Generic One-Loop Matching Conditions for Rare Meson Decays

Leptonic and semileptonic meson decays that proceed via flavour-changing neutral currents provide excellent probes of physics of the standard model and beyond. We present explicit results for the Wilson coefficients of the weak effective Lagrangian for these decays in any perturbative model in which these processes proceed via one-loop contributions. We explicitly show that our results are finite and gauge independent, and provide Mathematica code that implements our results in an easily usable form.

. e matrix U ij is the two-generation quark-mixing matrix. Note that since we only consider two fermion generations inside the loop, the charged vector couplings need to be speci ed only for one generation -see text for details.
read o . e Wilson coe cients depend on a minimal set of physical parameters and are guaranteed to be nite and gauge independent. ese properties follow from coupling constant sum rules derived from Slavnov-Taylor identities as outlined in Ref. [7].
As an example, consider the SM contribution to the Wilson coe cient C 9 in the weak e ective Lagrangian. 1 Generically, the minimal eld content in the loop that is required to obtain a nonzero, nite, result consists of two massive vector bosons, one charged and one neutral, two charged fermions, and one neutral fermion -see the le panel in Table 1. Once the couplings of these states are speci ed, and the sum rules among them are applied, Eq. (3.7) directly gives the nite and gaugeindependent result, where F L,BZ V and F γZ V are loop functions that, in the SM, only depend on m 2 t /m 2 W , see Eq. (4.4). Here, G F is the Fermi constant, e is the positron charge, s W ≡ sin θ W is the sine of the weak mixing angle, and V ij are the elements of the Cabibbo-Kobayashi-Maskawa matrix. e procedure is exactly the same for any extension of the SM, it's that simple! ere are two important points to note here. First, the unitarity of the quark-mixing matrix is guaranteed by the sum rule in Eq. (3.1). Furthermore, in the absence of tree-level avour-changing neutral currents (FCNCs), at least two fermion generations in the loop are required to give a non-zero contribution. Second, and more remarkable, the same sum rule, Eq. (3.1), xes the couplings of the Z boson to the internal and external fermions and, consequently, it is not necessary to specify them in the rst place. In this way, the Z penguin, photon penguin, and boxes are combined into gaugeindependent loop functions that generalise the penguin-box expansion of Ref. [8]. e penguin-box functions -X, Y , and Z of Ref. [8] -are directly related to our functions F L,B Z V , F L,BZ V and F γZ V in the SM limit. Apart from an overall normalisation, the only di erence is that F γZ V also incorporates the light particle contribution in the matching procedure. Our functions generalise X, Y , and Z to extensions of the SM with an arbitrary number of massive vectors, scalars, and fermions while remaining gauge independent.
General expressions for the photon dipole have already been presented in Refs. [9][10][11][12], while contributions of heavy new scalars and fermions to the b → s transition were considered in Refs. [13][14][15]. Here, we extend the discussion to the contributions of the photon and Z penguins to the semileptonic 1 e operators we will focus on in this paper are O bs 7 = m bs σ µν PRb Fµν , O 9 = (sγµPLb)(¯ γ µ ), and O 10 = (sγµPLb)(¯ γ µ γ5 ). Equation (2.1) then shows that C 9 = (C 23 LL + C 23 LR )/2, for example. Note that we use an effective Lagrangian, as opposed to an e ective Hamiltonian as in Ref. [3]. current-current operators, with a special focus on proving gauge invariance in the presence of heavy vectors, and eliminating couplings to unphysical scalars such as would-be Goldstone bosons. Moreover, we provide easy-to-use code to obtain the Wilson coe cients in general perturbatively unitary models, it is available at https://wellput.github.io . e paper is organized as follows. e generic interaction Lagrangian of the extended eld content is given and discussed in Sec. 2. e relevant sum rules are discussed in Sec. 3 along with the dipole and current-current Wilson coe cients. ere, we also explain the cancellation of the gauge dependent terms. In Sec. 4, we apply our setup to three models taken from the literature to illustrate how the one-loop matching contributions can be easily obtained. We conclude and summarize our work in Sec. 5 and give explicit expressions for the loop functions in App. A. e additional sum rules required for the renormalisation of the Z penguin are collected in App. B.
2 Generic Model and E ective Lagrangian e goal of this work is to provide the explicit form of the e ective Lagrangian relevant for leptonic, semileptonic, and radiative B, B s , and K meson decays for a generic renormalisable model. We write the ve-avour e ective Lagrangian that describes the d j → d i transition, obtained by integrating out the W and Z bosons, the top quark, as well as all heavy new particles at the electroweak scale, as e operators in the rst sum have the form of a product of a leptonic current and a FCNC. e second sum contains the photon dipole operators. Here, d i = d, s, b denote the down-type quark elds and the lepton elds. P L ≡ (1 − γ 5 )/2 and P R ≡ (1 + γ 5 )/2 are the chirality projection operators, and σ and σ denote the chiralities of the incoming quarks and leptons. We neglect all operators with mass dimension larger than six. e explicit results for the Wilson coe cients are given below in Eqs. (3.3) and (3.7) -(3.9).
In the following, we will determine the explicit form of the Wilson coe cients C ij σσ and D ij σ for a generic interaction Lagrangian of fermions (ψ), physical scalars (h), and vector bosons (V µ ) of the form (cf. Ref. [7]) where σ ∈ {L, R}. e indices f i , s i , and v i denote the di erent physical fermion, scalar, and vector elds, respectively, and run over all particles in a given multiplet of the gauge group U (1) EM × SU (3) color . Spinor indices are suppressed in our notation. e non-interacting part of the Lagrangian is given by the standard kinetic terms, an R ξ gauge xing term for each massive vector, and a 't Hoo -Feynman gauge-xing term for the photon eld. Here, φ v and ξ v denote the Goldstone boson and the gauge xing parameter associated with the vector eld V µ , while the coe cient σ v can have the values ±i for complex elds and ±1 for real elds. e kinetic term, furthermore, determines the trilinear interactions with the photon eld through the covariant derivatives D µ f = (∂ µ − ieQ f A µ )f that act on a eld f of charge Q f . With this choice we have g σ γf f = eQ f , g vvγ = eQ v and g γss = eQ s , where Q v and Q s denote the charges of the vector V v,µ and the scalar h s , respectively, and the bar denotes the coupling with a charge conjugated elds. 2 Without additional constraints, the Lagrangian of Eq. (2.2) does not describe a renormalisable quantum eld theory and cannot be used to derive predictions for physical processes that are nite and gauge independent. e necessary constraints arise from using the Slavnov Taylor Identities (STIs) derived in Ref. [7] from the vanishing Becchi-Rouet-Stora-Tyutin (BRST) [17,18] transformation of suitable vertex functions. ese STIs are su cient to constrain the relevant couplings for ∆F = 1 avour changing transitions that are generated at one-loop order. In addition, the STIs determine the unphysical Goldstone couplings in terms of the physical couplings. For instance, the Feynman rule of the photon interactions can be read of from the generic Lagrangian by replacing appropriate scalar elds s by φ and noting that the STIs derived in Ref. [7] imply g vvγ = g γφφ . is allows us to express all contributions of Goldstone bosons in terms of physical couplings. Hence, all following results include all relevant contributions from Goldstone bosons even if only physical coupling constants appear.
3 Results for the Wilson coe icients e Wilson coe cients of the e ective Lagrangian are functions of the couplings of the generic Lagrangian and the associated masses. ey are determined by calculating suitable Green's functions: e photon penguin diagrams ( Fig. 1) contribute in part to the dipole coe cients D ij σ , and in part to the current-current coe cients C ij σσ via the equations of motion of the photon eld. e Z-penguin and box diagrams ( Fig. 2) contribute to the current-current coe cients C ij σσ . In the remainder of this section, we spell out the details of this calculation, with a focus on obtaining a nite and gaugeindependent result.
We incorporate the constraints from the STIs by repeatedly applying the sum rules on the one-loop amplitudes. For the evaluation of the o -shell photon penguin d j → d i γ Green's function (see Fig. 1) we only need the "unitarity sum rule" [7] v 3 2 e QED interaction follows from the kinetic terms [16]: where the summation on the right hand side of the equation is over all possible fermions f 1 that satisfy the charge conservation conditions. Se ing Additionally, this implies that the charges of the fermions f 1 can either be which respectively contribute to the rst or second sum on the right hand side.
Since we only consider interactions where g v 3di d j = 0 for any neutral vector v 3 , we nd the following generalisation of the Glashow-Iliopoulos-Maiani (GIM) relation is relation can be used to eliminate the couplings of any one member of the set of fermions of charge Q f 1 that generate avour changing neutral currents through charged vector interactions. For de niteness, we always choose to eliminate the lightest of such fermions. is will simultaneously determine the Wilson coe cients of the dipole operators and the photon-penguin contribution to the ∆F = 1 current-current operators. e Wilson coe cients of the dipole operators are independent of the gauge xing parameters, while the photon-penguin contribution is not. 3 e Wilson coe cients then depend on the mass of the lightest fermion that can contribute in the loop, denoted above by the index f 0 . is particle could be either a light 4 standard-model fermion, such as an up quark, or a heavy fermion, such as a chargino. A fermion mass of f 0 that is considerably smaller than the matching scale requires an appropriate e ective theory counterpart that will account for the infrared logarithm generated in the limit m f 0 → 0, see App. A.2.
For the renormlisation of the Z penguin, two more sum rules are required [7]; we list them in App. B.

Dipole Operator Coe icients
Here and in the following, we write the Wilson coe cient of the ve-avour e ective Lagrangian as a product of the coupling constants and gauge-independent loop functions that depend on various mass ratios de ned by e matching coe cients of the dipole operators are immediately gauge independent. We nd if the generalised GIM mechanism of Eq. (3.2) has already been applied to the sum of fermions of Finally, let us note that we could further simplify the function F d V using the sum rule Eq. (B.1) if tree-level neutral current and scalar interactions are absent. In this limit we have when we set m d j = m d i = 0. Our results agree with Ref. [11] if we apply our generalised GIM mechanism to their results. Here we note that it is only possible to project the o -shell Green's function a er using the GIM mechanism. e coe cients D ij L can be recovered from D ij R by simply interchanging the chirality of all coupling constants, i.e. by replacing y L ··· ↔ y R ··· and g L ··· ↔ g R ··· in Eq. (3.3).

Neutral-Current Wilson Coe icient
Both the photon penguin diagrams of Fig. 1 and the Z penguin and box diagrams of Fig. 2 contribute to the matching conditions for the current-current Wilson coe cients. e analytic expression of each of the three diagram classes depends on the gauge xing parameters of the massive vector bosons in the loop. A renormalised result for the Z penguin was derived in Ref. [7] in 't Hoo -Feynman gauge using sum-rules derived from Slavnov-Taylor identities. Here we will show how to apply these same sum rules to combine the amplitudes of all three diagram classes into a nite and gauge-parameter independent result for the Wilson coe cients. To this end, we split our nal expression into three parts,C ij contains several gauge-independent loop functions. e functions F γZ V and F σ,B ( ) Z V are the gaugeindependent combinations of the photon penguin with the Z penguin and the photon penguin with the box-diagrams. While all of the above functions involve contributions from the lightest fermionic particle in the loop through our generalised GIM mechanism, only F γZ V will contain an infrared logarithm in the limit x f 0 v 1 → 0. is logarithm is reproduced by a light-quark loop involving f 0 in the e ective theory. In the standard model this corresponds to the leading logarithm associated with the mixing of the operator Q 2 into Q 9 of Ref. [8].
e loop function F γZ V reproduces this leading logarithm if the considered model of new physics has the same light-particle content as the standard model. It will then drop out in the di erence of the standard model and the new-physics contribution and we can consider the resulting di erence the leading new-physics contribution.
ere are two gauge-independent combinations for the Z-penguin and box diagram that are distinguished by their fermion ow. Charge conservation implies that the le box diagram in Fig. 2 con- In the SM, F σ,BZ V and F σ,B Z V will then contribute to b → sµ + µ − and s → dνν, respectively. e loop functions F Z V ( / ) are the M Z -independent parts of the functions evaluated in Ref. [7] and are only non-zero in physics beyond the standard model. In particular, we note that all contributions with diagonal Z couplings vanish since F Z V ( ) (x, x) = F Z V (x, y, 1) = 0. Finally, we give the contributions involving internal scalars, vectors, and fermions, and only scalars and fermions, where we in both cases we have a single box function that covers both fermion ow directions, albeit with a sign di erence.

Derivation of the pure vector part
In the following we will show how the combination of the results of Ref. [7] with our calculation of the photon penguin will lead to gauge independent results for the Wilson coe cients. Denoting the contribution of the photon penguin that involves a photon coupling to the internal fermion and vector boson by F γ and F γ , respectively, we write 5 where all functions are independent of the masses M Z arising from one-particle reducible diagrams involving neutral massive vector-particle propagators. e functions F Z V ( , ) have already been combined with the terms that originate from the o -diagonal eld renormalisation, as described in Ref. [7].
is combination is essential to arrive at a gauge-independent result. In this context it is interesting to note that we can further use the sum rules to write F Z V in a simpler and more symmetric form. e combination V agrees with the F V of Ref. [7] in the limit of 't Hoo -Feynman gauge; here the gauge-parameter dependent part has been split o into the loop function F (2) V . e dependence on the mass of the lightest fermion f 0 originates from the application of the generalised GIM mechanism, Eq. (3.2), to our result. It implies that the functions F e functions F γ and F γ have been calculated here for the rst time, while the box functions F Lσ B ( ) and F Lσ B ( ) are related to the expressions of Ref. [7] in the limit ξ v = 1 in the following manner:

(3.13)
For an arbitrary gauge-xing parameters ξ v , only the f d function contains ξ v -dependent terms. To combine the penguin and box contributions of (3.10) we specify the sum rule (3.1) to the interaction of leptons with vector bosons, which allows us to identify and Using the explicit form of the loop functions, it can then be shown that the resulting expressions are independent of the gauge-xing parameter.

Applications to Beyond the Standard Model Phenomenology
To exemplify our formalism we will apply it to models of new physics that address the current rare B-decay anomalies. In this context, it is standard to write vector and axial-vector current operators; the Wilson coe cients of this e ective Lagrangian, are related to our coe cients of Eq. (3.6) via the linear transformation e relation for the operators involving right-handed quarks can be inferred from the above relation, by replacing C 23 Lσ → C 23 Rσ . If we are interested in deviations from the standard model background, we have to subtract the standard model one-loop contribution from our complete new-physics calculation; hence, we de ne C NP 9/10 = C 9/10 − C SM 9/10 .
where we have used the fact that

A Z -Model with flavour o -diagonal couplings
To demonstrate the utility of the expressions derived in Sec. 3, we begin by applying them to a simple model [19] developed to address the b → s lepton avour non-universality anomaly. e model consists of a vector-like quark with up-type quantum numbers which is additionally charged under a hidden U (1) gauge group spontaneously broken by the vacuum expectation value of a scalar eld Φ. e relevant couplings of the mass eigenstates to the gauge bosons are given by where, s L/R and c L/R are the sine and cosine of the le -/right-handed t − T mixing angles andg is the U (1) gauge coupling. e U (1) charge of the top partner, T , is denoted by q and that of the muon by q ,V /A for the vectorial/axial couplings. With these couplings, Eq. (3.7) directly gives the contribution to the Wilson coe cients which are and where we have subtracted the SM contribution. To evade collider constraints, one furthermore assumes that m T m t . In this limit we nd: (4.8) where the log(x T W ) agrees with the result in Ref. [19], while the remaining terms are new and reduce the contribution to both C 9 and C 10 by 13(7)% for m T = 1(10) TeV.

A U (1) Lµ−Lτ model with Majorana fermions
e gauged U (1) Lµ−Lτ model was originally proposed in Refs. [20,21] and has been studied extensively in the context of lepton universality violation. Here we focus on the model of Ref. [22] where an additional Dirac fermion N and a coloured SU (2) L -doublet scalarq ≡ (ũ,d) T with hypercharge Y = 1/6 are introduced that are all charged under the L µ − L τ gauge group. A er spontaneous symmetry breaking the relevant interactions in terms of the mass eigenstates read 9) where N ± = (N ± N c ) / √ 2 is wri en in term of N and its charge conjugated eld N c , g X is the U (1) Lµ−Lτ gauge coupling, Q is the charge of N , and y s/b L are the Yukawa couplings of the SM bo om and strange quarks tod. e Z penguin does not involve any SM particles and is lepton universality violating by construction. e complete one-loop new physics contributions to C µ 9 can be read o from Eq. (3.9). Noting that the charge conjugated scalard c contributes in the sum of (3.9), we nd where the rst line represents the Z -penguin contribution and agrees with the results of Ref. [22]. e terms in the second line represent the lepton avour universal new physics contribution to C 9 from the photon-penguin and is new. Note that the photon-penguin decouples faster than the Z penguin in the limit of large scalar mass md. e Z coupling to the down quarks cancels with the Z couplings tod c in (3.9) so that the Z-penguin contribution cancels. e contribution to C NP 7,bs can be calculated from the general formula (3.3) and is given by where the operator O bs 7 is de ned in footnote 1. Note that only one of the terms is present since N is electrically neutral and therefore only the charged scalar,d c , contributes.

A model with vector-like fermions and neutral scalars
To give another application of our results, we consider a model that consists of SU (2) L doublet vector-like quarks and leptons in addition to one or two complex scalars that are neutral under the SM gauge group [23]. e interaction Lagrangian of interest reads Hermitian conjugation gives the le -handed Yukawa couplings, y L , which are related to the righthanded ones via where Φ ∈ {Φ L , Φ H }, Ψ ∈ {Ψ Q , Ψ }, and f ∈ {b, s, } as applicable. e expressions for the Yukawa couplings can be read o from Ref. [23] and we omit writing them explicitly. e NP contribution to C 9 and C 10 are, then, and, from Eq. (3.9), we have (4.15) Note that C 9 receives a lepton-avour-universal contribution from the photon penguin. is contribution breaks the relation C 9 = −C 10 but it is suppressed by fermion masses and is therefore subleading in the limit where the scalars are lighter. Substituting the couplings from Ref. [23] and translating the box functions, F B S , into their G functions gives perfect agreement with their result.

Summary and conclusions
In this work, we have presented nite and manifestly gauge-invariant matching contributions at the one-loop level onto the weak e ective Lagrangian in generic extensions of the SM. at is, we add to its eld content any number of massive vector bosons, physical scalars, and fermions. For a given eld content, only a minimal number of couplings needs to be speci ed because perturbative unitarity of the S-matrix implies that not all couplings can be independent. e constraints on the couplings are codi ed in the sum rules that arise from Slavnov-Taylor identities which are in turn obtained from the invariance of appropriate Green's functions under BRST transformations. e main results of this paper, the sum rules on the additional couplings and the nite and gaugeinvariant one-loop contribution, are implemented in a Mathematica package available for download from https://wellput.github.io . is package contains an example le that includes the SM contribution to the operators considered in this paper along with all three extensions discussed in Sec. 4. Speci cally, we considered three classes of extensions that demonstrate the three types of contributions in Eqs. Finally, the scope of this paper was to implement the matching onto the |∆F | = 1 dipole and current-current weak e ective Lagrangian Wilson coe cients. e extension to avour-conserving magnetic and electric dipole operators and to dimension-six scalar operators is already work-inprogress and will appear in the near future.

SC0011784.
is work was also supported by the Deutsche Forschungsgemeinscha (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC 2121 " antum Universe" -390833306 and the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. UM is supported by the Bolashak International Scholarship Programme.

A Loop Functions
In this appendix we collect the analytical expressions of all loop functions that appear in the nal results for the renormalised Wilson coe cients. ese functions depend on the masses of the particles inside the respective loop diagrams and on their electromagnetic charges.

A.1 Loop Functions for the Dipole Coe icients
In the limit where no particles are much lighter than the matching scale, we nd the functions involving scalars, and vectors, that contribute to the Wilson coe cient of the dipole operator in Eq. (3.3). As stated above, our results agree with Ref. [11] a er employing the relevant unitarity sum rule.

Limit of light internal particles
Light internal particles can in principle give a contribution from the e ective theory side of the matching equation. e scalar loop functions that multiplies Yukawa couplings of the same chirality must contain an odd number of chirality ips as explained above. is implies that the infrared logarithm √ x log(x) vanishes in the limit x → 0. Since we work at dimension ve for our dipole operators, the e ective theory contribution is vanishing in this limit and we do not have to consider the scalar functions further. e vector contributions of the dipole operator have no infrared logarithm in the limit of the lightest internal fermion mass tending to zero. Since F d V is multiplied with the internal fermion mass we only need to consider the limit x 0 → 0 for F d V and nd:

A.2 Loop Functions for the Neutral-Current Operators
We rst give the functions that contribute to the Wilson coe cient of the neutral-current operators in the scenario where no light internal particles are in the loop. We start with the rst term in Eq. (3.6) that comprises the contributions of internal vector bosons and fermions. We nd the following gaugeinvariant combination of the photon penguin and the Z Penguin