J. Phys. G : Nucl. Phys., Vol. 2, No. 7, 1976. Printed in Great Britain. @ 1976
A multispinor supersymmetric Lagrangian and spin
particle
Ali Chamseddine Physics Department, Imperial College, Prince Consort Road. London SW7 2BZ. U K
Received 26 October 1975. in final form 25 March 1976
Abstract. A new multispinor superfield of second rank is constructed which satisfies the Bargmann-Wigner equation and describes particles of spin ranging from 5 to 2, A supersymmetric Lagrangian is written with the help of an auxiliary superfield which vanishes in the absence of interaction. The resulting Green function for the spin f multispinor is the same as that obtained by Guralnik and Kibble and Chang. Finally an interaction with a scalar superfield is studied in perturbation theory and shown to be non-renormalizable.
1. Introduction The concept of global Fermi-Bose symmetry was first formulated by Wess and Zumino (1974a, b) and systemized by Salam and Strathdee (1974,1975) by defining superfields @(x, 6) over space-time and anticommuting Majorana spinors Oz. The scalar supermultiplet is the simplest case and describes particles of different spins ranging from spin 0 to spin 1. The spinor superfield with spins ranging between 0 and was studied by Adjei and Akyeampong (1975) who wrote a supersymmetric Lagrangian for it. The purpose of this paper is to study the second-rank bispinor superfield and to write a new supersymmetric Lagrangian for it. In S 2 the form of the Lagrangian and the equations of motion are found. In 4 3 the superfield propagators are calculated and the interaction Lagrangian of a scalarmultiplet superfield is studied in $4. Writing the multispinor in component form
4
+zB(x3 6) =
~,,(x) + Q , $ , + ~ i/ ~~e ~ , , c x+) W ~ Q G ~ , L+~aoiy,y5eA,,,(x) )
+ gwo,x,,/j + h(ee)20,,(x)
(1.1) where Aza(x),Fzp(x),Gzp(x),Dzp(x)are second-rank multispinors and $,,&x), x .u(x) are third-rank multispinors, A,,,(x) is a vector-bispinor, we see that the spin range of the supermultiplet varies from 0 to 2. This can be reduced as follows: 4 z p k N
P (C) 2l7-A
6) =
4 +,&,
6) +
4 - , / k 4 + +o&,
6)
(1.2) 445
446
A Chamseddine
by means of the projection operators (Salam and Strathdee 1974, 1975)
Eo = 1
1 + @(DD)’
where
The irreducible chiral superfields q5 i.,II(x,0) satisfy the conditions
CN
* iY,)D1;.44 ,,(x,0)
=0
and the non-chiral part q50,/J satisfies
In terms of components the resolution (1.2) takes the form
2. A supersymmetric Lagrangian It is well known that to find a Lagrangian for higher spin equations it is necessary to introduce auxiliary fields which are set equal to zero by the equations of motion
447
A multispinor supersymmetric Lagrangian
in the absence of interaction. We shall require that the multispinor superfields satisfy the Bargmann-Wigner (1946) equations. Let O,,,(x, 0) and LRzp(x,0) respectively be the symmetrical and antisymmetrical parts of &,,(x, e):
Then S2,,(x,O) will act as the auxiliary field in our Lagrangian, We shall verify that the free Lagrangian for the massive case is
+
2po = &DD)2(@f@-z(j R"$-,,)
+ Q"(i$l)@+z/J
+
- +(DD)(@Y(i$l - 2un)@+,,, @Y(i$l)R+xp
- Wf(ig1 - 2m)R+,p)+ (+ * -1.
(2.2)
We note the following: @!(VI
)@ k z l J = @!(ih)@
?rzg
WWl)Qkzp = -W(i$2)fikzB
(2.3)
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A Chamseddine
where equation (2.9) is to be symmetrized and equation (2.10) antisymmetrized wlth respect to a and j: (2.11) ( i r d - 2m)@,,ir - tDD@)s,lj+ (iAd)Q,,, = -vr(,/~)
-(ird
-
2m)Q2z,j- fDDQ,,,,
+ (iAd)@,,/r
= -V+[~/II.
(2.12)
Solving equation (2.11) gives
[(ira- 2m),
- (fDD)2]@i,ij
= -[(ird
- 2m)~,(,/~, + +DDv,,,,)I
-
(iAd)2@k,Ir- (iA4v,[xijl.
(2.13)
Making use of the identities
(2.6‘) equation (2.13) becomes 4m(m - ird)@,,,
=
-(iAd)vi[,/jl -
[(ire- 2m)vk(,p)+ SDDv, (x/j)l.
(2.14)
Similarly we could solve for Q ,,ii: 4m(m - ird)Q,,,
+ tDDv,[,,ll
= [(ird - 2~n)v+.[,/J]
-
(2.15)
(iAd)v,(,,,).
By setting the external sources to zero in these derivations we show at once that the free field variables satisfy the required Bargmann-Wigner equations. Equations (2.14) and (2.15) give (2.16)
(2.17) But a totally antisymmetric spinor cannot be a simultaneous eigenstate (Chang 1967) of both operators (i@)land (i& unless Qi&,
0) = 0.
(2.18)
Thus we have shown that the Lagrangian (2.2) yields the correct Bargmann-Wigner equations upon setting the auxiliary field equal to zero. The Lagrangian (2.2) takes the following form in terms of its component fields: y o=
Z;ij)(ira- 2m)F1,,,, + A‘,”/jl(ir8- 2m)~,,,/!, + F(f/jl(ir; + F$/’)(ira- 2m)A2,,,, - ij;(’/’)(i$- 2m):$,(j,,r) -
+
~ . , ‘ ( x / ’ ) ( i ~ ) ~ $F?/’)F,,,,,, ,,~/j)
-
2nz)Al(,i,,
+ FF/’)F,(,,, + A~/’)(iAi3)FI,,,,
+ Afol(iAd)F,,p, + F~P)(iAd)F,,,,l+ F‘;D)(iAd)A,I,ol -
+
$~(‘~)(i$)<,$j.(,/j) F~”IF,,,,,
+ FP”F,,,,,
- ij~[’~l(i$)~$j.[,/rl
449
A multispinor supersymmetric Lagrangian
+ $;[xiJI(ij? -
+ +
-
2m)~$;,[;,,il I[2,i l
- A[,"ijl(irc? - h)Fl[,/J]
Ayjjl(ira - 2m)F2[,j,1- Fyj'](ird - 2m)Al[,/J]- F[,""l(ird- 2m)A2[,,, A~"l(iAc?)F,,xjJ)Ai[;/j1(iAc')F2,,,, F[,"/j1(iAd)A1,,,) F[,""l(iA8)A2,,,,,- $7['j'1(i(3)i$.;(;/j) (2.19)
+
+
The equations of motion for the Lagrangian (2.19) allow the identification of component fields. Thus, equations (2.17) and (2.18) show that Al(z/,), A2(3,j)are DuffinKemmer fields satisfying (m - iTc')A,,,,,
=
(m -
=
(2.20)
0.
F2,2jj) are auxiliary fields which could be cancelled from the The fields Fl(x,J), Lagrangian by the substitution
(2.21) On the other hand the multispinor field satisfies the Bargmann-Wigner equation ( i r 8 - m)$.,.(xjj)= 0 -
2m)$.;,xlj) =
-(i@)&;(qj).
Therefore (m-
ij?)l$;(,jj)
=
(m- i$)2$7(x,1)=
-
i$)3$>(2/J)= 0.
$;,(xjj) can be resolved uniquely into a totally symmetric spinor of spin of mixed symmetry (2,l) of spin
4:
(2.22)
3 and a spinor (2.23)
Thus all the particles appearing in the Lagrangian are physical. The particle content here agrees with those obtained by an algebraic approach by Salam and Strathdee (1974,1975) where they have shown that in one of the representations of the supersymmetry algebra, characterized by a mass. a spin J and an intrinsic parity y, there are four irreducible representations of the poincare group. The spin-parity content is ( J - +)'I, ( J ) " ' , (J)-"', ( J + *)'I. The rest mass is common. Here J =' 1, y = +i; these states correspond respectively to &, A,,,,, F,,,), ~
7
w
3. The superfield propagators To study the renormalizability of the Lagrangian (2.2) interacting with external sources one has to find the corresponding superfield propagators. We start with the identity
450
A Chamseddine
Thus equations (2.14) and (2.15) could be written as
The propagators are given by equations like
and
Following the calculations of Salam and Strathdee (1974,1975) we get
A multispinor supersymmetric Lagrangian
x d(X1 -
XJ.
The formulae (3.5) to (3.9) summarize the propagators of all the field components A:(?P)= @ i a p / e = o
and so on for all other propagators. In particular we note from (3.1 1) that
45 1
(3.9)
452
A Chamseddine
(3.12) a result which agrees with that obtained by Guralnik and Kibble (1965) and later by Chang (1967). One easily passes to the momentum space expressions of the Green functions G,,(,/,,,,/J,,(PBl:PB,)
=
1 7exP(i$42i$Y,Ql2 - iOldQ2) 8in
(3.13)
4. Interaction Lagrangian of scalar-multispinor superfields
Because of the complicated form of our propagators we will consider the simplest interaction form-the multispinors 03/j(x.Q) and Rx,i(x,0) interacting with a scalar superfield $(x, 0) . The total Lagrangian is
2 = 9;
+ 98 +
(4.1)
A multispinor supersymmetric Lagrangian
P-9
453
P-9
Figure 1.
Figure 2.
2'8 is the Wess-Zumino Lagrangian
* ( W 2 ( 4 + 4 --)
2 40 --L -
ww:+ 42-1.
(44
9;'' +" is given by equation (2.2) and
y,,,, = (-+w(g@!@+x&+
+ g'P$2+zp++) + (+
* -).
(4.3) In the following we shall examine the renormalizability of our interaction Lagrangian, In the figures, the solid and curly curves will refer to the superfields @ and a, the broken curves to the scalar ones. The one-loop self-energy graphs E* * and ll, and the vertex graphs Ti e vanishes because (Delbourgo 1975, Capper and Leibbrandt 1975) for n 3 2 (ee'y = 0 (4.4) (e,,e:,)(e23e~3)...(e,, 8 ,') = 0. Applying the effective Feynman rules (Delbourgo 1975, Capper and Leibbrandt 1975) to figure l(u) we get
1 29' - 5m2 ( 2 ~ ( 4p ~ 4)2 - p 2 q2 - m2
I
Figure 3.
454
A Chamseddine I
I
I I
I
Figure 4.
which is quadratically divergent and requires two subtractions. Following the work of Delbourgo (1975) for qb4 interactions, the order g 2 counter Lagrangian is
6 9 1 = &(21- l)(DD)2(@'f'@-xp)+ Y l ( D D ) z ( ~ l , ~ ~ ' ~ ~+, @ (+- xc-t , f -) )
-
with
z1
1 + G2A2
Y,
-
G2 In A2.
The !2 interaction will give exactly the same divergence and need not be written. The self-energy of figure 2(a) is
=
4 p 2 g 2 1 240 (1 (2.14
+
q4
U P - 4)' - P21h2 - m 2 )
+ less divergent terms
which has divergences of fourth order. We need a counter term of the form
8 9 2 = $(z, - l)(DD)'($+ $ - )
+ Y , ( D D ) 2 ( J L l $?$-) + + Y3(DD)2(i?24+z24-).
Figures 3(a) and ( b ) will give fourth order divergences and could be cancelled by a counter term 6 9 3
23(-fDD)(@'!@++l/j$+) + (+ * -).
Figure 4(a) will give sixth-order divergences, and so on. Clearly the Lagrangian is non-renormalizable. 5. Conclusion
Our Lagrangian contains a massive spin 1 particle which causes non-renormalizability in QED. But since we are dealing with supersymmetry one may expect that cancellations take place between coupling of this particle with spin i and 3 particles in the theory. To see if this possibility occurs, we have examined the first-loop corrections. Cancellations have taken place but they are not enough to render the theory renormalizable. The existence of the sixth-order divergences shows that even in a more detailed analysis the divergences cannot be softened.
A multispinor supersymmetric Lagrangian
455
Acknowledgments I should like to thank Professor Abdus Salam for suggesting the problem. I am indebted to Dr R Delbourgo for his advice and many helpful discussions in the subject. I should like to thank the Lebanese University (Faculty of Science) for a maintenance grant. References Adjei S A and Akyeampong D A 1975 Nuovo Cirn. A 26 184 Bargmann V and Wigner E 1946 Proc. Nat. Acad. Sci. 34 21 1 Capper D and Leibbrandt G 1975 Nucl. Phys. B 85 492 Chang S J 1967 Phys. Ret.. 161 1316 Delbourgo R 1975 Nuouo Cim. A 25 646 Guralnik G and Kibble T W B 1965 Phys. Rev. 139 B712 Salam A and Strathdee J 1974 Nucl. Phys. B 70 39 -1975 Phys. Rev. D 11 1521 Wess J and Zumino B 1974a Nucl. Phys. B 70 39 -1974b Phys. Lett. 49B 52
