Computation of H→gg in FDH and DRED: renormalization, operator mixing, and explicit two-loop results
A. Broggio, Ch. Gnendiger, A. Signer, D. Stöckinger, A. Visconti
aa r X i v : . [ h e p - ph ] M a r PSI-PR-15-02ZU-TH 05/15
Computation of H → gg in FDH and DRED:renormalization, operator mixing, and explicit two-loopresults A. Broggio a , Ch. Gnendiger b , A. Signer a,c , D. St¨ockinger b , A. Visconti aa Paul Scherrer Institut,CH-5232 Villigen PSI, Switzerland b Institut f¨ur Kern- und Teilchenphysik,TU Dresden, D-01062 Dresden, Germany c Physik-Institut, Universit¨at Z¨urich,Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland
The H → gg amplitude relevant for Higgs production via gluon fusion is computed in thefour-dimensional helicity scheme ( fdh ) and in dimensional reduction ( dred ) at the two-loop level. The required renormalization is developed and described in detail, includingthe treatment of evanescent ǫ -scalar contributions. In fdh and dred there are additionaldimension-5 operators generating the Hgg vertices, where g can either be a gluon or an ǫ -scalar. An appropriate operator basis is given and the operator mixing through renor-malization is described. The results of the present paper provide building blocks for furthercomputations, and they allow to complete the study of the infrared divergence structureof two-loop amplitudes in fdh and dred . ontents H → gg
43 Genuine two-loop diagrams 64 Parameter and field renormalization in FDH and DRED 6 β functions 74.2 Anomalous dimensions 8 λ and λ ǫ
137 UV renormalized form factors of gluons and ǫ -scalars 14 A.1 Projectors and form factors of gluons and ǫ -scalars 18A.2 Feynman rules 19 Higgs production via gluon fusion is one of the most important LHC processes. Its com-putation at higher orders requires renormalization and factorization to cancel UV and IRdivergences. The renormalization is less trivial than the one of standard QCD processes dueto the required renormalization of non-renormalizable operators. The virtual correctionshave been computed in conventional dimensional regularization ( cdr ) [1–5]; the requiredtheory of operator renormalization in cdr has been developed in Ref. [6], based on generalwork in Refs. [7, 8].In the past years, several alternative regularization schemes have been developed.Purely four-dimensional schemes such as implicit regularization [9, 10] and fdr [11] have2een proposed and used to compute processes of practical interest such as H → γγ [12, 13]and H → gg [14]. The present paper is devoted to regularization by dimensional reduction( dred ) [15] and the related four-dimensional helicity ( fdh ) scheme [16]. Both schemesare actually the same regarding UV renormalization, but they differ in the treatment ofexternal partons related to IR divergences. There has been significant progress in the un-derstanding of fdh and dred : the equivalence to cdr [20, 21], mathematical consistencyand the quantum action principle [22], infrared factorization [23, 24] have been established— these results solved several problems that had been reported earlier, related to violationof unitarity [25], Siegel’s inconsistency [26], and the factorization problem of [27, 28]. Inaddition, explicit multi-loop calculations have been carried out [29–33].More recently, the multi-loop IR divergence structure of fdh and dred amplitudes hasbeen studied in Ref. [34]. It has been shown that IR divergences in fdh and dred can bedescribed by a generalization of the cdr formulas given in Refs. [35–38]. The descriptioninvolves IR anomalous dimensions γ i for each parton type i . In Ref. [34] they have beencomputed for the cases of quarks and gluons by comparing the general IR factorizationformulas with explicit results for the quark and gluon form factor. In fdh and dred ,however, the gluon can be decomposed into a D -dimensional gluon ˆ g and (4 − D ) additionaldegrees of freedom, so-called ǫ -scalars ˜ g . In dred , ǫ -scalars also appear as external states.The present paper is devoted to a detailed two-loop computation of the amplitude H → gg in fdh and dred . In dred , this involves the computations of H → ˆ g ˆ g and H → ˜ g ˜ g , since the external gluons can either be gauge fields or ǫ -scalars. The fdh resultis identical to the one for H → ˆ g ˆ g and has already been given in Ref. [34], but we willprovide further details here.This detailed computation is of interest for two reasons: First, it provides the basis forobtaining the remaining IR anomalous dimension for ǫ -scalars at the two-loop level. Second,it provides an example of the required renormalization in fdh and dred , including operatorrenormalization and operator mixing. The difficulty of renormalization in fdh and dred ,particularly in connection with H → gg , has been pointed out e. g. in Refs. [33, 39].The outline of the paper is as follows: Section 2 gives a brief description of the regular-ization schemes and of the relevant Lagrangian and operators. It ends with a detailed listof the required ingredients of the calculation. Apart from the actual two-loop computationand ordinary parameter and field renormalization that are described in Sections 3 and 4,respectively, the main difficulty lies in the renormalization and mixing of the operatorsgenerating H → gg . This is discussed in general in Section 5, and specific two-loop resultsare presented in Section 6. Section 7 then provides the final results for the on-shell ampli-tudes for H → ˆ g ˆ g and H → ˜ g ˜ g . The appendix contains details on our projection operatorsand gives Feynman rules for the different operator insertions. Parts of the literature, e. g. Refs. [17–19] used the term DR/dimensional reduction for what is called fdh here. dr hv fdh dred internal gluon ˆ g µν ˆ g µν g µν g µν external gluon ˆ g µν ¯ g µν ¯ g µν g µν Table 1 . Treatment of internal and external gluons in the four different regularization schemes,i.e. prescription which metric tensor has to be used in propagator numerators and polarizationsums. H → gg It is useful to distinguish the following regularization schemes [24]: conventional dimen-sional regularization ( cdr ), the ’t Hooft-Veltman ( hv ) scheme, the four-dimensional helic-ity ( fdh ) scheme, and dimensional reduction ( dred ). In all these schemes, momenta aretreated in D = 4 − ǫ dimensions (the associated space is denoted by QDS with metric ten-sor ˆ g µν ). In order to define the schemes, one also needs an additional quasi-4-dimensionalspace ( Q S , metric g µν ) and the original 4-dimensional space (4 S , metric ¯ g µν ). The treat-ment of gluons in the four schemes is given in Tab. 1. In the table, “internal” gluons aredefined as either virtual gluons that are part of a one-particle irreducible loop diagram or,for real correction diagrams, gluons in the initial or final state that are collinear or soft.“External gluons” are defined as all other gluons.Mathematical consistency and D -dimensional gauge invariance require that Q S ⊃ QDS ⊃ S and forbid to identify g µν and ¯ g µν . Details can be found in Refs. [22, 24, 34].The most important relations for the present paper are g µν = ˆ g µν + ˜ g µν , ˆ g µρ ˜ g ρν = 0 , ˆ g µρ ¯ g ρν = ¯ g µν , ˆ g µν ˆ g µν = D, ˜ g µν ˜ g µν = N ǫ , (2.1)where a complementary 2 ǫ -dimensional metric tensor ˜ g µν has been introduced. With thesemetric tensors we can decompose a quasi-4-dimensional gluon field A µ as A µ = ˆ g µν A ν + ˜ g µν A ν = ˆ A µ + ˜ A µ (2.2)into a D -dimensional gauge field ˆ A µ and an associated ǫ -scalar field ˜ A µ with multiplic-ity N ǫ = 2 ǫ . Correspondingly, there are two types of particles in the regularized theory: D -dimensional gluons ˆ g and ǫ -scalars ˜ g . The unregularized external gluons ¯ g of fdh are apart of ˆ g .The regularized Lagrangian of massless QCD then reads L QCD, regularized = −
14 ˆ F µνa ˆ F µν,a − ξ ( ∂ µ ˆ A µ,a ) + i ψ ˆ /Dψ + ∂ µ c a ˆ D µ c a + L ǫ , (2.3a) L ǫ = −
12 ( ˆ D µ ˜ A ν ) a ( ˆ D µ ˜ A ν ) a − g e ψ ˜ Aψ − (cid:0) g ǫ (cid:1) αβγδabcd ˜ A α,a ˜ A β,b ˜ A γ,c ˜ A δ,d . (2.3b) In many applications of fdh the dimensionality of Q S is left as a variable D s , which is eventually setto D s = 4. The multiplicity of ǫ -scalars is then N ǫ = D s − D . F µν and ˆ D µ = ∂ µ + ig s ˆ A µ denote the non-abelian field strength tensor and the covari-ant derivative in D dimensions; ψ and c are the quark and ghost fields. In Eq. (2.3b) thecoupling of ǫ -scalars to (anti-)quarks is given by the evanescent Yukawa-like coupling g e .This could in principle be set equal to the strong coupling g s . But, since both couplingsrenormalize differently this would only hold at tree-level and for one particular renormal-ization scale [20]; the same is true for the quartic ǫ -scalar coupling g ǫ . In Eq. (2.3b)we introduce an abbreviation that includes the appearing Lorentz and color structure: (cid:0) g ǫ (cid:1) αβγδabcd := g ǫ ( f abe f cde ˜ g αγ ˜ g βδ + perm.), where “perm.” denotes the 5 permutations aris-ing from symmetrization in the multi-indices ( a, α ) . . . ( c, γ ). In the following we use allcouplings in the form α i = g i π with i = s, e, ǫ .The process H → gg is generated by an effective Lagrangian which arises from in-tegrating out the top quark in the Standard Model. In cdr it contains only the term − λH ˆ F µνa ˆ F µν,a . In fdh and dred one again has to distinguish several gauge invariantstructures containing either D -dimensional gluons or ǫ -scalars. The effective Lagrangiancan be written as L eff = λHO + λ ǫ H ˜ O + X i λ ǫ,i H ˜ O ǫ,i , (2.4)with O = −
14 ˆ F µνa ˆ F µν,a , (2.5a)˜ O = −
12 ( ˆ D µ ˜ A ν ) a ( ˆ D µ ˜ A ν ) a . (2.5b)˜ O ǫ,i denote operators involving products of four ǫ -scalars. Such operators are not impor-tant in the present paper and will not be given explicitly. Like for α s , α e and α ǫ , thecouplings λ and λ ǫ can be set equal at tree-level, but they renormalize differently and havedifferent β functions.Our final goal is the calculation of the two-loop form factors for gluons and ǫ -scalars.This requires the on-shell calculation of the 3-point function Γ HA µ A ν ( q, − p, − r ). All mo-menta are defined as incoming, so q = p + r . The 3-point function can be separatedinto Γ H ˆ A µ ˆ A ν and Γ H ˜ A µ ˜ A ν , corresponding to the amplitudes for H → ˆ g ˆ g and H → ˜ g ˜ g ,respectively. In dred , both on-shell amplitudes are needed according to Tab. 1. In fdh ,only H → ¯ g ¯ g is needed, which however is identical to H → ˆ g ˆ g and will not be discussedseperately.The on-shell calculation requires the knowledge of the two-loop renormalization con-stants δZ λ and δZ λ ǫ . These in turn can be obtained from an off-shell calculation ofΓ HA µ A ν . Projectors extracting the required renormalization constants from the off-shellGreen functions and precisely defining the gluon and ǫ -scalar form factors are given inappendix A.1.We have now all ingredients to discuss the classes of Feynman diagrams that contributeto Γ HA µ A ν in fdh and dred : 5 H H H
Figure 1 . Sample two-loop diagrams for the process H → ˆ g ˆ g and H → ˜ g ˜ g in dred . ǫ -scalars aredenoted by dashed lines. The appearing coupling combinations from left to right are λα s , λ ǫ α e , λ ǫ α s , λ ǫ α ǫ .
1. Genuine two-loop diagrams Γ HA µ A ν . Some remarks concerning the calculation arepresented in Sec. 3.2. Counterterm diagrams Γ HA µ A ν and Γ HA µ A ν arising from one- and two-loop renor-malization of the fields, the gauge parameter ξ , and of the couplings α s , α e , and α ǫ .The required renormalization constants are presented in Sec. 4.3. Counterterm diagrams Γ HA µ A ν arising from one-loop renormalization of the effectiveLagrangian (2.4) at the one-loop level, which includes the renormalization of λ and λ ǫ . This is a major complication and will be presented in Sec. 5.4. Overall two-loop counterterm diagrams Γ HA µ A ν arising from the two-loop renormal-ization of the effective Lagrangian (2.4), equivalently from the renormalization con-stants δZ λ and δZ λ ǫ . These renormalization constants are generally defined by therequirement that the appropriate off-shell Green functions are UV finite after renor-malization. For the case of δZ λ , an elegant alternative determination is possible [6],but that method fails for δZ λ ǫ . The results for δZ λ and δZ λ ǫ are presented in Sec. 6. As mentioned above the Green function Γ HA µ A ν can be separated into Γ H ˆ A µ ˆ A ν and Γ H ˜ A µ ˜ A ν ,corresponding to H → ˆ g ˆ g and H → ˜ g ˜ g . Examples for genuine two-loop diagrams witheither external gluons or ǫ -scalars are shown in Fig. 1.All loop calculations have been performed using the following setup: the genera-tion of diagrams and analytical expressions is done with the Mathematica package Feyn-Arts [40]; to cope with the extended Lorentz structure in Q S we use a modified versionof TRACER [41]; all planar on-shell integrals are reduced and evaluated with an inple-mentation of an in-house algorithm that is based on integration-by-parts methods and theLaporta-algorithm [42]; all non-planar and off-shell integrals are reduced and evaluatedwith the packages FIRE [43] and FIESTA [44]. We now consider the counterterm contributions Γ HA µ A ν and Γ HA µ A ν . They are given bydiagrams exemplified in Fig. 2, where the counterterm insertions are generated by the usual6 H ✕ H ✕ H ✕ H Figure 2 . Sample one-loop counterterm diagrams originating from the renormalization of thecouplings α s , α e , α ǫ , and of the gauge parameter ξ , respectively. multiplicative QCD renormalization of the couplings and fields present in Eq. (2.3b). Inthe following we present the values of the required β functions and anomalous dimensions,which govern the renormalization constants. β functions The renormalization of the couplings α s , α e , and α ǫ is done by replacing the bare couplingswith the renormalized ones. As renormalization scheme we choose a modified version ofthe MS scheme: like in Ref. [34] we treat the multiplicity N ǫ of the ǫ -scalars as an initiallyarbitrary quantity and subtract divergences of the form (cid:0) N ǫ ǫ (cid:1) n . As a consequence, thecorresponding β functions depend on N ǫ : β i ≡ µ µ (cid:0) α i π (cid:1) = β i ( α s , α e , α ǫ , N ǫ ), with i = s, e, ǫ . They are given in Refs. [33, 34] and read: β s = − (cid:16) α s π (cid:17) " C A (cid:18) − N ǫ (cid:19) − N F − (cid:16) α s π (cid:17) " C A (cid:18) − N ǫ (cid:19) − C A N F − C F N F − (cid:16) α s π (cid:17) (cid:16) α e π (cid:17)" C F N F N ǫ + O ( α ) , (4.1a) β e = − (cid:16) α s π (cid:17)(cid:16) α e π (cid:17) C F − (cid:16) α e π (cid:17) " C A (2 − N ǫ ) + C F ( − N ǫ ) − N F + O ( α ) . (4.1b)The renormalization of the quartic coupling ( α ǫ ) αβγδabcd is more complicated since the tree-level color structure, f abe f cde , is not preserved under renormalization [20]. In the case ofan SU(3) gauge group one therefore has to introduce three quartic couplings, α ǫ,i with i = 1 , ,
3, each of them related to one specific color structure in a basis of color space.Examples for such a basis are given e. g. in Refs. [29, 30].In the present case of H → gg the renormalization constant for α ǫ only appears in7iagrams like the third of Fig. 2. Hence, only the following contracted β function is needed:( β ǫ ) αβγδabcd δ ab ˜ g αβ = ((cid:16) α s π (cid:17) C A (9 + 6 N ǫ ) + (cid:16) α s π (cid:17)(cid:16) α ǫ π (cid:17) C A (1 − N ǫ ) 12+ (cid:16) α e π (cid:17) h C A N F (4 − N ǫ ) + C F N F ( − − N ǫ ) i + (cid:16) α e π (cid:17)(cid:16) α ǫ π (cid:17) C A N F (1 − N ǫ )( − (cid:16) α ǫ π (cid:17) C A (1 − N ǫ )( − − N ǫ ) ) δ cd ˜ g γδ + O ( α ) . (4.2)This result is obtained from a direct off-shell calculation. It agrees with a general resultfrom [45]. For the off-shell calculation of Γ HA µ A ν also renormalization of the fields and of the gaugeparameter ξ is needed. The renormalization of ξ is fixed by the requirement that the gaugefixing term does not renormalize: ξ → Z ˆ A ξ . The anomalous dimensions γ i = µ ddµ ln Z i ofgluon and ǫ -scalar fields are obtained from a direct off-shell calculation of the respectivetwo-loop self energies. Their values up to two-loop level read: γ ˆ A = − (cid:16) α s π (cid:17)" C A (cid:18) − ξ − N ǫ (cid:19) − N F − (cid:16) α s π (cid:17) " C A (cid:18) − ξ − ξ − N ǫ (cid:19) − C A N F − C F N F − (cid:16) α s π (cid:17)(cid:16) α e π (cid:17) C F N F N ǫ + O ( α ) , (4.3a) γ ˜ A = − (cid:16) α s π (cid:17) C A (3 − ξ ) − (cid:16) α e π (cid:17)h − N F i − (cid:16) α s π (cid:17) " C A (cid:18) − ξ − ξ − N ǫ (cid:19) − C A N F − (cid:16) α s π (cid:17)(cid:16) α e π (cid:17)h − C F N F i − (cid:16) α e π (cid:17) " C A N F (cid:18) − N ǫ (cid:19) + C F N F (cid:18) N ǫ (cid:19) − (cid:16) α ǫ π (cid:17) C A (1 − N ǫ ) 34+ O ( α ) . (4.3b)Setting N ǫ and α e to zero in Eq. (4.3a) yields the well-known gluon anomalous dimensionin cdr , see e. g. [46]. The value of γ ˜ A agrees with the general result for the anomalousdimension of a scalar field [45], confirming the point of view that ǫ -scalars behave likeordinary scalar fields with multiplicity N ǫ . 8 Operator renormalization and mixing in FDH and DRED
The second type of counterterm contributions, denoted by Γ HA µ A ν and Γ HA µ A ν , originatesfrom the necessary renormalization of the effective Lagrangian (2.4), equivalently of theoperators O and ˜ O . One major difficulty is that multiplicative renormalization of theparameters λ and λ ǫ is not sufficient since the operators mix with further operators. Wewill show that the full operator mixing involving gauge non-invariant operators has tobe taken into account. The renormalization constants cannot be predicted from knownQCD renormalization constants but need to be determined from an off-shell calculation.The general theory of operator mixing in gauge theories and the classification of gaugeinvariant and gauge non-invariant operators has been developed long ago [7, 8, 47].In the following we briefly describe operator mixing in the much simpler case of cdr and then explain the cases of fdh and dred , which involve further operators. In cdr , a useful basis of scalar dimension-4 operators, which is closed under renormaliza-tion, is given in Ref. [6]: O = −
14 ˆ F µνa ˆ F µν,a , (5.1a) O = 0 , (5.1b) O = i ψ ←→ /D ψ, (5.1c) O = ˆ A νa ( ˆ D µ ˆ F µν ) a − g s ψ ˆ Aψ − ( ∂ µ c a )( ∂ µ c a ) , (5.1d) O = ( D µ ∂ µ c ) a c a . (5.1e)Operator O is gauge invariant and related to coupling renormalization; O = mψψ inRef. [6] and corresponds to the fermion mass renormalization; we set m = 0. All otheroperators are constrained by BRS invariance and Slavnov-Taylor identities [7, 8]; operators O and O are not gauge invariant. The basis is chosen such that O , O and O are relatedto field renormalization of ψ , ˆ A µ and c , respectively. In particular, the first two terms of O are generated by applying the functional derivativeˆ A νa ( x ) δδ ˆ A νa ( x ) (5.2)on the gauge invariant part of the QCD action; the remaining term is then required byBRS invariance and the non-renormalization of the gauge fixing term. See Refs. [8, 47] for more details; the full operator O can be obtained from evaluating W Y ˆ A νa ˆ A νa + W ( ∂ ν c a ) A ν,a , where W is the linearized Slavnov-Taylor operator and Y ˆ A νa is the source of the BRS trans-formation of ˆ A νa in the functional integral. Since W is nilpotent, this definition shows that O is compatiblewith BRS invariance and the Slavnov-Taylor identity and can appear in the operator mixing. O i → Z ij O j, bare , (5.3)where O j, bare arises from O j by replacing all parameters and fields by the respective barequantities. Following an elegant proof in Ref. [6] the nontrivial cdr renormalization matrix Z ij can be written in the form Z ij = δ ij + D i ln Z j . (5.4)Here, D i are derivatives with respect to parameters and Z j are combinations of ordinaryQCD renormalization constants. As a result, in particular the renormalization of Z isgiven by Z = 1 + α s ∂∂α s ln Z α s , (5.5)with the multiplicative renormalization constant of α s , Z α s . In this way the renormalizationof the parameter λ in the cdr version of L eff is related to the renormalization of α s . In fdh and dred , the basis of operators needs to contain additional terms involving ǫ -scalars. We use a basis constructed analogously to Eqs. (5.1) from gauge invariant operatorsand operators corresponding to field renormalization. Then there are two kinds of changes:there are modifications of the operators O and O , and there are additional basis elements.The new basis operators correspond to the ǫ -scalar kinetic term, ˜ O , to the new parameters α e and α ǫ , ˜ O and ˜ O ǫ,i , and to the field renormalization of ˜ A µ , ˜ O . The notation is chosensuch that in all cases O j and ˜ O j have a similar structure: O = −
14 ˆ F µνa ˆ F µν,a , (5.6a) O = 0 , (5.6b) O = i ψ ←→ /D ψ − g e ψ ˜ Aψ, (5.6c) O = ˆ A νa ( ˆ D µ ˆ F µν ) a + g s f abc ( ∂ µ ˜ A νa ) ˆ A µ,b ˜ A ν,c − g s ψ ˆ Aψ − ( ∂ µ c a ) ( ∂ µ c a ) , (5.6d) O = ( ˆ D µ ∂ µ c a ) c a , (5.6e)˜ O = −
12 ( ˆ D µ ˜ A ν ) a ( ˆ D µ ˜ A ν ) a , (5.6f)˜ O = g e ψ ˜ Aψ, (5.6g)˜ O = ˜ A νa ( ˆ D µ ˆ D µ ˜ A ν ) a , (5.6h)˜ O ǫ,i = O ( ˜ A ) . (5.6i)10ince we consider massless QCD there is no ǫ -scalar mass term. Like in Eq. (2.4), operatorsinvolving four ǫ -scalars are not needed explicitly.This set of operators differs in a crucial way from the cdr case. The difference be-tween operators ˜ O and ˜ O is related to the total derivative ✷ ˜ A µ ˜ A µ . Hence, the basis forspace-time integrated operators (zero-momentum insertions) does not coincide with theone for non-integrated operators (non-vanishing momentum insertions). As discussed bySpiridonov in Ref. [6], in such a case his method cannot be used. Therefore, in fdh and dred it is not possible to derive complete results for the operator mixing analogous toEqs. (5.4) and (5.5).This implies two difficulties: First, the two-loop renormalization of ˜ O and the corre-sponding parameter λ ǫ cannot be obtained from a priori known two-loop QCD renormal-ization constants but need to be determined from an explicit two-loop off-shell calculation.Second, the off-shell Green functions get contributions from unphysical, gauge non-invariantoperators, so the full operator mixing needs to be taken into account.We have carried out the explicit one-loop calculations to obtain all required one-loopresults for Z j and Z ˜1 j . The results are δZ = (cid:16) α s π (cid:17) (cid:20)(cid:16) −
113 + N ǫ (cid:17) C A + 23 N F (cid:21) ǫ , (5.7a) δZ = 0 , (5.7b) δZ = 0 , (5.7c) δZ = (cid:20)(cid:16) α s π (cid:17) ( − C A + (cid:16) α e π (cid:17) N F − (cid:16) α ǫ π (cid:17) (1 − N ǫ ) C A (cid:21) ǫ , (5.7d) δZ = 0 , (5.7e) δZ = (cid:16) α e π (cid:17) N ǫ C F ǫ , (5.7f) δZ = (cid:16) α s π (cid:17) C A ǫ , (5.7g) δZ = 0 , (5.7h) δZ = (cid:16) α s π (cid:17) (cid:16) − (cid:17) C A ǫ , (5.7i) δZ = (cid:16) α s π (cid:17)
12 (3 − ξ ) C A ǫ , (5.7j) δZ = 0 , (5.7k) δZ = 0 . (5.7l)Renormalization constants involving operators ˜ O or ˜ O ǫ,i are not needed for the calcula-tions in the present paper. The renormalization constants (5.7a)-(5.7d) agree with thosegiven in Ref. [34]. The only gauge-dependent quantity is Z . This is due to the fact thatoperator ˜ O is related to the field renormalization of the ǫ -scalars. In all other renormal-11 O ✕ O ✕ O ✕ O ✕ ˜ O ✕ ˜ O ✕ O ✕ O Figure 3 . Sample one-loop counterterm diagrams originating from operators O , O , ˜ O and O . ization constants related to field renormalization the gauge-dependent parts incidentallycancel out.With these results the bare effective Lagrangian can be written as L bareeff = H X j (cid:16) λ Z j O j, bare + λ ǫ Z ˜1 j O j, bare (cid:17) , (5.8)where the sum runs over all operators in Eqs. (5.6). Sometimes it is useful to write thisusing multiplicative renormalization constants for λ and λ ǫ as L bareeff = Z λ λHO , bare + Z λ ǫ λ ǫ HO ˜1 , bare + . . . , (5.9)suppressing operators not present at tree level, such that λZ λ = λZ + λ ǫ Z ˜11 and similarfor Z λ ǫ .The one-loop counterterm effective Lagrangian involving the renormalization constantsof Eqs. (5.7) is then given by L = H X j (cid:16) λ δZ j O j + λ ǫ δZ j O j (cid:17) . (5.10)We have now all ingredients for the one-loop counterterm diagrams Γ HA µ A ν relevant forthe computation of H → gg , where the gluons are either D -dimensional gauge fields or ǫ -scalars. These counterterm contributions arise from one-loop counterterm diagrams withone insertion of L . Sample diagrams are given in Fig. 3. They show insertions of oper-ators O , O , ˜ O and O . Feynman rules for operator insertions are given in appendix A.2.The calculation shows that all these operators generate non-vanishing contributionsto Γ HA µ A ν . However, in the extraction of the form factors and two-loop renormalizationconstants to be discussed in the next section there are cancellations, and O is the onlynew operator which contributes. 12 Two-loop renormalization constants of λ and λ ǫ Putting together the results from the previous three sections it is possible to calculatethe two-loop renormalization constants δZ λ and δZ λ ǫ appearing in Eq. (5.9). They canbe obtained from a complete off-shell two-loop calculation and the requirement that thecorresponding Green-functions are UV finite after renormalization:Γ HA µ A ν + Γ HA µ A ν + Γ HA µ A ν + Γ HA µ A ν + Γ HA µ A ν (cid:12)(cid:12)(cid:12)(cid:12) off-shellUV div. = 0 . (6.1)All ingredients except the last term are computed in the previous sections, and Eq. (6.1)is then used to extract δZ λ and δZ λ ǫ . The result for δZ λ is: δZ λ = (cid:16) α s π (cid:17) ( C A " − N ǫ + N ǫ ǫ + − + N ǫ ǫ + C A N F " − + N ǫ ǫ + 103 ǫ + C F N F ǫ + N F ǫ ) + (cid:16) α s π (cid:17)(cid:16) α e π (cid:17) C F N F (cid:0) − − λ ǫ λ (cid:1) N ǫ ǫ . (6.2)Since the off-shell calculations have been done numerically with the help of FIESTA [44]the analytical expressions have been obtained by rounding to a least common denominator.The numerical uncertainty is less than for the terms of the order O ( ǫ − ) and for theterms of the order O ( ǫ − ).Result (6.2) is not new; it agrees with Ref. [34], where it has been obtained usingSpiridonov’s method. The recalculation serves as a test of the setup and the results givenin the previous sections. At the same time a comparison with Ref. [34] confirms thatEq. (6.2) is actually exactly correct, in spite of numerical uncertainties.In the same way, we obtain the renormalization constant δZ λ ǫ : δZ λ ǫ = (cid:16) α s π (cid:17) ( C A " + N ǫ ǫ + − + N ǫ + λλ ǫ (cid:16) − N ǫ (cid:17) ǫ + C A N F " − ǫ + − λλ ǫ ǫ + (cid:16) α s π (cid:17)(cid:16) α e π (cid:17)( C A N F " − ǫ + + 3 λλ ǫ ǫ + C F N F " − ǫ + − λλ ǫ ǫ + (cid:16) α s π (cid:17)(cid:16) α ǫ π (cid:17) C A (1 − N ǫ ) " ǫ + − − λλ ǫ ǫ + (cid:16) α e π (cid:17) ( C A N F − + N ǫ ǫ + C F N F " − N ǫ ǫ + 3 − N ǫ ǫ + N F ǫ ) + (cid:16) α e π (cid:17)(cid:16) α ǫ π (cid:17) C A N F (1 − N ǫ ) " − ǫ + 32 ǫ + (cid:16) α ǫ π (cid:17) C A (1 − N ǫ ) " − − N ǫ ǫ + 158 ǫ . (6.3)13ompared to Eq. (6.2) this result is more complicated and includes all combinations ofthe three couplings α s , α e and α ǫ . This result is new; as described in Sec. 5 it cannotbe obtained using Spiridonov’s method. The numerical uncertainty is less than for allterms. A forthcoming comparison with a prediction of the infrared structure of H → ˜ g ˜ g will confirm that expression (6.3) is exactly correct [48]. ǫ -scalars Now that all renormalization constants are known it is possible to calculate the two-loopform factors of gluons and ǫ -scalars in the fdh and dred scheme. We present the resultsin two ways: First, we give results with independent couplings needed to determine the IRanomalous dimensions of gluons and ǫ -scalars; second, we give simplified results, where allcouplings are set equal. These can be viewed as the final results for the UV renormalizedbut IR regularized form factors. We give them including higher orders in the ǫ -expansion. The UV renormalized but IR divergent form factor for H → ˆ g ˆ g in dred is given at theone-loop and two-loop level by¯ F g ( α s , λ ǫ /λ, N ǫ )= (cid:16) α s π (cid:17)( C A " − ǫ + − + N ǫ ǫ + π λ ǫ λ N ǫ + ǫ (cid:16) − ζ (3) + 3 λ ǫ λ N ǫ (cid:17) + 2 N F ǫ ) + O ( ǫ ) , (7.1)¯ F g ( α s , α e , λ ǫ /λ, N ǫ )= (cid:16) α s π (cid:17) ( C A " ǫ + − N ǫ ǫ + − π − N ǫ (cid:16) λ ǫ λ (cid:17) + N ǫ ǫ + − − π − ζ (3) + N ǫ (cid:16) + π − λ ǫ λ (cid:17) + λ ǫ λ N ǫ ǫ + C A N F " − ǫ + − + N ǫ ǫ + + π + λ ǫ λ N ǫ ǫ + C F N F ǫ + 4 N F ǫ ) − (cid:16) α s π (cid:17)(cid:16) α e π (cid:17) C F N F N ǫ ǫ + O ( ǫ ) . (7.2)As mentioned in the beginning the ˆ g form factor in dred is identical to the gluon formfactor in fdh , and Eq. (7.2) agrees with the result given in Ref. [34].Since there are no external ǫ -scalars in diagrams related to the gluon form factorinternal ǫ -scalars have to be part of a closed ǫ -scalar loop or have to couple to a closed14ermion loop. Hence, the effective coupling λ ǫ always appears together with at least onepower of N ǫ in Eqs. (7.1) and (7.2).The ǫ -scalar form factor for H → ˜ g ˜ g in dred is given by¯ F g ( α s , α e , α ǫ , λ/λ ǫ , N ǫ )= (cid:16) α s π (cid:17) C A " − ǫ − ǫ − π λλ ǫ + ǫ (cid:16) − π
12 + 143 ζ (3) + 4 λλ ǫ (cid:17) + (cid:16) α e π (cid:17) N F ǫ + (cid:16) α ǫ π (cid:17) C A (1 − N ǫ ) " ǫ (cid:16) − π (cid:17) + O ( ǫ ) , (7.3)¯ F g ( α s , α e , α ǫ , λ/λ ǫ , N ǫ )= (cid:16) α s π (cid:17) ( C A " ǫ + − N ǫ ǫ + − π − N ǫ − λλ ǫ ǫ + − π − ζ (3) + N ǫ (cid:16) + π (cid:17) − λλ ǫ ǫ + C A N F " − ǫ − ǫ + + π ǫ + (cid:16) α s π (cid:17)(cid:16) α e π (cid:17)( C A N F " − ǫ − ǫ + − − π + 2 λλ ǫ ǫ + C F N F " − ǫ + 52 ǫ + (cid:16) α s π (cid:17)(cid:16) α ǫ π (cid:17) C A (1 − N ǫ ) " − ǫ + −
16 + π ǫ + (cid:16) α e π (cid:17) ( C A N F " − N ǫ ǫ + − N ǫ ǫ + C F N F " − N ǫ ǫ + − − N ǫ ǫ + N F ǫ ) + (cid:16) α e π (cid:17)(cid:16) α ǫ π (cid:17) C A N F (1 − N ǫ ) 2 ǫ + (cid:16) α ǫ π (cid:17) C A (1 − N ǫ ) − ǫ + O ( ǫ ) . (7.4)Compared to Eqs. (7.1) and (7.2) the result with external ǫ -scalars is more complicated andincludes all combinations of the couplings α s , α e and α ǫ . In this result, like in all previousresults, the evanescent coupling α e appears always together with at least one power of N F and the quartic coupling α ǫ is always accompanied by a factor (1 − N ǫ ). During the renormalization process the couplings α s , α e , α ǫ and λ , λ ǫ have to be distin-guished. After renormalization they can be set equal, giving a simpler form of the final15esult. The results for N ǫ = 2 ǫ at the one(two)-loop level up to order O ( ǫ ) ( O ( ǫ )) thenread:¯ F g = (cid:16) α s π (cid:17)( C A " − ǫ − ǫ + 13 + π ǫ ζ (3) + ǫ π + ǫ (cid:16) ζ (5) − π ζ (3) (cid:17) + ǫ (cid:16) π − ζ (3) (cid:17) + 2 N F ǫ ) + O ( ǫ ) , (7.5)¯ F g = (cid:16) α s π (cid:17) ( C A " ǫ + 776 ǫ + − π ǫ + − − π − ζ (3) ǫ + 5711162 + 179 π − ζ (3) − π + ǫ π − ζ (3) − π + 715 ζ (5) + 2318 π ζ (3) ! + ǫ − π − ζ (3) − π − ζ (5)+ 2571680 π − π ζ (3) + 9019 ζ (3) ! + C A N F " − ǫ − ǫ + + π ǫ − − π − ζ (3)+ ǫ − − π − ζ (3) − π ! + ǫ − π − ζ (3) − π − ζ (5) + 6127 π ζ (3) ! + C F N F " ǫ −
736 + 8 ζ (3) + ǫ − π + 923 ζ (3) + 427 π ! + ǫ − π + 12329 ζ (3) + 4681 π + 32 ζ (5) − π ζ (3) ! + 4 N F ǫ ) + O ( ǫ ) , (7.6) If the results of Sec. 7.1 were not desired for independent couplings, the genuine two-loop diagramscould have been computed in a simpler way, with all couplings set equal from the beginning — this is whatis done in many applications of fdh and dred in the literature. F g = (cid:16) α s π (cid:17)( C A " − ǫ − ǫ + 2 + π ǫ ζ (3) + ǫ π + ǫ (cid:16) ζ (5) − π ζ (3) (cid:17) + ǫ (cid:16) π − ζ (3) (cid:17) + N F ǫ ) + O ( ǫ ) , (7.7)¯ F g = (cid:16) α s π (cid:17) ( C A " ǫ + 272 ǫ + − π ǫ + − − π − ζ (3) ǫ + 605281 + 263108 π − ζ (3) − π + ǫ π − ζ (3) − π + 715 ζ (5) + 2318 π ζ (3) ! + ǫ π − ζ (3) − π − ζ (5)+ 2571680 π − π ζ (3) + 9019 ζ (3) ! + C A N F " − ǫ − ǫ + 15127 ǫ − − π − ζ (3)+ ǫ − − π − ζ (3) − π ! + ǫ − π − ζ (3) − π − ζ (5) + 20354 π ζ (3) ! + C F N F " − ǫ + 12 ǫ − − π ζ (3)+ ǫ − − π + 1963 ζ (3) + 29 π ! + ǫ − − π + 8683 ζ (3) + 6760 π + 48 ζ (5) − π ζ (3) ! + N F ǫ ) + O ( ǫ ) . (7.8)17 Conclusions
We have computed the H → gg amplitudes at the two-loop level in the fdh and dred scheme and presented the MS renormalized on-shell results up to the order ǫ . In dred ,this involves two different amplitudes for H → ˆ g ˆ g and H → ˜ g ˜ g with external gluons/ ǫ -scalars. The computation is motivated because it contains key elements which constituteimportant building blocks for further computations, and because it is essential for thecomplete understanding of the infrared divergence structure of fdh and dred amplitudes.The renormalization procedure has been described in detail. It is less trivial than inmany QCD calculations in cdr , since not only the strong coupling needs to be renormalizedbut also evanescent couplings of the ǫ -scalar. The computation provides a further exampleof the well-known fact that regardless of whether fdh or dred is used, the evanescentcouplings have to be renormalized independently.Further, the renormalization of the effective dimension-5 operators involves mixingwith new, ǫ -scalar dependent operators. A suitable basis of operators has been provided.One unavoidable fact is that the extended operator space contains operators which are totalderivatives. As a result the required operator mixing renormalization constants cannot beobtained in the same elegant way of Ref. [6] as in cdr . Instead, they had to be obtainedfrom explicit one- and two-loop off-shell calculations.The results for the UV renormalized but infrared divergent form factors can also beused to complete the study of the general infrared divergence structure of two-loop ampli-tudes in fdh and dred , begun in Ref. [33, 34]. From general principles it is known that allinfrared divergences can be expressed in terms of cusp and parton anomalous dimensions.The results of the present paper allow to extract the final missing two-loop anomalousdimension for external ǫ -scalars. This extraction, together with further checks and results,will be presented in a forthcoming paper [48], where the infrared structure will also beinvestigated by a SCET approach. Acknowledgments
We are grateful to M. Steinhauser and W. Kilgore for useful discussions. We acknowledgefinancial support from the DFG grant STO/876/3-1. A. Visconti is supported by the SwissNational Science Foundation (SNF) under contract 200021-144252.
A Appendix
A.1 Projectors and form factors of gluons and ǫ -scalars According to its Lorentz structure the on-shell Green-function Γ on-shell H ˆ A µ ˆ A ν can be representedas Γ on-shell H ˆ A µ ˆ A ν = a ( p · r ) ˆ g µν + b p ν r µ + c p µ r ν + d p µ p ν + e r µ r ν , (A.1)where the coefficients a . . . e are momentum-dependent quantities, and coefficient a is thegluon form factor. Due to QCD Ward-identities the relation a = − b holds, see e. g. Ref. [1].18ccordingly, the on-shell Green-function Γ on-shell H ˜ A µ ˜ A ν with external ǫ -scalars can be representedas Γ on-shell H ˜ A µ ˜ A ν = f ( p · r ) ˜ g µν , (A.2)where we refer to f as ǫ -scalar form factor. All coefficients of the covariant decompositioncan be extracted with appropriate projection operators that are given below.In the off-shell case the UV divergence structure of Γ H ˆ A µ ˆ A ν can be represented in amore specific way asΓ H ˆ A µ ˆ A ν (cid:12)(cid:12)(cid:12) off-shellUV div. = (cid:20) A + A ′ p + r ( p · r ) (cid:21) ( p · r ) ˆ g µν + B p ν r µ + C p µ r ν + D p µ p ν + E r µ r ν , (A.3)where the coefficients A . . . E are now momentum-independent. Since these divergences canbe absorbed by counterterms corresponding to operators O and O the relation A = − B again holds, see e. g. Feynman rules (A.6) and (A.12). Due to this there are two possibilitiesof extracting coefficient A , which corresponds to the desired renormalization constant δZ λ :The first one is to extract the coefficient of ( p · r ) ˆ g µν and neglect terms ∝ p , r ; the secondis to extract coefficient − B . We checked explicitly that the relations a = − b and A = − B hold throughout the paper.Again, the covariant decomposition with external ǫ -scalars is much simpler and reads:Γ H ˜ A µ ˜ A ν (cid:12)(cid:12)(cid:12) off-shellUV div. = (cid:20) F + F ′ p + r ( p · r ) (cid:21) ( p · r ) ˜ g µν . (A.4)The desired coefficient for the computation of δZ λ ǫ is F . Accordingly, we extract thecoefficient of ( p · r ) ˜ g µν and neglect terms ∝ p , r .The corresponding projection operators are: P µνg, ( p · r )ˆ g µν = n ˆ g µν (cid:2) ( p · r ) − p r (cid:3) − ( p ν r µ + p µ r ν )( p · r )+ p µ p ν r + r µ r ν p o D − p · r ) [( p · r ) − p r ] , (A.5a) P µνg,p ν r µ = n ˆ g µν ( p · r ) (cid:2) p r − ( p · r ) (cid:3) + p ν r µ (cid:2) ( p · r ) + p r ( D − (cid:3) + p µ r ν ( p · r ) ( D − p µ p ν r + r µ r ν p )( p · r )(1 − D ) o D −
2) [( p · r ) − p r ] , (A.5b) P µν ˜ g, ( p · r )˜ g µν = ˜ g µν N ǫ ( p · r ) . (A.5c) A.2 Feynman rules
In the following we give Feynman rules according to operators O , ˜ O , O and ˜ O that areneeded for the renormalization in the fdh and dred scheme. Feynman rules including four ǫ -scalars are not relevant in this paper and are not given explicitly.19 Feynman rules according to the Lagrangian term λHO : k k O H ˆ A αa ˆ A βb = iλ h ( k · k ) ˆ g αβ − k β k α i δ ab (A.6) k k k O H ˆ A αa ˆ A βb ˆ A γc = − λ g s f abc × ˆ g αβ ( k − k ) γ + ˆ g βγ ( k − k ) α + ˆ g γα ( k − k ) β (A.7) O H ˆ A αa ˆ A βb ˆ A γc ˆ A δd = − iλ g s × ˆ g αβ ˆ g γδ (cid:16) f ace f bde + f ade f bce (cid:17) + ˆ g αγ ˆ g βδ (cid:16) f abe f cde − f ade f bce (cid:17) − ˆ g αδ ˆ g βγ (cid:16) f abe f cde + f ace f bde (cid:17) (A.8) • Feynman rules according to the Lagrangian term λ ǫ H ˜ O k k ˜ O H ˜ A αa ˜ A βb = iλ ǫ h ( k · k ) ˜ g αβ i δ ab (A.9) k k k ˜ O H ˜ A αa ˜ A βb ˆ A γc = − λ ǫ g s f abc ˜ g αβ ( k − k ) γ (A.10) ˜ O H ˜ A αa ˜ A βb ˆ A γc ˆ A δd = − iλ ǫ g s ˜ g αβ ˆ g γδ (cid:16) f ace f bde + f ade f bce (cid:17) (A.11)20 Feynman rules according to the Lagrangian term HO : k k O H ˆ A αa ˆ A βb = − i h (cid:0) k + k (cid:1) ˆ g αβ − (cid:16) k α k β + k α k β (cid:17) i δ ab (A.12) k k k O H ˆ A αa ˆ A βb ˆ A γc = 3 g s f abc × ˆ g αβ ( k − k ) γ + ˆ g βγ ( k − k ) α + ˆ g γα ( k − k ) β (A.13) k k O H ˜ A αa ˜ A βb ˆ A γc = − g s f abc ˜ g αβ ( k − k ) γ (A.14) O H ˜ A αa q j q i = − ig s ˆ γ α ( T a ) ij (A.15) k k O H c a c b = i ( k · k ) δ ab (A.16) • Feynman rules according to the Lagrangian term H ˜ O : k k ˜ O H ˆ A αa ˆ A βb = − i (cid:0) k + k (cid:1) ˜ g αβ δ ab (A.17)21 k ˜ O H ˜ A αa ˜ A βb ˆ A γc = − g s f abc ˜ g αβ ( k − k ) γ (A.18) References [1] R. V. Harlander,
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