Two loop finiteness of Higgs mass and potential in the gauge-Higgs unification
aa r X i v : . [ h e p - ph ] S e p September 18, 2007 OU-HET 585/2007KOBE-TH-07-07TU-797SISSA 63/2007/EP
Two loop finiteness of Higgs mass and potentialin the gauge-Higgs unification
Y. Hosotani ∗ , N. Maru † , K. Takenaga ‡ and Toshifumi Yamashita § Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Department of Physics, Kobe University, Kobe 657-8501, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan Scuola Internazionale Superiore di Studi Avanzati, via Beirut 2-4, I-34014 Trieste, Italy
Abstract
The zero mode of an extra-dimensional component of gauge potentials serves asa 4D Higgs field in the gauge-Higgs unification. We examine QED on M × S anddetermine the mass and potential of a 4D Higgs field (the A component) at the twoloop level with gauge invariant reguralization. It is seen that the mass is free fromdivergences and independent of the renormalization scheme. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Introduction
In the standard model of electroweak interactions the Higgs boson is vital to induce theelectroweak symmetry breaking. It is one of the major goals in particle physics to discoverthe Higgs boson in the coming years. Its mass squared m H , in general, acquires O (Λ )radiative corrections where Λ is a cutoff scale which is as large as 10 GeV in grand unifiedtheories. In order to have m H = O (100) GeV, unnatural fine-tuning of parameters of thetheory is demanded.The supersymmetry naturally solves this gauge hierarchy problem to push down Λ to theTeV scale. It serves as a leading candidate for a model beyond the standard model, and isunder intensive study. There are alternative scenarios to have a naturally light Higgs boson,among which is the gauge-Higgs unification.[1]-[46] A 4D Higgs field is identified with apart of the extra-dimensional component of gauge potentials. When the extra-dimensionalspace is not simply connected, there appears a Wilson line phase θ H , an analogue of theAharonov-Bohm phase in quantum mechanics. The 4D Higgs field is nothing but a fielddescribing four-dimensional fluctuations of θ H . At the tree level the Higgs field appearsmassless, reflecting the nature of the Aharonov-Bohm phase. At the quantum level theeffective potential for the Wilson line phase, V eff ( θ H ), is generated radiatively. It has beenshown long ago that V eff ( θ H ) on M × S and the Higgs mass m H are finite at the one looplevel.[3, 5, 6]This has significant relevance in the context of the gauge-Higgs unification. Althoughhigher dimensional gauge theory is not renormalizable, the 4D Higgs mass can be predictedto be a finite value without afficting from the problem of of divergences. As the Higgs fieldis associated with the nonlocal Wilson line phase, it is commonly said that the Higgs massremains finite to all order as no gauge-invariant local counter term can be written. It isnot quite clear, however, whether this argument applies to non-renormalizable theories likethe one under consideration. It is desirable to have explicit evaluation and confirm thefiniteness of m H beyond one loop.There have been significant advances in the gauge-Higgs unification in the electroweaktheory in the last couple of years. In the early stage unification on orbifolds M × ( S /Z )and M × ( T /Z ) was pursued with chiral fermions.[10]-[24] It has been recognized thatunification on warped spacetime such as the Randall-Sundrum spacetime works much bet-ter for having phenomenologically viable models.[25]-[40] In all of such aspects as the Higgs2ass, the Kaluza-Klein mass scale, the gauge self-couplings, and the Weinberg angle, theunification in the Randall-Sundrum spacetime gives natural consistent results. Suppressionof the Higg-gauge couplings and Yukawa couplings has been predicted, which can be testedat LHC.[32, 33] Furthermore it has been shown recently that the gauge-Higgs unificationin the Randall-Sundrum spacetime is dual to the theory of holographic pseudo-Goldstoneboson.[34]-[38]In view of these developments it is appropriate and necessary to strengthen and confirmthe statement that the Higgs mass remains finite beyond one loop.[47]-[50] Its finitenesshas been investigated in the lattice simulation on orbifolds as well.[51] The calculabilityof the S and T parameters in the electroweak theory has been discussed at the one looplevel.[52]Evaluation of the Higgs mass at the two loop level is formidable in non-Abelian gaugetheory. To get insight in the problem it is instructive to examine, as the first step, QEDon M × S in which the zero mode of the extra-dimensional component A mimics the4D Higgs boson. It is called as a Higgs boson in the present paper.To evaluate the Higgs mass m H at the two loop level, renormalization at the one looplevel must be taken into account in due course. Two loop evaluation of the Higgs mass inQED on M × S has been previously attempted by Maru and Yamashita[49], where thevacuum polarization tensors Π MN are evaluated near θ H = 0 for the zero-modes, withoutpaying serious attention to the reguralization. In this article we evaluate both the effectivepotential V eff ( θ H ) and the vacuum polarization tensors in renormalized perturbation theoryin the dimensional regularization, maitaining the gauge invariance and the Ward-Takahashiidentities. The computation is carried out with an arbitrary value of θ H as a background.The effective potential V eff ( θ H ) is found to be minimized at θ H = π , and therefore thevacuum polarization tensors Π MN at θ H = π become relevant for determining m H .The paper is organized as follows. In the next section renormalized perturbation theoryfor QED in M × S is developed and renormalization conditions are given. In Section3 the effective potential V eff ( θ H ) is evaluated at the two loop level. Relevant integral-sums are evaluated in Appedix A. In Section 4 the vacuum polarization tensors Π MN aredetermined at the one loop level. Details of the computaion are given in Appendix B. Withthese results the Higgs mass is determined at the two loop level in Section 5. It is seenthat the Higgs mass thus evaluated is independent of the renormalization scheme. Section6 is devoted to a brief summary and discussions.3 QED in M × S S with a radius R . For the sake of simplicity we introduce only onefermion ψ with a mass m . Both the gauge potential A M (the photon field) and ψ aretaken to be periodic. Renormalization, at least at the two loop level, is done with thestandard renormalization procedure. Renormalized fields are defined by A (0) M = Z / A M and ψ (0) = Z / ψ . The renormalized coupling constant is defined by e (0) = Z Z − Z − / e .The renormalized mass of ψ is given by m (0) = m + δm . Here quantities with superscript(0) denote bare quantities. Renormalization conditions for Z , Z , Z and δm are specifiedbelow.We develop renormalized perturbation theory around the non-vanishing Wilson linephase θ H . The Lagrangian density is given by L = − F MN F MN −
12 ( ∂ M A M ) + ψ − ( iγ M D cM − m ) ψ − eA M ψ − γ M ψ − δ F MN F MN + ψ − ( δ iγ M ∂ M − δ m ) ψ − eδ A M ψ − γ M ψ . (2.1)Here D cM = ∂ M − δ M, ieA c where eA c = θ H / πR and the metric is η MN =diag (1 , − , − , − , −
1) Counter terms are defined as δ k = Z k − δ m = Z m (0) − m .We renormalize such that the Ward-Takahashi identity Z = Z is preserved so that θ H = e Z πR dy A = e (0) Z πR dy A (0)5 . (2.2)In other words the Wilson line phase is not renormalized.On M × S the fifth component of a momentum p is discretized. With θ H ≡ πa = 0, p = ( n − a ) /R for ψ and p = n/R for A M where n is an integer. We denote a five-momentum by ˆ p M = ( p µ , p ) ( µ = 0 ∼ p ; a, R ), that for the photon propagator byΠ MN (ˆ p ; a, R ), and the amputated fermion vertex function by Γ M (ˆ p, ˆ p ′ ; a, R ). In the R →∞ limit these functions take 5D Lorentz covariant form, and are denoted with a superscript(0); Σ(ˆ p ; a, R ) = Σ (0) (ˆ p/ ) + Σ (1) (ˆ p ; a, R ) , Π MN (ˆ p ; a, R ) = Π (0) MN (ˆ p ) + Π (1) MN (ˆ p ; a, R ) , M (ˆ p, ˆ p ′ ; a, R ) = Γ (0) M (ˆ p, ˆ p ′ ) + Γ (1) M (ˆ p, ˆ p ′ ; a, R ) , (2.3)where ˆ p/ = ˆ p M γ M and Σ (0) (ˆ p/ ) = lim R →∞ Σ(ˆ p ; a, R ) etc.. The vacuum polarization ten-sors Π MN (ˆ p ; a, R ) can be expressed in terms of two invariant functions Π(ˆ p ; a, R ) and F (ˆ p ; a, R ). The current conservation ˆ p M Π MN (ˆ p ; a, R ) = 0 implies thatΠ µν = ( η µν ˆ p − p µ p ν ) Π − p µ p ν p · ( p ) F , Π = − p ( Π + F ) , Π µ = Π µ = − p p µ ( Π + F ) , (2.4)where p = n/R ( n : an integer). We remark that F is finite and lim R →∞ F = 0. Thedivergent contributions appear only for Π (0) = lim R →∞ Π, which can be cancelled by thecounter term δ in (2.1). Also lim R →∞ Π MN = ( η MN ˆ p − ˆ p M ˆ p N ) Π (0) (ˆ p ).The full photon propagators D MN (ˆ p ) are given by iD µν = 1ˆ p (1 − Π) (cid:16) η µν − p µ p ν p (cid:17) + p µ p ν (ˆ p ) (cid:26) − ( p ) p (1 − Π − F ) (cid:27) ,iD µ = − p µ p (ˆ p ) Π + F − Π − F ,iD = − p (ˆ p ) (1 − Π − F ) + ( p ) (ˆ p ) . (2.5)In particular, for the zero modes ( p = 0) D µν | p =0 = − ip (1 − Π) (cid:16) η µν − p µ p ν p Π (cid:17) ,D µ | p =0 = 0 ,D | p =0 = ip (1 − Π − F ) , (2.6)where Π and F are evaluated at p = 0. It will be found that Π and F are expanded in p as Π | p =0 = c + · · · and F | p =0 = b − /p + b + · · · , respectively. Consequently theHiggs mass, or the mass of the zero mode of A , defined in the p expansion in the inversepropagator is given by m H = b − − c − b . (2.7)It slightly differs from the exact pole mass, but is convenient to relate to the effectivepotential V eff ( θ H ). The difference between the two is small in the weak coupling e / π ≪ e is the four-dimensional gauge coupling e = e / πR .5here are two typical ways to impose renormalization conditions. The most convenientway is to impose them in the R → ∞ limit, namely in M . In terms of Σ (0) , Π (0) andΓ (0) in (2.3) and (2.4) and the five-dimensional fermion mass m in M , renormalizationconstants Z j and δm are fixed byΣ (0) (ˆ p/ = m ) = 0 ,d Σ (0) d ˆ p/ (ˆ p/ = m ) = 0 , Π (0) (ˆ p = 0) = 0 , Γ (0) M (ˆ p = ˆ p ′ ) = γ M . (2.8)One can also adopt the mass-independent renormalization where m is set to be zero in theabove equations (2.8). An alternative prescription is to impose the conditions on shell in M × S for the 4D fermion with the lowest 4D mass ( ≡ m D phys ). As is seen below, V eff ( θ H )has a global minimum at θ H = π ( a = 1 / ≡ a ). Hence the alternative renormalizationconditions read S (ˆ p ) (cid:12)(cid:12)(cid:12) p = a R − = i ˆ p/ − m − Σ (cid:12)(cid:12)(cid:12)(cid:12) p = a R − ∼ ip/ + a R − γ − m near p = ( m D phys ) = m + a R , Π( p = 0 , p = 0) = 0 , Γ µ ( p = p ′ , p = p ′ = a R ) = γ µ . (2.9)Renormalization with other reference values of a is also possible. In all of these prescrip-tions the Ward-Takahashi identity Z = Z is preserved. The effective potential V eff for θ H = 2 πa is evaluated at the two loop level. We adopt thedimensional regularization method to maintain the gauge invariance. The evaluation isperformed in M d × S , and the d → V eff ( a ) in d + 1 dimensions at the one loop level is given by V eff ( a ) = − f ( d )2 Z d d p E (2 π ) d πR ∞ X n = −∞ ln (cid:16) p E + ( n − a ) R + m (cid:17) (3.1)6 + Figure 1: Diagrams contributing to the effective potential V eff ( θ H ). The second and thirddiagrams contain one loop counter term δ ˆ p/ − δ m and δ (ˆ p M ˆ p N − η MN ˆ p ), respectively.after Wick rotation. f ( d ) = 2 [( d +1) / . In the m → V eff ( a ) = f ( d )Γ (cid:16) d + 12 (cid:17) (2 π / R ) d +1 f d +1 ( a ) + constant . (3.2)where f k ( a ) is defined in (A.3). The 4D effective potential is V ( a ) = V eff ( a ) (cid:12)(cid:12) d =4 × πR so that V ( a ) = 316 π R f ( a ) + constant , (3.3)where the constant is divergent, but is independent of a .The effective potential is minimized at a = 1 / θ H = π . The effective potentialfor the zero mode of A , or the Higgs field φ H , is given by V ( a ) where a is replaced by e Rφ H . Here the four-dimensional coupling is given by e = e/ √ πR . Hence the Higgsmass m H at the one loop level is given, for m = 0, by m H (cid:12)(cid:12) = e R d da V ( a ) (cid:12)(cid:12)(cid:12) a = 12 = 9 e ζ R (3)16 π R (3.4)where ζ R ( z ) is the Riemann’s zeta function.At the two loop level the diagrams in fig. 1 contribute to V eff ( a ). The contribution fromthe first diagram (a) is given by − iV eff ( a ) = ( −
1) ( − ie ) Z d d p (2 π ) d πR X l Z d d q (2 π ) d πR X n × − iη MN (ˆ p/ − ˆ q/ ) + iǫ Tr i ˆ p/ − m + iǫ γ M i ˆ q/ − m + iǫ γ N , ˆ p M = (cid:16) p µ , l − aR (cid:17) , ˆ q M = (cid:16) q µ , n − aR (cid:17) , (3.5)so that V eff ( a ) = − e Z d d p E (2 π ) d d d q E (2 π ) d πR ) X l X n f ( d ) (cid:8) ( d − p E ˆ q E + ( d + 1) m (cid:9) (ˆ p E + m ) (ˆ q E + m ) (ˆ p E − ˆ q E ) , − e f ( d )2 Z d d p E (2 π ) d d d q E (2 π ) d πR ) X l X n (cid:20) m (ˆ p E + m ) (ˆ q E + m ) (ˆ p E − ˆ q E ) + d − (cid:26) p E + m )(ˆ p E − ˆ q E ) − p E + m ) (ˆ q E + m ) (cid:27)(cid:21) . (3.6)At this stage infinite sums over discrete momentum p E have to be evaluated. Wesummarize typical integral-sums in Appendix A. In terms of G j ( a ; m, d ) defined there, V eff ( a ) can be expressed as V eff ( a ) = − e f ( d )2 n m G ( a ; m, d )+ d − (cid:2) G ( a ; m, d ) G (0; 0 , d ) − G ( a ; m, d ) (cid:3)o . (3.7)In the mR → R kept fixed) it simplifies to V eff ( a ) = e ( d − f ( d )4 (cid:8) G ( a ; 0 , d ) − G (0; 0 , d ) (cid:9) = e ( d − f ( d )Γ (cid:16) d − (cid:17) (4 π ) d +1 ( πR ) d − (cid:8) f d − ( a ) − f d − (0) (cid:9) (3.8)up to an a -independent constant where f k ( a ) is defined in (A.3). Its contribution to the4D effective potential is given by V ( a ) = 3 e π (2 πR ) (cid:8) f ( a ) − f (0) (cid:9) . (3.9)Contributions from one-loop counter terms, namely from the second and third diagramsin fig. 1, either vanish for m = 0 or are a -independent.The effective potential at the two-loop level is given by (3.3) and (3.9). The globalminimum is located at a = 1 /
2. The two-loop contribution to the effective potential issuppressed by an order of the fine structure constant, as seen in Eq. (3.9). Hence, eventhough the two-loop contribution itself is minimized at a = 0, the effective potential isgoverned by the one-loop contribution (3.3) as long as the coupling e / π is small. Whenone needs a very small value of a for having a realistic model of the gauge-Higgs unification,the two-loop contribution may play an important role and affect the location of the globalminimum of the effective potential as discussed in ref. [19].8he second derivative of V ( a ) with respect to a at the global minimum of V ( a ) isrelated to the coefficient b − = − Π | p =0 ,p =0 introduced in Section 2, as easily confirmedby examining Feynman diagrams. Indeed, b − = e R d da V ( a ) (cid:12)(cid:12)(cid:12) a = 12 . (3.10)Hence to the two loop order we have b − = 9 e ζ R (3)16 π R − e ln 2 ζ R (3)128 π R . (3.11)We would like to note that e R ( d V , /da ) vanishes at a = 0, which is in conformitywith the result in ref. [49]. The vacuum polarization tensors Π MN in QED on M × S has been evaluated to thetwo loop order near a = 0 in Ref. [49]. We need to determine Π MN at a = 1 / b and c , or Π MN to the one loop order as b − has been already evaluated in (3.11). In this section Π MN (ˆ p/ ; a, R ) is evaluated at the oneloop level for an arbitrary value a . We note that Π MN in supersymmetric gauge theory onan orbifold M × ( T /Z ) with a vanishing Wilson line phase has been evaluated at theone loop level.[53]The evaluation is straightforward. The contribution from a fermion loop isΠ MN (ˆ p ) = ie Z d d q (2 π ) d πR X l Tr γ M q/ − m + iǫ γ N q/ + ˆ p/ − m + iǫ , ˆ p M = (cid:16) p µ , nR (cid:17) , ˆ q M = (cid:16) q µ , l − aR (cid:17) . (4.1)In the dimensional regularization scheme the gauge invariance is maintained so that Π MN satisfies the current conservation; ˆ p M Π MN (ˆ p ) = 0 . (4.2)The detailed evaluation of Π MN (ˆ p ) is given in Appendix B, which is summarized in(B.5) and (B.6). To find the coefficients b and c , we need Π and F at p = n/R = 0.9rom (B.6) and (B.7) it follows that Π F ! p =0 = Π (0) ( p )0 ! − e f ( d )(4 π ) ( d +1) / Z dx Z ∞ dt t (1 − d ) / e − t { m − x (1 − x ) p } × X ℓ =0 e − π R ℓ /t e πiℓa x (1 − x )1 p π ℓ R t . (4.3)The ℓ = 0 terms give finite contributions. At d = 4 and m = 0Π | p =0 = 3 e R
128 ( − p ) / − e f ( a )6 π + · · · ,F | p =0 = − e f ( a )4 π R p − e f ( a )12 π + · · · . (4.4)Note that the counter term δ = Π (0) (ˆ p = 0) vanishes at m = 0. We expand the invariantfunctions around p = 0 at the global minimum of the V eff ( a ), namely at a = 1 / | p =0 , a = 12 = c + · · · ,F | p =0 , a = 12 = b − p + b + · · · . (4.5)The coefficients at the one loop level are given by c = e ln 26 π ,b = e ln 212 π ,b − = 9 e ζ R (3)16 π R . (4.6)The coefficient b − coincides with the result from the effective potential (3.4) or (3.11) asit should.The gauge invariant mass for the zero mode of A appears as a pole 1 /p in F . Weremark that there is similarity to the gauge invariant mass in the Schwinger model (QEDin two dimensions) in which Π develops a pole from a fermion loop.[54] It differs in thepoint that F vanishes in the R → ∞ limit, or in M , whereas the pole remains in theSchwinger model in M . The effective potential for θ H in the Schwinger model on a circlehas the same structure as in the current model.[55] Its curvature at the minimum gives amass for photons. 10 The Higgs mass
The 4D effective action for the Higgs field φ H takes the formΓ eff [ φ H ] = Z d x n − V [ φ H ] + Z [ φ H ] ∂ µ φ H ∂ µ φ H + · · · o . (5.1)The Higgs mass in this approach is m H = 1 Z [ φ H ] ∂ V [ φ H ] ∂φ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ min H (5.2)where φ min H = (2 e R ) − is the location of the global minimum of V [ φ H ]. The effectivepotential V [ φ H ] is given by V ( a ) with a = e Rφ H . Its second derivative is related to b − by (3.10). Similarly Z [ φ min H ] is related to c and b by Z [ φ min H ] = 1 − c − b . Hence theHiggs mass defined by (5.2) coincides with the mass defined by (2.7). Inserting (3.11) and c , b in (4.6) there, one finds that in the massless fermion limit m = 0 m H = 9 e ζ R (3)16 π R (cid:26) − e ln 224 π (cid:27) . (5.3)The coupling constant e and the coefficients b − , b , c depend on the renormalizationscheme. However, the Higgs mass is a physical quantity so that it should not depend onthe renormalization scheme employed. This can be confirmed from the results at the twoloop level obtained above.Let e ′ , b ′− , b ′ , c ′ be the coupling constant and the coefficients in a second renormal-ization scheme. To be concrete, ( e, b − , b , c ) are defined in the renormalization in M as employed in the preceding sections, whereas ( e ′ , b ′− , b ′ , c ′ ) are defined in the on-shellrenormalization in M × S at a = 1 /
2. For the sake of simplicity we suppose that m = 0.At the one loop level we write Π(ˆ p ) = Π r (ˆ p ) − δ where Π r (ˆ p ) is the contribution from afermion loop. The counter terms are given by δ = lim R →∞ Π r (ˆ p ; a, R ) (cid:12)(cid:12) ˆ p =0 ,δ ′ = Π r ( p = 0 , p = 0; a = , R ) . (5.4)The difference between the two, δ − δ ′ , is finite. The coefficient c is defined by theexpansion of Π( p , p = 0) = c + · · · in p . Hence c ′ − c = δ − δ ′ . The Ward-Takahashiidentity Z = Z implies e (0) = Z − / e . It follows that e ′ = Z ′ Z e ≃ e − c + c ′ . (5.5)11e observed that F is finite at the one loop level in Section 4 and V eff ( a ) itselfis finite in the mR → b − = e b − + e b − + · · · and b = e b + · · · . Then the finiteness implies that b − = b ′− , b − = b ′− , and b = b ′ . From these identities one finds that( m H ) ′ = b ′− − c ′ − b ′ = e ′ · b ′− + e ′ b ′− − c ′ − e ′ b ′ ≃ e − c + c ′ · b − + e b − − c ′ − e b ≃ b − − c − b = m H . (5.6)The Higgs mass is independent of the renormalization scheme to this order as it should be. In this paper we have determined the Higgs mass m H at the two loop level, or to O ( e ), inthe QED gauge-Higgs unification model on M × S . The mass is shown to be independentof the renormalization scheme. The evaluation of the vacuum polarization tensors, orequivalently Z [ φ H ] in the effective action, at the one loop level is also required to find m H at the two loop level. The θ H -dependent part of the effective potential is found finite atthe two loop level. Divergences in Π MN (ˆ p ), which appear only in the Π part, but not inthe F part, are absorbed by the counter term δ . There is no need to introduce additionalcounter terms other than δ , δ , δ and δ m at this level.The fact that radiative corrections to the Higgs mass are finite and suppressed by apower of the four-dimensional gauge coupling constant with respect to the Kaluza-Kleinmass scale m KK = 1 /R has an important implication in the gauge-Higgs unification.The Higgs mass is not an input parameter of the theory, but is definitively predictedin terms of other fundamental constants such as the gauge coupling and the size of theextra dimension. Its value is stable against higher order corrections. In other words thegauge-Higgs unification yields a naturally light Higgs boson in four dimensions. We remarkthat in the gauge-Higgs unification in flat space, however, the Higgs mass becomes smallcompared with the W and Z boson masses unless the Wilson line phase θ H is sufficientlysmall.[18, 19, 20, 23] This problem can be naturally resolved in the gauge-Higgs unification12n the warped space.[28] Further the Higgs interactions with other fields and particles canbe predicted as well.[29, 32, 33, 38, 39]Although we considered, for the sake of simplicity, the massless fermion limit ( mR → m H in the present paper, the same features are expected to hold in the m = 0 case.Contributions of non-vanishing δ , δ and δ m have to be taken into account. In passing,we would like to point out that the massless fermion limit is well defined on M × S andon M × S . As the effective potential is minimized at a = 1 /
2, the fermion propagatordoes not vanish at ˆ p M = 0 with a = 1 / M × S is straightforward. To define a theoryand determine a Higgs mass at the two loop level, one must include additional counterterms such as ( ∂ L F MN ) in the original Lagrangian. Other than this the analysis remainsintact with the substitution d = 5.More important is the extension to the non-Abelian case and to higher order correctionsin the viewpoint of the gauge-Higgs unification. Not only propagators of gauge fields have θ H dependence, but also there appear a new interaction vertex proportional to θ H . It iscurious to see how the large gauge invariance ( θ H → θ H +2 π ) is maintained in perturbationtheory. Acknowledgments
This work was supported in part by Scientific Grants from the Ministry of Educationand Science, Grant No. 17540257(Y.H.), Grant No. 18204024(Y.H. and N.M.), and GrantNo. 19034007(Y.H.). K.T. is supported by the 21st Century COE Program at TohokuUniversity. One of the authors (Y.H.) would like to thank the CERN Theory Institute forits hospitality where a part of this work was done.
A Integrals and sums
We summarize useful formulas for the evaluation in Section 3. The first integral-sum is G ( a ; m, d ) = Z d d p E (2 π ) d πR ∞ X ℓ = −∞ p E + m ( p E = ℓ − aR )= Z ∞ dt e − tm (4 πt ) d/ πR ∞ X ℓ = −∞ e − t ( ℓ − a ) /R Z ∞ dt e − tm (4 πt ) ( d +1) / ∞ X ℓ = −∞ e − π R ℓ /t e − πiℓa = Γ (cid:16) − d (cid:17) m d − (4 π ) ( d +1) / + m d − (2 π ) d ∞ X ℓ =1 πℓa ) K ( d − / (2 πℓmR )( ℓmR ) ( d − / . (A.1)In the third equality the Poisson resummation formula has been employed. In the lastexpression K ν ( z ) is the modified Bessel function. For small mR one finds G ( a ; m, d ) = Γ (cid:16) − d (cid:17) m d − (4 π ) ( d +1) / + 2Γ (cid:16) d − (cid:17) f d − ( a )(4 π ) ( d +1) / ( πR ) d − − (cid:16) d − (cid:17) f d − ( a ) m (4 π ) ( d +1) / ( πR ) d − + · · · (A.2)where f k ( a ) = ∞ X ℓ =1 cos 2 πℓaℓ k . (A.3)The second integral-sum is G ( a ; m, d ) = Z d d p E (2 π ) d d d q E (2 π ) d πR ) ∞ X ℓ = −∞ ∞ X n = −∞ p E + m ) (ˆ q E + m ) (ˆ p E − ˆ q E ) (cid:16) p E = ℓ − aR , q E = n − aR (cid:17) . (A.4)Introducing Feynman parameters, exponentiating the denominator, integrating over p E and q E , and making repeated use of the Poisson resummation formula, one finds G ( a ; m, d ) = 12 Z Ω dxdy Z ∞ dt t e − t ( x + y ) m (4 πt ) d +1 h ( x, y ) ( d +1) / × ∞ X ℓ = −∞ ∞ X n = −∞ e − πi ( ℓ + n ) a exp (cid:26) − π R S ℓn ( x, y ) t h ( x, y ) (cid:27) , Ω = { ( x, y ); 0 ≤ x, y, x + y ≤ } ,h ( x, y ) = (1 − x )(1 − y ) − (1 − x − y ) ,S ℓn ( x, y ) = (1 − x ) ℓ + (1 − y ) n + 2(1 − x − y ) ℓn . (A.5)The ( ℓ, n ) = (0 ,
0) term gives contributions in M , which is independent of a . For small mR one finds G ( a ; m, d ) = Γ(2 − d ) m d − π ) d +1 Z Ω dxdy ( x + y ) d − h ( x, y ) ( d +1) /
14 Γ( d − π ) d +1 ( πR ) d − X ( ℓ,n ) =(0 , e − πi ( ℓ + n ) a Z Ω dxdy h ( x, y ) ( d − / S ℓn ( x, y ) d − − Γ( d − m π ) d +1 ( πR ) d − X ( ℓ,n ) =(0 , e − πi ( ℓ + n ) a Z Ω dxdy h ( x, y ) ( d − / S ℓn ( x, y ) d − + · · · . (A.6) B Evaluation of Π M N
Evaluation of the vacuum polarization tensors proceeds as follows. Performing the tracein (4.1) and introducing a Feyman parameter, one obtainsΠ MN (ˆ p ) = ie f ( d ) Z dx Z d d q (2 π ) d πR X l S MN [ˆ q − m + x (ˆ p + 2ˆ q ˆ p ) + iǫ ] ,S MN = 2ˆ q M ˆ q N + ˆ q M ˆ p N + ˆ p M ˆ q N − ˆ q (ˆ q + ˆ p ) η MN + m η MN . (B.1)We shift the integration variable q µ → q ′ µ = q µ + xp µ , exponentiate the denominator, andintegrate over Wick-rotated q ′ E to find Π µν Π Π µ = − e f ( d )(4 π ) d/ Z dx Z ∞ dt t − ( d/ πR ∞ X ℓ = −∞ e − t [( q + xp ) + m − x (1 − x )ˆ p ] × (cid:8) ( d − t − + x (1 − x ) p + q ( q + p ) + m (cid:9) η µν − x (1 − x ) p µ p ν − t − − x (1 − x ) p + q ( q + p ) − m p µ (cid:8) (1 − x ) q − xp (cid:9) . (B.2)Recalling q = ( ℓ − a ) /R and p = n/R , one employ the Poisson resummation formula tofind Π µν Π Π µ = − e f ( d )(4 π ) ( d +1) / Z dx Z ∞ dt t (1 − d ) / e − t { m − x (1 − x )ˆ p } ∞ X ℓ = −∞ e − π R ℓ /t e πiℓ ( a − xn ) × n d − t − π R ℓ t + (1 − x ) iπnℓt + x (1 − x )ˆ p + m o η µν − x (1 − x ) p µ p ν − d − t − π R ℓ t + (1 − x ) iπnℓt − x (1 − x ) (cid:8) ( p ) + p (cid:9) − m p µ n (1 − x ) iπRℓt − x (1 − x ) p o . (B.3)15he expression is simplified with identities Z ∞ dt n − d t − m + x (1 − x )ˆ p + π R ℓ t o t (1 − d ) / e − t { m − x (1 − x )ˆ p } e − π R ℓ /t = Z ∞ dt ∂∂t n t (1 − d ) / e − t { m − x (1 − x )ˆ p } e − π R ℓ /t o = 0 , Z dx n (1 − x )ˆ p − πiℓnt o e tx (1 − x )ˆ p − πiℓnx = Z dx t ∂∂x e tx (1 − x )ˆ p − πiℓnx = 0 , (B.4)to Π µν Π Π µ = − e f ( d )(4 π ) ( d +1) / Z dx Z ∞ dt t (1 − d ) / e − t { m − x (1 − x )ˆ p } ∞ X ℓ = −∞ e − π R ℓ /t e πiℓ ( a − xn ) × x (1 − x ) (cid:0) ˆ p η MN − ˆ p M ˆ p N (cid:1) + (1 − x ) iπℓt nη µν np µ R − π R ℓ t . (B.5)Notice that only the ℓ = 0 term survives in the R → ∞ limit. Π µν and Π are even in p = n/R , whereas Π µ is odd after the integration over x .As a consequence of the current conservation Π MN can be expressed in terms of thetwo invariant functions Π and F in (2.4). Comparison of the two expressions for Π andΠ µ in (B.5) shows that in the integrand − π R ℓ /t and (ˆ p /p )(1 − x ) R ( iπℓ/t ) givethe same contribution in (B.5).The invariant functions are given by Π F ! = − e f ( d )(4 π ) ( d +1) / Z dx Z ∞ dt t (1 − d ) / e − t { m − x (1 − x )ˆ p } × ∞ X ℓ = −∞ e − π R ℓ /t e πiℓ ( a − xn ) x (1 − x ) + 1ˆ p (1 − x ) n iπℓtp (ˆ p ) π ℓ R t . (B.6)In the R → ∞ limit, namely in M d +1 , F vanishes andΠ (0) (ˆ p ) = lim R →∞ Π= − e f ( d )Γ (cid:16) − d (cid:17) (4 π ) ( d +1) / Z dx x (1 − x ) (cid:2) m − x (1 − x )ˆ p (cid:3) ( d − / . (B.7)16t d = 4 ( D = 5), Π (0) (0) is removed by the counter term δ so that the renormalized Πis Π(ˆ p ) − Π (0) (0). At d = 5 ( D = 6) an additional counter term ( ∂ L F MN ) is necessary toremove the divergence proportional to ˆ p in (B.7). References [ 1 ] D.B. Fairlie,
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