JOURNAL OF MATHEMATICAL PHYSICS 47, 063504 共2006兲

Scale invariance in the spectral action Ali H. Chamseddine Physics Department, American University of Beirut, Lebanon

Alain Connes College de France, 3 rue Ulm, F-75005, Paris, France, I.H.E.S. Bures-sur-Yvette, France, and Department of Mathematics, Vanderbilt University, Nashville, TN 37240 共Received 19 January 2006; accepted 15 March 2006; published online 16 June 2006兲

The arbitrary mass scale in the spectral action for the Dirac operator is made dynamical by introducing a dilaton field. We evaluate all the low-energy terms in the spectral action and determine the dilaton couplings. These results are applied to the spectral action of the noncommutative space defined by the standard model. We show that the effective action for all matter couplings is scale invariant, except for the dilaton kinetic term and Einstein-Hilbert term. The resulting action is almost identical to the one proposed for making the standard model scale invariant as well as the model for extended inflation and has the same low-energy limit as the Randall-Sundrum model. Remarkably, all desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2196748兴

I. INTRODUCTION

It is known that the standard model of strong and electroweak interactions is classically almost scale invariant, and that the only terms that break the dilatation symmetry are the mass terms in the Higgs sector. Scale invariance of the classical Lagrangian can be achieved by introducing a compensating dilaton field.1–3 Breaking of scale invariance occurs after the electroweak symmetry is broken spontaneously through the generation of radiative corrections to the scalar potential. The dilaton mass scale is much larger than the weak scale and could be as large as the GUT scale or Planck scale. This leads naturally to consider the coupling of the dilaton to gravity. The dilaton is always part of the low energy spectrum in string theory. Historically it first appeared in the Jordan-Brans-Dicke theory of gravity which corresponds to one particular coupling of the dilaton to the metric.2 The dilaton plays a fundamental role in models of inflation.4 It also appears in the gravitational couplings of the noncommutative Connes-Lott formulation of the standard model,5–8 where the dilaton is the scalar field that couples the two sheets of space time. The resulting matter interactions in this case are also scale invariant, and the gravitational couplings are different than the Jordan-Brans-Dicke theory. More recently, a scalar field, the radion field, appeared in the Randall-Sundrum 共RS兲 scenario of compactification9 which is related to the question of masses and scales in physics. The RS scenario was shown to be equivalent to the results derived from noncommutative geometry,10 which is not too surprising, because both the Connes-Lott model and the RS model, describe a system with two branes. At present, and within the noncommutative geometric picture, the spectral action gives the most elegant formulation of the standard model.11,12 All details of the standard model as well as its unification with gravity are achieved by postulating the action Trace F共D2/m2兲 + 具⌿兩D兩⌿典, where D is the Dirac operator of a certain noncommutative space and ⌿ is a spinor in the Hilbert space of the observed quarks and leptons. However, the dilaton field does not appear in the spectral action, which is to be contrasted with the Connes-Lott formulation of the noncommutative action where the dilaton field is part of the gravitational interactions. This suggests that the Dirac 0022-2488/2006/47共6兲/063504/19/$23.00

47, 063504-1

© 2006 American Institute of Physics

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063504-2

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine and A. Connes

operator used in the construction of the spectral action should be modified in order to take into account the presence of the dilaton. The appearance of the dilaton field in physical models is related to the question of mass and scales. It is therefore natural to consider replacing the mass parameter in the ratio D2 / m2 by a function of the dilaton, thus introducing a sliding scale, as the Dirac operator have the dimension of mass. This is also relevant when dealing with noncompact manifolds where D no longer has discrete spectrum and the counting of eigenvalues requires a localization. Let the dynamical scale factor ␳ be written in the form

␳ = me␾ , where we assume that ␾ is dimensionless. The dilaton can be related to a scalar field ␴ of dimension one by writing 1 ␾ = ␴, f where f is the dilaton decay constant. The mass scale m can be absorbed by the redefinition

␾ → ␾ − ln m, and therefore we can assume, without any loss in generality, that ␳ = e␾. One can always recover the scale m by performing the opposite transformation ␾ → ␾ + ln m. Now using ␳ instead of the scalar m in the counting of eigenvalues: N共m兲 = Dim兵D2 艋 m2其 → N共␳兲 = Dim兵D2 艋 ␳2其 is equivalent to replacing the operator D2 / m2 in Ref. 11 by P = e −␾D 2e −␾ . If we insist that the metric g␮␯ be dimensionless to insure that its flat limit be the Minkowski metric, then the scale m will explicitly appear in the action after rescaling e␾ → me␾. Otherwise we can absorb this mass scale by assuming that the metric has the dimension of mass. The aim of this article is to determine the interactions of the dilaton field ␾ with all other fields present in the spectral action formulation of the standard model. Because of the spectral character of the action, it is completely determined from the form of P and there is no room for fine tuning the results. It is then very reassuring to find that the resulting interactions are identical to those constructed in the literature by postulating a hidden scale invariance of the matter interactions.3 These are also equivalent to the interactions of the radion field in the RS model.9 All of these results now support the conclusion that space time at high energies reveals its discrete structure, and is governed by noncommutative geometry. The plan of this article is as follows. In Sec. II we briefly review the derivation of the spectral action and comment on the modifications needed to include the dilaton. In Sec. III we derive the Seeley-de Witt coefficients of the spectral action in presence of the dilaton. In Sec. IV we give the full low-energy spectral action including dilaton interactions specialized to the noncommutative space of the standard model. In Sec. V we compare our results with those obtained by imposing scale invariance on the standard model interactions, to the RS model and the model of extended inflation. Section six is the conclusion. The appendices contain detailed proofs of some identities used. II. A SUMMARY OF SPECTRAL ACTION

We begin by summarizing the results of Ref. 11. The square of the Dirac operator appearing in the spectral triple of a noncommutative space is written in the following form suitable to apply the standard local formulas for the heat expansion 共see Ref. 13 Sec. 4.8兲.

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063504-3

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

D2 = − 共g␮␯I⳵␮⳵␯ + A␮⳵␮ + B兲,

共1兲

where g␮␯ plays the role of the inverse metric, I is the unit matrix, and A␮ and B are matrix functions computed from the Dirac operator. The bosonic part of the spectral action can be expanded in a power series as a function of the inverse scale m, and is given in dimension 4 by Trace共F共D2/m2兲兲 ⯝

f nan共D2/m2兲, 兺 n艌0

where F is a positive function and f0 =

冕

⬁

F共u兲u du,

f2 =

0

冕

⬁

f 2n+4 = 共− 1兲nF共n兲共0兲,

F共u兲du,

n 艌 0.

共2兲

0

The positivity of the function F will insure that the actions for gravity, Yang-Mills, Higgs couplings are all positive and the Higgs mass term is negative. We will comment on the positive sign of the cosmological constant at the end of the article. The first few Seeley-deWitt coefficients an共D2 / m2兲 are given 共see Ref. 13, Theorem 4.8兲 by 关according to our notations the scalar curvature R is negative for spheres 共see Ref. 13 Sec. 2.3兲 and the space is Euclidean兴 a0共D2/m2兲 =

a2共D2/m2兲 =

a4共D2/m2兲 =

1 1 16␲2 360 +

60E;␮␮ +

冕

M

m4 16␲2

m2 16␲2

冕

冕

d4x冑g Tr共1兲,

共3兲

M

冉

d4x冑g Tr −

M

冊

R +E , 6

共4兲

d4x冑g Tr共− 12R;␮␮ + 5R2 − 2R␮␯R␮␯ + 2R␮␯␳␴R␮␯␳␴ − 60RE + 180E2

30⍀␮␯⍀␮␯兲,

共5兲

whereas the odd ones all vanish a2n+1共D2/m2兲 = 0. The notations are as follows, one lets ⌫␮␳ ␯共g兲 be the Christoffel symbols of the Levi-Civita connection of the metric g and lets ⌫␳共g兲 = g␮␯⌫␮␳ ␯共g兲. ¯ , its curvature ⍀, and the endomorphism E are then defined by 共see Ref. 13 The connection form ␻ Sec. 4.8兲 ¯ ␮ = 21 g␮␯共A␯ + ⌫␯共g兲I兲, ␻

共6兲

¯ ␯ − ⳵ ␯␻ ¯ ␮ + 关␻ ¯ ␮, ␻ ¯ ␯兴, ⍀ ␮␯ = ⳵ ␮␻

共7兲

¯␯ + ␻ ¯ ␮␻ ¯ ␯ − ⌫␮␳ ␯共g兲␻ ¯ ␳兲. E = B − g ␮␯共 ⳵ ␮␻

共8兲

To understand algebraically the dependence in the operator D it is convenient to express the previous coefficients as residues and this is done as follows in the generality that we need. One lets P be a second order elliptic operator with positive scalar principal symbol and defines a zeta function as

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063504-4

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine and A. Connes

␨ P共s兲 = Trace共P−s/2兲. One then gets in the required generality for our purpose the equality Trace共F共P/m2兲兲 ⬃

m4 m2 Ress=4␨ P共s兲f 0 + Ress=2␨ P共s兲f 2 + ␨ P共0兲f 4 + ¯ , 2 2

which, using the Wodzicki residue which is given independently of D by

冕

− T = Ress=0 Trace 共T共D2兲−s/2兲,

共9兲

can be written as Trace 共F共P/m2兲兲 ⬃

冕

冕

m4 m2 f 0 − P−2 + f 2 − P−1 + f 4␨ P共0兲 + ¯ . 2 2

共10兲

We want to compute the spectral action associated with the operator P = e−␾D2e−␾, i.e., to determine the dependence of the spectral action on the dilaton field ␾. The first term −兰P⫺2 is a Dixmier trace and one can permute the functions with the operators without altering the result since the Dixmier trace vanishes on operators of order ⬍−4. One thus gets, for any test function h,

using the trace property of the residue and again we get an overall factor of e2␾ multiplying a2共x , D2兲. Note that the result remains valid when the test function h is taken with values in endomorphisms of the vector bundle on which P is acting. This suggests that the invariance of the a2 term 共up to the e2␾ scale factor兲 takes place before taking the fiberwise trace. The direct computation as follows in Eq. 共18兲 will confirm this point. The term f 4␨ P共0兲 is more tricky to analyze and we shall only give now a heuristic argument explaining why it should be independent of ␾. We shall then check it by a direct calculation. The formal argument proceeds as follows. First one lets P共t兲 = e−t␾D2e−t␾ so that P共0兲 = D2 and P共1兲 = P with the previous notations. Let then Y共t兲 = log P共t兲 − log P共0兲. Using the equality 共a ⬎ 0兲

冕冉 ⬁

log a =

0

冊

1 1 d␭, − ␭+1 ␭+a

applied to P共t兲 one obtains the relation d Y共t兲 = − dt

冕

⬁

共P共t兲 + ␭兲−1共␾ P共t兲 + P共t兲␾兲共P共t兲 + ␭兲−1d␭.

共11兲

0

One then shows that

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063504-5

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

d Y共t兲 = − 2␾ + 关P共t兲,C共t兲兴, dt

共12兲

where C共t兲 is a pseudodifferential operator 共see Appendix A兲. Thus one gets a similar expression Y = Y共1兲 = − 2␾ +

冕

1

关P共t兲,C共t兲兴dt.

共13兲

0

Next one uses the expansional formula ⬁

e

=兺

A+B −A

e

0

冕

0艋t1艋¯艋tn艋1

B共t1兲B共t2兲 . . . B共tn兲 兿 dti ,

where B共t兲 = etABe−tA . One lets A = −s log P共0兲 and B = −sY. This gives an equality of the form P−s = D−2s − s

冕

1

␴−st共Y兲dtD−2s

0

⬁

+ 兺 共− s兲n 2

冕

0艋t1艋¯艋tn艋1

␴−st1共Y兲␴−st2共Y兲 ¯ ␴−stn共Y兲 兿 dtiD−2s ,

where

␴u共T兲 = 共D2兲uT共D2兲−u . One infers from this equality and the absence of poles of order ⬎1 in the zeta functions of the form Tr共QD−2s兲 that the terms of order n ⬎ 1 in s will not contribute to the value at s = 0. Thus the following should hold:

冕

1 ␨ P共0兲 − ␨D2共0兲 = − − Y 2 and using 共13兲 one gets

冕

␨ P共0兲 − ␨D2共0兲 = − ␾ = 0, as the residue vanishes on differential operators. It would take a lot more care to really justify the previous manipulations. Instead, in the next section, we shall show by a brute force calculation that a4 is independent of ␾ so that the above-mentioned identity is valid. We thus see that in the first few terms of the spectral action, the only modification we expect when the operator D2 is replaced by P is to get an overall factor of e共4−n兲␾ multiplying an共x , D2兲: 6

Trace共F共P兲兲 ⯝ 兺 f n n=0

冕

d4x冑ge共4−n兲␾an共x,D2兲 + ¯ .

Also as will be shown in Appendix B, we have the identity an共x,e−␾D2e−␾兲 = an共x,D2e−2␾兲 = an共x,e−2␾D2兲. It is easy to check that by applying the inverse transformation ␾ → ␾ + ln m one recovers all the m scaling factors obtained in Ref. 11. In the next section, we shall confirm this result by directly evaluating the spectral action associated with the operator P and in particular the low-energy terms

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063504-6

A. H. Chamseddine and A. Connes

J. Math. Phys. 47, 063504 共2006兲

a0, a2, and a4. We will not attempt to evaluate higher order terms as these are not needed in our analysis. III. DILATON AND SEELEY-deWITT COEFFICIENTS

We compare quite generally the Seeley-deWitt coefficients of an operator P0 = D2 given by 共1兲 and those of the rescaled operator P = e−␾D2e−␾. We use the rescaled metric G in the Einstein frame, where the dilaton factor is absorbed in the metric. First we write P = e−␾D2e−␾ = − 共G␮␯⳵␮⳵␯ + A␮⳵␮ + B兲,

共14兲

where G␮␯ = e−2␾g␮␯ , A␮ = e−2␾A␮ − 2G␮␯⳵␯␾ , B = e−2␾B + G␮␯共⳵␮␾⳵␯␾ − ⳵␮⳵␯␾兲 − e−2␾A␮⳵␮␾ . The Seeley-deWitt coefficients for an共P兲 are expressed in terms of E and ⍀␮␯ defined by 共6兲 so that, E = B − G␮␯共⳵␮␻⬘␯ + ␻⬘␮␻⬘␯ − ⌫␮␳ ␯共G兲␻⬘␳兲,

␻⬘␮ = 21 G␮␯共A␯ + ⌫␯共G兲兲, ⍀␮␯ = ⳵␮␻⬘␯ − ⳵␯␻⬘␮ + 关␻⬘␮, ␻⬘␯兴. These relations imply that

␻⬘␮ = 21 g␮␯A␯ − ⳵␮␾ + 21 G␮␯⌫␯共G兲. The conformal transformations of the Christoffel connection give ␮ ␮ ⌫␯␳ 共G兲 = ⌫␯␳ 共g兲 + 共␦␯␮⳵␳␾ + ␦␳␮⳵␯␾ − g␯␳g␮␴⳵␴␾兲,

⌫␮共G兲 = e−2␾⌫␮共g兲 − 2e−2␾g␮␯⳵␯␾ . Using these relations we finally get ¯ ␮ − 2 ⳵ ␮␾ , ␻ ⬘␮ = ␻

共15兲

E = e−2␾共E + g␮␯共ⵜ␮g ⵜ␯g␾ + ⳵␮␾⳵␯␾兲兲,

共16兲

where the covariant derivative ⵜ␮g is taken with respect to the metric g. It is quite striking that the perturbation is only a scalar multiple of the identity matrix and does not involve the endomorphisms A␮ at all. The term a0 only involves 冑G Tr共1兲 which, when expressed in terms of the metric g gives 冑ge4␾Tr共1兲. The a2 term is proportional to

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063504-7

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

冕

冉

冊

1 d4x冑G Tr E − R共G兲 , 6

where the curvature scalar is constructed as function of the metric G. We now use 共16兲 and the relation R共G兲 = e−2␾共R共g兲 + 6g␮␯共ⵜ␮g ⵜ␯g␾ + ⳵␮␾⳵␯␾兲兲

共17兲

to obtain at the level of endomorphisms 共before taking the fiberwise trace兲 E − 61 R共G兲 = e−2␾共E − 61 R共g兲兲 .

共18兲

This of course implies the required rescaling of the a2 term in the required generality, but it is more precise since it holds before taking the trace. We shall use this more precise form in the proof of the invariance of the a4 term. The term a4共P兲 is given by 1 1 16␲2 360

冕

d4x冑G Tr 关共5R2共G兲 − 2R␮␯共G兲R␮␯共G兲 + 2R␮␯␳␴共G兲R␮␯␳␴共G兲兲 − 60R共G兲E + 180E2 + 30⍀␮␯⍀␳␴G␮␳G␯␴ 兴 ,

¯␮ where we have omitted the total derivative terms 12共−R共G兲 + 5E兲;␮␮. As the modification from ␻ to ␻⬘␮ is Abelian 共15兲 we get ⍀ ␮␯ = ⍀ ␮␯ and ⍀␮␯⍀␳␴G␮␳G␯␴ = e−4␾⍀␮␯⍀␳␴g␮␳g␯␴ . Next we group the terms 180E2 − 60ER共G兲 + 5R2共G兲 = 180共E − 61 R共G兲兲2 , which yields upon using Eq. 共18兲 180e−4␾共E − 61 R共g兲兲2 . We are left with the terms 30⍀␮␯⍀␳␴G␮␳G␯␴ − 2R␮␯共G兲R␮␯共G兲 + 2R␮␯␳␴共G兲R␮␯␳␴共G兲. We now use 30 Tr共⍀␮␯⍀␳␴G␮␳G␯␴兲 = 30e−4␾Tr共⍀␮␯⍀␳␴g␮␳g␯␴兲. − 2R␮␯共G兲R␮␯共G兲 + 2R␮␯␳␴共G兲R␮␯␳␴共G兲 = − R共G兲*R*共G兲 + 3C␮␯␳␴共G兲C␮␯␳␴共G兲. In deriving the last relation we made use of the two identities R*R* − R2 = R␮2 ␯␳␴ − 4R␮2 ␯ , C␮2 ␯␳␴ − 31 R2 = R␮2 ␯␳␴ − 2R␮2 ␯ , where C␮␯␳␴ is the conformal tensor and

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063504-8

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine and A. Connes

R *R * =

1

4 冑g

␥␦ ⑀␮␯␳␴⑀␣␤␥␦R␮␣␤␯ R␳␴ .

These imply R␮2 ␯␳␴ = 2C␮2 ␯␳␴ − R*R* + 31 R2 , 1 1 R␮2 ␯ = C␮2 ␯␳␴ − R*R* + 31 R2 . 2 2 The square of the conformal tensor is known to be conformal invariant

冕

d4x冑GC␮␯␳␴共G兲C␮␯␳␴共G兲 =

冕

d4x冑gC␮␯␳␴共g兲C␮␯␳␴共g兲.

The topological Gauss-Bonnet term is metric independent and therefore conformal invariant

冕

d4x冑GR共G兲*R*共G兲 =

冕

d4x冑gR共g兲*R*共g兲,

and this can be rewritten as 1 4

冕

d 4x

1

冑g ⑀

␮␯␳␴

␥␦ ⑀␣␤␥␦R␮␣␤␯ R␳␴ .

This shows that the a4 term has the expected invariance under the rescaling of the operator P0 → P = e −␾ P 0e −␾. IV. SPECTRAL ACTION WITH DILATON

We now use the result of the previous section to compute the spectral action with dilaton as a function of the rescaled metric G in the Einstein frame, where the dilaton factor is absorbed in the metric. The lowest term in the spectral action is given by 45 f0 4␲2

冕

d4x冑ge4␾ =

45 f0 4␲2

冕

d4x冑G.

共Note that the dimension of the bundle on which the operator is acting is 4 ⫻ 3 ⫻ 15 where the 4 is the dimension of spinors, 3 the number of generations, and 15= 4 ⫻ 3 + 3 is the content of each generation兲. The next term in the spectral action with dilaton of the standard model is, in terms of the original metric g: 3 f2 4␲2

冕

d4x冑ge2␾

冉

冊

5 R共g兲 − 2y 2H*H . 4

共19兲

We can transform this back to the Einstein frame with metric G␮␯ so that the curvature scalar term has no scale factors in front of it. Using Eq. 共17兲 with g → G and ␾ → −␾ the curvature R共g兲 is R共g兲 = e2␾共R共G兲 + 6G␮␯共− ⵜ␮Gⵜ␯G␾ + ⳵␮␾⳵␯␾兲兲, and we obtain

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063504-9

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

冕

d4x冑ge2␾R共g兲 = =

冕 冕

d4x冑G共R共G兲 + 6G␮␯共− ⵜ␮Gⵜ␯G␾ + ⳵␮␾⳵␯␾兲兲 d4x冑G共R共G兲 + 6G␮␯⳵␮␾⳵␯␾兲

after integrating by parts. The a2 term 共19兲 thus becomes 3 f2 4␲2

冕

d4x冑G

冉

冊

15 5 R共G兲 + G␮␯⳵␮␾⳵␯␾ − 2y 2H⬘*H⬘ , 4 2

共20兲

where we have defined H = e ␾H ⬘ , so that the only appearance of the dilaton ␾ is through its kinetic energy. Let us pause a bit and discuss signs at this point. For a positive test function F the coefficients f 0, f 2, f 4 are all positive. It is important that the Einstein term 兰d4x冑GR共G兲 appears in 共20兲 with the correct sign for the Euclidean functional integral, and that the kinetic term for ␾ namely 兰d4x冑GG␮␯⳵␮␾⳵␯␾ appears with a positive coefficient in 共20兲. The next term coming from a4共x , P兲 is unchanged for the spectral action with dilaton, and thus given independently of ␾ by Ref. 11,

冕

f4 4␲2

d 4 x 冑g

冉

冉

1 共11R共g兲*R*共g兲 − 18C␮␯␳␴共g兲C␣␤␥␦共g兲g␮␣g␯␤g␳␥g␴␦兲 32

1 + 3y 2 D␮H*D␯Hg␮␯ − R共g兲H*H 6

冉

冊冊冉

冊

冊

5 i ␣ + g23G␮i ␯G␳␴ + g22F␮␣ ␯F␳␴ + g21B␮␯B␳␴ g␮␳g␯␴ + 3z2共H*H兲2 , 3 where we have omitted total derivatives as they only contribute to boundary terms. Let us show that we can rewrite this term in the following way as a function of the metric G␮␯ by making use of the conformal invariance of a4: f4 4␲2

冕

d 4 x 冑G

冉

冉

1 共11R共G兲*R*共G兲 − 18C␮␯␳␴共G兲C␣␤␥␦共G兲G␮␣G␯␤G␳␥G␴␦兲 32

1 + 3y 2 D␮H⬘*D␯H⬘G␮␯ − R共G兲H⬘*H⬘ 6

冉

冊冉 冊

冊

冊

5 ␣ i + g21B␮␯B␳␴ G␮␳G␯␴ + 3z2共H⬘*H⬘兲2 . + g23G␮i ␯G␳␴ + g22F␮␣ ␯F␳␴ 3 The terms which only involve the metric are conformal by construction. The same holds for the terms which involve the gauge fields since the Yang-Mills action is conformal. Thus we need only to take care of the terms that involve the Higgs fields. We have to show that the following expression is unchanged by g → G and H → H⬘; 3f 4y 2 4␲2

冕

冉

冊

1 d4x冑g g␮␯D␮H*D␯H − R共g兲H*H . 6

To see this we first rescale the kinetic energy of the Higgs field

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063504-10

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine and A. Connes

冑gg␮␯D␮H*D␯H = 冑GG␮␯e−2␾D␮共e␾H⬘*兲D␯共e␾H⬘兲 = 冑GG␮␯共D␮H⬘*D␯H⬘ + ⳵␮␾H⬘*D␯H⬘ + D␮H⬘*H⬘⳵␯␾ + H⬘*H⬘⳵␮␾⳵␯␾兲. The conformal coupling of the Higgs field to the scalar curvature transforms as

−

1 冑gR共g兲H*H = − 1 冑GH⬘*H⬘共R共G兲 − 6G␮␯共共ⵜ␮ⵜ␯兲G␾ − ⳵␮␾⳵␯␾兲兲. 6 6

After integrating by parts the term

冕

d4x冑GH⬘*H⬘共ⵜ␮ⵜ␯兲G␾G␮␯ ,

we find that all cross terms cancel, thus obtaining

冕

冉

冊冕

1 d4x冑g g␮␯D␮H*D␯H − R共g兲H*H = 6

冉

冊

1 d4x冑G G␮␯D␮H⬘*D␯H⬘ − R共G兲H⬘*H⬘ . 6

The quartic Higgs interactions are evidently scale invariant

冕

d4x冑g共H*H兲2 =

冕

d4x冑G共H⬘*H⬘兲2 .

Collecting all terms, the low-energy bosonic part of the spectral action with dilaton is given by

Ib =

45 f0 4␲2 +

f4 4␲2

冕 冕

d4x冑G +

冉 冊冉 冉

d 4 x 冑G

1 − R共G兲H⬘*H⬘ 6

3 f2 4␲2

冕

d4x冑G

冉

15 5 R共G兲 + G␮␯⳵␮␾⳵␯␾ − 2y 2H⬘*H⬘ 4 2

冊 冉

冊

1 共11R共G兲*R*共G兲 − 18C␮␯␳␴共G兲C␮␯␳␴共G兲兲 + 3y 2 D␮H⬘*D␯H⬘G␮␯ 32

冊

冊

5 i ␣ + g23G␮i ␯G␳␴ + g22F␮␣ ␯F␳␴ + g21B␮␯B␳␴ G␮␳G␯␴ + 3z2共H⬘*H⬘兲2 . 3 共21兲

For higher order terms one expects a scaling factor of the form e共4−n兲␾ to be present, but derivatives of the dilaton field ␾ may also occur. Therefore in the Einstein frame, one does not expect the dilaton field ␾ to acquire a potential. As will be discussed later, this will change when quantum corrections are taken into account and the dilaton acquires a potential of the ColemanWeinberg type.14 Fermionic interactions take the simple form 具⌿兩D兩⌿典 =

冕

¯ D⌿, d4x冑g⌿

where the metric g␮␯ is used to insure hermiticity of D. We will now show that the fermions will not feel the dilaton. To see this we first redefine the spinors by ⌿ = e共3/2兲␾⌿⬘ , then we have, for the parts not involving the Higgs or gauge fields,

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063504-11

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

具⌿兩D兩⌿典 =

冕

冉

冊

¯ ⬘␥ce␾E␮ ⳵ + 1 ␻ab共e兲␥ 共e共3/2兲␾⌿⬘兲, d4x冑Ge−4␾e共3/2兲␾⌿ ␮ ab c 4 ␮

where the rescaled vierbein is E␮a = e␾e␮a . We have to express the spin connection ␻␮ab共e兲 in terms of the spin connection of the rescaled vierbein ⍀␮ab共E兲. To do this we use the equations

⳵␮e␯a − ⳵␯e␮a − ␻␮ab共e兲e␯b + ␻␯ab共e兲e␮b = 0, ⳵␮E␯a − ⳵␯E␮a − ⍀␮ab共E兲E␯b + ⍀␯ab共E兲E␮b = 0, and these imply that

␻␮ab共e兲 = ⍀␮ab共E兲 + 共e␮a e␯b − e␮a e␯b兲⳵␯␾ . Therefore

␥cec␮共 23 ⳵␮␾ + 41 ␻␮ab共e兲␥ab兲 = ␥ce␾Ec␮共 41 ⍀␮ab共E兲␥ab兲 , and the fermionic action reduces to the nice form

冕

冉

冊

1 d4x冑G ⌿⬘␥cEc␮ ⳵␮ + ⍀␮,ab共E兲␥ab ⌿⬘ , 4

which is independent of the dilaton. Finally the parts involving interactions between the fermions and the Higgs or gauge fields could be written in the forms ¯ ␥ H⌿ = d4x冑g⌿ 5

冕

d4x冑G ⌿⬘␥5H⬘⌿⬘ ,

¯ ␥ ae ␮A ⌿ = d4x冑g⌿ a ␮

冕

d4x冑G ⌿⬘␥aEa␮A␮⌿⬘ .

冕 冕

The fermionic interactions are I f = 具Q兩Dq兩Q典 + 具L兩Dl兩L典, where

Q=

冢冣冢冣 uL dL , dR uR

␯L eL eR

and these take exactly the same form as those without dilaton when expressed in terms of the metric G␮␯. Rewriting this in terms of the fermionic fields Q⬘ = e−共3/2兲␾Q,

L⬘ = e−共3/2兲␾L,

and the Higgs field H⬘ we obtain If =

冕

d4x冑G共L⬘Dl⬘L⬘ + Q⬘Dq⬘Q⬘兲,

where11

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063504-12

Dl⬘ =

Dq⬘ =

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine and A. Connes

冢

冉

␥␮ 丢 共D␮ 丢 12 − 2i g2A␮␣␴␣ + 2i g1B␮ 丢 12兲 丢 13

␥5 丢 ke 丢 H⬘

␥5 丢 k*e 丢 H*⬘

␥␮ 丢 共D␮ + ig1B␮兲 丢 13

␥␮ 丢 ⵜ␮共1,2兲 丢 13


⬘ ␥5 丢 ku 丢 H

␥5 丢 kd 丢 H⬘

冉

i ␥ ␮ 丢 D ␮ + g 1B ␮ 3

␥5 丢 k*d 丢 H*⬘
⬘ * ␥5 丢 k*u 丢 H

冊

丢

13

0

冉

␥␮ 丢 D␮ −

0

冉

冊

冊

冊

2i g 1B ␮ 丢 1 3 3

i + ␥␮ 丢 14 丢 13 丢 − g3V␮i ␭i , 2

冣

and

␥␮ = ␥aEa␮ , D␮ = ⳵␮ + 41 ⍀␮ab共E兲␥ab , i i ⵜ␮共1,2兲 = D␮ 丢 12 − g2A␮␣␴␣ − g1B␮ 丢 12 . 2 6 From the previous considerations we deduce that the only effect of the dilaton on the low-energy terms of the spectral action is that the dilaton gets a kinetic term with no other interactions. This confirms that all matter interactions in the above-mentioned Lagrangian are scale invariant when expressed in the rescaled fields G␮␯, H⬘, and ⌿⬘. Only the Einstein term and the dilaton kinetic energy are not scale invariant. Note that the invariance of the action for the Fermions, that is the equality 具⌿兩D兩⌿典 = 具⌿⬘兩D⬘兩⌿⬘典⬘ ,

共22兲

where D⬘ corresponds to the metric G and the fields H⬘, does not mean that the operators D and D⬘ are the same. Indeed the transformation ⌿ → ⌿⬘ is not unitary and one has 具⌿⬘兩⌿⬘典⬘ = 具⌿兩e␾兩⌿典,

共23兲

D⬘ ⬃ e−␾/2De−␾/2 .

共24兲

which gives the unitary equivalence

One might then be tempted to conclude that the square of e−␾/2De−␾/2 should be unitarily equivalent to P = e−␾D2e−␾ but this does not hold precisely because of the additional kinetic term in the spectral action with dilaton. Indeed one can prove 共Appendix C兲 in the general framework of spectral triples, with a minimum amount of hypothesis, the identity

冕

冕

− e2␾D−2 = − 共e−␾/2De−␾/2兲−2 +

冕

1 − 关D,e␾兴关D,e␾兴*D−4 2

共25兲

with D⬘ = e−␾/2De−␾/2 the last term gives the canonical kinetic energy of the dilaton

冕

1 − 关D⬘␾兴关D⬘, ␾兴*D⬘−4 2 with the correct sign.

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063504-13

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

V. APPLICATIONS

We have shown that the dilaton interactions of the spectral action are almost the same as the ones proposed in the literature,2,3 the difference lies in the derivative couplings of the dilaton field. These were obtained by requiring the standard model matter sector to be scale invariant by introducing a compensating dilaton field. The origin of the dilatational symmetry breaking are the mass terms of the Higgs potential, and these are scaled with the dilaton field to make them scale invariant. In a curved space time all fields couple to gravity, and the dilaton. The proposed action for the gravity-dilaton-Higgs sectors, in our notation, was derived to be 共this expression is in the conventions of Ref. 4 and is in Minkowski space兲3,4

I=

冕

冉

d 4 x 冑G −

冉

冊

冊

1 1 6 1 + 2 2 G␮␯⳵␮␾⳵␯␾+ G␮␯D␮H⬘*D␯H⬘ − V0共H⬘*H⬘兲 . 2R + 2 2␬ ␬ f

There it was shown that in curved space-time this corresponds to the Jordan-Brans-Dicke theory of gravity. The only difference between this action and the spectral action is that the latter has the conformal coupling

⬘

1 * 6 R共G兲共H H

⬘兲 2 ,

which is necessary to make the matter couplings scale invariant. A slight modification was also proposed in the study of models of extended inflation, by also considering the possibility of modifying the Higgs sector by taking4 e共2/f兲␾G␮␯D␮H*D␯H − e共4/f兲␾V0共H*H兲. This differs from the spectral action by the appearance of derivative couplings of the form G ␮␯D ␮H ⬘*H ⬘⳵ ␯␾ . It is amusing to note that this alternative proposed action is exactly the same action as the one derived for the Connes-Lott gravitational interactions.6,10 Therefore the two models proposed in the literature for making the Higgs sector scale invariant are the same as the interactions obtained for the noncommutative standard model, either for the spectral action formulation, or the ConnesLott formulation. We also note that scale invariance of the action is broken by the Einstein term and by the kinetic term for the dilaton. This is remarkable because it was shown that if the full action is scale invariant, then the couplings will not lead to a model with extended inflation. Quantum corrections and renormalization conditions break scale invariance in the matter sector of the standard model and lead to an exponentially large hierarchy between the mass scale f where ␾ = 共1 / f兲␴ and the electroweak scale without fine tuning. The scale f is normally of the order of the Planck scale. The dilaton mass obtained depends on the Higgs mass, but should be constrained to be smaller than 10−6 eV. The noncommutative space of the standard model is obtained by taking the product of a four-dimensional Riemannian manifold times a discrete space dictated by the symmetries of the Hilbert space spanned by the quarks and leptons. The presence of left- and right-handed fermions provides the intuitive picture where these fermions are placed on different sheets. The gauge fields in the discrete dimensions are the Higgs fields, with the inverse of the distance between the sheets interpreted as the electroweak energy scale. This picture is similar to the RS scenario where the four-dimensional space is embedded into a five-dimensional space as a three-brane positioned at the points x5 = 0 and x5 = ␲rc, where rc is the compactification radius. The action for the Higgs sector in the RS model9 was obtained to be

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063504-14

A. H. Chamseddine and A. Connes

冕 =

冕

J. Math. Phys. 47, 063504 共2006兲

d4x冑g共g␮␯D␮H*D␯H − ␭共兩H兩2 − v20兲兲

¯ ␮␯D␮H⬘*D␯H⬘ − ␭共兩H⬘兩2 − e−2krc␲v20兲兲, d4x冑¯g共g

where g␮␯ = e−2krc␲¯g␮␯ , H⬘ = e2krc␾H, in the visible sector located at x5 = ␲rc. The physical mass scales are set by the symmetry breaking scale v = e−krc␲v0 so that m = m0e−krc␲. The bare symmetry breaking scale v0 is taken to be of the order of the Planck scale at 1019 GeV and the scaling factor ekrc␲ tuned to be of the order of 1015 so that the low-energy masses are of the order of TeV. The hierarchy problem is only partially solved in a technical sense because the tuning could not be maintained at the quantum level. A choice of krc = 10 can generate the large scale 1015 GeV. Comparing the Higgs sectors in the RS action with that in the spectral action we immediately see that they are identical provided we identify the expectation value of the dilaton field 具␾典 with krc␲. VI. CONCLUSIONS

The Dirac operator being a differential operator has the dimensions of mass. The spectral action in noncommutative geometry is defined as a function of a dimensionless operator which is taken to be the Dirac operator divided by some arbitrary large mass scale. The arbitrariness of the mass scale naturally suggests to make this scale dynamical by introducing a dilaton field in the Dirac operator of the noncommutative space defined by the standard model. To understand the appearance of the mass scales of the spectral action, we evaluated all interactions of the dilaton with the matter sector in the standard model. We found the remarkable result that the low-energy action, when evaluated in the Einstein frame, is scale invariant except for the Einstein-Hilbert term and the dilaton kinetic term. The resulting model is almost identical to the one proposed in the literature.2–4 The main motivation in these works is the observation that the standard model is classically almost scale invariant, with the symmetry only broken by the mass term in the Higgs potential. The symmetry is restored by the use of a dilaton field. When coupled to gravity, neither the dilaton kinetic energy nor the scalar curvature are scale invariant, leading to a Jordan-BransDicke theory of gravity. The vacuum expectation value of the Higgs field is then dependent on the dilaton and is classically undetermined. Quantum corrections break the scale invariance of the scalar potential and change the vacuum expectation value of the Higgs field. The dilaton acquires a large negative expectation value given by −m and a small mass. The hierarchy in mass scales is due to the large Yukawa coupling of the top quark. The dilaton expectation value can range between the GUT scale of 1015 GeV to the Planck scale of 2.4⫻ 1018 GeV. The hierarchy in mass scales is not possible if the dilaton kinetic energy and the gravitational action were scale invariant. It is remarkable that all the essential features of building a scale invariant standard model interactions to generate a mass hierarchy and predict the Higgs mass are naturally included in the spectral action without any fine tuning. It is worth mentioning that the scalar potential of exactly the same model considered here was shown to admit extended inflation and a metastable ground state. It also evades the problems of the original version of extended inflation. The vacuum expectation value of the dilaton field is determined by getting contributions from classical and radiative corrections to the vacuum energy density. One does not obtain naturally a vanishing cosmological constant. There are two possibilities to cure this problem. The first is to determine the low-energy value of the cosmological constant as determined by the renormalization group equations and then fine tune this value to cancel the contributions of the Coleman-Weinberg

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063504-15

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

potential. The second possibility to cure this problem is to fix the total invariant volume. This restricts general relativity to the form considered in Ref. 15, where the volume form is held fixed. There it was shown that this picture is consistent both at the classical and quantum levels.16–18 This fixes the total invariant volume and eliminates the scalar mode of the metric tensor g␮␯. This is done at the expense of introducing the dilaton mode ␾.3 In noncommutative geometry the volume is fixed by a Hochschild cycle c whose compatibility with the Dirac operator D is a basic constraint on the Hilbert space representation giving the metric.19 One applies the representation to monomials

␲共f 0, f 1, f 2, f 3, f 4兲 = f 0关D, f 1兴关D, f 2兴关D, f 3兴关D, f 4兴, and requires that when applied to the Hochschild cycle c it gives

␲共c兲 = ␥5 . The cosmological constant becomes determined by the initial conditions of the theory. To summarize, we have shown that the spectral action includes naturally a dilaton field which guarantees the scale invariance of the standard model interactions, and provides a mechanism to generate mass hierarchies. This is in addition to the advantages obtained previously in Ref. 11 which are now well known.12 There it was shown that all the correct features of the standard model are obtained without any fine tuning, such as unification with gravity, unification of the three gauge coupling constants and relating the Higgs to the gauge couplings. These results should be taken to support the idea that all the geometric information about the physical space is captured by the knowledge of the Dirac operator of an appropriate noncommutative space. ACKNOWLEDGMENT

The research of A. Chamseddine is supported in part by the National Science Foundation under Grant No. Phys-0313416. APPENDIX A

In this appendix we shall prove formula 共12兲. Given an elliptic positive invertible second order operator Q and a differential operator T we use the notation ⵜ共T兲 = 关Q , T兴 and the following identity 共for n 艌 0兲: n

Q−1T = 兺 共− 1兲kⵜk共T兲Q−k−1 + 共− 1兲n+1Q−1ⵜn+1共T兲Q−n−1 .

共A1兲

0

We apply this to Q = P共t兲 + ␭, T = ␾ P共t兲 + P共t兲␾. The operator ⵜk共T兲 is independent of ␭ and is a differential operator of order 艋2 + k since ⵜk共␾兲 is at most of order k. Thus the operator ⵜk共T兲Q−k−1 is pseudodifferential of order at most 2 + k − 2共k + 1兲 = −k and the remainder in 共A1兲 is of order at most −n − 1. This shows that when working modulo operators of order less than −n we have n

共P共t兲 + ␭兲 T ⬃ T共P共t兲 + ␭兲 + 兺 共− 1兲kⵜk共T兲共P共t兲 + ␭兲−k−1 −1

−1

1

so that n

共P共t兲 + ␭兲−1T共P共t兲 + ␭兲−1 ⬃ T共P共t兲 + ␭兲−2 + 兺 共− 1兲kⵜk共T兲共P共t兲 + ␭兲−k−2 . 1

But one has

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063504-16

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine and A. Connes n

兺1 共− 1兲kⵜk共T兲共P共t兲 + ␭兲−k−2 = 关P共t兲,A共t,␭兲兴, where n

A共t,␭兲 = 兺 共− 1兲kⵜk−1共T兲共P共t兲 + ␭兲−k−2 . 1

Thus integrating from ␭ = 0 to ⬁ and using 共11兲 we get that, modulo operators of order less than −n, d Y共t兲 ⬃ − dt

冕

⬁

共␾ P共t兲 + P共t兲␾兲共P共t兲 + ␭兲−2d␭ + 关P共t兲,A共t兲兴

0

where

A共t兲 = −

冕

⬁

A共t,␭兲d␭.

0

This thus gives d Y共t兲 ⬃ − 共␾ P共t兲 + P共t兲␾兲P共t兲−1 + 关P共t兲,A共t兲兴 = − 2␾ + 关P共t兲,C共t兲兴, dt

C共t兲 = A共t兲 − ␾ P共t兲−1 .

It is important to note that all of the previous manipulations hold in the general context of spectral triples with simple dimension spectrum. Moreover one can prove fairly strong properties of the spectral action in this general context.

APPENDIX B

In this appendix we shall prove the identity an共x, P兲 = an共x, P1兲 = an共x, P2兲, where P = e−␾D2e−␾ and P1 = D2e−2␾, P2 = e−2␾D2. It can then be used to simplify some of the computations of Sec. III. One simply writes P = e −␾ P 1e ␾ so that Trace 共P−s兲 = Trace 共P−s 1 兲. From this the identity an共x , P兲 = an共x , P1兲 immediately follows. One can also do a direct check as follows, one first writes P = − 共G␮␯I⳵␮⳵␯ + A␮⳵␮ + B兲, P1 = − 共G␮␯I ⳵␮⳵␯ + A1␮⳵␮ + B1兲, where

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063504-17

J. Math. Phys. 47, 063504 共2006兲

Scale invariance

G␮␯ = e−2␾g␮␯ , B = e−2␾B + G␮␯共⳵␮␾⳵␯␾ − ⳵␮⳵␯␾兲 − e−2␾A␮⳵␮␾ , A␮ = e−2␾A␮ − 2G␮␯⳵␯␾ , A1␮ = e−2␾A␮ − 4G␮␯⳵␯␾ , B1 = e−2␾B + 2G␮␯共2⳵␮␾⳵␯␾ − ⳵␮⳵␯␾兲 − 2e−2␾A␮⳵␮␾ . These relations imply A1␮ = A␮ − 2G␮␯⳵␯␾ , B 1 = B + G ␮␯共 ⳵ ␮␾ ⳵ ␯␾ − ⳵ ␮⳵ ␯␾ 兲 − A ␮⳵ ␮␾ . We also have

␻⬘1␮ = 21 G␮␯共A1␯ + ⌫␯共G兲兲 = ␻ ⬘␮ − ⳵ ␮␾ , so that E1 = B1 − G␮␯共⳵␮␻⬘1␯ + ␻⬘1␮␻⬘1␯ − ⌫␮␳ ␯共G兲␻⬘1␳兲=E. Similarly ⍀ 1␮␯ = ⍀ ␮␯ and the equality of the Seely-de Witt coefficients follow from the fact that these depend only on E, ⍀␮␯ and the curvature tensors are functions of the same metric G␮␯. APPENDIX C

In this appendix we shall show 共25兲 and the appearance of the kinetic term in the general framework of spectral triples using the following manipulations. One has

冕

冕

− 共e−␾/2De−␾/2兲−2 = − e␾D−1e␾D−1 .

共C1兲

D−1e␾ = e␾D−1 − D−1关D,e␾兴D−1 ,

共C2兲

Also

which allows to write 共C1兲 as

冕

冕

冕

− 共e−␾/2De−␾/2兲−2 = − e2␾D−2 − − e␾D−1关D,e␾兴D−2 ,

共C3兲

and using 共C2兲 again,

The first of the two terms vanishes since the residue is a trace. The second is given by

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063504-18

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine and A. Connes

冕

冕

− − D−1关D,e␾兴D−1关D,e␾兴D−2 = − 关D,e␾兴2D−4 + R,

共C4兲

where R is the remainder

冕

R = − − D−2关D2,e␾兴D−1关D,e␾兴D−2 . Let us show that R=−

冕

1 − 关D2,e␾兴2D−6 . 2

共C5兲

To see this write

冕

R = − − D−2关D2,e␾兴DD−2关D,e␾兴D−2 ,

共C6兲

and note that the commutator of D with 关D2 , e␾兴 is equal to 关D2 , 关D , e␾兴兴 and has order 1 so that

Thus moving D to the left and using the trace property of the residue one gets R=−

冕

1 − 关D2,e␾兴D−2共D关D,e␾兴 + 关D,e␾兴D兲D−4 , 2

and one obtains 共C5兲. Summarizing, we have shown the equality

Thus to obtain 共C1兲 one just needs to prove the equality

冕

冕

− 关D,a兴2D−4 = − 关D2,a兴2D−6 ,

共C7兲

and apply it to a = e␾. One can check 共C7兲 directly in the Riemannian case by computing the residue as the integral of the principal symbols on the unit sphere bundle. The factor 22 from the Poisson brackets 关D2 , a兴 is compensated by the integral of ␰␮2 on the sphere which gives 41 . In the general framework of spectral triples one gets 共C7兲 from the general hypothesis

Note that the commutator 关D , e␾兴 is skew adjoint and in particular 关D , e␾兴* = −关D , e␾兴. Thus we get the correct sign in

冕

冕

− e2␾D−2 = − 共e−␾/2De−␾/2兲−2 +

冕

1 − 关D,e␾兴关D,e␾兴*D−4 . 2

共C8兲

S. Coleman, Dilatations in Aspects of Symmetry 共Cambridge University Press, Cambridge, MA, 1985兲, p. 67. W. Buchmüller and N. Dragon, Phys. Lett. B 195, 417 共1987兲. 3 W. Buchmüller and C. Busch, Nucl. Phys. B, Proc. Suppl. 18, 295 共1990兲. 4 R. Holman, E. Kolb, S. Vadas, and Y. Wang, Phys. Rev. D 43, 3833 共1991兲. 5 A. Connes and J. Lott, Nucl. Phys. B 349, 71 共1991兲. 1 2

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Scale invariance

J. Math. Phys. 47, 063504 共2006兲

A. H. Chamseddine, G. Felder, and J. Fröhlich, Commun. Math. Phys. 155, 109 共1993兲. A. H. Chamseddine, J. Fröhlich, and O. Grandjean, J. Math. Phys. 36, 6255 共1995兲. 8 A. Chamseddine and J. Fröhlich, Phys. Lett. B 314, 308 共1993兲. 9 L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 共1999兲. 10 F. Lizzi, G. Mangano, and G. Miele, Mod. Phys. Lett. A 16, 1 共2001兲. 11 A. H. Chamseddine and A. Connes, Phys. Rev. Lett. 77, 4868 共1996兲; Commun. Math. Phys. 186, 731 共1997兲. 12 D. Kastler, Lect. Notes Phys. 543, 131 共2000兲. 13 P. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem 共Publish or Perish, Wilmington, DE, 1984兲. 14 S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 共1973兲. 15 W. Buchmüller and N. Dragon, Phys. Lett. B 207, 292 共1988兲. 16 Y. Ng and H. van Dam, Phys. Rev. Lett. 65, 1972 共1990兲. 17 W. Buchmüller and N. Dragon, Phys. Lett. B 223, 313 共1987兲. 18 Y. Ng, Int. J. Mod. Phys. D 1, 145 共1992兲. 19 A. Connes, J. Math. Phys. 41, 3832 共2000兲. 6 7

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