Patterns of flavour violation in models with vector-like quarks

Journal of High Energy Physics, Apr 2017

We study the patterns of flavour violation in renormalisable extensions of the Standard Model (SM) that contain vector-like quarks (VLQs) in a single complex representation of either the SM gauge group GSM or G SM ′ ≡ GSM ⊗ U(1)L μ  − L τ . We first decouple VLQs in the M = (1 − 10) TeV range and then at the electroweak scale also Z, Z ′ gauge bosons and additional scalars to study the phenomenology. The results depend on the relative size of Z- and Z ′-induced flavour-changing neutral currents, as well as the size of |ΔF | = 2 contributions including the effects of renormalisation group Yukawa evolution from M to the electroweak scale that turn out to be very important for models with right-handed currents through the generation of left-right operators. In addition to rare decays like \( P\to \ell \overline{\ell},P\to {P}^{\prime}\ell \overline{\ell},P\to {P}^{\prime}\nu \overline{\nu} \) with P = K, B s , B d and |ΔF | = 2 observables we analyze the ratio ε ′ /ε which appears in the SM to be significantly below the data. We study patterns and correlations between these observables which taken together should in the future allow for differentiating between VLQ models. In particular the patterns in models with left-handed and right-handed currents are markedly different from each other. Among the highlights are large Z-mediated new physics effects in Kaon observables in some of the models and significant effects in B s,d -observables. ε ′ /ε can easily be made consistent with the data, implying then uniquely the suppression of \( {K}_L\to {\pi}^0\nu\ \overline{\nu} \). Significant enhancements of \( Br\left({K}^{+}\to {\pi}^{+}\nu \overline{\nu}\right) \) are still possible. We point out that the combination of NP effects to |ΔF | = 2 and |ΔF | = 1 observables in a given meson system generally allows to determine the masses of VLQs in a given representation independently of the size of VLQ couplings.

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Patterns of flavour violation in models with vector-like quarks

Received: October avour violation in models with vector-like quarks Christoph Bobeth 0 1 2 3 4 6 Andrzej J. Buras 0 1 3 4 6 Alejandro Celis 0 1 3 5 Martin Jung 0 1 2 3 6 0 Open Access , c The Authors 1 Boltzmannstr. 2, D-85748 Garching , Germany 2 Excellence Cluster Universe, Technische Universitat Munchen 3 Arnold Sommerfeld Center for Theoretical Physics , 80333 Munchen , Germany 4 Physik Department, TU Munchen, James-Franck-Stra e , D-85748 Garching , Germany 5 Ludwig-Maximilians-Universitat Munchen, Fakultat fur Physik 6 TUM Institute for Advanced Study , Lichtenbergstr. 2a, D-85748 Garching , Germany We study the patterns of avour violation in renormalisable extensions of the Standard Model (SM) that contain vector-like quarks (VLQs) in a single complex representation of either the SM gauge group GSM or G0SM GSM the relative size of Z- and Z0-induced avour-changing neutral currents, as well as the size of j F j = 2 contributions including the e ects of renormalisation group Yukawa evolution from M to the electroweak scale that turn out to be very important for models with right-handed currents through the generation of left-right operators. In addition to rare ArXiv ePrint: 1609.04783 quarks; Beyond Standard Model; CP violation; Heavy Quark Physics; Kaon Physics - decouple VLQs in the M = (1 10) TeV range and then at the electroweak scale also Z; Z0 gauge bosons and additional scalars to study the phenomenology. The results depend on decays like P ! ``, P ! P 0``, P ! P 0 patterns and correlations between these observables which taken together should in the future allow for di erentiating between VLQ models. In particular the patterns in models with left-handed and right-handed currents are markedly di erent from each other. Among the highlights are large Z-mediated new physics e ects in Kaon observables in some of the the data, implying then uniquely the suppression of KL ! ) are still possible. We point out that the combination of NP e ects to . Signi cant enhancements the masses of VLQs in a given representation independently of the size of VLQ couplings. 1 Introduction 2 The VLQ models 2.2.1 Scalar sectors Decoupling of VLQs VLQ representations Yukawa interactions of VLQs Yukawa couplings of several representations Tree-level decoupling and Z and Z0 e ects GSM-models G0SM-models Decoupling at one-loop level Renormalisation group evolution 4 Implications for the down-quark sector 4.1 j F j = 2 4.2 j F j = 1: semi-leptonic dj ! di + (``; 4.3 j F j = 1: hadronic dj ! diqq and "0=" Patterns of avour violation 5.1 j F j = 2 5.2 j F j = 1 GSM models G0SM( ) model Determination of M Kaon and B-meson systems Summary and conclusions A Scalar sectors of G0SM-models A.1 G0SM(S) models A.2 G0SM( ) models B VLQ decoupling and RG e ects 2'2D operators 2'3 operators B.3 Top-Yukawa RG e ects C Master formulae for K and B decays C.1 j F j = 2 C.4 dj ! di qq and "0=" D Statistical approach and numerical input Introduction Among the simplest renormalisable extensions of the Standard Model (SM) that do not introduce any additional ne tunings of parameters are models in which the only new particles are vector-like fermions. Such fermions can be much heavier than the SM ones as they can acquire masses in the absence of electroweak symmetry breaking. If in the process of this breaking mixing with the SM fermions occurs, the generation of avour-changing neutral currents (FCNC) mediated by the SM Z boson is a generic implication. If in addition the gauge group is extended by a second U(1) factor, a new heavy gauge boson Z0 is present and additional heavy scalars are necessary to provide mass for the Z0 and to break the extended gauge-symmetry group down to the SM gauge group. There is a rich literature on FCNCs implied by the presence of vector-like quarks (VLQs), see in particular [1{12]. The goal of the present paper is an extensive study of patterns of avour violation in models with VLQs that are based on the following gauge groups: The choice of the particular symmetry group U(1)L L [13, 14] is phenomenologically motivated by the fact that it allows in a simple manner to address successfully the LHCb anomalies [9, 15], while being anomaly-free and containing less parameters than general Z0 models [16]. In our paper we will be guided by the analyses in refs. [3, 11, 17] which identi ed all renormalisable models with additional fermions residing in a single vector-like complex representation of the SM gauge group with a mass M . It turns out that there are 11 models where new fermions have the proper quantum numbers so that they can couple in a renormalisable manner to the SM Higgs and SM fermions, thereby implying new sources of avour violation. Our analysis will concentrate on FCNCs in the K, Bd and Bs systems, therefore only the ve models with couplings to down quarks are relevant for us, as speci ed in section 2. We call this class of models GSM-models. Consequently the models based on the gauge group G0SM are called G0SM-models. The VLQs in these models belong to the same representations under GSM as in GSM-models, but are additionally charged under U(1)L L . These models also contain new heavy scalars. As we will discuss in detail in section 2 and section 5, the patterns of avour violation in GSM-models and G0SM-models di er signi cantly from each other: In GSM-models Yukawa interactions of the SM scalar doublet H involving ordinary quarks and VLQs imply avour-violating Z couplings to ordinary quarks, which transitions is much more involved and depends on whether right-handed (RH) or left-handed (LH) avour-violating quark couplings to the Z are present. If they are RH the e ects of renormalisation group (RG) evolution from M (the common VLQ mass) down to the electroweak scale, EW, generate left-right operators [18] via top-Yukawa induced mixing. These operators are strongly enhanced through QCD RG e ects below the electroweak scale and in the case of the K system through chirally enhanced hadronic matrix elements. They dominate then new physics (NP) contributions to "K , but in the Bs;d meson systems for VLQ-masses above 5 TeV they have to compete with contributions from box diagrams with VLQs [11]. If they are LH the Yukawa enhancement is less important, because left-right operators are not present and box diagrams play an important role both in the Bs;d and K systems. In G0SM-models the pattern of avour violation depends on the scalar sector involved. We consider only models in which at least one of the additional scalars is charged under U(1)L L in such a way that Yukawa couplings between the given VLQ and ordinary quarks are allowed. If this is the case for a new scalar which is just a singlet S under the SM group, the latter imply avour-violating Z0 couplings to ordinary quarks without any FCNCs mediated by the Z. In the following we refer to these models as G0SM(S)-models. If, on the other hand, such a Yukawa coupling requires the scalar to be a doublet , both tree-level Z0 and Z contributions to avour observables will be present. Their relative size depends on the model parameters, speci cally the Z0 mass. In these cases we introduce again an additional scalar singlet, but without Yukawa couplings, since otherwise the Z0 mass would have to be of the order of the electroweak scale, which is phenomenologically very di cult to achieve. In the following we refer to these models as G0SM( )-models. In this manner we will consider three classes of VLQ models with rather di erent patterns of avour violation: G0SM(S) ; 5 TeV by box diagrams with VLQs and scalar exchanges, while in the G0SM(S) models also tree-level Z0 exchanges can play an important, sometimes dominant, role. A particular feature of GSM models are the topare absent in G0SM models. In [11] an extensive analysis of the GSM-models has been performed and a subset of G0SM-models has been analyzed in [9, 15]. Therefore it is mandatory for us to state what is new in our article regarding these models: The authors of [11] concentrated on the derivation of bounds on the Yukawa couplings as functions of M but did not study the correlations between various avour observables which is the prime target of our paper. Similar comments apply to [9]. NP contributions to avour observables depend in each model on the products of complex Yukawa couplings s d b s for s ! d, b ! d and b ! s transitions, respectively, as well as the VLQ mass M . This structure allows to set one of the q-phases to zero, such that each model depends on only ve Yukawa parameters and M , implying a number of correlations between avour observables. The strongest correlations are, however, still found between observables corresponding to the same avour-changing transition, and we concentrate our analysis on them. The correlations between observables with di erent transitions are weaker, but could turn out to be useful in the future when the data and theory improve, in particular in the context of models for Yukawa couplings. An important novelty of our paper, relative to [9, 11, 15], is the inclusion of the signi cantly above its SM prediction [19{22]; it is hence of interest to see which of the models analyzed by us, if any, are capable of addressing this tension and what the consequences for other observables are. general is the inclusion of the e ects of RG top-Yukawa evolution from M to the electroweak scale that turn out to be very important for models with RH currents through the generation of left-right operators contributing to these transitions as mentioned above. This changes markedly the pattern of avour violation in such models relative to models with LH currents where no left-right operators are generated. Our paper is organized as follows. In section 2 we present the particle content of the considered VLQ models, together with the gauge interactions, Yukawa interactions and the scalar sector. In section 3 we perform the decoupling of the VLQs and construct the e ective eld theory (G(S0M)-EFT) for each model for scales EW < < M . Section 4 is devoted to the the scale EW. This results in explicit avour-violating couplings of the Z and Z0 to the SM quarks. These enter the e ective Lagrangians for the various avour-changing processes, from which we derive the explicit formulae for the considered observables. In section 5 we describe the patterns of avour violation expected in di erent models, summarizing them with the help of two DNA tables. In section 6, after formulating our strategy for the phenomenology, we present numerical results of our study. We conclude in section 7. Several appendices collect additional information on the models, the decoupling of VLQs, RG equations in the GSM-EFT, the considered decays, some technical details and the input and statistical procedure used in the numerical analysis. The VLQ models Throughout the article we focus on models with vector-like fermions residing in complex representations, either of the the SM gauge group GSM or its extension by an additional L ) symmetry, U(1)L L . For both models we adapt the usual SM fermion standard scalar SU(2)L doublet H. The gauged (L L ) symmetry is anomaly-free in the SM [13, 14]. The only non L ) charges of the SM fermions are introduced as Q0(L2L) = Q0( R) = Q0`; Q0(L3L) = Q0( R) = right-handed singlets. We normalize the (L L ) charges of the leptons without loss of gauge boson Z0. However, such couplings are generated in G0SM models through Yukawa interactions of SM quarks with VLQs that couple directly to Z0. VLQ representations As we are mainly interested in the phenomenology of down-quark physics, we will restrict our analysis to SU(3)c triplets and consider the following ve models with SU(2)L singlets, doublets and triplets: D(1; 1=3; X); where the transformation properties are indicated as (SU(2)L; U(1)Y; U(1)L denotes the charge under U(1)L L . It is implied that in GSM-models the U(1)L charge should be omitted. The representations D, QV , Qd, Td, Tu correspond to the models V, IX, XI, VII, VIII introduced in ref. [11], where a complete list of renormalisable models with vector-like fermions under GSM can be found, see also [3, 17]. Concerning studied rst in [9]. The kinetic and gauge interactions of the new VLQs are given by Lkin = D(i D= MD)D + Qa(i D= MQa ) Qa + Tr T a(i D= with appropriate covariant derivatives D and we follow [11] for the triplet representations as given in (2.13) and (2.14) of that paper. The masses M of the VLQs introduce a new scale, which we will assume to be signi cantly larger than all other scales. The covariant derivative is, omitting the SU(3)c part, = @ with the gauge couplings g2;1 and g0 of SU(2)L, U(1)Y and U(1)L L , respectively, and charges Y and Q0 of U(1)Y and U(1)L L . The Pauli-matrices are denoted by \hat" on Z^0 indicates that we deal here with the gauge eigenstate and not mass eigenstate, Yukawa interactions of VLQs The scalar sector consists of the SM scalar doublet H with its usual scalar potential. The VLQs interact with SM quarks (qL; uR; dR) via Yukawa interactions LYuk(H) = iD HyDR + iTd HyT dR + iTu He yT uR q i 2H . The complex-valued Yukawa couplings iVLQ give rise to mixing with the SM quarks and avour-changing Z-couplings, which have been worked out in detail [3, 11] and are discussed in section 3.1. G0SM(S) In models with an additional U(1)L L the scalar sector has to be extended in order to generate the mass of the corresponding gauge boson Z0. A complex scalar S(1; 0; X) (SU(3)c singlet) is added in the minimal version. As VLQs are charged under U(1)L their Yukawa couplings with the SM doublet H are forbidden, but the ones involving S are allowed for Q0S = Q0VLQ and given by [9] LYuk(S) = iD diR DL + iV QV R qLi S + h.c. : models as G0SM(S)-models. The special feature of these models is that because of the absence of tree-level Z contributions tree-level Z0 exchanges dominate F = 1 transitions and in some part of the parameter space can also compete with contributions from box diagrams with VLQs and scalars in the case of F = 2 transitions. For VLQs with GSM quantum numbers di erent from one of the SM quark elds, the simple extension by a scalar singlet is not possible. In a next-to-minimal version we therefore add to the scalar sector an additional scalar SU(2)L doublet (2; +1=2; X), besides the SMYukawa couplings | see for example [23] | and in consequence there are no LFV Z0 couplings, which are subject to strong constraints at low energies. The vacuum expectation value (VEV) of gives an unavoidable contribution to the Z0 mass of the order of the electroweak scale, contributes to the mass of H and generates potentially large Z mass mixing e ects. The latter would be strongly constrained by electroweak precision tests [24], in particular there would be sizeable corrections to the Z couplings to muons. In order to avoid these di culties, is accompanied by an additional complex scalar singlet S(1; 0; Y ), which breaks the U(1)L L symmetry at the TeV scale. The L L charge of scalar sector and to forbid Yukawa couplings of S with SM fermions and VLQs. The Yukawa interactions of the VLQs with LYuk( ) = Td yT dR + iTu eyT uR qLi + iQd eydiR QdL + h.c.; and we will refer to these models as G0SM( )-models. We note that the structure of couplings equals the one of GSM models given in eq. (2.5) upon H models FCNCs are mediated by both Z and Z0 but in the case of F = 2 transitions box diagrams with VLQs and scalars play the dominant role for su ciently large M . For ease of notation, we will sometimes refrain below from explicitly labelling the i by the VLQ representation, as should be done if several of them are considered simultaneously. Yukawa couplings of several representations In our numerics we will consider one VLQ representation at a time as this simpli es the analysis signi cantly. In particular the number of parameters is quite limited. Still it is useful to make a few comments on the structure of avour-violating interactions and at various places in our paper to state how our formulae would be modi ed through the presence of several VLQ representations in a given model. We plan to return to the phenomenology of such models in the future. When admitting several VLQ representations F m and F n simultaneously, potentially additional locally gauge-invariant Yukawa couplings to be included in the case of GSM-models [3]. They give rise to avour-changing neutral Higgs currents at tree level. In the G0SM-models the U(1)L L -charges of the additional have been chosen following the criteria explained above, which xes in turn the emnF L 'mnFRn with 'mn = H have 'mn = S; G0SM(S) models, only the particular choice of the U(1)L L charges Q0QV = ity to replace QV R qi L ! qLi QV R in eq. (2.6), which maintains gauge invariance since S is a singlet. On the other hand, in G0SM( ) models such couplings arise for Qd with D and Td. are not perL -charge. In Q0D [9] forbids Another important consequence of the presence of several representations is the gendiagrams discussed in section 3.2, which is the case when singlets or triplets together with doublets are present. In the case of a single representation such operators can also be generated in models with doublets through the top-Yukawa RG evolution from M to the electroweak scale, see section 3.3. Scalar sectors which provides masses to gauge bosons and standard fermions in the course of spontaneous symmetry breaking of SU(2)L U(1)Y ! U(1)em via the VEV v ' 246 GeV, where hHi = (0; v=p2)T : MZ20 = g02vS2X2: Z0-gauge boson In G0SM( )-models the doublet 2 h 0ai = pva ; v = appendix A.2. given model. Further details on the scalar sectors of the G0SM(S) and G0SM( ) models are collected in appendix A.1 and A.2, respectively. In table 1 we summarize all G0SM-models and indicate Decoupling of VLQs The VLQ models are characterised by the masses M of the VLQs, the various Yukawa couplings i see section 2.3. The present lower bound on M from the LHC is in the ballpark of 1 TeV, while the lower bounds on MZ0 are typically close to 3 TeV if Z0 has a direct coupling to light quarks. But as emphasized in [9, 15, 25], Z0 of U(1)L L does not have such couplings, implying a much weaker lower bound on its mass, which could in fact be as low 1This convention corresponds to that of the Type I 2HDM. VLQ Representation Scalar Singlet Da(3; 1; 1=3; X) S(1; 1; 0; X) Scalar Doublets H(1; 2; 1=2; 0) Db(3; 1; 1=3; X) S(1; 1; 0; X=2) 1(1; 2; 1=2; X), 2(1; 2; 1=2; 0) QV (3; 2; +1=6; +X) S(1; 1; 0; X) Qd(3; 2; 5=6; X) S(1; 1; 0; X=2) Td(3; 3; 1=3; X) S(1; 1; 0; X=2) Tu(3; 3; +2=3; +X) S(1; 1; 0; X=2) H(1; 2; 1=2; 0) 1(1; 2; 1=2; X), 2(1; 2; 1=2; 0) 1(1; 2; 1=2; X), 2(1; 2; 1=2; 0) 1(1; 2; 1=2; X), 2(1; 2; 1=2; 0) and j F j = 2 transitions for M as the electroweak scale and even lower. While it could also be as heavy as the VLQ mass, we will assume the hierarchy MZ . MZ0 or equivalently v . vS in order to simplify the analysis. It is then natural to decouple rst the VLQs and to consider EFTs for GSM and G0SM valid between the scales are subsequently matched in one step onto SU(3)c U(1)em-invariant phenomenological mb, where mb denotes the bottom mass. The coe cients determined in the process will indicate which operators are the most important. In principle one could consider an intermediate EFT which is constructed by integrating out Z0 and the new scalars before integrating out top quark, W and Z, but from the point of view of renormalisation group e ects, integrating out all these heavy elds simultaneously appears to be an adequate approximation. at the scale In this section we present the results from the decoupling of the VLQs that are important for our phenomenological applications within the framework of the G(S0M)-EFTs. EW is given in section 4. The Lagrangian of the G(S0M)-EFT consists of the dimension-four interactions of the light elds and dimension six interactions generated by the decoupling of VLQs which are invariant under either GSM or G0SM, depending on the model. Thus in GSMmodels Ldim 4 coincides with the SM Lagrangian and the corresponding non-redundant set of operators of dimension six has been classi ed in ref. [26]. In G0SM-models operators that are invariant under G0SM must be added, which involve the Z0-boson and the additional scalar singlets and/or doublets. The Wilson coe cients Ca2 are e ective couplings, which 2The Wilson coe cients of G(S0M)-EFTs are denoted with calligraphic Ci, whereas the ones of phenomenological EFTs with Ci. when decoupling VLQs. The decoupling proceeds either by explicit matching calculations starting at tree-level and including subsequently higher orders or by integrating them out in the path integral method [3]. The tree-level decoupling has been known for a long time for GSM models [3] and is given for G0SM(S) models in ref. [9]. Within the EFT, RG equations allow to evolve the Wilson coe cients from M down to EW. In leading logarithmic approximation and retaining only the rst logarithm (1stLLA) it has the approximate solution Ca( EW) = which holds as long as the second term remains small compared to the rst. The anomalous dimension matrices (ADM) ab depend in general on couplings of the gauge, Yukawa and scalar sectors and are known for the GSM-EFT [27{29]. Largest contributions might be expected for the case of ab / YuyYu t2 mixing due to the top-quark Yukawa coupling generated at 1stLLA order.3 In particular, as we will see below, in the case of models with right-handed neutral currents left-right operators can be generated in this manner j F j = 1 observables. The VLQs have a very limited set of couplings to light elds, which are either via gauge interactions (2.3) to the gauge bosons or via Yukawa interactions (2.5){(2.7) to light | At tree-level, this particular structure of interactions can give rise only to avour-changing Z and Z0 couplings, whereas all other decoupling e ects are loop-suppressed [30]. The decoupling of the VLQs proceeds in the unbroken phase of SU(2)L U(1)Y, hence take place within the G(S0M)-EFTs and the transformation from quark elds are avour-eigenstates and neutral components of scalar elds are without VEV at this stage. After the RG evolution from M to EW, spontaneous symmetry breaking will avour- to mass-eigenstates for fermions and gauge bosons can be performed, accounting for the dimension six part in Tree-level decoupling and Z and Z0 e ects The couplings of the VLQs permit at tree level only a dimension six contribution from the generic 4-point diagram in gure 1a. Since its dimension- ve contribution vanishes [3], it is equivalent to consider the 5-point diagram gure 1b, where either SU(2)L or U(1)Y gauge bosons in GSM-models or in addition a Z^0 in G0SM-models is radiated o the VLQ [3, 9]. As a consequence, in GSM- and G0SM-models only operators of the type 3Note that the 1stLLA neglects \secondary mixing" e ects that are present in LLA, i.e. summing all large logarithms, because although operator OA might not have ADM entry with operator OB (no \direct mixing"), it can still contribute to the Wilson coe cient CB( EW), if it mixes directly with some operator OC that in turn mixes directly into OB. = (qL; uR; dR). The gauge boson G depends on the representation. Tree-level graph (c) requires two representations Fm;n with a Yukawa coupling via 'c and give rise to 2'3 operators. (S S)[qLiujRHe ] (He yiD H)[uiR (HyH)[qLiujRHe ] (HyH)[qLidjRH] We follow the de nitions of [26] for 2'2D operators, except for the signs of gauge couplings in the covariant derivatives, and ( 2'3 + h.c.) operators in the case of GSM models and elds denote the generations. These are all operators that could arise from tree-level decoupling of VLQs, depending on the model. are projected in part onto 2'3-type operators via equation of motions (EOM) [26, 31]. We list the corresponding de nitions of the operators in table 2, following the notation of [26] in the case of the GSM-EFT and extending it to G0SM-EFTs. After spontaneous symmetry breaking the 2'3 operators contribute to the quark = u; d) at the scale EW via mij = p which allows to substitute Yukawa couplings Y in terms of measured m and new physics parameters C 2'3 / Y C 2'2D, see appendix B.2. If several representations of VLQs are present in a given model and two of them Fm;n couple to a scalar 'c4 via Yukawa couplings emn, a third possibility is allowed at tree-level depicted in gure 1c, which contributes 2'3 operators and gives rise to avour-changing neutral H i j interactions at contributions. Their diagonalisation proceeds as usual for the quark elds with the help of 3 unitary rotations in avour space: R = mdiag; V = (VLu)yVLd ; basis for the VLQ Yukawa couplings i with diagonal up- and down-quark masses mdiag and the unitary quark-mixing matrix V . In the limit of vanishing dimension-six contributions, V will become the Cabibbo-KobayashiMaskawa (CKM) matrix of the SM. Throughout we will assume for down quarks the weak xes also the de nition of the Wilson coe cients C 2'2D (for more details see [32]) and the After spontaneous symmetry breaking the 2'2D operators give rise to (Z0) = f i h iLj (Z0) For completeness, we provide the matching conditions for the Wilson coe cients in appendix B. We note that RG e ects have been neglected in (3.7) and (3.8) since they are only due to self-mixing of 2'2D operators as listed in appendix B.3. SM as corrections from NP to them are in GSM-models one-loop suppressed. This is also the case of G0SM(S) models where Z does not play any role in FCNCs. In G0SM( ) models modi cations of the Zf f couplings come from Z Z0 mixing. These shifts are relevant for leptons in partial widths of Z ! `` (see appendix A.2) and could be of relevance in consistency in G0SM( ) models, although they are negligible in comparison to other e ects. GSM-models In the case of GSM-models, the decoupling of VLQs gives the results for L;R(Z) couplings collected for down-quarks in table 3, where Except for the sign in the case of Tu, our results agree with those in [11]. Furthermore, also non-zero couplings to up-type quarks arise [11] but they will not play any role in our paper. 4As discussed above 'c = H in GSM and G0SM-models. ij =2 mn(V y)nj =2 models. Here Vij is the CKM matrix and u = G0SM-models In the G0SM-models, the (L L ) symmetry xes the Z0 coupling to leptons to be `L`(Z0) = `R`(Z0) = ` ` (Z0) = g0Q0`; with Q0` = f0; +1; 1g for ` = fe; ; g . Here we have neglected Z Z0 mixing e ects existing in G0SM( )-models. However, for consistency we have to include these e ects in the couplings of the Z to leptons `L`(Z) = `R`(Z) = gZ s2W + g0Q0` ZZ0 ; to rst order in the small mixing angle ZZ0 (see appendix A.2 for details). On the other hand, the gauge couplings to quarks are model dependent. In G0SM(S)-models the scalar sector of S and H generates only non-zero quark couplings to Z0, whereas in G0SM( )-models the scalar sector of S, H and couplings of SM quarks to both Z0 and Z. We de ne gives rise to non-zero 2 Xigj0 MMZ220 ; ij de ned in eq. (3.9) and the Z Z0 mixing angle [see (A.9)] Kij in eq. (3.12). cos is a parameter associated with the scalar sector (see (2.10)) of G0SM( )Z0 mixing, which is phenomenologically constrained to be small, ZZ0 < 0:1, due to constraints from the Z-boson mass, MZ , and partial widths Z ! `` measured at LEP, as described in more detail in appendix A.2. The down- and up-quark couplings to Z0 and Z are collected for these models in table 4. We con rm previous ndings [9] for the G0SM(S)-models. We note that the Z0 couplings are suppressed/enhanced by the ratio r0 w.r.t. the Zcouplings. Enhancement takes place for 2 g0X > gZ 0:75, such that for example r0 can be reached with g0X 1:1, still within the perturbative regime. The couplings of Td and Tu di er just by a sign and factors 1/2. In distinction to Z-contributions in GSMmodels, both Z- and Z0-contributions in G0SM( ) models decouple with large tan , see r0 VimKmn(V y)nj u r0VimKmn(V y)nj=2 r0Kij=2 [1 r0 ZZ0] Kij=2 [1 r0 ZZ0] VimKmn(V y)nj=2 models. Here Vij is the CKM matrix. interactions with scalars ' = H; S; and SM quarks = (qL; dR; uR). The crossed graph appears Decoupling at one-loop level All other decoupling processes proceed via loops. Those that would lead to non-canonical kinetic terms in the G(S0M)-EFTs can be absorbed by a suitable choice of wave-function renormalisation constants in the full theory above the scale M , resulting in non-minimal renormalisation of interactions and giving rise to nite threshold e ects of coupling constants. In G0SM-models this is the case for kinetic mixing of B and Z^0 , which enters our analysis only as a higher order e ect. All other e ects enter as dimension six operators. The ones with four quarks are most important for quark- avour phenomenology. They involve only VLQ-Yukawa interactions, as depicted in gure 2a and gure 2b, and give rise to 4-type operators, among which thereby the intermediate matching to the GSM-invariant form.5 Still, we outline this step for completeness here. In the VLQ models considered, there are four relevant 4 operators in G(S0M)-EFTs at the VLQ scale via RG evolution from M to EW. These are the (LL)(LL) operators M and a fth operator is generated due to QCD mixing [Oq(1q)]ijkl = [qLi [Oq(3q)]ijkl = [qLi the (LL)(RR) operators and the (RR)(RR) operator [Oq(1d)]ijkl = [qLi [Oq(8d)]ijkl = [qLi [Odd]ijkl = [diR electroweak scale EW [32] as CVijLL = CLijR;1 = CVijRR = A ; CLijR;2 = Nij 1[Cq(8d) where Nij is given in (C.2). Here we anticipate this matching to the VLQ scale there are no RG e ects of phenomenological importance for the discussion of B-meson and Kaon sectors. For more details see section 3.3, where also QCD mixing is given for these operators. Since the Wilson coe cients of these operators are generated at M at one-loop, their interplay with other sectors in quark- avour physics due to RG mixing are considered higher order and hence beyond the scope of our work. via box diagrams (see gures 2a and 2b), which contain two heavy VLQ propagators (H+; H0)T . These box diagrams yield the general structure of the Wilson coe cients Caij ( M ) = at the scale EFT, see (C.2). The function f1(Mm; Mn) = f1(Mm; Mm) = 5Note that the set of 4-type operators is the same in all G(S0M) models and a non-redundant set can be found in ref. [26]. (Fm; Fn) LR1, 1=4 VLL, 1=8 VRR, 1=4 LR1, 3=8 LR1, 3=8 VLL, 1=8 followed by corresponding mn. depends on the VLQ masses of representations Fm;n. The couplings imj are imj = ( im) j imj = im( jm) Fm = D; Td; Tu; Fm = Qd; QV : The index a of the operator and the numerical factors mn are collected in table 5. Note crossed, which gives rise to an additional sign w.r.t. the diagram with non-crossed scalar XI) we nd an additional factor of 2. Concerning QV (model IX) we nd a contribution to OVRR instead of OVLL and also opposite sign. For completeness we provide also the results for Fm 6= Fn. In G0SM(S) models we consider only VLQs D and QV and their interference D : CVRR = CVLL = CLR1 = which agrees with [9] except for a minus sign from crossed scalar propagators in the interference term D The results for G0SM( ) models can be found straight-forwardly from the ones of the GSM models, bearing in mind that (2.5) and (2.7) are equivalent up to the replacement The VLQ tree-level exchange in the considered VLQ scenarios generates only 2'2D- and 2'3-type operators at the scale M with nonvanishing Wilson coe cients (see appendix B) G0SM(S) : depending on the VLQ scenario.6 The RG evolution from M down to EW can induce via operator mixing leading logarithmic contributions also to other classes of operators in G(S0M) EFTs at the scale and thus imply additional potential constraints. EW. These operators are possibly related to a variety of processes The largest enhancements can appear if the ADM ab in (3.3) is proportional to the strong coupling 4 1:4 or the top-Yukawa coupling yt 1. Note that QCD mixing is avour-diagonal and hence can not give rise to new genuine phenomenological e ects, i.e. one can not expect qualitative changes. On the other hand, Yukawa couplings are the main source of avour-o -diagonal interactions and we will focus on these here. The SU(2)L gauge interactions induce via ADMs ab / g22 [29] only intra-generational mixing between u iL and are parametrically smaller than yt-induced e ects, such that we do not consider them here. The U(1)Y gauge interactions are only avour-diagonal and numerically even more suppressed. Concerning G0SM models, RG e ects due to top-Yukawa couplings are absent for 2'2D 2'3 operators, because ' = S; do not have Yukawa couplings to qL; uR; dR, which are forbidden by their additional U(1)L L charge. Hence RG e ects as discussed below are not present in these scenarios. and we collect the ones involving the Wilson coe cients (3.22) in appendix B.3. The RG equations of these Wilson coe cients are also coupled with those of SM couplings, such as the quartic Higgs coupling and quark-Yukawa couplings [27], but in 1stLLA they decouple. The modi cation of SM couplings due to dim-6 e ects can be neglected when discussing the RG evolution of dim-6 e ects themselves in rst approximation. Moreover, the quartic Higgs coupling is irrelevant for the processes discussed here and the quark masses are determined from low-energy experiments, i.e. much below EW. Hence phenomenologically most interesting are RG e ects of mixing of 2H2D and 2H3 operators into other operator classes that do not receive tree-level matching contributions at M . Those classes are H4D2 (2) ; where we list in parentheses the number of operators.7 We focus on the 4 operators, which all turn out to be four-quark operators, because they are most relevant for processes 7Implying footnote 6. also CHu and CHud must be considered. of down-type quarks considered here. We comment shortly on the H6 and H4D2 classes in appendix B.3. The RG equation (3.3) implies for a speci c a 2 Ca( EW) = 4, see also [18], novel chiral structure of the return to this point in section 4.3. 2H2D contributions. Three of the 4 operators (Oq(1q;3) and Oq(1d)) can mediate downin both SMEFT and phenomenological EFTs, therefore receiving another suppression in where it competes with the direct one-loop box contribution in VLQ models discussed in ated directly by 2H2D operators in the next matching step of GSM to phenomenological EFTs at EW (see section 4 and gure 4), which are therefore enhanced in these processes compared to the 1stLLA contributions discussed here. Consequently, the 1stLLA is one4 operators enhances a speci c hadronic observable. We will Under the transformation from weak to mass eigenstates for up-type quarks (3.5) the corresponding ADMs of 4 operators in appendix B.3 transform as 2 VLumdUiagV uy = [YuyYu]ij = [YuYuy]ij = with up-type quark mass mk and the de nition of CKM-products i(jt) given in (4.1). Since the ADMs are needed here for the evolution of dim-6 Wilson coe cients themselves, we have used tree-level relations derived from the dim-4 part of the Lagrangian only, thereby neglecting dim-6 contributions, which would constitute a dim-8 corrections in this context. In the sum over k only the top-quark contribution is relevant (mu;c mt), if one assumes that the unitary matrix V is equal to the CKM matrix up to dim-6 corrections.8 The j F j = 2 mediating 4 operators involve the combination (3.29). We obtain via (3.26) and explicit matching conditions (B.1) Caij ( EW) = EW are entirely negligible. and the VLQ-model-dependent factor a = VLL a = LR; 1 Fm = D; Td; Tu ; Fm = Qd; QV ; m = Fm = (D; Qd; QV ; Td; Tu): We note the relations m Mij2 = [CH(1q) (Fm = D; Td; Tu); where the relative sign comes from relative signs in (B.23) and (B.24) when inserted in (3.17) and m Mij2 = [CHd]ij ; (Fm = Qd; QV ) : We point out the di erent avour structure of the 1stLLA contribution (3.30) compared to the one of the direct box-contribution (3.18) discussed in the previous section section 3.2: showing linear versus quadratic dependence on the product of VLQ Yukawa couplings ij . A detailed comparison of both contributions is given in section 5. CVLL(VRR)( EW) = 6 CLR;1( EW) = 6 CLR;1( M ) ; CLR;2( EW) = The initial conditions of Caij ( M ) from box-diagrams are collected in (3.18) and (3.21). s(6)( M )= s(6)( EW). Implications for the down-quark sector one-loop level at the scale M has been presented, including the most important e ects from the RG evolution down to the electroweak scale EW. In this section we discuss the decoupling of degrees of freedom of the order of EW by matching onto phenomenological for avour physics. In fact, as we have shown, large NP e ects in avour observables can be present for MVLQ = 10 TeV and in the avour-precision era one is sensitive to even VLQs in a given representation independently of the size of Yukawa couplings. Acknowledgments Genon for providing us an update of a tree-level CKM t from CKM tter [94]. This research was done and nanced in the context of the ERC Advanced Grant project \FLAVOUR"(267104) and was partially supported by the DFG cluster of excellence \Origin and Structure of the Universe". The work of A.C. is supported by the Alexander von Humboldt Foundation. This work is supported in part by the DFG SFB/TR 110 \Symmetries and the Emergence of Structure in QCD". Scalar sectors of G0SM-models G0SM(S) models The scalar sector in G0SM(S)-models with one complex scalar S(1; 0; X) and the SM doublet H(2; +1=2; 0) is given by L = jD Hj2 + jD Sj2 with the potential V = m2HyH + We parametrise the SM Higgs doublet and the complex scalar as H = v + h0 + iG0 = 2 S = The neutral mass-eigenstates are given by (h; H)T ' (h0; R0)T with approximate masses Kinetic mixing of Z and Z0 is caused by VLQ-exchange and depends on the VLQ masses M and the U(1)L L -gauge coupling. It will be neglected in the following, see ref. [9]. Mass mixing does not occur in G0SM(S) models. G0SM( ) models doublets 1 (2; +1=2; X) and H(2; +1=2; 0) is given by L = jD with the potential V = m2a ya a + We neglect kinetic mixing and parametrise the mass mixing via @Z^0 A = @ After partial diagonalization of the neutral gauge boson system, the Z and Z0 masses and their mass mixing are given by [96] M^ Z20 = (g0X)2 S 2 = with e = p = g2s^W = g1c^W = gZ s^W c^W . The Z Z0 mixing angle tan 2 ZZ0 = = c2 4Xg0 is small unless X becomes large. The diagonalisation of the neutral gauge boson mass matrix gives mass eigenvalues MZ2;Z0 = M^ Z20 + M^ Z2 lution for which MZ < MZ0 , i.e. throughout we will implicitly impose that the lighter mass eigenstate couples predominantly SM-like to quarks and leptons. As a consequence a lower bound on g0 will be obtained. On the other hand, the decoupling limit g0 ! 0 is not excluded, but it will lead to MZ0 < MZ , i.e. that the heavier mass-eigenstate couples predominantly to SM-like fermions. The tan dependence of MZ0 becomes irrelevant once vS & 0:5 TeV. The mixing angle ZZ0 can be suppressed with large tan and MZ0 , since we work in the part of the parameter space, where the other possibility of g0 ! 0 is not an option. In G0SM( )-models we make use of the fact that photon- and W -interactions to leptons are SM-like in order to determine the values of the fundamental gauge couplings g1;2 and the VEV v from e(MZ ), GF and the W -boson pole mass MW . As the remaining free parameters we choose tan , g0, X and vS, whereas dependent parameters are MZ;Z0 and ZZ0 . Note that the latter depend only on the product g0X, such that there are e ectively only three parameters. We will restrict this parameter space to 2 TeV: (A.11) The lower bound on tan guarantees perturbativity of the top-quark Yukawa coupling [64], whereas vS is bounded from above by the requirements (3.1) and yields MZ0 . 1:5 TeV within the above limits. Constraints on these parameters arise from the measured value of MZ , which we impose with an error of constraining the new physics contributions of the Z-lepton couplings (3.11) that depend on the ZZ0 and g0 due to gauge mixing. We nd a small mixing angle ZZ0 . 0:1 in the above speci ed parameter space of tan , g0X and vS if we impose the bound on new physics contributions to the partial widths of Z ! `` from LEP [24], allowing for 5 from the measured central values, together with the bound on MZ . This justi es the expansion in the small mixing angle as done in table 4. VLQ decoupling and RG e ects This appendix contains results of the Wilson coe cients of 2'2D and 2'3 operators in G(S0M)-EFTs after the tree-level decoupling of VLQs at the scale M . We provide further the relations to avour-changing Z and Z0 couplings (3.7) and (3.8) after spontaneous symmetry breaking at the scale EW (neglecting self-mixing). 2'2D operators contributions for The matching in GSM models at the scale M of order of the VLQ mass yields nonvanishing D : [CH(1q)]ij = [CH(3q)]ij = Td : [CH(1q)]ij = 3 [CH(3q)]ij = Tu : [CH(1q)]ij = 3 [CH(3q)]ij = 4 M 2 8 M 2 Qd : [CHd]ij = QV : [CHd]ij = 2 M 2 ; [CHu]ij = ; [CHud]ij = in agreement with [3], and analogously for G0SM( ) models with H ! G0SM(S) models for VLQs D and QV yields nonvanishing Wilson coe cients . The matching of D : [CSd]ij = 2 M 2 QV : [CSq]ij = 2 M 2 The avour-changing Z and Z0 couplings (3.7) and (3.8) after spontaneous symmetry breaking are given in terms of the Wilson coe cients at the scale EW. In the case of GSM-models, the tree-level calculation of the process fifj Z from GSM-EFT (3.2) yields G0SM(S)-models with the scalar sector of S and H generates only non-zero couplings to Z0. We nd for with the EFT-coe cients Ci given in (B.1) and FS m2Z0 =(g0X). The variant of G0SM( )models with the scalar sector of S, H and generates non-zero couplings to Z0 and Z. The results for G0SM( ) models are similar to GSM models, with the di erence that they Z0 mixings: where V = Z; Z0 and 2'3 operators We de ne the SM Yukawa couplings of quarks as in [26] LYuk = qL Yd H dR + qL Yu He uR + h.c.: Nonvanishing Wilson coe cients are generated also for 2'3 operators (see table 2 for de nitions) as a consequence of the application of equations of motion (EOM) in the treelevel decoupling of VLQs in section 3.1. Due to the application of EOMs, these Wilson coe cients scale with the corresponding Yukawa coupling as [CuH ]ij = Yu CHyuD + (CH(1q)D Note the matrix multiplications w.r.t. the generation indices of Yu;d with the respective coe cients CH D inside the brackets. The tree-level matching in GSM-models gives nonvanishing contributions at D : [CH(1q)D + CH(3q)D]ij = Td : [CH(1q)D + CH(3q)D]ij = Tu : [CH(1q)D + CH(3q)D]ij = [CHdD]ij = [CHdD]ij = 2 M 2 4 M 2 2 M 2 CH(3q)D]ij = 0; CH(3q)D]ij = CH(3q)D]ij = 2 M 2 4 M 2 [CHuD]ij = in agreement with [3]. Analogous Wilson coe cients in G0SM( ) are found by H ! In G0SM(S) models analogous relations [CuS]ij = Yu CSyuD + CSqD Yuiij hold with nonvanishing [CSdD]ij = [CdS]ij = Yd CSydD + CSqD Yd ij [CSqD]ij = Top-Yukawa RG e ects This appendix collects the ADM entries of the GSM-EFT proportional to the up-type quark Yukawa coupling Yu from [28], i.e. neglecting contributions from Yd;e. We list them only for operators that receive leading logarithmic contributions at the scale EW from the initial Wilson coe cients at the scale 2H3 operators in the 1stLLA via direct mixing, see footnote 3. For convenience of the reader we keep here also CHu and CHud, which are absent in the VLQ models D; Tu; Td; Qd, but contribute in QV for Vu 6= 0. 2 dCH = 12 Tr CuH YuyYuYuy + YuYuyYu CuyH ; and leads to a shift of the VEV [29]. The H4D2-operators OH = 6 Tr h (1) C_HD = 24 Tr hCH(1q)YuYuy Their Wilson coe cients contribute to the Higgs-boson mass and the electroweak precision observable T = 16Note that if the generation indices are not given explicitly on Yukawa couplings and Wilson coe cients then a matrix multiplication is implied. CuH = 2 CH(1q)YuYuyYu + 2 YuYuyYuCHu + 6 Tr CuH Yuy Yu + 9 Tr YuYuy CuH + 5 CuH YuyYu + CdH = 12 Tr CH(3q)YuYuy]Yd + 6 CH(3q)YuYuyYd 2 YuYuyYuCHud 2 YuCuyH Yd CuH YuyYd + 9 Tr YuYuy CdH CeH = 12 Tr CH(3q)YuYuy]Ye + 6 Tr YuCuH Y y 2 YuYuyCdH ; have self-mixing for CuH;dH , and CuH mixes also into CdH;eH . They receive also contributions from C 2H2D. The C 2H3 enter fermion-mass matrices (3.4) and lead also to fermion-Higgs couplings that are in general avour-o -diagonal. 2H2D-operators (see table 2) YuYuyCH(3q) + CH(3q)YuYuy CHud = 6 Tr[YuYuy]CHud + 3YuyYuCHud C_H(3q) = 6 Tr YuYuy]CH(3q) + YuYuyCH(3q) + CH(3q)YuYuy YuYuyCH(1q) + CH(1q)YuYuy ; CHd = 6 Tr YuYuy]CHd ; CHu = 2YuyCH(1q)Yu + 6 Tr[YuYuy]CHu + 4 YuyYuCHu + CHuYuyYu ; show a mixing pattern among CHq (1;3) as well as CH(1q) and CHu. The latter implies that the LH scenarios D; Tu; Td will generate via mixing also a RH coupling CHu via CHq, which is however a one-loop e ect compared to the e ects of CHq. Both CHd and CHud have only In the case of 4-operators there are (LL)(LL) operators [C_q(1q)]ijkl = + [C_q(3q)]ijkl = [YuYuy]ij [CH(1q)]kl + [CH(1q)]ij [YuYuy]kl ; [YuYuy]ij [CH(3q)]kl + [CH(3q)]ij [YuYuy]kl ; the (LL)(RR) operators and the (RR)(RR) operators [C_q(1u)]ijkl = [YuYuy]ij [CHu]kl [C_q(1d)]ijkl = [YuYuy]ij [CHd]kl; [C_uu]ijkl = [C_u(1d)]ijkl = [YuyYu]ij [CHu]kl 2[YuyYu]ij [CHd]kl; [CHu]ij [YuyYu]kl; under the assumption CHu = 0. F j = 2 Master formulae for K and B decays The e ective Lagrangian for neutral meson mixing in the down-type quark sector (dj di ! dj di with i 6= j) can be written as [34] where the normalisation factor and the CKM combinations are H F =2 = Nij X Caij Oaij + h.c.; Nij = OVijLL = [di OLijR;1 = [di OLijR;2 = [diPLdj ][diPRdj ]; OSijLL;1 = [diPLdj ][diPLdj ]; OSijLL;2 = which are built out of colour-singlet currents [di : : : dj ][di : : : dj ], where ; denote colour indices. The chirality- ipped sectors VRR and SRR are obtained from interchanging PL $ PR in VLL and SLL. Note that the minus sign in QSLL;2 arises from di erent de nitions of ]=2 in ref. [34] w.r.t. = i~ used here. The ADM's of the 5 distinct sectors (VLL, SLL, LR, VRR, SRR) have been calculated in refs. [33, 34] at NLO in QCD, and numerical solutions are given in ref. [97]. The NLO ADM's are also available for an alternative In the SM only CVijLL( EW)jSM = S0(xt); S0(x) = 11x + x2) is non-zero at the scale EW, depending on the ratio xt W -boson masses. mt2=M W2 of the top-quark and are given in table 14. The j F j = 2 observables of interest MK; Bd; Bs , K and sin(2 d;s) derive all from the complex-valued o -diagonal elements M1ij2 of the mass-mixing matrices of the neutral mesons [99, 100]. For the latter we use the full higher-order SM expressions in combination with the LO new physics contributions. In particular for M1d2s, we make use of NLO and in part NNLO QCD corrections cc; tt; ct collected in table 13 and for the hadronic matrix QCD corrections B to the SM and use for the hadronic matrix elements the latest results The e ective Lagrangian for dj ! di (i 6= j) is adopted from ref. [91], = fe; ; g In the SM only It is given by has non-vanishing contribution at the scale C depend on the ratio xt EW, whereas CR = 0. The functions B and the gauge-independent linear combination X0(xt) 4B(xt) [101, 102], OLij;(R) = [di SM = X0(x) = X0 ! XLSM = 1:481 The Br(K+ the local OL charm quark at c 2), where e ective Hamiltonian of the SM as [108] the \top"-sector, when decoupling heavy degrees of freedom at EW, which yields directly mc, which is enhanced due to the strong CKM hierarchy ( (std) / ) receives in the SM the numerically leading contribution from when including higher order QCD and electroweak corrections [103{106] as extracted in ref. [107] from original papers. The theoretical predictions for b ! s observables de ned in eq. (6.5) are based on formulae given in ref. [92]. These expressions account for the lepton-non-universal contribution of VLQ's w.r.t. the neutrino avour in G0SM models. However, the particular structure of the gauged U(1)L L (2.1) leads to a cancellation of the numerically leading interference contributions of the SM and new physics [9]. He = N The NP contributions in VLQ-models cannot compete with the SM contribution to the tree-level processes entering the \charm"-sector, since they are suppressed by an additional Xt = XLSM + XLsd; + XRsd; + = rK+ e e) = 0:5173(25) contains the experimental value Br(K ! e e) and the isospin correction rK+ and has been evaluated in ref. [110] (table 2) including various corrections. Further for Emax 20 MeV [110]. If one takes into account the di erent value of s2 EM = W = 0:231 taken in ref. [110] compared to our value in table 13, then + = 0:5150 10 10 ( =0:225)8. The sum (C.12) contains the SM contribution and further the interference of SM and NP NP. Besides Pc at NNLO in the SM contribution, the NLO numerical values Xce = 10:05 Xc = 6:64 The branching fraction of KL ! is obtained again by averaging over the three neutrino avours ) = L = rKL KL = 2:231(13) with XLsd;R; given in eq. (4.8), such that the top-sector becomes neutrino- avour dependent. The experimental measurement averages over the three neutrino avours, ; (C.12) with the assumption that (scd)Xc is real. The NNLO QCD results of the functions Xc [108] together with long distance contributions [109] are combined into Pc = 3 Xce + 3 Xc The numerical value is from ref. [110] (table 2) and it decreases to L = 2:221 Ld!d`` = O9ij(;`90) = [di PL (R)dj ][` `]; O1ij0;`(100) = [di PL (R)dj ][` whereas scalar OS`;P(S0;P0) and tensorial operators OT`(T5) are not generated in the context of VLQ models. In the SM the only non-zero Wilson coe cients, SM = C1ij0;` SM = are lepton- avour universal and also universal w.r.t. down-type quark transitions, as the CKM elements have been factored out. All other Wilson coe cients vanish at the scale EW. The functions B; C; D depend again on the ratio xt mt2=M W2 of the top-quark and W -boson masses and give two gauge-independent combinations Y0(xt) and Z0(xt) Y0(x) = Z0(x) = 163x3 + 259x2 15x2 + 18x ln x: (C.23) In the predictions of Br(Bd;s ! ) and the mass-eigenstate rate asymmetry ) we include for the SM contribution the NNLO QCD [112] and NLO EW [39] corrections, whereas NP contributions are included at LO. The values of the decay constants FBd;s are collected in table 13. The branching fractions Br(B+ are predicted within the framework outlined in refs. [113{115]. We neglect contributions from QCD penguin operators, which have small Wilson coe cients and the NLO QCD corrections to matrix elements of the charged-current operators [116, 117], but include the VubVud(s). The form factors and their uncertainties are adapted from lattice calculations [118, 119] for B ! and [120] for B ! K with a summary given in [121]. We add additional relative uncertainties of 15% for missing NLO QCD corrections and 10% ) at high dilepton invariant mass q2 for possible duality violation [114] in quadrature. The predictions for observables of B ! K are based on refs. [89] and [122] for lowand high-q2 regions, respectively. The corresponding results for B ! K form factors in the two regions are from the LCSR calculation [123] and the lattice calculations [124, 125]. The measurement of Br(KL ! ) provides important constraints on its shortdistance (SD) contributions, despite the dominating long-distance (LD) contributions inducing uncertainties that are not entirely under theoretical control. In particular there is the issue of the sign of the interference of the SD part SD of the decay amplitude with the LD parts. Allowing for both signs implies a conservative bound 3:1 [74]. Relying on predictions of this sign based on the quite general assumptions stated in [74, 126, 127] one nds 1:7 which we employ in most of this work. Note, however, that a di erent sign is found17 in [126, 128], implying In light of this situation, we comment on the impact of the more conservative choice where appropriate, which includes both sign choices. 17We thank G. D'Ambrosio and J-M. Gerard for the discussion on this point. last column gives Pa for B(1=2) = 0:57 and B(3=2) = 0:76. 6 8 dj ! di qq and "0=" de nition of the operators can be found and here we restrict ourselves to s ! d, i.e. ij = sd. At the scale EW (Nf = 5) it reads Ld!dqq = + X( va + vaNP)Oa + X va0Oa0 where O1(c;2) denote current-current operators. The sum over a extends over the QCD- and 5]. Thereby we assume that VLQ contributions to other operators are strongly suppressed. The Wilson coe cients are denoted as za, va(NP) and va0, taken at the scale EW. For the SM-part, CKM unitarity was used, and we introduced a new physics contribution vaNP as shown above, which is related to the VLQ-contribution (4.13) as va0 = Cas0d: The RG evolution at NLO in QCD and QED leads to the e ective Hamiltonian at a after decoupling of b- and c-quarks at scales b;c [129], where ya za and all Wilson coe cients are at the scale . z1O1 + z2O2 + X[za + ya + vaNP]Oa + X va0Oa0 + h.c.; (C.27) The coe cients are where the minus sign is due to h( )I jOajKi = )I jOa0jKi for the pseudo-scalar pions nal state [130]. For the readers convenience we provide a semi-numerical formula 2:58 + 24:01B6(1=2) 12:70B8(3=2)i Pa = p(a0) + p(a6)B(1=2) + p(a8)B(3=2) 6 8 B(3=2)( ) = 0:76. For this purpose with p(an) given in table 12, where the last column gives Pa for B(1=2)( ) = 0:57 and 6 EW = MW , b = mb(mb), c = 1:3 GeV and compared to 1:9 10 4 in [20] due to di erent numerical inputs. Statistical approach and numerical input The input quantities included in our analysis are collected in table 13 and table 14. The CKM parameters have to be determined independently of contributions from the VLQs. The \tree-level" t carried out by the CKM tter collaboration achieves such a determination, taking only measurements into account that are una ected in our NP scenarios, i.e. (semi-)leptonic tree-level decays, tree-level determinations of as a constraint on . The results of this t are again quoted in table 13. As a statistical procedure, we choose a frequentist approach. The ts include as parameters of interest the VLQ couplings and in addition nuisance parameters, which constitute theoretical uncertainties. The nuisance parameters are listed in table 13 and consist of CKM parameters from a \tree-level" t;18 The 1- and 2-dimensional con dence regions (CL) of parameters are obtained by pro ling over the remaining parameters, i.e. maximisation of the likelihood function over the subspace of remaining parameters for a xed value of the (pair of) parameter(s) of interest. Similarly, correlation plots for pairs of observables are obtained by pro ling over all parameters and imposing in addition the speci c values for the pair observables. The 2-dimensional 68% and 95% con dence regions are determined then for two degrees of freedom. The SM predictions of observables are found in the same way by setting VLQ contributions to zero and pro ling only over the CKM and hadronic nuisance parameters. 18We thank Sebastien Descotes-Genon for providing us an update of a tree-level CKM t from CKM t= 0:22544(+3238) = 0:125(+3108) mK = 497:614(24) MeV FK =F = 1:194(5) BK = 0:750(15) ct = 0:496(47) mBd = 5279:61(16) MeV mBs = 5366:79(23) MeV FBd = 190:5(42) MeV FBd (B^Bd )1=2 = 229:4(93) MeV B = 0:55(1) A = 0:8207(7)(13) = 0:382(+2128) F = 130:41(20) MeV tt = 0:5765(65) cc = 1:87(76) = 1:638(4) ps Bd = 1:520(4) ps Bs = 1:505(4) ps MW = 80:385(15) GeV GF = 1:16638(1) (MZ ) = 1=127:9 MZ = 91:1876(21) GeV sin2 W = 0:23126(13) s(MZ ) = 0:1185(6) quark masses md(2 GeV) = 4:68(16) MeV [131] ms(2 GeV) = 93:8(24) MeV [131] mc(mc) = 1:275(25) GeV mb(mb) = 4:18(3) GeV mt(mt) = 163(1) GeV s= s = 0:124(9) : Calculated by demanding that the uncertainty of the ratio of the decay constants given above should equal the uncertainty given explicitly for the ratio, also given in ref. 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Christoph Bobeth, Andrzej J. Buras, Alejandro Celis, Martin Jung. Patterns of flavour violation in models with vector-like quarks, Journal of High Energy Physics, 2017, 79, DOI: 10.1007/JHEP04(2017)079