Renormalization-group evolution of new physics contributions to (semi)leptonic meson decays
aa r X i v : . [ h e p - ph ] S e p CERN-TH-2017-103
Renormalization-group evolution of new physicscontributions to (semi)leptonic meson decays
Mart´ın Gonz´alez-Alonso a , Jorge Martin Camalich b , Kin Mimouni ca IPN de Lyon/CNRS, Universite Lyon 1, Villeurbanne, France b CERN, Theoretical Physics Department, Geneva, Switzerland c Institut de Th´eorie des Ph´enom`enes Physiques, EPFL, Lausanne, Switzerland
Abstract
We study the renormalization group evolution (RGE) of new physics contributionsto (semi)leptonic charged-current meson decays, focusing on operators involving a chi-rality flip at the quark level. We calculate their evolution under electroweak and electro-magnetic interactions, including also the three-loop QCD running and provide numeri-cal formulas that allow us to connect the values of the corresponding Wilson coefficientsfrom scales at the TeV to the low-energy scales. The large mixing of the tensor oper-ator into the (pseudo)scalar ones has important phenomenological implications, suchas the RGE of new physics bounds obtained from light quark decays or in b → cℓν transitions. For instance, we study scenarios involving tensor effective operators, whichhave been proposed in the literature to address the B -decay anomalies, most notablythose concerning the R D ( ∗ ) ratios. We conclude that the loop effects are importantand should be taken into account in the analysis of these processes, especially if theoperators are generated at an energy scale of ∼ Introduction
Charged-current transitions represent the dominant mechanism for the weak decays of most hadronsand are the main source of experimental inputs to determine the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements. Precision studies of these decays provide benchmarks to test the flavorstructure of the Standard Model (SM) and to look for New Physics (NP). A remarkable exampleis the analysis of (semi)leptonic nuclear, pion and kaon decays, where the very precise database,and exquisite understanding of the SM corrections, lead to sub-permille level tests of the SM thatare sensitive to NP scales in the 1 −
100 TeV range [1–3].Moreover, several tensions with the SM have appeared in the semileptonic decays of B mesonsinduced by the b → ( c, u ) ℓν transitions that could be pointing to the effects of NP. The mostsignificant one corresponds to the τ -to- ℓ rate ratios R D ( ∗ ) = Γ( B → D ( ∗ ) τ ν ) / Γ( B → D ( ∗ ) ℓν ) (where ℓ is a muon or electron) that exhibit enhancements of a 30% with respect to the SM expectationswith a significance of ∼ σ [4–9] (see [10] for a review). Furthermore, inclusive and exclusive determinations of the CKM matrix element | V cb | and | V ub | obtained with the light-lepton modeshave disagreed with each other with a significance in the range of ∼ − σ [11–15].These results have triggered an intense activity in the theory community, both through model-building efforts and through Effective Field Theory (EFT) studies. However, these analyses usuallywork at tree level, neglecting Renormalization-Group Evolution (RGE) effects. These necessarilyappear when the predictions of the underlying model at the high NP scale are connected withthe low scale associated with the experimental measurements. It was recently pointed out theimportance of such RGE effects for chirality-conserving operators addressing the B anomalies,since they could generate one-loop contributions to Z and τ decays, which are very preciselymeasured [16, 17]. Although chirality-flipping operators have also been considered in the literatureto explain the various anomalies [18–29], a study of the RGE effects in such scenarios is absentin the literature. We amend this limitation in this work, calculating the one-loop QED andEW evolution of the complete low-energy EFT Lagrangian, including also the three-loop QCDrunning. We provide numerical formulas that can be trivially implemented in future analyses,and we illustrate the importance of these effects with several simple applications. We find inparticular a large mixing of the tensor operator into the (pseudo)scalar ones, which has importantphenomenological implications, including for instance NP scenarios that have been proposed inthe literature to address several B -decay anomalies.This work is organized as follows. In Section 2 we introduce the low- and high-energy EFTLagrangians, and we calculate the associated anomalous dimension matrices. In Section 3 we usethese results to connect the effective operators at the low-energy scale with the NP scale Λ. Wepresent some phenomenological implications of our results in Section 4, whereas Section 5 containsour conclusions. 1 EFTs and anomalous dimensions
Assuming Lorentz invariance and heavy NP physics, d j → u i ℓ ¯ ν ℓ transitions are described at lowenergies by the following effective Lagrangian L eff = − G F V ij √ " S ew (cid:16) ǫ ijℓL (cid:17) ¯ ℓγ µ P L ν ℓ · ¯ u i γ µ P L d j + ǫ ijℓR ¯ ℓγ µ P L ν ℓ ¯ u i γ µ P R d j + ǫ ijℓS L ¯ ℓP L ν ℓ · ¯ u i P L d j + ǫ ijℓS R ¯ ℓP L ν ℓ · ¯ u i P R d j + ǫ ijℓT ¯ ℓσ µν P L ν ℓ · ¯ u i σ µν P L d j + h . c ., (2.1)where P L/R = (1 ∓ γ ) / i, j ( ℓ ) label the quark (lepton)families and G F is the Fermi constant. Notice that the G F value extracted from muon decay withina SM framework can also be affected by NP effects: G expF = G F + δG F . The S ew factor encodes theuniversal short distance corrections to the semileptonic transitions in the SM [30, 31], and the ǫ Γ coefficients encode the leading NP contributions to this process. Both S ew and the ǫ Γ coefficientsdisplay renormalization-scale dependence that is to be canceled in the observables by the oppositedependence in the quantum corrections to the matrix elements of the decays. Namely, at oneloop, we have d ~ǫ ( µ ) d log µ = (cid:18) α em ( µ )2 π γ T em + α s ( µ )2 π γ Ts (cid:19) ~ǫ ( µ ) , (2.2)where ~ǫ = ( S ew , ǫ R , ǫ S R , ǫ S L , ǫ T ), and α em and α s are the electromagnetic and strong structureconstants. We omit flavor indices in the ǫ Γ coefficients, as both QED and QCD conserve flavor.The matrices γ em and γ s are the corresponding one-loop anomalous dimensions, and the superindex T simply indicates matrix transposition.Figure 1: One-loop QED radiative corrections contributing to d ¯ u → ℓ ¯ ν effective operators.Calculating the diagrams in Fig. 1, we obtain γ T em = y − x de − x du + 4 x ue y − x de − x du + x ue y − x de − x du + x ue y − x de − x du + x ue − x de − x ue − x de − x ue y − x de + 3 x ue The only exception is the ǫ L coefficient, which is not scale-dependent because it is defined with respect to theSM contribution. y = P d,u,e Q i / x ij = Q i Q j (with Q f denoting the electric charge of the fermion f ).Numerically, γ T em = − −
40 0 0 − , γ s = − − / . (2.3)The γ s matrix given above is the QCD one-loop result [32], which is actually known to threeloops (see below). Concerning the electromagnetic anomalous dimension, our result agrees withthe well-known SM calculation of the [ γ em ] element [30,31], as well as with the partial calculationin Ref. [33], where the [ γ em ] element was calculated.If the nonstandard particles are not only heavier than the hadronic scale but also much heavierthan the electroweak scale, and assuming that the electroweak symmetry breaking is linearlyrealized, then the low-energy EFT of Eq. (2.1) must be matched to the so-called Standard ModelEFT (SMEFT) [34] L SMEFT = X w i v O i , (2.4)where v = ( √ G F ) − / ≃
246 GeV. In this work we will use the so-called Warsaw basis of Ref. [35] .Table 1 lists the relevant operators for this work, which contribute at tree level to the processes d j → u i ℓ ¯ ν ℓ and µ → e ¯ ν e ν µ .Table 1: Operators in the Warsaw basis [35] that are relevant for this work. For further detailsabout the conventions, we refer the reader to the original publication [35]. O (3) ϕl ( ϕ † i ↔ D Iµ ϕ )(¯ lτ I γ µ l ) O (3) ϕq ( ϕ † i ↔ D Iµ ϕ )(¯ qτ I γ µ q ) O ϕud i ( e ϕ † D µ ϕ )(¯ uγ µ d ) O ll (¯ lγ µ l )(¯ lγ µ l ) O (3) lq (¯ lγ µ τ I l )(¯ qγ µ τ I q ) O ledq (¯ l j e )( ¯ dq j ) O ℓequ (¯ l j e ) ǫ jk (¯ q k u ) O (3) ℓequ (¯ l j σ µν e ) ǫ jk (¯ q k σ µν u )Once again, the Wilson coefficients w i are scale-dependent quantities. Here we focus on thecoefficients ~w = n w ledq , w ℓequ , w (3) ℓequ o , which are particularly interesting from a phenomenologicalpoint of view, as we will explain below. Their running above the EW scale is described by d ~w ( µ ) d log µ = (cid:18) π γ Tw ( µ ) + α s ( µ )2 π ˆ γ Ts (cid:19) ~w ( µ ) , (2.5)where ˆ γ s refers to the 3 × γ s corresponding to the running of ( ǫ S R , ǫ S L , ǫ T ). Wenoticed that the result of Ref. [36] for the electroweak anomalous dimension matrix γ w disagreeswith a previous calculation [37]. We obtain One difference is that for operators with the SU (2) L singlet contraction of fermionic currents we omit thesuperscript (1) . T w = − g Y − g Y g Y + 9 g L g Y + g L g Y − g L ≈ − .
169 0 00 − .
232 5 . . − . , (2.6)which agrees with Ref. [36] neglecting terms suppressed by Yukawa couplings. Let us note that thedifference with Ref. [37] does not have any significant impact on RGE calculations, since it onlyaffects the diagonal elements, whose running is fully dominated by QCD. The RHS of Eq. (2.6)shows for illustration the numerical value of the electroweak anomalous dimension matrix for µ = m Z . One can see that the mixing is one order of magnitude larger than in the QED case. Higher loop corrections to the QCD running of the Wilson coefficients can give a sizable impactwhen evolving to or from low-energy scales. At three loops we have d ~ǫ ( µ ) d log µ = (cid:18) α s ( µ )2 π γ (1) s + α s ( µ ) π γ (2) s + α s ( µ ) π γ (3) s (cid:19) ~ǫ ( µ ) , (2.7)where γ ( n ) s is the n -loop anomalous dimension matrix. The 2- and 3-loop results have the samestructure as the 1-loop matrix γ (1) s in Eq. (2.3), with the following non-zero entries (see Ref. [38]and references therein)[ γ (2) s ] = [ γ (2) s ] = 29 ( −
303 + 10 n f )[ γ (3) s ] = [ γ (3) s ] = 181 (cid:0) − ζ (3)) n f + 140 n f (cid:1) [ γ (2) s ] = 227 (543 − n f )[ γ (3) s ] = 181 (cid:0) − ζ (3) − ζ (3)) n f − n f (cid:1) . (2.8)The running from low- to high-energy renormalization scales might require to cross variousheavy-quark thresholds. The so-called threshold corrections that relate the Wilson coefficients inthe n f − and n f effective theories have the following form ǫ ( n f − ( µ th ) = ǫ ( n f )Γ ( µ th ) " X n ξ Γ ,n α ( n f ) s ( µ th ) π ! n (Γ = S, P, T ) . (2.9)The (pseudo)scalar Wilson coefficients have the same QCD running as the quark masses [39] ξ S/P, = 0 , ξ S/P, = 89432 . (2.10)The matching coefficients for the tensor Wilson coefficient ξ T,n are the same as for the product m q C , where C is the Wilson coefficient of the operator Q ∼ m b (¯ s L σ µν b R ) F µν in the b → sγ effective Lagrangian. Combining the matching coefficients of the quark masses given above [39]and the two-loop matching coefficient of C given in Ref. [40], we find ξ T, = 0 , ξ T, = − . (2.11)4 RG running and SMEFT matching
Next, we solve the coupled differential RGE equations presented in the previous section, i.e. work-ing at three-loop in QCD and one-loop in QED/EW. We work at the same order for the corre-sponding couplings constants α s , α em and g L,Y [41]. Finite threshold corrections are also takeninto account, both for the couplings and the Wilson coefficients.
RG running below the weak scale.-
For the decays of light quarks, the low-energy Wilsoncoefficients ǫ Γ are typically extracted from the experiment at µ = 2 GeV, simply because that is therenormalization scale chosen usually to extract the necessary lattice form factors. The subsequentrunning to the weak scale is given by ǫ L ǫ R ǫ S ǫ P ǫ T ( µ = 2 GeV)= . .
72 2 . × − − . . × − . − . − . × − − . × − . ǫ L ǫ R ǫ S ǫ P ǫ T ( µ = Z )(3.12)where we took into account the bottom quark threshold. In order to illustrate the numericalimportance of the QCD loops, let us focus on the (3 ,
3) element. Working at n = { , , , } loopswe find { . , . , . , . } . The 2-loop correction is clearly non-negligible, whereas the 3-loopone is comparable to the error introduced by α s ( m Z ) [12].In the decay of bottom quarks, the reference scale is instead the b quark mass m b . Thereforethe corresponding running to the weak scale is a bit smaller and it does not involve crossing anythreshold. We find: ǫ L ǫ R ǫ S ǫ P ǫ T ( µ = m b )= . .
46 1 . × − − . . × − . − . − . × − − . × − . ǫ L ǫ R ǫ S ǫ P ǫ T ( µ = Z ) . (3.13) Matching equations at the weak scale.-
Once the coefficients ǫ Γ are evolved to the weakscale they can be matched to the proper EFT or model relevant at that scale. Here we matchedthem to SMEFT at order Λ − and at tree level. This matching was first given in Ref. [42] usinga specific operator basis, which was a slightly modified version of the Buchmuller-Wyler basis [34]with all relevant redundancies taken care of. Here we show the same equations, but this time using For phenomenological applications it is convenient to use the (pseudo)scalar coefficients ǫ S/P = ǫ S L ± ǫ S R ratherthan their chiral counterparts. Their anomalous dimension matrix can be obtained by a trivial transformation ofthe matrices eq. (2.3). δG F G F = [ w (3) ϕl ] + [ w (3) ϕl ] −
12 [ w ll ] , (3.14) V ij · ǫ ijℓL = V ij [ w (3) ϕl ] ℓℓ + [ w (3) ϕq V ] ij − [ w (3) lq V ] ℓℓij ,V ij · ǫ ijℓR = 12 [ w ϕud ] ij , (3.15) V ij · ǫ ijℓS/P = −
12 [ w † ℓequ V ± w † ledq ] ℓℓij , (3.16) V ij · ǫ ijℓT = −
12 [( w (3) ℓequ ) † V ] ℓℓij . (3.17)Operators containing flavor indices are defined in the flavor basis where the up-quark Yukawamatrices are diagonal. Here repeated indices i, j, ℓ are not summed over; and transposition andmatricial notation only affect quark indexes. RG running above the weak scale.-
In order to match to the underlying NP model, or tomake contact with searches performed at high-energy colliders such as the LHC, it is necessary toperform an additional running from the weak scale to higher scales. Focusing once again on thechirality-breaking operators we find: w ledq w ℓequ w (3) ℓequ ( µ = m Z ) = .
19 0 . . . . − . . − . . w ledq w ℓequ w (3) ℓequ ( µ = 1 TeV) , (3.18)where we took into account the top quark threshold. In Ref. [3] the d ( s ) → uℓ ¯ ν ℓ transitions, such as nuclear, baryon and meson decays, were studiedwithin the SMEFT and obtained bounds for 14 combinations of effective low-energy couplingsbetween light quarks and leptons ( ǫ duℓ Γ ) at the hadronic scale µ = 2 GeV. Using the results obtainedin the previous section we can evolve them to the weak scale where they can be matched to theSMEFT coefficients. The resulting bounds and correlations are given in Appendix A.In order to illustrate the importance of the QED mixing, let us investigate a NP scenario whereonly the tensor interaction ǫ T is generated at the weak scale. Fig. 2 shows the bounds on suchan interaction obtained from the phenomenological analysis of Ref. [3] neglecting and includingthe electromagnetic mixing. As we can see, the inclusion of the mixing increases the bound byseveral orders of magnitude in some cases. Needless to say, the effect is even larger if one choosesa higher scale (instead of the weak scale) to assume the dominance of the tensor interaction, notonly because of the larger running but also because the EW mixing is larger than the QED one. Explicit models that produce tensor operators are for example those including tree-level leptoquark-mediatedinteractions, although in this case (pseudo)scalar operators are also generated [21, 43–45]. ǫ T ( m Z ) for different flavor structures. Theyellow (blue) bars show the results neglecting (including) the QED mixing. The QCD runningis included in both cases. Only one operator at a time is present. The ǫ udµT ( m Z ) coefficient isunbounded if the mixing is neglected, which is represented by an unlimited vertical bar.The reason of this is well known: the bound obtained at the low-energy scale on the pseudoscalarcoupling ǫ P is usually much stronger than the bounds over the scalar or tensor couplings. This isso because the pseudoscalar coupling contributes at tree level to the leptonic meson decay withoutthe helicity suppression that affects the SM contribution, namely [3]Γ( P → ℓν ℓ )Γ SM = (cid:12)(cid:12)(cid:12)(cid:12) ǫ ijℓL − ǫ ijℓR − m P ± m ℓ ( m j + m i ) ǫ ijℓP (cid:12)(cid:12)(cid:12)(cid:12) (4.19)where ( i, j ) = ( u, d ) , ( u, s ) for P = π, K respectively. The NP sensitivity of these channelsis further enhanced by very precise measurements and theoretical predictions, especially in thelepton-universality ratios R P = Γ( P e γ ) ) / Γ( P µ γ ) ) [12, 46, 47]. This makes possible to set limitsat the 10 − level on the electronic couplings ǫ udeP and ǫ useP [3], corresponding to a NP effectivescale of O (500) TeV. Such enormous sensitivity largely compensates for the 1-loop suppressionthat affects the tensor coefficient ǫ T and makes possible to set the strong bounds shown in Fig. 2.Furthermore, as we will see below, this data can also be used to obtain indirect bounds on operatorsprobing different flavor structures such as those in b -quark decays, where the high sensitivity tothe pseudoscalar Wilson coefficient compensates now for CKM-suppression factors. New physics scenarios involving (pseudo)scalar and tensor operators to explain the B -decaysanomalies in the b → ( c, u ) ℓν transitions are phenomenologically viable and have been exploredextensively in the literature [18–29]. In this section we show that the effect of the RGE mixing The tensions in | V cb | could be solved within the SM by reanalyzing the parametrizations of the form factors inthe exclusive B → D ∗ decay mode, as has been shown recently in refs. [14, 15]. Note that our analysis in the EFTof NP is completely general; we will adopt in the following, and for the sake of the argument, the results in ref. [24]as a benchmark for a tensor NP scenario contributing to the B → D ( ∗ ) ℓν decays. b → cℓν transitions. The R D ( ∗ ) and | V cb | anomalies canbe accommodated with the contributions ǫ cbτT ( m b ) ≃ .
38 [48], ǫ cb ( e, µ ) T ( m b ) ≃ . O (3) ℓequ in Tab. 1. For simplicity,we assume that only the entry in flavor space connecting the second and third quark families ofthis operator is nonzero. Taking into account the running we find that these contributions to thetensor operators at µ = m b can be generated by[ w (3) ℓequ ] ττ (Λ) = − . , [ w (3) ℓequ ] ℓℓ (Λ) = − . , (4.21)at the NP scale Λ = 1 TeV. However, even if only that operator was generated in the matchingto the NP model, it will induce by mixing under EW interactions ( cf. Eq. (3.18)) the followingcontribution to the scalar operators at the scale of the Z -mass[ w ℓequ ] ττ ( m Z ) = 0 . , [ w ℓequ ] ℓℓ ( m Z ) = 0 . . (4.22)Matching the SMEFT and the low-energy EFT at that scale using Eqs. (3.17) and running downto µ = m b using Eq. (3.13), we find the contributions ǫ cbτS ( m b ) = ǫ cbτP ( m b ) = − . , ǫ cbℓS ( m b ) = ǫ cbℓP ( m b ) = − . . (4.23)These contributions to the scalar operators at low energies are significant. Let us first look attheir effects on the lepton-universality ratios R D and R D ∗ , which are summarized in the followingtable: expt. SM Mixing neglected Mixing included R D . .
310 0 .
458 0 . R D ∗ . .
252 0 .
312 0 . R D ( ∗ ) values generated by the tensor contribution in the left-hand side ofEq. (4.21), taking into account the QCD running but neglecting the QED and EW mixing,[ w (3) ℓequ ] ττ (1 TeV) = − . → ( ǫ cbτS , ǫ cbτP , ǫ cbτT )( m b ) = (0 , , . . (4.24)Finally, the fourth column shows the values generated by the same initial operator at the highenergy scale (1 TeV), but this time including also the effect of the (pseudo)scalar interactionsproduced through the QED and EW mixing, cf. Eq. (4.23),[ w (3) ℓequ ] ττ (1 TeV) = − . → ( ǫ cbτS , ǫ cbτP , ǫ cbτT )( m b ) = ( − . , − . , . . (4.25)The R D ( ∗ ) values including non-standard effects in the above-given table are obtained followingRef. [23]. As we can see, the impact of the RGE mixing in the B → Dτ ν predictions is importantdue to the well known sensitivity of this channel to the scalar operator, contrary to the case ofthe B → D ∗ τ ν channel, which has little sensitivity to the pseudoscalar contribution. Interestingly8nough, we find that the inclusion of mixing effects improves significantly the agreement of thetensor scenario with data. Models for which these mixing effects are relevant include leptoquarks,such as those in Refs. [19, 21, 25, 29]. New physics producing scalar contributions at the TeVor electroweak scales, such as two-Higgs doublet models (see e.g. Ref. [49]), experience mixinginto the tensor operator. However, as it can be concluded from the corresponding entries in thematrices (3.13) and (3.18), this has a very small impact in the phenomenology.On the other hand, the RGE-induced pseudoscalar operator also modifies the branching fractionof the decay B c → τ ν with a contribution that does not suffer the helicity suppression, namely BR ( B c → τ ν ) ≈ . , (4.26)which is a factor 2.5 larger than the SM branching fraction, BR ( B c → τ ν ) | SM = 2 . f B c lattice result from Ref. [50]. This is an importantenhancement, although not as large as to enter in conflict with the bounds that can be obtainedfrom the lifetime of the B c and the calculations of the corresponding inclusive decay width inQCD [48].Next we investigate the effects of the tensor contribution on the light-lepton operators, i.e. theright-hand side of Eq. (4.21)[ w (3) ℓequ ] ℓℓ (1 TeV) = − . → ( ǫ cbℓS , ǫ cbℓP , ǫ cbℓT )( m b ) = ( − . , − . , . . (4.27)In contrast to the B c → τ ν decay, and more in line with the pion and kaon decays above, thecontributions of the RGE-induced pseudoscalar operator to B c → ℓν can be much more amplifieddue to the chiral enhancement. Namely BR ( B c → µν ) ≈ . × − , BR ( B c → eν ) ≈ . × − , (4.28)which correspond to an enhancement of one order of magnitude for the muonic decays with respectto the SM ( BR ( B c → µν ) | SM = 1 . × − ) and no less than six orders of magnitude for theelectronic case ( BR ( B c → eν ) | SM = 2 . × − ). Although the branching fraction is smaller thanin the τ channel, the experimental signal is cleaner in this case. All in all, the leptonic decay modesof the B c meson represent an interesting tool to probe the tensor scenarios, with predictions thatcould be at reach for the LHC in the near future. The leptonic decays of the light-quarks can probe SMEFT coefficients with flavor structures involv-ing third quark generation even if their contribution is suppressed by CKM factors arising whentransforming to the mass basis. In order to illustrate this let us now focus on possible contributionsof chirality-flipping NP operators in the b → uℓν transitions, which could be advocated to, e.g.,solve the inclusive versus exclusive discrepancy on | V ub | [11]. Let us take, for example, a NP modelwhose main contribution is the (real) Wilson coefficient [ w ℓequ ] ee , in our SMEFT flavor basis.Such an operator will generate, according to the matching equations (3.17), a leading contributionto the b → ueν decays. Namely, ǫ ubeS/P ( m Z ) = − V tb V ub [ w ℓequ ] ee ( m Z ) . (4.29)9owever it also gives a contribution to s → ueν e transitions via, ǫ useS/P ( m Z ) = − V ts V us [ w ℓequ ] ee ( m Z ) , (4.30)which, despite the strong CKM suppression, can lead to very strong bounds from kaon decaydata, as discussed in Section 4.1. For instance, the process B → eν has an experimental limit of ≤ − at 90% confidence level in the branching fraction [12]. Taking into account the running,this translates into the following bound for the underlying SMEFT Wilson coefficient | [ w ℓequ ] ee ( m Z ) | ≤ . × − , (4.31)where we use the N f = 2 + 1 + 1 lattice result f B = 0 . w ℓequ ] ee ( m Z ) = (2 . ± . × − , (4.32)at 90% C.L. Similar conclusions can be derived for tensor operators through the mixing discussedin the previous sections.Let us stress, to finish, that the purpose of this simple exercise, with only one coefficient([ w ℓequ ] ee ) at the electroweak scale, is illustrative. The specific numbers will certainly change ifmore operators or flavor-space entries are included, which will be probably the case for any realisticNP model or in scenarios advocating specific flavor symmetries. Nonetheless, the main messageremains: NP models (or SMEFT benchmark scenarios) addressing the so-called B anomalies couldbe potentially constrained by light quark decays. We studied the renormalization group evolution of (pseudo)scalar and tensor new physics contribu-tions to the (semi)leptonic charged-current meson decays. We calculated the mixing and evolutionunder electroweak and electromagnetic interactions, including also the three-loop running in QCDand taking into account the crossing of quark thresholds. We provided numerical formulas, cf.
Eqs. (3.12), (3.13) and (3.18), which connect the values of the corresponding Wilson coefficientsat the high-energy scales, relevant for collider physics and model-building, with their values at thelow energies where they are phenomenologically accessible in meson decays. These results can bebe trivially implemented in public codes [3, 52–54] and in future analysis of these transitions. Asan example, we used our results to evolve the bounds obtained in the recent global fit of Ref. [3]using nuclear, pion and kaon decay data at the low-energy scale to the electroweak scale.From the phenomenological point of view, the most important effect is related to the largemixing of the tensor operator into the scalar ones, both through QED and electroweak one-loopeffects. We first illustrated the importance of this mixing for light quark decays, discussing thestringent bounds that can be derived on tensor operators by exploiting the mixing and the factthat the induced (pseudo)scalar operators lift the helicity suppression that purely-leptonic decaysof pions and kaons receive in the SM.We also showed how this helicity suppression can also be exploited in tensor scenarios thataddress the exclusive vs. inclusive determinations of | V cb | and | V ub | . Indeed, the RGE-induced10seudoscalar operators would produce enhancements with respect of the SM that in the case ofthe B c → eν can reach several orders of magnitude.Furthermore, we showed that the RGE mixing has an important impact on the tensor explana-tions of the R D ( ∗ ) anomalies. The sizable scalar contributions generated radiatively at low energiesmodify importantly the phenomenology and, interestingly enough, improves the agreement of thisscenario with data. Finally, we discussed the indirect bounds that can also be derived on operatorsconnecting to third-quark generation using the very stringent bounds stemming again from theleptonic pion and kaon decays.We conclude by noting that other finite radiative corrections are expected in the maching ofspecific NP models to the SMEFT, the matching between the SMEFT and the low-energy EFT,and from the radiative corrections to the physical processes. In comparison, the pieces computedin this work receive logarithmic enhancements which do not represent a large amplification if theNP scale is not very high. Our work shows that radiative corrections can be important for thelow-energy phenomenology of the chirally-flipping operators and should be taken into account innew physics studies of the (semi)leptonic meson decays. Note added.-
As this work was being completed, Ref. [55] appeared, including the 1-loop QEDanomalous dimension that agrees with our result of Eq. (2.3). Our work in progress was presentedby one of us (M.G.-A.) at the Portoroz 2017 workshop on April 20, 2017.
Acknowledgements
We thank R. Alonso, V. Mateu, M. Misiak and A. Pich for useful discussions, and A. Falkowskifor his valuable collaboration in the early stage of this work. M.G.-A. is grateful to the LABEXLyon Institute of Origins (ANR-10-LABX-0066) of the Universit´e de Lyon for its financial supportwithin the program ANR-11-IDEX-0007 of the French government.
A Results of the fit to light quarks at µ = m Z Here we present the result of the fit to nuclear, hyperon, pion and kaon decay data of Ref. [3]evolved from µ = 2 GeV to µ = m Z using eq. (3.12). We refer the reader to this reference for11etails on notation and definitions. We find ∆ CKM ǫ usµL − ǫ useL ǫ udµL − ǫ udeL − ǫ udµP + 0 . ǫ udµT ǫ udeR ǫ useR ǫ udeS ǫ udeP ǫ udeT ǫ useS ǫ useP ǫ useT ǫ usµS ǫ usµP ǫ usµT µ = m Z = − . ± . . ± . − . ± . − . ± . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . − . ± . . ± . × ∧ − − − − − − − − − − − − − − (1.33)where ∆ CKM = 2 | V ud | ( ǫ udeL + ǫ udeR ) + 2 | V us | ( ǫ useL + ǫ useR ) − δG F G F . (1.34)The correlation matrix given by ρ = . − .
07 0 .
01 0 . . .
73 0 . . . . . − .
11 0 .
02 0 . − . . . . . . . . . . .
14 0 .
04 0 . − − . − .
87 0 . .
01 0 .
32 0 . . . . .
04 0 .
09 0 . − − − . . . − .
27 0 . . . . . . . − − − − . . . . . − .
04 0 . . − .
98 0 . − − − − − . .
02 0 .
02 0 . . . . . . − − − − − − . .
95 0 . . . .
01 0 .
03 0 . − − − − − − − . . . . . . . − − − − − − − − . .
16 0 .
16 0 . . . − − − − − − − − − . . . .
04 0 . − − − − − − − − − − . . . . − − − − − − − − − − − . . . − − − − − − − − − − − − . . − − − − − − − − − − − − − . (1.35) References [1] M. Antonelli et al.,
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