Anomaly-free local horizontal symmetry and anomaly-full rare B-decays
Rodrigo Alonso, Peter Cox, Chengcheng Han, Tsutomu T. Yanagida
IIPMU17-0068CERN-TH-2017-094
Anomaly-free local horizontal symmetry and anomaly-full rare B -decays Rodrigo Alonso, ∗ Peter Cox, † Chengcheng Han, ‡ and Tsutomu T. Yanagida
2, 3, § CERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba 277-8583, Japan Hamamatsu Professor
The largest global symmetry that can be made local in the Standard Model + 3 ν R while beingcompatible with Pati-Salam unification is SU (3) H × U (1) B − L . The gauge bosons of this theorywould induce flavour effects involving both quarks and leptons, and are a potential candidate toexplain the recent reports of lepton universality violation in rare B meson decays. In this letter wecharacterise this type of models and show how they can accommodate the data and naturally bewithin reach of direct searches. INTRODUCTION
Lepton flavour universality (LFU) violation in rare B meson decays provides a tantalising hint for new physicswhose significance has recently increased [1]. A consis-tent picture may be beginning to emerge, with LHCbmeasurements [1, 2] of the theoretically clean ratios [3] R ( ∗ ) K = Γ (cid:0) B → K ( ∗ ) µ + µ − (cid:1) Γ (cid:0) B → K ( ∗ ) e + e − (cid:1) , (1)in a combined tension of order 4 σ [4–9] with the Stan-dard Model (SM). Several phenomenologically motivatedmodels have been proposed to explain this discrepancy(see [5] for a review), one such possibility being a new U (1) gauge symmetry [10]. In this letter, we propose acomplete model which gives rise to a type of U (1) sym-metry that can accommodate the observed low-energyphenomenology.The characteristics of the new physics that might beresponsible for the observed discrepancy with the Stan-dard Model follow quite simply from the particles in-volved in the decay: a new interaction that (i) involvesboth quarks and leptons and (ii) has a non-trivial struc-ture in flavour space. This profile is fit by well-motivatedtheories that unify quarks and leptons and have a gaugedhorizontal [11] –i.e. flavour– symmetry to address points (i) and (ii) respectively.Let us address first the latter point, that is, horizon-tal symmetries. Given the representations of the fiveSM fermion fields – q L , u R , d R , (cid:96) L , e R – under the non-abelian part ( SU (3) c × SU (2) L ) of the gauge group,for one family of fermions there is only a single abeliancharge assignment possible for a gauge symmetry. Thisis precisely U (1) Y , hence the Standard Model local sym-metry, G SM = SU (3) c × SU (2) L × U (1) Y . On theother hand, a global U (1) B − L has only a gravitationalanomaly; promoting B − L to be gravity-anomaly free and a local symmetry can be done in one stroke by in-troducing right-handed (RH) neutrinos, otherwise wel-come to account for neutrino masses [12] and baryoge-nesis through leptogenesis [13]. The ‘horizontal’ direc-tion of flavour has, on the other hand, three replicas of each field and the largest symmetry in this sector is then SU (3) . Anomaly cancellation without introducing anymore fermion fields nevertheless restricts the symmetrywhich can be made local to SU (3) Q × SU (3) L . It is worthpausing to underline this result; the largest anomaly-free local symmetry extension that the SM +3 ν R admitsis SU (3) Q × SU (3) L × U (1) B − L . However, now turn-ing to point ( i ), one realises that the horizontal symme-tries above do not connect quarks and leptons in flavourspace. Although it is relatively easy to break the twonon-abelian groups to the diagonal to satisfy (i) , the de-sired structure can arise automatically from a unified the-ory; one is then naturally led to a Pati-Salam [14] model SU (4) × SU (2) L × SU (2) R × SU (3) H , which also solvesthe Landau pole problem of U (1) B − L and U (1) Y .Explicitly: G = SU (4) × SU (2) L × SU (2) R × SU (3) H (2) ψ L = u L d L ν L e L ψ R = u R d R e R ν R (3)where ψ L ∼ (4 , ,
1) and ψ R ∼ (4 , ,
2) under Pati-Salam,and both are in a fundamental representation of SU (3) H .The breaking of the Pati-Salam group, however, oc-curs differently from the usual SU (4) × SU (2) → G SM ;instead we require SU (4) × SU (2) → G SM × U (1) B − L .This can be done breaking separately SU (4) → SU (3) c × U (1) B − L and SU (2) R → U (1) with U (1) being right-handed isospin –we recall here that hyper-charge is Q Y = Q B − L / σ R . This breaking would require two scalarfields in each sector to trigger the breaking; the detaileddiscussion of this mechanism nevertheless is beyond thescope of this work and will not impact the low energyeffective theory. THE MODEL
Having discussed the Pati-Salam motivation for ourhorizontal symmetry, we shall now walk the steps downto the low energy effective theory and the connection withthe SM. At energies below unification yet far above the a r X i v : . [ h e p - ph ] O c t SM scale we have the local symmetry: G = G SM × SU (3) H × U (1) B − L . (4)The breaking SU (3) H × U (1) B − L → U (1) h occurs asone goes down in scale with the current of the unbrokensymmetry being: J hµ = ¯ ψγ µ (cid:0) g H c θ T H C S + g B − L s θ Q B − L (cid:1) ≡ g h ¯ ψγ µ T hψ ψ (5) T hψ = T H C S + t ω Q B − L , (6)where T H C S is an element of the Cartan sub-algebra of SU (3), i.e. the largest commuting set of generators(which we can take to be the diagonal ones), ψ is theDirac fermion ψ L + ψ R with the chiral fields given inEq. (3), and θ is an angle given by the representation(s)used to break the symmetry. Before proceeding any fur-ther, it is useful to give explicitly the basis-invariant re-lations that the generators of this U (1) h satisfy:Tr fl (cid:0) T h T h (cid:1) = 12 + 3 t ω Q B − L , (7)Tr fl (cid:0) T h (cid:1) = 3 t ω Q B − L , (8)where the trace is only over flavour indices, there is agenerator T h for each fermion species including RH neu-trinos, and the sign of the traceless piece of T h is thesame for all fermion representations.The one condition we impose on the flavour breaking SU (3) H × U (1) B − L → U (1) h is that the unbroken U (1) h allows for a Majorana mass term for RH neutrinos, suchthat they are heavy and can give rise to leptogenesis andsmall active neutrino masses via the seesaw formula. Ahigh breaking scale is further motivated by the need tosuppress FCNC mediated by the SU (3) H gauge bosons.The desired breaking pattern can be achieved by intro-ducing fundamental SU (3) H scalar fields, which at thesame time generate the Majorana mass term. Let usbriefly sketch this: we introduce two scalars φ , φ in(3 , −
1) of SU (3) H × U (1) B − L , so that we can write:¯ ν cR λ ij φ ∗ i φ † j ν R + h.c. (9)This implies two generations of RH neutrinos have a largeMajorana mass ( ∼ GeV), which is the minimum re-quired for leptogenesis [15] and to produce two mass dif-ferences for the light neutrinos ν L –one active neutrinocould be massless as allowed by data. The third RH neu-trino requires an extra scalar field charged under U (1) h to get a mass; depending on the charge of the scalar fieldthis might be a non-renormalisable term, making the RHneutrino light and potentially a dark matter candidate. These scalars can each be embedded in a (4 , ,
2) multiplet underthe SU (4) × SU (2) L × SU (2) R Pati-Salam group.
The second role of these scalar fields is symmetrybreaking; in this sense two fundamentals of an U (3) sym-metry can at most break it to U (1), this makes our U (1) h come out by default. To be more explicit, with all gener-ality one has (cid:104) φ (cid:105) = ( v H , , , (cid:104) φ (cid:105) = v (cid:48) H ( c α , s α ,
0) andthen for s α (cid:54) = 0 there is just one unbroken U (1) whosegauge boson Z h is the linear combination that satisfies: D µ (cid:104) φ , (cid:105) = (cid:0) g H T A Hµ − g B − L A B − Lµ (cid:1) (cid:104) φ , (cid:105) = 0 . (10)Given the v.e.v. alignment, the solution involves T in SU (3) H and via the rotation A H, = c θ Z h − s θ A (cid:48) , A B − L = s θ Z h + c θ A (cid:48) , where A (cid:48) is the massive gaugeboson, we find that the solution to Eq. (10) is: t θ = 12 √ g H g B − L , t ω = t θ g B − L g H = 12 √ , (11)with g h = g H c θ , in close analogy with SM electroweaksymmetry breaking (EWSB). This solution implies, forleptons T hL = T H − t ω = 12 √ − , (12)whereas for quarks T hQ = T H + 13 t ω = 12 √
43 43 − . (13)At this level the current that the U (1) h couples to isdifferent for quarks ( T hQ ) and leptons ( T hL ) but vectorial for each of them. On the other hand, most previous Z (cid:48) explanations for the LFU anomalies have consideredphenomenologically motivated chiral U (1) symmetries.Of course, the above charge assignment is one of severalpossibilities that can be obtained from a bottom-up ap-proach [16]; however, as we have shown, this particularflavour structure is well-motivated by the underlying UVtheory.The last step to specify the low energy theory is torotate to the mass basis of all fermions. In this regardsome comments are in order about the explicit generationof masses and mixings in this model. Charged fermionmasses would require the introduction of scalar fieldscharged under both the electroweak and the horizontal Additional assumptions on the rotation matrices in [16] lead todifferent mass-basis couplings from those we consider. group. At scales above the U (1) h breaking the fieldscan be categorised according to their U (1) h charge ; onewould need at least a charge 3, a charge − g h / √
3– ‘Higgs’ transforming as (2 , /
2) un-der SU (2) × U (1) Y ; a linear combination of these threemuch lighter than the rest would emerge as the SM Higgsdoublet.An additional SM singlet scalar is also required tobreak U (1) h and should simultaneously generate a Ma-jorana mass for the third RH neutrino. If this scalarhas U (1) h charge 3, such a term is non-renormalisableand if suppressed by a unification-like scale yields a keVmass, which is interestingly in a range where this fermioncould be dark matter [18]. Alternatively, a charge 6 scalarwould generate a mass of order a few TeV.The main focus of this work is, however, the effect ofthe gauge boson associated with the U (1) h . In this sense,however generated, the change to the mass basis impliesa chiral unitary rotation. This will change the vectorialnature of the current to give a priori eight different gen-erators T hf for each of the eight chiral fermion speciesafter EWSB: f = u R , u L , d L , d R , ν R , ν L , e L , e R . How-ever, before performing the chiral rotations, it is goodto recall that the vectorial character of the interaction isencoded in the basis-invariant relations:Tr fl (cid:0) T hf T hf (cid:1) = 12 + 14 Q B − L , Tr fl (cid:0) T hf (cid:1) = √ Q B − L , (14)which applies to both chiralities of each fermion field f .As mentioned before a priori all fields rotate when go-ing to the mass basis f = U f f (cid:48) , however we only haveinput on the mixing matrices that appear in the chargedcurrents: V CKM = U † u L U d L and U P MNS = U † e L U ν L ,which involve only LH fields. Hence, for simplicity, weassume that RH fields are in their mass bases and neednot be rotated. The CKM matrix is close to the iden-tity, whereas the lepton sector possesses nearly maxi-mal angles; following this lead we assume the angles in U u L , U d L are small so that there are no large cancella-tions in U † u L U d L , whereas U e L and U ν L have large angles.Phenomenologically however, not all angles can be largein U e L since they would induce potentially fatal µ − e Alternatively, the effective Yukawa couplings can be generated byassuming a horizontal singlet Higgs doublet at the electroweakscale and introducing two pairs of Dirac fermions for each ofthe six fermion fields, q L , u R , d R , l L , e R and ν R at the SU (3) H breaking scale, and one pair of these fermions at the U (1) h break-ing scale. The extra fermions are all SU (3) H singlets. See [17]for a similar mechanism. Ultimately, these three Higgs belong to H (2 , / , ) and H (2 , / , ) under SU L × U (1) Y × SU (3) H . To realise massmatrices for the quarks and leptons requires three H (2 , / , )and one H (2 , / , ) at the scale of G SM × SU (3) H × U (1) B − L . flavour transitions. Hence we restrict U e L to rotate onlyin the 2 − U d L . To make our assumptions explicit: U e L = R ( − θ l ) , U ν L = R ( θ − θ l ) R ( θ ) R ( θ ) ,U u L = , U d L = V CKM , (15)where R ij ( θ ab ) is a rotation matrix in the ij sector withangle θ ab . Hence, T h (cid:48) f L = U † f L T hf U f L , T h (cid:48) f R = T hf R , (16)and the current reads: J hµ = g h (cid:88) f (cid:0) ¯ f γ µ T (cid:48) hf L f L + ¯ f T (cid:48) hf R f R (cid:1) . (17)We have now made all specifications to describe the inter-actions of Z h ; all in all only two free parameters, θ l and g h , control the couplings to all fermion species. For thoseprocesses well below the Z h mass ( ∼ TeV), the effects aregiven at tree level by integrating the Z h out: S = (cid:90) d x (cid:26) Z µh (cid:0) ∂ + M (cid:1) Z h,µ − g h Z µh J hµ (cid:27) (18) On-shell Z h = (cid:90) d x (cid:18) − g h J h M + O (cid:0) ∂ /M (cid:1)(cid:19) (19)with J hµ as given in (12, 13, 15-17), so that the effectiveaction depends on θ l and M/g h . LOW ENERGY PHENOMENOLOGY
The most sensitive probes of Z h effects come fromflavour observables, in particular the FCNC producedin the down sector. An important consequence of therotation matrices in Eq. (15) is that these FCNC havea minimal flavour violation (MFV) [20, 21] structure:¯ d i γ µ V ∗ ti V tj d j . Additionally, there can be charged leptonflavour violation (LFV) involving the τ − µ transition.Even after allowing for these constraints, the Z h couldalso potentially be accessible at the LHC. Effects on otherpotentially relevant observables including the muon g − Z -pole measurements at LEP, and neutrino trident pro-duction are sufficiently suppressed in our model. Belowwe discuss the relevant phenomenology in detail. Semi-leptonic B decays The relevant Lagrangian for semi-leptonic B s decays is L B s = − g h M ( V tb V ∗ ts ¯ sγ µ b L ) (cid:0) J µl L + J µl R + J µν L (cid:1) + h.c. , (20)where for simplicity we have assumed all three RH neu-trinos are not accessible in B decays and we have J ρl L = s θ l ¯ µγ ρ µ L + c θ l ¯ τ γ ρ τ L + s θ l c θ l ¯ µγ ρ τ L + h.c. , (21) J ρl R = ¯ τ R γ ρ τ R , (22) J ρν L = ¯ ν i γ ρ ( U ∗ ν L ) i ( U ν L ) j ν jL . (23)In recent times, a number of measurements of b → sµµ processes have shown discrepancies from their SM predic-tions, most notably in the theoretically clean LFU violat-ing ratios R K and R ∗ K . Global fits to LFU violating datasuggest that the observed discrepancies can be explainedvia a new physics contribution to the Wilson coefficients C l , , with the preference over the SM around 4 σ [4–9].The effective Hamiltonian is defined as H eff = − G F √ V tb V ∗ ts (cid:0) C l O l + C l O l + C ν O ν (cid:1) , (24)where O l = α π (¯ sγ µ b L ) (cid:0) ¯ lγ µ l (cid:1) , (25) O l = α π (¯ sγ µ b L ) (cid:0) ¯ lγ µ γ l (cid:1) , (26) O ijν = α π (¯ sγ µ b L ) (cid:16) ¯ ν i γ µ ν jL (cid:17) . (27)In our model, separating the Wilson coefficients into theSM contribution ( C SM ) and the Z h piece ( δC ) we have,for muons: δC µ = − δC µ = − πα √ G F g h M s θ l . (28)In fitting the observed anomalies we use the results ofRef. [4], which for the relevant scenario δC µ = − δC µ gives δC µ ∈ [ − . − .
48] ([ − . , − . σ .The fully leptonic decay B s → µµ provides an additionalconstraint on δC µ ; the current experimental value [22]is consistent with the above best-fit region.There is also a contribution to decays involving neu-trinos, B → K ( ∗ ) ν ¯ ν , where we now have: δC ijν = δC ν ( U ∗ ν L ) i ( U ν L ) j , δC ν = − πα √ G F g h M , (29)so that the ratio to the SM expectation reads: R ν ¯ ν ≡ ΓΓ SM = 1 + 23 (cid:18) δC ν C νSM (cid:19) + 13 (cid:18) δC ν C νSM (cid:19) , (30)where C νSM ≈ − .
35 [23]. Notice that this is independentof the mixing in the lepton sector, and the rate is alwaysenhanced. The current experimental bound on this ratiois R ν ¯ ν < . B → K ( ∗ ) τ µ can also be significantly enhanced, whereas there is an irreduciblecontribution to B → K ( ∗ ) τ τ from the RH currents inEq. (23); both of these contributions nevertheless lie wellbelow the current experimental bounds [26, 27].Finally, one might also expect similar contributions in b → d and s → d transitions, the latter leading to effectsin K decays. However, given our assumptions on themixing matrices, the MFV structure in the down quarkcouplings means that these contributions are sufficientlysuppressed. In particular, the otherwise stringent boundfrom K → πν ¯ ν [28, 29] is found to be comparable, yetstill sub-dominant, to that from B → Kν ¯ ν . ¯ B – B Mixing
The Z h gives a tree-level contribution to ¯ B s – B s and¯ B d – B d mixing, which provide some of the most stringentconstraints on the model. The relevant Lagrangian is L ∆ B =2 = − g h M (cid:0) V tb V ∗ ti ¯ d i γ µ b L (cid:1) . (31)This leads to a correction to ∆ m B given by C B ≡ ∆ m B ∆ m SM B = 1 + 4 π G F m W ˆ η B S ( m t /m W ) 38 g h M c ( M ) , (32)where S ( m t /m W ) ≈ .
30 is the Inami-Lim function [30],ˆ η B (cid:39) .
84 accounts for NLO QCD corrections [31, 32],and c ( M ) ≈ . M down to m B using the NLO anomalous dimension calculated inRefs. [33, 34]. This observable is tightly constrained,yielding 0 . < C B s < .
252 and 0 . < C B d < . K – K mixing are well below currentbounds. In this case the SM prediction for ∆ m K alsosuffers from theoretical uncertainties. Lepton Flavour Violation in τ → µ There is a contribution to the cLFV decay τ → µ : L LFV = − g h M s θ l c θ l ¯ τ γ ρ µ L ¯ µγ ρ µ L , (33)resulting in a branching ratioBR( τ → µ ) = m τ π Γ τ g h M s θ l c θ l . (34)The current experimental bound is BR( τ → µ ) < . × − at 90% CL [36]. This restricts the allowed values ofthe mixing angle θ l . Collider Searches
Depending on its mass, the Z h may be directly pro-duced at the LHC. The large U (1) h charge in the leptonsector results in a potentially sizeable branching ratiointo muons: BR( Z h → µµ ) (cid:39) . s θ l . The strongestbounds on a spin-1 di-muon resonance are from the AT-LAS search at √ s = 13 TeV with 36 fb − [37]. Fur-thermore even for very large masses, M (cid:38) Perturbativity
The one-loop beta function for U (1) h is β ( g h ) = 26936 g h (4 π ) , (35)where we have assumed the U (1) h breaking scalar hascharge 3. The gauge coupling g h then encounters a Lan-dau pole at the scaleΛ = exp (cid:18) π g h ( M ) (cid:19) M . (36)This scale should at least be larger than the SU (3) H × U (1) B − L → U (1) h breaking scale. Assuming that thebreaking occurs at 10 GeV –so that the RH neutrinosobtain a sufficiently large mass for viable leptogenesis–leads to the bound g h (10 TeV) (cid:46) .
9. Also note thatdepending on the specific UV mechanism for generat-ing the fermion mass matrices, SU (3) H may not remainasymptotically free, in which case there can be additionalconstraints from perturbativity. DISCUSSION
In Fig. 1 we combine the above constraints and showthe region of parameter space which can explain the ob-served LFU anomalies. It is clear that this scenario is al-ready tightly constrained by the existing measurements,in particular ¯ B – B mixing and LHC searches. Requiringperturbativity up to the scale of the right-handed neutri-nos ( (cid:38) GeV) provides an additional upper bound onthe gauge coupling, leaving a small region of parameterspace consistent with the best fit value of C µ at 1 σ . The2 σ region for C µ –still a significant improvement overthe SM– opens up substantially more viable parameterspace.The dependence on the mixing angle in the lepton sec-tor is shown in Fig. 2. Consistency with the 2 σ best-fit re-gion for the anomalies and the bounds from ¯ B – B mixing requires θ l (cid:38) π/
4. There is also a potentially importantadditional constraint from τ → µ . In the M − g h plane,the situation remains similar to Fig. 1; however the best-fit regions for the anomaly move towards smaller massesas θ l is reduced. Let us also comment briefly on themixing in the quark sector. For simplicity, in Eq. (15)we made the assumption U d L = V CKM . Allowing in-stead for an arbitrary angle, one obtains the upper bound θ (cid:46) .
08; this is qualitatively similar to the case wehave considered ( | V ts | (cid:39) . θ below this value,¯ B – B mixing can be alleviated, but the bounds from LHCsearches and perturbativity become more severe.One consequence of the relatively strong experimentalconstraints is that this model can be readily tested inthe relatively near future. Improved precision for ∆ m B would either confirm or rule out this model as a potentialexplanation for the LFU anomalies. On the other hand,improvements in the LHC limit, when combined withthe perturbativity bounds, would force one to considerlower SU (3) H × U (1) B − L → U (1) h breaking scales. Inaddition, the LFV decay τ → µ provides an importantcomplementary probe of the mixing angle in the leptonsector. Similarly, the decay B → K ( ∗ ) τ µ can be signif-icantly enhanced and could be observable in the future.In this sense it is good to note that the vectorial characterof the U (1) h reveals itself in the sum rules (cid:88) l δC ll = 0 , (cid:88) l δC ll = 2 (cid:88) i δC iiν , (37) (cid:88) ll (cid:48) (cid:16) | δC ll (cid:48) | + | δC ll (cid:48) | (cid:17) = 4 (cid:88) ij | δC ijν | , (38)which is basically a manifestation of Eq. (14).Finally, we have focused on the specific case of a G SM × SU (3) H × U (1) B − L symmetry, however there ex-ist other related scenarios which provide equally inter-esting possibilities. For example, if one instead assumes G SM × SU (3) Q × SU (3) L × U (1) B − L , it is possible toobtain T hL ∼ diag(0,0,-3) and T hQ ∼ diag(0,0,1). This isnothing other than a U (1) B − L under which only the thirdgeneration is charged. The LHC bounds would be signifi-cantly weakened in such a scenario; g h could then remainperturbative up to the Planck scale. Another possiblesymmetry is G SM × SU (3) Q × SU (3) L if a bifundamen-tal Higgs (3 , ∗ ) condenses at low energies, since it mixestwo U(1) gauge bosons. A merit of this model is that onecan give heavy Majorana masses to all right-handed neu-trinos by taking the unbroken U (1) h as diag(0,1,-1) forleptons [39], and diag(1,1,-2) for quarks. The low energyphenomenology of a U (1) with similar flavour structurewas previously considered in [40, 41], the latter based onanother non-abelian flavour symmetry [42]. We leave thedetailed investigation of such related scenarios for futurework, but application of our analysis is straightforward. FIG. 1. The best-fit region to the LFU anomalies at 1 σ (solidlines) and 2 σ (dashed lines). The shaded regions are excludedby existing measurements at 95% CL. The dotted lines corre-spond to upper bounds on the SU (3) H × U (1) B − L breakingscale from perturbativity. We have fixed θ l = π/ θ l . CONCLUSION
If confirmed, the violation of lepton flavour universal-ity would constitute clear evidence for new physics. Inthis letter, we have proposed a complete, self-consistentmodel in which the observed anomalies are explainedby the presence of a new U (1) h gauge symmetry link-ing quarks and leptons. We have shown how such asymmetry can naturally arise from the breaking of an SU (3) H × U (1) B − L horizontal symmetry. Furthermore,within the SM+3 ν R , this is the largest anomaly-free sym- metry extension that is consistent with Pati-Salam uni-fication. The model is readily testable in the near futurethrough direct searches at the LHC, improved measure-ments of ¯ B – B mixing and charged LFV decays. Acknowledgements
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