Exact Top Yukawa corrections to Higgs boson decay into bottom quarks
Amedeo Primo, Gianmarco Sasso, Gabor Somogyi, Francesco Tramontano
ZZU-TH 48/18
Exact Top Yukawa corrections to Higgs boson decay into bottom quarks
Amedeo Primo ∗ Department of Physics, University of Z¨urich, CH-8057 Z¨urich, Switzerland
Gianmarco Sasso † Universit`a di Napoli Federico II,Complesso di Monte Sant’Angelo, Edificio 6via Cintia I-80126, Napoli, Italy
G´abor Somogyi ‡ MTA-DE Particle Physics Research Group, H-4010 Debrecen, PO Box 105, Hungary
Francesco Tramontano § Universit`a di Napoli Federico II and INFN sezione di Napoli,Complesso di Monte Sant’Angelo, Edificio 6via Cintia I-80126, Napoli, Italy (Dated: December 20, 2018)In this letter we present the results of the exact computation of contributions to the Higgs bosondecay into bottom quarks that are proportional to the top Yukawa coupling. Our computationdemonstrates that approximate results already available in the literature turn out to be particularlyaccurate for the three physical mass values of the Higgs boson, the bottom and top quarks. Further-more, contrary to expectations, the impact of these corrections on differential distributions relevantfor the searches of the Higgs boson decaying into bottom quarks at the Large Hadron Collider israther small.
INTRODUCTION
The discovery of the Higgs boson [1, 2] by the AT-LAS [3] and CMS [4] experiments at CERN has usheredin a new era in particle physics phenomenology. TheStandard Model (SM) of elementary particles is now com-plete and there is no decay or scattering phenomenon atlow energies that significantly deviates from what is pre-dicted by the SM. Still, we know that the SM cannot bethe ultimate theory, if not for the lack of consistency withmainly cosmological observations, but for the fact thatit contains quite a large number of parameters, whichmakes it unreasonable to think of it as a fundamentaltheory. The LHC is guiding the experimental commu-nity towards the study of the Higgs potential and theHiggs boson direct couplings with all the other particles,an experimentally previously completely unexplored sec-tor of the SM Lagrangian. The SM makes very precisepredictions for all the vertices and couplings of the Higgsboson and the verification of these predictions is amongthe fundamental questions addressed at the LHC.In particular, the gluon fusion production mechanismhas given direct access to Higgs boson decay into vectorbosons and an indirect access the top Yukawa coupling.More recently, also the direct coupling of the Higgs bosonto the top quark has been observed [5, 6]. Measurementsof the Higgs coupling to the tau lepton have been ex-tracted by combining all production modes [7, 8], whilethe direct coupling to the bottom quark has been ob-served by exploiting the features of the VH (V = W or Z) associated production mechanism [9, 10]. The decayto bottom quarks is quite special, because it is the onewith the largest branching ratio. The decay width ofthe Higgs boson into bottom quarks has been computedat up to four loops in QCD [11–18] using an approxi-mated treatment of the bottom quark mass, up to oneloop including electroweak corrections [19, 20] and alsoincluding mixed QCD-electroweak effects [21]. The ex-act bottom mass corrections have been computed up totwo loops [22]. A relatively large component of the twoloop computation is represented by the diagrams in whichthe Higgs boson couples to a top quark loop, see Figs. 1and 2. These two sets of diagrams are both UV and IRfinite separately, and their contributions have been com-puted only approximately in [16], finding a very compactformula that should be considered valid for values of themasses such that m b (cid:28) m H (cid:28) m t . Comparing this for-mula with the rest of the two loop contributions, it turnsout that these pieces, proportional to the top Yukawacoupling y t , account for about 30% of the total two loopresult.The aim of this letter is twofold. First, we want toassess the impact of the neglected terms in the expan-sion of [16], that in principle could be of the order of( m H /m t ) ∼
20% (see Eq. (3) in [16]). We do this bycomputing the full analytic result for the contributions tothe Higgs boson decay into bottom quarks that are pro-portional to the top Yukawa coupling, including the exactdependence on the top and bottom quark masses. Fur-thermore, recently two groups have computed the fullydifferential decay width of the Higgs boson into bottom a r X i v : . [ h e p - ph ] D ec H bbt H bbt
FIG. 1. Virtual O ( α s y t ) contributions to H → b ¯ b decay. quarks up to two loops for the massless case, and mergedthis computation to the two loop corrections to the as-sociated production in hadronic collisions [23, 24]. Thecorrections to key distributions like the transverse mo-mentum and the mass spectra of the Higgs boson (recon-structed using the two hardest b-jets in the final state)are found to be very large. y t contributions to the Higgsdecay into bottom quarks are not included in the differ-ential results mentioned above and so it is natual to askwhat the impact of these corrections is (see for exam-ple [24]). We answer this second question by presentingdifferential results which include the y t contributions tothe decay and retain the full mass dependence on the topand bottom quark masses. CALCULATIONDouble virtual
The decay H ( q ) → b ( p ) + ¯ b ( p ) receives O ( α s y t ) con-tributions from the interference |M y t ,b ¯ b | of the Born am-plitude with two loop virtual corrections that involve aclosed top quark loop, |M y t ,b ¯ b | ≡ M (0) † b ¯ b M (2) y t ,b ¯ b , (1)where M (2) y t ,b ¯ b is given by the two diagrams shown inFig. 1. By evaluating the Feynman diagrams, we de-composed |M y t ,b ¯ b | as |M y t ,b ¯ b | = α s y t C A C F (cid:88) (cid:126)a c a ··· a ( (cid:15), m i ) I a ··· a ( (cid:15), m i ) , (2)with a j ∈ Z and i = t, b, H . In Eq. (2), the c a ··· a arerational coefficients and I a ··· a are two loop integrals ofthe type I a ··· a ( (cid:15),m i ) = (cid:90) d d k (2 π ) d d d k (2 π ) d D a D a D a D a D a D a D a , (3)defined by the set of inverse propagators: D = k − m t , D = k , D = ( k − k ) − m t ,D = ( k + q ) − m t , D = ( k + q ) ,D = ( k + q − p ) − m b D = ( k + q − p ) , (4) with kinematics m H = q = ( p + p ) , p = p = m b .We computed the loop integrals through the consoli-dated differential equations (DEs) method [25–28]. First,we used integration-by-parts identities (IBPs) [29–31],generated with the help of Reduze2 [32], in order toreduce the integrals that appear in |M y t ,b ¯ b | to a set of20 independent master integrals (MIs) (cid:126) I = ( I I , . . . , I ).Subsequently, we derived the analytic expression of theMIs by solving the system of coupled first-order DEs inthe kinematic ratios m H /m t and m b /m t . The structureof the DEs, and hence of their solutions, is simplified byparametrizing such ratios in terms of the variables x and y , defined by m H m t = − (1 − x ) x , m b m t = (1 − x ) (1 − y ) y x , (5)and by using the Magnus exponential method [33–38] inorder to identify a basis of MIs that fulfil a system ofcanonical DEs [39],d (cid:126) I = (cid:15) d A (cid:126) I , with d f = (cid:88) z = x,y d z ∂∂z f . (6)In Eq. (6), the coefficient matrix d A is a dlog-form thatcontains 12 distinct letters,d A = M dlog ( x ) + M dlog (1 + x ) + M dlog (1 − x )+ M dlog (1 + x ) + M dlog ( y ) + M dlog (1 + y )+ M dlog (1 − y ) + M dlog (1 + y )+ M dlog ( x + y ) + M dlog ( x − y )+ M dlog (1 + xy ) + M dlog (1 − xy ) , (7)with M i ∈ M × ( Q ). Since all letters are algebraically-rooted polynomials, we could derive the (cid:15) -expansionof the general solution of Eq. (6) in terms of two-dimensional generalized polylogarithms (GPLs) [40–44],by iterative integration of the dlog-form, which we per-formed up O ( (cid:15) ), i.e. to GPLs of weight four. In order tofully specify the analytic expression of the MIs, we com-plemented the general solution of DEs with a suitableset of boundary conditions. The latter were obtainedby demanding the regularity of the MIs at the pseudo-thresholds m H = 0 and m H = 4 m b that appear as un-physical singularities of the DEs.The expression of the MIs obtained in this way is validin the Euclidean region m H < ∧ < m b < m t , wherethe logarithms of Eq. (7) have no branch-cuts and, hence,the MIs are real. The values of the MIs for positive val-ues of the Higgs squared momentum, and in particularfor the decay region m H > m b , are obtained throughanalytic continuation, by propagating the Feynman pre-scription m H → m H + i + to the kinematic variables x and y . All results have been numerically validated with GiNaC [45] against the results of
SecDec3 [46], bothin the Euclidean and in the physical regions.
H bbt
H bbt
FIG. 2. Real-virtual y t contributions to H → b ¯ b decay. Upon inserting the expressions of the MIs into Eq. (2),we observed the expected analytic cancellation of all (cid:15) -poles and obtained a finite result for |M y t ,b ¯ b | , |M y t ,b ¯ b | = α s π C A C F y t y b m t m b Re C ( x, y ) , (8) with C ( x, y ) being a polynomial combination, with al-gebraic coefficients, of 256 distinct GPLs. The explicitexpression of C ( x, y ), as well as of the newly computedMIs, can be released by the authors upon request. Real-virtual
The real-virtual part of the computation involves theinterference |M y t ,b ¯ bg | of the tree level amplitude for H ( q ) → b ( p ) + ¯ b ( p ) + g ( p ) with the loop diagramsin Fig. 2 containing a closed top quark loop, |M y t ,b ¯ bg | ≡ M (0) † b ¯ bg M (1) y t ,b ¯ bg . (9)We used standard techniques to evaluate the one loopamplitude. As expected, |M y t ,b ¯ bg | is finite (in (cid:15) ) andcan be written as |M y t ,b ¯ bg | = 32 α s C A C F y t y b m t m b (cid:32) s + 2 m b )( s + s ) + s + s − m b s s (cid:33) × (cid:40) (cid:34)(cid:115) m t m H − (cid:32) arctan (cid:115) m t m H − − π (cid:33) − (cid:115) m t s + 2 m b − (cid:32) arctan (cid:115) m t s + 2 m b − − π (cid:33)(cid:35) + s + s s + 2 m b (cid:34) − (cid:32) m t s + s − (cid:33)(cid:34)(cid:32) arctan (cid:115) m t m H − − π (cid:33) − (cid:32) arctan (cid:115) m t s + 2 m b − − π (cid:33) (cid:35)(cid:35)(cid:41) , (10)plus terms that vanish in four dimensions. In Eq. (10), s ij denotes twice the dot-product of momenta, s ij ≡ p i · p j .We integrated the real-virtual contribution over thewhole phase space both analytically and numerically us-ing Monte Carlo integration, finding perfect agreement.The analytic computation was performed by direct inte-gration of |M y t ,b ¯ bg | over the three-particle phase space.The phase space measure for the decay H ( q ) → b ( p ) +¯ b ( p ) + g ( p ) readsd P S = 2 − (cid:15) π − (cid:15) ( q ) − (cid:15) (∆ ) − (cid:15) × Θ(∆ ) δ ( q − m b − s − s − s ) × dΩ d − dΩ d − d s d s d s , (11)where ∆ is given by∆ = s s s − m b (cid:0) s + s (cid:1) . (12)The integral is finite in four dimensions and was evalu-ated in terms of GPLs after suitable transformations ofthe integration variables. In particular, square roots in-volving the integration variables appear at intermediatestages of the calculation (both from the one loop matrix element and from resolving the phase space constraintimplied by the positivity of ∆ ) and must be linearized,e.g. by using the techniques of [47]. The full result is rep-resented in terms of a formula with 1841 distinct GPLsand can be released by the authors upon request.The numerical integration of the real-virtual contribu-tion is straightforward and has been used to validate theanalytic computation. It also allows to build Monte Carlosimulations with acceptance cuts and has been used toobtain the differential result of the next section. RESULTS
We begin the presentation of our results by discussingthe inclusive decay rate. In Table I, we compare our exactformula, obtained from the sum of the double virtual andreal-virtual contributions described in the previous sec-tion, to the approximated one of Ref. [16]. The numbersin the table are obtained with the following formula forthe relative discrepancy among exact and approximatedresults: d = 100 (cid:32) − Γ Approxy t Γ Exacty t (cid:33) . (13)The agreement is excellent for the physical mass values,proving for the first time and in a completely independentway the validity of the approximated formula, and thefact that it works much better then expected.We now turn to the second question regarding the im-pact of the y t contribution at differential level. To thisextent, we present results for Higgs boson associated pro-duction and to avoid the contamination from initial stateradiation we consider pp → W + ( l + ν l ) H ( b ¯ b ) at leadingorder and add the corrections to the decay process atthe next-to-leading order. Then, we compare this resultwith the one obtained by adding also the y t contribution.Note that in both cases we normalize the cross section tothe total Higgs boson decay width into bottom quarksreported in the Yellow Report of the Higgs Cross SectionWorking Group [48] (HXSWG), that includes higher or-der corrections. So, effectively, we are comparing theshapes of distributions. To obtain our results, we usethe SM parameters recommended by the HXSWG andthe NNPDF3.0 [49] parton distribution functions. Fur-thermore, we impose the following typical lepton accep-tance cuts: selected events must have a missing trans-verse momentum greater than 30 GeV, the charged lep-ton is required to have a transverse momentum greaterthan 15 GeV and an absolute rapidity smaller than 2 . . . TABLE I. The discrepancy d between our result and the ap-proximate formula in [16], we fix m b = 4 .
92 GeV. m t m H
20 75 125 180100 2.123 0.075 1.025 6.704125 2.329 0.011 0.335 2.107175 2.452 -0.019 0.018 0.355250 2.566 -0.024 -0.055 -0.035350 2.656 -0.023 -0.069 -0.113
In Figs. 3 and 4 we present the transverse momen-tum and mass distributions of the two b-jet system from
W H ( bb ) production at the 14 TeV LHC. The error barsin the figures represent the statistical uncertainty asso-ciated with Monte Carlo integration. We observe that � � � � � � � � � � � �� � � � ����� � � � � � � � ��� ��� � �������� ��������������� �� �� �� �� �� �� �� � ��� � ����� ����� � �������������� � ��� � ���� � � � ���� � � � ��� � ��� � ��� � ��� � ��� FIG. 3. Transverse momentum distribution of the two b-jetsystem from
W H ( bb ) production in proton proton collisionsat 14 TeV. Only corrections to the Higgs decay into bottomquarks are included. � � � � � � � � � � � �� � � � ����� � � � � � � � ��� �� � �������� ������������� � ����� � ���� � ��� � � � �� � ����� ����� � �������������� � ��� � ���� � � � ���� � �� � �� � �� � �� � �� � ��� � ��� � ��� � ��� FIG. 4. Mass distribution of the two b-jet system from
W H ( bb ) production in proton proton collisions at 14 TeV.Only corrections to the Higgs decay into bottom quarks areincluded. the impact of the y t contribution on both the transversemomentum distribution and the mass distribution of theputative Higgs boson is extremely small with at most a5% effect in the low energy tail of the transverse momen-tum distribution. These corrections are much smallerthan the scale variation uncertainty of the computation. CONCLUSIONS
In this letter we presented the results of the full an-alytic computation of the top Yukawa contribution tothe Higgs boson decay width into bottom quarks. First,we demonstrated that the approximate formula used sofar in the literature works exceedingly well for physicalvalues of the masses. This nice behaviour was not pre-dictable a priori and, with respect to a possible estimateof about a 20% error, we have instead found smaller thanper mill deviations of this formula from the exact result.Then, we showed that the impact of this contributionat the differential level is very small and Monte Carlosimulations performed so far are not affected by an ad-ditional significant source of uncertainty due to the ne-glected terms proportional to y t .This work has been supported by the Swiss NationalScience Foundation under grant number 200020-175595(AP), the Italian Ministry of Education and ResearchMIUR, under project n o ∗ [email protected] † [email protected] ‡ [email protected] § [email protected][1] P. W. Higgs, Phys. Lett. , 132 (1964).[2] F. Englert and R. Brout, Phys. Rev. Lett. , 321 (1964),[,157(1964)].[3] G. Aad et al. (ATLAS), Phys. Lett. B716 , 1 (2012),arXiv:1207.7214 [hep-ex].[4] S. Chatrchyan et al. (CMS), Phys. Lett.
B716 , 30 (2012),arXiv:1207.7235 [hep-ex].[5] M. Aaboud et al. (ATLAS), Phys. Lett.
B784 , 173(2018), arXiv:1806.00425 [hep-ex].[6] A. M. Sirunyan et al. (CMS), JHEP , 066 (2018),arXiv:1803.05485 [hep-ex].[7] A. M. Sirunyan et al. (CMS), Phys. Lett. B779 , 283(2018), arXiv:1708.00373 [hep-ex].[8] M. Aaboud et al. (ATLAS), Submitted to: Phys. Rev.(2018), arXiv:1811.08856 [hep-ex].[9] M. Aaboud et al. (ATLAS), Phys. Lett.
B786 , 59 (2018),arXiv:1808.08238 [hep-ex].[10] A. M. Sirunyan et al. (CMS), Phys. Rev. Lett. ,121801 (2018), arXiv:1808.08242 [hep-ex].[11] S. G. Gorishnii, A. L. Kataev, S. A. Larin, and L. R.Surguladze, Mod. Phys. Lett. A5 , 2703 (1990).[12] S. G. Gorishnii, A. L. Kataev, S. A. Larin, and L. R.Surguladze, Phys. Rev. D43 , 1633 (1991).[13] A. L. Kataev and V. T. Kim, Mod. Phys. Lett. A9 , 1309(1994). [14] L. R. Surguladze, Phys. Lett. B341 , 60 (1994),arXiv:hep-ph/9405325 [hep-ph].[15] S. A. Larin, T. van Ritbergen, and J. A. M. Vermaseren,Phys. Lett.
B362 , 134 (1995), arXiv:hep-ph/9506465[hep-ph].[16] K. G. Chetyrkin and A. Kwiatkowski, Nucl. Phys.
B461 ,3 (1996), arXiv:hep-ph/9505358 [hep-ph].[17] K. G. Chetyrkin, Phys. Lett.
B390 , 309 (1997),arXiv:hep-ph/9608318 [hep-ph].[18] P. A. Baikov, K. G. Chetyrkin, and J. H. Kuhn, Phys.Rev. Lett. , 012003 (2006), arXiv:hep-ph/0511063[hep-ph].[19] A. Dabelstein and W. Hollik, Z. Phys. C53 , 507 (1992).[20] B. A. Kniehl, Nucl. Phys.
B376 , 3 (1992).[21] L. Mihaila, B. Schmidt, and M. Steinhauser, Phys. Lett.
B751 , 442 (2015), arXiv:1509.02294 [hep-ph].[22] W. Bernreuther, L. Chen, and Z.-G. Si, JHEP , 159(2018), arXiv:1805.06658 [hep-ph].[23] G. Ferrera, G. Somogyi, and F. Tramontano, Phys. Lett. B780 , 346 (2018), arXiv:1705.10304 [hep-ph].[24] F. Caola, G. Luisoni, K. Melnikov, and R. Rntsch, Phys.Rev.
D97 , 074022 (2018), arXiv:1712.06954 [hep-ph].[25] A. Kotikov, Phys.Lett.
B254 , 158 (1991).[26] E. Remiddi, Nuovo Cim.
A110 , 1435 (1997), arXiv:hep-th/9711188 [hep-th].[27] T. Gehrmann and E. Remiddi, Nucl. Phys.
B580 , 485(2000), arXiv:hep-ph/9912329 [hep-ph].[28] M. Argeri and P. Mastrolia, Int.J.Mod.Phys.
A22 , 4375(2007), arXiv:0707.4037 [hep-ph].[29] F. V. Tkachov, Phys. Lett. , 65 (1981).[30] K. Chetyrkin and F. Tkachov, Nucl.Phys.
B192 , 159(1981).[31] S. Laporta, Int.J.Mod.Phys.
A15 , 5087 (2000),arXiv:hep-ph/0102033 [hep-ph].[32] A. von Manteuffel and C. Studerus, (2012),arXiv:1201.4330 [hep-ph].[33] M. Argeri, S. Di Vita, P. Mastrolia, E. Mirabella,J. Schlenk, et al. , JHEP , 082 (2014),arXiv:1401.2979 [hep-ph].[34] S. Di Vita, P. Mastrolia, U. Schubert, and V. Yundin,JHEP , 148 (2014), arXiv:1408.3107 [hep-ph].[35] R. Bonciani, S. Di Vita, P. Mastrolia, and U. Schubert,JHEP , 091 (2016), arXiv:1604.08581 [hep-ph].[36] S. Di Vita, P. Mastrolia, A. Primo, and U. Schubert,JHEP , 008 (2017), arXiv:1702.07331 [hep-ph].[37] P. Mastrolia, M. Passera, A. Primo, and U. Schubert,JHEP , 198 (2017), arXiv:1709.07435 [hep-ph].[38] S. Di Vita, S. Laporta, P. Mastrolia, A. Primo, andU. Schubert, JHEP , 016 (2018), arXiv:1806.08241[hep-ph].[39] J. M. Henn, Phys.Rev.Lett. , 251601 (2013),arXiv:1304.1806 [hep-th].[40] A. Goncharov, Proceedings of the International Congreeof Mathematicians , 374 (1995).[41] E. Remiddi and J. Vermaseren, Int.J.Mod.Phys. A15 ,725 (2000), arXiv:hep-ph/9905237 [hep-ph].[42] T. Gehrmann and E. Remiddi, Comput.Phys.Commun. , 296 (2001), arXiv:hep-ph/0107173 [hep-ph].[43] T. Gehrmann and E. Remiddi, Comput.Phys.Commun. , 200 (2002), arXiv:hep-ph/0111255 [hep-ph].[44] J. Vollinga and S. Weinzierl, Comput.Phys.Commun. , 177 (2005), arXiv:hep-ph/0410259 [hep-ph].[45] C. W. Bauer, A. Frink, and R. Kreckel, (2000),arXiv:cs/0004015 [cs-sc]. [46] S. Borowka, G. Heinrich, S. P. Jones, M. Kerner,J. Schlenk, and T. Zirke, Comput. Phys. Commun. ,470 (2015), arXiv:1502.06595 [hep-ph].[47] M. Besier, D. Van Straten, and S. Weinzierl, (2018),arXiv:1809.10983 [hep-th]. [48] D. de Florian et al. (LHC Higgs Cross SectionWorking Group), (2016), 10.23731/CYRM-2017-002,arXiv:1610.07922 [hep-ph].[49] R. D. Ball et al. (NNPDF), JHEP04