New light mediators for the $R_K$ and $R_{K^*}$ puzzles

The measurements of $R_{K}$ and $R_K^{*}$ provide hints for the violation of lepton universality. However, it is generally difficult to explain the $R_{K^*}$ measurement in the low $q^2$ range, $0.045 \le q^2 \le 1.1$ GeV$^2$. Light mediators offer a solution by making the Wilson coefficients $q^2$ dependent. We check if new lepton nonuniversal interactions mediated by a scalar ($S$) or vector particle ($Z^\prime$) of mass between $10-200$ MeV can reproduce the data. We find that a 25 MeV $Z^\prime$ with a $q^2$-dependent $b-s$ coupling and that couples to the electron but not the muon can explain all three anomalies in conjunction with other measurements. A similar 25 MeV $S$ provides a good fit to all relevant data except $R_{K^*}$ in the low $q^2$ bin. A 25 MeV $Z^\prime$ with a $q^2$-dependent $b-s$ coupling and that couples to the muon but not the electron provides a good fit to the combination of the $R_K$ and $R_{K^*}$ data, but does not fit $R_{K^*}$ in the low $q^2$ bin well.

0.045 ≤ q 2 ≤ 1.1 GeV 2 , although the predictions are consistent with measurements within 1.5σ. A resolution to this problem may be possible if the new physics is light.
In models with light mediators [30-32, 39, 40], the new physics cannot be integrated out, resulting in a q 2 dependence of the Wilson coefficients (WCs). If the light mediator mass is between m B and twice the lepton mass, and the mediator width is narrow, then it is observable as a resonance in the dilepton invariant mass. To avoid constraints from the search for such states, one generally takes the mediator mass to be m B or less than 2m ℓ . In this paper we study a light scalar mediator denoted by S and a light vector mediator denoted by Z ′ .

II. LIGHT SCALAR
We start our discussion with a light scalar S with mass in the 10 − 200 MeV range. For this scenario, we assume the following flavor-changing bsS vertex, where F (q 2 ) is a form factor. 1 The matrix elements for the processes b → sℓ + ℓ − and the mass difference in B s mixing are bs P L + g S ′ bs P R )b l (g ℓℓ L P L + g ℓℓ R P R )ℓ , where we have used Ref. [42] for B 0 s -B 0 s mixing. The mass difference in the SM for the B s system is [43] ∆M SM Bs = (17.4 ± 2.6) ps −1 , which is consistent with experimental measurement [44], ∆M Bs = (17.757 ± 0.021) ps −1 .
We will choose the new physics contribution, ∆M N P Bs , to be as large as the uncertainty in the SM prediction. We now consider b → sℓ + ℓ − transitions. For light scalars coupling to muons, R K and R K * are generally increased from their SM values in contradiction with experiment. Moreover, the measured B s → µ + µ − rate also puts strong constraints on new scalar couplings to muons.
We therefore suppose the scalar couples mainly to electrons in which case the matrix element for b → se + e − from Eq. (5) is where g S ee ≡ (g ee L + g ee R )/2 and g S ′ ee ≡ (g ee R − g ee L )/2. In the following discussion, we chose different structures for the form factor F (q 2 ). 1 In our effective theory approach, the structure in Eq. (3) is of the general form consistent with the assumed symmetries. As an illustration of how a flavor changing vertex with a q 2 -dependent form factor may occur, consider the following Lagrangian at the b-quark mass scale in the gauge basis: where χ is a hidden sector fermion (which may serve as a dark matter candidate) of mass mχ < ∼ m b , and we have suppressed all Lorentz structures in the Lagrangian. (In the context of Section III, for a light vector mediator Z ′ , one may consider a similar Lagrangian of the form, gχχγ µ χZ ′ µ .) The first term in the Lagrangian represents an effective coupling between the b and χ fields that might arise via the exchange of a heavy mediator of mass Λ ≫ m b , which has been integrated out of the theory at the m b scale. Although there is no direct coupling between b and S (or Z ′ ), abbS (orbbZ ′ ) vertex with a q 2 -dependent coupling will be generated by a χ loop. Transforming the b quark from the gauge to the mass basis then generates asbS (orsbZ ′ ) coupling. In the case of the scalar mediator the form factor contains terms of the form, m 2 χ Λ 2 and q 2 Λ 2 . For the latter term to dominate, q 2 ≫ m 2 χ , which implies that mχ < ∼ 30 MeV for the q 2 values of interest. For the Z ′ case, the leading term in the form factor goes as q 2 due to the conserved vector current [41]. We note that the situation is similar to the SM case where χ is replaced by the charm quark and S (or Z ′ ) by the photon. In this case the first term in the Lagrangian, of the form g First, we consider the situation in which the bsS vertex is generated either at tree level or at loop level with internal particles with masses much greater than the b quark mass. Then, the form factor F (q 2 ) ≡ 1, and to avoid a pole contribution to the measurements of B(B 0 → K * 0 e + e − ) in the dielectron invariant mass range, m ee = [30 − 1000] MeV [45], we choose M S = 25 MeV.
Note that the BaBar [46] and Belle [47,48] measurements require m ee to be larger than 30 MeV [49] and 140 MeV, respectively. We fix g S ee = 2.0 × 10 −4 , which is the largest value allowed by the anomalous magnetic moment of the electron [50] for M S = 25 MeV at the 2σ CL. Then we perform a χ 2 -fit to the theoretically clean observables R K and R K * , and the new physics contribution to the B s mass difference, ∆M N P s = 0 ± 2.6 ps −1 . In Ref. [51] the lepton flavor dependent angular observables Q 4,5 were measured but since the errors in the measurements are large we do not use them in our fit. We use flavio [52] to calculate the theoretical values of the observables O th . We then compute where O exp are the experimental measurements of the observables, and the total covariance matrix C is the sum of theoretical and experimental covariance matrices. The SM gives a very poor fit to the R K and R K * measurements with The best fit values of the couplings g S bs and g S ′ bs along with predictions at the best fit point, for M S = 25 MeV and g S ee = 2.0 × 10 −4 , are provided in Table I. As a good fit is obtained in this case, we check if these values are consistent with the various measured branching ratios in b → se + e − modes. If S can decay to e + e − with a branching ratio ∼1 then the decays B → K ( * ) e + e − will be dominated by the two-body decays, B → K ( * ) S, with S decaying to e + e − .
For the two body B → KS decay, the branching ratio is where the form factor f 0 (z) can be found in Ref. [53]. For the two body B → K * S decay, the branching ratio is where τ B is the lifetime of B meson, | p K * | = λ 1/2 (m 2 B , m 2 K * , m 2 S )/2m B , and the form factor A 0 is taken from Ref. [54]. To bound the NP coupling constants g S bs and g S ′ bs , we require the B → K ( * ) S branching ratio to be less than 1%. This choice is consistent with uncertainties in the calculation of the B meson width [55]. For M S between 10 − 200 MeV, B(B 0 → K * 0 e + e − ) and B(B 0 → K 0 e + e − ) impose the constraints shown in Table II. The best-fit values of the coupling given in Table I are in contradiction with these constraints. Hence, a light scalar with form factor F (q 2 ) ≡ 1 is ruled out.
Now we consider a q 2 -dependent form factor F (q 2 ) = 1 which may be loop induced. For momentum transfer q 2 ≪ m 2 B , F (q 2 ) can be expanded as [39] F (q 2 ) = a bs + b bs where m B is the B-meson mass. We do not include the B s mass difference and B(B s → e + e − ) as constraints since F (q 2 ) is unknown for q 2 ∼ m 2 B . We assume that S does not couple to neutrinos so that B → Kνν [56,57] does not constrain a bs . Redefining a bs g S bs as g S bs , and a bs g S ′ bs as g S ′ bs , we perform a χ 2 -fit to the theoretically clean observables R K and R K * . The best fit values of the couplings and the predictions for R K and R K * are shown in Table I. Taking into account the constraints on g S bs and g S ′ bs from Table II along with the constraints on g ee from the anomalous magnetic moment of the electron, we see that the best fit values O(10 −8 ) cannot be achieved in this case.
To avoid the strong constraints from the two-body decays we set a bs = 0 in Eq. (13) (thereby also evading the B → Kνν constraint if the mediator couples to neutrinos [39]), and absorbing the factor b bs to redefine g S bs and g S ′ bs , the matrix element for b → se + e − is given by With the form factor q 2 /M 2 B , requiring B(B 0 → K * 0 e + e − ) and B(B 0 → K 0 e + e − ) to be less than 1% gives the constraints on g S bs and g S ′ bs in Table II. The best-fit values can be found in Table I. A reasonable fit is obtained in this case with a pull of 4.4. We see that R K and R K * values in the central q 2 bin can be reasonably accommodated, while the effect on R K * in the low q 2 bin is small in this case. We also evaluated the branching ratios for various b → se + e − observables; see Table III [59], the discrepancy is about 2.3σ.
The prediction for the inclusive mode B(B → X s e + e − ) [1.0−6.0] , which suffers from less hadronic uncertainties, is consistent with measurement. Finally, we considered the case with a pseudoscalar coupling of the electron and find similar results to that of the scalar coupling.

III. LIGHT Z ′
A Z ′ with mass less than 2m µ was recently proposed in Ref. [39] to simultaneously explain the measurements of R K and the anomalous magnetic moment of the muon, with implications for nonstandard neutrino interactions. Such a Z ′ may potentially explain R K * in the low q 2 bin [31]. A Z ′ with a mass in the few GeV range was discussed recently [30,32] but the q 2 dependence of the WC is not strong enough to explain the R K * at low q 2 [32] . Here we focus on an MeV Z ′ .
We assume the flavor-changing bsZ ′ vertex to have the form, The matrix elements for b → sℓ + ℓ − and the mass difference in B s mixing are where we have used Ref. [42] for B 0 s -B 0 s mixing. Also, we define g ℓℓ ≡ (g ℓℓ L + g ℓℓ R )/2 and g ′ ℓℓ ≡ (g ℓℓ R − g ℓℓ L )/2 for convenience.

A. Z ′ with muon coupling
We begin with the case where the Z ′ couples to muons and not to the electrons.
We first assume that F (q 2 ) ≡ 1 and consider the case g µµ L = g µµ R = g µµ , so the leptonic term is a purely vector current. We perform a fit to the R K and R K * data, and the new physics contribution to the B s mass difference. We choose M Z ′ = 25 MeV and fix g µµ = 8.0 × 10 −4 , which is the 2σ upper bound from the anomalous magnetic moment of the muon. The fit results are shown in Table I. We see that the overall improvement over the SM is insignificant because g S bs and g S ′ bs are suppressed by B s mixing.

F (q 2 ) = 1
Now we consider F (q 2 ) = 1 and assume an expansion as in Eq. (13). Keeping only the leading a bs term, we perform a fit to the observables R K and R K * for M S = 25 MeV. We do not employ the new physics contribution to the B s mass difference as a constraint since F (q 2 ) is unknown for q 2 ∼ m 2 B . The fit results are shown in Table I. The overall improvement over the SM is poor, with a pull of 2.4. Clearly, a light Z ′ with pure vector coupling to the muon is unable to explain the R K We next consider the case with a bs = 0 and the Z ′ also has nonzero axial vector coupling with the muons, i.e., g ℓℓ L = g ℓℓ R . To keep the number of new couplings unchanged, we take either g ′ bs = 0 or g bs = 0. This case also does not give a good fit to the data; see Table I.
As can be seen from Table I, overall two of the scenarios with a bs = 0 provide good fits except to the R K * measurement in the low q 2 bin. Morevover, a Z ′ with purely vector muon coupling is easily compatible with other b → sℓ + ℓ − observables [32].

B. Z ′ with electron coupling
We now consider the case where the Z ′ couples to electrons and not to muons.
We first assume that F (q 2 ) ≡ 1 and we start by considering the case g ee L = g ee R = g ee so the leptonic term is a purely vector current. We perform a fit to the R K and R K * data, and the new physics contribution to the B s mass difference. We fix g ee = 2.5 × 10 −4 , which is within the 90% CL upper limit from NA48/2 [60]. The fit results are shown in Table I. The fit to R K and R K * is close to the SM predictions because of B s mixing.

F (q 2 ) = 1
Now we consider F (q 2 ) = 1. We fit to the observables R K and R K * only since F (q 2 ) is unknown for q 2 ∼ m 2 B . The best fit results are shown in Table I. While a good fit to R K and R K * is obtained, we need to check if these couplings are consistent with other measurements. As in the scalar case there is a two-body contribution to B(B → K ( * ) e + e − ) from B → K ( * ) Z ′ and Z ′ decaying to e + e − with a branching ratio ∼1.
The branching ratio for B → KZ ′ is [61,62], where is a form factor. For B → K * Z ′ the branching ratio is given by, where the helicity amplitudes are defined as, and V , A 1 and A 2 are form factors [53,54] and ξ = (m 2 . Assuming the decay rate of B → KZ ′ and B → K * Z ′ to be less than 1% of the B 0 width, we obtain the constraints shown in Table II. Since g ee is constrained to be less than 2.5 × 10 −4 at the 90% CL for M Z ′ = 25 MeV [60], the constraints in Table II exclude the best-fit values to explain the R K and R K * measurements in this case. We next consider the case when Z ′ also has nonzero axial vector coupling with the electrons, i.e., g ee L = g ee R . The best-fit results are shown in Table I. While a good fit to R K and R K * is obtained, the best-fit values do not satisfy the two-body constraints of Table II along with the constraint on g ee . Now, to avoid the two-body constraint, like in the scalar case, we set a bs = 0 in Eq. (13). In this case, assuming g ee L = g ee R = g ee , i.e., pure vector coupling to the electron, and for M Z ′ = 25 MeV, we fit the product g ee g bs and g ee g ′ bs to the R K and R K * data. The results are summarized in Table I. Clearly, at the best fit point the predictions for R K and R K * are within the 1σ range of the measurements. Requiring B(B 0 → K 0 e + e − ) < 1% and B(B 0 → K * 0 e + e − ) < 1%, we get the constraints shown in Table II. The best fit satisfies all constraints on g bs , g ′ bs and g ee . From Table I, we see that R K and R K * values in all measured q 2 bins can be reasonably accommodated. We also checked that the predictions for the branching ratios to electron modes are consistent with the various observables; see Table III. Our prediction for B(B → Ke + e − ) [1.0−6.0] is somewhat higher than the measurement and this tension could become significant with a reduction in the theoretical and experimental uncertainties. The prediction for the inclusive mode B(B → X s e + e − ) [1.0−6.0] , which suffers from less hadronic uncertainties, is consistent with measurement.
Next we consider the case when Z ′ also has nonzero axial vector coupling with the electrons, i.e., g ee L = g ee R . Again, we either set g ′ bs = 0 or g bs = 0. The best-fit values shown in Table I satisfy the constraints on the NP couplings, and the R K and R K * values in all measured q 2 bins can be reasonably accommodated. The corresponding branching ratios with electron modes are provided in Table III.

IV. SUMMARY
In this work we have addressed the recent measurement of R K * with particular attention to the low q 2 bin, 0.045 ≤ q 2 ≤ 1.1 GeV 2 . This measurement has been difficult to explain with new physics above the GeV scale. For mediators in the 10 − 200 MeV mass range, we find: 1. A (pseudo)scalar that only couples to muons cannot explain the R K and R K * measurements as the predicted values are larger than in the SM, in conflict with experiment. An S coupling to only electrons can reproduce the R K [1.0−6.0] , R K * [0.045−1.1] and R K * [1.1−6.0] data, but the desired values of the couplings are not consistent with the measurements of the branching ratios B(B → K ( * ) e + e − ). A q 2 -dependent flavor changing b − s coupling to the scalar can produce compatibility with B(B → K ( * ) e + e − ) and gives a good fit to R K and R K * in the central q 2 bin, but the deviation of R K * from the SM in the low q 2 bin is small.

2.
A Z ′ with general vector and axial vector couplings to the muon and a q 2 -dependent b − s coupling provides a good fit to the combination of the three R K and R K * measurements, but does not fit R K * [0.045−1.1] well.
3. A Z ′ with general vector and axial vector couplings to the electron can explain R K and R K * data in all measured bins but the desired values of the couplings are not consistent with the measurements of B(B → K ( * ) e + e − ). However, a q 2 -dependent flavor changing b − s coupling to the vector is compatible with B(B → K ( * ) e + e − ) and gives good fits to R K and R K * ; of the cases we considered, the case with purely vector electron coupling provides the best agreement with the data with a pull of 4.8.