A light $Z^\prime$ for the $R_K$ puzzle and nonstandard neutrino interactions

We show that the $R_K$ puzzle in LHCb data and the discrepancy in the anomalous magnetic moment of the muon can be simultaneously explained if a 10 MeV mass $Z^\prime$ boson couples to the muon but not the electron, and that clear evidence of the nonstandard matter interactions of neutrinos induced by this coupling may be found at DUNE.

There are several perplexing anomalies related to the muon including its anomalous magnetic moment [1] and the charge radius of the proton extracted from muonic hydrogen [2]. In B physics, data from b → s decays indicate evidence of lepton flavor universality -the so called R K puzzle. The LHCb Collaboration has found a hint of lepton non-universality in the ratio R K ≡ B(B + → K + µ + µ − )/B(B + → K + e + e − ) = 0.745 ± 0.097 in the dilepton invariant mass-squared range 1 GeV 2 ≤ q 2 ≤ 6 GeV 2 [3]. We take the view that the R K puzzle may also be a consequence of new physics (NP) affecting the muon. There is also an anomaly in one of the angular observables in B → K * µ + µ − decay [4] which may be subject to large hadronic uncertainties [5]. However, unlike the R K puzzle lepton non-universal new physics is not necessary to explain the anomaly [6].
Here we focus on the R K puzzle which is a clean probe of the Standard Model (SM) due to very small hadronic uncertainties. Several NP models with heavy mediators have been considered to explain the R K puzzle. We consider a simple NP scenario with a Z lighter than the muon. The Z has flavor conserving coupling to quarks and leptons and in addition we assume that there is a flavor-changing bsZ vertex. The Z couplings to the lepton generations are non-universal to solve the R K puzzle. In particular we assume the Z has suppressed couplings to first generation leptons but has non-negligible couplings to second and third generation leptons. We constrain the bsZ coupling using B → Kνν and B s mixing and then from R K we fix the Z coupling to muons. We check that the coupling to muons is consistent with the muon a µ ≡ (g − 2) µ /2 measurement, ∆a µ ≡ a exp µ − a SM µ = (29±9)×10 −10 [7]. B → Kνν does not fix the Z coupling to neutrinos, but assuming SU(2) invariance we set the Z neutrino couplings to the charged lepton couplings. Estimates of the Z couplings to light quarks are obtained from non-leptonic b → sqq transitions where q = u, d, s. After we obtain the constraints from B physics, we study their implications for nonstandard neutrino interactions (NSI) at DUNE [8].
bsZ vertex. We assume there is a light Z with mass of order 10 MeV. The most general form of the bsZ vertex with vector type coupling is where the form factor F (q 2 ) can be expanded as where m B is the B meson mass and the momentum transfer q 2 m 2 B . The leading order term a bs is constrained by B → Kνν to be smaller than 10 −9 [9]. As will become clear below, the solution to the R K puzzle would then require the Z coupling to muons to be O(1) or larger which is in conflict with the (g − 2) µ measurement. The absence of flavor-changing neutral currents forces a bs ∼ 0, so that where g bs is assumed to be real. B → Kνν. Assuming Gaussian errors, the 95% C.L. upper limit for B → Kνν is [10] B(B → Kνν) ≤ 1.9 × 10 −5 .
From Ref. [11], the SM prediction is The SM Hamiltonian for each neutrino generation is where In the SM, the Wilson coefficient is determined by box and Z-penguin loop diagrams computation which gives, where the loop function X can be found e.g. in Ref. [12]. Now we introduce a Z coupling only to left-handed neutrinos. We further simplify by assuming only flavor conserving couplings but do not assume the couplings to be generation-independent. We write for generation α = µ, τ , Equations (3) and (8) lead to the Hamiltonian for b → sν ανα decays, We get B(B → Kνν) = 3.96 × 10 −6 for the SM. From Eq. (4) we obtain the 2σ constraint, Note that this constraint does not dependent on g νν as the NP contribution is dominated by the two body b → sZ transition. In principle, we can also consider B → K * νν but only certain helicity amplitudes are affected by NP. Furthermore at low q 2 the NP amplitudes are suppressed.
Hence this decay provides a weaker constraint than B → Kνν. B s mixing. Absent knowledge of F (q 2 ) for q 2 ∼ m 2 B , we assume that effects of the longitudinal polarization of the Z are compensated by the form factor so that the Hamiltonian responsible for B s mixing can be written as The correction to B s mixing is given by Using the vacuum insertion approximation [13] and the fact that m Bs ≈ m b + m s , The mass difference in the SM is given by where In the above, x t ≡ m 2 t /m 2 W , η Bs = 0.551 is the QCD correction [14] andB Bs is the bag parameter. Taking [16,17], and m t = 160 GeV [16,18], the SM prediction is [19] ∆M SM s = (17.4 ± 2.6) ps −1 .
This is to be compared with the experimental measurement [20], which is consistent with the SM prediction. To bound the NP coupling g bs we take the NP contribution to be at most the 1σ uncertainty in the SM contribution, i.e., ∆M N P s ∼ 2.6 ps −1 . With the B s decay constant f Bs from Ref. [21], and assuming m Bs m Z , Eq. (13) yields This is consistent with the bound obtained on g bs from B → Kνν. R K puzzle. Here we follow the discussions in Ref. [22,23]. Within the SM, the effective Hamiltonian for the quark-level transition b → sµ + µ − is [24] where correspond to the P i in Ref. [25], and m b = m b (µ) is the running b-quark mass in the MS scheme. We use the SM Wilson coefficients as given in Ref. [26]. Introducing a Z coupling to leptons Equations (3) and (19) lead to the Hamiltonian for b → s decays We can rewrite this as, where We assume the Z does not couple to electrons and so B(B + → K + e + e − ) is described by the SM, while B(B + → K + µ + µ − ) is modified by NP. We scan the parameter space of g bs and g µµ for values that are consistent with the experimental measurement of R K ; see Fig. 1.
Muon magnetic moment. The light Z also explains the discrepancy in the muon magnetic moment measurement. From Ref. [27], we have For m Z = 10 MeV, the measured value of ∆a µ gives g µµ as in Fig. 1. Other constraints. We now check that the result is consistent with other b → sµ + µ − transitions. Note that our light Z cannot be produced as a resonance in b → sµ + µ − decays. Also, as we have a vector coupling in Eq. (21) there is no contribution toB 0 s → µ + µ − . The BaBar Collaboration measures B(B → X s µ + µ − ) = (0.66 ± 0.88) × 10 −6 in the range 1 GeV 2 ≤ q 2 ≤ 6 GeV 2 [28]. The differential branching ratio forB 0 d → X s µ + µ − with SM and the general NP operators can be found in Ref. [22]. We find that the NP contribution to B(B 0 d → X s µ + µ − ) is only 7% of the SM prediction for 1 GeV 2 ≤ q 2 ≤ 6 GeV 2 . Given the current experimental uncertainties, the constraint from this decay is not stringent.
The branching fractions forB 0 d →Kµ + µ − ,B 0 d → K * µ + µ − and the corresponding electron modes are known for the entire kinematical range. However due to the long distance contributions we do not use them to directly constrain NP.
Finally, the NP amplitude forB 0 d →K * µ + µ − in the low q 2 region is suppressed relative to the leading SM amplitudes by √ q 2 m B and so this decay does not provide any constraints on the NP coupling. We note in passing that constraints from b → sτ + τ − decays are very weak [19] and do not produce a meaningful constraint on the NP coupling g τ τ .
b → sqq. We now consider the Z coupling to light quarks with a focus on the up and down quarks: It is reasonable for the Z coupling to quarks to be of the same size as the coupling to the charged leptons, i.e., ∼ 10 −4 . Decays like B → Kπ can constrain the Z coupling to light quarks. In spite of the hadronic uncertainties approximate bounds are obtainable from these decays. Equations (3) and (24) lead to the Hamiltonian for b → sqq decays, which is similar to Eq. (20) with replaced by q.
The NP can add to the electroweak contribution in the SM. It is interesting to speculate if such NP can resolve the so called K − π puzzle [29]. This is the difference in the direct CP asymmetry in the decays B + → π 0 K + and B 0 → π − K + . It is puzzling that the leading amplitudes in both decays are the same in the SM while the former decay also gets contributions from a small color and CKM suppressed tree amplitude and the electroweak penguins for the two decays are different. It is possible that new contributions to the electroweak penguins may resolve the puzzle. However, the situation is a bit complicated. First, there are two other relevant decays, B + → π + K 0 and B 0 → π 0 K 0 , and one has to fit to all the decays. Since these are non-leptonic decays one has to account for hadronic uncertainties.
In naive factorization, our NP does not contribute at leading order to B + → π 0 K + as the vector quark current does not produce a pion but can produce a ρ and will thus contribute to B + → ρ 0 K + and B 0 → ρ − K + . We can always change the chiral structure of the Z coupling to quarks to get a leading order contribution to B → πK. Our intention here is not to resolve the K − π puzzle but we can estimate the Z qq coupling in the following way. A reasonable assumption is that NP produces effects of the size of about 10% of the SM electroweak penguin.
Both color allowed and color suppressed electroweak penguins are possible in the decay B 0 → ρ 0 K 0 , and we can compare these with the NP amplitude. The ratio of the NP amplitude to the color allowed penguin is where H SM EW is the color allowed SM electroweak Hamiltonian. Using naive factorization, where we have assumed real couplings. The factor a 9 = C 9 + C10 Nc where C 9,10 are the Wilson's coefficients and N c = 3 is the number of colors. The ratio of the NP amplitude to the color suppressed electroweak penguin is −1.22α em and a 10 (µ = m b ) = 0.04α em [30] and requiring |r| ∼ 0.1 we find For |g bs | ∼ 10 −5 we get |g uu − g dd | ∼ 10 −3 . As we discuss next, this leads to nonstandard neutrino interactions that are too large. On the other hand requiring |s| ∼ 0.1 gives In this case |g uu − g dd | ∼ 10 −5 . We will assume that g uu is the same size as g dd and take these couplings to be ∼ 10 −5 to discuss neutrino NSI. NSI at DUNE. The light Z couplings to neutrinos and first generation quarks affect the neutrino propagation in matter. The matter NSI can be parameterized by the effective Lagrangian [31], where α = µ, τ , C = L, R, q = u, d, and qC αα are dimensionless parameters that represent the strength of the new interaction in units of G F . Since neutrino propagation in matter is affected by coherent forward scattering, q αα ≡ qL αα + qR αα , can be written as regardless of the Z mass. For propagation in the earth, neutrino oscillation experiments are only sensitive to the combination, We now use the light Z couplings obtained from B physics to study signatures at neutrino oscillation experiments. We assume g νµνµ = g µµ , which is motivated by an SU(2) invariant realization of Eq. (30). We fix g µµ = 5.4 × 10 −4 and g bs = 1.3 × 10 −5 to explain both the R K and muon g − 2 anomalies; this set of couplings is marked by a cross in Fig. 1. To avoid a finetuned cancellation, we take g uu = 1.2 × 10 −5 and g dd = −1.0 × 10 −5 , which satisfies the relation in Eq. (29). For m Z = 10 MeV, these couplings satisfy a plethora of constraints [32]. From Eqs. (31) and (32), we get µµ = 1.0. To satisfy constraints from current neutrino oscillation data [33], we assume τ τ = µµ .
Following the procedure in Ref. [34], we simulate 300 kt-MW-years of DUNE data with the normal neutrino mass hierarchy, the neutrino CP phase δ = 0, and µµ = 1.0. We scan over both the mass hierarchies, the neutrino oscillation parameters and µµ . The expected sensitivity of DUNE to reject the SM scenario is shown in Fig. 2. We see that the SM scenario with µµ = 0 is ruled out at the 3.6σ C.L. at DUNE.
Summary. We showed that the R K puzzle in LHCb data can be explained by a light Z . The resulting coupling of the Z to muons also reconciles the muon g−2 measurement. After carefully examining various constraints from B physics, we find that this Z could yield large NSI in neutrino propagation. We further demonstrated that evidence of NSI induced by the light Z coupling may be found at DUNE. A scattering experiment at CERN will also search for such a boson [35].