Reduction Schemes in Cutoff Regularization and Higgs Decay into Two Photons
aa r X i v : . [ h e p - ph ] J a n Preprint typeset in JHEP style - HYPER VERSION
Reduction Schemes in Cutoff Regularizationand Higgs Decay into Two Photons
Hua-Sheng Shao
Department of Physics and State Key Laboratory of Nuclear Physics andTechnology, Peking University, Beijing 100871, ChinaEmail: [email protected]
Yu-Jie Zhang
Key Laboratory of Micro-nano Measurement-Manipulation and Physics(Ministry of Education) and School of Physics, Beihang University, Beijing100191, ChinaEmail: [email protected]
Kuang-Ta Chao
Department of Physics and State Key Laboratory of Nuclear Physics andTechnology, Peking University, Beijing 100871, ChinaCenter for High Energy Physics, Peking University, Beijing 100871, ChinaEmail: [email protected]
Abstract:
We present a new systematic method to evaluate one-loop tensor inte-grals in conventional ultraviolet cutoff regularization. By deriving a new recursiverelation that describes the momentum translation variance of ultraviolet integrals,we implement this relation in the Passarino-Veltman reduction method. With thismethod, we recalculated the Higgs boson decay into two photons process at one-looplevel in the Standard Model. We reanalyze this process carefully and clarify someissues arisen recently in cutoff regularization.
Keywords:
Higgs Physics,Electromagnetic Processes and Properties, StandardModel. ontents
1. Introduction 12. A New Recursive Relation 23. Modified Passarino-Veltman Reduction Schemes 54. Higgs Decay into Two Photons 95. Summary 14A. Derivation of Expressions for J Nµ µ ...µ s
1. Introduction
Recently, the ATLAS[1] and CMS[2] collaborations have renewed efforts to searchfor the Higgs boson at the CERN LHC with data integrated up to O ( f b − ). Theexcluded mass region for the Standard Model(SM) Higgs boson has been extendedto most of the region between 145 and 466 GeV. In the low mass region of the Higgsboson, the two-photon mode of Higgs decay plays a crucial role in experimentalstudies.R. Gastmans et al. recently recalculated H → γγ via W-boson loop [3, 4],which yielded a result in contradiction with the old ones in the literature[5, 6, 7, 8].Their computation was carried out in four-momentum cutoff regularization ratherthan dimensional regularization (DREG). To reduce the number of Feynman dia-grams, Gastmans et al. chose unitary gauge. In their treatment, the new result,which satisfies the decoupling theorem[9], was favored by the authors. Later, sev-eral authors[10, 11, 12, 13] have pointed out that the old results are still correctand decoupling theorem is violated by Hφ + φ − in this case. However, we are stillunsatisfactory with the explanations about the problems with the calculations ofR.Gastmans et al. , since there have never been doubts about the correctness of theiralgebra. In order to clarify this problem, we develop a new method to do one-loopcalculation in cutoff regularization.Although DREG has proven its superiority and achieved the most widely usagein phenomenological applications, cutoff regularization, the oldest regularization, still– 1 –as some advantages compared with DREG theoretically. For instance, in DREG,one is unable to obtain the correct divergent terms higher than logarithmic diver-gences, which means that quadratic divergent terms of SM Higgs self-energy diagramsdisappeared in DREG. Pauli-Villars regularization is flawed because it violates chi-ral symmetry, while the symmetry is preserved in cutoff regularization. Moreover,from Wilsonian effective field theory viewpoint, cutoff regularization scheme is also amore intuitive and straightforward scheme. Therefore, the introduction of an explicitcutoff is sometimes advantageous.However, there are still many drawbacks in this four-momentum regularizationthat should be mentioned.Considering the truncation in momentum modes, this reg-ularization is flawed because it violates gauge invariance and translation invarianceregarding the loop momentum. The latter condition signifies that the results mayambiguously depend on the manner how the propagators are written. Hence, in thepresent paper a new recursive relation for loop momentum translation is derived first.Then the Passarino-Veltman reduction method [14, 15] is modified to reduce the ten-sor integrals in this regularization. One can follow Dyson’s prescription[16, 17] toobtain a gauge invariant result, just as shown in our calculations of H → γγ .As an example, we reconsider the process H → γγ in this four-dimensionalmomentum cutoff regularization with our proposed approach. Given that the processof H → γγ is free from infrared and mass singularities, only the ultraviolet cutoff isconsidered here. Readers who need to handle infrared or mass singularities shouldturn to the mass regularization scheme demonstrated in the literature e.g.[18].The present paper is organized as follows. In Section 2, a new recursive relationfor the loop momentum translation in cutoff regularization is demonstrated. Then,it is implemented in the Passarino-Veltman reduction schemes in Section 3. Withthis approach, calculations and analysis of H → γγ are performed in Section 4. Ourconclusion is present in Section 5. In AppendexA, the expressions for J Nµ ...µ s used inSection 2 is derived. Finally, some scalar integrals can be found in Appendix B.
2. A New Recursive Relation
In this section, we will show how to calculate I ∆ µ ...µ s ( b, a ) ≡ Z d k k µ . . . k µ s ( − ( k − b ) + a ) n − Z d k ( k + b ) µ . . . ( k + b ) µ s ( − k + a ) n . (2.1)This integration have a superficial divergence degree ∆ ≡ s + 4 − n . Note that, integration by parts (IBP) reduction methods are not valid in this case due tononvanishing surface terms. – 2 – negative ∆ evidently simplifies the calculation, because the limits of the inte-grals in Eq.(2.1) can be set to infinity and the translation shift k → k + b does notchange these limits. Therefore, I ∆ µ ...µ s completely vanishes when ∆ <
0. However, re-sults may vary when the integrals in Eq.(2.1) are ultraviolet divergent because thereis an artificial four-momentum cutoff scale Λ in these integrals. These conditions arethen considered following. I ∆ µ ...µ s can be rewritten as I ∆ µ ...µ s ( b, a ) = (cid:18)Z d k k µ . . . k µ s ( − ( k − b ) + a ) n − Z d k k µ . . . k µ s ( − k + a ) n (cid:19) − Z d k f rem ( b )( − k + a ) n , (2.2)with the remainder f rem ( b ) ≡ ( k + b ) µ . . . ( k + b ) µ s − k µ . . . k µ s . Using the identity1 A n − B n = Z d x n ( B − A )( xA + (1 − x ) B ) n +1 , (2.3)one arrived at I ∆ µ ...µ s ( b, a ) = Z d k Z d x n ( − b · k + b ) k µ . . . k µ s ( − ( k − c ) + d ) n +1 − Z d k f rem ( b )( − k + a ) n , (2.4)where c ≡ x b, d ≡ a − b x (1 − x ). The integral momentum k to k + c in the firstintegral of the previous equation is shifted, so that I ∆ µ ...µ s ( b, a ) = − n b µ Z d xI ∆ − µ µ ...µ s ( c, d ) + n b Z d xI ∆ − µ ...µ s ( c, d )+ Z d k Z d x n ( − b · k + b − b · c ) k µ . . . k µ s ( − k + d ) n +1 + Z d k Z d x n ( − b · k + b − b · c ) f rem ( c )( − k + d ) n +1 − Z d k f rem ( b )( − k + a ) n – 3 – − n b µ Z d xI ∆ − µ µ ...µ s ( c, d ) + n b Z d xI ∆ − µ ...µ s ( c, d ) − n b µ Z d k Z d x k µ k µ . . . k µ s ( − k + d ) n +1 − n Z d k Z d x b · kf rem ( c )( − k + d ) n +1 − Z d k Z d x ∂f rem ( c ) ∂x ( − k + d ) n . (2.5)In the sixth line of Eq.(2.5), a spurious part that is proportional to (1 − x ) inthe numerator of the integrand is removed. What’s more, integration by parts isperformed at the end of Eq.(2.5).Aside from the terms expressed in I ∆ − and I ∆ − , Eq.(2.5) can be simplifiedfurther using the formulae J Nµ ...µ s ( a ) ≡ R d k k µ k µ ...k µs ( − k + a ) N given in appendix A. Afterexpanding the terms proportional to x j in the integrands and implementing theexpressions for J N , a lot of terms are canceled. Thus, the final result is I ∆ µ ...µ s ( b, a ) = − n b µ Z d xI ∆ − µ µ ...µ s ( c, d ) + n b Z d xI ∆ − µ ...µ s ( c, d ) − ⌊ s +12 ⌋− n +2 X t =max(0 , − n ) , even { g s + t − ∆ b ∆ − t } µ ...µ s iπ ( − − s + t − ∆2 Γ( s + t − ∆2 + 3) h ( t, n, ∆) , (2.6)where the notation { g s + t − ∆ b ∆ − t } µ ...µ s defined in appendix A, n = s +4 − ∆2 , ⌊ y ⌋ is aGaussian function ( the greatest integer that is not larger than y ), and the functionis h ( t, n, ∆) ≡ t X l =0 C ln − l n ,n ,n ≥ X n + n + n = l ( − n + n (∆ − t )2 n + n + ∆ − t l ! n ! n ! n ! ( b ) n + n ( a ) n Λ t − l . (2.7)We should make some remarks about above equation before going forward.In therecursive relation Eq. (2.6), there are two integrals left. However, since all of the I ∆ µ ...µ s ( b, a ) are only polynomials of b and a ,i.e.,they are only polynomials of integralvariable x , the explicit expressions for this recursive relation can be easily obtainedwith the help of computers. Especially, the explicit expressions for I ∆ µ ...µ s (∆ =0 , , ,
3) are – 4 – µ ...µ s ( b, a ) = 0 ,I µ ...µ s ( b, a ) = −{ g s − b } µ ...µ s iπ ( − − s − Γ( s +52 ) ,I µ ...µ s ( b, a ) = (cid:0) n b { g s } µ ...µ s + (4 n + 2) θ ( s − { g s − b } µ ...µ s (cid:1) iπ ( − − n Γ( n + 2) , with n = s + 22 ,I µ ...µ s ( b, a ) = − θ ( s − { g s − b } µ ...µ s iπ ( − − n (3 n + 6 n + 2)Γ( n + 3) −{ g s − b } µ ...µ s iπ ( − − n Γ( n + 2) (cid:18) Λ − n a − n ( n + 1) n + 2 b (cid:19) , with n = s + 12 . (2.8)
3. Modified Passarino-Veltman Reduction Schemes
It is known that the one-loop tensor integrals can be reduced to a linear combinationof up to four-point scalar integrals[14]. In this section, a generic one-loop integral T Nµ ...µ s ≡ Z d k k µ . . . k µ s D D . . . D N − , (3.1)with propagators D i ≡ ( k + p i ) − m i + iε and p = 0 is considered. As mentionedin the previous sections, the integrals T Nµ ...µ s may not be translation invariant be-cause of the finite integral limits when they are ultraviolet divergent. Therefore, theexpressions for∆ L Nµ ...µ s ≡ Z d k k µ . . . k µ s D . . . D N − Z d k ( k − p ) µ . . . ( k − p ) µ s ˜ D . . . ˜ D N , (3.2)where propagators D i ≡ ( k + p i ) − m i + iε but p = 0 and ˜ D i ≡ ( k + p i − p ) − m i + iε should be calculated. After the conventional Feynman parameterization, ∆ L Nµ ...µ s can be reexpressed as∆ L Nµ ...µ s ≡ ( − ) N Γ( N ) "Z simplex N Y i =1 d u i I s +4 − Nµ ...µ s (˜ b − p , ˜ a ) − Z simplex N Y i =1 d u i s X i =0 ( − ) s − i { p s − i I i +4 − N, i (˜ b, ˜ a ) } µ ...µ s , (3.3)– 5 –here ˜ b ≡ − P Ni =1 u i p i + p , ˜ a ≡ − P Ni =1 u i ( p i − m i ) + P Ni,j =1 u i u j p i · p j , and thenotation { p s − i I i +4 − N,i (˜ b, ˜ a ) } µ ...µ s is defined in appendix A with the divergencedegree of I i +4 − N, i defined in Section 2 is i + 4 − N . The polynomial dependenceof b, a in I ∆ ( b, a ) µ ...µ s makes the simplex integration in Eq.(3.3) straightforwardusing Z simplex N Y i =1 d u i N Y i =1 u r i − i = Q Ni =1 Γ( r i )Γ( P Ni =1 r i ) . (3.4)Next, several notations similar to that given in ref.[15] are reintroduced here∆ L Nµ ...µ s ≡ n ,n ,...,n N ≥ X n + n + ... + n N = s { g n p n . . . p n N N } µ ...µ s ∆ L N . . . | {z } n ... N . . . N | {z } nN ,T Nµ ...µ s ≡ n ,n ,...,n N − ≥ X n + n + ... + n N − = s { g n p n . . . p n N − N − } µ ...µ s T N . . . | {z } n ... ( N − . . . ( N − | {z } nN − ,T Nµ ...µ s (0) ≡ Z d k k µ . . . k µ s D . . . D N ,T Nµ ...µ s ( k ) ≡ Z d k k µ . . . k µ s D . . . ˆ D k . . . D N , ˜ T Nµ ...µ s (0) ≡ Z d k k µ . . . k µ s ˜ D . . . ˜ D N ,T Nµ ...µ s (0) ≡ n ,n ,...,n N ≥ X n + n + ... + n N = s { g n p n . . . p n N N } µ ...µ s T N . . . | {z } n ... N . . . N | {z } nN (0) , ˜ T Nµ ...µ s (0) ≡ n ,n ,...,n N − ≥ X n + n + ... + n N − = s { g n ( p − p ) n . . . ( p N − p ) n N − } µ ...µ s ˜ T N . . . | {z } n ... ( N − . . . ( N − | {z } nN − (0) , (3.5)with D . . . ˆ D k . . . D N ≡ D . . . D k − D k +1 . . . D N and employing the caret ” ^ ” toindicate the indices omitted.Thus in this cutoff regularization, Eq.(2.9) in ref.[15] should be replaced by T N . . . | {z } n . . . | {z } k i n + k +1 ...i s (0) = ( − ) k k X l =0 C lk N − X i ,...,i l =1 ˜ T N , . . . , | {z } n ,i ,...,i l , i n + k +1 − ,..., i s − (0)+∆ L N . . . | {z } n . . . | {z } k i n + k +1 ...i s , i n + k +1 , . . . , i s > . (3.6)– 6 –hen the determinant of Gram matrix Z ( N ) = p · p . . . p · p N ... . . . ...2 p N · p . . . p N · p N (3.7)for ( N + 1)-point functions is non-vanishing, the reduction can be continued as T N i ...i s = 12(3 + s − N ) " T N − i ...i s (0) + 2 m T Ni ...i s + N − X j =1 f j T Nji ...i s +∆ L N − i ...i s + N − X j =1 ∆ L N − ji ...i s ,T Ni ...i s = N − X j =1 ( Z ( N − ) − i j S sji ...i s − s X r =2 δ ji r T N i ... ˆ i r ...i s ! , i = 0 , (3.8)where some notations are defined in ref.[15] f k ≡ p k − m k + m , ¯ δ ij ≡ − δ ij , ( i r ) k ≡ (cid:26) i r , k > i r i r − , k < i r ,S ski ...i s ≡ T N − i ) k ... ( i s ) k ( k )¯ δ ki . . . ¯ δ ki s − T N − i ...i s (0) − f k T Ni ...i s . (3.9)Otherwise, when the Gram determinant is zero, there is at least one non-vanishingelement ˜ Z ( N ) kl in the adjoint matrix of Z ( N ) ˜ Z ( N ) kl ≡ ( − ) k + l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p . . . p p l − p p l +1 . . . p p N ... . . . ... ... . . . ...2 p k − p . . . p k − p l − p k − p l +1 . . . p k − p N p k +1 p . . . p k +1 p l − p k +1 p l +1 . . . p k +1 p N ... . . . ... ... . . . ...2 p N p . . . p N p l − p N p l +1 . . . p N p N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.10)and one non-zero element ˜ X ( N )0 j in the adjoint matrix of the following matrix– 7 – ( N ) ≡ m f . . . f N f p p . . . p p N ... ... . . . ... f N p N p . . . p N p N . (3.11)One-loop reduction can be applied using the following equations T Ni ...i s = − X ( N − j N − X n =1 ˜ Z ( N − jn ˆ S s +1 ni ...i s − s X r =1 δ ni r T N i ... ˆ i r ...i s ! ,T N i ...i s = 12(6 + s − N + P sr =1 ¯ δ i r ) ˜ Z ( N − kl h ˜ Z ( N − kl S s +200 i ...i s + N − X n =1 (cid:16) ˜ Z ( N − nl ˆ S s +2 nki ...i s − ˜ Z ( N − kl ˆ S s +2 nni ...i s (cid:17) − N − X n,m =1 ˜˜ Z ( N − kn )( lm ) f n ˆ S s +1 mi ...i s + 2 s X r =1 δ ni r ˆ S s +2 m i ... ˆ i r ...i s − f n f m T Ni ...i s − s X r =1 ( f n δ mi r + f m δ ni r ) T N i ... ˆ i r ...i s − s X r,t =1 ,r = t δ ni r δ mi t T N i ... ˆ i r ... ˆ i t ...i s ! . (3.12)Some notations in Eq.(3.12) should be recalled, i.e.˜˜ Z ( N )( ik )( jl ) ≡ ( − ) i + j + k + l sgn( i − k )sgn( l − j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p . . . p p j − p p j +1 . . . p p l − p p l +1 . . . p p N ... . . . ... ... . . . ... ... . . . ...2 p i − p . . . p i − p j − p i − p j +1 . . . p i − p l − p i − p l +1 . . . p i − p N p i +1 p . . . p i +1 p j − p i +1 p j +1 . . . p i +1 p l − p i +1 p l +1 . . . p i +1 p N ... . . . ... ... . . . ... ... . . . ...2 p k − p . . . p k − p j − p k − p j +1 . . . p k − p l − p k − p l +1 . . . p k − p N p k +1 p . . . p k +1 p j − p k +1 p j +1 . . . p k +1 p l − p k +1 p l +1 . . . p k +1 p N ... . . . ... ... . . . ... ... . . . ...2 p N p . . . p N p j − p N p j +1 . . . p N p l − p N p l +1 . . . p N p N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,N > , ˜˜ Z (2)( ik )( jl ) ≡ δ il δ kj − δ ij δ kl , ˆ S ski ...i s ≡ T N − i ) k ... ( i s ) k ( k )¯ δ ki . . . ¯ δ ki s − T N − i ...i s (0) . (3.13)– 8 –or det( Z ( N − ) = 0,det( X ( N − ) = 0 and all ˜ X ( N − k = 0 but ˜ Z ( N − kl = 0 and˜ X N − ij = 0,following equations T N . . . | {z } r l . . . l | {z } n i ...i m = 12( n + 1) ˜ Z ( N − kl − m X j =1 ˜ Z ( N − ki j T N . . . | {z } r l . . . l | {z } n +1 i ... ˆ i j ...i m + N − X j =1 ˜ Z ( N − kj ˆ S r + n + mj . . . | {z } r − l . . . l | {z } n +1 i ...i m , i , . . . , i m = 0 , l,T Ni ...i s = 1˜ X ( N − ij h ˜ Z ( N − ij (cid:0) s − N ) T N i ...i s − T N − i ...i s (0) − ∆ L N − i ...i s − N − X n =1 ∆ L N − ni ...i s ! + N − X m,n =1 ˜˜ Z ( N − in )( jm ) f n ˆ S s +1 mi ...i s − s X r =1 δ mi r T N i ... ˆ i r ...i s ! (3.14)can be used. Other details in the derivation of these equations can be found in [15].
4. Higgs Decay into Two Photons
In this section, one-loop reduction as illustrated in the previous section is applied tothe process H → γγ .In unitary gauge, the three diagrams via the W-boson loop that contribute tothis process with a specific loop momentum configuration are shown in Fig.(1). A di-rect calculation of amplitude (dropping the polarization vectors of external photons)yields M µν unitary = − e m W π m H s w (cid:2) k µ k ν (cid:0) iπ − ( m H − m W ) C (0 , , m H , m W , m W , m W ) (cid:1) − k · k g µν (cid:0) iπ − m H − m W ) C (0 , , m H , m W , m W , m W ) (cid:1)(cid:3) = 3 ie m W π m H s w (cid:20) − k µ k ν (cid:18) m H + 4( m H − m W ) f ( m H m W ) (cid:19) + k · k g µν (cid:18) m H + 4( m H − m W ) f ( m H m W ) (cid:19)(cid:21) , (4.1)with f ( x ) ≡ ( arcsin( √ x ) , x ≤ − h ln ( √ − x − −√ − x − ) − iπ i , x > , (4.2)– 9 –nd the scalar integral C is given in appendix B. However, gauge invariance isspoiled in this four-momentum cutoff regularization. Therefore, a term should besubtracted from the above expressions to recover gauge invariance. In this gauge, arequirement of M µν ( k = k = 0) = 0 should be made. However, M µν unitary ( k = k = 0) = − ie m W π s w g µν = 0 . (4.3)Following Dyson’s prescription[16, 17], gauge invariance is recovered after makingsubtraction from Eq.(4.3), and the final result is M µν unitary = − ie π m W s w ( k µ k ν − g µν k · k ) (cid:0) τ − + (2 τ − − τ − ) f ( τ ) (cid:1) (4.4)with τ = m H m W following the notations of refs.[3, 4]. Eq.(4.4) is the same as thosein refs.[3, 4] up to a factor of − i from the symmetry factor of loops and differentconventions of Feynman rules. However, in this gauge, there are high degrees of ul-traviolet divergence in each diagram. The expressions for amplitude may be differentunder different choices of loop momentum. One may suspect that the discrepancybetween Eq.(4.4) and the result given in DREG M µν DREG = − ie π m W s w ( k µ k ν − g µν k · k ) (cid:0) τ − + 3(2 τ − − τ − ) f ( τ ) (cid:1) (4.5)is originated from the bad loop momentum choices in Eq.(4.4). However, in ourcalculation we find that the terms∆ M µν ( p ) = − ie π m W s w (cid:2) ( k µ p ν − p µ k ν ) (cid:0) − − m H − m W + ( k + k ) · p − p (cid:1) +2 g µν ( k − k ) · p (cid:0) − − m H + 3 m W + ( k + k ) · p − p (cid:1)(cid:3) (4.6)should be added to Eq.(4.4) if loop momentum k is shifted to k + p . From thesymmetric consideration of k , k , a proper choice of p is k + k which is the same asthat presented in refs.[3, 4]. Since ∆ M µν ( k + k ) = 0 in Eq.(4.6), the result in Eq.(4.4)remains unchanged. From Eq.(4.6), it seems hopeless that the difference betweenEq.(4.4) and Eq.(4.5) can be eliminated through shifting the integral momentum k .In ’t Hooft-Feynman gauge ( ξ = 1), the amplitude with one-loop diagrams shownin Fig.(2) is – 10 – µνξ =1 = e π m H m W s w (cid:8) k µ k ν (cid:2) − iπ ( m H + 6 m W )+6 m W ( m H − m W ) C (0 , , m H , m W , m W , m W ) (cid:3) + k · k g µν (cid:2) iπ ( m H + 6 m W ) − m W (cid:0) m H − m W (cid:1) C (0 , , m H , m W , m W , m W ) (cid:3)(cid:9) = ie π m H m W s w (cid:2) − k ν k µ (cid:0) m H + 6 m H m W + 12 m W (cid:0) m H − m W (cid:1) f ( τ ) (cid:1) + k · k g µν (cid:0) m H + 6 m H m W + 24 m W ( m H − m W ) f ( τ ) (cid:1)(cid:3) . (4.7)Following a similar procedure to obtain a gauge invariant result, the amplitude at k = k = 0 is calculated as M µνξ =1 ( k = k = 0) = − ie m W π s w g µν , (4.8)which is the same as Eq.(4.4). However, the gauge invariant amplitude is non-vanishing at k = k = 0 because of the contributions of diagrams ( g ) and ( h ) inFig.(2) in this gauge. These contributions are M µνξ =1 , ( g,h ) ( k = k = 0) = ie m H π m W s w g µν . (4.9)Therefore, the subtracted terms should be M µνξ =1 ( k = k = 0) − M µνξ =1 , ( g,h ) ( k = k = 0) instead of M µνξ =1 ( k = k = 0). The final result is M µνξ =1 = − ie π m W s w ( k µ k ν − g µν k · k ) (cid:0) τ − + 3(2 τ − − τ − ) f ( τ ) (cid:1) = M µν DREG . (4.10)The term generated by the contributions of the Goldstone triangle diagrams ( d, e )in Fig.(2) spoils the decoupling theorem, as pointed out by Shifman et al. recently[10]. Given that there are only logarithmic divergences under this covariant gauge,the result in Eq.(4.10) is unique with a different loop momentum chosen. To bestof our knowledge, it is the first derivation in the ’t Hooft-Feynman gauge in cutoffregularization.It seems there are some problems with unitary gauge in this cutoff regularization.Given that the top quark loop ( Fig.(3)) does not suffer from any ambiguities in thegauge or loop momentum choices, the diagrammatic expressions are expected to bethe same in DREG and in this cutoff regularization. These conditions have beenverified following the same procedures. The result is as follows– 11 – µν top = ie N c π m W s w ( k µ k ν − k · k g µν ) (cid:0) χ − + ( χ − − χ − ) f ( χ ) (cid:1) , (4.11)where χ = m H m t .The authors of refs.[11, 12] have also calculated this Higgs decay process in Pauli-Villars regularization and dimensional regularization respectively, and obtained thesame result as the old ones[5, 6, 7, 8]. Their statement about this issue is that theintegral(in Euclidean) I µν ≡ Z k g µν − k µ k ν ( k + m ) (4.12)is vanishing in cutoff regularization, while it is nonzero in DREG, which is alsopointed out by R.Gastmans [3, 4]. They also argued that I µν contained the differenceof two logarithmic divergencies and should be regulated. Therefore, the integral thatviolats electromagnetic gauge invariance may suffer from some ambiguities. Actually,this issue was first discussed by R.Jackiw[21] in a more general case. However,wethink that the vanishing of I µν in 4 dimensions is just a result of the fact that theintegral intervals are symmetric about the origin even when there is a cutoff Λ , anda replacement of k µ k ν → g µν k in the integrand is also proper.Moreover, very recently R.Jackiw also pointed out that by combining the twoterms in the integrand of I µν one can avoid infinities but the difference of the integralsin these two regularization schemes is still the same, thus both evaluations are math-ematically defensible[22]. So, what are the physical reasons for these ambiguities?In the following, we will try to clarify this issue.Considering that the diagrammatic expressions in unitary gauge are not well-defined in the four-momentum cutoff regularization, how to recover the correct resultunder this condition may be still an open question. We investigate the Lagrangianfor the Standard Model in unitary gauge, similar to the treatments in ref.[19]. Thecovariant terms for scalars are L scalar ≡ (D µ Φ) † (D µ Φ) − V(Φ) , with V(Φ) ≡ − µ Φ † Φ + λ (Φ † Φ) , D µ Φ ≡ (cid:18) ∂ µ − i τ i W iµ − i ′ B µ (cid:19) Φ . (4.13)The scalar doublet produces the vacuum expectation value through the Higgs mech-anism as – 12 – Φ i = v √ ! , v = (cid:18) µ λ (cid:19) / . (4.14)Therefore, the scalar fields can be redefined asΦ = φ +v √ + h + iφ ! . (4.15)There are terms like m W W + µ W − ,µ + im W (cid:0) W − µ ∂ µ φ + − W + µ ∂ µ φ − (cid:1) = m W (cid:18) W + µ + im W ∂ µ φ + (cid:19) (cid:18) W − ,µ − im W ∂ µ φ − (cid:19) − ∂ µ φ + ∂ µ φ − (4.16)after expanding the Lagrangian given in Eq.(4.13), where W ± µ ≡ W µ ∓ iW µ √ . By follow-ing the prescription in ref.[19], the W-boson fields in unitary gauge can be redefinedas ˜W + µ ≡ W + µ + im W ∂ µ φ + , ˜W − µ ≡ W − µ − im W ∂ µ φ − . (4.17)In this gauge there are no kinetic term ∂ µ φ + ∂ µ φ − and mass term for the Goldstone φ + , φ − because of the cancelation between the last term in Eq.(4.16) and the originalkinetic term of the W-boson’s Goldstone in L scalar . However, terms such as hφ + φ − still exist in the original Lagrangian. In DREG, φ + = φ − = 0 can be set safely, similarto a previous work by Grosse-Knetter [20], because all the momentum modes can beincluded in this regularization . Hence, the conventional Lagrangian in unitary gaugeonly with physical fields is obtained. However, the results are in contrast to those ofthe four-momentum cutoff regularization, because an artificial scale Λ is introducedin the Lagrangian. The absence of a kinetic term for φ + , φ − does not mean that theseGoldstone fields are vanishing intuitively, but because the theory does not provideany information above Λ in this regularization. If the mass of φ + , φ − is assumed tobe O ( Λ ), there are still finite contributions from the Goldstone triangle diagramswhen Λ → ∞ . From this viewpoint, the cutoff regularization in unitary gauge is Note that the limits of loop integrals are taken to be infinity – 13 – γγ k , µk , νWk k − k k − k − k ( a ) H γγ k , µk , νWk k − k k − k − k ( b ) H γγW k , µk , νk k − k − k ( c ) Figure 1:
Feynman diagrams via W-boson loop in unitary gauge for H → γγ . problematic. The violation of the property in gauge invariance can also be attributedto the absence of large momentum modes. Therefore, the old results in the literaturefor H → γγ are still valid.In fact, dimensional and Pauli-Villars regularization schemes are free of missinglarge momentum modes, and can maintain gauge invariance. Therefore, these resultsare correct also in unitary gauge. From the evaluations of R.Jackiw, the integral I µν can be dealt without any infinities, and the only difference is from surface terms(i.e.,large momentum region), which also verifies our conclusion.
5. Summary
A method for systematical evaluations of one-loop tensor integrals in cutoff regular-ization is proposed by deriving a new recursive relation Eq.(2.6) and implementingit in the Passarino-Veltman reduction method. The result has been expressed ina form that can be directly translated into computer codes. Similar to the meth-ods presented in ref.[15], our results are also numerical stable for up to four-pointintegrals. Surely, our method can be extended to deal with high-point integralsstraightforwardly.With this approach, we have calculated the amplitudes for Higgs decay into twophotons via the W-boson loop and the top-quark loop. The correctness of the methodhas been confirmed by evaluating these processes and checking other programs, andit is certainly useful in both theoretical and phenomenological aspects. Moreover,we also reanalyze the Higgs decay process and make our efforts to find the physicalreasons for some puzzles appeared in the calculations of this process.– 14 – γγ k , µk , νWWW ( a ) H γγ k , µk , νWWW ( b ) H γγ k , µk , νWW ( c ) H γγ k , µk , νGGG ( d ) H γγ k , µk , νGGG ( e ) H γγ k , µk , νGG ( f ) H γγ k , µk , νWGG ( g ) H γγ k , µk , νWGG ( h ) H γγ k , µk , νG W ( i ) Figure 2:
Some representative Feynman diagrams via W-boson loop in ’t Hooft-Feynmangauge for H → γγ . H γγ k , µk , νttt ( a ) H γγ k , µk , νttt ( b ) Figure 3:
Feynman diagrams via top-quark loop for H → γγ . Acknowledgments
We thank R.Jackiw for providing us with his evaluations for some loop integrals.This work was supported by the National Natural Science Foundation of China (No10805002, 10847001, 11021092, 11075002, 11075011), the Foundation for the Authorof National Excellent Doctoral Dissertation of China (Grant No. 201020), and theMinistry of Science and Technology of China (2009CB825200).– 15 – . Derivation of Expressions for J Nµ µ ...µ s In this appendix, the general formulae for J Nµ µ ...µ s ≡ R d k k µ k µ ...k µs ( − k + a ) N used in section2 are derived first. Obviously, J Nµ µ ...µ s is vanishing until s is even. Therefore, s should be an even and non-negative integer and N should be positive in the followingcontext.A notation (similar to but a little different from that in Ref.[15]) is introducedfirst in order to write down the tensor decomposition in a concise way. We usecurly braces to denote symmetrization with respect to Lorentz indices, where all non-equivalent permutations of the Lorentz indices on metric tensor g and momenta p areimplicitly understood. A generic notation { g n p n . . . p n k k } µ ...µ t with t = P kl =1 n l +2 n means a sum that the of Lorentz indices µ , . . . , µ t are distributed to n metric tensors g while n l of them are distributed to n l momenta p l with equalweights. For instance, { g } µνρσ ≡ g µν g ρσ + g µρ g νσ + g µσ g νρ , { g p } µνρ ≡ g µν p ρ + g νρ p µ + g ρµ p ν , { p p } µνρ ≡ p µ p ν p ρ + p µ p ρ p ν + p ν p ρ p µ . (A.1)The Lorentz covariance ensures us to make the following replacement k µ k µ ...k µ s −→ { g s } µ µ ...µ s ( k ) s Γ( s + 2)2 s (A.2)in the integral J Nµ ...µ s , which can be proven by the induction of the integer s . Afterthis replacement and subsequent Wick rotation, spherical coordinate system trans-formation and some trivial variable substitutions, one arrived J Nµ ...µ s = { g s } µ ...µ s iπ ( − − s Γ( s + 2) Z Λ d K K s +22 ( K + a ) N , (A.3)where Λ was denoted as the ultraviolet cutoff scale.Eq.(A.3) can be solved directly when a = 0, i.e. J Nµ ...µ s = { g s } µ ...µ s iπ ( − − s Γ( s + 2) 2Λ ∆ ∆ (A.4)– 16 –hen superficial degree of ultraviolet divergence ∆ ≡ s − N + 4 >
0. In the caseof ∆ ≤
0, Eq.(A.3) encounters infrared divergence, which is not considered in thisarticle. When a = 0, result becomes a little more complicated than the previousone. However, this problem can be resolved after implementing the tricks of usingintegration by parts and following integral formulae Z d x x n ln( x + a ) = 1 n + 1 x n +1 ln( x + a ) − n X k =0 ( − a ) k x n +1 − k n + 1 − k − ( − a ) n +1 ln( a + x ) (cid:1) , n ∈ N (A.5)into Eq.(A.3). The explicit expressions for J Nµ ...µ s can also be obtained, i.e. a = 0and superficial degree of divergence ∆ ≡ s − N + 4 ≥ J Nµ ...µ s = { g s } µ ...µ s ( − − s iπ Γ( N ) − N − X k =1 Γ( N − k )Γ( s + 3 − k ) ∆2 X l =0 C lN − k + l − ( − a ) l Λ ∆ − l + Γ(1)Γ( ∆2 + 1) ∆2 − X k =0 ( − a ) k ∆ − k ∆ − k + ( − a ) ∆2 ln( Λ a ) , (A.6)while a = 0 but ∆ < J Nµ ...µ s = { g s } µ ...µ s ( − − s iπ Γ( − ∆2 )Γ( N ) a ∆ . (A.7) B. Some Scalar Integrals
After the reduction of one-loop integrals using the modified Passarino-Veltman re-duction formulas given in the section 2, every tensor integral can be expressed as alinear combination of up to four-point scalar integrals. In this appendix, the ana-lytical expressions for some scalar integrals are listed below. Some of them may beused in the one-loop calculations of the process Higgs decay to two photons.First of all, the conventions for the scalar integrals used in this article are fixedas follows T N ≡ Z d k D D . . . D N − . (B.1)– 17 –or one-point functions, A (0) = − iπ Λ ,A ( m ) = iπ m (cid:18) ln( Λ m ) − Λ m (cid:19) ,A . . . | {z } n ( m ) = ( − ) n +1 iπ Γ( n + 2)2 n n +1 X i =1 ( − ) n +1 − i i Λ i m n +2 − i + ( − ) n +1 m n +20 ln( Λ m ) ! . (B.2)Two-point functions can be easily verified as B ( p , m , m ) = iπ ln( Λ p ) + 1 + X i =1 [ γ i ln( γ i − γ i ) − ln( γ i − ! , with γ , = p − m + m ± q ( p − m + m ) − p m p ,B (0 , , m ) = iπ ln( Λ m ) ,B ( p , ,
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