Hovhannisyan, Artyom

Nierste, Ulrich

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JHEP06(2022)090Published for SISSA by SpringerReceived: April 25, 2022Accepted: April 30, 2022Published: June 16, 2022Addendum: Towards next-to-next-to-leading-logaccuracy for the width difference in the Bs−B̄ssystem: fermionic contributions to order (mc/mb)0and (mc/mb)1Artyom Hovhannisyana and Ulrich NierstebaYerevan Physics Institute,Alikhanian Br. 2, 0036 Yerevan, ArmeniabInstitut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology,Engesserstraße 7, 76128 Karlsruhe, GermanyE-mail: artyom@yerphi.am, ulrich.nierste@kit.eduAddendum to: JHEP10(2017)191Abstract: We calculate the three-loop master integrals of ref. [1] in analytic form. Thisallows us to present the fermionic contributions to the ∆B = 2 Wilson coefficients of theB–B̄ decay matrix in next-to-next-to-leading order of QCD with full analytic dependenceon the mass of the charm quark in the fermionic loops.ArXiv ePrint: 2204.11907Open Access, c© The Authors.Article funded by SCOAP3. https://doi.org/10.1007/JHEP06(2022)090JHEP06(2022)090Contents1 Introduction 12 Updated master integrals 13 Analytic results for ∆B = 2 coefficients at order α2sNf 21 IntroductionThe off-diagonal element Γ12 of the 2× 2 decay matrix of the B–B̄ mixing problem, whereB represents Bs or Bd, must be calculated to predict the width difference ∆Γ betweenthe B meson mass eigenstates and the CP asymmetry in flavour-specific B decays. Thecontributions with fermionic loops of the next-to-next-to-leading order (NNLO) predictioncalculated in ref. [1] are functions of z = m2c/m2b , where mc and mb are the masses of charmand bottom quark, respectively. The three-loop master integrals have been derived as anexpansion in z and all but two integrals are given in analytic form, while the remaining twointegrals involve numerically calculated coefficients. In this Addendum, we present analyticresults for these as well and arrive at concise analytic results for the contributions of current-current operators to the ∆B = 2 Wilson coefficients entering Γ12. The correspondingcontributions with penguin operators are also known in analytic form [2].2 Updated master integralsOur updated analytic results concern the imaginary parts originating from four-particle-cuts of the master integrals in eqs. (B.2) and (B.6) of ref. [1]. The expansion in z = m2c/m2bup to O(z4) in the MS scheme reads:Im(4)∫[dk] 1(k22−m2b)k23 ((k1−pb)2−m2c)((k1−k2)2−m2c)(k2−k3)2= (µ/mb)6ε−28192π5[−72 +π23 −z(4+ 2π23 +4log(z))+z2(272 −7log(z)+log2(z))+z3(−113 +2log(z))+z4( 724 +12 log(z))+ε(−1754 +3π22 +22ζ(3)+z(−56+ 2π23 −44ζ(3)−32log(z)+2log2(z))+ 32π23 z3/2+z2(−1874 +20π23 +12ζ(3)+(32−4π23)log(z)+ 92 log2(z)− log3(z))− 32π215 z5/2– 1 –JHEP06(2022)090+z3(163 −2π2− 43 log(z))− 32π2105 z7/2 +z4(−66911800−π22 +4310 log(z)))]+O(z5,ε2), (2.1)Im(4)∫[dk] 1k21(k1−pb)2((k2−pb)2−m2b)(k1−k2)2(k23−m2c)((k2−k3)2−m2c)= (µ/mb)6ε8192π5[−4+ π23 +z(2− 2π23 +12ζ(3)−(2− 2π23)log(z)+log2(z)− 13 log3(z))+z2(−192 +3log(z))+z3(−16 +12 log(z))+z4( 67216 +16 log(z))]+O(z5,ε1), (2.2)with the loop measure defined as∫[dk] =∫dkd1(2π)d∫dkd2(2π)d∫dkd3(2π)d . (2.3)The eq. (2.1) was obtained with the differential-equation method, while for eq. (2.2) wehave used the formulas for the calculation of four-particle phase space integrals derivedin [3].3 Analytic results for ∆B = 2 coefficients at order α2sNfThe results above entail the following updated analytic functions F (2),Nvij (z) and F(2),NvS,ij (z)(determining the NNLO charm-loop contribution to the ∆B = 2 Wilson coefficients),superseding eqs. (4.8)–(4.13) of ref. [1]:F(2),NV11 (z) = −3869 log(µ1mb)+ 1769 log(µ2mb)− 403 log(µ1mb)log(µ2mb)+ 203 log2(µ2mb)−268954 +5π29 +32ζ(3)−4π2√z+z (37−24log(z))−4π2z3/2+z2(22199 +8π29 −192ζ(3)−5729 log(z)+2log2(z))+z3(467142025 +128π227 +4846135 log(z)−1289 log2(z))+O(z4), (3.1)F(2),NV12 (z) =55427 log(µ1mb)+ 35227 log(µ2mb)− 809 log(µ1mb)log(µ2mb)+ 683 log2(µ1mb)+409 log2(µ2mb)+ 3473324 +10π227 +64ζ(3)3 −8π23√z−z(3343 +16log(z))−8π23 z3/2 +z2(734527 −86π227 −128ζ(3)−318427 log(z)+553 log2(z))+z3(1895126075 +256π281 +8468405 log(z)−25627 log2(z))+O(z4), (3.2)– 2 –JHEP06(2022)090F(2),NV22 (z) =(23627 +4π29)log(µ1mb)+ 5827 log(µ2mb)− 329 log(µ1mb)log(µ2mb)+ 203 log2(µ1mb)+169 log2(µ2mb)+ 3911324 +13π254 +16ζ(3)3 −16π23√z−z(943 −4π23 +32log(z))+32π29 z3/2 +z2(703 +40π227 −20ζ(3)−89954 log(z)−176 log2(z))+z3(−283696075 +40π281 +3764405 log(z)−4027 log2(z))+O(z4), (3.3)F(2),NVS,11 (z) = −809 log(µ1mb)+ 3209 log(µ2mb)+ 1283 log(µ1mb)log(µ2mb)− 643 log2(µ2mb)−47027 +56π29 +32ζ(3)−16π2√z+136z−16π2z3/2+z2(19649 +80π29 −192ζ(3)−6809 log(z)+8log2(z))+z3(437442025 +128π227 +5296135 log(z)−1289 log2(z))+O(z4), (3.4)F(2),NVS,12 (z) =46427 log(µ1mb)+ 64027 log(µ2mb)+ 2569 log(µ1mb)log(µ2mb)+ 323 log2(µ1mb)−1289 log2(µ2mb)+ 73481 +112π227 +64ζ(3)3 −32π23√z+ 803 z−32π23 z3/2+z2(529627 +112π227 −128ζ(3)−232027 log(z)+403 log2(z))+z3(1327046075 +256π281 +10016405 log(z)−25627 log2(z))+O(z4), (3.5)F(2),NVS,22 (z) =(70427 −32π29)log(µ1mb)− 32027 log(µ2mb)− 1289 log(µ1mb)log(µ2mb)+323 log2(µ1mb)+ 649 log2(µ2mb)+201881 −148π227 −32ζ(3)3 +32π23√z−z(2083 +32π23)+ 608π29 z3/2+z2(−55529 +352π227 −32ζ(3)+638027 log(z)−1723 log2(z))+z3(−2435926075 +64π281 +11552405 log(z)−6427 log2(z))+O(z4). (3.6)The contribution of any light quark u, d, s can be obtained by setting z=0 in eqs. (3.1)–(3.6).The new analytic results presented in this addendum are in excellent agreement withthe previous numerical results presented in ref. [1].– 3 –JHEP06(2022)090AcknowledgmentsThe work of A.H. has been supported by the State Committee of Science of ArmeniaProgram, Grant No. 21AG-1C084. U.N. is supported by BMBF under grant Verbundprojekt05H2021 (ErUM-FSP T09) — Belle II: Theoretische Studien für Belle II und LHCb and byproject C1b of the DFG-funded Collaborative Research Center TRR 257, “Particle PhysicsPhenomenology after the Higgs Discovery”.Open Access. This article is distributed under the terms of the Creative CommonsAttribution License (CC-BY 4.0), which permits any use, distribution and reproduction inany medium, provided the original author(s) and source are credited.References[1] H.M. Asatrian, A. Hovhannisyan, U. Nierste and A. Yeghiazaryan, Towardsnext-to-next-to-leading-log accuracy for the width difference in the Bs − B̄s system: fermioniccontributions to order (mc/mb)0 and (mc/mb)1, JHEP 10 (2017) 191 [arXiv:1709.02160][INSPIRE].[2] H.M. Asatrian, H.H. Asatryan, A. Hovhannisyan, U. Nierste, S. Tumasyan andA. Yeghiazaryan, Penguin contribution to the width difference and CP asymmetry in Bq–B̄qmixing at order α2sNf , Phys. Rev. D 102 (2020) 033007 [arXiv:2006.13227] [INSPIRE].[3] H.M. Asatrian, A. Hovhannisyan and A. Yeghiazaryan, The phase space analysis for three andfour massive particles in final states, Phys. Rev. D 86 (2012) 114023 [arXiv:1210.7939][INSPIRE].– 4 –

Addendum to: Towards next-to-next-to-leading-log accuracy for the width difference in the Bs−B¯s system: fermionic contributions to order (mc/mb)⁰ and (mc/mb)¹

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Addendum to: Towards next-to-next-to-leading-log accuracy for the width difference in the Bs−B¯s system: fermionic contributions to order (mc/mb)⁰ and (mc/mb)¹

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