The helicity suppression mv

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1 Electroweak Corrections to Bs,d --+.e.+.e.m. GORBAHN Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom The rare decay Bs,d -+ t+l- plays a central role in current particle physics phenomenology. In this proceeding we discuss the recent progress in the calculation of two-loop electroweak corrections, which helped to remove a 7% uncertainty in the theory prediction. We will put a special emphasis on the different renormalisation schemes used for the two-loop electroweak matching calculation. 1 Introduction The rare decays of Bq --+ c+c- with q = d, s and C = e, µ, T are highly suppressed in the standard model and exhibit a particular sensitivity to physics beyond the standard model. Two suppression mechanisms are at work in the standard model: the decays are generated first at one-loop through W-box and Z-penguin diagrams and the respective matrix element is helicity suppressed by mv The helicity suppression mv of the branching ratio can be lifted in models where scalar flavour changing operators are generated - one typical example are models with extra Higgs doublets. A precise measurement of these decay modes can constrain models with scalar masses reaching a few TeV provided the theory uncertainty is comparable to, or less then, the experimental one. The current experimental world average for the time-integrated branching ratio 1 (1) M. M. had been obtained combining the results of CMS 2 and LHCb. 3 The experimental uncertainty is expected to be significantly reduced for the Bs --+ µ+ µ- decay in the next decade. To keep up with this experimental progress NLO EW 4 and NNLO QCD 5 contributions were recently calculated and combined. 6 In this proceeding we will discuss the impact of higher order electroweak (EW) and QED corrections to the SM one-loop W-box and Z-penguin diagrams. The calculations of the respective corrections are best organised using an effective field theory framework. Here the effect of heavy particles, i.e. the weak gauge bosons, the top-quark and the Higgs boson, is encoded in an effective Lagrangian comprising only light fields and higher dimensional operators. The resulting effective coupling constants are called Wilson coefficients and are suppressed by powers of heavy masses. At the renormalisation scale µb "" mb, close to the scale of the B-Meson mass, the decay Bq --+ c+c- is governed to a very good approximation by the effective t:;.b = 1 Lagrangian (2) Here V;j denotes the relevant elements of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, while Q 10 = [<1L /µ bl] [l1µ/5 ] is the only relevant operator in the standard model and C10 its Wilson coefficient. The Wilson coefficients of the scalar operators Qs = [<1L bl] [l ] and

2 Qp = [<1L bl] [l15 ] are highly suppressed in the standard model, while the matrix element of all other standard model current-current and penguin type operators vanish as long as QED interactions are neglected in the evaluation of the respective matrix elements. Noting that since the helicity suppression cannot be lifted at O (a2 ) in the electromagnetic coupling constant, it is clear that the only enhanced QED or electroweak corrections can originate from higher order corrections to the Wilson coefficient Cw at the scale µb. 2 Branching Ratio For the effective Lagrangian of Eq. (2) the fully time-integrated branching ratios read Br -= -- yq ICw 1 2, 1 N- where N= TBq MB3 q 1B2 q m2 I v;bv;q 1 2 e S 7r M'JJ. * Vl - 4 me2;mb 2 (3) is a normalisation factor, while the factor of Yq =.6.I'q/(2I'q) incorporates the effects of time integration, which is particularly important for the large decay width difference ti.r8 of the B8 system. 7 The normalisation factor exhibits the helicity suppression due to the lepton mass me and depends on the lifetime TB and the mass MB. of the Bq meson. Moreover, a sin gle hadronic parameter enters, the Bq decay constant fb., which is defined via the relation (OIQJµf5 b!bq(p)) = ifbqpµ; where Pµ, denotes the four momentum of the Bq meson. The deter mination of the decay constant is nowadays subject to lattice calculations with errors at a few percent level, eliminating this previously major source of uncertainty The uncertainties due to fbq, TB and Yq approach a level of below in the branching ratio. Given the small parametric uncertainties, perturbative corrections and the estimation of higher order contributions become particularly important. At leading order we have G}Mfir _ Cw = --- CW ' where cw = -Yo (Xt ). 7r 2 Here Yo (x) ( ) = -x8 xx (x 3-x1 )2 ln x (4) (5) is a gauge independent function 14 of the ratio of the top-quark and W-boson mass Xt = m'f/mfir. At lowest order in the EW interactions the above normalisation is equivalent due to the tree-level relation Gp = 7rll<e/ ( v'2mfirs ) to the traditional normalisation --, ae Yo (xt) 47r s where cw = -- (6) which incorporates the sine of the weak mixing sw = sin Ow and the electromagnetic coupling constant 'e The top-quark mass is the only parameter that is renormalised if we consider pure QCD corrections, while the Fermi constant Gp, ae, sw, Mw and mt depend on the electroweak renormalisation scheme. 3 Renormalisation Schemes Before discussing the electroweak scheme dependence, we note that the strong dependence of Cw on the choice of the renormalisation scheme for Mt is removed after including NLO and NNLO 5 QCD corrections. The QCD corrections are particularly small if we renormalise the top-quark mass in the MS-scheme at the scale of the MS renormalised top-quark mass, i.e. if we use mt(mt) = for the QCD renormalisation. Here mt denotes the top-quark mass, where QCD corrections are MS-renormalised, but EW corrections are considered in the on-shell scheme. In the case that the latter are also MS-renormalised, we shall choose the

3 notation ffit. The additional shift from mt -+ ffit, while numerically quite significant yielding = GeV, is dominated by the contribution of tadpole diagrams. The tadpole induced shift cancels in the ratio Xt mr IM entering the LO Wilson coefficient. Yet, there is an additional finite shift in the top-quark mass beyond the tadpole contribu tion which will change Xt and the LO prediction of the Wilson coefficient C10 in the MS-scheme accordingly. The resulting shift in the numerical value of the LO Wilson coefficient will be com pensated by a modification in the analytic expression of the two-loop electroweak contribution in the MS-scheme, such that the numerical value of the sum of the two will be invariant under a change of scheme up to residual higher-order corrections. This property extends to other shifts in the renormalisation and we will employ different renormalisation schemes to estimate the size of potential higher-order corrections. In particular the shift due to the non-tadpole contribu tions in the top-quark mass will result at LO in a 2.53 shift in the branching ratio. Similarly a change of scheme of sw in (6) from MS to the on-shell definition will shift the branching ratio by 73. While the inclusion of two-loop electroweak corrections 19 in the large mt limit reduces these shifts to 13 and 53 respectively only after incorporating the full two-loop results 4 can we reduce the uncertainty due to higher order electroweak corrections to less then 13. For the two-loop calculation - for sample diagrams see Figure 1 four different renormalisation schemes have been employed to estimate the size of the residual theory uncertainty. Yet, it is essential ffit(ffit) = - b q Figure 1 - Two-loop diagrams in the SM contributing to the b -+ qt+e- at NLO in EW interactions. that the same physical input is used for the numerical evaluation. Only then will the numerical results agree up to residual higher order corrections and we can use the numerical difference to estimate the size of potential higher order contributions. To be specific we choose (7) as the physical input in all schemes for the numerical evaluation, where a8 denotes the strong coupling constant and MH the mass of the standard model scalar Boson. Having fixed the physical input, we define the different renormalisation schemes and discuss the relation of their renormalised parameters to the physical input in Eq. (7). In all schemes we use MS renormalisation for O:e and the top-quark mass under QCD, whereas additional finite terms are included into the field renormalisation constants in order to obtain canonical kinetic terms in the effective theory. Therefore, our schemes differ only by finite EW renormalisations of the parameters appearing at LO in c10 or cio respectively. For c10 these are sw, Mt and Mw and we define the 3 following schemes: 1. On-shell scheme: In the on-shell scheme, at the order we consider, the on-shell masses of Z boson and top quark coincide with their pole masses. The mass of the W boson is a dependent quantity for our choice of physical input. We calculate it including radiative corrections following Ref. 20 This relation introduces a mild Higgs-mass dependence of C10 at LO. The weak mixing angle in the on-shell scheme is defined by (8)

4 Therefore, the only finite counterterms necessary are omi, oma, and OMt at one-loop, they are given in Refs We also treat tadpoles as in Refs : we include tadpole diagrams (see Fig. 1), and a renormalisation ot to cancel the divergence and the finite part of the one-loop tadpole diagram. This way we ensure that all renormalisation constants apart from wave function renormalisations are gauge invariant MS scheme: In the MS scheme the fundamental parameters are those of the "unbroken" SM Lagrangian v, and (9) Yt Here 93, 92 and 91 are the couplings of the SM gauge group SU(3)c x SU(2)L x U(l)y, v is the vacuum expectation value of the Higgs field and >. its quartic self-coupling, whereas Yt is the top-yukawa coupling. The parameters are renormalised by counterterms subtracting only divergences and log (4n) 'YE terms, i.e., they are running MS parameters. We do not treat tadpoles differently in this respect, only their divergences are subtracted by the counterterm for v. By expressing the parameters of the LO Wilson coefficients in terms of the "unbroken" -phase parameters s& = 9if (9i + 9 ), 4na:e = 9foV(9i + 9 ), (10) - Mw = v92 /2 ' Xt = 2yz /9 ' we iteratively fix the values of the "unbroken" parameters at the matching scale. To this end, we use the renormalisation group equations to evolve the MS parameters to the respective scale of the top-quark, the Higgs-boson and weak-bosons and require that the physical input in Eq. (7) is reproduced using one-loop relations for all parameters but the Higgs-mass. a 3. Hybrid scheme: When sw appears at LO, we may adopt yet another scheme. We renormalise the couplings O:e and sw in the MS scheme and the masses in Xt on-shell. Effectively this corresponds to including the on-shell counterterms for masses and using Eq. (10) instead of Eq. (8) for sw. Correspondingly, we use sw, O:e, Mt, Mw and MH as fundamental parameters for the hybrid scheme. This scheme is a better-behaved alterna tive to the on-shell scheme, in which the counterterm for sw receives large top-quark mass dependent corrections. For c10 we employ only one renormalisation scheme: 4. On-shell scheme 2: In the c10 normalisation a:e/ s& is absorbed in the additional factor GpMa, and both coupling constants do not appear at LO. Hence only the electroweak parameters mi, Mw have to be renormalised and we use the on-shell scheme for their definition as described in scheme 1. In all four schemes the parameter Gp denotes the Fermi constant extracted from muon life-time experiments. 4 Numerics We will now consider the size and the reduction of the scheme dependences in C10 at the matching scale (11) "Since the LO result does not depend on the Higgs-mass we employ the tree-level relation for its mass at the scale µ = MH.

5 for the single- and quadratic-cf normalization respectively, after including the NLO EW cor rections ci 2) and zj_ 2) respectively. To separate the effects of the EW calculation we fix the QCD renormalisation of the top-quark mass by setting the corresponding QCD scale to mt ( mt ) and using on-shell mass under EW renormalization, as far as the OS-1, OS-2 and HY schemes are concerned. In the MS scheme we perform the additional shift mt -+ mt using the value of mt (mt ) as input value. As noted above, for this choice of scale the omitted higher order QCD corrections are particularly small, i.e. the LO result accounts for the dominant part of the higher-order QCD correction. 0 :: x 10. OS-2 D Ed OS-1 s HY MS DO Figure 2 - Comparison of the matching scale,, dependence of C10 at the scale in four renormalisation schemes (OS-2, OS-1, HY and MS) at LO (dotted) and with NLO EW corrections (solid). The LO and (LO + NLO EW) results are depicted in Figure 2 for the four renormalization schemes. For -independent top-quark mass the LO C10 is independent in the OS-2 scheme, whereas the replacement CF -+ ae()/(s\'.v-shell)2 introduces a dependence in OS-1 and a quite significant shift of about 43 with respect to OS-2, which translates into a 83 change of the LO branching ratio. Although based on the same single-cf normalization, the MS and HY schemes exhibit relatively large shifts with respect to OS-1 and a modified dependence due to the MS renormalization of sw in both, HY and MS, schemes and additionally the EW MS renormalization of the top-quark and W mass in the MS scheme. The overall uncertainty due to EW corrections at LO may be estimated from the variation of C10 given by all four schemes ranging in the interval C10(µ0) E [-8.9, -8.2] 10-8 for E [50, ] GeV corresponding to a ±83 uncertainty on the level of the branching ratio. The inclusion of the NLO EW corrections eliminates this large uncertainty, as all four schemes yield aligned (LO + NLO EW) results and show the same dependence - up to residual higher order corrections. It is noteworthy that only a mild renormalisation scale dependence remains after the inclusion of the two-loop matching corrections. This small scale dependence will be canceled up to higher orders if we go beyond LO in QED through the resulting operator mixing. Yet, at LO in the effective theory there is no renormalization group mixing of C10 and the dependence may be used directly as an uncertainty. Hence the strong reduction of the dependence in 2 is due to the inclusion of NLO corrections in the relation of EW parameters. For technical reasons 4 we will use the difference of the HY and the OS-2 scheme to es timate the residual electroweak uncertainties. After including the NLO matching corrections the resulting effective Lagrangian is run to the scale µb using the anomalous dimension as already discussed in a previous analysis of Bq -+ c+c The solution of the RGE involves the mixing of current-current, QCD penguin operators and Q9 = [ih 1'µ blj ll1'µ CJ - further details can be found in the literature4. Yet, only the matrix-element of Qio for Bq -+ c+c- is known in pure QCD. Beyond LO in QED also other operators could contribute, e.g. Qg or current-current operators. To estimate the size of these missing contributions we use the residual µb dependence for the fixed value = 160 GeV of C10(µb) It is shown in Figure 3 at LO, NLO QCD and NLO (QCD + EW) in the OS-2 and HY schemes. As our final result we choose for the central value the OS-2 scheme with scale settings

6 = 160 GeV and µb = 5 GeV C10 = (-8.34 ± 0.04) 10-8, (12) where we have estimated higher-order corrections of EW origin from the scale variations of E [50, ] GeV and µb E [2.5, 10] GeV in two schemes, OS-2 and HY, and added linearly the two errors. This result has to be combined 6 with the NNLO QCD corrections 5 - for further details see another contribution to these proceedings 27. µ0-7.5 x 10-s OS x 10-s HY >-.:. ::: :.:: :.:. :. :: :.: :.: :.: :.: ;ii µb µb 10.0 Figure 3 - The µb dependence of the Wilson coefficient C10 (µb ) for fixed = 160 GeV in two renormalisation schemes (OS-2, HY) at LO (dotted), NLO QCD (dashed) and NLO (QCD + EW) (solid). 5 Conclusions We presented results of the next-to-leading (NLO) electroweak (EW) corrections to the Wilson coefficient C10 that governs the rare decays Bq --t +R- in the Standard Model. Incorporating these corrections leads to a drastic reduction in the perturbative uncertainty. This was made explicit by studying the renormalisation scheme dependence in different electroweak renormali sation schemes. The result removes an uncertainty of about ±73 at the level of the branching ratio. Acknowledgments Martin Gorbahn would like to thank the organisers for the invitation to the XLIXth Rencontres de Moriond session devoted to Electroweak Interactions and Unified Theories and acknowledges partial support by the UK Science & Technology Facilities Council (STFC) under grant number ST /G00062X/1. References 1. CMS and LHCb Collaborations, EPS-HEP 2013, Conference Report No. CMS-PAS-BPH13-007, LHCb-CONF , 2. S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. Lett. 111, (2013) [arxiv:l ]. 3. R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett , (2013) [arxiv: ]. 4. C. Bobeth, M. Gorbahn and E. Stamou, Phys. Rev. D 89, (2014) [arxiv:l ]. 5. T. Hermann, M. Misiak and M. Steinhauser, JHEP 1312, 097 (2013) [arxiv: ].

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