Interpreting Hints for Lepton Flavor Universality Violation

Wolfgang Altmannshofer,1, ∗ Peter Stangl,2, † and David M. Straub2, ‡

1Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA2Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany

We interpret the recent hints for lepton flavor universality violation in rareB meson decays. Based ona model-independent effective Hamiltonian approach, we determine regions of new physics parameterspace that give a good description of the experimental data on RK and RK∗ , which is in tensionwith Standard Model predictions. We suggest further measurements that can help narrowing downviable new physics explanations. We stress that the measured values of RK and RK∗ are fullycompatible with new physics explanations of other anomalies in rare B meson decays based on theb→ sµµ transition. If the hints for lepton flavor universality violation are first signs of new physics,perturbative unitarity implies new phenomena below a scale of ∼ 100 TeV.

Introduction. The wealth of data on rare leptonicand semi-leptonic b hadron decays that has been accu-mulated at the LHC so far allows the Standard Model(SM) CKM picture of flavor and CP violation to betested with unprecedented sensitivity. Interestingly, cur-rent data on rare b → s`` decays show an intriguingpattern of deviations from the SM predictions both forbranching ratios [1–3] and angular distributions [4, 5].The latest global fits find that the data consistentlypoints with high significance to a non-standard effectthat can be described by a four fermion contact inter-action C9 (sγνPLb)(µγνµ) [6] (see also earlier studies [7–9]). Right now the main obstacle towards conclusivelyestablishing a beyond-SM effect is our inability to ex-clude large hadronic effects as the origin of the apparentdiscrepancies (see e.g. [10–15]).

In this respect, observables in b→ s`` transitions thatare practically free of hadronic uncertainties are of partic-ular interest. Among them are lepton flavor universality(LFU) ratios, i.e. ratios of branching ratios involvingdifferent lepton flavors such as [16–18]

RK =B(B → Kµ+µ−)

B(B → Ke+e−), RK∗ =

B(B → K∗µ+µ−)

B(B → K∗e+e−).

(1)In the SM, the only sources of lepton flavor universalityviolation are the negligibly small neutrino masses, themasses of the charged leptons and their interactions withthe Higgs. Higgs interactions do not lead to any ob-servable effects in rare b decays and lepton mass effectsbecome relevant only for a very small di-lepton invari-ant mass squared close to the kinematic limit q2 ∼ 4m2

` .Over a very broad range of q2 the SM accurately pre-dicts RK = RK∗ = 1, with theoretical uncertainties ofO(1%) [19]. Deviations from the SM predictions can beexpected in various models of new physics (NP), e.g. Z ′

models based on gauged Lµ−Lτ [20–22] or other gaugedflavor symmetries [23–25], models with partial compos-iteness [26–28], and models with leptoquarks [29–34].

A first measurement of RK by the LHCb collabora-tion [35] in the di-lepton invariant mass region 1 GeV2 <

q2 < 6 GeV2,

R[1,6]K = 0.745+0.090

−0.074 ± 0.036 , (2)

shows a 2.6σ deviation from the SM prediction. Veryrecently, LHCb presented first results for RK∗ [36],

R[0.045,1.1]K∗ = 0.660+0.110

−0.070 ± 0.024 , (3)

R[1.1,6]K∗ = 0.685+0.113

−0.069 ± 0.047 , (4)

where the superscript indicates the di-lepton invariantmass bin in GeV2. These measurements are in tensionwith the SM at the level of 2.4 and 2.5σ, respectively.Intriguingly, they are in good agreement with the recentRK∗ predictions in [6] that are based on global fits ofb → sµµ decay data, assuming b → see decays to beSM-like.

In this letter we interpret the RK(∗) measurements us-ing a model-independent effective Hamiltonian approach(see [37–43] for earlier model independent studies of RK).We also include Belle measurements of LFU observablesin the B → K∗`+`− angular distibutions [5]. We donot consider early results on RK(∗) from BaBar [44] andBelle [45] which, due to their large uncertainties, havelittle impact. We identify the regions of NP parameterspace that give a good description of the experimentaldata. We show how future measurements can lift flat di-rections in the NP parameter space and discuss the com-patibility of the RK(∗) measurements with other anoma-lies in rare B meson decays.Model independent implications for new physics. We

assume that NP in the b→ s`` transitions is sufficientlyheavy such that it can be model-independently describedby an effective Hamiltonian, Heff = HSM

eff +HNPeff ,

HNPeff = −4GF√

2VtbV

∗ts

e2

16π2

∑i,`

(C`iO`i + C ′ `i O

′ `i ) + h.c. ,

(5)with the following four-fermion contact interactions,

O`9 = (sγµPLb)(¯γµ`) , O′ `9 = (sγµPRb)(¯γµ`) , (6)

O`10 = (sγµPLb)(¯γµγ5`) , O′ `10 = (sγµPRb)(¯γµγ5`) , (7)

arX

iv:1

704.

0543

5v1

[he

p-ph

] 1

8 A

pr 2

017

2

Coeff. best fit 1σ 2σ pull

Cµ9 −1.59 [−2.15, −1.13] [−2.90, −0.73] 4.2σ

Cµ10 +1.23 [+0.90, +1.60] [+0.60, +2.04] 4.3σ

Ce9 +1.58 [+1.17, +2.03] [+0.79, +2.53] 4.4σ

Ce10 −1.30 [−1.68, −0.95] [−2.12, −0.64] 4.4σ

Cµ9 = −Cµ10 −0.64 [−0.81, −0.48] [−1.00, −0.32] 4.2σ

Ce9 = −Ce10 +0.78 [+0.56, +1.02] [+0.37, +1.31] 4.3σ

C′µ9 −0.00 [−0.26, +0.25] [−0.52, +0.51] 0.0σ

C′µ10 +0.02 [−0.22, +0.26] [−0.45, +0.49] 0.1σ

C′ e9 +0.01 [−0.27, +0.31] [−0.55, +0.62] 0.0σ

C′ e10 −0.03 [−0.28, +0.22] [−0.55, +0.46] 0.1σ

TABLE I. Best-fit values and pulls for scenarios with NP inone individual Wilson coefficient.

and the corresponding Wilson coefficients C`i , with ` =e, µ. We do not consider other dimension-six operatorsthat can contribute to b → s`` transitions. Dipole oper-ators and four-quark operators [46] cannot lead to vio-lation of LFU and are therefore irrelevant for this work.Four-fermion contact interactions containing scalar cur-rents would be a natural source of LFU violation. How-ever, they are strongly constrained by existing measure-ments of the Bs → µµ and Bs → ee branching ra-tios [47, 48]. Imposing SU(2)L invariance, these boundscannot be avoided [49]. We have checked explicitly thatSU(2)L invariant scalar operators cannot lead to any ap-preciable effects in RK(∗) (cf. [50]).

For the numerical analysis we use the open source codeflavio [51]. Based on the experimental measurementsand theory predictions for the LFU ratios RK(∗) andthe LFU differences of B → K∗`+`− angular observ-ables DP ′4,5

(see below), we construct a χ2 function thatdepends on the Wilson coefficients and that takes intoaccount the correlations between theory uncertainties ofdifferent observables. The experimental uncertainties arepresently dominated by statistics, so their correlationscan be neglected. For the SM we find χ2

SM = 24.4 for 5degrees of freedom.

Tab. I lists the best fit values and pulls, defined as the√∆χ2 between the best-fit point and the SM point for

scenarios with NP in one individual Wilson coefficient.The plots in Fig. 1 show contours of constant ∆χ2 ≈2.3, 6.2, 11.8 in the planes of two Wilson coefficients forthe scenarios with NP in Cµ9 and Cµ10 (top), in Cµ9 andCe9 (center), or in Cµ9 and C ′µ9 (bottom), assuming theremaining coefficients to be SM-like.

The fit prefers NP in the Wilson coefficients corre-sponding to left-handed quark currents with high sig-nificance ∼ 4σ. Negative Cµ9 and positive Cµ10 decreaseboth B(B → Kµ+µ−) and B(B → K∗µ+µ−) while pos-

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

ReCµ9

−1.0

−0.5

0.0

0.5

1.0

1.5

ReCµ 10

flavio v0.21

LFU observables

b→ sµµ global fit

all

all, fivefold non-FF hadr. uncert.

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

ReCµ9

−1.0

−0.5

0.0

0.5

1.0

1.5

ReCe 9

flavio v0.21

LFU observables

b→ sµµ global fit

all

−3 −2 −1 0 1 2 3

ReCµ9

−2

−1

0

1

2

ReC′µ 9

flavio v0.21

RKR∗K

LFU observables

b→ sµµ global fit

FIG. 1. Allowed regions in planes of two Wilson coefficients,assuming the remaining coefficients to be SM-like.

3

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

q2 [GeV2]

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

DP′ 4

flavio v0.21

SM

Cµ9 = −1.6

Ce9 = +1.6

Cµ10 = +1.3

Ce10 = −1.3

Cµ9 = −Cµ10 = −0.7

Ce9 = −Ce10 = 0.7

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

q2 [GeV2]

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

DP′ 5

flavio v0.21

SM

Cµ9 = −1.6

Ce9 = +1.6

Cµ10 = +1.3

Ce10 = −1.3

Cµ9 = −Cµ10 = −0.7

Ce9 = −Ce10 = 0.7

FIG. 2. The B → K∗`+`− LFU differences DP ′4 and DP ′5 in the SM and various NP benchmark models as functions of q2.The error bands contain all theory uncertainties including form factors and non-factorisable hadronic effects. In the region ofnarrow charmonium resonances, only the short-distance contribution is shown, without uncertainties.

itive Ce9 and negative Ce10 increase both B(B → Ke+e−)and B(B → K∗e+e−), allowing a good description of thedata in each case. Also along the direction C`9 = −C`10

that corresponds to a left-handed lepton current, we findexcellent fits to the data.

The primed Wilson coefficients, that correspond toright-handed quark currents, cannot improve the agree-ment with the data by themselves. As is well known [18],the primed coefficients imply RK∗ > 1 given RK < 1 andvice versa. The complementary sensitivity of RK∗ andRK to right-handed currents is illustrated in the bottomplot of Fig. 1 for the example of Cµ9 vs. C ′µ9 . In com-bination with sizable un-primed coefficients, the primedcoefficients can slightly improve the fit.

Among the un-primed Wilson coefficients, there areapproximate flat directions. We find that a good de-scription of the experimental results is given by

Cµ9 − Ce9 − Cµ10 + Ce10 ' −1.4 , (8)

unless some of the individual coefficients are much largerthan 1 in absolute value. The flat direction is clearlyvisible in the top and center plot of Fig. 1. In manyNP models one has relations among these coefficients. Inmodels with leptoquarks one finds C`9 = ±C`10 [29, 52],models based on gauged Lµ − Lτ predict Ce9 = C`10 =0 [20], while in some Z ′ models one finds C`9 = aC`10,where a is a constant of O(1) (see e.g. [53]).

We find that a non-standard C`10 (C`9) leads to slightlylarger (smaller) effects in RK∗ than in RK . Therefore,RK∗ . RK < 1 is best described by a non-standard C`10.The opposite hierarchy, RK . RK∗ < 1, would lead to aslight preference for NP in C`9.

A more powerful way to distinguish NP in C`9 fromNP in C`10 is through measurements of LFU differencesof angular observables [22, 54, 55]. We find that the

observables

DP ′4= P ′4(B → K∗µ+µ−)− P ′4(B → K∗e+e−) , (9)

DP ′5= P ′5(B → K∗µ+µ−)− P ′5(B → K∗e+e−) , (10)

are particularly promising (for a definition of the observ-ables P ′4,5 see [56]). Predictions for the observables DP ′4,5

as functions of q2 in the SM and various NP scenariosare shown in the plots of Fig. 2. The SM predictions areclose to zero with very high accuracy across a wide q2

range. In the presence of NP, DP ′4,5show a non-trivial q2

dependence. If the discrepancies in RK(∗) are explainedby NP in C`9, we predict a negative DP ′4

∼ −0.1 at low

q2 . 2.5 GeV2 and a sizable positive DP ′5∼ +0.5. With

NP in C`10 we predict instead a positive DP ′4∼ +0.15

and a small negative DP ′5∼ −0.1. We observe that DP ′5

has even the potential to distinguish between NP in Ce9and Cµ9 . For q2 & 5 GeV2, a negative Cµ9 leads to asizable increase of P ′5(B → K∗µ+µ−), while a positiveCe9 can decrease P ′5(B → K∗e+e−) only slightly, as theSM prediction for P ′5 in this q2 region is already closeto its model-independent lower bound of −1. The re-

cent measurements by Belle, D[1,6]P ′4

= +0.498±0.553 and

D[1,6]P ′5

= +0.656±0.496 [5], have still sizable uncertainties

and are compatible with NP both in C`9 and in C`10. Theyslightly favor NP in C`9. We note that, while the SM pre-diction for these observables has a tiny uncertainty, forfixed values of LFU violating Wilson coefficients, formfactor and other hadronic uncertainties do play a role, asalso shown in Fig. 2. However, these uncertainties arestill so small that sufficient experimental precision couldallow a clean identification of the underlying NP contactinteraction.

We stress that the NP contact interactions in (5) leadalso to a characteristic q2 shape in the LFU ratios RK(∗) .

4

0 5 10 15 20

q2 [GeV2]

0.5

0.6

0.7

0.8

0.9

1.0RK

flavio v0.21

SM

Cµ9 = −1.6

Ce9 = +1.6

Cµ10 = +1.3

Ce10 = −1.3

Cµ9 = −Cµ10 = −0.7

Ce9 = −Ce10 = 0.7

LHCb

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

q2 [GeV2]

0.5

0.6

0.7

0.8

0.9

1.0

RK∗

flavio v0.21

SM

Cµ9 = −1.6

Ce9 = +1.6

Cµ10 = +1.3

Ce10 = −1.3

Cµ9 = −Cµ10 = −0.7

Ce9 = −Ce10 = 0.7

LHCb

FIG. 3. The LFU ratios RK(∗) in the SM and two NP benchmark models as function of q2. Conerning the error bands, thesame comments as for Fig. 2 apply.

In Fig. 3 we show RK(∗) as functions of q2 in the SM andin the same NP scenarios as in Fig. 2. In the SM, RK(∗)

are to an excellent approximation q2 independent. Forvery low q2 ' 4m2

µ they drop to zero, due to phase spaceeffects. NP contact interactions lead to an approximatelyconstant shift in RK . The ratio RK∗ , on the other hand,shows a non-trivial q2 dependence in the presence of NP.In contrast to B → K``, the B → K∗`` decays at low q2

are dominated by the photon pole, which gives a leptonflavor universal contribution. The effect of NP is there-fore diluted at low q2. Given the current experimentaluncertainties, the measured q2 shape of RK∗ is compati-ble with NP in form of a contact interaction. Significantdiscrepancies from the shapes shown in Fig. 3 would im-ply the existence of light NP degrees of freedom aroundor below the scale set by q2 and a breakdown of the ef-fective Hamiltonian framework.

Assuming that the description in terms of contactinteractions holds, we translate the best fit values ofthe Wilson coefficients into a generic NP scale. Repa-rameterizing the effective Hamiltonian (5) as HNP

eff =−∑

iOi/Λ2i , one gets

Λi =4π

e

1√|VtbV ∗ts|

1√|Ci|

v√2' 35 TeV√

|Ci|. (11)

Based on perturbative unitarity we therefore predict theexistence of NP degrees of freedom below a scale ofΛNP ∼

√4π × 35 TeV/

√|Ci| ∼ 100 TeV.

Compatibility with other rare B decay anomalies. It isnatural to connect the discrepancies in RK(∗) to the otherexisting anomalies in rare decays based on the b → sµµtransition. In the plots of Fig. 1 we show in dotted graythe 1, 2, and 3σ contours from our global b→ sµµ fit thatdoes not take into account the measurements of the LFUobservables RK(∗) and DP ′4,5

[6]. We observe that theblue regions prefered by the LFU observables are fully

compatible with the b → sµµ fit. We have also per-formed a full fit, taking into account all the observablesfrom the b→ sµµ fit, the branching ratio of Bs → µ+µ−

(assuming it not to be affected by scalar NP contribu-tions), and the BaBar measurement of the B → Xse

+e−

branching ratio [57]. This fit, shown in red, points toa non-standard Cµ9 ' −1.2 with very high singificance.Wilson coefficients other than Cµ9 are constrained by theglobal fit.

Compared to the LFU observables, the global b→ sµµfit depends more strongly on estimates of hadronic uncer-tainties in the b → s`` transitions. To illustrate the im-pact of a hypothetical, drastic underestimation of theseuncertainties, we also show results of a global fit whereuncertainties of non-factorisable hadronic contributionsare inflated by a factor of 5 with respect to our nominalestimates. In this case, the global fit becomes dominatedby the LFU observables, but the b → sµµ observablesstill lead to relevant constraints. For instance, the best-fit value for Cµ10 in Tab. I would imply a 50% suppresionof the Bs → µ+µ− branching ratio, which is already intension with current measurements [47], barring cancel-lations with scalar NP contributions.

Conclusions. The discrepancies between SM predic-tions and experimental results in the LFU ratios RK andRK∗ can be explained by NP four-fermion contact inter-actions (sb)(¯ ) with left-handed quark currents. Futuremeasurements of LFU differences of B → K∗`+`− angu-lar observables can help to identify the chirality struc-ture of the lepton currents. If the hints for LFU vio-lation in rare B decays are first signs of NP, perturba-tive unitarity implies new degrees of freedom below ascale of ΛNP ∼ 100 TeV. These results are robust, i.e.they depend very mildly on assumptions about the sizeof hadronic uncertainties in the B → K(∗)`+`− decays.

Intriguingly, the measured values of RK and RK∗ are

5

fully compatible with NP explanations of various addi-tional anomalies in rare B meson decays based on theb→ sµµ transition. A combined fit singles out NP in theWilson coefficient Cµ9 as a possible explanation.

Acknowledgments

WA acknowledges financial support by the Universityof Cincinnati. The work of PS and DS was supported bythe DFG cluster of excellence “Origin and Structure ofthe Universe”.

∗ altmanwg@ucmail.uc.edu† peter.stangl@tum.de‡ david.straub@tum.de

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