Linear flavour violation and anomalies in B physics

Journal of High Energy Physics, Jun 2016

We propose renormalizable models of new physics that can explain various anomalies observed in decays of B-mesons to electron and muon pairs. The new physics states couple to linear combinations of Standard Model fermions, yielding a pattern of flavour violation that gives a consistent fit to the gamut of flavour data. Accidental symmetries prevent contributions to baryon- and lepton-number-violating processes, as well as enforcing a loop suppression of new physics contributions to flavour violating processes. Data require that the new flavour-breaking couplings are largely aligned with the Yukawa couplings of the SM and so we also explore patterns of flavour symmetry breaking giving rise to this structure.

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Linear flavour violation and anomalies in B physics

JHE avour violation and anomalies in 0 Wilberforce Road , Cambridge, CB3 0WA, U.K 1 J.J. Thomson Avenue , Cambridge, CB3 0HE, U.K 2 DAMTP, University of Cambridge 3 Cavendish Laboratory, University of Cambridge We propose renormalizable models of new physics that can explain various anomalies observed in decays of B-mesons to electron and muon pairs. The new physics states couple to linear combinations of Standard Model fermions, yielding a pattern of avour violation that gives a consistent t to the gamut of avour data. Accidental symmetries prevent contributions to baryon- and lepton-number-violating processes, as well as enforcing a loop suppression of new physics contributions to Data require that the new avour-breaking couplings are largely aligned with the Yukawa couplings of the SM and so we also explore patterns of avour symmetry breaking giving rise to this structure. B-Physics; Beyond Standard Model - Linear physics HJEP06(21)83 2 3 4 5 1 Introduction The models Phenomenological analysis 3.1 Indirect searches processes 3.2 3.3 Direct searches Parameter space plots Flavour symmetries Conclusions 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 b ! s b ! s Semileptonic four-fermion operators Four-quark operators Anomalous magnetic moment of the muon A SU(2)L decompositions B Photon and Z-boson mediated contributions consistency is the particular (even peculiar) structure of the SM, which leads to many subtle and delicate cancellations in avour-changing and CP-violating processes. As examples at tree level (see, e.g. [1] for more details), the representation content of the SM fermions (which is such that any two equivalent irreducible representations under the unbroken SU(3)c U(1)em are also equivalent under the full SU(3)c izable; indeed, if we relax this requirement, we quickly run into con ict with data, unless the UV cut-o of the theory is rather high. In the last year or so, cracks have begun to appear in the SM edi ce, in the form of a variety of anomalies associated with B-meson decays. These include e ects seen in angular observables [2{10] of the decay B ! K [11], the observable RK [12], and a series of branching ratios with muons in the nal state (e.g. [13, 14]). Whilst it is surely far too soon to claim that the end of the SM is nigh, it is perhaps worthwhile to explore how the SM might be modi ed in such a way that these anomalies in the data can be accommodated. Of course, this must be done in such a way as not to spoil the various delicate cancellations that occur in the SM, since this would lead to gross contradictions with data elsewhere. As such, it makes sense to look for theory which is renormalizable (or has a cut-o which is much larger than the scale of new physics, which amounts to the same thing). Then we can at least hope that, just like in the SM, dangerous avour-changing processes can be kept under control. A number of models explaining the anomalies have already been proposed. These can be sub-divided into models which generate the anomalies via tree- vs. loop-level corrections. Models in the rst category include leptoquarks [15{23] and Z0s [24{39]. Not all of these models are renormalizable. Ref. [17], for example, describes a well-motivated model in which the electroweak hierarchy problem is solved by Higgs compositeness and SM fermions are partially composite. A light leptoquark can arise naturally as a Goldstone boson [40]. But if the compositeness scale is su ciently high, the renormalizable limit is approximately recovered. There is one recently-proposed model in the second category, which generates contributions that explain the B anomalies and the anomalous magnetic moment of the muon, via a Z0 with loop-induced couplings to muons [41]. In this work, we wish to propose a second type of renormalizable model in the category of those generating the anomalies at loop level. In order to obtain the avour structure that seems to be required (in the quark sector at least) in an automatic way, we insist that the new avour-violating couplings be linear in the SM fermions. We then survey the various possibilities for the BSM elds and nd that basic phenomenological considerations (such as the insistence on an accidental symmetry stabilising the proton), together with a minimality criterion, lead to just two possible models. The models both feature two new scalar elds and a single fermion eld, which couple to linear combinations of the SM fermions via Yukawa interactions. The two models are very similar in their phenomenology and so we discuss only one in detail. We nd that, for suitable values of the parameters, a satisfactory t to the anomalies can be obtained. The t to the data requires that the new avour-violating couplings be strongly hierarchical (at least in the quark sector; in the lepton sector there is more room to manoeuvre) and moreover largely aligned with the avour breaking already present in the Yukawa couplings of the SM. Thus, even more than in the SM, the low-energy theory seems to be crying out for a fundamental theory of avour. Rather than attempt to nd an explicit theory of avour that does the job, we content ourselves with showing the plausibility of the existence of such a theory, by exhibiting patterns of avour symmetry breaking that { 2 { can give rise to the required structure. We nd that a variety of symmetries are possible. Interestingly, they give rise to a pattern of couplings similar to that obtained in theories featuring partial compositeness. The models have another feature of interest. By construction, they yield explicit UV completions that generate the non-renormalizable avour structure that was identi ed in [ 42 ] as a viable one for explaining the anomalies. The fact that such UV completions exist is a desirable thing, because the alignment of the avour-violating new physics couplings in [ 42 ] is not preserved by the SM RG ow. Thus, in the absence of an unexplained ne tuning in the couplings, the scale of new physics completing the e ective lagrangian in [ 42 ] should be light, in order that the picture makes sense. Our models provide exan analysis of possible avour breaking patterns, and our conclusions are given in section 5. 2 The models Recent experimental data in avour physics, in particular measurements by the LHCb collaboration [11, 12, 14], suggest possible e ects of New Physics (NP) in semileptonic decays of B-mesons. In particular, considering the b ! s quark avour transitions, the most signi cant departures from the Standard Model predictions are observed in: (i) the so-called decays [11] (ii) a series of branching ratios P50 angular observable of the B ! K + B ! Ke+e of B-decays with muons in the nal states (like the recently measured Bs0 ! (iii) the observable RK [12] de ned as the ratio of branching ratios of B ! K + + [14]) and in the low q2 (lepton pair invariant mass squared) region. It is perhaps premature to claim evidence for NP. Indeed the discrepancies in (i) and (ii) could be due to underestimates of hadronic uncertainties,1 while the discrepancy in (iii), despite the observable being theoretically very clean in the SM, could simply be a statistical uctuation. However, if we allow for an interpretation of these experimental results in terms of NP, it is quite remarkable that model-independent approaches based on higher-dimension operators [5, 7, 8, 15, 24, 45, 49{54] give a simple and consistent t to the data. In general, we expect that NP will couple chirally to the matter elds; assuming a coupling purely to either right- or left-handed currents, the ts nd that the operator bL sL L L is preferred. Motivated by this, we seek renormalizable models that couple the NP to quark and lepton doublets. Moreover, for reasons that will become clear, we demand that the SM fermions appear linearly in the NP couplings. In what follows, the SM fermions are written as QL; UR; DR; LL; ER while the beyond SM (BSM) fermions and scalars are denoted by i and i. The Higgs doublet is denoted by H and transforms as (1; 2; 12 ). The possible linear interactions of the new BSM elds with the SM doublets (QiL 1For discussions on the size of the hadronic uncertainties on these observables we refer the reader to e.g. [6, 8, 10, 43{49]. { 3 { and LiL) can be classi ed in the following way: or or or q i q i q i qQiL q i qQiL i ` QL q + i i QL q + i ` c i i LL ` + h:c:; LL ` + h:c:; + i `LiL ` + i `LL ` c i + h:c:; + h:c: (2.1) (2.2) (2.3) (2.4) Notice that one new state appears simultaneously in both the quark and lepton interactions (this is needed in order to draw a diagram contributing to b ! s ). In the rst two cases the common mediator is while in the last two the common mediator is . There are in nitely many combinations with suitable gauge quantum numbers to yield these interactions. In order to reduce the possibilities for the SM quantum numbers of the new states, we will impose conditions on them based on the following considerations: terms of the form ( y )(HyH) and ( yT a )(HyTHa H) are never forbidden by gauge symmetry (though the second one is absent if is a SU(2)L singlet). These terms are phenomenologically viable, but other quartic and trilinear interactions of a scalar with the Higgs, like HH, or HHH, could give rise to a violation of the custodial symmetry at the tree level and/or could modify the observed Higgs phenomenology. To be safe from these unwanted e ects, we choose the quantum numbers of the new scalars such that these dangerous interactions are prohibited. (c) Direct searches, coloured particles New particles in the loops will need to be rather light to create a measurable e ect in B decays, so it is convenient to choose quantum numbers such that their masses are less constrained by direct searches. The quark interaction requires at least one state that transforms non-trivially under the colour group. A coloured scalar will have weaker bounds on its mass than a coloured fermion, since in the latter case the production cross section is higher for a xed gauge quantum number. Selecting a scalar to be the only new coloured particle leads us to consider only the rst two cases in the above list of new interactions, namely the ones with a single fermion mediator and two di erent scalars q and ` . { 4 { (d) Direct searches, BSM Lightest Particle (LP) The Yukawa interactions above are manifestly invariant under a U(1) transformation that acts non trivially only on the BSM states. This symmetry is respected by the gauge-kinetic terms of the new states too. We look for irreps such that this transformation is an accidental symmetry of the whole renormalizable model. This has the advantage that all NP avour-violating processes are loop suppressed. But it also implies that the lightest NP state is stable; in order to evade strong constraints on coloured and/or electrically charged stable states coming from colliders and cosmology we look for a colourless irrep containing a neutral particle. The gauge quantum numwe require that all new irreps have dimensionality fewer than 5. Let us now identify the irreps that could contain the LP. From (d) and (e) we obtain a nite list of candidates. Most of them are excluded because of the presence of renormalizable interactions that violate conditions (a) and/or (b). The excluded cases are listed in table 1. We are left with four cases, each of which has a single fermion, , with SM quantum numbers (1; 4; 1=2) or (1; 4; 3=2). However, radiative corrections split the values of the particle masses in the multiplet and it turns out that, for the quantum number (1; 4; 1=2), the LP is not the neutral one. We conclude that, since we are demanding a neutral LP, the LP can only be contained in the fermion eld with quantum numbers (1; 4; 32 ). Imposing condition (e) on the eld q we are left with just two models: (1; 4; + 32 ); q (3; 3; 43 ), ` (1; 3; 2) with Yukawa interactions as (1; 4; 32 ); q (3; 3; 53 ), ` (1; 3; 2) with Yukawa interactions as Model A. in (2.1): Model B. in (2.2): The two models have very similar implications for the phenomenology that we are interested in here. Henceforth, we discuss only Model A. The quantum numbers of the SM and NP elds under the gauge and global symmetries (to be discussed below) are summarised in table 2 and the most general renormalizable { 5 { q i q i QL q + i` i i LL ` + h.c. QL q + i` c i i LL ` + h.c. (2.5) (2.6) (1; 1; 0) U(1)Y Interactions ` neutral particle but which are rejected because they could give rise to unwanted renormalizable interactions, as listed in the last column. Irreps with negative hypercharges are related to these ones by charge conjugation. lagrangian is given by L = LSM + L + L + Lyuk; L L lagrangian. Before considering the breaking coming from Llin it is easy to show that the Lagrangian is invariant under a global U(1)7. Indeed, the SM alone has accidental global symmetry U(1)B U(1)e U(1) elds have global symmetry U(1) U(1) , while the gauge kinetic terms of the new BSM U(1) q U(1) ` . Moreover, it is easy to prove that the most general renormalizable scalar potential V ( H ; q; `) is invariant under U(1)7. Now consider the e ect of Llin. For a generic choice of the couplings ` and q there is always an unbroken U(1)3 U(1)B0 U(1) , de ned as follows. Under the U(1)B0 the SM elds have their usual baryon number while q has charge -1/3. Similarly, under the U(1)L0 the SM elds have their usual lepton number while ` has charge -1. Finally the SM elds are uncharged under U(1) , while the BSM elds have charge unity.2 2This symmetry makes our neutral and colourless LP stable. Hence, the LP is a potential dark matter (DM) candidate. However, even if its mass and couplings could be xed in order to reproduce the right ( 13 ; 0; 0) ( 13 ; 0; 0) ( 13 ; 0; 0) (3; 2; 16 ) (3; 1; 23 ) metry (second column), and under the accidental global symmetries of the theory (third column). Thus, the model retains analogues of the accidental baryon and lepton number symmetries of the SM, which su ce to stabilize the proton and to prevent contributions to numerous unobserved lepton- and baryon-number violating processes. Moreover, the model features an additional accidental U(1) symmetry, under which SM elds are uncharged. An immediate consequence of this is that all NP-generated processes involving only SM particles in the initial and nal states are loop-suppressed. This is certainly an advantage from the point of view of the vast majority of avour-violating observables, where no deviation from the SM is observed. It might be regarded as a disadvantage from the point of the view of the B-physics anomalies, where a sizable NP e ect is needed. But this is o set somewhat by the desirable structure of linear NP avour violation that results. We shall see in the sequel that the anomalies can be reproduced even for values of the NP couplings that are of order unity or smaller. 3 Phenomenological analysis In this section we discuss the phenomenology of Model A. In an obvious notation, we denote the masses of the new states as M , Mq, and M`. In a basis where the left-handed quark doublet is de ned as QiL = (VCKM uiL; diL)T , the minimal set of couplings that are needed to t the b ! s`` anomalies are q 3 2` (i.e. couplings involving b, s and ). To begin with, we will assume that only these couplings are non-zero and investigate the processes induced. In this section we collect relevant formula on indirect searches, and investigate direct production bounds. In subsection 3.3 we use this information to nd allowed parameter space regions. In section 4 we will discuss relaxing the assumptions on the couplings and propose more motivated avour structures. 3.1 Indirect searches As described above, the accidental global U(1) symmetry under which the new particles are charged implies that contributions to processes containing only SM particles in the { 7 { initial and nal states are only induced at loop level. Here we investigate the size of these contributions. s The relevant processes, given the assumption on couplings described above, are b ! processes, Bs mixing, b ! s , and the anomalous magnetic moment of the muon. The process b ! s``, important for the LHCb B meson anomalies, is induced at loop level by the diagram in gure 1.3 The SU(2)L structure of the NP-induced semileptonic fourfermion interaction can be derived from the discussion in appendix A, using the lagrangian (eq. (A.6)) written explicitly in terms of SU(2)L components. The resulting e ective NP HJEP06(21)83 2 em em 4 4 5 9 ; K(x) K(x; y) 1 x + x2 log x K(x) (x x 1)2 K(y) y : He = 4GF (VtsVtb) X C`( ) Oi`( ) ; p i i lagrangian is Le K(xq; x`) i j m n q q ` ` MM`22 . The loop function K(xq; x`) can be obtained by the following The e ective hamiltonian relevant to b ! s`` transitions is where Oi` are a basis of SU(3)C U(1)Q-invariant dimension-six operators giving rise to the avour-changing transition. The superscript ` denotes the lepton avour in the nal state (` 2 fe; ; g), and the important operators for our process, Oi`, are given in a standard basis by O9 `(0) = O10 `(0) = s s PL(R)b (` PL(R)b (` `) ; 5`): Comparing equations (3.1) and (3.2) we nd the NP contribution to the Wilson coe cients relevant to b ! s is 9 2 3 2` 2 : 3There are also Z and photon penguin diagrams which contribute, with a NP loop connecting the quarks and joining to the leptons via a Z= propagator. These penguin diagrams are discussed in appendix B and are found to be very suppressed relative to both the SM contribution and the diagram in gure 1, and hence are neglected here. { 8 { The most recent best t ranges on this combination of Wilson coe cients are taken from [49] and are given by 9 9 Interactions between four quarks are induced at loop level by diagrams like those in gure 2. These interactions can lead to meson mixing; in particular, if the process b ! s is present, then inevitably Bs mixing must also be induced. This process can therefore introduce important constraints on the masses and couplings of the new particles. The four quark e ective operator induced by the NP is Le K0(xq) i j m n q q q q ~QjL m QL ~QnL where K0(x) is the rst derivative of K(x). The SU(2)L structure of the e ective operator is similar to that of eq. (3.1) and can again be derived from the discussion in appendix A. Projecting the quark doublet along the down components we nd that for Bs mixing the relevant operator is Le 7 bL)(sL bL) + h.c.: The Wilson coe cient is easily extracted at high energy = where the BSM particles are dynamical elds. We x = 1 TeV in what follows. At this energy we have C1bs( ) = 576 2 K0(xq) 2 3 { 9 { (3.5) (3.6) ; (3.7) (3.8) (3.9) In order to place bounds on the parameters of our model, we take into account QCD e ects using the results and procedure of [55]. Using the anomalous dimension of this work we found that the running of Wilson coe cient from the scale of the New Physics ( ) to the scale of the process (mb) is given by C1bs(mb) = V LLC1bs( ) with V LL = 0:78. For the evaluation of the relevant matrix element we used the lattice result of [56]. These lead to a constraint (at 95% con dence level) on the coe cient HJEP06(21)83 which translates into C1bs( ) . 1:8 2 3 q q 2 < 1:8 Thus the measurement of Bs mixing produces a bound on the hadronic couplings involved in the b ! s this bound and the b ! s process, viz. respect the model is similar to Z0 models | the couplings involved factorize into leptonic couplings and hadronic couplings which can be set independently. This factorization does not occur in leptoquark models. 3.1.3 b ! s The radiative process b ! s will also be induced by the diagram in gure 3. The couplings involved are the same as those for Bs mixing. However, the amplitudes will scale di erently with the parameters q and Mq between the two processes. Constraints from b ! s could therefore provide complementary information. At the mass of the b quark, the process b ! s is described by the following e ective hamiltonian: He = 4GF (VtsVtb) C7(mb)O7(mb) + C70(mb)O70(mb) ; p where O7 (0) = 16e 2 mb s cient of the dipole operator; 2 PR(L)b F . At the matching scale M , we get an additional contribution from the NP to the coefC7NP = GpF2 VtsVtb 1 q q The 2 allowed range for this parameter has been tted recently in [49], giving F1(x) = 12(x 1 1)4 x 3 6x2 + 3x + 2 + 6x log x : C7NP(mb) 2 [ 0:10; 0:02] (at 2 ): 3.1.4 Anomalous magnetic moment of the muon Although it is somewhat peripheral to our discussion, let us remark that loops of and `, as shown in gure 4, generate a 1-loop contribution to the magnetic moment of the muon, which may be able to resolve the long-standing experimental discrepancy therein [57]. The NP contribution is given by which should be compared to the observed discrepancy [58] aNP = ` As we will show in section (3.3), it is possible to t the anomalous magnetic moment in this model. However, it requires a large value of 2`, which is problematic, since it can lead to large corrections to electroweak precision observables at the Z-pole. 3.1.5 b ! s processes Contributions to B ! K gure 1 with the muons replaced with muon neutrinos (as well as Z penguin diagrams | see the comment in section 3.1.1). A detailed analysis of NP contributions to this process is given in [59], and we use their results here. Current measurements give bounds on the ratio of total (NP+SM) to SM branching ratios to be are expected in the model, due to a diagram RK RK BR(B ! K BR(B ! K BR(B ! K BR(B ! K ) )SM ) )SM < 4:3 < 4:4 (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) at 90% con dence level. An expression for RK( ) in the presence of NP with couplings to left-handed SM fermions is RK = RK = 1 X jCL`j2 3 ` jCLSMj2 where CL` is the coe cient of the operator OL = e2 4GF VtbVts 16 2 (s p2 in the e ective Hamiltonian. The SM Wilson coe cient CSM is known to be quite accurately CSM = L 7:65. In the case of our model, HJEP06(21)83 CNP; L = p 2GF VtbVts PLb) ( ` 5) `) (1 L 9 M 2 4 K(xq; x`) 2q 3q j 2j where the de nitions of K(x; y), xq and x` are as in section 3.1.1. Thus bounds exist on the parameters of the model due to b ! s processes: 24:9 < CNP; L < 30:0; 55:4 < K(xq; x`) 2q 3q j 2j leptonic coupling in the model, then we nd the relation CLNP = 2 CNP; Therefore, for values of the Wilson coe cients required to t the b ! s (eq. (3.5)), the NP contributions to the branching ratios for the b ! s below the bounds, adding approximately 5% to the SM values. 3.2 Direct searches The particles of the three new multiplets, q, `, and , will be directly produced at the LHC if their masses are within kinematic reach. In this subsection, we outline current limits on their masses from direct searches, and identify promising channels to search for them. It will be convenient to label the SU(2)L components of the multiplets by a superscript denoting their respective electric charges; the full list of NP particles is, therefore, q=( q+7=3; q The collider phenomenology of the new particles will depend on the mass spectrum. As before, we assume that M < M`; Mq, since we require the LP to be neutral. We further assume that M` < Mq, since this minimises contributions to Bs mixing (it also maximises contributions to the muonic g 2). The three multiplets can, generically, be well-separated in mass, but within each SU(2)L multiplet there may also be signi cant mass splittings. For the scalar multiplets, there are tree level mass splittings due to the (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) anomalies processes are well presence of direct couplings with the Higgs; for the fermion multiplet, there are only small radiative mass splittings between the components. In the limit that the common mass is much larger than the electroweak scale v, the radiative mass splitting between the di erent charge eigenstates is [60, 61] mrad = mQ+1 mQ 166MeV 1 + 2Q + (3.25) 2Y cos W ; a formula which holds for both scalars and fermions. According to eq. (3.25) the lightest particle within the fermion multiplet will be uncharged, as desired. As the fermion multiplet has the lowest common mass of all the new states, and due to the U(1) symmetry within the new sector, the lightest state within the multiplet will be stable. The small radiative mass splitting means that heavier fermion components will decay to the lightest (neutral) component by emission of one or more soft charged pions or leptons, which will not be energetic enough to be reconstructed in the detector. Thus if any particle is produced at the LHC, it will appear as missing transverse momentum, similarly to the Wino-like dark matter described in [62].4 We thus neglect henceforth the soft undetectable pions or leptons emitted in the decays of heavier components of . Therefore, q and ` particles (being heavier than the states) will e ectively decay to a SM particle plus missing transverse energy. Furthermore, due to the U(1) symmetry, NP particles will always be pair-produced at the LHC. This means that searches for (R-parity conserving) supersymmetry should be sensitive to q and `. We will now discuss each of the elds q, ` and in turn. The fermions X can be pair produced via a photon or a W=Z, or through the decay of q;` . By the arguments above, they will always behave as uncharged weakly interacting particles. Limits can be set on these from mono-x searches, from constraints on the invisible width of the Z boson, and from LEP searches for charginos that are almost degenerate in mass with the neutralinos. A detailed analysis of all of these has been performed in [63]; the last-mentioned has been found to be the most constraining, implying a bound of m > 90 GeV. Each component ( q+7=3; q +4=3; q +1=3) of the coloured scalar multiplet q will be strongly pair produced and will decay with a similar signature to that of a squark; i.e. to a quark and (either directly or via emission of soft pions or leptons) the stable neutral component of the fermion . Note that there is the possibility that two q particles produced each decay to a di erent avour of quark. An example decay chain displaying this property is illustrated in gure 5. This complicates the re-interpretation of SUSY search limits. The avour considerations discussed earlier only constrain the product of couplings q q 3 2, without constraining their quotient. The strongest constraints from existing LHC searches will hold for situations where one of the couplings 2q or 3q is much larger than 4Note that our setup is subtly di erent from that described in [62]. There, strong constraints can be put on the mass of the new fermion multiplet from disappearing tracks searches, since the lifetime of a charged fermion decaying to the neutral fermion can be long enough to create a disappearing track in the detector. Here, these searches are not constraining because the lifetime is too short for a track to be visible. di erent quarks. Since the pion produced when the 1=3 appears similar to that of a (anti-)sbottom, whereas the decay of the q q 1=3 appears similar to that of a stop. the other, so that the branching ratio to a particular generation dominates. If the q particles will decay like sbottoms and stops, whereas if q 3 2q they will decay q 3 q 2 like second-generation squarks. The branching ratio to an up-type quark as opposed to a down-type within a particular generation is determined by the SU(2)L structure. We focus on the case q 3 2q, since this is motivated by avour considerations, as explained in the next section. In this limit, one can show that the total branching ratio times cross-section for a pair of q particles to be produced and to decay to a pair of tops is 7=8 of that for direct stop pair production. The most recent limits on direct stop pair production are given in [64, 65]. As a conservative estimate, given that we have the limit m > 90 GeV, we can take the limits on direcly pair-produced stops decaying to tops and 90 GeV neutralinos to apply to our q. This gives a limit of Mq & 750 GeV. Likewise, the total branching ratio times cross-section for a pair of q particles to be produced and to decay to a pair of b quarks is 7=8 of that for direct sbottom pair production. Latest limits on direct sbottom pair production are given in [66], and again taking these limits to apply to our q particles, we nd that Mq & 720 GeV for a of mass 90 GeV. The SU(2)L components ( `+3, `+2, `+1) of the scalar ` will, if they have only muonic couplings, always decay to either a muon or a muon neutrino, together with a particle. So they will sometimes decay in the same way as a smuon in supersymmetry. However, the production cross-sections and branching ratios will di er. Results of recent LHC slepton searches are given in [67, 68]. These rule out left-handed smuons, pair produced directly via a W=Z= and decaying to a muon and a neutralino, up to a maximum mass of roughly 300 GeV (for a massless neutralino). We used Feynrules [69, 70] and Madgraph5 aMC@NLO [71] to calculate electroweak (EW) pair production cross-sections of the ` particles, and then multiplied by the branching ratios in order to reinterpret the limits on cross-section given in [67]. The CMS limit plot, with our model superimposed tralino, taken from [67]. The region under the blue line shows the exclusion on our model found by reinterpreting the exclusion plot in terms of direct pair production of `s decaying to a lepton and a . For our model the x-axis should be taken to mean the mass of the ` , M`, while the y-axis means the mass of the , M . The dotted part of the blue line is extrapolated. in blue, is shown in gure 6. If the mass of the particle is greater than about 150 GeV, there are no bounds on the mass of the ` (other than the assumption that its mass is greater than that of ). 3.3 Parameter space plots muons BSM 2` and the combination q 2 q 3 elds to make these parameters real. In this subsection we show allowed regions in the parameter space of the model considering the observables described above; b ! s``, B ! Xs , Bs meson mixing and the anomalous magnetic moment of the muon. The relevant parameters entering the expressions of these observables are the masses of the new states (M ; Mq and M`) as well as the coupling to . Without loss of generality, we can re-de ne the In order to t the b ! s`` anomalies without being in disagreement with the measured Bs mixing rate, the muonic coupling 2` must be rather large. In order to have an idea of the typical values of the parameters needed in our model, in gure 7 we show parameter space regions assuming that 2` = 1:2 while parametrizing the masses in terms of one single scale M assuming the following hierarchy M = M; M` = M + 200 GeV; Mq = M + 700 GeV. In this way we are left with two parameters only (M and 2q 3q). The B ! Xs allowed region is not shown because it yields weaker constraints than Bs mixing does. For this hierarchy of masses, the only relevant direct production constraint is the bound on the mass of , M > 90 GeV. There is an overlap between the allowed Bs mixing region and the 1 model can preferred region for the b ! s`` measurements | so with these parameters, the t the b ! s`` anomalies. The value of 2` can be further lowered to be . 1, in this case the values of M ; M` and Mq are close to present bounds coming from direct searches. For example we veri ed that a t to the data with 0:8 could be achieved ` 2 when M = 150 GeV, M` = 200 GeV and Mq = 800 GeV. α3qα2q 0.2 0.1 0.0 = M; M` = M + 200 GeV; Mq = M + 700 GeV. For this value of 2`, it is not possible to explain the anomalous magnetic moment of the muon whilst tting the other constraints. However, if we also wish to t the anomalous magnetic moment of the muon, the muonic coupling 2` must be larger. We show in gure 8 the relevant parameter space regions when this coupling is set to 2` = 2:5, with the same hierarchy of masses as before. If we want to take this explanation of the (g 2) anomaly seriously, then we should consider possible bounds from the shift of the EW gauge couplings Z ; Z and W + (see also the discussion in section IV of [41]). The corrections are non-universal and so a global t to EW data is required to establish the precise constraints on the couplings. Though such a t is beyond the scope of our work, nave arguments suggest that O(1) values of 2` are not problematic. 4 Flavour symmetries terms is U(3)5 the following way; with In this section we establish a possible connection between the avour violation present in the SM and in the NP sector. In the SM and in the limit of vanishing Yukawa couplings, the largest group of unitary eld transformations that commutes with the gauge group and leaves invariant the kinetic U(1)H . Adopting notation similar to [72] we can decompose this group in GK SU(3)q3 SU(3)`2 U(1)B U(1)L U(1)Y U(1)P Q U(1)ER U(1)H ; SU(3)q3 = SU(3)QL SU(3)`2 = SU(3)LL (4.1) (4.2) = M; M` = M + 200 GeV; Mq = M + 700 GeV. With this large value of 2` there is an overlap between the regions that t the B anomalies (in blue), and the anomalous magnetic moment of the muon (in green). The U(1) factors can be identi ed with the baryon (B) and lepton (L) numbers, the hypercharge (Y ), a transformation (P Q) acting non trivially and in the same way only on DR and ER, and nally a universal rotation for the elds ER and a U(1) global symmetry associated to the Higgs doublet. We would like now to make connections with the avour structure of the SM and the possible e ects coming from NP. In order to do that a rst step is to identify (i) a avour symmetry and (ii) a set of irreducible symmetry-breaking terms. The avour symmetry group GF GK has to be broken in order to reproduce the observed pattern of fermion masses and mixing. In order to do that a set of symmetry-breaking spurions are introduced to formally restore the symmetry GF . We will now consider 3 explicit examples and we will focus on the quark sector. This is the case of Minimal Flavour Violation [72]. The spurion elds are the three YU (3; 3; 1) YD (3; 1; 3); (4.3) where the quantum numbers are speci ed with respect to the direct product of groups SU(3)QL This is the avour symmetry of the quark sector if only the Yukawa couplings yt and yb are non-vanishing. So to a good level this is an approximate symmetry of the SM. 1. GF = U(3)q3 Yukawa couplings 2. GF = U(2)q3 Recent works [73{75] considered the following set of irreducible spurions; u (2; 2; 1); d (2; 1; 2); V (2; 1; 1); (4.4) This case mimics partial compositeness. The irreducible spurions are connected to the Yukawa couplings in the following way; (YU )ij iq ju; (YD)ij iq jd: (4.5) HJEP06(21)83 With these speci c cases in mind we are now ready to discuss avour violation induced by operators of the form q i QiL , iu U Ri and d i DRi . These operators break the avour symmetry and in order to restore it we could assume that the vectors F are again spurions with de nite transformation rules under the avour symmetry. We could now assume minimality of avour violation in the following sense: the using the irreducible spurions used to construct the SM Yukawa couplings. Following this iF can be expressed procedure we obtain the following results. To recover avour invariance the F have to transform in the following way; q (3; 1; 1); u (1; 3; 1); d (1; 1; 3): (4.6) However, it can be proved using triality properties of the SU(3) irreps that tensor products of YU ; YUy; YD; YDy can never give rise to any of the F . Furthermore, the structure of the Yukawas cannot be reproduced by combinations of the s, since this can only lead to Yukawa matrices of rank 1, meaning two of the up-type quarks and two of the down-type quarks would have zero mass. We therefore conclude that this structure cannot work. To recover avour invariance the F have to transform in the following way; Atleading order in the number of spurion elds we have that ( 1q; 2q ( 1u; 2u ( 1d; 2d ) ) ) (2; 1; 1); (1; 2; 1); (1; 1; 2); q 3 u 3 d 3 (1; 1; 1); (1; 1; 1); (1; 1; 1): ( 1q; 2q)i = aq Viy; ( 1u; 2u)i = au (Vy u)i; ( 1d; 2d)i = ad (Vy d)i; (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) while 3F = bF with aF ; bF order one numbers and i = 1; 2. Doing a spurion analysis with this setup, we nd that in the basis in which QiL = (generation index) and Q3L = ei t tL; bL q is found to be VCyiKj M uj ; di T for i = 1; 2 T , the coupling vector in the quark sector ~q = (y12Vtd; y12Vts; y3) ; (4.13) where y12 and y3 are each (generally complex) numbers of O(1). up to a factor of O(1)). The natural size of 3q 2q is O(jVtsj) with the allowed regions in gures 7 and 8. This is very similar to the case of Partial Compositeness, except that there is slightly less freedom in the couplings; the ratio of 2q to 1q is exactly xed (rather than xed where ciF are order one numbers. It is easy to show that link between the iF and iF is simply given by iF = ci i F F Thus, a number of patterns of avour symmetry breaking can give rise to the hierarchical structure and alignment with the SM Yukawa couplings that is needed to explain the anomalies. In all cases, the pattern of couplings is similar to that arising in models of partial compositeness (see for example [76]). Although our phenomenological analysis in the previous section was done with more restrictions on the couplings, we can be sure that a partial compositeness-like avour structure in the quark sector is phenomenologically viable, since a full analysis of existing bounds on this setup has recently been performed in [17]. An analogous discussion could be repeated in the leptonic sector, starting from the global symmetry U(3)LL U(3)ER of the gauge kinetic terms associated with left- and right-handed leptons. However, a complete understanding of the leptonic avour requires the knowledge of the mechanism that generates neutrino masses. Here we remain agnostic as to the possible avour orientation of the spurions ` and e . For recent works that consider possible links between the avour violation in the neutrino sector and the physics of LFV transitions in b ! s``0, we refer the reader to [19, 42, 77{79]. 5 Conclusions We have presented renormalizable extensions of the SM that can explain several anomalies observed in B-meson decays. Renormalizability (which amounts to the assumption that further NP is heavy and decoupled) allows us to introduce deviations from the SM, coming from NP, in a controlled way. This is almost a sine qua non, given that we observe just a handful of anomalies in data, while many thousands of other observations agree with the SM. We have surveyed the possible NP elds that allow for a coupling to linear combinations of left-handed SM fermions, since this generates, at one-loop, an operator of the form q q ` `QiL i j k l QjLLk L LlL. Coupled with the plausible assumption that the linear avourviolating spurions, q;` are roughly aligned with the Yukawa couplings of the SM, we end up with a good t to the anomalies, without contradicting other data. As a spectacular example of the control that renormalizability brings, the models that we identify feature 3 accidental global symmetries, corresponding to conservation of (generalized) baryon and lepton numbers and to a `NP number'. The consequences of these accidental symmetries are manifold. Not only is the proton stabilized, but also all other baryon- and lepton-number violating processes (e.g. neutron-antineutron oscillations, ), many of which are strongly constrained, are forbidden automatically. The NP number leads to a generic suppression of NP avour-violating processes, since these can only occur at loop level. Yet another advantage of the models is that NP can only couple to left-handed SM fermions at the renormalizable level, meaning that contributions to processes requiring a helicity ip, such as ! e , are further suppressed. The accidental symmetries of the models, while su cient to prevent many dangerous processes, are quite di erent from the accidental symmetries of the SM, namely baryon and individual lepton family numbers. This is a crucial feature, since it allows us to have large violations of lepton universality. This is precisely what is needed to t the anomalies. The models are not panace , in that there is a further anomaly in B physics that cannot be explained, arising in decays to D( ) [80{82]. But it seems hard to ex plain this anomaly in any NP model, for the simple reason that the SM contribution, with which it needs to be comparable, is so large (being a tree-level e ect with minimal Cabibbo suppression).5 The main weakness of the models is arguably that they require a rather large value of the coupling 2` in order to explain the anomalies. While this coupling can be rather smaller than other couplings in the avour sector (i.e. the top quark Yukawa coupling), some readers may be alarmed that such a large coupling should appear in the light lepton sector, where (at least in the SM) all other couplings are small. It is important to note, however, that not only does this coupling not cause phenomenological problems per se,6 but also that, provided that it is suitably aligned, it does not lead to large avour violations in the light leptons via renormalization group ow. This follows immediately from the observation that there exists a basis in which the SM leptonic Yukawa couplings are diagonal. Nevertheless, the necessary alignment is aesthetically disturbing; we have shown that it is plausible from the point of view of avour symmetries, but it would be nice to have an explicit model of avour in which it is realised dynamically. Acknowledgments This work has been partially supported by the Galileo Galilei Institute for Theoretical Physics, STFC grant ST/L000385/1, and King's College, Cambridge. We thank members of the Cambridge SUSY Working Group for discussions. 5However it is not impossible, see for example [21, 83, 84]. 6In fact, as shown in [85], one can even put O(1) couplings among the light quark generations without necessarily getting into trouble. By denoting the generators in the fundamental representation of SU(2)L as T a = a=2 (with a being the Pauli matrices and a = 1; 2; 3), we de ne their action on the (2j + 1)dimensional completely symmetric tensor i1i2:::i2j (i1; i2; : : : ; i2j = 1; 2) as a ( i1i2:::i2j ) = Tia1k ki2:::i2j + Tia2k i1k:::i2j + : : : + Tia2jk i1i1:::k : (A.1) In general, we arrive at the following embedding of the properly normalized electric charge eigenstates: . . . where the superscripts denote the electric charge of the eld, Bn;k is the binomial factor n! Bn;k = k!(n k)! and the normalization of the states is such that i1i2:::i2j i1i2:::i2j = j j j2 + j j 1j2 + : : : + j j+1j2 + j j j2 : In the following, we provide the SU(2)L decomposition for the BSM elds ( ; q and ` ) introduced in the Model A (2.5): = (1; 4; 32 ) 111 = 112 = p1 122 = p1 222 = 3 3 3 0 2 1 q = (3; 3; 43 ) ( q)11 = ( q)12 = p1 ( q)22 = 7=3 q 2 q 1=3 q 4=3 ` = (1; 3; 2) ( `)11 = ( `)12 = p1 ( `)22 = 3 ` 1 ` 2 2 ` The relevant linear interactions (2.10) introduced in Model A can be rewritten in the following way Llin = q i RQL q + i RLiL ` + h.c. i ` = q ` i ( R)k1k2k3 (QiL)k1 ( q)k2k3 + i ( R)k1k2k3 (LiL)k1 ( `)k2k3 + h.c.; where k1; k2; k3 = f1; 2g are SU(2)L fundamental indices. More explicitly we get R 1 1=3 q uiL + h.c. ! ! R Photon and Z-boson mediated contributions An explicit calculation of the photon penguin diagrams leads to the following (lepton avour universal) contribution to the Wilson coe cient of the leptonic vector current at low energy; where C mM2q2 . This function is normalised in such a way that f (1) = 1. In our model, the contribution of the Z-boson mediated penguin diagram is suppressed compared to that of the photon mediated one. Indeed the Z-boson exchange is enhanced only in diagrams containing a source of explicit SU(2)L breaking;7 such diagrams are not present in our case. When there is no explicit SU(2)L breaking, the contribution from the Z-boson penguin diagram is suppressed by a factor Mm2BZ2 one; this factor is given simply by the ratio of the propagators in the two cases. 3 10 3 compared to the photon Neglecting the Z-boson contribution, we now quantitatively show that the photon contribution is suppressed at the percent level when compared to the one in eq. (3.4). 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Ben Gripaios, Marco Nardecchia, Sophie Renner. Linear flavour violation and anomalies in B physics, Journal of High Energy Physics, 2016, 83, DOI: 10.1007/JHEP06(2016)083