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

TL;DR

This paper investigates light scalar and vector mediators around 25 MeV as potential explanations for anomalies in lepton universality measurements $R_K$ and $R_{K^*}$, proposing models that fit the data better than previous explanations.

Contribution

It introduces light scalar and vector mediator models with specific couplings that can simultaneously explain the $R_K$ and $R_{K^*}$ anomalies and other measurements, addressing the low $q^2$ challenge.

Findings

A 25 MeV $Z'$ with $q^2$-dependent $b-s$ coupling and electron coupling explains all anomalies.

A 25 MeV scalar provides a good fit except for low $q^2$ $R_{K^*}$.

A 25 MeV $Z'$ with muon coupling fits $R_K$ and $R_{K^*}$ data but not low $q^2$ $R_{K^*}$.

Abstract

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.

Tables

Table 1. Table 1: The fit results and the predictions for R K subscript 𝑅 𝐾 R_{K} and R K ∗ subscript 𝑅 superscript 𝐾 R_{K^{*}} at the best fit point for three scenarios of a light mediator with a mass of 25 25 25  MeV.

Case			$R_{K^{\ast }}^{}{}_{\left[0.045-1.1\right]}{}^{}$	$R_{K^{\ast }}^{}{}_{\left[1.1-6.0\right]}{}^{}$	$R_K^{}{}_{\left[1.0-6.0\right]}{}^{}$	pull
Experimental results			$0.66\pm 0.09$	$0.69\pm 0.10$	$0.75\pm 0.09$
Standard model predictions			0.93	0.99	1.0
(i) Light scalar with electron coupling
$F\left(q^2\right)\equiv 1$, $g_{ee}^S=2.0\times {10}^{-4}$	$g_{bs}^S{g}_{ee}^S=\left(12.6\pm 2.2\right)\times {10}^{-9}$	$g_{bs}^{S^{\prime }}{g}_{ee}^S=\left(4.0\pm 1.6\right)\times {10}^{-9}$	0.70	0.91	0.69	4.3
$a_{bs}\ne 0$	$g_{bs}^S{g}_{ee}^S=\left(-1.3\pm 2.1\right)\times {10}^{-9}$	$g_{bs}^{S^{\prime }}{g}_{ee}^S=\left(-13.1\pm 2.1\right)\times {10}^{-9}$	0.58	0.85	0.75	4.7
$a_{bs}=0$	$g_{bs}^S{g}_{ee}^S=\left(2.7\pm 2.6\right)\times {10}^{-8}$	$g_{bs}^{S^{\prime }}{g}_{ee}^S=\left(-15.5\pm 2.6\right)\times {10}^{-8}$	0.89	0.65	0.75	4.4
(ii) Light vector with muon coupling
$F\left(q^2\right)\equiv 1$, $g_L^{\mu \mu }=g_R^{\mu \mu }=8.0\times {10}^{-4}$	$g_{bs}{g}_{\mu \mu }=\left(2.3\pm 2.0\right)\times {10}^{-10}$	$g_{bs}^{\prime }{g}_{\mu \mu }=\left(1.3\pm 2.2\right)\times {10}^{-10}$	0.93	0.99	0.96	1.4
$a_{bs}\ne 0$, $g_L^{\mu \mu }=g_R^{\mu \mu }$	$g_{bs}{g}_{\mu \mu }=\left(6.5\pm 3.5\right)\times {10}^{-10}$	$g_{bs}^{\prime }{g}_{\mu \mu }=\left(1.6\pm 3.6\right)\times {10}^{-10}$	0.93	0.96	0.92	2.4
$a_{bs}\ne 0$, $g_{bs}^{\prime }=0$, $g_L^{\mu \mu }\ne g_R^{\mu \mu }$	$g_{bs}{g}_{\mu \mu }=\left(5.7\pm 2.3\right)\times {10}^{-10}$	$g_{bs}{g}_{\mu \mu }^{\prime }=\left(0.2\pm 0.1\right)\times {10}^{-11}$	0.89	0.95	0.93	2.9
$a_{bs}\ne 0$, $g_{bs}=0$, $g_L^{\mu \mu }\ne g_R^{\mu \mu }$	$g_{bs}^{\prime }{g}_{\mu \mu }=\left(-3.2\pm 2.5\right)\times {10}^{-10}$	$g_{bs}^{\prime }{g}_{\mu \mu }^{\prime }=\left(-0.1\pm 0.1\right)\times {10}^{-11}$	0.85	0.97	1.05	1.6
$a_{bs}=0$, $g_L^{\mu \mu }=g_R^{\mu \mu }$	$g_{bs}{g}_{\mu \mu }=\left(4.4\pm 1.4\right)\times {10}^{-8}$	$g_{bs}^{\prime }{g}_{\mu \mu }=\left(-1.9\pm 1.4\right)\times {10}^{-8}$	0.86	0.72	0.76	4.6
$a_{bs}=0$, $g_{bs}^{\prime }=0$, $g_L^{\mu \mu }\ne g_R^{\mu \mu }$	$g_{bs}{g}_{\mu \mu }=\left(3.9\pm 1.8\right)\times {10}^{-8}$	$g_{bs}{g}_{\mu \mu }^{\prime }=\left(4.4\pm 4.2\right)\times {10}^{-11}$	0.87	0.80	0.69	4.4
$a_{bs}=0$, $g_{bs}=0$, $g_L^{\mu \mu }\ne g_R^{\mu \mu }$	$g_{bs}^{\prime }{g}_{\mu \mu }=\left(-0.5\pm 5.6\right)\times {10}^{-9}$	$g_{bs}^{\prime }{g}_{\mu \mu }^{\prime }=\left(0.0\pm 1.5\right)\times {10}^{-11}$	0.92	0.99	1.01	0.1
(iii) Light vector with electron coupling
$F\left(q^2\right)\equiv 1$, $g_L^{ee}=g_R^{ee}=2.5\times {10}^{-4}$	$g_{bs}{g}_{ee}=\left(-0.6\pm 1.0\right)\times {10}^{-10}$	$g_{bs}^{\prime }{g}_{ee}=\left(-0.4\pm 1.1\right)\times {10}^{-10}$	0.93	0.99	0.99	0.7
$a_{bs}\ne 0$, $g_L^{ee}=g_R^{ee}$	$g_{bs}{g}_{ee}=\left(-1.9\pm 0.6\right)\times {10}^{-9}$	$g_{bs}^{\prime }{g}_{ee}=\left(-0.8\pm 0.5\right)\times {10}^{-9}$	0.62	0.92	0.74	4.5
$a_{bs}\ne 0$, $g_{bs}^{\prime }=0$, $g_L^{ee}\ne g_R^{ee}$	$g_{bs}{g}_{ee}=\left(-4.4\pm 5.9\right)\times {10}^{-10}$	$g_{bs}{g}_{ee}^{\prime }=\left(7.5\pm 3.3\right)\times {10}^{-10}$	0.55	0.86	0.84	4.5
$a_{bs}\ne 0$, $g_{bs}=0$, $g_L^{ee}\ne g_R^{ee}$	$g_{bs}^{\prime }{g}_{ee}=\left(3.9\pm 4.2\right)\times {10}^{-10}$	$g_{bs}^{\prime }{g}_{ee}^{\prime }=\left(12.4\pm 2.6\right)\times {10}^{-10}$	0.58	0.98	0.81	4.0
$a_{bs}=0$, $g_L^{ee}=g_R^{ee}$	$g_{bs}{g}_{ee}=\left(-3.9\pm 1.0\right)\times {10}^{-8}$	$g_{bs}^{\prime }{g}_{ee}=\left(1.4\pm 1.0\right)\times {10}^{-8}$	0.78	0.60	0.75	4.8
$a_{bs}=0$, $g_{bs}^{\prime }=0$, $g_L^{ee}\ne g_R^{ee}$	$g_{bs}{g}_{ee}=\left(-3.2\pm 2.3\right)\times {10}^{-8}$	$g_{bs}{g}_{ee}^{\prime }=\left(0.4\pm 1.4\right)\times {10}^{-8}$	0.83	0.70	0.67	4.6
$a_{bs}=0$, $g_{bs}=0$, $g_L^{ee}\ne g_R^{ee}$	$g_{bs}^{\prime }{g}_{ee}=\left(4.6\pm 1.5\right)\times {10}^{-8}$	$g_{bs}^{\prime }{g}_{ee}^{\prime }=\left(2.0\pm 0.3\right)\times {10}^{-8}$	0.80	0.58	0.77	4.7

Table 2. Table 2: Constraints from ℬ ​ ( B 0 → K 0 ​ e + ​ e − ) ℬ → superscript 𝐵 0 superscript 𝐾 0 superscript 𝑒 superscript 𝑒 \mathcal{B}(B^{0}\rightarrow K^{0}e^{+}e^{-}) and ℬ ​ ( B 0 → K ∗ 0 ​ e + ​ e − ) ℬ → superscript 𝐵 0 superscript 𝐾 absent 0 superscript 𝑒 superscript 𝑒 \mathcal{B}(B^{0}\rightarrow K^{*0}e^{+}e^{-}) . See the text for details.

	$ℬ\left(B^0\to K^0{e}^{+}{e}^{-}\right)$	$ℬ\left(B^0\to K^{\ast 0}{e}^{+}{e}^{-}\right)$	Combined
$S$, $a_{bs}\ne 0$	$\left|g_{bs}^S+g_{bs}^{S^{\prime }}\right|\lesssim 9.9\times {10}^{-7}$	$\left|g_{bs}^S-g_{bs}^{S^{\prime }}\right|\lesssim 9.0\times {10}^{-7}$	$\left|g_{bs}^S\right|,\left|g_{bs}^{S^{\prime }}\right|\lesssim 9.5\times {10}^{-7}$
$S$, $a_{bs}=0$	$\left|g_{bs}^S+g_{bs}^{S^{\prime }}\right|\lesssim 4.4\times {10}^{-2}{\left(\frac{25\mathrm{MeV}}{M_S}\right)}^2$	$\left|g_{bs}^S-g_{bs}^{S^{\prime }}\right|\lesssim 4.0\times {10}^{-2}{\left(\frac{25\mathrm{MeV}}{M_S}\right)}^2$	$\left|g_{bs}^S\right|,\left|g_{bs}^{S^{\prime }}\right|\lesssim 4.2\times {10}^{-2}{\left(\frac{25\mathrm{MeV}}{M_S}\right)}^2$
$Z^{\prime }$, $a_{bs}\ne 0$	$\left|g_{bs}+g_{bs}^{\prime }\right|\lesssim 5.8\times {10}^{-9}\left(\frac{M_{Z^{\prime }}}{25\mathrm{MeV}}\right)$	$\left|g_{bs}-g_{bs}^{\prime }\right|\lesssim 5.4\times {10}^{-9}\left(\frac{M_{Z^{\prime }}}{25\mathrm{MeV}}\right)$	$\left|g_{bs}\right|,\left|g_{bs}^{\prime }\right|\lesssim 5.6\times {10}^{-9}\left(\frac{M_{Z^{\prime }}}{25\mathrm{MeV}}\right)$
$Z^{\prime }$, $a_{bs}=0$	$\left|g_{bs}+g_{bs}^{\prime }\right|\lesssim 2.6\times {10}^{-4}\left(\frac{25\mathrm{MeV}}{M_{Z^{\prime }}}\right)$	$\left|g_{bs}-g_{bs}^{\prime }\right|\lesssim 2.4\times {10}^{-4}\left(\frac{25\mathrm{MeV}}{M_{Z^{\prime }}}\right)$	$\left|g_{bs}\right|,\left|g_{bs}^{\prime }\right|\lesssim 2.5\times {10}^{-4}\left(\frac{25\mathrm{MeV}}{M_{Z^{\prime }}}\right)$

Table 3. Table 3: The experimental results for various b → s ​ e + ​ e − → 𝑏 𝑠 superscript 𝑒 superscript 𝑒 b\to se^{+}e^{-} observables, along with predictions for the SM and four new physics cases that fit the R K subscript 𝑅 𝐾 R_{K} and R K ∗ subscript 𝑅 superscript 𝐾 R_{K^{*}} data and satisfy the ℬ ​ ( B → K ( ∗ ) ​ e + ​ e − ) ℬ → 𝐵 superscript 𝐾 superscript 𝑒 superscript 𝑒 \mathcal{B}(B\rightarrow K^{(*)}e^{+}e^{-}) constraints. The light mediator mass is 25 MeV, F ​ ( q 2 ) ≠ 1 𝐹 superscript 𝑞 2 1 F(q^{2})\neq 1 and a b ​ s = 0 subscript 𝑎 𝑏 𝑠 0 a_{bs}=0 .

	$R_K^{}{}_{\left[0.045-1.0\right]}{}^{}$	$ℬ{\left(B\to K{e}^{+}{e}^{-}\right)}_{\left[1.0-6.0\right]}$	$ℬ{\left(B\to X_s{e}^{+}{e}^{-}\right)}_{\left[1.0-6.0\right]}$	$B{\left(B^0\to K^{\ast 0}{e}^{+}{e}^{-}\right)}_{\left[{0.03}^2-1\right]}$
Experimental results	-	$\left(1.56\pm 0.18\right)\times {10}^{-7}$ RKexpt	$\left(1.93\pm 0.55\right)\times {10}^{-6}$ Lees:2013nxa	$\left(3.1\pm 0.9\right)\times {10}^{-7}$ Aaij:2013hha
Standard model predictions	0.98	$1.69\times {10}^{-7}$	$1.74\times {10}^{-6}$	$2.6\times {10}^{-7}$
Light scalar $g_{bs}^S{g}_{ee}^S=2.7\times {10}^{-8}$,  $g_{bs}^{S^{\prime }}{g}_{ee}^S=-15.5\times {10}^{-8}$	0.93	$2.5\times {10}^{-7}$	$2.3\times {10}^{-6}$	$2.6\times {10}^{-7}$
Light vector $g_{bs}{g}_{ee}=-3.9\times {10}^{-8}$, $g_{bs}^{\prime }{g}_{ee}=1.4\times {10}^{-8}$	0.73	$2.4\times {10}^{-7}$	$2.6\times {10}^{-6}$	$2.8\times {10}^{-7}$
Light vector, $g_{bs}^{\prime }=0$ $g_{bs}{g}_{ee}=-3.2\times {10}^{-8}$, $g_{bs}{g}_{ee}^{\prime }=0.4\times {10}^{-8}$	0.66	$2.7\times {10}^{-7}$	$2.5\times {10}^{-6}$	$2.7\times {10}^{-7}$
Light vector, $g_{bs}=0$ $g_{bs}^{\prime }{g}_{ee}=4.6\times {10}^{-8}$,  $g_{bs}^{\prime }{g}_{ee}^{\prime }=2.0\times {10}^{-8}$	1.04	$2.4\times {10}^{-7}$	$2.5\times {10}^{-6}$	$2.8\times {10}^{-7}$

Equations

$$R_K^{\mathrm{expt}}=0.745_{-0.074}^{+0.090}(\mathrm{stat})\pm 0.036(\mathrm{syst}),1\le q^2\le 6.0{\mathrm{GeV}}^2.$$

$$R_{K^{*}}^{\rm expt}=\left\{\begin{array}[]{cc}0.660^{+0.110}_{-0.070}~{}{\rm(stat)}\pm 0.024~{}{\rm(syst)}~{},{}{}&0.045\leq q^{2}\leq 1.1~{}{\rm GeV}^{2}~{},\ \ \ (\rm{low}\ q^{2})\\ 0.685^{+0.113}_{-0.069}~{}{\rm(stat)}\pm 0.047~{}{\rm(syst)}~{},{}{}&1.1\leq q^{2}\leq 6.0~{}{\rm GeV}^{2}~{},\ \ \ (\rm{central}\ q^{2})\,.\end{array}\right.$$

$$F(q^2)\bar{s}\left[g_{bs}^S{P}_L+g_{bs}^{S^{\prime }}{P}_R\right]bS,$$

$$\mathcal{L}$$

$$M_{b\to s{\ell }^{+}{\ell }^{-}}$$

$$\Delta M_{B_s}^{NP}$$

$$\Delta M_{B_s}^{SM}=(17.4\pm 2.6){\mathrm{ps}}^{-1},$$

$$\Delta M_{B_s}=(17.757\pm 0.021){\mathrm{ps}}^{-1}.$$

$$\displaystyle M_{b\to se^{+}e^{-}}^{S,S^{\prime}}=\frac{g_{ee}^{S}}{q^{2}-M_{S}^{2}}F(q^{2})\left[g_{bs}^{S}(\bar{s}P_{L}b)+{g_{bs}^{S^{\prime}}}(\bar{s}P_{R}b)\right](\bar{e}e)+\frac{g_{ee}^{S^{\prime}}}{q^{2}-M_{S}^{2}}F(q^{2})\left[g_{bs}^{S}(\bar{s}P_{L}b)+{g_{bs}^{S^{\prime}}}(\bar{s}P_{R}b)\right](\bar{e}\gamma_{5}e)\,,\$$

$${\chi }^2(g_{bs}^S,g_{bs}^{S^{\prime }})=\sum _{R_K,R_{K\ast },\Delta M_s^{NP}}({\mathcal{O}}_{th}(g_{bs}^S,g_{bs}^{S^{\prime }})-{\mathcal{O}}_{exp}{)}^T{\mathcal{C}}^{-1}({\mathcal{O}}_{th}(g_{bs}^S,g_{bs}^{S^{\prime }})-{\mathcal{O}}_{exp}),$$

$${\chi }_{\text{SM}}^2/\text{dof}=25.5/3.$$

$$\mathcal{B}(B\to KS)=\frac{(g_{bs}^S+g_{bs}^{S^{\prime }}{)}^2|{\vec{p}}_K|(m_B^2-m_K^2{)}^2{f}_0^2(m_S^2/m_B^2){\tau }_B}{32\pi m_b^2{m}_B^2},$$

$$\mathcal{B}(B\to K^{\ast }S)=\frac{(g_{bs}^S-g_{bs}^{S^{\prime }}{)}^2|{\vec{p}}_{K^{\ast }}{|}^3{A}_0^2(m_S^2){\tau }_B}{8\pi m_b^2},$$

$$F(q^2)$$

$$\displaystyle M_{b\to se^{+}e^{-}}^{S,S^{\prime}}=\frac{q^{2}}{m_{B}^{2}}\frac{g_{ee}^{S}}{q^{2}-M_{S}^{2}}\left[g_{bs}^{S}(\bar{s}P_{L}b)+{g_{bs}^{S^{\prime}}}(\bar{s}P_{R}b)\right](\bar{e}e)+\frac{q^{2}}{m_{B}^{2}}\frac{g_{ee}^{S^{\prime}}}{q^{2}-M_{S}^{2}}\left[g_{bs}^{S}(\bar{s}P_{L}b)+{g_{bs}^{S^{\prime}}}(\bar{s}P_{R}b)\right](\bar{e}\gamma_{5}e)\,.\$$

$$F(q^2)\bar{s}{\gamma }^{\mu }\left[g_{bs}{P}_L+g_{bs}^{\prime }{P}_R\right]b{Z}_{\mu }^{\prime }.$$

$$M_{b\to s{\ell }^{+}{\ell }^{-}}$$

$$\Delta M_{B_s}^{NP}$$

$$\mathcal{B}(B\to K{Z}^{\prime })=\frac{|g_{bs}+g_{bs}^{\prime }{|}^2}{64\pi }\frac{m_B^2{\beta }_{BK{Z}^{\prime }}^3}{M_{Z^{\prime }}^2{\Gamma }_B}{\left[f_{+}^{BK}(M_{Z^{\prime }}^2)\right]}^2,$$

$$\mathcal{B}(B\to K^{\ast }{Z}^{\prime })=\frac{{\beta }_{B{K}^{\ast }Z}}{16\pi m_B{\Gamma }_B}\left(|H_0{|}^2+|H_{+}{|}^2+|H_{-}{|}^2\right),$$

$$H_0=(g_{bs}-g_{bs}^{\prime })\left[-\frac{1}{2}(m_B+M_{K^{\ast }})\xi A_1(M_{Z^{\prime }}^2)+\frac{M_{K^{\ast }}{M}_{Z^{\prime }}}{m_B+M_{K^{\ast }}}\sqrt{{\xi }^2-1}{A}_2(M_{Z^{\prime }}^2)\right],$$

$$H_{\pm }=\frac{1}{2}(g_{bs}-g_{bs}^{\prime })\left[(m_B+M_{K^{\ast }})A_1(M_{Z^{\prime })}^2\right]\pm (g_{bs}+g_{bs}^{\prime })\frac{M_{K^{\ast }}{M}_{Z^{\prime }}}{m_B+M_{K^{\ast }}}\sqrt{{\xi }^2-1}V(M_{Z^{\prime }}^2).$$

Full text

I Introduction

The search for new physics in $B$ decays is an ongoing endeavor. Recently, anomalies in semileptonic $B$ decays have received a lot of attention. These anomalies are found in the charged current $b\to c\tau^{-}{\bar{\nu}}_{\tau}$ and neutral current $b\to s\ell^{+}\ell^{-}$ transitions. Here we focus on the neutral current anomalies though the anomalies might be related datta_shiv . Other anomalies appear in $B\to K^{*}\mu^{+}\mu^{-}$ where the LHCb BK*mumuLHCb1 ; BK*mumuLHCb2 and Belle BK*mumuBelle Collaborations find deviations from the Standard model (SM) predictions, particularly in the angular observable $P^{\prime}_{5}$ P'5 . The ATLAS BK*mumuATLAS and CMS BK*mumuCMS Collaborations have also made measurements of the $B\to K^{*}\mu^{+}\mu^{-}$ angular distribution with results consistent with LHCb. Further, the LHCb has made measurements of the branching ratios and angular distributions in $B_{s}^{0}\to\phi\mu^{+}\mu^{-}$ BsphimumuLHCb1 ; BsphimumuLHCb2 which are at variance with SM predictions based on lattice QCD latticeQCD1 ; latticeQCD2 and QCD sum rules QCDsumrules .

The measurements discussed above are subject to unknown hadronic uncertainties silves making it necessary to construct clean observables to test for new physics (NP). One such observable is $R_{K}\equiv{\cal B}(B^{+}\to K^{+}\mu^{+}\mu^{-})/{\cal B}(B^{+}\to K^{+}e^{+}e^{-})$ hiller1 ; hiller2 , which has been measured by LHCb RKexpt :

[TABLE]

This differs from the SM prediction, $R_{K}^{\rm SM}=1\pm 0.01$ IsidoriRK by $2.6\sigma$. Note, the observable $R_{K}$ is a measure of lepton flavor universality and requires different new physics for the muons versus the electrons, while it is possible to explain the anomalies in the angular observables in $b\to s\mu^{+}\mu^{-}$ in terms of lepton flavor universal new physics Datta:2013kja .

Recently, the LHCb Collaboration reported the measurement of the ratio $R_{K^{*}}\equiv{\cal B}(B^{0}\to K^{*0}\mu^{+}\mu^{-})/{\cal B}(B^{0}\to K^{*0}e^{+}e^{-})$ in two different ranges of the dilepton invariant mass-squared $q^{2}$ RK*expt :

[TABLE]

These differ from the SM predictions by 2.2-2.4$\sigma$ (low $q^{2}$) and 2.4-2.5$\sigma$ (central $q^{2}$), which further strengthens the hint of lepton non-universality observed in $R_{K}$.

Lepton universality violating new physics may occur in $b\to s\mu^{+}\mu^{-}$ and/or $b\to se^{+}e^{-}$ transitions. The fact that the measurement of ${\cal B}(B^{+}\to K^{+}e^{+}e^{-})$ is found to be consistent with the prediction of the SM may lead one to conclude that NP is more likely to be in $b\to s\mu^{+}\mu^{-}$. However, the branching ratios suffer from hadronic uncertainties bslltheorerror unlike the ratios $R_{K}$ and $R_{K^{*}}$ and so new physics in $b\to s\mu^{+}\mu^{-}$ and/or in $b\to se^{+}e^{-}$ is still allowed.

Since the announcement of the $R_{K^{*}}$ result, a number of papers have analyzed the new measurements, mostly in terms of new physics with heavy mediators Capdevila:2017bsm ; Altmannshofer:2017yso ; DAmico:2017mtc ; Hiller:2017bzc ; Geng:2017svp ; Ciuchini:2017mik ; Celis:2017doq ; Becirevic:2017jtw ; DiChiara:2017cjq ; Sala:2017ihs ; Ghosh:2017ber ; datta_rk ; jacky_rk ; wang ; Bonilla:2017lsq . The general conclusion is that there is a significant disagreement with the SM, possibly as large as $\sim 6\sigma$, and that theoretical hadronic uncertainties BK*mumuhadunc1 ; BK*mumuhadunc2 ; BK*mumuhadunc3 are insufficient to understand the data. However, with heavy new physics it is difficult to understand the $R_{K^{*}}$ measurement in the very low $q^{2}$ bin $0.045\leq q^{2}\leq 1.1~{}{\rm GeV}^{2}$, although the predictions are consistent with measurements within $1.5\sigma$. A resolution to this problem may be possible if the new physics is light.

In models with light mediators ZMeV ; Sala:2017ihs ; Ghosh:2017ber ; datta_rk ; Bishara:2017pje , 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 $2m_{\ell}$. In this paper we study a light scalar mediator denoted by $S$ and a light vector mediator denoted by $Z^{\prime}$.

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,

[TABLE]

where $F(q^{2})$ is a form factor.111In 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:

$\displaystyle\cal{L}$ $\displaystyle=$ \displaystyle\frac{g}{\Lambda^{2}}\bar{b}b\bar{\chi}\chi+g_{\chi}\bar{\chi}\chi S\,,\

(4) where $\chi$ is a hidden sector fermion (which may serve as a dark matter candidate) of mass m_{\chi}\mathrel{\raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}}m_{b}, and we have suppressed all Lorentz structures in the Lagrangian. (In the context of Section III, for a light vector mediator $Z^{\prime}$, one may consider a similar Lagrangian of the form, $g_{\chi}\bar{\chi}\gamma^{\mu}\chi Z^{\prime}_{\mu}$.) The first term in the Lagrangian represents an effective coupling between the $b$ and $\chi$ fields that might arise via the exchange of a heavy mediator of mass $\Lambda\gg m_{b}$, which has been integrated out of the theory at the $m_{b}$ scale. Although there is no direct coupling between $b$ and $S$ (or $Z^{\prime}$), a $\bar{b}bS$ (or $\bar{b}bZ^{\prime}$) vertex with a $q^{2}$-dependent coupling will be generated by a $\chi$ loop. Transforming the $b$ quark from the gauge to the mass basis then generates a $\bar{s}bS$ (or $\bar{s}bZ^{\prime}$) coupling. In the case of the scalar mediator the form factor contains terms of the form, $\frac{m_{\chi}^{2}}{\Lambda^{2}}$ and $\frac{q^{2}}{\Lambda^{2}}$. For the latter term to dominate, $q^{2}\gg m_{\chi}^{2}$, which implies that m_{\chi}\mathrel{\raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}}30 MeV for the $q^{2}$ values of interest. For the $Z^{\prime}$ case, the leading term in the form factor goes as $q^{2}$ due to the conserved vector current FF . We note that the situation is similar to the SM case where $\chi$ is replaced by the charm quark and $S$ (or $Z^{\prime}$) by the photon. In this case the first term in the Lagrangian, of the form $\frac{g}{M_{W}^{2}}\bar{b}s\bar{c}c$, is just one of the terms in the SM effective Lagrangian after integrating out the $W$ boson. The charm loop then induces an effective $\bar{b}s\gamma^{*}$ vertex which yields $\bar{b}s\ell^{+}\ell^{-}$ via $\gamma^{*}\to\ell^{+}\ell^{-}$. The matrix elements for the processes $b\to s\ell^{+}\ell^{-}$ and the mass difference in $B_{s}$ mixing are

[TABLE]

where we have used Ref. soni_2HDM for $B_{s}^{0}$-${\bar{B}}_{s}^{0}$ mixing. The mass difference in the SM for the $B_{s}$ system is newpaper_datta

[TABLE]

which is consistent with experimental measurement HFAG ,

[TABLE]

We will choose the new physics contribution, $\Delta M_{B_{s}}^{NP}$, to be as large as the uncertainty in the SM prediction.

We now consider $b\to s\ell^{+}\ell^{-}$ 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 $B_{s}\to\mu^{+}\mu^{-}$ 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 $b\to se^{+}e^{-}$ from Eq. (5) is

[TABLE]

where $g_{ee}^{S}\equiv(g_{L}^{ee}+g_{R}^{ee})/2$ and $g_{ee}^{S^{\prime}}\equiv(g_{R}^{ee}-g_{L}^{ee})/2$. In the following discussion, we chose different structures for the form factor $F(q^{2})$.

II.1 $F(q^{2})\equiv 1$

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 $F(q^{2})\equiv 1$, and to avoid a pole contribution to the measurements of ${B(B^{0}\to K^{*0}e^{+}e^{-})}$ in the dielectron invariant mass range, $m_{ee}=[30-1000]$ MeV Aaij:2013hha , we choose $M_{S}=25$ MeV.

Note that the BaBar Aubert:2008ps and Belle adachi ; wei measurements require $m_{ee}$ to be larger than 30 MeV flood and 140 MeV, respectively. We fix $g_{ee}^{S}=2.0\times 10^{-4}$, which is the largest value allowed by the anomalous magnetic moment of the electron ae for $M_{S}=25$ MeV at the 2$\sigma$ CL. Then we perform a $\chi^{2}$-fit to the theoretically clean observables $R_{K}$ and $R_{K^{*}}$, and the new physics contribution to the $B_{s}$ mass difference, $\Delta M_{s}^{NP}=0\pm 2.6$ ps*-1*. In Ref. Wehle:2016yoi 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 flavio to calculate the theoretical values of the observables $\mathcal{O}_{th}$. We then compute

[TABLE]

where $\mathcal{O}_{exp}$ are the experimental measurements of the observables, and the total covariance matrix $\mathcal{C}$ 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

[TABLE]

The best fit values of the couplings $g_{bs}^{S}$ and $g_{bs}^{S^{\prime}}$ along with predictions at the best fit point, for $M_{S}=25$ MeV and $g_{ee}^{S}=2.0\times 10^{-4}$, are provided in Table 1. As a good fit is obtained in this case, we check if these values are consistent with the various measured branching ratios in $b\to se^{+}e^{-}$ modes. If $S$ can decay to $e^{+}e^{-}$ with a branching ratio $\sim$1 then the decays $B\to K^{(*)}e^{+}e^{-}$ will be dominated by the two-body decays, $B\to K^{(*)}S$, with $S$ decaying to $e^{+}e^{-}$.

For the two body $B\to KS$ decay, the branching ratio is

[TABLE]

where the form factor $f_{0}(z)$ can be found in Ref. Ball:2004ye .

For the two body $B\to K^{*}S$ decay, the branching ratio is

[TABLE]

where $\tau_{B}$ is the lifetime of $B$ meson, $|\vec{p}_{K^{*}}|=\lambda^{1/2}(m_{B}^{2},m_{K^{*}}^{2},m_{S}^{2})/2m_{B}$, and the form factor $A_{0}$ is taken from Ref. Ball:2004rg . To bound the NP coupling constants $g_{bs}^{S}$ and $g_{bs}^{S^{\prime}}$, we require the $B\to K^{(*)}S$ branching ratio to be less than 1%. This choice is consistent with uncertainties in the calculation of the $B$ meson width Lenz_lifetime . For $M_{S}$ between $10-200$ MeV, $\mathcal{B}(B^{0}\rightarrow K^{*0}e^{+}e^{-})$ and $\mathcal{B}(B^{0}\rightarrow K^{0}e^{+}e^{-})$ impose the constraints shown in Table 2. The best-fit values of the coupling given in Table 1 are in contradiction with these constraints. Hence, a light scalar with form factor $F(q^{2})\equiv 1$ is ruled out.

II.2 $F(q^{2})\neq 1$

Now we consider a $q^{2}$-dependent form factor $F(q^{2})\neq 1$ which may be loop induced. For momentum transfer $q^{2}\ll m_{B}^{2}$, $F(q^{2})$ can be expanded as ZMeV

[TABLE]

where $m_{B}$ is the $B$-meson mass. We do not include the $B_{s}$ mass difference and ${\mathcal{B}(B_{s}\to e^{+}e^{-}})$ as constraints since $F(q^{2})$ is unknown for $q^{2}\sim m_{B}^{2}$. We assume that $S$ does not couple to neutrinos so that $B\to K\nu\bar{\nu}$ BKnunubarBaBar ; BKnunubarBelle does not constrain $a_{bs}$. Redefining $a_{bs}g_{bs}^{S}$ as $g_{bs}^{S}$, and $a_{bs}g_{bs}^{S^{\prime}}$ as $g_{bs}^{S^{\prime}}$, we perform a $\chi^{2}$-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 1. Taking into account the constraints on $g_{bs}^{S}$ and $g_{bs}^{S^{\prime}}$ from Table 2 along with the constraints on $g_{ee}$ from the anomalous magnetic moment of the electron, we see that the best fit values $\mathcal{O}(10^{-8})$ 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 $B\to K\nu\bar{\nu}$ constraint if the mediator couples to neutrinos ZMeV ), and absorbing the factor $b_{bs}$ to redefine $g_{bs}^{S}$ and $g_{bs}^{S^{\prime}}$, the matrix element for $b\to se^{+}e^{-}$ is given by

[TABLE]

With the form factor $q^{2}/M_{B}^{2}$, requiring $\mathcal{B}(B^{0}\rightarrow K^{*0}e^{+}e^{-})$ and $\mathcal{B}(B^{0}\rightarrow K^{0}e^{+}e^{-})$ to be less than 1% gives the constraints on $g_{bs}^{S}$ and $g_{bs}^{S^{\prime}}$ in Table 2. The best-fit values can be found in Table 1. 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 $b\to se^{+}e^{-}$ observables; see Table 3. Our prediction for ${\mathcal{B}(B\to Ke^{+}e^{-}})_{[1.0-6.0]}$ is somewhat in tension with the experimental result. Allowing for a 10% uncertainty in the theoretical prediction lat , the discrepancy is about $2.3\sigma$. The prediction for the inclusive mode ${\mathcal{B}(B\to X_{s}e^{+}e^{-})}_{[1.0-6.0]}$, 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 $Z^{\prime}$

A $Z^{\prime}$ with mass less than $2m_{\mu}$ was recently proposed in Ref. ZMeV to simultaneously explain the measurements of $R_{K}$ and the anomalous magnetic moment of the muon, with implications for nonstandard neutrino interactions. Such a $Z^{\prime}$ may potentially explain $R_{K^{*}}$ in the low $q^{2}$ bin Ghosh:2017ber . A $Z^{\prime}$ with a mass in the few GeV range was discussed recently datta_rk ; Sala:2017ihs but the $q^{2}$ dependence of the WC is not strong enough to explain the $R_{K^{*}}$ at low $q^{2}$ datta_rk . Here we focus on an MeV $Z^{\prime}$.

We assume the flavor-changing $bsZ^{\prime}$ vertex to have the form,

[TABLE]

The matrix elements for $b\to s\ell^{+}\ell^{-}$ and the mass difference in $B_{s}$ mixing are

[TABLE]

where we have used Ref. soni_2HDM for $B_{s}^{0}$-${\bar{B}}_{s}^{0}$ mixing. Also, we define $g_{\ell\ell}\equiv(g_{L}^{\ell\ell}+g_{R}^{\ell\ell})/2$ and $g^{\prime}_{\ell\ell}\equiv(g_{R}^{\ell\ell}-g_{L}^{\ell\ell})/2$ for convenience.

III.1 $Z^{\prime}$ with muon coupling

We begin with the case where the $Z^{\prime}$ couples to muons and not to the electrons.

III.1.1 $F(q^{2})\equiv 1$

We first assume that $F(q^{2})\equiv 1$ and consider the case $g_{L}^{\mu\mu}=g_{R}^{\mu\mu}=g_{\mu\mu}$, 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 $M_{Z^{\prime}}=25$ MeV and fix $g_{\mu\mu}=8.0\times 10^{-4}$, which is the 2$\sigma$ upper bound from the anomalous magnetic moment of the muon. The fit results are shown in Table 1. We see that the overall improvement over the SM is insignificant because $g_{bs}^{S}$ and $g_{bs}^{S^{\prime}}$ are suppressed by $B_{s}$ mixing.

III.1.2 $F(q^{2})\neq 1$

Now we consider $F(q^{2})\neq 1$ 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 $q^{2}\sim m_{B}^{2}$. The fit results are shown in Table 1. The overall improvement over the SM is poor, with a pull of 2.4. Clearly, a light $Z^{\prime}$ with pure vector coupling to the muon is unable to explain the ${R_{K}}_{[1.0-6.0]}$, ${R_{K^{*}}}_{[0.045-1.1]}$ and ${R_{K^{*}}}_{[1.1-6.0]}$ anomalies simultaneously. However, on removing ${R_{K^{*}}}_{[0.045-1.1]}$ from the fit, one can easily accommodate the measured values of ${R_{K}}_{[1.0-6.0]}$ and ${R_{K^{*}}}_{[1.1-6.0]}$, and a pull of around 4.0 is obtained.

We next consider the case with $a_{bs}\neq 0$ and the $Z^{\prime}$ also has nonzero axial vector coupling with the muons, i.e., $g_{L}^{\ell\ell}\neq g_{R}^{\ell\ell}$. To keep the number of new couplings unchanged, we take either $g_{bs}^{\prime}=0$ or $g_{bs}=0$. This case also does not give a good fit to the data; see Table 1.

As can be seen from Table 1, 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 $Z^{\prime}$ with purely vector muon coupling is easily compatible with other $b\to s\ell^{+}\ell^{-}$ observables datta_rk .

III.2 $Z^{\prime}$ with electron coupling

We now consider the case where the $Z^{\prime}$ couples to electrons and not to muons.

III.2.1 $F(q^{2})\equiv 1$

We first assume that $F(q^{2})\equiv 1$ and we start by considering the case $g_{L}^{ee}=g_{R}^{ee}=g_{ee}$ 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 $g_{ee}=2.5\times 10^{-4}$, which is within the 90% CL upper limit from NA48/2 Batley:2015lha . The fit results are shown in Table 1. The fit to $R_{K}$ and $R_{K^{*}}$ is close to the SM predictions because of $B_{s}$ mixing.

III.2.2 $F(q^{2})\neq 1$

Now we consider $F(q^{2})\neq 1$. We fit to the observables $R_{K}$ and $R_{K^{*}}$ only since $F(q^{2})$ is unknown for $q^{2}\sim m_{B}^{2}$. The best fit results are shown in Table 1. 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 $\mathcal{B}(B\to K^{(*)}e^{+}e^{-})$ from $B\to K^{(*)}Z^{\prime}$ and $Z^{\prime}$ decaying to $e^{+}e^{-}$ with a branching ratio $\sim$1.

The branching ratio for $B\to KZ^{\prime}$ is Fuyuto:2015gmk ; Oh:2009fm ,

[TABLE]

where $\beta_{XYZ}=\lambda^{1/2}(1,M_{Y}^{2}/M_{X}^{2},M_{Z}^{2}/M_{X}^{2})$ and $f_{+}^{BK}$ is a form factor. For $B\to K^{*}Z^{\prime}$ the branching ratio is given by,

[TABLE]

where the helicity amplitudes are defined as,

[TABLE]

and

[TABLE]

$V$, $A_{1}$ and $A_{2}$ are form factors Ball:2004ye ; Ball:2004rg and $\xi=(m_{B}^{2}-M_{K^{*}}^{2}-M_{Z^{\prime}}^{2})/(2M_{K^{*}}M_{Z^{\prime}})$.

Assuming the decay rate of $B\to KZ^{\prime}$ and $B\to K^{*}Z^{\prime}$ to be less than 1% of the $B^{0}$ width, we obtain the constraints shown in Table 2. Since $g_{ee}$ is constrained to be less than $2.5\times 10^{-4}$ at the 90% CL for $M_{Z^{\prime}}=25$ MeV Batley:2015lha , the constraints in Table 2 exclude the best-fit values to explain the $R_{K}$ and $R_{K^{*}}$ measurements in this case.

We next consider the case when $Z^{\prime}$ also has nonzero axial vector coupling with the electrons, i.e., $g_{L}^{ee}\neq g_{R}^{ee}$. The best-fit results are shown in Table 1. 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 2 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 $g_{L}^{ee}=g_{R}^{ee}=g_{ee}$, i.e., pure vector coupling to the electron, and for $M_{Z^{\prime}}=25$ MeV, we fit the product $g_{ee}g_{bs}$ and $g_{ee}g_{bs}^{\prime}$ to the $R_{K}$ and $R_{K^{*}}$ data. The results are summarized in Table 1. Clearly, at the best fit point the predictions for $R_{K}$ and $R_{K^{*}}$ are within the $1\sigma$ range of the measurements. Requiring $\mathcal{B}(B^{0}\rightarrow K^{0}e^{+}e^{-})<1\%$ and $\mathcal{B}(B^{0}\rightarrow K^{*0}e^{+}e^{-})<1\%$, we get the constraints shown in Table 2. The best fit satisfies all constraints on $g_{bs}$, $g_{bs}^{\prime}$ and $g_{ee}$. From Table 1, 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 3. Our prediction for ${\mathcal{B}(B\to Ke^{+}e^{-}})_{[1.0-6.0]}$ 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 ${\mathcal{B}(B\to X_{s}e^{+}e^{-})}_{[1.0-6.0]}$, which suffers from less hadronic uncertainties, is consistent with measurement.

Next we consider the case when $Z^{\prime}$ also has nonzero axial vector coupling with the electrons, i.e., $g_{L}^{ee}\neq g_{R}^{ee}$. Again, we either set $g_{bs}^{\prime}=0$ or $g_{bs}=0$. The best-fit values shown in Table 1 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 3.

IV Summary

In this work we have addressed the recent measurement of $R_{K^{*}}$ with particular attention to the low $q^{2}$ bin, $0.045\leq q^{2}\leq 1.1~{}{\rm GeV}^{2}$. 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:

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 $\mathcal{B}(B\to K^{(*)}e^{+}e^{-})$. A $q^{2}$-dependent flavor changing $b-s$ coupling to the scalar can produce compatibility with $\mathcal{B}(B\to K^{(*)}e^{+}e^{-})$ 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 $Z^{\prime}$ 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 $Z^{\prime}$ 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 $\mathcal{B}(B\to K^{(*)}e^{+}e^{-})$. However, a $q^{2}$-dependent flavor changing $b-s$ coupling to the vector is compatible with $\mathcal{B}(B\to K^{(*)}e^{+}e^{-})$ 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.

Acknowledgments. We thank W. Altmannshofer, T. Browder, A. Denig, A. Dighe, T. Gershon, D. Ghosh, K. Flood, D. McKeen, G. Miller, L. Piilonen and D. Straub for discussions. A.D. thanks the Institute for the Physics and Mathematics of the Universe for hospitality and partial support. D.M. thanks the Mainz Institute for Theoretical Physics (MITP) for its hospitality and partial support during the completion of this work. This research was supported by the U.S. NSF under Grant No. PHY-1414345 and by the U.S. DOE under Grant No. DE-SC0010504.