Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT) 7618 Karlsruhe, Germany E-mail: joshua.davies@kit.edu Matthias Steinhauser Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT) 7618 Karlsruhe, Germany E-mail: matthias.steinhauser@kit.edu David Wellmann Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT) 7618 Karlsruhe, Germany E-mail: david.wellmann@kit.edu We compute corrections to the decay of the Standard Model Higgs boson to hadrons, to the fourth order in the strong coupling constant α s. We use an effective theory in which the Higgs boson couples directly to bottom quarks and to gluons, via top quark mediated effective couplings. Numerically, our results are of a comparable size to the previously-known massless contributions and complete the order α 4 s corrections to the hadronic decay of the Higgs boson. In these proceedings we also provide an independent cross check of the gluonic Higgs boson decay at order α 5 s. PoS(DIS017)95 XXV International Workshop on Deep-Inelastic Scattering and Related Subjects -7 April 017 University of Birmingham, UK Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/
1. Introduction In the coming years, it will be an important task to precisely determine the Higgs boson couplings to other Standard Model particles. To this end, it is important to have a good theoretical understanding of the decay rate of the Higgs boson into bottom quarks. Combined with the Higgs boson s decay rate into gluons, it forms some 70% of the hadronic decay width and therefore influences all Higgs boson branching ratios. QCD corrections to the partial width Γ(H b b) have been studied for some time. At one and two loops they are known with exact bottom quark mass dependence [1,,, 4, 5]. Beyond this order, at three and four loops, only the massless approximation was known [6, 7, 8]. In Ref. [9] the order α 4 s corrections were completed. The calculation will be described in these proceedings. For a summary of further corrections, the reader is referred to [10, 11, 1]. For the calculation in the limit M t M H, we consider an effective Lagrange density in which the top quark has been integrated out. It consists of QCD and effective couplings between the Higgs boson and bottom quarks and gluons, L eff = H0 v 0 ( C1 [O 1] +C [O ] ) + L QCD, (1.1) where primed quantities refer to the five-flavour effective theory. The bare Higgs field H 0 and vacuum expectation v 0 may be identified with their renormalized quantities since we neglect electroweak effects in this calculation. The top quark mass dependence is provided by the coefficient functions C 1 and C, which are known to fifth order in α s [1, 14, 15]. The bare effective operators are given by O 1 = (G0 a,µν) and O = m0 b b 0 b 0, and [O 1 ], [O ] are their renormalized counterparts. Correlators formed by the above operators can be related, via the optical theorem, to the total decay rate of the Higgs boson. With the correlator Π i j (q ) = i dxe iqx 0 T [O i,o j] 0, (1.) we define the quantities 11 = Im [ Π 11 (M H) ] /(πm 4 H), 1 = Im [ Π 1 (M H) + Π 1 (M H) ] /(6πM Hm b ), = Im [ Π (M H) ] /(6πM Hm b ), (1.) in terms of which, Γ(H hadrons) = A b b [(C ) (1 + ) +C 1 C 1 ] + A gg (C 1 ) 11 (1.4) PoS(DIS017)95 where A = G b b FM H mb (µ)/(4π ) and A gg = 4G F MH /(π ). Π 11 and Π are already known at four [16] and five [8] loops, providing αs 5 and αs 4 corrections to the decay rate, respectively. The mixed contribution Π 1 (= Π 1 ), however, was known only at three loops [7]. It has now been computed at four loops [9], completing the αs 4 QCD corrections to the decay rate. Additionally, we compute new two-, three- and four-loop corrections to Π 11 which are suppressed by one power of (mb /M H ); they are numerically small, but are of the same order in (mb /M H ) as the leading contributions from Π 1 and Π. The results presented in these proceedings (Eq. (.7)) provide an independent cross check of the four-loop (mb /M H )0 terms of Π 11. Corrections suppressed by powers of (MH /M t ) are small, and not considered here. 1
. Calculation The calculation was performed by making use of a well-tested collection of software. We use qgraf [17] to generate the diagrams. qe and exp [18, 19, 0] are used to map the diagrams into a set of two-point integral topologies. FORM [1] is used to perform the Dirac traces and to re-write all integrals as linear combinations of scalar integrals. The resulting scalar integrals are reduced to master integrals by FIRE 5. [, ]; we find 8 four-loop master integrals, which have been computed in Refs. [4, 5, 6]. We do not discuss the details of the renormalization of the bare expressions here, and refer the reader to Ref. [9]. Using this setup, we have computed one-, two-, three- and four-loop corrections to each of the correlators Π 11, Π 1 and Π. In the case of Π 11 we compute both the leading (mb /M H )0 and subleading (mb /M H ) contributions. In all cases we employ a generic gauge parameter ξ and expand the amplitudes to linear order in ξ before integral reduction. All dependence on ξ drops out after reduction to a minimal set of master integrals.. Results To investigate the numerical size of the new corrections, we re-write Eq. (1.4) in the form ( Γ(H hadrons) = A b b 1 + light + top + m ) b=0 gg, (.1) with a common prefactor. light gives the terms for C 1 = 0 and C = 1, i.e., terms with no top quark mass dependence. m b=0 gg collects the leading terms from 11, which are proportional to (mb /M H )0. The remaining terms are collected inside top. For completeness, these are defined in terms of the symbols of Eq. (1.4) as light =, m b=0 gg = 16M H mb (C 1 ) m b=0 11, top = [ (C ) 1 ] (1 + ) +C 1 C 1 + 16M H mb (C 1 ) m b 11. (.) PoS(DIS017)95 Numerically, we find light 5.6667a s + 9.1467a s + 41.7576a s 85.7466a 4 s 0.0 + 0.075 + 0.00199 0.00168, (.) top a s [.5556 1 + 0.9895 ] + a s [0. 11 + 4.166 1 + 1.0855 ] + a 4 s [8.99 11 + 8.901 1 + 50.646 ] + a 5 s [17.145 11 ] 0.0090 1 + 0.00174 + 0.0000106 11 + 0.001947 1 + 0.000604 + 0.000018 11 + 0.0005616 1 + 0.0000890 + 0.00000815 11, (.4)
m b=0 gg M H ( a 7mb s + 17.9167a s + 15.091a 4 s + 9.6176a 5 ) s 0.09699 + 0.065 + 0.01911 + 0.001759, (.5) in which the expansion parameter is a s = α s (5µ)/π and we have used the values α s (5M H ) = 0.117, m b (M H ) =.77 GeV, M H = 15.09 GeV, M t = 17.1 GeV. The subscripts 11, 1 and in Eq. (.4) tag the origin of each number. m b=0 gg includes the a 5 s term from Ref. [16] and top the new a 5 s (mb /M H ) term computed here. We observe that the a 4 s term of light is only about 0% smaller than the a s term. For this reason it is important to consider also the complete a 4 s term of top, as we have done here. It is of the same order of magnitude, but appears with an opposite sign. Its inclusion significantly improves the convergence of the sum of all (mb /M H ) terms, 1 + light + top 1 + 0.0 + 0.0408 + 0.004490 0.0007090, (.6) to which the a 4 s term now provides a correction of just 0.0567%. The (mb /M H ) corrections to top are very small. We now present the a s corrections to 11, which have not been fully presented before, in terms of their colour factors and including the scale-dependence. The (mb /M H ) terms are new. We find ( a 11 = C 1540961 A 46656 704 9 ζ 4459 4 ζ + 85 [ 96585 4 ζ 5 + 5184 11 7 ζ 605 ] 4 ζ L H s + 5 9 L H + 11 4 L H n l CA 66767 888 01 48 ζ 44 144 ζ + 1 5 [ 1706 ζ 5 + 178 11 1 ζ 11 ] ζ L H + 01 96 L H + 11 7 L H 1 n l C F C A 576 14 48 ζ 41 16 ζ 5 ζ 5 [ 8569 + 576 11 ] ζ L H + 14 96 L H + n l CF 1 19 + ζ 5ζ 5 + L H + nl 107 C A 888 17 4 ζ + 7 7 [ 9187 ζ + 576 11 6 ζ + 6 1 ] ζ L H + 17 48 L H + 6 11 L H + nl 55 C F 8 1 4 ζ 15 [ ] 767 4 ζ + 88 ζ L H + 48 1 L H nl 717 58 18 7 ζ 7 1 [ 17 ζ + 16 1 ] 9 ζ L H + 6 7 L H + 54 1 ) L H + m b { ( MH CA 991 11 ζ 17 4 ζ 5 [ ] 675 8 ζ 5 + 1 + ζ L H + 11 4 L H 14891 +C F C A 19 1415 8 ζ 881 4 ζ 685 [ 4767 4 ζ 5 + 96 99 4 ζ 189 ] 4 ζ L H + 1415 16 L H + ) 8 L H ( +CF 18485 64 69 8 ζ 675 8 ζ + 105 [ 519 4 ζ 5 + 81 ] 4 ζ 7ζ L H + 69 16 L H + 7 8 L H 015 n l C A 144 ζ + 67 [ ] 845 4 ζ + 1 + 4ζ L H + 11LH 1991 n l C F 96 11 4 ζ 1 ζ + 10 [ 475 ζ 5 + 48 9 ] ζ 9ζ L H + 11 8 L H + 4 L H + nl 157 18 ζ )} + 5L H + LH. (.7) PoS(DIS017)95 Here, C A and C F are the quadratic Casimir invariants of SU(N), L H = log(µ /M H ), n l is the number of light quarks running in loops and ζ n is the Riemann Zeta function. Eq. (.7) agrees with the
expression presented in [16] if one sets L H = 0, C A =, C F = 4/ and (mb /M H ) = 0. We also find agreement with [7] if one sets L H = 0 and (mb /M H ) = 0, which contains an independent calculation of all but the ζ terms of 11 a s. 4. Conclusions By computing the previously-unknown αs 4 corrections to 1, we complete the QCD corrections to the hadronic decay of the Higgs boson at this order. The computation is performed in an effective theory, however additional corrections proportional to powers of (MH /M t ) are known to be small. The new contributions to Γ(H hadrons) are of the same order of magnitude as the previously-known contributions, but come with an opposite sign; they do much to improve the convergence of the QCD perturbative series. Additionally, we provide an independent cross check of the four-loop Higgs boson decay to gluons. Computer-readable files containing the analytic results of these proceedings and Ref. [9] can be found at [8]. Acknowledgements This work is supported by the BMBF grant 05H15VKCCA. References [1] E. Braaten and J. P. Leveille, Phys. Rev. D (1980) 715. doi:10.110/physrevd..715 [] S. G. Gorishnii, A. L. Kataev, S. A. Larin and L. R. Surguladze, Mod. Phys. Lett. A 5 (1990) 70. doi:10.114/s0177900015 [] K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, Nucl. Phys. B 505 (1997) 40 doi:10.1016/s0550-1(97)00481-1 [hep-ph/970554]. [4] R. Harlander and M. Steinhauser, Phys. Rev. D 56 (1997) 980 doi:10.110/physrevd.56.980 [hep-ph/970446]. [5] K. G. Chetyrkin, R. Harlander and M. Steinhauser, Phys. Rev. D 58 (1998) 01401 doi:10.110/physrevd.58.01401 [hep-ph/98014]. [6] K. G. Chetyrkin, Phys. Lett. B 90 (1997) 09 doi:10.1016/s070-69(96)0168-8 [hep-ph/960818]. [7] K. G. Chetyrkin and M. Steinhauser, Phys. Lett. B 408 (1997) 0 doi:10.1016/s070-69(97)00779-x [hep-ph/970646]. [8] P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Phys. Rev. Lett. 96 (006) 0100 doi:10.110/physrevlett.96.0100 [hep-ph/051106]. [9] J. Davies, M. Steinhauser and D. Wellmann, Nucl. Phys. B 90 (017) 0 doi:10.1016/j.nuclphysb.017.04.01 arxiv:170.0988 [hep-ph]]. [10] D. de Florian et al. [LHC Higgs Cross Section Working Group], arxiv:1610.079 [hep-ph]. [11] M. Spira, arxiv:161.07651 [hep-ph]. [1] A. Djouadi, J. Kalinowski and M. Spira, Comput. Phys. Commun. 108 (1998) 56 doi:10.1016/s0010-4655(97)001-9 [hep-ph/9704448]. PoS(DIS017)95 4
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