B-decay discrepancies after Moriond 2019

TL;DR

Recent measurements of B decay ratios suggest potential new physics, favoring models with lepton flavor universality violation, especially involving semi-leptonic Wilson coefficients and leptoquark scenarios, with some models nearing exclusion limits.

Contribution

The paper provides a comprehensive analysis of B decay anomalies post-Moriond 2019, highlighting the preference for semi-leptonic new physics contributions and the viability of the U_1 leptoquark model.

Findings

Data favors new physics in semi-leptonic Wilson coefficients.

A muonic contribution with C_9 = -C_10 is preferred.

LFU shifts in C_9 improve fit and can originate from high-scale operators.

Abstract

Following the updated measurement of the lepton flavour universality (LFU) ratio R_K in B -> Kll decays by LHCb, as well as a number of further measurements, e.g. R_K* by Belle and B_s -> mu mu by ATLAS, we analyse the global status of new physics in b -> s transitions in the weak effective theory at the b-quark scale, in the Standard Model effective theory at the electroweak scale, and in simplified models of new physics. We find that the data continues to strongly prefer a solution with new physics in semi-leptonic Wilson coefficients. A purely muonic contribution to the combination C_9 = -C_10, well suited to UV-complete interpretations, is now favoured with respect to a muonic contribution to C_9 only. An even better fit is obtained by allowing an additional LFU shift in C_9. Such a shift can be renormalization-group induced from four-fermion operators above the electroweak scale, in particular from semi-tauonic operators, able to account for the potential discrepancies in b -> c transitions. This scenario is naturally realized in the simplified U_1 leptoquark model. We also analyse simplified models where a LFU effect in b -> sll is induced radiatively from four-quark operators and show that such a setup is on the brink of exclusion by LHC di-jet resonance searches.

Figures

Tables

Table 1. Table 1: Best-fit values, 1 and 2 ​ σ 2 𝜎 2\sigma ranges, and pulls (cf. Eq. ( 12 )) between the best-fit point and the SM point for scenarios with NP in a single Wilson coefficient (or Wilson coefficient combination). For the scalar Wilson coefficients, we show the SM-like solution, while also a sign-flipped solution is allowed, see [ 66 ] .

Coeff.	best fit	$1\sigma$	$2\sigma$	pull
$C_9^{bs\mu \mu }$	$-0.97$	[$-1.12$, $-0.81$]	[$-1.27$, $-0.65$]	$5.9\sigma$
$C_9^{\prime bs\mu \mu }$	$+0.14$	[$-0.03$, $+0.32$]	[$-0.20$, $+0.51$]	$0.8\sigma$
$C_{10}^{bs\mu \mu }$	$+0.75$	[$+0.62$, $+0.89$]	[$+0.48$, $+1.03$]	$5.7\sigma$
$C_{10}^{\prime bs\mu \mu }$	$-0.24$	[$-0.36$, $-0.12$]	[$-0.49$, $+0.00$]	$2.0\sigma$
$C_9^{bs\mu \mu }=C_{10}^{bs\mu \mu }$	$+0.20$	[$+0.06$, $+0.36$]	[$-0.09$, $+0.52$]	$1.4\sigma$
$C_9^{bs\mu \mu }=-C_{10}^{bs\mu \mu }$	$-0.53$	[$-0.61$, $-0.45$]	[$-0.69$, $-0.37$]	$6.6\sigma$
$C_9^{bsee}$	$+0.93$	[$+0.66$, $+1.17$]	[$+0.40$, $+1.42$]	$3.5\sigma$
$C_9^{\prime bsee}$	$+0.39$	[$+0.05$, $+0.65$]	[$-0.27$, $+0.95$]	$1.2\sigma$
$C_{10}^{bsee}$	$-0.83$	[$-1.05$, $-0.60$]	[$-1.28$, $-0.37$]	$3.6\sigma$
$C_{10}^{\prime bsee}$	$-0.27$	[$-0.57$, $-0.02$]	[$-0.84$, $+0.26$]	$1.1\sigma$
$C_9^{bsee}=C_{10}^{bsee}$	$-1.49$	[$-1.79$, $-1.18$]	[$-2.05$, $-0.79$]	$3.2\sigma$
$C_9^{bsee}=-C_{10}^{bsee}$	$+0.47$	[$+0.33$, $+0.59$]	[$+0.20$, $+0.73$]	$3.5\sigma$
$\left(C_S^{bs\mu \mu }=-C_P^{bs\mu \mu }\right)\times \text{GeV}$	$-0.006$	[$-0.009$, $-0.003$]	[$-0.014$, $-0.001$]	$2.8\sigma$
$\left(C_S^{\prime bs\mu \mu }=C_P^{\prime bs\mu \mu }\right)\times \text{GeV}$	$-0.006$	[$-0.009$, $-0.003$]	[$-0.014$, $-0.001$]	$2.8\sigma$

Table 2. Table 3: Absolute χ 2 superscript 𝜒 2 \chi^{2} values and p 𝑝 p -values for the global fit and various subsets of observables.

Observables	$N_{\text{obs}}$	${\chi }^2$	$p$-value [%]
$b\to s\ell \ell$	127	126.7	49
$+R_{K^{(\ast )}}+D_{P_i^{\prime }}^{\mu e}$	138	149.9	23
$+B_s\to {\mu }^{+}{\mu }^{-}+b\to s\gamma$	143	155.9	22
$+\Delta F=2$	149	193.4	0.8
$+b\to c\tau \nu$	218	264.7	1.7
all quark flavour	258	301.9	3.1
all low-energy	276	308.6	8.6
global	313	361.4	3.1

Equations

$$R_K=\frac{\text{BR}(B\to K\mu \mu )}{\text{BR}(B\to Kee)}=0.846{}_{-0.054}^{+0.060}{}_{-0.014}^{+0.016},\text{for}1.1{\text{GeV}}^2<q^2<6{\text{GeV}}^2,$$

$$R_{K^{\ast }}=\frac{\text{BR}(B\to K^{\ast }\mu \mu )}{\text{BR}(B\to K^{\ast }ee)}=\{\begin{array}{l}0.90_{-0.21}^{+0.27}\pm 0.10,\text{for}0.1{\text{GeV}}^2<q^2<8{\text{GeV}}^2,\\ 1.18_{-0.32}^{+0.52}\pm 0.10,\text{for}15{\text{GeV}}^2<q^2<19{\text{GeV}}^2.\end{array}$$

$$R_{K^{\ast }}=\frac{\text{BR}(B\to K^{\ast }\mu \mu )}{\text{BR}(B\to K^{\ast }ee)}=\{\begin{array}{l}0.66_{-0.07}^{+0.11}\pm 0.03,\text{for}0.045{\text{GeV}}^2<q^2<1.1{\text{GeV}}^2,\\ 0.69_{-0.07}^{+0.11}\pm 0.05,\text{for}1.1{\text{GeV}}^2<q^2<6{\text{GeV}}^2,\end{array}$$

$${\mathcal{H}}_{\text{eff}}^{bs\ell \ell }={\mathcal{H}}_{\text{eff, SM}}^{bs\ell \ell }+{\mathcal{H}}_{\text{eff, NP}}^{bs\ell \ell },$$

$$\mathcal{H}_{\text{eff, NP}}^{bs\ell\ell}=-\mathcal{N}\bigg{(}C^{bs}_{7}O^{bs}_{7}+C^{\prime bs}_{7}O^{\prime bs}_{7}\\ +\sum_{\ell=e,\mu}\sum_{i=9,10,S,P}\left(C^{bs\ell\ell}_{i}O^{bs\ell\ell}_{i}+C^{\prime bs\ell\ell}_{i}O^{\prime bs\ell\ell}_{i}\right)\bigg{)}+\text{h.c.}\,,$$

$$\mathcal{N}=\frac{4G_F}{\sqrt{2}}{V}_{tb}{V}_{ts}^{\ast }\frac{e^2}{16{\pi }^2}.$$

$$O_7^{bs}$$

$$O_9^{bs\ell \ell }$$

$$O_{10}^{bs\ell \ell }$$

$$O_S^{bs\ell \ell }$$

$$O_P^{bs\ell \ell }$$

$$\text{pull}=\sqrt{\Delta {\chi }^2},\text{where}-\frac{1}{2}\Delta {\chi }^2=\ln L(\vec{0})-\ln L({\vec{C}}_{\text{best fit}}).$$

$$C_S^{bs\mu \mu }$$

$$C_9^{bs\mu \mu }$$

$$C_9^{bsee}$$

$$C_{10}^{bs\mu \mu }$$

$$C_{10}^{bsee}$$

$$2\mathcal{N}{C}_9^{bs{\ell }_i{\ell }_i}$$

$$2\mathcal{N}{C}_{10}^{bs{\ell }_i{\ell }_i}$$

$$c_Z=[C_{\phi q}^{(1)}{]}_{23}+[C_{\phi q}^{(3)}{]}_{23},$$

$$[O_{qe}{]}_{23ii}=({\bar{q}}_2{\gamma }_{\mu }{q}_3)({\bar{e}}_i{\gamma }^{\mu }{e}_i),$$

$$[O_{lq}^{(1)}{]}_{ii23}=({\bar{l}}_i{\gamma }_{\mu }{l}_i)({\bar{q}}_2{\gamma }^{\mu }{q}_3),$$

$$[O_{\phi q}^{(1)}{]}_{23}=({\phi }^{†}i{\overleftrightarrow{D}}_{\mu }\phi )({\bar{q}}_2{\gamma }^{\mu }{q}_3),$$

$$[O_{qq}^{(1)}{]}_{23ii}=({\bar{q}}_2{\gamma }_{\mu }{q}_3)({\bar{q}}_i{\gamma }^{\mu }{q}_i),$$

$$[O_{qu}^{(1)}{]}_{23ii}=({\bar{q}}_2{\gamma }_{\mu }{q}_3)({\bar{u}}_i{\gamma }^{\mu }{u}_i),$$

$$[C_{qu}^{(1)}{]}_{2311},$$

$${\mathcal{L}}_{U_1}\supset g_{lq}^{ji}\left({\bar{q}}^i{\gamma }^{\mu }{l}^j\right)U_{\mu }+\text{h.c.}$$

$$[C_{lq}^{(1)}{]}_{ijkl}=[C_{lq}^{(3)}{]}_{ijkl}=-\frac{g_{lq}^{jk}{g}_{lq}^{il\ast }}{2M_U^2}.$$

$$[O_{dB}{]}_{ij}=({\bar{q}}_i{\sigma }^{\mu \nu }{d}_j)\phi B_{\mu \nu },$$

$$[O_{dG}{]}_{ij}=({\bar{q}}_i{\sigma }^{\mu \nu }{T}^A{d}_j)\phi G_{\mu \nu }^A.$$

$${\mathcal{L}}_{U_1}\supset -\frac{1}{2}{U}_{\mu \nu }^{†}{U}^{\mu \nu }+i{g}_s{k}_s{U}_{\mu }^{†}{T}^A{U}_{\nu }{G}^{A,\mu \nu }+i{g}^{\prime }\frac{2}{3}{k}_Y{U}_{\mu }^{†}{U}_{\nu }{B}^{\mu \nu },$$

$$U_{\mu \nu }=D_{\mu }{U}_{\nu }-D_{\nu }{U}_{\mu }\text{with}{D}_{\mu }={\partial }_{\mu }+i{g}_s{T}^A{G}_{\mu }^A+i{g}^{\prime }\frac{2}{3}{B}_{\mu }.$$

$$[C_{dW}{]}_{23}$$

$$[C_{dB}{]}_{23}$$

$$[C_{dG}{]}_{23}=Y_b\frac{g_s}{16{\pi }^2}\left(-\frac{5}{12}\right)\frac{g_{lq}^{i2}{g}_{lq}^{i3\ast }}{M_U^2},[C_{dG}{]}_{32}=Y_s\frac{g_s}{16{\pi }^2}\left(-\frac{5}{12}\right)\frac{g_{lq}^{i3}{g}_{lq}^{i2\ast }}{M_U^2}.$$

$$g_{lq}^{32}=0.6,g_{lq}^{33}=0.7,$$

Full text

1 Introduction

In recent years, several deviations from Standard Model (SM) expectations have been building up in $B$-decay measurements. While each of them could be a first sign of physics beyond the SM, statistical fluctuations or underestimated experimental or theoretical systematic uncertainties cannot be excluded at present. These deviations – or “anomalies” – can be grouped into four categories that have very different experimental and theoretical challenges:

1. (i)

Apparent suppression of various branching ratios of exclusive decays based on the $b\to s\mu\mu$ flavour-changing neutral current (FCNC) transition [1, 2]. The uncertainties are dominated by the limited knowledge of the $B$ to light meson hadronic form factors [3, 4, 5]. 2. (ii)

Deviations from SM expectations in $B\to K^{*}\mu^{+}\mu^{-}$ angular observables [6, 7, 8, 9] (also based on the $b\to s\mu\mu$ transition), where form factor uncertainties are much smaller than for the branching ratios, but hadronic uncertainties are nevertheless significant [10, 11]. 3. (iii)

Apparent deviations from $\mu$-$e$ universality in $b\to s\ell\ell$ transitions in the processes $B\to K\ell\ell$ and $B\to K^{*}\ell\ell$ (via the $\mu/e$ ratios $R_{K}$ [12] and $R_{K^{*}}$ [13], respectively). Here the theoretical uncertainties are negligible [14] and the sensitivity is only limited by statistics at present. 4. (iv)

Apparent deviations from $\tau$-$\mu$ and $\tau$-$e$ universality in $b\to c\ell\nu$ transitions [15, 16, 17, 18, 19, 20, 21]. Uncertainties are dominated by statistics, with non-negligible experimental systematics but small theoretical uncertainties [22, 23, 24, 25]. (Note that $e$-$\mu$ universality in $b\to c\ell\nu$ transitions is tested to hold at the percent level [26, 27, 28].)

While the deviations in (i) and (ii) could be alleviated by more conservative assumptions on the hadronic uncertainties, it is tantalizing that a simple description in terms of a single non-standard Wilson coefficient of a semi-muonic operator like $(\bar{s}\gamma^{\rho}P_{L}b)(\bar{\mu}\gamma_{\rho}\mu)$ or $(\bar{s}\gamma^{\rho}P_{L}b)(\bar{\mu}\gamma_{\rho}P_{L}\mu)$ leads to a consistent description of (i), (ii), and (iii), with a best-fit point that improves the fit to the data by more than five standard deviations compared to the SM (for a single degree of freedom) [29, 30, 31, 32, 33, 34]. Moreover, it was shown that simplified models with a single tree-level mediator can not only explain (i), (ii), and (iii), but even all four categories of deviations simultaneously without violating any other existing constraints [35, 36, 37, 38, 39, 40].

Taken together, these observations explain the buzz of activity around these deviations and the anticipation of improved measurements of the theoretically clean ratios $R_{K}$ and $R_{K^{*}}$. The purpose of this article is to examine the status of the tensions after inclusion of a number of updated or newly available measurements, in particular:

• •

The new measurement of $R_{K}$ by the LHCb collaboration combining Run-1 data with 2 fb*-1* of Run-2 data (corresponding to about one third of the full Run-2 data set). The updated measurement finds [41]111In our numerical analysis, we use the full one-dimensional numerical likelihood provided in [41], which is markedly non-Gaussian, rather than symmetrizing the uncertainties in (1).

[TABLE]

where the first uncertainty is statistical and the second systematic, and $q^{2}$ is the dilepton invariant mass squared. The SM predicts lepton flavour universality, i.e. $R_{K}^{\text{SM}}$ is unity with uncertainties that are well below the current experimental sensitivities. While the updated experimental value is closer to the SM prediction than the Run-1 result [12], the reduced experimental uncertainties imply a tension between theory and experiment at the level of $2.5\sigma$, which is comparable to the situation before the update.

• •

The new, preliminary measurement of $R_{K^{*}}$ by Belle [42]. Averaged over $B^{\pm}$ and $B^{0}$ decays, the measured $R_{K^{*}}$ values at low and high $q^{2}$ are

[TABLE]

Given their sizable uncertainties, these values are compatible with both the SM predictions ($R_{K^{*}}^{\text{SM}}$ approximately unity) and previous results on $R_{K^{*}}$ from LHCb [13]

[TABLE]

that are in tension with the SM predictions by $\sim 2.5\sigma$ in both $q^{2}$ bins.

• •

One further, important piece of information included in our study is the 2018 measurement of $B_{s}\to\mu\mu$ by the ATLAS collaboration [43], that we combine with the existing measurements by CMS and LHCb [44, 45, 46].

In this paper we will explore the implications of all these, as well as other data, to be described in fuller detail in the next section, in the context of global fits to model-independent new physics scenarios, identify those that lead to a good description of the data, and discuss possible realizations in terms of simplified new-physics models.

Our numerical analysis is entirely based on open-source software, notably the global likelihood in Wilson coefficient space provided by the smelli package [47], built on flavio [48] and wilson [49]. As such, our analysis is easily reproducible and modifiable.

The rest of this work is organized as follows.

• •

In Section 2, we describe our statistical approach and the experimental measurements we employ in our numerical analysis.

• •

In Section 3, we perform a model-independent global analysis of $b\to s\ell\ell$ transitions, first in the weak effective theory (WET) below the electroweak (EW) scale, then in the SM effective field theory (SMEFT) above the EW scale, which allows us to extend the discussion to the charged-current deviations and to incorporate constraints from electroweak precision tests and other precision measurements.

• •

In Section 4, we discuss a number of specific simplified new-physics (NP) models that are favoured by the current data, assuming the deviations to be due to NP.

• •

Section 5 contains our conclusions.

2 Setup

Our numerical analysis is based on a global likelihood function in the space of the Wilson coefficients of the WET valid below the EW scale, or the SMEFT valid above it. Theoretical uncertainties (for observables where they cannot be neglected) are treated by computing a covariance matrix of theoretical uncertainties within the SM and combining it with the experimental uncertainties (approximated as Gaussian). The main assumption in this approach is that the sizes of theory uncertainties are weakly dependent on NP, which we checked for the observables included. This approach was first applied to $b\to s\ell\ell$ transitions in [50]. The theoretical uncertainties in exclusive $B$-decay observables stem mainly from hadronic form factors, which we take from [3] for $B$ to light vector meson transitions and from [5] for $B\to K$, as well as unknown non-factorizable effects that are parametrized as in [50, 3, 48] (and are compatible with more sophisticated approaches [10, 11]). Additional parametric uncertainties (e.g. from CKM matrix elements) are based on flavio v1.3 with default settings [48]. For more details on the statistical approach and the list of observables and measurements included, we refer the reader to [47].

Here we highlight the changes in observables sensitive to $b\to s$ transitions included with respect to the recent global analyses [29, 30] by some of us.

• •

We include the LHCb update of $R_{K}$ [41] (cf. footnote 1) and the new, preliminary measurement of $R_{K^{*}}$ by Belle [42]. The Belle results are available for various $q^{2}$-bin choices, separately for $B^{\pm}$ and $B^{0}$ decays and in an isospin averaged form. In our numerical analysis we use the $0.1\,$GeV${}^{2}<q^{2}<8\,$GeV2 and $15\,$GeV${}^{2}<q^{2}<19\,$GeV2 bins, separately for $B^{\pm}$ and $B^{0}$ decays.

• •

We include the new ATLAS measurement of $B_{s}\to\mu^{+}\mu^{-}$ [43], that we combine with the CMS and LHCb measurements [44, 45, 46]. This combination is discussed in detail in appendix A. Our combination is in slight tension with the SM prediction of BR$(B_{s}\to\mu^{+}\mu^{-})$ by approximately $2\sigma$.

• •

We include the updated LHCb measurement of forward-backward asymmetries in $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ [51] as well as its branching ratio [52]. For the theory predictions of the baryonic decay we follow [53, 54].

• •

Here we are working with the global likelihood described in [47] (i.e. including as many observables sensitive to the Wilson coefficients as possible), while in [29] we focused on observables sensitive to the $b\to s\ell\ell$ transition only. This means e.g. that we also include all the observables sensitive to the $b\to s\gamma,g$ dipole transitions studied in [55]. In addition, the global likelihood also includes observables that do not directly depend on the Wilson coefficients of interest but whose theory uncertainties are strongly correlated with those of the directly dependent observables. This is in particular relevant for the $b\to s\mu\mu$ observables. In our figures, we indicate the set of observables consisting of $b\to s\mu\mu$, $b\to s\gamma,g$, and other correlated observables as “$b\to s\mu\mu$ & corr. obs.”.

Like in the previous analysis [30] by some of us, we again include the LFU differences of angular observables222$D_{P_{4,5}^{\prime}}=P_{4,5}^{\prime}(B\to K^{*}\mu^{+}\mu^{-})-P_{4,5}^{\prime}(B\to K^{*}e^{+}e^{-})$. The observables $P_{4,5}^{\prime}$ are defined in [56]. $D_{P_{4}^{\prime}}$ and $D_{P_{5}^{\prime}}$. To this end, we have added them to the global likelihood in version 1.3.0 of the smelli package.

3 Model-independent numerical analysis

Having at hand the global likelihood in the space of NP Wilson coefficients, $L(\vec{C})$, we perform a model-independent numerical analysis by studying it in simple one- and two-coefficient scenarios. This analysis proceeds in two steps:

We first investigate the Wilson coefficients of the weak effective theory at the $b$-quark mass scale. This analysis can be seen as an update of earlier analyses (see e.g. [29, 30, 31, 32, 33, 34]) and is completely general, barring new particles lighter than the $b$ quark (see e.g. [57, 58, 59, 60]). 2. 2.

Next, we embed these results into the SMEFT at a scale $\Lambda$ above the electroweak scale. This is based on the additional assumptions that there are no new particles beneath $\Lambda$ and that EW symmetry breaking is approximately linear (see e.g. [61]). This allows us to correlate NP effects in $b\to s\ell\ell$ model-independently with other sectors like EW precision tests or $b\to c$ transitions (cf. [62, 63, 47, 64]).

3.1 $b\to s\ell\ell$ observables in the WET

We start by investigating the constraints on NP contributions to the $|\Delta B|=|\Delta S|=1$ Wilson coefficients of the WET at the $b$-quark scale $\mu_{b}\approx m_{b}$ that we take to be $4.8\,\text{GeV}$. We work with the effective Hamiltonian

[TABLE]

where the first term contains the SM contributions to the Wilson coefficients. The second term reads

[TABLE]

with the normalization factor

[TABLE]

The dipole operators are given by333The sign of the dipole coefficients $C_{7}^{(\prime)}$ are fixed by our convention for the covariant derivative $D_{\mu}\psi=\partial_{\mu}+ieQ_{\psi}A_{\mu}+ig_{s}T^{A}G^{A}_{\mu}$.

[TABLE]

where $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, and the semi-leptonic operators

[TABLE]

We have omitted from $\mathcal{H}_{\text{eff, NP}}^{bs\ell\ell}$ semi-leptonic tensor operators, which are not generated at dimension 6 in theories that have SMEFT as EW-scale limit, as well as chromomagnetic and four-quark operators. Even though the latter can contribute via one-loop matrix elements to $b\to s\ell\ell$ processes, their dominant effects typically stem from renormalization group (RG) evolution above the scale $\mu_{b}$, and we will discuss these effects in the SMEFT framework in the next section. For the same reason, we have constrained the sum over lepton flavours to $e$ and $\mu$: semi-tauonic WET operators can contribute via QED RG mixing, but their direct matrix elements are subleading [65].

3.1.1 Scenarios with a single Wilson coefficient