Electroweak Corrections to B s,d → ℓ + ℓ −
FFLAVOUR(267104)-ERC-54LTH 991
Electroweak Corrections to B s,d → (cid:96) + (cid:96) − Christoph Bobeth,
1, 2, ∗ Martin Gorbahn,
1, 3, † and Emmanuel Stamou
1, 2, 4, ‡ Excellence Cluster Universe, Technische Universit¨at M¨unchen, D–85748 Garching, Germany Institute for Advanced Study, Lichtenbergstrasse 2a,Technische Universit¨at M¨unchen, D–85748 Garching, Germany Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom Department of Particle Physics and Astrophysics,Weizmann Institute of Science, Rehovot 76100, Israel
We calculate the full two-loop electroweak matching corrections to the operator governing thedecay B q → (cid:96) + (cid:96) − in the Standard Model. Their inclusion removes an electroweak scheme andscale uncertainty of about ±
7% of the branching ratio. Using different renormalization schemesof the involved electroweak parameters, we estimate residual perturbative electroweak and QEDuncertainties to be less than ±
1% at the level of the branching ratio.
I. INTRODUCTION
The rare decays of B q → (cid:96) + (cid:96) − with q = d, s and (cid:96) = e, µ, τ are helicity suppressed in the Standard Model(SM) and can be predicted with high precision, whichturns them into powerful probes of nonstandard inter-actions. In November 2012, LHCb [1] reported first ex-perimental evidence of the decay B s → µ + µ − with asignal significance of 3 . σ and the time integrated andCP-averaged branching ratioBr( B s → µ + µ − ) = (cid:0) . +1 . − . (stat) +0 . − . (sys) (cid:1) · − , (1)well in agreement with SM predictions. More recently,the signal significance was raised to 4 . σ after analyzingthe currently available data set of 1 fb − at √ s = 7 TeVand 2 fb − at √ s = 8 TeV, with the result [2]Br( B s → µ + µ − ) = (cid:0) . +1 . − . (stat) +0 . − . (sys) (cid:1) · − . (2)CMS confirmed this independently utilizing the completedata set of 5 fb − at √ s = 7 TeV and 20 fb − at √ s = 8TeV [3] obtainingBr( B s → µ + µ − ) = (cid:0) . +0 . − . (stat) +0 . − . (sys) (cid:1) · − (3)and the slightly better signal significance of 4 . σ .The large decay width difference ∆Γ s of the B s systemimplies that the instantaneous branching ratio at time t = 0, Br [ t =0] ( B q → (cid:96) + (cid:96) − ), deviates from Br. Neglect-ing for a moment cuts on the lifetime in the experimentaldetermination of Br, the fully time-integrated and the in-stantaneous branching ratios are related in the SM as [4]Br = Br [ t =0] − y q , where y q = ∆Γ q q . (4)LHCb has measured y s = 0 . ± .
014 [5, 6] and estab-lished a SM-like sign for ∆Γ s [7]. By 2018, the experi-mental accuracy in Br is expected to reach 0 . · − and ∗ [email protected] † [email protected] ‡ [email protected] with 50 fb − . · − [8], the latter corresponding tothe level of about 5% error with respect to the currentcentral value. Results of comparable precision may beexpected from CMS, and perhaps also from ATLAS.Motivated by the experimental prospects, this workpresents a complete calculation of the next-to-leading(NLO) electroweak (EW) matching corrections in theSM, supplemented with the effects of the QED renor-malization group evolution (RGE). Thereby, we removea sizable uncertainty which has often been neglected inthe past and became one of the major theoretical un-certainties after the considerable shrinking of hadronicuncertainties from recent progress in lattice QCD.After decoupling the heavy degrees of freedom of theSM – the top quark, the weak gauge bosons and theHiggs boson – at lowest order in EW interactions, thedecay B q → (cid:96) + (cid:96) − is governed by an effective ∆ B = 1Lagrangian L eff = V tb V ∗ tq C P + L (5)QCD × QED + h.c. (5)with a single operator P = [¯ q L γ µ b L ][¯ (cid:96) γ µ γ (cid:96) ] and itsWilson coefficient C , as well as the QCD × QED inter-actions of leptons and five light quark flavors. V ij de-notes the relevant elements of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. Here we deviatefrom the usual convention to factor out combinations ofEW parameters , such as Fermi’s constant, G F , the QEDfine structure constant, α e , the W -boson mass, M W , orthe sine of the weak mixing angle s W ≡ sin( θ W ). Themost common normalizations are C = 4 G F √ c , C = G F M W π (cid:101) c , (6)with the LO Wilson coefficients c = − α e π Y ( x t ) s W , (cid:101) c = − Y ( x t ) . (7) Since we shall not vary the EW renormalization scheme of theCKM factor V tb V ∗ tq , we prefer to keep it as a prefactor, to havea universal C for both q = d, s . a r X i v : . [ h e p - ph ] N ov They depend on the gauge-independent function Y [9],where x t = ( M t /M W ) denotes the ratio of top-quark to W -boson masses. We will frequently refer to the choice c and (cid:101) c as the “single- G F ” and “quadratic- G F ” nor-malization, respectively. The former choice is the stan-dard convention of the ∆ B = 1 effective theory in theliterature, whereas the latter choice removes the depen-dence on α e and s W in favor of G F and M W [10]. Atlowest order in the EW interactions both normalizationsmay be considered equivalent due to the tree-level rela-tion G F = πα e / ( √ M W s W ). In practice, however, largedifferences arise once numerical input for the EW pa-rameters is used that corresponds to different renormal-ization schemes. For example, a noticeable 7% change ofthe branching ratio is caused by choosing s W = 0 . s W = 0 . C depends on the choice of normalization as well asthe EW renormalization scheme of the involved param-eters. Especially the power of G F affects the match-ing, whereas the choice of EW renormalization schemeimplies different finite counterterms for the parameters.Thereby, the overall numerical differences among the dif-ferent choices of normalizations and EW renormalizationschemes become much smaller, removing the large uncer-tainty present at lowest order.The instantaneous branching ratio takes the formBr [ t =0] ( B q → (cid:96) + (cid:96) − ) = N (cid:12)(cid:12) C (cid:12)(cid:12) , (8)with the normalization factor N = τ B q M B q f B q π | V tb V ∗ tq | m (cid:96) M B q (cid:113) − m (cid:96) /M B q . (9)It exhibits the helicity suppression due to the lepton mass m (cid:96) and depends on the lifetime τ B q and the mass M B q of the B q meson. Moreover, a single hadronic parameterenters, the B q decay constant f B q , (cid:10) | ¯ q γ µ γ b | ¯ B q ( p ) (cid:11) = if B q p µ . (10)It is nowadays subject to lattice calculations with errorsat a few percent level, eliminating this previously majorsource of uncertainty [12–15]. The uncertainties due to f B q , τ B q and y q approach a level of below 3% [16] in Br.Concerning perturbative uncertainties, the strong depen-dence of C on the choice of the renormalization schemefor M t is removed when including the NLO QCD contri-bution in the strong coupling α s [17–20]. So far the fullNLO EW corrections have not been calculated and in thiswork we close this gap as previously done for the analo-gous corrections to s → dν ¯ ν [21]. Being usually ignoredin the budget of theoretical uncertainties of Eq. (8), theimportance of a complete calculation has recently beenemphasized [22]. There, the NLO EW corrections in thelimit of large top-quark mass have been employed, whichis known to be insufficient at the level of accuracy aimed at Ref. [21] and the residual EW uncertainties were esti-mated to be at least 5% on the branching ratio.In Sec. II we describe the calculation of the NLO EWcorrection to C adopting the two choices of normaliza-tion and using different renormalization schemes for theinvolved EW parameters. In Sec. III, we summarize thesolution of the RGE and obtain C at the low-energyscale of the order of the bottom-quark mass at the NLOin EW interactions. Finally, in Sec. IV we discuss the re-duction of the EW renormalization-scheme dependencesin C after the inclusion of NLO EW corrections. Inthe accompanying appendices A and B we collect addi-tional technical information on the matching calculationand the RGE, respectively. Some supplementary detailsof Sec. IV have been relegated to App. C. II. MATCHING CALCULATION OF NLOELECTROWEAK CORRECTIONS
We obtain the EW NLO corrections to the Wilson co-efficient C by matching the effective theory of EW in-teractions to the Standard Model. For this purpose weevaluate one-light-particle irreducible Greens functionswith the relevant external light degrees of freedom upto the required order in the EW couplings in both theo-ries. The Wilson coefficients are determined by requiringequality of the renormalized Greens functions order byorder A full ( µ ) ! = A eff ( µ ) (11)at the matching scale µ . It is chosen of the order of themasses of the heavy degrees of freedom to minimize oth-erwise large logarithms that enter the Wilson coefficients.The Wilson coefficients have the general expansion C i ( µ ) = C (00) i + ˜ α s C (10) i + ˜ α s C (20) i + ˜ α e (cid:16) C (11) i + ˜ α s C (21) i + ˜ α e C (22) i (cid:17) + . . . , (12)in the strong and electromagnetic ˜ α s,e ≡ α s,e / (4 π ) run-ning couplings of the effective theory at the scale µ ,where we follow the convention of Ref. [23]. This expan-sion starts with tree-level contributions denoted by thesuperscript (00), has higher-order QCD corrections ( m m >
0, pure QED corrections ( mm ) with m > mn ) with m > n >
0, allof which depend explicitly on µ except for (00). For C the non-zero matching corrections start at order ˜ α e , i.e.,for n ≥
1. The C (11)10 [9] and C (21)10 [17–20] contributionsare known and here we calculate C (22)10 . Above, Eq. (12)has to be understood as the definition of the components C ( mn ) i that complies with the single- G F normalization inthe literature [23]. Comparison with Eqs. (6) and (7)yields C (11)10 = 4 G F √ c (11)10 = − G F √ Y ( x t ) s W (13)and C (11)10 = G F M W π ˜ α e (cid:101) c (11)10 = − G F M W π ˜ α e Y ( x t ) (14)showing that this convention introduces an artificial fac-tor 1 /α e into the components in the case of the quadratic- G F normalization. However, we will organize the renor-malization group evolution (see Sec. III) such that thesefactors are of no consequence, as should be.Although the operator P does not mix with other∆ B = 1 operators under QCD, at higher order in QEDinteractions such a mixing does take place [23, 24]. Asa consequence the effective Lagrangian (5) has to be ex-tended C P −→ (cid:88) i C i P i , (15)where the term ∼ V ub V ∗ uq [ C ( P c − P u ) + C ( P c − P u )]does not contribute to the order considered here. Theoperators relevant to B q → (cid:96) + (cid:96) − at the considered or-der in strong and EW interactions comprise the current-current operators ( i = 1 , i = 3 , , ,
6) and the semi-leptonic operator ( i = 9 , α e / (4 π ) in P , . This factor isincluded in the matching conditions of the Wilson coeffi-cients at the matching scale µ in Eq. (12). In the match-ing calculation only P and P as defined in App. A 1are needed, whereas the remaining operators enter in therenormalization group evolution discussed in Sec. III.We describe the calculation of A full and A eff in Sec-tions II A and II B, respectively. In the SM calculationof A full , we apply different EW renormalization schemesfor the involved parameters to demonstrate in Sec. IVthat the renormalization scheme dependence is reducedto sub-percent effects when including C (22)10 . The schemesdiffer by finite parts of the counterterms that renormalizethe bare parameters of the Lagrangian or equivalently theparameters appearing in the LO Wilson coefficient. Nev-ertheless, we use the same physical input in all schemesfor the numerical evaluation that we have chosen to be G F , α e ( M pole Z ) , α s ( M pole Z ) ,V ij , M pole Z , M pole t , M pole H . (16) G F is the Fermi constant as extracted from muon life-time experiments. It is itself a Wilson coefficient of theeffective theory and plays thus a special role in the calcu-lation of EW corrections; we postpone further discussionto Section II B. The couplings α e and α s are the MS cou-plings at the scale of the Z pole mass in the SM withdecoupled top quark . V ij are elements of the CKM ma-trix. M pole Z , M pole t and M pole H are the pole masses of Z boson, top quark and Higgs boson, respectively. Thenumerical values are summarized in Tab. I. I.e. W and Z bosons are still dynamical degrees of freedom. Parameter Value Ref. G F .
166 379 · − GeV − [11] α s ( M pole Z ) ( N f = 5) 0 . ± . α e ( M pole Z ) ( N f = 5) (127 . ± . − [11] M pole Z (91 . ± . M pole t (173 . ± .
9) GeV [11, 25, 26] M pole H (125 . ± .
4) GeV [11, 27, 28]∆ α (5) e, hadr ( M pole Z ) 0 . ± . α s,e are the running MScouplings of the five-flavor theory at µ = M Z . Masses are theexperimentally measured pole masses. A. Standard Model Calculation
We keep only the leading contributions of the expan-sion in the momenta of external states, in which case thefull amplitude for b → q(cid:96) + (cid:96) − takes the form A full = (cid:88) i A full , i ( µ ) (cid:104) P i ( µ ) (cid:105) (0) . (17) (cid:104) P i ( µ ) (cid:105) (0) denote the tree-level matrix elements of op-erators mediating b → q(cid:96) + (cid:96) − , i.e., i = 9 ,
10 as well asevanescent operators defined in App. A 1. The A full ,i ’sare coefficient functions with the electroweak expansion A full , i = A (0)full , i + ˜ α e A (1)full , i + ˜ α e A (2)full , i + . . . , (18)with α e of the SM, i.e. six active quark flavors as wellas heavy weak gauge bosons and the Higgs boson. Inthe case of the single- G F normalization, A (0)full , i = 0 for b → q(cid:96) + (cid:96) − mediating operators, whereas A (0)full , i (cid:54) = 0 forthe quadratic- G F normalization due to the substitution α e /s W → G F .Our focus here is the calculation of the two-loop con-tribution to A full , and some parts of A full , i at one-loop that involve evanescent operators E and E (seeApp. A 1). For this purpose, we calculate all two-loopEW Feynman diagrams and the corresponding one-loopdiagrams with inserted counterterms, Fig. 1 depicts someexamples. We proceed as in Ref. [21] and perform allcalculations in the Feynman gauge ξ = 1 using two in-dependent setups. Similarly to Ref. [21] also here wefind contributions from electroweak gauge bosons thatare 1 /s W enhanced. In App. A 2 we discuss the moretechnical aspects of the calculation, e.g. γ -algebra in d -dimensions and loop-integrals. Here, we concentrate onthe electroweak renormalization conditions.Having fixed the physical input, we define threerenormalization schemes and discuss the relation oftheir renormalized parameters to the physical inputin Eq. (16). In all three schemes we use MS renor-malization for α e and the top-quark mass under QCD,whereas additional finite terms are included into the fieldrenormalization constants as explained in more detail in t ℓZ γG − ℓb ℓq G + bZb W + tℓb ℓq tW − Z H tℓb ℓq
FIG. 1. Two-loop diagrams in the SM contributing to the b → q(cid:96) + (cid:96) − at NLO in EW interactions. App. A 2. Therefore, our schemes differ only by finiteEW renormalizations of s W , M t and M W appearing atLO in c . For (cid:101) c , s W is absorbed in the additionalfactor G F and needs no further specification. In the on-shell scheme, at the order we consider, the on-shell masses of Z boson and top quark coincide with theirpole masses. The mass of the W boson is a dependentquantity for our choice of physical input. We calculate itincluding radiative corrections following Ref. [29]. Thisrelation introduces a mild Higgs-mass dependence of C at LO. The weak mixing angle in the on-shell scheme isdefined by s W ≡ ( s on-shell W ) = 1 − (cid:0) M on-shell W /M on-shell Z (cid:1) . (19)Therefore, the only finite counterterms necessary are δM Z , δM W and δM t at one-loop, they are given inRefs. [30, 31]. We also treat tadpoles as in Refs. [30, 31]:we include tadpole diagrams (see Fig. 1), and a renor-malization δt to cancel the divergence and the finitepart of the one-loop tadpole diagram. This way weensure that all renormalization constants apart fromwave function renormalizations are gauge invariant [32]. In the MS scheme the fundamental parameters arethose of the “unbroken” SM Lagrangian g , g , g , v, λ and y t . (20)Here g , g and g are the couplings of the SM gaugegroup SU (3) c × SU (2) L × U (1) Y , v is the vacuum ex-pectation value of the Higgs field and λ its quartic self-coupling, whereas y t is the top-Yukawa coupling. Theparameters are renormalized by counterterms subtract-ing only divergences and log(4 π ) − γ E terms, i.e., theyare running MS parameters. We do not treat tadpolesdifferently in this respect, only their divergences are sub-tracted by the counterterm for v . By expressing the pa-rameters of the LO Wilson coefficients in terms of the“unbroken”-phase parameters s W = g / ( g + g ) , πα e = g g / ( g + g ) ,M W = vg / , x t = 2 y t /g , (21)we iteratively fix the values of the “unbroken” parame-ters at the matching scale µ . To this end, we require that the physical input in Eq. (16) is reproduced toone-loop accuracy. For Eq. (7), where s W appears at LO, we may adoptyet another scheme. We renormalize the couplings α e and s W in the MS scheme and the masses in x t on-shell. Effectively this corresponds to including the on-shell counterterms for masses and using Eq. (21) insteadof Eq. (19) for s W . Correspondingly, we use s W , α e , M t , M W and M H as fundamental parameters for the hybridscheme. This scheme is a better-behaved alternative tothe on-shell scheme, in which the counterterm for s W re-ceives large top-quark mass dependent corrections. (seeApp. C).Having fixed all renormalization conditions we evaluate A (2)full , . In practice we calculate the MS amplitude andinclude the appropriate counterterms in A (1)full , to shiftfrom the MS to the on-shell or hybrid scheme. The fullexpression for A (2)full , is too lengthy to be included here . B. Effective Theory Calculation
The effective theory is described by the effective La-grangian in Eqs. (5) and (15) with canonically normal-ized kinetic terms for all fields. To simplify the nota-tion we drop any indices indicating an expansion in ˜ α s throughout this Section. The fields and couplings areMS-renormalized via the redefinitions of bare quantities d → (cid:112) Z d d , (cid:96) → (cid:112) Z (cid:96) (cid:96) , C j → (cid:88) i C i ˆ Z i,j , (22)where d denotes down-type quark fields and (cid:96) denotescharged-lepton fields. The renormalization constant ofthe Wilson coefficients is the matrix ˆ Z i,j arising fromoperator mixing. It has an expansion in ˜ α e ˆ Z i,j = δ i,j + ˜ α e ˆ Z (1) i,j + ˜ α e ˆ Z (2) i,j + . . . (23) We attach the complete analytic two-loop EW contribution inthe on-shell scheme for the quadratic- G F normalization, (cid:101) c (22)10 ,to the electronic preprint. analogously to the expansion of the renormalization con-stants of the fields and couplings given in Eq. (A12).All loop diagrams in the effective theory vanish, sincewe set all light masses to zero, expand in external mo-menta and employ dimensional regularization. Accord-ingly, the effective theory amplitude is entirely deter-mined through the product of tree-level matrix elements (cid:104) P j (cid:105) (0) , the Wilson coefficients C i and appropriate renor-malization constants. The renormalized amplitude reads A eff ( µ ) = (cid:88) i A eff , i ( µ ) (cid:104) P i ( µ ) (cid:105) (0) = V tb V ∗ tq (cid:88) i,j C i ( µ ) ˆ Z i,j Z j (cid:104) P j ( µ ) (cid:105) (0) . (24)As mentioned above, both the Wilson coefficients C i and the renormalization constants are expanded in ˜ α e as given in Eqs. (12) and (23), respectively. The Z j ’ssummarize products of field- and charge-renormalizationconstants of the operator in question, i.e. for P Z = Z d Z (cid:96) , (25)which is the one required up to two-loop level in ˜ α e .Only a few physical operators contribute to the part ofthe amplitude in Eq. (24) proportional to (cid:104) P (cid:105) (0) sinceonly a few mix either at one-loop or two-loop level into P and have, at the same time, a non-zero Wilson coef-ficient at one-loop or tree-level, respectively. These are:the operator P having a non-zero Wilson coefficient C (00)2 as well as an entry in ˆ Z (2)2 , and P that mixes at one-loopinto P and have a non-vanishing C (11) i . Apart from thephysical operators also one evanescent operator, i.e. E contributes. We give the definition of the operators inApp. A 1 and present some details on the calculation ofthe renormalization constants in the five-flavor theory inApp. A 3. All contributing mixing renormalization con-stants of physical operators can be extracted from theanomalous dimension in the literature [24]. We collectall constants and discuss the mixing of evanescent oper-ators in App. A 3. Finally, at the two-loop level A (2)eff , = V tb V ∗ tq (˜ α e ) n (cid:20) C (22)10 + C (11)10 Z (1)10 + C (00)2 ˆ Z (2)2 , + (cid:88) i =9 ,E C (11) i ˆ Z (1) i, (cid:21) (26)with the power n = 2 and n = 1 for the single- andquadratic- G F normalization, respectively. In this equa-tion α e is the electromagnetic coupling constant in the∆ B = 1 effective theory. It differs from the one in Tab. Iby threshold corrections due to W and Z gauge bosonsand from the one in the SM in Eq. (18) by the addi-tional top-quark threshold corrections as explained aboveEq. (A9). Note that the renormalization constant ˆ Z (2)2 , ,see Eq. (A14), implies the existence of a quadratic loga-rithm that will be resummed with the help of the RGEin Sec. III. The one-loop Wilson coefficients in Eq. (26), multipliedwith renormalization constants, contribute finite termsto the matching through their O ( (cid:15) ) terms. We repro-duce the finite and O ( (cid:15) ) parts of C (11)9 , in [33]. For C (11) E only the finite term is needed, we give it in App. A 3.For this purpose we have matched also the one-loop am-plitudes proportional to the (cid:104) P , , E (cid:105) (0) keeping O ( (cid:15) )terms when required.The Fermi constant, G F , is very precisely measured inmuon decay and provides a valuable input for the deter-mination of the EW parameters. Following [21], we define G F to be proportional to the Wilson coefficient G µ of theoperator Q µ = (¯ ν µ L γ ρ µ L )(¯ e L γ ρ ν e L ) that induces muondecay in the effective Fermi theory G F ≡ √ G µ = 12 √ (cid:16) G (0) µ + ˜ α e G (1) µ + . . . (cid:17) , (27)with the tree-level matching relation G (0) µ = 2 πα e s W M W = 2 v (28)and the NLO EW correction G (1) µ . Since we work atNLO in EW interactions, G (1) µ enters the effective theoryamplitude in Eq. (24). Moreover, the power of G F inthe normalization of the effective Lagrangian affects thematching contribution of G (1) µ /G (0) µ × C (11) i to C (22) i , incontrast to the leading EW components C (11) i that remainunchanged when using different powers. This can be bestunderstood by the explicit ˜ α e expansion for the single- G F normalization C ∼ G F c ∼ (cid:2) G (0) µ + ˜ α e G (1) µ (cid:3)(cid:2) c (11)10 + ˜ α e c (22)10 (cid:3) (29)= G (0) µ (cid:34) c (11)10 + ˜ α e (cid:32) c (22)10 + G (1) µ G (0) µ c (11)10 (cid:33)(cid:35) + O (˜ α e )and the quadratic- G F normalization C ∼ ( G (0) µ ) (cid:34)(cid:101) c (11)10 + ˜ α e (cid:32)(cid:101) c (22)10 + 2 G (1) µ G (0) µ (cid:101) c (11)10 (cid:33)(cid:35) , (30)which receives an additional factor of 2. Depending onthe choice of normalization, the according contributionproportional to G (1) µ /G (0) µ × C (11) i enters Eq. (26).The merit of defining G F to be itself a Wilson coeffi-cient at the matching scale is that the large uncertaintiesfrom the scale dependence of the vacuum expectationvalue in G (0) µ do not appear at all at LO in the Wilsoncoefficient.This way, we obtain C (22)10 , which has been known onlyin the large top-quark-mass limit [34, 35], by matchingthe parts of A eff ∼ (cid:104) P (cid:105) (0) and A full ∼ (cid:104) P (cid:105) (0) at NLOorder in ˜ α e and verify the explicit cancellations of allleft-over divergences. III. RENORMALIZATION GROUPEVOLUTION
This section summarizes the results of the evolution ofthe Wilson coefficients under the renormalization groupequations from the matching scale µ down to the lowscale µ b . The matching scale µ is of the order ofthe masses of the decoupled heavy degrees of freedom ∼
100 GeV and µ b ∼ B = 1 effective theory, including NLO EW cor-rections, are given in Ref. [24] and the RGE is solvedin Ref. [23] for the single- G F normalized Lagrangian inEqs. (5) and (7) including the running of α e . These cor-rections have already been considered in Ref. [10] in theanalysis of B q → (cid:96) + (cid:96) − .The evolution operator U ( µ b , µ ) relates the Wilsoncoefficients at the matching scale, see Eq. (12), to theones at µ b : C i ( µ b ) = (cid:88) j U ( µ b , µ ) ij C j ( µ ) . (31)At the low-energy scale the Wilson coefficients mayagain be expanded in α s ( µ b ) and the small ratio κ ≡ α e ( µ b ) /α s ( µ b ): C i ( µ b ) = (cid:88) m,n =0 [˜ α s ( µ b )] m [ κ ( µ b )] n C i, ( mn ) . (32)We obtain the explicit expressions for the components C i, ( mn ) ( µ b ) from the solution given in Ref. [23] with fur-ther details and the solution for i = 10 presented inApp. B.In the single- G F normalization the Wilson coefficient c ( µ b ) starts at order α e with the following non-zerocontributions c ( µ b ) = ˜ α e (cid:0) c , (11) + ˜ α s c , (21) (cid:1) + ˜ α e (cid:18) c , (02) ˜ α s + c , (12) ˜ α s + c , (22) (cid:19) . (33)The components c i, ( mn ) are functions of the ratio η ≡ α s ( µ ) /α s ( µ b ) and the high-scale components c ( mn ) j ofEq. (12). For illustration, we give here numerical resultsfor the exemplary values µ = 160 GeV and µ b = 5 GeV, yielding η = 0 . c , (11) = c (11)10 ,c , (21) = η c (21)10 ,c , (02) = 0 . c (00)2 ,c , (12) = 0 . c (00)2 + 0 . c (10)1 − . c (10)4 + 0 . c (11)9 + 1 . c (11)10 ,c , (22) = 0 . c (10)1 + 0 . c (10)4 + 0 . c (20)1 + 0 . c (20)2 + 0 . c (20)3 − . c (20)4 + 0 . c (20)5 − . c (20)6 − . c (11)9 − . c (11)10 + 0 . c (21)9 + 0 . c (21)10 + c (22)10 . (34)We give the explicit solution for arbitrary values of η inApp. B 2. Furthermore, the c , ( mn ) depend on the initialmatching conditions of the Wilson coefficients, the c ( mn ) i in Eq.(12), at various orders: tree-level for i = 2, one-loop in α s for i = 1 , α e for 9 ,
10 and two-loop in α s for i = 1 , . . . , α e α s for i = 9 ,
10 [33] as wellas the two-loop NLO EW correction for i = 10 presentedin Sec. II.We derive the equivalent expressions for the case of thequadratic- G F normalization from the single- G F normal-ization in Eq. (32) (cid:101) c i ( µ b ) = (cid:88) m,n =0 [˜ α s ( µ b )] m − [ κ ( µ b )] n − (cid:101) c i, ( mn ) . (35)For i = 10 the lowest-order non-zero terms (cid:101) c ( µ b ) = (cid:101) c , (11) + ˜ α s (cid:101) c , (21) + ˜ α e (cid:18) (cid:101) c , (02) ˜ α s + (cid:101) c , (12) ˜ α s + (cid:101) c , (22) (cid:19) , (36)already start at order α e . The components of the initialWilson coefficients in Eq. (12) are related as (cid:101) c ( mn ) i = s W c ( mn ) i for n < , (37)where a factor ˜ α e ( µ ) has been pulled out and substitutedby ˜ α e ( µ b ). For cases n ≥
2, which is here only of concernfor C , an additional shift has to be taken into accountexplicitly in the matching analogously to the discussionbelow Eq. (27). Eventually, the downscaled components (cid:101) c i, ( mn ) in Eq. (35) are given by Eq. (34) with the replace-ment c ( mn ) i → (cid:101) c ( mn ) i and by omitting the contributionsof (cid:101) c (11)10 in (cid:101) c , (12) as well as (cid:101) c (11)10 and (cid:101) c (21)10 in (cid:101) c , (22) , asexplained in more detail in App. B. IV. NUMERICAL IMPACT OF NLO EWCORRECTIONS
In Sec. II we presented the details of the calculation ofthe complete NLO EW matching corrections to the Wil-son coefficient C in the SM and in Sec. III the effects ofthe renormalization group evolution within the ∆ B = 1effective theory from the matching scale µ to the lowenergy scale µ b . In this section, we discuss the numericalimpact of these corrections on C at both scales and as-sess the reduction of theoretical uncertainties associatedwith the different choices of the renormalization scheme.Finally, we shall briefly comment on the branching ratioBr ∝ |C | .Throughout, we use the four-loop β function for α s in-cluding the three-loop mixed QCD × QED term given inRef. [23]. When crossing the N f = 5 to N f = 6 thresholdat the matching scale µ , we include the three-loop QCDthreshold corrections using the pole-mass value for thetop-quark mass M pole t (see Tab. I). The running of α e is implemented including the two-loop QED and three-loop mixed QED × QCD terms presented in [23], wherethe threshold corrections have been omitted when cross-ing the N f = 5 to N f = 6 threshold entering the evolu-tion of α s . We list the initial conditions for the couplingconstants in Tab. I and remark that the value of α e givenin Ref. [11] refers to the coupling within the SM with thetop quark decoupled. From this value we determine α e at µ = M Z in the SM with N f = 6 with the help ofthe decoupling relation of Eq. (A9) thereby omitting theconstant and logarithmic term from the gauge boson con-tribution and determine the dependent EW parametersas described in Sec. II A. The value of α e in the effec-tive theory is found as described below the decouplingrelation of Eq. (A9).We determine the running top-quark mass in the MSscheme with respect to QCD from M pole t with the aidof the three-loop relation , m t ( m t ) = 163 . m t denotes the top-quark mass, where QCD correctionsare MS-renormalized, but EW corrections are consideredin the on-shell scheme. In the case that the latter are alsoMS-renormalized, we shall choose the notation m t . Theadditional shift from m t → m t , while numerically quitesignificant yielding m t ( m t ) = 172 . x t = m t /M W enteringthe LO Wilson coefficient.As already emphasized in Sec. II, once consideringhigher EW corrections, the different choices of normal-ization of the effective Lagrangian from Eq. (6) affects The choice of the matching scale that determines the N f = 5to N f = 6 threshold has a numerically negligible impact for µ ∈ [50 , differently the NLO EW matching corrections of C . Asrenormalization schemes (RS) we consider the on-shellscheme, the MS scheme and the hybrid scheme intro-duced in Sec. II A, which we abbreviate in the followingas RS = OS, MS and HY. We apply both, the single- G F and the quadratic- G F normalization for the on-shellscheme denoted as RS = OS-1 and OS-2, respectively.For RS = MS and HY we use only the single- G F normal-ization.We first consider the size and the reduction of thescheme dependences in C at the matching scale C ( µ ) = G F √ α e ( µ ) (cid:104) c (11)10 + ˜ α e ( µ ) c (22)10 ( µ ) (cid:105) G F M W π (cid:104)(cid:101) c (11)10 + ˜ α e ( µ ) (cid:101) c (22)10 ( µ ) (cid:105) , (38)for the single- and quadratic- G F normalization respec-tively, after including the NLO EW corrections C (22)10 .To separate the effects of the EW calculation, we firstswitch off any QCD dependence. Namely, we omit theNLO QCD correction C (21)10 and neglect the µ depen-dence of the top-quark mass under QCD by fixing theQCD scale and using m t ( m t ) as the on-shell top-quarkmass under EW renormalization, as far as OS-1, OS-2and HY schemes are concerned. In the MS scheme weperform the additional shift m t → m t using the valueof m t ( m t ) as input value. Note, that for the choice ofscale of m t in the running QCD top mass, the omittedNLO QCD correction C (21)10 is particularly small [18–20],i.e. the LO result C (11)10 accounts for the dominant part ofthe higher-order QCD correction.The LO and (LO + NLO EW) results are depictedin Fig. 2 for the four renormalization schemes. For µ -independent top-quark mass the LO C is µ inde-pendent in the OS-2 scheme, whereas the replacement G F → α e ( µ ) / ( s on − shell W ) introduces a µ dependencein OS-1 and a quite significant shift of about 4% withrespect to OS-2, which translates into a 8% change ofthe LO branching ratio. Although based on the samesingle- G F normalization, the MS and HY schemes ex-hibit relatively large shifts with respect to OS-1 and amodified µ dependence due to the MS renormalizationof s W in both, HY and MS, schemes and additionally theEW MS renormalization of the top-quark and W massin the MS scheme. The overall uncertainty due to EWcorrections at LO may be estimated from the variationof C given by all four schemes ranging in the inter-val C ( µ ) ∈ [ − . , − . · − for µ ∈ [50 , ±
8% uncertainty on the level of thebranching ratio. The inclusion of the NLO EW correc-tions eliminates this large uncertainty, as all four schemesyield aligned (LO + NLO EW) results and the µ depen-dence cancels to large extent in all schemes. The residualuncertainty due to EW corrections is now confined to thesmall interval of C ( µ ) ∈ [ − . , − . · − at the
100 200 300 µ [GeV] − . − . − . G F M W π ˜ c × − OS-2
100 200 300 µ [GeV] G F √ c OS-1
100 200 300 µ [GeV] HY
100 200 300 µ [GeV] MS FIG. 2. Comparison of the matching scale, µ , dependence of C at the scale µ in four renormalization schemes (OS-2, OS-1,HY and MS) at LO (dotted) and with NLO EW corrections (solid). See text for more details. scale µ , it is less than ± .
4% corresponding to ± . µ dependence in Fig. 2 is due to the inclusion of NLO cor-rections in the relation of EW parameters, which are for-mally not part of the effective theory and hence cannotbe cancelled by the RGE in the effective theory. At LOin the effective theory there is no renormalization groupmixing of C and the µ dependence may be used directlyas an uncertainty. As discussed in Sec. III, beyond LO inQED the operator mixing will reduce the remaining µ dependence even further.Before proceeding, we comment on the OS-1 and MSscheme and why we shall discard them for the estimate ofresidual higher-order uncertainties. The OS-1 scheme ex-hibits the worst perturbative behavior of all four schemes,as seen in Fig. 2. The s W -on-shell counterterm inducesthis, for an electroweak correction, unnaturally large shiftat two-loop. As further discussed in App. C, the top-quark mass dependence of the s W -on-shell countertermimplies a significant higher-order QCD scale dependence,which we consider artificial. On the other hand, the OS-2and HY schemes do not exhibit this strong dependenceon the top-quark mass and the estimate of the size ofhigher-order QCD contributions by varying the scale of m t indicates much smaller corrections. In view of this, werestrict ourselves to schemes with reasonable convergenceproperties and leave OS-1 aside. In the case of the MSscheme, the application of RG equations is required forthe iterative determination of the EW parameters fromthe input given in Eq. (16). For the purpose of Fig. 2, thepresence of QCD could be ignored and lowest-order RGequations were sufficient. However, in the general casethe solution of the according RG equations are rather in-volved and we prefer to use the comparison of the HYand OS-2 scheme to estimate higher order EW × QCDcorrections.In the following, we include QCD effects and discuss C at the low-energy scale µ b after applying the RGErunning presented in Sec. III. We express the Wilson co-efficient C ( µ b ) as a double series in the running cou- plings ˜ α s and ˜ α e , see Eqs. (32) and (34), with five rel-evant contributions C , ( mn ) , ( mn = 11 , , , , µ . So far, only the LO ≡ ( mn = 11) and the NLO QCD ≡ ( mn = 11 + 21)contributions were known. Now, we can include the fullNLO EW correction with the additional contributions( mn = 11 + 21 + 02 + 12 + 22) ≡ NLO (QCD + EW) .For this purpose, also the scale dependence of m t thatoriginates from QCD will be taken into account whenvarying the matching scale µ . Note that C ( µ b ) is in-dependent of the matching scale µ up to the consideredorders in couplings due to the inclusion of the RGE evo-lution. However, the residual µ b dependence will only becancelled by the according µ b dependence of the matrixelements of the relevant operators.Fig. 3 shows the µ dependence of C ( µ b = 5 GeV) atLO, NLO QCD and NLO (QCD + EW) in the OS-2 andHY schemes. It is clearly visible that the dependenceon the renormalization scale of m t reduces when goingfrom LO to NLO QCD and that the LO results coincidewith the ones at NLO QCD at the scale µ ≈
150 GeV.A further reduction of this scheme dependence requiresthe inclusion of NNLO QCD corrections [36]. The NLOQCD result is quite different in the OS-2 and HY schemecomprising values of C ( µ b ) ∈ [ − . , − . · − .The NLO (QCD + EW) result shows again rather largeshifts with respect to NLO QCD and a clear convergenceof both schemes towards the same value. The resultsof the OS-2 and HY schemes are now confined within C ( µ b ) ∈ [ − . , − . · − reducing the combineduncertainty due to scheme dependencies of both QCDand EW interactions to ± These corrections were discussed in the large top-quark-masslimit including the RGE effects in Ref. [10], whereas RGE effectswere neglected in Ref. [22] for ( mn = 02 , ,
100 200 300 µ [GeV] − . − . − . − . G F M W π ˜ c × − OS-2
100 200 300 µ [GeV] − . − . − . − . G F √ c × − HY FIG. 3. The µ dependence of the Wilson coefficient C ( µ b = 5 GeV) in two renormalization schemes (OS-2, HY) at LO(dotted), NLO QCD (dashed) and NLO (QCD + EW) (solid). See text for more details.
100 200 300 µ [GeV]0 . . C [ HY ] C [ O S - ] HY/OS-2
FIG. 4. The µ dependence of the ratio of the Wilson coeffi-cient C ( µ b = 5 GeV) in the HY and the OS-2 scheme at LOand NLO QCD (dashed) and NLO (QCD + EW) (solid). LOand NLO QCD curves coincide. the uncertainty due to higher-order EW and QCD cor-rections to our two-loop EW result from 1) the ratio ofthe results of the HY to the OS-2 scheme, thereby elim-inating the numerically leading QCD µ -dependence of m t , and 2) by varying the scale µ only in m t of thetwo-loop EW matching corrections c (22)10 (or (cid:101) c (22)10 ). Ascan be seen in Fig. 4, at the level of NLO QCD the ratiodeviates quite strongly from 1 whereas at NLO (QCD+ EW) the deviations are less than 0 . µ dependence of the OS-2and HY results (about ± m t of the EW two-loop matching correction. Wechoose the OS-2 scheme with µ = 160 GeV to predictthe central value of C = − . · − , the HY scheme yields − . · − , and we assign an error due to higher-order EW corrections from the variation of µ of about ± .
3% as suggested by the comparison of the OS-2 andHY schemes.We now turn to the discussion of the residual µ b depen-dence for the fixed value µ = 160 GeV. As already men-tioned above, including the according matrix elementsof the involved operators shall decrease this dependencefurther, however, for the moment it remains an addi-tional source of uncertainty. Fig. 5 shows C ( µ b ) at LO,NLO QCD and NLO (QCD + EW) in the OS-2 and HYschemes. Whereas the values of C ( µ b ) are quite differ-ent in all three schemes at NLO QCD, the inclusion ofNLO (QCD + EW) corrections in the form of the renor-malization group evolution yields a convergence towardsthe same value and a very small residual µ b dependencein each scheme of less than ± ± ± µ b ∈ [2 . ,
10] GeV. We wouldlike to note, that the non-perturbative uncertainty due tounknown QED corrections in the evaluation of the ma-trix elements is an additional source of uncertainty, notincluded in the above estimate.The dependence of the EW corrections on the Higgsmass is entirely negligible. Varying M H ∈ [120 , C of less than ± . µ = 160 GeV and µ b =5 GeV C = ( − . ± . · − , (39)where we have estimated higher-order corrections of EWorigin from the scale variations of µ ∈ [50 , µ b ∈ [2 . ,
10] GeV in two schemes, OS-2 and HY, andadded linearly the two errors. We have not included intothe error budget the residual errors associated to higherQCD corrections that can be removed by means of theNNLO QCD calculation [36] nor any of the parametric0 . . . . µ b [GeV] − . − . − . − . G F M W π ˜ c × − OS-2 . . . . µ b [GeV] − . − . − . − . G F √ c × − HY FIG. 5. The µ b dependence of the Wilson coefficient C ( µ b ) for fixed µ = 160 GeV in two renormalization schemes (OS-2,HY) at LO (dotted), NLO QCD (dashed) and NLO (QCD + EW) (solid). See text for more details. errors listed in Tab. I. To show the improvements of ourfinal result (39), we quote for comparison the results atNLO QCD C OS − = − . · − , C HY10 = − . · − (40)taken from the according curves of the OS-2 and HYschemes in Fig. 3.Finally, we compare our prediction with the previousestimate [22], which was obtained using the large- m t ap-proximation of C (22)10 and neglecting the effects of theRGE evolution. In particular, the authors found in theHY scheme BR [ t =0] = 3 . · − in Table 2 of their work.Adopting the same numerical input ( f B s = 227 MeV, τ B s = 1 .
466 ps − , M B s = 5 . | V tb V ∗ ts | =0 . m µ = 105 . ⇒ N = 4 . · ) andEq. (39), our result BR [ t =0] = 3 . · − is about 5%lower, mainly due to the above mentioned approxima-tions. Furthermore, the authors of Ref. [22] argued thatNLO EW corrections in the HY scheme should be smalland suggested a procedure, based on LO expressions, thatlead to the preliminary value of BR [ t =0] = 3 . · − (see Eq. (17) in Ref. [22]), which is closer to our re-sult and deviates only by 3%. In particular it was sug-gested to use EW parameters α e and s W in the MSscheme at the scale M Z ≈
90 GeV and the LO expression c (11)10 ∼ Y ( x t ) with m t ( m t ) with an additional correctionfactor η Y to account for higher-order QCD correctionsfrom c (21)10 . We find from Fig. 2, 3rd panel for the HYscheme, at µ = 90 GeV a deviation of about 1.5% be-tween the LO result and the NLO EW one. We wouldlike to close this comparison with the remark that theauthors of Ref. [22] work at LO in the EW couplings al-lowing them to combine values of the input parameterswhich are dependent beyond the LO, where as in our casecertain EW parameters, especially M W and s W , do de-pend on the input quantities of our choice in Eq. (16). Asa consequence, a straightforward numerical comparison is not possible, however, adopting the suggested proce-dure using our numerical values of dependent quantitieswe obtain a slightly larger value BR [ t =0] = 3 . · − in-stead of 3 . · − . For definiteness we give here our value M on − shell W = (80 . ± . M PDG W = (80 . ± . M pole t by ± . s W and obtain s W ( M Z ) = 0 . . V. CONCLUSIONS
We have calculated the next-to-leading (NLO) elec-troweak (EW) corrections to the Wilson coefficient C that governs the rare decays B q → (cid:96) + (cid:96) − in the Stan-dard Model. To assess the size of higher-order correc-tions, the numerical analysis has been performed withinthree different renormalization schemes of the involvedEW parameters, described in Sec. II A, and two differ-ent normalizations of the effective Lagrangian, given inEq. (6). The inclusion of NLO EW corrections stronglyreduced the scheme dependences present at LO for allconsidered schemes. We identified the two schemes withthe better convergence behavior and estimated the uncer-tainty from missing beyond NLO EW corrections to beabout ± C . The first renormalization schemeis based on a new normalization [10] that eliminates theratio α e /s W → G F in favor of Fermi’s constant. Thesecond is based on the MS scheme for both quantitiesentering the ratio α e /s W [21].Apart from the NLO EW matching corrections to C ,we took into account the effects of the renormalizationgroup running of C caused by operator mixing at higherorder in QED in the effective theory. As we do not in-1clude QED corrections to the matrix elements of the rel-evant operators we estimated the remaining perturbativeuncertainty due to the variation of the low-energy scale µ b and found an about ± .
2% uncertainty for C .In the error budget, we do not include uncertaintiesdue to higher-order QCD corrections, which are removedby the NNLO QCD calculation [36], nor parametric un-certainties of C and the branching ratio, which are dis-cussed in detail in Ref. [37].Our calculation removes an uncertainty of about ± − µ and low-energy scale µ b . The combination of bothresults in uncertainties of ± .
5% at the level of C andconsequently ±
1% on the branching ratio.
ACKNOWLEDGMENTS
We would like to thank Joachim Brod and An-drzej J. Buras for many valuable explanations and sug-gestions, Bernd Kniehl for useful correspondence andThomas Hermann, Matthias Steinhauser and Miko-laj Misiak for extensive discussions and careful readingof the manuscript. Martin Gorbahn acknowledges par-tial support by the UK Science & Technology FacilitiesCouncil (STFC) under grant number ST/G00062X/1.Christoph Bobeth received partial support from the ERCAdvanced Grant project “FLAVOUR” (267104).
Appendix A: Details on the Matching Calculation1. Operator Basis
Throughout, we use the same definition of the opera-tors as in Ref. [23]. The RGE evolution from the match-ing scale µ down to µ b involves the operators mentionedin Sec. III, whereas here, we list only operators whoseWilson coefficients contribute to the matching of theNLO EW correction to C in Sec. II. They are the phys-ical operator P and the according evanescent operator E that mediate b → q ¯ ccP = (¯ q L γ µ c L ) (¯ c L γ µ b L ) , (A1) E = (¯ q L γ µνρ c L ) (¯ c L γ µνρ b L ) , (A2) Actually, E does not contribute to the matching, but only be-cause it does not mix in P at one-loop, i.e. ˆ Z (1) E , = 0. as well as P , P and the according evanescent operators E and E [24] that mediate b → q (cid:96) + (cid:96) − P = (¯ q L γ µ b L ) (cid:88) (cid:96) (¯ (cid:96)γ µ (cid:96) ) , (A3) P = (¯ q L γ µ b L ) (cid:88) (cid:96) (¯ (cid:96)γ µ γ (cid:96) ) , (A4) E = (¯ q L γ µνρ b L ) (cid:88) (cid:96) (¯ (cid:96)γ µνρ (cid:96) ) − P + 6 P , (A5) E = (¯ q L γ µνρ b L ) (cid:88) (cid:96) (¯ (cid:96)γ µνρ γ (cid:96) ) + 6 P − P . (A6)The evanescent operators vanish algebraically in d = 4dimensions. Above γ µνρ ≡ γ µ γ ν γ ρ and γ µνρ ≡ γ µ γ ν γ ρ .In our case, there are no equation-of-motion vanishingoperators with a projection on (cid:104) P (cid:105) (0) to contribute tothe matching.
2. Details on the Standard Model Calculation
The two-loop EW SM calculation is very similar to theanalogous calculation for the K → πν ¯ ν decays [21]. Thecalculation comprises of generating and calculating alltwo-loop topologies for the transition b → q(cid:96) + (cid:96) − (Fig. 1).We perform two independent calculations, in the firstwe use FeynArts [38] to generate the topologies and aself-written
Mathematica program to evaluate them andin the second
QGRAF [39] and a self-written
FORM [40] pro-gram, respectively.By setting the external momenta and the masses of allfermions except for the top quark to zero all diagrams re-duce to massive tadpoles with maximally three differentmasses. We reduce them to a few known master integralsusing the recursion relations from Refs. [33, 41].We work in dimensional regularization, which raisesthe question of how to treat γ in d (cid:54) = 4 dimensions.The naive anticommutation relation (NDR) { γ , γ µ } = 0can lead to algebraic inconsistencies in the evaluation oftraces with γ ’s. Yet, the algebraically consistent defini-tion of γ by ’t Hooft-Veltman (HV) [42] leads to spuriousbreaking of the axial-current Ward identities and as suchrequires the incorporation of symmetry-restoring finitecounterterms. Diagrams that are free of algebraic incon-sistencies in the NDR scheme yield the same finite resultafter the appropriate counterterms are added. This triv-ially holds for all diagrams free of internal fermion loopsas well as for diagrams that involve traces with an evennumber of γ matrices if the γ matrices are eliminatedthrough naive anticommutation from the relevant traces[43]. Since selfenergy diagrams involving a single axialcoupling vanish, diagrams involving fermionic loops onbosonic propagators also correspond to the same finiteexpression in both schemes after appropriate renormali-sation. Accordingly, special care has to be taken only fordiagrams involving a fermion-triangle loop and comingwith an odd number of γ matrices. We use the HV pre-scription for these type of diagrams, since in particular2the diagram with three γ matrices cannot be simply cal-culated in the NDR scheme. Here we note that the finiterenormalization, which will restore the axial-anomaly re-lation of diagrams involving fermion traces, will drop outin our calculation after the sum over the complete setof standard model fermions is performed. This followsfrom the fact that Standard Model is anomaly free andcan be understood by noting that e.g. the difference ofthe singlet and non-singlet counterterm in Ref. [43] hasopposite sign for up-type and down-type quarks. Yet,one subtlety could arise from charged W and Goldstonebosons connecting the fermion-triangle diagram with theexternal fermion line. The axial couplings on the externalline could in principle result in a spurious breaking of theaxial-current Ward identity if treated in the HV scheme.Yet, only the 4-dimensional part of this coupling con-tributes if the fermion triangle contains an odd numberof γ matrices, since the corresponding diagrams are ei-ther finite after GIM or their traces vanish. Accordingly,we can safely use the HV scheme in these circumstanceswithout the need of an extra finite renormalisation andcalculate all other diagrams in the NDR scheme. The ef-fective theory calculation does not involve fermion traceswith γ and for this reason can be performed completelyin the NDR scheme.In the SM, the renormalization scheme of the fermionfields f = q, (cid:96) , i.e. quarks and leptons, is chosen such thatthe kinetic terms in the effective theory remain canoni-cally normalized at NLO in EW interactions. As a con-sequence, Wilson coefficients of dimension three b → s mediating operators in the effective theory are zero. Thebare SM fields, f (0) , with flavor type i and of chirality-type a are renormalized f (0) i,a = (cid:18) δ ij + 12 Z aij (cid:19) f j,a (A7)with the help of the matrix-valued field renormalizationconstant Z a . The latter is determined from one-loop f → f (cid:48) two-point functions such that the matching relation forthe fields in the SM and effective theory f full = f eff , (A8)holds, implying that tree-level matrix elements of oper-ators, (cid:104) P i (cid:105) (0) , are the same in the SM and effective the-ory amplitude, see Eqs. (17) and (24) respectively. Forthis purpose, the two-point functions are evaluated inan expansion up to first order in external momenta andmasses over heavy masses. The heavy particle contribu-tions yield finite parts to Z a , whereas light particle con-tributions eventually drop out in the matching and thusmay be discarded in the calculation. In addition, theflavor off-diagonal quark-field renormalization constant Z bq is determined at two-loop level from the two-pointfunction b → q .The counterterm of the CKM matrix is entirely deter-mined by the field renormalization constants Z L of theup- and down-quark fields. This renormalization pre-scription corresponds to a definition of the CKM elements in the effective theory where the kinetic terms of all lightquark fields are canonical.Since we renormalize both the couplings α full e and α eff e of the full and effective theory, respectively, in the MSscheme, the α e threshold corrections have to be includedin the case of the single- G F normalization. In the thresh-old corrections, ∆ α e , α full e = α eff e (cid:20) α eff e π ∆ α e (cid:21) , ∆ α e = − −
14 ln µM W + 329 ln µM t (A9)the first two terms arise from the decoupling of the elec-troweak gauge bosons and the last term from the topquark at the scale µ . Since the definition of α e ( M Z ) inTab. I compiled by the particle data group [11] alreadyimplies a decoupled top quark, we determine α eff e from α e ( M Z ) using only the gauge boson contribution and find α eff e ( M Z ) = 1 / .
751 that we use in our numerical eval-uations.In order to match consistently, we apply Eq. (A9) tosubstitute the α full e → α eff e , which affects the matching atnext-to-leading order due to an additional contribution inthe amplitude of the full theory from the lower order partin Eq. (18) (omitting here the subscript A full , → A )˜ α full e A (1) + (cid:0) ˜ α full e (cid:1) A (2) =˜ α eff e A (1) + (cid:0) ˜ α eff e (cid:1) (cid:104) A (2) + ∆ α e A (1) (cid:105) . (A10)
3. Details on the Effective Theory Calculation
Before being able to evaluate the two-loop b → q(cid:96) + (cid:96) − amplitude in the effective theory we need to know all Wil-son coefficients and renormalization constants appearingin Eq. (26). The tree-level contribution C (00)2 and theone-loop results C (11)9 and C (11)10 are given in Ref. [33] in-cluding the O ( (cid:15) ) terms for the latter two. Here we givein addition the Wilson coefficients of the two evanescentoperators c (11) E = c (11) E =116 s W x t ( x t − (1 − x t + log x t ) + O ( (cid:15) ) . (A11)The O ( (cid:15) ) terms of c (11) E and c (11) E do not contribute tothe matching as the mixing renormalization constantsˆ Z (1) E , and ˆ Z (1) E , carry no divergent terms, only finiteones. The operator E does not contribute to the matching at allbecause ˆ Z (1) E , = 0. Z i = 1 + ˜ α e Z (1) i + . . . (A12)with Z (1) d = − (cid:15) , Z (1) (cid:96) = − (cid:15) . We proceed similarly for the constants governing the mix-ing of operators into P . We calculate the UV poles ofall one-loop insertions of a given operator, project on thetree-level matrix element of P and absorb the left-overpole in the mixing renormalization constant.For the case of physical operators mixing into phys-ical ones we absorb only the divergences into the con-stants ˆ Z P,P . For evanescent operators this is not the case.Evanescent operators are unphysical in four dimensionsand at each order in perturbation theory their operatorbasis needs to be extended. To ensure that the Wilsoncoefficients at a given fixed order are independent fromthe choice of evanescent operators in some higher orderwe include finite terms in ˆ Z E,P and completely cancel themixing of evanescent to physical operators.We have calculated all contributing one-loop mixingrenormalization constants including the mixing of evanes-cent to physical operators. The mixing of physical opera-tors can also be extracted from the anomalous dimensionmatrices in Refs. [23, 24]. Here we report the relevantnon-zero constantsˆ Z (1)9 , = − (cid:15) , ˆ Z (1) E , = 323 . (A13)We extract the 1 /(cid:15) -part of the one two-loop renormaliza-tion constant we need from the corresponding anomalousdimension in Ref. [24] and calculated the 1 /(cid:15) -termˆ Z (2)2 , = 49 (cid:15) − (cid:15) . (A14) Appendix B: Details on the RGE1. General
The dependence of the Wilson coefficients C i on therenormalization scale µ is governed by the anomalousdimension matrix ˆ γµ ddµ C i ( µ ) = (cid:2) ˆ γ T ( µ ) (cid:3) ij C j ( µ ) (B1)with the expansion in the couplingsˆ γ ( µ ) = (cid:88) m,n =0 m + n ≥ ˜ α s ( µ ) m ˜ α e ( µ ) n ˆ γ ( mn ) , (B2) which is known up to and including relevant entries in( mn ) = (30) and (21). It has been solved as an expansionin terms of the small quantities [23] ω ≡ β s ˜ α s ( µ ) , (B3) λ ≡ β e β s ˜ α e ( µ )˜ α s ( µ ) = β e β s κ ( µ ) (B4)in which case the evolution operator in Eq. (31) takes theform U ( µ b , µ ) = (cid:88) m,n ≥ ω m λ n U ( mn ) , (B5)excluding the term ( mn ) = (22) that requires the knowl-edge of higher-order contributions to the anomalous di-mension matrix. The U ( mn ) can be read off from Eq. (47)of Ref. [23], whereas the initial Wilson coefficients (in thesingle- G F normalization) at the scale µ have the expan-sion c i ( µ ) = c (00) i + ω c (10) i β s + ω c (20) i (2 β s ) + ωλ c (11) i β e + ω λ c (21) i β e β s + ω λ c (22) i ( β e ) . (B6)The components C i, ( mn ) of the downscaled Wilson coeffi-cients in Eq. (32) are then obtained from the reexpansionof Eq. (31) in the new parameters ˜ α s ( µ b ) ω = 2 β s η ˜ α s ( µ b ) , (B7)and κ ( µ b ) λ = β e β s κ ( µ b ) η (cid:104) κ ( µ b ) A ( η )+ ˜ α s ( µ b ) κ ( µ b ) A ( η ) + O (cid:0) κ , ˜ α s (cid:1) (cid:105) (B8)after inserting Eqs. (B5) and (B6). The coefficients A , ( η ) are given in Eq. (67) of Ref. [23].
2. Solution
Here the solution of the components c , ( mn ) inEq. (31) of the single- G F normalization from Eq. (5) atthe low scale µ b are given in terms of η = α s ( µ ) /α s ( µ b )and their initial components c ( mn ) i in Eq. (12) at thematching scale µ . The derivation of the according re-sults (cid:101) c , ( mn ) for the quadratic- G F normalization wasgiven in Sec. III.The numerical diagonalization of the leading-orderanomalous dimension yields the exponents a i = ( − , − , − . , − . , − . , . , . , . . (B9)The components read4 i b i . . − . − . − . . . − . d (2 a ) i . . . . − . − . d (2 b ) i − . . . − . − . − . − . − . d (1) i . − . . . . . . − . d (4) i − . − . . − . − . − . e (1 a ) i − . − . − . . − . − . e (1 b ) i . . − . . . − . − . . e (4 a ) i − . . − . − . e (4 b ) i . − . . . − . . e (1) i . − . . . . . . − . e (2) i . . − . − . − . . . − . e (3) i . − . . . . − . e (4) i − . − . . − . − . − . e (5) i . − . . . . − . e (6) i − . . . − . . − . b i , d ( j ) i and e ( j ) i entering (B10). c , (11) = c (11)10 , c , (21) = η c (21)10 , c , (02) = (cid:88) i =1 b i η a i c (00)2 ,c , (12) = (cid:88) i =1 η a i +1 (cid:104) (cid:16) d (2 a ) i η − + d (2 b ) i (cid:17) c (00)2 + d (1) i c (10)1 + d (4) i c (10)4 (cid:105) − . ηη c (00)2 + (cid:0) η − − (cid:1) (cid:16) . c (11)9 + 1 . c (11)10 (cid:17) ,c , (22) = (cid:88) i =1 η a i +2 (cid:16) e (1 a ) i η − + e (1 b ) i (cid:17) c (10)1 + (cid:16) e (4 a ) i η − + e (4 b ) i (cid:17) c (10)4 + (cid:88) j =1 e ( j ) i c (20) j + (cid:16) . c (10)1 + 0 . c (10)4 + 2 . c (11)9 + 3 . c (11)10 (cid:17) ln η + (1 − η ) (cid:16) . c (21)9 + 1 . c (21)10 (cid:17) + c (22)10 , (B10)with the coefficients b i , d ( j ) i and e ( j ) i given in Tab. II. Appendix C: Numerical study of C in OS-1 scheme In this appendix we estimate higher-order correctionsin the OS-1 scheme and supplement in this context thediscussion of the OS-2 and HY schemes from Sec. IV. Forthis purpose, we proceed as in Fig. 3 and Fig. 4 and vary the matching scale µ , which allows to estimate higher-order QCD corrections via the dependence on the run-ning top-quark mass. The result is shown in Fig. 6 atNLO QCD and NLO (EW + QCD) order normalized tothe OS-2 result at the respective orders. To understandthe different µ dependence of the NLO QCD result for5
100 200 300 µ [GeV]0 . . C [ O S - ] C [ O S - ] OS-1/OS-2
FIG. 6. The µ dependence of the ratio of the Wilson coef-ficient C ( µ b = 5 GeV) in OS-1 and OS-2 schemes. The LOand NLO QCD result coincide (dashed). The full µ depen-dence of NLO (QCD + EW) (solid) and partial µ dependencefor fixed m t (160 GeV) in the s W -on-shell counterterm (dasheddotted). the OS-1 and OS-2 schemes, we remind that they in-volve different normalizations (see Eq. (7)), which beara µ dependence due to their m t dependence when de- termining values of M on-shell W and consequently s on-shell W ,see Eq. (19) and the input in Eq. (16). As mentioned inSec. II A, we calculate M on-shell W with the aid of the re-sult in Ref. [29], which incorporates various higher-ordercorrections that contribute beyond the NLO EW calcula-tion of C performed in this work, especially those thatrequire the choice of a particular renormalization schemefor the top-quark mass. Throughout we use the pole topmass as numerical input as in Ref. [29].At NLO (EW + QCD) the OS-1 scheme exhibits a verydifferent µ dependence with respect to OS-2 and HYschemes, which is increased compared to NLO QCD. Themain reason being the large EW two-loop correction to c (22)10 from the s W -on-shell counterterm as already men-tioned in connection with Fig. 2. The counterterm hasa strong top-quark-mass dependence. To illustrate thelatter, we present in Fig. 6 additionally the NLO (EW +QCD) result (dashed-dotted line) when keeping the scaleof the running top-quark mass in the counterterm con-tribution fixed at µ = 160 GeV. Hence, the large shiftcaused by the electroweak two-loop correction in the OS-1 scheme is accompanied with an artificially large top-quark-mass dependence. As a consequence we do notconsider the OS-1 scheme in our estimate of higher-orderuncertainties. It would increase the estimate due to µ variation of about ± .
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