Decoupling Relations to O(α 3 s) and their Connection to Low-Energy Theorems

Transcription

1 MPI/PhT/ hep ph/ July 1997 Decoupling Relations to Oα 3 s and their Connection to Low-Energy Theorems K.G. Chetyrkin, B.A. Kniehl, and M. Steinhauser Max-Planck-Institut für Physik Werner-Heisenberg-Institut, Föhringer Ring 6, Munich, Germany Abstract If quantum chromodynamics QCD is renormalized by minimal subtraction MS, at higher orders, the strong coupling constant α s and the quark masses m q exhibit discontinuities at the flavour thresholds, which are controlled by so-called decoupling constants, ζ g and ζ m, respectively. Adopting the modified MS MS scheme, we derive simple formulae which reduce the calculation of ζ g and ζ m to the solution of vacuum integrals. This allows us to evaluate ζ g and ζ m through three loops. We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m h of ζ g and ζ m, respectively. Fully exploiting present knowledge of the anomalous dimensions of α s and m q, we thus calculate these effective couplings through four loops. Finally, we perform a similar analysis for the coupling of the Higgs boson to photons. PACS numbers: Me, 1.38.Bx, Bn Permanent address: Institute for Nuclear Research, Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 11731, Russia.

2 1 Introduction As is well known, in renormalization schemes based on the method of minimal subtraction MS [1], including the modified MS MS scheme [], which is routinely used in quantum chromodynamics QCD, the Appelquist-Carazzone decoupling theorem [3] does not hold true in its naïve sense. Let us consider QCD with n l = n f 1 light quarks flavours q and one heavy flavour h. Then, the contributions of h to the Green functions of the gluons and light quarks expressed in terms of the renormalized parameters of the full theory do not exhibit the expected 1/m h suppression, where m h is the mass of h. The reason for this is that the β and γ m functions governing the running of the strong coupling constant α s µ and the light-quark masses m q µ with the renormalization scale µ do not depend on any mass. The standard procedure to circumvent this problem is to render decoupling explicit by using the language of effective field theory, i.e. h is integrated out. Specifically, one constructs an effective n l -flavour theory by requiring consistency with the full n f -flavour theory at the heavy-quark threshold µ n f = Om h [4,5]. This leads to nontrivial matching conditions between the couplings and light-quark masses of the two theories. Although, α n l s m h =α n f s m h andm n l q m h =m n f q m h at leading tree-level and next-toleading one-loop order, these identities do not generally hold at higher orders in the MS scheme. Starting at next-to-next-to-leading two-loop order, they are broken by finite corrections, of Oαs, as was noticed in the pioneering works of Refs. [4,5]. The relations between the couplings and light-quark masses of the full and effective theories are called decoupling relations; the proportionality constants that appear in these relations are denoted decoupling constants, ζ g and ζ m. In this paper, ζ g and ζ m are computed through next-to-next-to-next-to-leading three-loop order, Oαs 3. They have to be applied whenever a flavour threshold is to be crossed. If the µ evolutions of α n f s µ andm n f q µare to be performed at N loops, then consistency requires that the decoupling relations be implemented at N 1 loops. Then, the residual µ dependence of physical observables will be of the N + 1-loop order. If our new results are combined with the recently evaluated four-loop coefficients of the β [6] and γ m [7] functions, then it is possible to consistently describe QCD-related observables through Oαs[8]. 4 The dominant subprocess for the production of the standard-model SM Higgs boson at the CERN Large Hadron Collider LHC will be the one via gluon fusion. Therefore, an important ingredient for the Higgs-boson search will be the effective coupling of the Higgs boson to gluons, usually called C 1. At the two-loop level, C 1 has been known since long [9,10]. Recently, the three-loop correction to C 1 has been obtained through a direct diagrammatic calculation [11]. In Ref. [11], also a low-energy theorem LET which allows for the computation of C 1 from the knowledge of ζ g has been used. In a similar way, the effective couplings of the Higgs boson to light quarks may be treated as well. The corresponding coefficient function, which is usually denoted by C, has a similar connection to ζ m. In this paper, we establish the relationships between C 1,C and ζ g,ζ m to all orders by formulating appropriate LET s. With their help, we compute C 1 and C through four loops, i.e. Oαs 4.

3 The outline of the paper is as follows. In Section, we derive simple formulae which allow us to determine the decoupling constants ζ g and ζ m for α s µandm q µ, respectively, by simply evaluating vacuum integrals. With the help of these formulae, in Section 3, we calculate ζ g and ζ m up to the three-loop order. In Section 4, we derive LET s, valid to all orders, for the coefficient functions C 1 and C, which comprise the virtual top-quark effects on the interactions of the Higgs boson with gluons and quarks, respectively. Exploiting theselet s,wecomputec 1 and C through four loops. In Section 5, we extend our analysis to also include the interaction of the Higgs boson with photons. In Section 6, we explore the phenomenological significance of our results and present our conclusions. In Appendix A, we display the N c dependence of our key results, adopting the general gauge group SUN c. In Appendix B, we list the decoupling constants for the quark and gluon fields in the covariant gauge. Formalism Let us, in a first step, fix the notation and present the framework for our calculation. Throughout this paper, we work in the MS renormalization scheme []. In the main part of the paper, we concentrate on the QCD gauge group SU3, i.e. we put N c =3. Theβ and γ m functions of QCD are defined through 1 m n f q µ d α nf dµ s = βn f α n f s = µ d dµ mn f q = γ n f m β n f N 1 N=1 α n f s = γ n f N=1 m,n 1 αn f s αn f s N+1, 1 N, where N denotes the number of loops. Recently, the four-loop coefficients β n f 3 and γ n f m,3 have become available [6,7]. According to present knowledge, we have β n f 0 = 1 [ 4 β n f 1 = 1 [ 16 β n f = 1 64 β n f 3 = n f ], ] 3 n f, [ n f n f [ ζ γ n f m,0 =1, γ n f m,1 = 1 16 ], ζ3 n f n3 f [ ] 9 n f, 3 ], ζ3 n f

4 γ n f m, = 1 [ 64 γ n f m,3 = ζ3 n f 140 [ ζ3 8800ζ n f ζ n f ], ζ ζ4 n f ζ ζ ζ3 n 3 f where ζ is Riemann s zeta function, with values ζ = /6, ζ , ζ4 = 4 /90, and ζ The relations between the bare and renormalized quantities read gs 0 = ε Z g g s, m 0 q = Z mm q, ξ 0 1=Z 3 ξ 1, ψq 0 = Z ψ q, G 0,a µ Z = 3 G a µ, c0,a = Z3 c a, 4 where g s = 4α s is the QCD gauge coupling, µ is the renormalization scale, D =4 ε is the dimensionality of space time, ψ q is a quark field with mass m q, G a µ is the gluon field, and c a is the Faddeev-Popov-ghost field. For simplicity, we do not display the colour indices of the quark fields. The gauge parameter, ξ, is defined through the gluon propagator in lowest order, i g µν + ξ qµ q ν. 5 q + iɛ q The index 0 marks bare quantities. We should mention that relations 4 hold true both in the full n f -flavour and effective n l -flavour theories. Only the renormalization constants Z g and Z m are relevant for our purposes. They are known through order Oαs 4 [6,7]. The other relations are only listed in order to fix the notation. In a similar way, we can write down the relations between the parameters and fields of the full and effective theories. From the technical point of view, it is simpler to first consider these relations for bare quantities and to construct those for the renormalized ones afterwards. The bare decoupling constants are defined through the following equations: gs 0 = ζgg 0 s, 0 m 0 q = ζmm 0 0 q, ξ 0 1=ζ3ξ 0 0 1, ψq 0 = ζψ 0 q, 0 Gµ 0,a = ζ3g 0 0,a µ, c 0,a = ζ0 3c 0,a, 6 where the primes mark the quantities of the effective n l -flavour theory. Combining Eqs. 4 and 6, we obtain relations between the renormalized coupling constants and masses of the full and effective theories, viz α sµ= Zg ζ 0 Z g g α s µ=ζgα s µ, 7 m qµ= Z m ζ Z mm 0 m q µ=ζ m m q µ. 8 ], 3 4

5 Here, it is understood that the right-hand sides are expressed in terms of the parameters of the full theory. The unknown quantities in these decoupling relations are ζg 0 and ζ m 0. In a next step, let us introduce the effective Lagrangian, L, which depends on the decoupling constants ζi 0 and on the bare parameters and light fields of the full theory. It is obvious from gauge invariance that the most general form of L is simply the one of usual QCD, L QCD, retaining, however, only the light degrees of freedom. Specifically, the definition of L reads L gs,m 0 0 q,ξ 0 ;ψq,g 0 0,a µ,c 0,a ; ζ i 0 = L QCD gs 0,m 0 q,ξ 0 ;ψq 0,Gµ 0,a,c 0,a, 9 where q represents the n l light quark flavours and ζi 0 collectively denotes all bare decoupling constants of Eq. 6. Exploiting the circumstance that the result for some n-particle Green function of light fields obtained from L QCD in the full theory agrees, up to terms suppressed by inverse powers of the heavy-quark mass, with the corresponding evaluation from L in the effective theory, we may derive relations which allow us to determine the decoupling constants ζi 0. As an example, let us consider the massless-quark propagator. Up to terms of O1/m h, we have 1 p [1 + Σ 0 V p ] = i dx e ip x Tψq 0 x ψ q 0 0 = i dx e ip x Tψ 0 0 ζ 0 q x ψ q 0 = 1 1 ζ 0 p [1 + Σ 0 V p ], 10 where the subscript V reminds us that the QCD self-energy of a massless quark only consists of a vector part. Note that Σ 0 V p only contains light degrees of freedom, whereas Σ 0 V p also receives virtual contributions from the heavy quark h. As we are interested in the limit m h, we may nullify the external momentum p, which entails an enormous technical simplification because then only tadpole integrals have to be considered [1]. Within dimensional regularization, one also has Σ 0 V 0 = 0. Thus, we obtain ζ 0 =1+Σ0h V 0, 11 where the superscript h indicates that only the hard part of the respective quantities needs to be computed, i.e. only the diagrams involving the heavy quark contribute. In a similar fashion, it is possible to derive formulae for the decoupling constants ζ 0 m, ζ 0 3,and ζ 0 3 as well. These read ζm 0 = 1 Σ0h S 0 1+Σ 0h V 0, 1 ζ3 0 =1+Π 0h G0, 13 ζ 3 0 =1+Π 0h c 0, 14 5

6 where Σ V p andσ S p are the vector and scalar components of the light-quark selfenergy, defined through Σp = pσ V p +m q Σ S p, and Π G p andπ c p are the gluon and ghost vacuum polarizations, respectively. Specifically, Π G p andπ c p are related to the gluon and ghost propagators through { δ ab g µν } p [1 + Π 0 G p ] + terms proportional to pµ p ν = i dx e ip x TG 0,aµ xg 0,bν 0, δ ab p [1 + Π 0 cp ] = i dx e ip x Tc 0,a x c 0,b 0, 15 respectively. Finally, we extract an expression for ζg 0 from the relationship between the bare G cc-vertex form factors of the full and effective theories, [ gs 0 1+Γ 0 G cc p, k] = g0 s ζ 3 0 ζ3 0 [ 1+Γ 0 G cc p, k ], 16 by nullifying the external four-momenta p and k. Here, Γ 0 G ccp, k is defined through the one-particle-irreducible 1PI part of the amputated G cc Green function as { p µ g [ s 0 if abc 1+Γ 0 G cc p, k] + other colour structures } = i dxdy e ip x+k y Tc 0,a x c 0,b 0G 0,cµ y 1PI, 17 where p and k are the outgoing four-momenta of c and G, respectively, and f abc are the structure constants of the QCD gauge group. We thus obtain ζg 0 = ζ 1 0, 18 ζ 3 0 ζ3 0 where ζ 1 0 =1+Γ0h G cc 0, In contrast to the renormalization constants Z i in Eq. 4, the decoupling constants ζi 0 also receive contributions from the finite parts of the loop integrals. Thus, at Oαs, 3 we are led to evaluate three-loop tadpole integrals also retaining their finite parts. 3 Decoupling relations In this section, we compute the bare decoupling constants ζg 0 and ζm 0 and combine them with the well-known Oαs 3 results for Z g and Z m to obtain the finite quantities ζ g and ζ m. As may be seen from Eq. 1, the computation of ζm 0 requires the knowledge of the hard part of the light-quark propagator. There are no one-loop diagrams contributing to Σ 0h V 0 and Σ 0h S 0. At two loops, there is just one diagram. Even at three loops, there is only a moderate number of diagrams, namely 5. Typical specimen are depicted in 6

7 Fig. 1. Actually, through three loops, Eq. 1 simplifies to ζ 0 m =1 Σ 0h V 0 Σ 0h S 0. It should be noted that the vector and scalar parts separately still depend on the QCD gauge parameter ξ, but ξ drops out in their sum, which is a useful check for our calculation. We use the program QGRAF [13] to generate the relevant diagrams. The computation is then performed with the help of the program MATAD [14], which is based on the technology developed in Ref. [15] and written in FORM [16]. Using Eq. 8, we finally obtain ζ m =1+ αn f s µ ζ B n l αn f s µ h + 1 µ m h 1 + m h ζ3 m h 53 ζ m h n l s µ 3 [ ζ3 αn f µ 43 m h µ ] m h αn f s µ h µ m h 3, 0 where [15] 1 B 4 =16Li 4 13 ζ4 4ζ , 1 with Li 4 being the quadrilogarithm, is a constant typical for three-loop tadpole diagrams, n l = n f 1 is the number of light-quark flavours q, andm h =m n f h µisthems mass of the heavy quark h. For the numerical evaluation in the last line of Eq. 0, we have chosen µ = µ h,whereµ h =m n f h µ h. The Oαs term of Eq. 0 agrees with Ref. [5]; the Oαs 3 term represents a new result. The generalization of Eq. 0 appropriate for the gauge group SUN c is listed in Appendix A. For the convenience of those readers who prefer to fix the matching scale µ in units of the pole mass, M h, we substitute in Eq. 0 the well-known relation between m n f h µ and M h [17] to obtain ζ OS m =1+ αn f s µ Mh ζ B ζ3 Mh n l ζ αn f s M h 43 M h + Mh µ n l 7 s µ 3 [ ζ3 αn f µ Mh ] Mh αn f s M h µ M h 3.

8 Figure 1: Typical three-loop diagrams pertinent to Σ 0h V 0 and Σ 0h S 0. Solid, bold-faced, and loopy lines represent massless quarks q, heavy quarks h, and gluons G, respectively. According to Eq. 18, three ingredients enter the calculation of ζg 0, namely the hard parts of the gluon and ghost propagators and the gluon-ghost vertex correction. At one loop, only one diagram contributes, namely the diagram where the gluon splits into a virtual pair of heavy quarks. one to Π 0h c 0. In the case of Γ 0h At two loops, three diagrams contribute to Π 0h G 0 and G cc0, 0, there are five diagrams, which, however, add up to zero. To this order, the three contributions are still separately independent of the gauge parameter ξ, so that the ξ independence of their combination does not provide a meaningful check for our calculation. The situation changes at Oαs, 3 where all three parts separately depend on ξ and only their proper combination is ξ independent as is required for a physical quantity. At this order, the numbers of diagrams pertinent to Π 0h G 0, Π 0h c 0, and Γ 0h G cc0, 0 are 189, 5, and 8, respectively. Typical representatives are shown in Fig.. The complexity of the problem at hand necessitates the use of powerful analytic technology [13 16] to organize the calculation. Inserting into Eq. 7 the result for ζg 0 thus obtained finally leads to the following answer in the MS scheme: + m h f ζg =1+αn s µ αn f 3 [ s µ n l s µ µ 4 m h 36 m h αn f ζ ] 1 36 µ m h + 53 µ m h µ m h 16 3 m h m h αn f s µ h n l αn f 3 s µ h. 3 The Oα 3 s term in Eq. 3 has recently been published [8]. Leaving aside this term, Eq. 3 agrees with Ref. [18], while the constant term in Oα s slightly differs from the result published in Ref. [5]. In the meantime, the authors of Ref. [5] have revised [19] their original analysis and have found agreement with Ref. [18]. The SUN c version of 8

9 Figure : Typical three-loop diagrams pertinent to Π 0h G 0, Π0h c 0, and Γ0h G cc 0, 0. Boldfaced, loopy, and dashed lines represent heavy quarks h, gluonsg, and Faddeev-Popov ghosts c, respectively. Eq. 3 may be found in Appendix A. Introducing the pole mass M h leads to ζ OS α n f s µ g = αn f 3 [ s µ Mh αn f + Mh s µ αn f ζ Mh s M h Mh n l ζ n l ζ ] Mh αn f 3 s M h Mh M h. 4 Notice that the Oα 3 stermsofζ m and ζ g depend on the number n l of light massless quark flavours. However, this dependence is feeble. In the framework of the QCD-improved parton model, quarks and gluons appear as external particles, so that the knowledge of the decoupling constants ζ and ζ 3,which emerge as by-products of our analysis, is actually of practical interest for higher-order calculations. In contrast to ζ m and ζ g, ζ and ζ 3 are ξ dependent. For future applications, we list ζ and ζ 3 in Appendix B. 4 Low-energy theorems for the ggh and q qh interactions The interactions of the SM Higgs boson with gluons and light quarks are greatly affected by the virtual presence of the top quark. In fact, the Higgs-boson coupling to gluons is essentially generated by a top-quark loop alone. In general, the theoretical description of such interactions is very complicated because there are two different mass scales involved, 9

10 M H and M t. However, in the limit M H M t, the situation may be greatly simplified by integrating out the top quark, i.e. by constructing a heavy-top-quark effective Lagrangian. The starting point of our consideration is the bare Yukawa Lagrangian of the full theory, L = H0 n f m 0 v 0 q ψ0 i qi ψq 0 i, 5 which governs the interactions of the neutral CP-even Higgs boson H with all n f quark flavours, including the heavy one. Here, v is the Higgs vacuum expectation value. The heavy-quark effective Lagrangian describing the interactions of H with the gluon G and the n l light-quark flavours may be written in bare form as L eff = H0 5 C v 0 i 0 O i. 6 The operators O i are only constructed from light degrees of freedom and read [9,0,1] O 1 = G 0,a µν, n O = l m 0 q ψ0 i q i ψq 0 i, n O 3 = l ψ q 0 i i D 0 m 0 q i ψ 0 q i, n O 4 = l G0,a ν ab µ G0,bµν + gs 0 ψ 0 λ a q i γν ψq 0 i µ c 0,a µ c 0,a, O 5 = ab µ µ c 0,b c 0,a, 7 where G a µν is the colour field strength, D µ = µ iµ ε g s λ a /G a µ and ab µ = δ ab µ iµ ε g s f abc G c µ are the covariant derivatives acting on the quark and gluon/ghost fields, respectively, and λ a are the Gell-Mann matrices. The residual dependence on the mass m h of the heavy quark h is contained in the coefficient functions Ci 0. In phenomenological applications, one is mainly interested in the renormalized coefficient functions C 1 and C, since only these contribute to physical observables. The renormalization of Ci 0 and O i i =1, has been explained in Ref. []. For the reader s convenience, we repeat here the key results. Denoting the renormalized operators by square brackets, we have [0,1] [ ] α [O 1]= 1+ s α Z g O 1 4 s Z m O, α s α s [O ]=O. 8 Note that O 3, O 4,andO 5 do not mix with O 1 and O. On the other hand, the coefficient functions are renormalized according to 1 C 1 = C 0 1+α s / α sz g 1, 9 10

11 C = Consequently, the physical part of L eff takes the form L phys eff 4α s / α s Z m C 0 1+α s / α sz g 1 +C = H0 v C 0 1 [O 1 ]+C [O ]. 31 C i and [O i ], are individually finite, but, with the exception of [O ], they are not separately renormalization-group RG invariant. In Ref. [], a RG-improved version of Eq. 31 has been constructed by exploiting the RG-invariance of the trace of the energymomentum tensor. The ratio H 0 /v 0 receives a finite renormalization factor, which is of OG F Mt. Its two- and three-loop QCD corrections have been found in Refs. [3] and [4], respectively. The derivation of formulae to compute the coefficient functions is very similar to the case of the decoupling relations. As an example, let us consider the derivation of a formula involving C1 0 and C0 4. The starting point is the 1PI Green function of two gluons which contains a zero-momentum insertion of the composite operator O h = m 0 h h 0 h 0. In momentum space, it reads in bare form δ ab Γ 0,µν GGO h p=i dxdy e ip x y TG 0,aµ xg 0,bν yo h 0 1PI = δ ab [ g µν p Γ 0 GGO h p + terms proportional to p µ p ν], 3 where p denotes the four-momentum flowing along the gluon line. In the limit m h, O h may be written as a linear combination of the effective operators given in Eq. 7, so that Γ 0,µν GGO h p= i dxdy e ip x y TG 0,aµ xg 0,aν y C1 0 8 O 1 + 1PI C0 4 4 O +... = i 8 ζ0 3 dxdy e ip x y TG 0,aµ xg 0,aν y C1 0 O 1 + 1PI C0 4 4 O +... = g µν p ζ3 0 4C0 1 +C higher orders +..., 33 where the ellipses indicate terms of O1/m h and terms proportional to p µ p ν. In the second step, we have used Eq. 6 together with the fact that Γ 0,µν GGO h p represents an amputated Green function. If we consider the coefficients of the transversal part in the limit p 0, we observe that the contributions due to the higher-order corrections on the r.h.s. of Eq. 33 vanish, as massless tadpoles are set to zero in dimensional regularization. The contribution due to the other operators vanish for the same reason. On the l.h.s., only those diagrams survive which contain at least one heavy-quark line. Consequently, the hard part of the amputated Green function is given by Γ 0h GGO h 0 = ζ3 0 4C 0 1 +C

12 It is convenient to generate the diagrams contributing to the l.h.s. of Eq. 34 by differentiating the gluon propagator w.r.t. the heavy-quark mass, so that we finally arrive at ζ 0 3 4C 0 1 +C 0 4= 1 0 hπ 0h G0, 35 where 0 h =[m0 h / m 0 h ]. The last equation results from the fact that the Yukawa coupling of the Higgs boson to the heavy quark is proportional to m 0 h. In a similar way, we obtain four more relationships, namely ζ 0 C 0 3 = 1 0 hσ 0h V 0, ζ 0 mζ 0 C 0 C 0 3=1 Σ 0h S hσ 0h S 0, ζ 0 3 C0 4 + C0 5 =1 0 h Π0h c 0, ζ 0 1C 0 5 = 1 0 hγ 0h G cc0, We may now solve Eqs. 35 and 36 for the coefficient functions C 0 i i =1,...,5. It is tempting to insert Eqs into Eqs. 35 and 36, so as to express the bare coefficient functions in terms of derivatives of the decoupling constants w.r.t. m 0 h. Solving the three equations involving C 0 1, C 0 4,andC 0 5 for C 0 1,weobtain C 0 1 = 0 hζ 0 g. 37 Next, we express ζ 0 g through renormalized quantities. Using 0 h = h, we find C 0 1 = h ζ 0 g = h α0 s αs 0 = h Z g + h α s [ = 1+ α s ] Z α s g h α s. 38 Identifying the renormalization factor of Eq. 9, we obtain the amazingly simple relation C 1 = h α s = h ζ g. 39 This relation opens the possibility to compute C 1 through Oαs, 4 since one only needs to know the logarithmic pieces of ζ g in this order, which may be reconstructed from its lower-order terms in combination with the four-loop β [6] and γ m [7] functions. It is possible to directly relate C 1 to the β and γ m functions of the full and effective theories. 1

13 Exploiting the relation β α s= dα s dµ = where α s = α sµ, α s,m h, we find C 1 = [ µ µ + βα s α s + γ m α s m h m h ] α s, 40 [ ] β α α s s [1 γ m α s ] βα s α s. 41 α s InthecaseofC, we may proceed along the same lines to obtain C =1+ h ζ m [ =1 γ m 1 γ m α s α s γ mα s βα s 1 m q m ] q, 4 α s where m q = m qµ, α s,m h. It should be stressed that Eqs. 41 and 4 are valid to all orders in α s. Fully exploiting present knowledge of the β [6] and γ m [7] functions, we may evaluate C 1 and C through Oαs 4 via Eqs. 41 and 4. In the pure MS scheme, we find C 1 = 1 α n f s µ {1+ αn f s µ 1 + αn f [ s µ 81 3[ + αn f s µ m h m + n h 36 m l 67 h m h 859 ζ3 88 m h ] µ µ m h 16 3 m h n l ζ m h 88 m h + n l ]} 178 m h 18 m h 1 α n f s µ h [ αn f s µ h n l αn f s µ h n l 0.07 n αn f 3 ] s µ h l, 43 C =1+ αn f s µ m h 13

14 + αn f 3 [ s µ ζ3 9 µ m h 36 + n m l h µ ] 18 m h 4[ + αn f s µ ζ3 ζ4 ζ B ζ3 119 m 301 h 16 m 3 h 144 m h + n l ζ ζ B m + 5 h 48 m 3 h 18 m h n l ζ3 1 µ ] m h m h αn f s µ h n l αn f 3 s µ h n l n αn f 4 s µ h l, 44 where, for simplicity, we have chosen µ = µ h in the approximate expressions. The Oα 3 s term of Eq. 44 may also be found in Ref. [5]. The corresponding expressions written with the pole mass M h read C1 OS = 1 1 α n f s µ {1+ αn f + αn f [ s µ 693 3[ + αn f s µ Mh n l s µ Mh Mh µ M h + n l ] 3 Mh ζ ζ µ 144 Mh ζ µ 16 3 Mh 109 ζ µ 96 Mh Mh + n l ]} 178 Mh 18 Mh 1 α n f s M h [ αn f s M h n l αn f s M h n l 0.07 n αn f 3 ] s M h l, 45 14

15 C OS =1+ αn f s µ Mh + αn f 3 [ s µ n l ] 18 Mh 4[ + αn f s µ ζ B ζ3 Mh + n l ζ3 108 Mh ζ ζ µ µ 144 Mh 31 ζ3 59 Mh n l αn f s M h 34 M h ζ4 1 M h 9 36 Mh M h 36 B Mh n l n l n αn f l s M h ] αn f ζ3 ζ ζ Mh s 3 M h 4, 46 wherewehaveputµ=m h in the numerical evaluations. The Oαs 3 term of Eq. 45 may also be found in Ref. [11], where the hadronic decay width of the SM Higgs boson has been calculated through Oαs 4. 5 Low-energy theorem for the γγh interaction In this section, we extend the formalism developed in Sections and 4 to include the Higgs-boson interactions with photons. From the technical point of view, this means that we are now concerned with QCD corrections to Green functions where the external particles are photons instead of gluons. In the following, this is indicated by an additional subscript γ. Furthermore, we do not need to consider Green functions involving external ghost lines any more. The Abelian versions of Eqs. 4, 6, and 7 emerge via the substitutions g ē = 4ᾱ, G a µ A µ, G a µν F µν, λ a / 1, and f abc 0, where ē is the gauge coupling of quantum electrodynamics QED, ᾱ is the fine-structure constant, A µ is the photon field, and F µν is the electromagnetic field strength. We continue to work in the MS renormalization scheme. The system of composite operators 7 is reduced 15

16 to the Abelian counterparts of the gauge-invariant operators O 1, O,andO 3. The other formulae derived in Sections and 4 simplify accordingly. Similarly to Eq. 7, the decoupling relation for the renormalized MS fine-structure constant reads Using ᾱ µ = Zgγ ζ 0 Z gγ gγ ᾱµ =ζgγᾱµ. 47 Z gγ = 1 Z3γ, ζ 0 gγ = 1 ζ 0 3γ, 48 we thus obtain ζgγ = Z 3γ. 49 Z 3γ ζ3γ 0 To three loops in QCD, the photon wave-function-renormalization constant Z 3γ may be extracted from Ref. [6], while the bare decoupling constant ζgγ 0 for ᾱµ is determined by the hard part of the three-loop photon propagator, which may be found in Refs. [7,8]. Putting everything together, we find in the pure MS scheme ζgγ =1+ᾱnf { µ Q h + αn f s µ Q m h 13 + h 1 m h [ + αn f s µ Q h ζ m h 4 m h + n l m h 1 m h n l + Q q i µ ]} 7 m h 1, 50 m h where Q q is the fractional electric charge of quark flavour q. Introducing the pole mass M h, this becomes ζ OS gγ ᾱ nf { µ =1+ Q h + αn f [ s µ Q h n l + M h M h 31 4 µ M h Q q i Mh s µ + αn f Q h ζ n l ζ ]} 1 Mh M h ζ3 M h + 1 µ 1 M h

17 Next, we turn to the coefficient function C 1γ of the heavy-quark effective γγh coupling. Equation 41 undergoes obvious modifications to become C 1γ = 1 h ζ gγ = ᾱ [1 γ m α s ] [ ] β γᾱ,α s β γ ᾱ, α s ᾱ ᾱ βα s ᾱ, 5 α s where β γ is the β function governing the running of the MS fine-structure constant ᾱµ. An expression for β γ may be extracted from the well-known QCD corrections to the photon propagator [9]. Restricting ourselves to the leading order in ᾱ, we find µ d ᾱ nf dµ = β n f γ ᾱn f,α n f s where N denotes the number of loops and β n f 0γ = β n f 1γ = β n f γ = 1 64 β n f 3γ = n f Q q i, n f 500 Q q i n f { nf [ Q q i 7 nf 1760 Q qi, ᾱn f = β n f N 1,γ N=1 αn f s N 1, ζ3 + n f ζ3 13 ] 43 n f ζ3 }. 54 We are now in the position to evaluate C 1γ up to the four-loop level. In the pure MS scheme, we find C 1γ = 1 ᾱ n f µ {Q h αn f s µ + n l n l + 6 m h 3[ + αn f s µ Q h Q h + αn f [ s µ Q h Q q i ] 6 m h ζ n l ζ m h 36 m h + n l µ 54 m h 36 m h m h m h m h 17

18 n l 53 + Q q i ζ nf nl + Q qi Q qi + 9 m + n h 7 m l 449 h m h ζ3 ]}. 55 The corresponding expression written with the pole mass M h reads C OS 1γ = 1 ᾱ n f µ {Q h αn f s µ Q h + αn f [ s µ Q h n l n l + Q 6 Mh q i ] 6 Mh 3[ + αn f s µ Q h ζ Mh n l ζ Mh 36 Mh + n l Mh 36 Mh n l Q q i ζ µ Mh 7 Mh + nf Q qi Q qi nl M h Mh + n l µ 36 Mh ζ3 ]}. 56 The Oᾱα s term of Eq. 56 agrees with Ref. [30]. By comparing Eqs. 55 and 56, we notice that only the Oᾱαs 3 terms depend on the definition of the top quark mass. The Oᾱαs contributions due to the diagrams where the Higgs boson and the photons are connected to the same quark loop agree with Ref. [8], where the corresponding vertex diagrams have been directly calculated. In Ref. [8], also power-suppressed terms of Oᾱαs have been considered. 6 Discussion and conclusions In the context of QCD with MS renormalization, the decoupling constants ζ m and ζ g given in Eqs. 0 and 3, respectively, determine the shifts in the light-quark masses m q µ and the strong coupling constant α s µ that occur as the threshold of a heavy-quark flavour h is crossed. In Eqs. 0 and 3, the matching scale µ, at which the crossing is implemented, is measured in units of the MS mass m h µ ofh. The corresponding decoupling relations, formulated for the pole mass M h, are given in Eqs. and 4. Equations 0 and 3 are valid through three loops, i.e. they extend the results of Refs. [5,18] by one order. These results will be indispensible in order to relate the QCD 18

19 predictions for different observables at next-to-next-to-next-to-leading order. Meaningful estimates of such corrections already exist [31]. We wish to stress that our calculation is based on a conceptually new approach that directly links the decoupling constants to massive tadpole integrals. This is crucial because the presently available analytic technology does not permit the extension of the methods employed in Refs. [5,18] to the order under consideration here. In fact, this would require the evaluation of three-loop diagrams with nonvanishing external momenta in the case of Ref. [5], or even four-loop diagrams in the case of Ref. [18]. We are now in a position to explore the phenomenological implications of our results. To that end, it is convenient to have perturbative solutions of Eqs. 1 and for fixed n f in closed form. Iteratively solving Eq. 1 yields [8] α s µ = 1 β 0 L b 1 L β 0 L + 1 [ ] b β 0 L 3 1 L L 1 + b [ + 1 β 0 L 4 b L + 5 L +L 1 3b 1 b L + b ] 3, 57 where b N = β N /β 0 N =1,,3, L =µ /Λ, and terms of O1/L 5 have been neglected. The asymptotic scale parameter, Λ, is defined in the canonical way, by demanding that Eq. 57 does not contain a term proportional to const./l []. Equation 57 extends Eq. 9.5a of Ref. [3] to four loops. Combining Eqs. 1 and, we obtain a differential equation for m q µ as a function of α s µ. It has the solution [17] m q µ m q µ 0 = cα sµ/ cα s µ 0 /, 58 with [7] { cx=x c 0 1+c 1 b 1 c 0 x+ 1 [ ] c1 b 1 c 0 +c b 1 c 1 +b 1c 0 b c 0 x [ c 1 b 1 c c 1 b 1 c 0 c b 1 c 1 +b 1c 0 b c ] } c3 b 1 c +b 3 1c 1 b c 1 b 3 1c 0 +b 1 b c 0 b 3 c 0 x 3, 59 where c N = γ N /β 0 N =0,...,3 and terms of Ox 4 have been neglected. Going to higher orders, one expects, on general grounds, that the relation between α n l s µ andα n f s µ, where µ µ n f µ, becomes insensitive to the choice of the matching scale, µ n f,aslongasµ nf =Om h. This has been checked in Ref. [33] for three-loop evolution in connection with two-loop matching. Armed with our new results, we are in a position to explore the situation at the next order. As an example, we consider the crossing of the bottom-quark threshold. In particular, we wish to study how the µ 5 dependence of the relation between α s 4M τ andα s 5M Z is reduced as we implement four-loop evolution with three-loop matching. Our procedure is as follows. We first calculate α s 4 µ 5 by exactly integrating Eq. 1 with the initial condition α s 4 M τ = 19

20 α 5 s M Z µ 5 /M b Figure 3: µ 5 dependence of α s 5 M Z calculated from α s 4 M τ =0.36 and M b =4.7GeV using Eq. 1 at one dotted, two dashed, three dot-dashed, and four solid loops in connection with Eqs. 7 and 4 at the respective orders [33], then obtain α s 5µ5 from Eqs. 7 and 4 with M b =4.7 GeV, and finally compute α s 5 M Z with Eq. 1. For consistency, N-loop evolution must be accompanied by N 1-loop matching, i.e. if we omit terms of Oαs N+ on the right-hand side of Eq. 1, we need to discard those of Oαs N in Eq. 4 at the same time. In Fig. 3, the variation of α s 5 M Z withµ 5 /M b is displayed for the various levels of accuracy, ranging from one-loop to four-loop evolution. For illustration, µ 5 is varied rather extremely, by almost two orders of magnitude. While the leading-order result exhibits a strong logarithmic behaviour, the analysis is gradually getting more stable as we go to higher orders. The four-loop curve is almost flat for µ 5 > 1 GeV. Besides the µ5 dependence of α s 5 M Z, also its absolute normalization is significantly affected by the higher orders. At the central matching scale µ 5 = M b, we encounter a rapid, monotonic convergence behaviour. 0

21 m c M Z µ 5 10 /M b Figure 4: µ 5 dependence of m 5 c M Z calculated from µ c = m 4 c µ c =1.GeV,M b = 4.7GeV,andα s 5 M Z =0.118 using Eq. 59 at one dotted, two dashed, three dotdashed, and four solid loops in connection with Eqs. 8 and at the respective orders. Similar analyses may be performed for the light-quark masses as well. For illustration, let us investigate how the µ 5 dependence of the relation between µ c = m 4 c µ c and m 5 c M Z changes under the inclusion of higher orders in evolution and matching. As typical input parameters, we choose µ c =1.GeV,M b =4.7GeV,andα s 5M Z= We first evolve m 4 c µ from µ = µ c to µ = µ 5 via Eq. 59, then obtain m 5 c µ 5 from Eqs. 8 and, and finally evolve m 5 c µ from µ = µ 5 to µ = M Z via Eq. 59. In all steps, α n f s µ is evaluated with the same values of n f and µ as m n f c µ. In Fig. 4, we show the resulting values of m 5 c M Z corresponding to N-loop evolution with N 1- loop matching for N = 1,...,4. Similarly to Fig. 3, we observe a rapid, monotonic convergence behaviour at the central matching scale µ 5 = M b. Again, the prediction for N = 4 is remarkably stable under the variation of µ 5 as long as µ 5 > 1GeV. An interesting and perhaps even surprising aspect of the decoupling constants ζ g and 1

22 ζ m is that they carry the full information about the virtual heavy-quark effects on the Higgs-boson couplings to gluons and light quarks, respectively. In fact, the relevant coefficient functions C 1 and C in the renormalized version of the effective Lagrangian 6 emerge from ζ g and ζ m, respectively, through logarithmic differentiation w.r.t. the heavyquark mass m h. In the LET s 39 and 4, these relationships have been established to all orders. By virtue of these LET s, we have succeeded in obtaining the four-loop Oαs 4 corrections to C 1 and C from the three-loop Oαs 3 expressions for ζ g and ζ m complemented by the four-loop results for the β [6] and γ m [7] functions. The pure MS expressions for C 1 and C may be found in Eqs. 43 and 44, respectively; the corresponding versions written in terms of the heavy-quark pole mass M h are listed in Eqs. 45 and 46. From the phenomenological point of view, C 1 is particularly important, since it enters the theoretical prediction for the cross section of the gg H subprocess, which is expected to be the dominant production mechanism of the SM Higgs boson H at the CERN protonproton collider LHC. The Oαs term of Eq. 43 is routinely included in next-to-leadingorder calculations within the parton model [34]. Recently, a first step towards the nextto-next-to-leading order has been taken in Refs. [35,36] by considering the resummation of soft-gluon radiation in pp H + X. The Oαs 3 formula 13 for κ h,h in the revised version of Ref. [35] agrees with our Eq. 43 for C n f 1 µ to this order, if we identify C 1 µ t,m H = α5 s µ t β 5 α s 5M H α s 5 M H β 5 5 α s µ t C 6 1 µ t = 1 α s 5M H κh,h The RG improvement of C 1 in the first line of Eq. 60 is adopted from Eq. 30 of Ref. [], where it has been employed in the context of three-loop Oαs G F Mt corrections to hadronic Higgs-boson decays. We stress that the particular choice of scales in Eq. 60 is essential in order to recover Eq. 13 of Ref. [35] from the general framework elaborated here. The µ dependence of κ h,h is not displayed in Ref. [35]. We note that the Oαs 3 result for C 1 was originally obtained in Ref. [11]. In their Eq. 8, the authors of Ref. [35] also present an alternative version of a generic formula for C 1 ; see also Eq. 5 of Ref. [36]. A central ingredient of this formula is the quantity β t α s, which they paraphrase as the top-quark contribution to the QCD β function at vanishing momentum transfer. According to Ref. [37], β t α s maybeobtained from some gauge-independent variant of the top-quark contribution to the bare gluon selfenergy at vanishing momentum transfer. Since we are unable to locate in the literature, including Refs. [35,36] and the papers cited therein, a proper general definition of β t α s in terms of the familiar quantities of QCD, it remains unclear whether Eq. 8 of Ref. [35] agrees with our Eq. 41 in higher orders. If we were to fix β t α s sothateq.8of Ref. [35] reproduces our Eq. 41 after the RG-improvement of Eq. 60, the result would

23 be β t µ, α 6 s [,m6 t = β 5 α 5 s β 6 α 6 s α 5 s α s 6 ], 61 where α s 5 = α s 5 µ, α 6 s,m6 t. We emphasize that, in order to derive Eq. 41, we found it indispensable to consider the effective Lagrangian 6 comprising a complete basis of scalar dimension-four operators. As a result, Eq. 41 relates C 1 to basic, gauge-invariant quantities of QCD in a simple way, and is manifestly valid to all orders. The formalism developed for QCD in Sections and 4 naturally carries over to QED. In particular, it allows us to evaluate the two-loop Oᾱα s and three-loop Oᾱαs corrections to the decoupling constant ζ gγ for the fine-structure constant ᾱµ renormalized according to the MS scheme. The results formulated in terms of m h µ andm h may be found in Eqs. 50 and 51, respectively. An appropriately modified LET relates the coefficient function C 1γ of the heavy-quark effective γγh coupling to ζ gγ. Similarly to the case of the ggh coupling, this enables us to calculate C 1γ through four loops, i.e. to Oᾱαs nwith n=0,...,3. The final results written with m h µ andm h are listed in Eqs. 55 and 56, respectively. Note added After the submission of this manuscript, a preprint [38] has appeared in which α s 5M Z is related to α s 3M τ with four-loop evolution [6] and three-loop matching [8] at the charm- and bottom-quark thresholds. In particular, the theoretical error due to the evolution procedure is carefully estimated. In Ref. [38], also the logarithmic terms of the three-loop decoupling constant ζ g, which was originally found in Ref. [8], are confirmed using standard RG techniques. These techniques were also employed in Ref. [8] in order to check the logarithmic part of the full result, which was obtained there through explicit diagrammatical calculation. Acknowledgements We thank Werner Bernreuther for a clarifying communication regarding Ref. [5] and Michael Spira for carefully reading this manuscript and for his comments. The work of K.G.C. was supported in part by INTAS under Contract INTAS ext. 3

24 Appendix A Results for the gauge group SUN c In the following, we list the decoupling constants ζ m and ζ g appropriate for the general gauge group SUN c. The results read ζ m =1+ αn f s µ 1 N c N c + αn f 3 { s µ 1 Nc m h 3 m h ζ ζ B N c ζ ζ3 1 + N c ζ3 + Nc [ Nc ζ N c Nc 576N c N c N c N c N c N c 3 m h 1 + n l N c 137 N c f ζg =1+αn s µ m h αn f [ s µ 13 + αn f 3 { s µ 1 Nc ζ3 + N c Nc N c N c 1 19N c N c 3 B ζ ζ4 864 N c Nc µ m h 53 ζ m 3 h 88 m h ] ζ3 3 m h } N c 48 N c + 1 µ ] m h 36 m h ζ3 + 1 N c ζ3 + N c 5 19N c N c 1169 [ 41 + n l N c N c N c 1 m h N c N c 3456 N c ζ ζ3 m h 3 m h m h,

25 1 + 1 ]} 96N c 96 N c. A.1 m h For N c = 3, we recover Eqs. 0 and 3. B Decoupling relations for the quark and gluon fields In analogy to Eqs. 7 and 8, the renormalized decoupling constants ζ and ζ 3 for the quark and gluon fields, respectively, arise from the relations ψ q = Z ζ 0 Z ψ q = ζ ψ q, G a µ = Z3 ζ 0 Z 3 3 G a µ ζ = 3 G a µ. B.1 Of course, ζ and ζ 3 are both gauge dependent. Restricting ourselves to the case N c =3, we find in the covariant gauge 5 ζ =1+ αn f s µ ξ + m h 3 [ s µ ζ3 αn f + 49 m 1 35 h 576 m 3 + n h 96 m l h ζ µ m h µ m h ζ 3 =1+ αn f s µ αn f s µ 3 [ m h + αn f s µ 91 m h ] m h m h 3 m h ζ ζ4 1 3 B ζ m h 304 m 3 h 768 m h n l ζ m 1 h 36 m 3 h 96 m h + ξ ζ m + 3 ] h 3 m 3. B. h 56 m h, 5

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