One constraint to kill them all?

Many new physics models that explain the intriguing anomalies in the $b$-quark flavour sector are severely constrained by $B_s$-mixing, for which the Standard Model prediction and experiment agreed well until recently. The most recent FLAG average of lattice results for the non-perturbative matrix elements points, however, in the direction of a small discrepancy in this observable. Using up-to-date inputs from standard sources such as PDG, FLAG and one of the two leading CKM fitting groups to determine $\Delta M_s^{\rm SM}$, we find a severe reduction of the allowed parameter space of $Z'$ and leptoquark models explaining the $B$-anomalies. Remarkably, in the former case the upper bound on the $Z'$ mass approaches dangerously close to the energy scales already probed by the LHC. We finally identify some model building directions in order to alleviate the tension with $B_s$-mixing.


I. INTRODUCTION
Direct searches for new physics (NP) effects at the LHC have so far shown no discrepancies from the Standard Model (SM), while we have an intriguing list of deviations between experiment and theory for flavour observables. In particular b → s + − transitions seem to be in tension with the SM expectations: branching ratios of hadronic b → sµ + µ − decays [1][2][3] and the angular distributions for B → K ( * ) µ + µ − decay [2][3][4][5][6][7][8][9][10][11] hint at a negative, beyond the SM (BSM) contribution to C 9 [12][13][14][15][16][17][18][19][20][21][22][23]. The significance of the effect is still under discussion because of the difficulty of determining the exact size of the hadronic contributions (see e.g. [24][25][26][27][28][29][30]). Estimates of the combined significance of all these deviations range between three and almost six standard deviations. A theoretically much cleaner observable is given by the lepton flavour universality (LFU) ratios R K and R K * [31,32], where hadronic uncertainties drop out to a very large extent. Here again a sizeable deviation from the SM expectation is found by LHCb [33,34]. Such an effect might arise for instance from new particles coupling to bs and µ + µ − , while leaving the e + e − -coupling mainly unchanged (see e.g.  for an arbitrary set of papers investigating Z models). Any new bs-coupling immediately leads to tree-level contributions to B s -mixing, which is severely constrained by experiment. For quite some time the SM value for the mass difference ∆M s of neutral B s mesons -triggering the oscillation frequency -was in perfect agreement with experiment, see e.g. [65] or [66]. Taking, however, the most recent lattice inputs, in particular the new average provided by the Flavour Lattice Averaging Group (FLAG) one gets a SM value considerably above the measurement. In this paper we investigate the drastic consequences of this new theory prediction. In Section II we review the SM prediction of B s -mixing, whose consequences for BSM models trying * luca.di-luzio@durham.ac.uk † m.j.kirk@durham.ac.uk ‡ alexander.lenz@durham.ac.uk to explain the B-anomalies are studied in Section III. We conclude in Section IV. In the Appendices we give further details of the SM prediction as well as a more critical discussion of the theoretical uncertainties.

II. Bs-MIXING IN THE SM
The mass difference of the mass eigenstates of the neutral B s mesons is given by The calculation of the box diagrams in Fig. 1  SM value for M s 12 , see e.g. [65] for a brief review, and one gets with the Fermi constant G F , the masses of the W boson, M W , and of the B s meson, M Bs . Using CKM unitarity one finds only one contributing CKM structure λ t = V * ts V tb . The CKM elements are the only place in Eq. (2) where an imaginary part can arise. The result of the 1-loop diagrams given in Fig. 1 is denoted by the Inami-Lim function [67] [68] of the top quark. Perturbative 2-loop QCD corrections are compressed in the factorη B ≈ 0.83798, they have been calculated by [69]. In the SM calculation of M s 12 one four quark ∆B = 2 operator arises 2 The hadronic matrix element of this operator is parametrised in terms of a decay constant f Bs and a bag parameter B: We also indicated the renormalisation scale dependence of the bag parameter; in our analysis we take µ = m b (m b ). Sometimes a different notation for the QCD corrections and the bag parameter is used in the literature (e.g. by FLAG: [70]), (η B ,B) instead of (η B , B) witĥ The parameterB has the advantage of being renormalisation scale and scheme independent. A commonly used SM prediction of ∆M s was given by [65,66] ∆M SM, 2011 ∆M SM, 2015 Both predictions agreed very well with the experimental measurement [71] ∆M Exp s = (17.757 ± 0.021) ps −1 .
In 2016 Fermilab/MILC presented a new calculation [72], which gave considerably larger values for the nonperturbative parameter, resulting in values around 20 ps −1 for the mass difference [72][73][74][75][76] and being thus larger than experiment. An independent confirmation of these large values would of course be desirable; a first step in that direction has been done by the HQET sum rule calculation of [77] which is in agreement with Fermilab/MILC for the bag parameters.
Using the most recent numerical inputs listed in Appendix A we predict the mass difference of the neutral B s mesons to be 1 Here the dominant uncertainty still comes from the lattice predictions for the non-perturbative parameters B and f Bs , giving a relative error of 5.8%. The uncertainty in the CKM elements contributes 2.1% to the error budget. The CKM parameters were determined assuming unitarity of the 3 × 3 CKM matrix. The uncertainties due to m t , m b and α s can be safely neglected at the current stage. A detailed discussion of the input parameters and the error budget is given in Appendix A and Appendix B, respectively. The new central value for the mass difference in Eq. (10) is 1.8 σ above the experimental one given in Eq. (9). This difference has profound implications for NP models that predict sizeable positive contributions to B s -mixing. The new value for the SM prediction depends strongly on the non-perturbative input as well as the values of the CKM elements. We use the averages that are provided by the lattice community (FLAG) and by one of the two leading CKM fitting groups (CKMfitter) -see Appendix C and Appendix D for a further discussion of these inputs.

III. Bs-MIXING BEYOND THE SM
To determine the allowed space for NP effects in B smixing we compare the experimental measurement of the mass difference with the prediction in the SM plus NP: (11) For this equation we will use in the SM part the CKM elements, which have been determined assuming the validity of the SM. In the presence of BSM effects the CKM elements used in the prediction of M SM 12 could in general differ from the ones we use -see e.g. the case of a fourth chiral fermion generation [78]. In the following, we will assume that NP effects do not involve sizeable shifts in the CKM elements. A simple estimate shows that the improvement of the SM prediction from Eqs.
where Λ NP denotes the mass scale of the NP mediator and κ is a dimensionful quantity which encodes NP couplings and the SM contribution. If κ > 0, which is often the case in many BSM scenarios for B-anomalies considered in the literature, and since ∆M SM where δ∆M SM s denotes the 1σ error of the SM prediction. Hence, in models where κ > 0, the limit on the mass of the NP mediators is strengthened by a factor 5. On the other hand, if the tension between the SM prediction and ∆M Exp s increases in the future, a NP contribution with κ < 0 would be required in order to accommodate the discrepancy.
A typical example where κ > 0 is that of a purely LH vector-current operator, which arises from the exchange of a single mediator featuring real couplings, cf. Section III A. 2 In such a case, the short-distance contribution to B s -mixing is described by the effective Lagrangian where In the following, we will show how the updated bound from ∆M s impacts the parameter space of simplified models (with κ > 0) put forth for the explanation of the recent discrepancies in semi-leptonic B-physics data (Section III A) and then discuss some model-building directions in order to achieve κ < 0 (Section III B).

Z'
A paradigmatic NP model for explaining the B-anomalies in neutral currents is that of a Z dominantly coupled via LH currents. Here, we focus only on the part of the Lagrangian relevant for b → sµ + µ − transitions and B smixing, namely where d i and α denote down-quark and charged-lepton mass eigenstates, and λ Q,L are hermitian matrices in flavour space. Of course, any full-fledged (i.e. SU (2) L × U (1) Y gauge invariant and anomaly free) Z model attempting an explanation of R K ( * ) via LH currents can be mapped into Eq. (20). After integrating out the Z at tree level, we obtain the effective Lagrangian Matching with Eq. (17) and (14) we get and where η LL (M Z ) encodes the running down to the bottom mass scale using NLO anomalous dimensions [80,81].
Here we consider the case of a real coupling λ Q 23 , so that C LL bs > 0 and δC µ 9 = −δC µ 10 is also real. This assumption is consistent with the fact that nearly all the groups performing global fits [12][13][14][15][16][17][18][19][20][21][22][23] (see however [82] for an exception) assumed so far real Wilson coefficients in Eq. (17) and also follows the standard approach adopted in the literature for the Z models aiming at an explanation of the b → sµ + µ − anomalies (for an incomplete list, see ). In fact, complex Z couplings can arise via fermion mixing, but are subject to additional constraints from CP-violating observables (cf. Section III B). √ 4π, which saturates the perturbative unitarity bound [85,86], we find that the updated limit from B s -mixing requires M Z 8 TeV for the 1σ explanation of R K ( * ) . Whether a few TeV Z is ruled out or not by direct searches at LHC depends however on the details of the Z model. For instance, the stringent constraints from di-lepton searches [87] are 4 For m Z 1 TeV the coupling λ L 22 is bounded by the Z → 4µ measurement at LHC and by neutrino trident production [83]. See for instance Fig. 1 in [84] for a recent analysis. tamed in models where the Z couples mainly third generation fermions (as e.g. in [63]). This notwithstanding, the updated limit from B s -mixing cuts dramatically into the parameter space of the Z explanation of the b → sµ + µ − anomalies, with important implications for LHC direct searches and future colliders [88].

Leptoquarks
Another popular class of simplified models which has been proposed in order to address the b → sµ + µ − anomalies consists in leptoquark mediators (see e.g. [89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104][105][106]). Although B s -mixing is generated at one loop [107,108], 5 and hence the constraints are expected to be milder compared to the Z case, the connection with the anomalies is more direct due to the structure of the leptoquark couplings. For instance, let us consider the scalar leptoquark S 3 ∼ (3, 3, 1/3), 6 with the Lagrangian where σ a (for a = 1, 2, 3) are Pauli matrices, = iσ 2 , and we employed the quark Q i = (V * ji u j L d i L ) T and lepton L α = (ν α L α L ) T doublet representations (V being the CKM matrix). The contribution to the Wilson coefficients in Eq. (17) arises at tree level and reads while that to B s -mixing in Eq. (14) is induced at one loop [110] C LL bs = where the sum over the leptonic index α = 1, 2, 3 is understood. In order to compare the two observables we consider in Fig. 3 the case in which only the couplings y QL 32 y QL *

22
(namely those directly connected to R K ( * ) ) contribute to B s -mixing and further assume real couplings, so that we can use the results of global fits which apply to real δC µ 9 = −δC µ 10 . The bound on M S3 from B s -mixing is strengthened by a factor 5 thanks to the new determination of ∆M s , which yields M S3 22 TeV, in order to explain R K ( * ) at 1σ (cf. Fig. 3). On the other hand, in flavour models predicting a hierarchical structure for the leptoquark couplings one rather expects y QL i3 y QL i2 , so that the dominant contribution to ∆M s is given by y QL 33 y QL * 23 . For example, y QL i3 /y QL i2 ∼ m τ /m µ ≈ 4 in the partial compositeness framework of Ref. [90], so that the upper bound on M S3 is strengthened by a factor y QL 33 y QL * 23 /y QL 32 y QL *

22
∼ 16. The latter can then easily approach the limits from LHC direct searches which imply M S3 900 GeV, e.g. for a QCD pair-produced S 3 dominantly coupled to third generation fermions [111].

Combined R K ( * ) and R D ( * ) explanations
Another set of intriguing anomalies in B-physics data is that related to the LFU violating ratios R D ( * ) ≡ B(B → D ( * ) τν)/B(B → D ( * ) ν) (here, = e, µ), which turn out to be larger than the SM [112][113][114]. Notably, in this case NP must compete with a tree-level SM charged current, thus requiring a sizeably larger effect compared to neutral current anomalies. The conditions under which a combined explanation of R K ( * ) and R D ( * ) can be obtained, compatibly with a plethora of other indirect constraints (as e.g. those pointed out in [115,116]), have been recently reassessed at the EFT level in Ref. [117]. Regarding B s -mixing, dimensional analysis (see e.g. Eq. (6) in [117]) shows that models without some additional dynamical suppression (compared to semi-leptonic operators) are severely constrained already with the old ∆M s value. For instance, solutions based on a vector triplet V ∼ (1, 3, 0) [118], where B s -mixing arises at tree level, are in serious tension with data unless one invokes e.g. a percent level cancellation from extra contributions [117].
The updated value of ∆M s in Eq. (10) makes the tuning required to achieve that even worse. On the other hand, leptoquark solutions (e.g. the vector U µ 1 ∼ (3, 1, 2/3)) comply better with the bound due to the fact that B smixing arises at one loop, but the contribution to ∆M s should be actually addressed in specific UV models whenever calculable [104]. at about 2σ, it is interesting to speculate about possible ways to obtain a negative NP contribution to ∆M s , thus relaxing the tension between the SM and the experimental measurement. Sticking to the simplified models of Section III A (Z and leptoquarks coupled only to LH currents), an obvious solution in order to achieve C LL bs < 0 is to allow for complex couplings (cf. Eq. (23) and Eq. (26)). For instance, in Z models this could happen as a consequence of fermion mixing if the Z does not couple universally in the gaugecurrent basis. A similar mechanism could be at play for vector leptoquarks arising from a spontaneously broken gauge theory, while scalar-leptoquark couplings to SM fermions are in general complex even before going in the mass basis. Extra phases in the couplings are constrained by CPviolating observables, that we discuss in turn. In order to quantify the allowed parameter space for a generic, complex coefficient C LL bs in Eq. (14), we parametrise NP effects in B s -mixing via where The former is constrained by ∆M Exp s /∆M SM s = |∆|, while the latter by the mixing-induced CP asymmetry [65,119] where A mix CP = −0.021±0.031 [71], β s = 0.01852±0.00032 [120], and we neglected penguin contributions [65]. The combined 2σ constraints on the Wilson coefficient C LL bs are displayed in Fig. (4). For Arg(C LL bs ) = 0 we recover the 2σ bound C LL bs /R loop SM 0.014, which basically corresponds to the case discussed in Section III A where we assumed a nearly real C LL bs (up to a small imaginary part due to V ts ). On the other hand, a non-zero phase of C LL bs allows to relax the bound from ∆M s , or even accommodate ∆M s at 1σ (region between the two solid red curves in Fig. 4), compatibly with the 2σ allowed region from A mix CP (blue shaded area in Fig. 4). For Arg(C LL bs ) ≈ π values of C LL bs /R loop SM as high as 0.21 are allowed at 2σ, relaxing the bound on the modulus of the Wilson coefficient by a factor 15 with respect to the Arg(C LL bs ) = 0 case. Note, however, that the limit Arg(C LL bs ) = π corresponds to a nearly imaginary δC µ 9 = −δC µ 10 which would presumably spoil the fit of R K ( * ) , since the interference with the SM contribution would be strongly suppressed. Nevertheless, it would be interesting to perform a global fit of R K ( * ) , together with ∆M s and A mix CP while allowing for non-zero values of the phase, in order to see whether a better agreement with the data can be obtained. Nonzero weak phases can potentially reveal themselves also via their contribution to triple product CP asymmetries in B → K ( * ) µ + µ − angular distributions [82]. This is however beyond the scope of the present paper and we leave it for a future work.
An alternative way to achieve a negative contribution for ∆M NP s is to go beyond the simplified models of Section III A and contemplate generalised chirality structures. Let us consider for definiteness the case of a Z coupled both to LH and RH down-quark currents Upon integrating out the Z one obtains The LR vector operator can clearly have any sign, even for real couplings. Moreover, since it gets strongly enhanced by renormalisation-group effects compared to LL and RR vector operators [121], it can easily dominate the contribution to ∆M NP s . Note, however, that λ d 23 contributes to R K ( * ) via RH quark currents whose presence is disfavoured by global fits, since they break the approximate relation R K ≈ R K * that is observed experimentally (see e.g. [22]). Hence, also in this case, a careful study would be required in order to assess the simultaneous explanation of R K ( * ) and ∆M s .

IV. CONCLUSIONS
In this paper, we have updated the SM prediction for the B s -mixing observable ∆M s (Eq. (10)) using the most recent values for the input parameters, in particular new results from the lattice averaging group FLAG. Our update shifts the central value of the SM theory prediction upwards and away from experiment by 13%, while reducing the theory uncertainty compared to the previous SM determination by a factor of two. This implies a 1.8 σ discrepancy from the SM. We further discussed an important application of the ∆M s update for NP models aimed at explaining the recent anomalies in semi-leptonic B s decays. The latter typically predict a positive shift in the NP contribution to ∆M s , thus making the discrepancy with respect to the experimental value even worse. As a generic result we have shown that, whenever the NP contribution to ∆M s is positive, the limit on the mass of the NP mediators that must be invoked to explain any of the anomalies is strengthened by a factor of five (for a given size of couplings) compared to using the 2015 SM calculation for ∆M s . In particular, we considered two representative examples of NP models featuring purely LH current and real couplings -that of a Z with the minimal couplings needed to explain R K ( * ) anomalies, and a scalar (SU (2) L triplet) leptoquark model. For the Z case we get an upper bound on the Z mass of 2 TeV (for unit Z coupling to muons, cf. Fig. 2), an energy scale that is already probed by direct searches at LHC. On the other hand, the bounds on leptoquark models from B s -mixing are generically milder, being the latter loop suppressed. For instance, taking only the contribution of the couplings needed to fit R K ( * ) for the evaluation of ∆M s we find that the upper bound on the scalar leptoquark mass is brought down to about 20 TeV (cf. Fig. 3). This limit gets however strengthened in flavour models predicting a hierarchical structure of the leptoquark couplings to SM fermions and can easily approach the region probed by the LHC. Trying in addition to solve the deviations in R D ( * ) implies very severe bounds from B s -mixing as well, since the overall scale of NP must be lowered compared to the case of only neutral current anomalies. Given the current status of a higher theory value for ∆M s compared to experiment, we also have looked at possible ways in which NP can provide a negative contribution that lessens the tension. A non-zero phase in the NP couplings is one such way, and we have shown how extra constraints from the CP violating observable A mix CP in B s → J/ψφ decays cuts out parameter space where otherwise a significant NP contribution could be present. However, a large phase can potentially worsen the fit for R K ( * ) -here a global combined fit of ∆M s , A mix CP and R K ( * ) seems to be an important next step. Another possibility is to consider NP models with a generalised chirality structure. In particular, ∆B = 2 LR vector operators, which are renormalisation-group enhanced, can accommodate any sign for ∆M NP s , even for real couplings. Large contributions from RH currents are however disfavoured by the R K ( * ) fit, hence also here a more careful analysis is needed. Finally, a confirmation of our results, by further lattice groups confirming the large FNAL/MILC results for the four quark matrix elements, as well as a definite solution of the V cb puzzle, would give further confidence in the extraordinary strength of the bounds presented in this paper.

ACKNOWLEDGMENTS
We thank Sébastien Descotes-Genon for providing the unpublished tree-level only CKM values obtained by CKMfitter, Tomomi Ishikawa and Andreas Jüttner for advise on lattice inputs and Marco Nardecchia for helpful feedback on the BSM section. This work was supported by the STFC through the IPPP grant.

Appendix A: Numerical input for theory predictions
We use the following input for our numerical evaluations. The values in Table I are taken from the PDG [122], from non-relativistic sum rules (NRSR) [123,124], from the CKMfitter group (web-update of [120] -similar values can be taken from the UTfit group [125]) and the non-perturbative parameters from FLAG (web-update of [70]). For α s we use RunDec [126] with 5-loop accuracy [127][128][129][130][131], running from M Z down to the bottom mass scale. At the low scale we use 2-loop accuracy to determine Λ (5) . with the ones given in 2015 by [65], in 2011 by [66] and 2006 by [119]. The numbers are given in Table II. We observe a considerable improvement in accuracy and a sizeable shift compared to the 2015 prediction, mostly stemming from the new lattice results for f Bs √ B, which still is responsible for the largest error contribution of about 6%. The next important uncertainty is the accuracy of the CKM element V cb , which contributes about 2% to the error budget. If one gives up the assumption of the unitarity of the 3×3 CKM matrix, the uncertainty can go up. The uncertainties due to the remaining parameters play a less important role. All in all we are left with an overall uncertainty of about 6%, which has to be compared to the experimental uncertainty of about 1 per mille.
Appendix C: Non-perturbative inputs As a word of caution we present here a wider range of non-perturbative determinations of the matrix elements of the four-quark operators including also the corresponding predictions for the mass differences, see Table III: HPQCD presented in 2014 preliminary results for N f = 2 + 1 in [132] and for our numerical estimate in Table (III) we had to read off the numbers from Fig. 3   their proceedings [132]. When finalised, this new calculation will supersede the 2006 [137] and 2009 [134] values.
The ETMC N f = 2 number stems from 2013 [133], it is obtained with only two active flavours in the lattice simulation. The Fermilab/MILC N f = 2 + 1 number stems from 2016 [72] and it supersedes the 2011 value [139]. This precise value is currently dominating the FLAG average. The numerical effect of these new inputs on mixing observables was e.g. studied in [74]. The previous FLAG average from 2013 [135] was considerably lower. There is also a large N f = 2 + 1 value from RBC-UKQCD presented at LATTICE 2015 (update of [138]). However, this number is obtained in the static limit and currently missing 1/m b corrections are expected to be very sizeable. 8 The HQET sum rules estimate for the Bag parameter [77] can also be combined with the decay constant from lattice.
Here clearly a convergence of these determinations, in particular an independent confirmation of the Fer- The second most important input parameter for the prediction of ∆M s is the CKM parameter V cb . There is a longstanding discrepancy between the inclusive determination and values obtained from studying exclusive B decays, see [122]. Recent studies found, however, that the low exclusive value might actually be a problem originating in the use of a certain form factor parametrisation in the experimental analysis. 9 Using the BGL parametrisation one finds values that lie considerably closer to the inclusive one, see [142][143][144][145]. Currently, there are various V B→D * , BGL cb = 0.0419 +0.0020 −0.0019 [142] .
In Fig. 5 we plot the dependence of the SM prediction of ∆M s on V cb , and show the regions predicted by the above inclusive and exclusive determinations. We use the CKMfitter result for V cb (see Table I) for our new SM prediction of ∆M s (see Eq. (10) and the (upper) horizontal dashed line denoted with "SM"), the corresponding er-ror band is shown in orange. The predictions obtained by using the inclusive value of V cb only is given by the blue region. For completeness we show also the regions obtained by using the exclusive extractions of V cb . The disfavoured CLN values result in much lower values for the mass difference (hatched areas), while the BGL value agrees well with the inclusive region, albeit with a higher uncertainty. The experimental value of ∆M s is shown by the (lower) horizontal dashed line denoted with "Exp". The preference for the inclusive determination agrees also with the value obtained from the CKM fit (which we use in our SM estimate), as well as with the fit value that is found if the direct measurements of V cb are not included in the fit V CKM-fitter (no direct) cb = 0.04235 +0.00074 −0.00069 [120] . (D5) We also note that the CKMfitter determinations take into account loop-mediated processes, where potentially NP can arise. Taking only tree-level inputs, they find: 10 V us = 0.22520 +0.00012 −0.00038 , which shows an overall consistency with the prediction in Eq. (10).