0
Figure
I
I
49: Scale dependence
I
I
I 10
I ,
I
I
/I Ecevl
of the bottom
I
I
I
I
,
1 25
20
15
I
quark cross section in second and third
order direction,
pl = p:.
direction,
pr = p;.
and the lower incoming
parton
to be directed
In the small qa region the ‘plus’ component
along the minus’
of q is small, because
the lower final state gluon is on shell, (pp - qy = 0,
since in the centre of mass system p: x p; % ,/5. component
of q is determined
from the condition (pl + q)l x 472,
q- is therefore
also small in the fragmentation
(7.23)
qf = &,
In the low q2 region the ‘minus’
that production
is close to threshold,
q- x $
region in which p: z 4.
We therefore
-107-
a) Example
of flavour
excitation
containing
spin-one
b) Graphs
Figure
50: Graphs relevant
find that in the fragmentation
exchange
in the
of flavour
region of the upper incoming 9T ’ qT =
-qT
t-channel
excitation
hadron, (7.25)
’ qT
J to which the exchanged gluon of momentum
by the upper part of the three diagrams. component
graph
for discussion
9’ = q+q-The current
FERMILAB-Conf-90/164-T
In the fragmentation
q couples is determined
region only the ‘plus’
is large. qw~p = q+ J-
where the Ward identity term proportional in the amplitude
+ q-J+
is a property
to qr in the amplitude squared.
- qT. JT = 0,
Jc
X -9T ’ JT 9-
of the sum of all three diagrams. shows that one power of the l/q’
(7.26) The explicit is cancelled
-106-
This cancellation
only occurs when the soft approximation
requires the terms quadratic denominators
FERMILAB-Conf-90/164-T
to J+ is valid.
This
in q to be small compared to the terms linear in q in the
in the upper parts of the diagrams
in Fig.(BOb).
The momentum
q-
must not be too small, pa < 2p+q-
We therefore expect the soft approximation when q* < ma. For further ref. (841 provides
7.4
to be valid and some cancellation
to occur
details we refer the reader to ref. [82]. The calculation
an explicit
Results
(7.27)
zz ma.
verification
of this cancellation
on the production
of charm
The value of the heavy quark maas is the principal
in the total cross section.
and bottom
parameter
of
controlling
cross section. This dependence is much more marked than the l/nil
quarks the size of the
dependence in the
short distance cross section expected from Eq.(7.16).
As the mass decreases, the value
of I at which the parton
becomes smaller (cf
distributions
are evaluated
and the cross section rises because of the growth The approach
which
we shall take to the estimate
quark cross sections is a+ follows Eq.(1.46)
with corresponding
variations
choose to vary the parameter
sensitivity
to II. Lastly,
theoretical
certainly
of theoretical
errors in heavy
of the gluon distribution p in the range m/2
function.
We shall
< ~1 < 2m to test the
we shall consider quark masses in the ranges, 1.2
<
1.8 GeV
4.5
< m(, <
5.0 GeV.
the extremum
of all these variations
(7.28) to give an estimate
of the
error.
We immediately Variations
flux.
[86]. We shall take A to run in the range given by
arbitrarily
We shell consider
of the parton
Eq.(7.11))
encounter
of p down to m/2 do not trust
a difficulty
with this procedure
in the case of charm.
will carry us into the region p < 1 GeV in which we
perturbation
theory.
A estimate
charm production
cross sections is therefore
not possible.
charm production
We have taken the lower limit
of the theoretical In preparing
on p variations
error on
the curve for
to be 1 GeV.
The dependence on the value chosen for the heavy quark mass is particularly
acute
-109-
1000
1
y’
-+ D/DtX
I
I
I
FERMILAB-Conf-90/164-T
I
I
I
I
I-1
-I
’ ’ /.1.,--’I, ; T ho---
--p---j
100
z 3
30
b
10
3
1 10
0
Figure
20
50
In fact, variations
of D/D
compared
due to plausible
are bigger than the uncertainties
We shall therefore
due to variations
in the other parameters.
take the aim of studies of the hadroproduction
value for the charm quark
of the data on hadroproduction? charm production. compilation
In Fig.(51)
and photoproducquestion.
can accommodate
we show the theoretical
Note the large spread in the prediction.
Is there a the majority
prediction
for
Also shown plotted
is a
of data taken from ref. [87] which suggests that a value of m, = 1.5 GeV
gives a fair description of the O(oi)
mass which
with theory
changes in the quark mass,
tion of charm to be the search for an answer to the following reasonable
70
60
[Gely
51: Data on hadroproduction
for the case of charm. Eq.(7.28),
30 4s
corrections,
of the data on the hadroproduction the data can be explained
of D’s.
without
After
inclusion
recourse to very small
values of the charmed quark mass [86]. This conclusion
is further
reinforced
by consideration
of the data on photopro-
-llO-
2
I
I
,
I
I
I
I
I
I
Photoproduction
FERMILAB-Conf-90/164-T
I 8 I
Figure 52: Data on photoproduction
been considered predictions beam.
The higher
in ref. [88].
so we have plotted
O(or$)
After inclusion
uncertainty
in Fig.(52),
indicates
acceptable
explanation
In conclusion,
I
I I
I
with theoretical
corrections
lower limits
to photoproduction
of these higher
have
order terms we obtain
of the energy of the tagged photon
derives from the value of the heavy quark
mass,
cross section
A and
which is obtained
by varying
the range 1 GeV < /J < 2m for three values of the charm quark
mass. The comparison
the D/D
of charm compared
order
the minimum
the scale p within
I
(GeV)
for the total cross section as a function
The principal
I
300 q
of charm.
I
of charm
200
duction
I
with the data on the photoproduction
that charm quark masses smaller than 1.5 GeV do not give an of the data.
within
production
of charm [89,90], shown
the large uncertainties
present in the theoretical
data presented here can be explained
estimates,
with a charm quark mass
of the order of 1.5 GeV. This is not true of all data on the hadroproduction
of charm,
-lll-
especially
the older experiments.
FERMILAB-Conf-90/164-T
For a review of the experimental
situation
we refer
the reader to ref. [91]. As emphasised collider
above, the theoretical
energies is very uncertain.
prediction
present collider energies the bottom at values of z < 10-a.
quark production
The cause of this large uncertainty
the very small value of c at which the parton function
for bottom
distributions
at
is principally
are probed.
In fact, at
cross section is sensitive to the gluon distribution
Needless to say the gluon distribution
function
has
not been measured at such small values of +. An associated problem is the form of the short distance cross section in the large d region.
The lowest order short distance cross
sections, 7(s), tend to zero in the large i region [84]. This is a consequence of the fact that they involve
at most spin i exchange in the t-channel
higher order corrections they involve Fig(50b).
to 99 and gq processes have a different
spin 1 exchange in the t-channel. In the high energy limit
of energy [84]. Naturally of energetic
The relevant
they yield a constant
these high j contributions
gluons in the parton
makes a sizeable contribution behaviour
flux,
to the bottom
At fixed target theoretically to Fig.(48) lation
more reliable.
indicates
results
[92] and estimates
shown in Table 6. The experimental reactions
of experimental
study
is still in its infancy,
cross section at collider
shape of the transverse
momentum
is shown in Fig.(53),
of bottom
which
form similar
errors.
A compi-
theoretical of bottom
error is quarks in
the limited
the pi and rapidity Although
energy is uncertain, theory.
quarks is
number
cross sections.
cross sections.
the form found in lowest order pertubation this conjecture
function.
of the production
and rapidity
> m
between the size of
of the associated
of ref. [84] also allow us to examine
of the one heavy quark inclusive
the total bottom
an interplay
of the theoretical
production
4
of the size of this
but Table 6 also includes
results on total bottom
The calculations butions
the sensitivity
plot has a characteristic
and it is possible to make estimates
are shown in
The fact that this constant
of the gluon distribution
The p dependence
because
cross section, independent
energies the cross section for the production
of theoretical
hadronic
diagrams
energies the region
cross section.
term to the value chosen for p. There is therefore and the small 2 behaviour
behaviour
The
are damped by the small number
but at collider
is present in both 7(‘1 and 7”’
this term
as shown in Fig(46).
that
of
that the
is well described
The supporting
demonstrates
the prediction
it is plausible
distributions
distri-
by
evidence
[97] for
the inclusion
of the
-112-
g (theory)
mb
[GeV]
fi
= 41 GeV, pp
Theoretical
4.5
23 nb
$21 -15
5.0
9 nb
$8.4 -5.9
fi
error
FERMILAB-Conf-90/164-T
Experimental
data
= 62 GeV, pp
4.5
142 nb
+98 -80
5.0
66 nb
i-47 -38
Ji
BCF[93],
150 < o < 500 nb
= 630 GeV. v~
;:;
1 :; :;
fi
1 UA1[94],
1 i-i-O-8
10.2f3.3
pb
= 24.5 GeV, xN
4.5
7.6 nb
+4.7 -3.8
1 WA78[95],
&=
24.5 GeV, 4.8 %0.6&1.5 nb
5.0
3.1 nb
1-1.5 -1.5
NA10[96],
&=
23 GeV, 14+7-6 nb
Table 6: Cross section for bottom first non-leading momentum
correction
and rapidity
of one another
production
does not significantly
distributions.
modify
by a constant
hold also for the shape of the top quark distribution the transverse
comparison
of the full oi prediction
a function
the shape of the transverse
At a fixed value of p, the two curves lie on top
if the lowest order is multiplied
investigated
at various energies.
distribution
[97]. The UAl
of the produced with UAl
of the lower cutoff pk(min)
factor.
bottom
Similar
collaboration quarks.
results have
In Fig.(54)
data is made. The data are plotted
on the transverse
momentum
as
of the b quark.
The agreement is satisfactory. The corresponding at the Tevatron
7.5
The
prediction
for the shape of the bottom
production
is shown in Fig.(55).
search
for the
top
quark
The belief that the top quark must exist is based both on theoretical evidence.
The theoretical
cancellation partner
cross section
of anomalies
motivation
is that
in the currents
complete
which
of the b, r and v, must exist to complete
families
and experimental are required
couple to gauge fields. the third
family.
for the
Hence the
FERMILAB-Conf-90/164-T
-113-
mb = 5 GeV y = 0, 3, 4 -
LO + NLO
----
LO times
kr
are destroyed
occurs in a theory by quantum
gence of a current anomalous,
Anomalies axial
vector
because symmetries
which are vital
present at the classical level contributions
to the diver-
for the proof that
are
the gauge theory
are destroyed.
occur in the simple triangle current.
Elimination diagram
free, even after the inclusion
diagram
of the anomalies
is sufficient
of more complicated
fields, and to the matrices
with two vector currents for a particular
to ensure that the current
at the three corners of the triangle
the left-handed
involve
quark production
which is conserved at the classical level. If the gauge currents
lowest order triangle interact
for bottom
effects. They typically
the Ward identities,
is renormalisable,
25
[GeVl
Figure 53: The shape of the cross-section An anomaly
20
15
10
5
2.5
diagrams.
current
in the
remains anomaly
If the currents
couple to the matrices
and one
which
L”, Lb and L’ for
R”, Rb and R’ for the right-handed
fields,
FERMILAB-Conf-90/164-T
-114-
e
I
I
I
I
I
pp collisions,
\/S
I
I
= .63 TeV, -
m,=4.75
I
I
ly1<1.5,
I
pT>p\
GeV, A+=260
1
(min) MeV,
DFLM, pco = \/(mss+p,*)
1
OHigh
-
n
mess
Muon-jet
dimuons
samples
10
0
(1990) 50
40
20
60
p” (miny[GeV]
Figure
54: The cross-section
the vector-vector-axial
for bottom
vector triangle
quark production
anomaly
A = Tr [Iz”{Rb,R=}]
is proportional
family
to, (7.29)
- Tr [P{Lb,LC}].
For the specific case of the SU(2) L x U(1) theory (GSW) we have the following
at CERN energy
of Glashow,
weak isospin and hypercharge
Weinberg
assignments
and Salam for the third
(Q = Ts + Y), tL., T3 = ;,YL
= ;,
bL, T3 = -;,Y,
= ;,
VL, T3 = f,Y,
= -a
TV, T3 = -i,YL
= -I 19
2 tR, T3 = 0, YB = -, 3 b.q, T3 = 0, YR = -5,
Y-R, T3 = 0, Yn = -1.
(7.30)
-115-
FERMILAB-Conf-90/164-T
10 -
57 3 z 2 t ‘2 Y
MeV,
x11.=4.75 Gel’, &=260
LFLM. /& = qm.*+h*)
1 .1
.Ol
,001
.OOOl 10
0
20 ~~(rninl;“[Ge\r;O
Figure 55: The cross-section Substituting
these couplings
into Eq.(7.29),
trices T” or the U(1) matrices currents of the GSW theory. each fermion
for bottom
Y we obtain
‘O
quark production
‘O
at FNAL
with all combinations
energy
of the SU(2) ma-
the form of the anomaly
Two of the resulting
for the gauge
traces of the couplings
vanish for
separately, Tr Ta(T”,
TC} = 0,
Tr T”{YL,
YL}
The other two traces vanish only for a complete family
It should be noted that there are still anomalies the GSW model.
For example
the normal
[SE]
isospin current
(7.32)
= 0.
in global
(in the absence of quark masses) is anomalous.
(7.31)
= 0.
Tr (YR” - Yj) = 0, Tr Y,{T”,T*}
symmetry
7o
(non-gauged) corresponding
currents
in
to a global
It is this anomaly
which is
-116-
responsible
for K” decay.
Tke experimental
reason to believe in the existence
surement of the weak isospin of the bottom of b-jets in e+e- annihilation coupling
FERMILAB-Conf-90/164-T
to the electron
quark.
[99] is controlled
and the b quark.
of the top quark is the mea-
The forward backward
by A.&
the product
The produced
asymmetry
of the axial vector
b and 6 quarks
are identified
by the sign of the observed muons to which they decay. The measurement subject to a small correction to the electron
due to P-F
has its standard
mixing.
Assuming
is therefore
that the axial coupling
value the measured weak isospin of the left-handed
b
quark is [99], T3 = -0.5
The simplest
hypothesis
top quark, although
is that the bottom
more complicated
rt 0.1.
(7.33)
quark is in an SU(2)
schemes are certainly
doublet
with
the
possible.
Thus assured that the top quark exists, it only remains to find it. The expected cross section for the process (7.34)
p+p--+t+t+x
is shown in Fig.(56).
The cross section is calculated
of [84] and the method of theoretical (c$ [86]). In addition, also shown. energies.
Note the differing
At fi
gluon annihilation mainly
production
= l.S(O.63)
of top quarks through
TeV the tf production
sections,
the decay chain W + t6 is
of the two modes at CERN
at both energies.
is due predominantly
This explains
and FNAL to gluoncomes
the more rapid growth
shown in Fig.(56).
the range of top quark masses which can be investigated
can be derived.
will be produced
error estimate described in the previous
proportions
with energy of the tt production
experiments
calculation
for mt < lOO(40) GeV. On the other hand the W production
from qn annihilation
From Fig.(56)
using the full O(ai)
In a sample of 5 inverse picobarns
in current
about 2500 tf pairs
if the top quark has a mass of 70 GeV. One can observe the decays
of the top quark to the ep channel or to the e+ jets channel. the numbers of events expected Number Number
With
a perfect detector
is,
of ep events =
2 x .ll
x .ll
of e + jet events =
2 x .ll
x .66 x 2500 z 360.
x 2500 zz 60 (7.35)
The e plus jets channel gives a more copious signal and does not require muon detec-
-117-
E”““‘““‘“““3 Top quark DFLM
p
production
= m/Z, = 2 m,
p
I
I
!I
I
in
o(W’ g(W’
\
I
->
tii),
2, ), (NDE)
I
d/s=13
I
I
I
I
TeV
I
l-I==4 200
[GeVl
56: The cross section for top quark production
at CERN and FNAL
is larger due to the process pp -t W+jets.
may become less severe with increasing
:
TeV
150 mtop
tion, but the background
curves)curves)
ti$,dS=O.63
->
100
50
Figure
0(
As = 250 MeV (upper As = 90 MeV (lower
-----.---.
I
FERMILAB-Conf-90/164-T
This background
top mass as the jets present in top decay be-
come more energetic. The current efficiency
lower limit
of extracting
top quark,
by increasing
of 10. Note however that for a heavier
W+jets
top quark.
the limit
the luminosity the efficiency
no extra
production,
by an additional
accumulated
If the
of the e+ multi-jets
price in coupling
in the detector constants.
discussed in the previous
40 GeV above the
at the Tevatron
As the mass of the top quark
in its decay will be recognised
occurs with
is 69 GeV [66].
the signal from the data does not change with the mass of the
we. can expect to improve
present limit,
occurring
on the mass of the top quark
channels
will increase
increases the b quark jets as fully-fledged
The background
lecture,
by a factor
is suppressed
jets.
This
due to normal by a power of
-118-
FERMILAB-Conf-90/164-T
3
1 80
100
Figure 57: Required
luminosity
LY.~for each extra jet. and/or
160
m, [Gevl
180
are shown in Fig.(57),
7.6
in jets
Another
question
are found amongst containing
of experimental
interest
the decay products
heavy quarks have appreciable
events as a signature heavy quark production
branching
Since hadrons
ratios such events
If we wish to use lepton
for new physics we must understand and decay.
are based on
with which heavy quarks
quark or gluon jet.
semi-leptonic
will often lead to final states with leptons in jets.
The limits
for top
CDF detectors.
is the frequency
of a light
an electron
study of the prospects
taken from ref. [loo].
of the DO and upgraded
quark
in the channel with
The results of a detailed
the expected performance
Heavy
240
220
200
to discover top at 1.8 TeV in various decay modes
It will become less important
three and four jets.
quark discovery
140
120
the background
plus jet due to
-119-
Figure This issue is logically above, the total
FERMILAB-Conf-90/164-T
58: Heavy quark production
unrelated
to the total heavy quark cross section.
cross section is dominated
by events with
of the order of the quark mass. Jet events inhabit since they contain
a cluster of transverse
region gives a small contribution
in jets
a different
heavy quark
decaying into a heavy quark pair must have a virtuality methods should be applicable per gluon jet is calculable The calculation
has two parts.
gluons of off-shellness needs the transition heavy quarks.
for a sufficiently [loll
probability
k’
heavy quark.
using diagrams
energy
region of phase space This latter kinematic cross section.
A gluon
> 4mr so perturbative The number of Qa pairs
such as the one shown in Fig.(58).
First one has to calculate
k’ inside the original
a small transverse
energy ET > m.,ms.
to the total
As discussed
n,(E2,
gluon with off-shellness
of a gluon with off-skellness
k’), El.
the number
of
Secondly, one
k2 to decay to a pair of
-120-
The number
of gluons of mass squared k’ inside a jet of virtuality
[W
[ 1 ln(EZ/A1)
ns(E’,k’)
=
-;1+ and b is the first order The correct calculation
El
is given by
(7.36)
’
(7.37)
[
coefficient
in the expansion
of the p function,
of the growth of the gluon multiplicity
of the angular
of the emitted
-exp J[(2N/rb)ln(Ea/Aa)] exp J[(2N/xb)ln(k’/hr)]
In(kz,Az)
where
imposition
FERMILAB-Conf-90/164-T
ordering
constraint
Eq(7.36)
Eq.(7.20). requires the
which takes into account the coherence
soft gluons [102] as discussed in the second lecture.
Define R,Q to be the number of Qv pairs per gluon jet. Ignoring gluon branching
calculated
for the moment
above, we obtain
Rsp=~~::~a(k’)~~=[r’+(l-=)‘+~]
where the integration
limits
term (z’ + (1 - z)r)/2
is recognisable
for massless quarks.
&on
are given by z+ = (l&p)/2
with/3
= J(l-4m’/k’).
The
as P zS., branching probability .- the Altarelli-Parisi over the longitudinal momentum fraction z we obtain,
Integrating
The final result including
(7.38)
gluon branching
for the number
of heavy quark pairs per
jet is, 1 RQG = G The predicted
$-w(k’)
number
[1 + g]
,/iz
n,(Ea,
of charm quark pairs per jet is plotted
k’).
in Fig.(59)
(7.40) using a
value of A@) = 300 MeV and three values of the charm quark mass. Also shown is the prediction
for the number
point shows the number
of bottom of D’
and by the CDF collaboration used for the branching the Mark III collaboration Data Group.
ratios
quarks per jet with
A(‘1 = 260 MeV. The data
per jet as measured by the UAl [104]. Note that (D’
these results
--t DA) and (D -+ Krr).
[IO51 whereas UAl
collaboration
[103]
depend on the values CDF uses the values of
uses the values quoted by the Particle
In order compare these numbers with the cs pair rates, a model of the
.3
I
I
-121-
FERMILAB-Conf-90/164-T
I
I
I
. UA1 0 CDF
.1
,
0 20
0
Figure relative
40
E [GeV]
59: Heavy quarks in jets compared
rates of D and D’ production
80
60
with UAl
equally
one would expect the charged D’
D production
rate. The points in Fig. 59 need to be corrected
before they can be compared
and CDF data
is also needed. For example,
are produced
100
if aII spin states
rate to be 75% of the total for unobserved
with the curves for the total CE pair rate.
modes
FERMILAB-Conf-90/164-T
-122-
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