Frugal $U(1)_{X}$ models with non-minimal flavor violation for $b\to s\ell\ell$ anomalies and neutrino mixing

The origin of the CKM matrix is in the diagonalization of up-type quark mass matrix $M_{u}$ and the down-type quark matrix $M_{d}$ by the bi-unitary transformations

The CKM matrix is then given by $V_{\text{CKM}}=V_{uL}^{\dagger}V_{dL}$.

In the presence of a new $U(1)_{X}$ gauge symmetry, the flavor-changing neutral currents (FCNC) induced by the new gauge boson $Z^{\prime}$ would affect the neutral meson mixings by giving additional tree-level contributions to the box diagram in the SM. We focus on the constraints from CP violation in $K-\bar{K}$ mixing ($\epsilon_{K}$), and the mass splitting as well CP-violation in $B_{d}-\overline{B}_{d}$ as well as $B_{s}-\overline{B}_{s}$ mixing. We ignore constraints from $\Delta m_{K}$, since its value is dominated by long distance effects Grinstein:2015nya . We also do not incorporate constraints from $D-\overline{D}$ mixing for the same reason. The NP contribution may be calculated by writing down the Lagrangian for left-handed d-type quarks $D_{L}\equiv(d_{L},s_{L},b_{L})^{T}$ in their mass basis:

where $\mathbb{X}\equiv{\rm diag}(X_{u},X_{c},X_{t})={\rm diag}(\alpha_{1}/3,\alpha_{2}/3,\alpha_{3}/3)$. As shown in ref. prev-paper , the mixing in the right-handed d-quark sector may be chosen to be small, so that the contributions due to right-handed currents stay subdominant. The relevant matrix elements that control the dominant NP contributions in the $K$, $B_{d}$ and $B_{s}$ systems may be written as

where the unitarity of $V_{dL}$ has been used. The choice $X_{u}=X_{c}$, and the small values of $[V_{dL}]_{td}$ and $[V_{dL}]_{ts}$, allow us to minimize the strong constraints from the $K$ sector and somewhat weaker constraints from the $B_{d}$ and $B_{s}$ sectors. The condition $X_{u}=X_{c}$ also implies an underlying $U(2)^{3}$ flavor symmetry present in the Lagrangian, which is broken only by the Yukawa interactions isidori-straub . This has also been referred to as “less-minimal flavor violation” less-MFV ; less-MFV-crivillin . The additional choice $V_{uL}={\mathbb{I}}$ (or equivalently, $V_{dL}=V_{CKM}$) made in ref. prev-paper makes the scenario “MFV-like”, wherein the combinations of CKM elements contributing to the mixing in the $B_{d}$ and $B_{s}$ sectors are the same as those in the SM. It also ensures that the NP contribution from the second term to $K-\overline{K}$ mixing is suppressed by $|[V_{\rm CKM}]_{td}[V_{\rm CKM}]_{ts}|\sim{\cal O}(\theta_{C}^{~{}5})$, where $\theta_{C}\approx 0.2$ is the Cabibbo angle.

We continue to use the condition $X_{u}=X_{c}$ in this paper. Later, for non-minimal scenarios, we will relax the condition $V_{dL}=V_{\rm CKM}$, however the smallness of $[V_{dL}]_{td}$ will still be valid, keeping in mind the stringent constraints from kaon oscillation data.

The condition $X_{u}=X_{c}$ (i.e. $\alpha_{1}=\alpha_{3}$) also impacts the structure of the Yukawa matrices. Since in our framework, the SM Higgs doublet is uncharged under $U(1)_{X}$, the only nonzero elements in the SM Yukawa matrix can be the three diagonal elements and the off-diagonal elements in the first two generations. This would force the $2$-$3$ and $1$-$3$ mixings in the CKM matrix to be zero. In order to prevent this, the SM needs to be augmented with an additional doublet $\Phi_{\text{NP}}$ whose $X$-charge equals $X_{u}-X_{t}$, or equivalently, $(\alpha_{1}-\alpha_{3})/3$. This would result in the NP contribution to the Yukawa matrices of up-type and down-type quarks in the form

where $\times$ denotes nonzero elements. These off-diagonal elements give rise to the required mixing in the 2-3 and 1-3 sector, to reproduce the CKM matrix prev-paper . These Yukawa matrices can now give rise to the mass matrices

where $\langle v_{SM}\rangle=v\sin\beta$ and $\langle v_{NP}\rangle=v\cos\beta$ are the vacuum expectation values of the SM Higgs $\Phi_{\rm SM}$ and the NP Higgs $\Phi_{\rm NP}$, respectively. The matrices $M_{u}$ and $M_{d}$ are diagonalized by the unitary matrices $V_{dL}$, $V_{dR}$, $V_{uL}$ and $V_{uR}$, as shown in eq. (4). Note that the requirement of $X_{u}=X_{c}$ is instrumental in generating the CKM matrix with only one additional Higgs doublet.

Among the two Higgs doublets $\Phi_{\rm SM}$ and $\Phi_{\rm NP}$, the former is a singlet under $U(1)_{X}$, to ensure nonzero diagonal elements in the flavor basis. The latter has an $X$-charge equal to $(\alpha_{1}-\alpha_{3})/3$, as seen above.

The absence of a massless pseudoscalar, which would be created due to the breaking of a global $U(1)_{A}$ symmetry in the Lagrangian, necessitates the introduction of an extra scalar $S$, which has the same $X$-charge as $\Phi_{\rm NP}$ prev-paper . It allows a term $S\Phi_{\text{NP}}^{\dagger}\Phi_{\text{SM}}$ in the scalar sector, which yields a mass for the pseudoscalar after the breaking of $U(1)_{X}$ where $S$ gets a vacuum expectation value. The $X$-charge of $S$ also needs to be $X_{S}=(\alpha_{1}-\alpha_{3})/3$.

The global fits to neutral-current flavor anomalies and other $b\to s$ data in the $\left(C_{9\mu}^{\text{NP}},C_{9e}^{\text{NP}}\right)$ plane strongly indicate $|C_{9e}^{\text{NP}}|<|C_{9\mu}^{\text{NP}}|$, which indicate $|\alpha_{e}|<|\alpha_{\mu}|$ as seen in figure 1. We therefore take this to be one of the conditions on our model. The value of $\alpha_{\tau}$ remains unconstrained from the current measurements.

We determine the $X$-charges of the leptons which can explain the patterns of neutrino mixing well. In particular, we desire that the leptonic mixing arises completely in the neutrino sector, where the neutrino mass is generated by the Type-I seesaw mechanism. However, since the $X$-charges of the three lepton generations are, in general, different, it would not be possible to generate off-diagonal elements in the neutrino mass matrix, which are needed for the large neutrino mixing observed. In our model, this can be achieved without the need for any additional particle, but by using the interactions of the neutrinos with the scalar $S$ that is already present prev-paper . The terms contributing to neutrino mass are:

where $[m^{\nu}_{D}]$ is the Dirac matrix, $[m^{\nu}_{R}]$ is the Majorana mass matrix of right-handed neutrinos, and $i,j$ are flavor indices. The effective Marorana mass matrix after the symmetry breaking becomes

where $v_{S}$ is the vacuum expectation value of $S$. Thus the mass matrix $M^{\nu}_{R}$ gets off-diagonal elements, which further lead to the mixing of left-handed neutrinos, through Type-I seesaw formula

In order for the above neutrino mass matrix $[m_{\nu}]$ to reproduce the observed neutrino mixing pattern, only certain texture-zero patterns of $[M^{\nu}_{R}]$ are allowed Grimus:2004hf ; neutrino-textures . A subset of these patterns may be created by appropriate choices of the values of $\alpha_{e},\alpha_{\mu},\alpha_{\tau}$, and $X_{S}$ prev-paper . A further subset satisfies the requirement $|\alpha_{e}|<|\alpha_{\mu}|$. The leptonic symmetries $(\alpha_{e}L_{e}+\alpha_{\mu}L_{\mu}+\alpha_{\tau}L_{\tau})$ that satisfy all these criteria are:

• •

  $a\left(L_{\mu}-L_{\tau}\right)$ or $aL_{\mu}$, with $X_{S}=\pm a$,

• •

  $a\left(L_{e}-3L_{\mu}+L_{\tau}\right)$ or $a\left(L_{e}\pm 3L_{\mu}-L_{\tau}\right)$, with $X_{S}=\pm 2a$.

Here $a$ is the overall multiplicative factor.

Inferring the $X$-charges of leptons from the allowed leptonic symmetries, and using the conditions $\alpha_{1}=\alpha_{2}$ and $X_{S}=(\alpha_{1}-\alpha_{3})/3$, the $X$-charges of all the other particles are fixed automatically by demanding the theory to be anomaly free. The $X$-charges of all leptons, in turn, are fixed up to an overall multiplying factor $a$ as seen in the last section. We fix the normalization by choosing $a$ so as to make $\alpha_{\mu}=1$. All the $U(1)_{X}$ scenarios thus determined are listed in Table 2. We further categorize them depending on their values of $X_{S}$ and $\alpha_{e}$. This ensures that flavor constraints for scenarios belonging to the same category are identical. Note that the categories A, B and C listed in table 2, with negative $X_{S}$ values, are the same as given in prev-paper . The category D from ref. prev-paper is not present in the current version because of the imposition of $|\alpha_{e}|<|\alpha_{\mu}|$. In addition, have also included the categories AA, BB and CC with positive $X_{S}$ values. This inclusion completes the set of scenarios allowed by conditions in sections 2.1, 2.2, 2.3. Note that for the categories in each pair (A, AA), (B, BB), and (C, CC), the leptonic symmetries are identical, but the sign of $X_{S}$ is different.

The NP in our class of models influences $R_{K^{(*)}}$ primarily through the Wilson coefficients $C_{9\ell}$. The contributions to $C_{9\ell}^{\prime}$ are small due to the small mixing angles in $V_{dR}$ (see ref. prev-paper and section 4.2). The tree-level contributions to $C_{10\ell}^{(\prime)}$ operators are zero. They may arise due the $Z-Z^{\prime}$ mixing, and vanish in the small $Z-Z^{\prime}$ mixing limit. The effective Hamiltonian relevant for the process $B\to K^{(*)}\ell\ell$ is

Since $C_{9\ell}=C_{9\ell}^{\rm SM}+C_{9\ell}^{\rm NP}$, the above equation is equivalent to

Note that the WCs have scale dependence, however the qualitative inferences in this section do not change while running from the scale $M_{Z^{\prime}}$ to $m_{b}$. From eq. (15), the two relevant Wilson coefficients are related by $C_{9e}^{\text{NP}}(m_{b})=\alpha_{e}C_{9\mu}^{\text{NP}}(m_{b})$.

From the global fits, we have seen that the sign of $C_{9\mu}^{\text{NP}}$ needed to explain the observed $b\to s$ anomalies has to be negative, for the NP to destructively interfere with the SM, where $C_{9\mu}^{\text{SM}}$ is positive WC-SM . This leads to

i.e., either the charge $X_{S}$ is negative, or the ratio

is negative. In ref. prev-paper , the assumption of $V_{uL}=\mathbb{I}$ led to $V_{dL}=V_{\rm CKM}$, so that $\mathcal{R}_{\rm mix}$ was always unity. As a result, only those symmetry combinations where $X_{S}<0$ had been selected. These are the categories A, B, C shown in Table 2. In this paper, we follow a generalized approach, without the assumption $V_{uL}=\mathbb{I}$. This allows three additional categories, viz. AA, BB and CC, as shown in Table. 2, where the sign of $X_{S}$ is positive.

The large amount of data being collected at the LHC strongly constrains any new physics that couples to light quarks. Our scenarios in table 2 necessarily have non-zero couplings of $Z^{\prime}$ to the first two generations of quarks. Therefore, the $Z^{\prime}$ particle will be produced at the LHC for low $M_{Z^{\prime}}$ and high enough $g_{Z^{\prime}}$ values. Furthermore, even a $Z^{\prime}$ that couples dominantly to the third generations may be produced in a pp collision, albeit with a smaller cross section due to the smaller parton fraction in the proton. The non-observation of any such particle so far puts severe constraints on model parameters. The main observations that would constrain our class of models are:

• •

  Top-quark pair production limits from $pp\rightarrow Z^{\prime}\rightarrow t\bar{t}$ CMS:2020tvq ; ATLAS:2020aln ; ATLAS:2020lks

• •

  Dijet limits from $pp\rightarrow Z^{\prime}\rightarrow q\bar{q}$, including $b\bar{b}$ ATLAS:2019fgd

• •

  Dilepton limit from $pp\rightarrow Z^{\prime}\rightarrow e^{+}e^{-},\,\mu^{+}\mu^{-}$ atlas-dilepton ; cms-dilepton (including the non-resonant shape of the $m_{\mu\mu}$ distribution tail altas-non-resonant ; cms-non-resonant )

As we show below, the most stringent limits come from dimuon searches. A comparison of the limits from all above observables can be seen in figure 2.

For $t\bar{t}$ searches, currently there is a measurement of $t\bar{t}$ pair production cross section with 35.9 fb^-1 data from CMS CMS:2020tvq and 139 fb^-1 data from ATLAS ATLAS:2020aln . Limits can be calculated either by using the total cross section or the invariant mass spectrum shape in addition. The best measurement for the total cross section is currently $830\pm 39$ pb ATLAS:2020aln , and is completely consistent with the calculated SM cross section. We therefore require that the contribution from NP to the total $t\bar{t}$ cross section keeps the prediction less than two sigma away from the measured value. Further incorporating the $t\bar{t}$ spectral shape, a lower bound $M_{Z^{\prime}}\gtrsim 3.2$ TeV is obtained for $g_{Z^{\prime}}=1$, as can be seen in figure 2.

For $M_{Z^{\prime}}\lesssim 3$ TeV, the background for dijet searches is high as compared to that for the $t\bar{t}$ searches, and hence the sensitivity of $t\bar{t}$ searches is better. However, at higher $Z^{\prime}$ masses, the dijet searches give slightly stronger constraints. For example, $M_{Z^{\prime}}\gtrsim 4.5$ TeV for $g_{Z^{\prime}}=1$, as seen in figure 2.

Our condition $|\alpha_{e}|<|\alpha_{\mu}|$ implies that the constraints from the di-electron searches would always be weaker than those from the dimuon searches. Hence, we focus on the dimuon channel. Experiments provide measurements of the invariant mass spectrum in the dimuon final state. Due to the simplicity of the final state, this may be interpreted in terms of a 95% upper limit on the production cross-section of $Z^{\prime}$ (with minimal fiducial cuts). The parameter space ($M_{Z^{\prime}},g_{Z^{\prime}}$) can then be constrained in any given scenario by comparing the theoretical production cross-section with the experimental 95% confidence limits. Such upper limits are available from the ATLAS experiment atlas-dilepton for $M_{Z^{\prime}}<6$ TeV, and from the CMS experiment cms-dilepton for $M_{Z^{\prime}}<5.5$ TeV. We use the constraints from CMS, which are slightly stronger than those from ATLAS, to represent the dimuon limits. As we see from figure 2, dimuon constraints are much stronger than either $t\bar{t}$ or dijet constraints for the scenario A1 (or equivalently, AA1). This observation remain true for all scenarios listed in table 2.

Using the number of observed events with high invariant mass, we can extend the dimuon limits to higher values of $M_{Z^{\prime}}$ for which the calculated limits have not been published by the experimental analyses (see figure 2). The details of our calculations are explained in appendix A. The published bounds from CMS are available for $M_{Z^{\prime}}\leq 5.5$ TeV. Our calculated limits agree with these limits in the region $3.5\;{\rm TeV}\leq M_{Z^{\prime}}\leq 5.5$ TeV, thereby justifying the method used for extrapolation. For the rest of this study, we use the published CMS dimuon limits upto $M_{Z^{\prime}}\leq 5.5$ TeV, and our extrapolation for masses $5.5\;\text{TeV}<M_{Z^{\prime}}\leq 10$ TeV.

Note that a 10 TeV $Z^{\prime}$ can be excluded for high-enough coupling for all of our scenarios. On the other hand, requiring at least three events as a threshold for detection puts the LHC reach for the discovery of $Z^{\prime}$ to $M_{Z^{\prime}}\approx 4-6$ TeV, depending on the scenario in table 2.

Apart from mediating tree-level $b\to s\ell\ell$ transitions, the additional $Z^{\prime}$ particle would also be responsible for generating tree-level mixing in $B_{d}-\overline{B}_{d}$, $B_{s}-\overline{B}_{s}$ and $K-\overline{K}$ sectors. These new physics contributions are heavily constrained from data utfit . Since the mixing constraints are not taken into consideration in global fits matias-2019 ; mahmoudi-2019 ; matias-2021 ; mahmoudi-2021 , one has to incorporate them separately. Additionally, the new physics contributions generated by $Z^{\prime}$ only affect the operators with left handed quark currents, as the right handed mixing is smaller in comparison to the left handed mixing (see ref. prev-paper and section 4.2). Hence $Z^{\prime}$ contributes to the same operators as in the SM. We get

where $C^{\rm NP}_{P}$ at the $M_{Z^{\prime}}$ scale are given as

Here $P$ generically refers to one of the $B_{d}$, $B_{s}$ or $K$ meson. Note that while the CKM factors explicitly appear for $B-\overline{B}$ mixings, they are conventionally absorbed in $C^{\text{SM}}_{K}(\mu)$. After incorporating the running of the effective operators at one-loop order in QCD at $M_{W}$ scale buras-review , the WCs are obtained as

where $P$ stands for $K,B_{d}$, or $B_{s}$. Note that the running of SM and NP is identical after the $M_{W}$ scale, hence we have taken the running here only upto $W$-mass scale. These additional contributions to $P-\overline{P}$ mixing get constrained from the measurements. The constraints on $\Delta m$ and CP-violating phases are parameterized utfit in terms of

which can be studied in the plane of $(M_{Z^{\prime}},g_{Z^{\prime}})$ for a given symmetry and a given $V_{dL}$. Note that as mentioned in section 2, we do not consider the constraint from $\Delta m_{K}$ as it is dominated by long distance corrections utfit . For constraining our model parameter, we shall require that the allowed parameter space lies within 2$\sigma$ uncertainties for all these five observables, viz. $C_{\epsilon_{K}}$, $C_{B_{d}}$, $C_{B_{s}}$, $\phi_{B_{d}}$, and $\phi_{B_{s}}$.

When $V_{dL}=V_{\rm CKM}$, the CKM factors in eq. (15) cancel, and the $C_{9\mu}^{\text{NP}}$ Wilson coefficient simplifies to

The desired negative value of $C_{9\mu}^{\rm NP}$ is obtained if $X_{S}<0$. This points towards the scenarios belonging to the categories A, B and C listed in table 2.

We now subject these scenarios to the experimental constraints discussed in section 3. The results are presented in fig 3. Note that for scenarios belonging to the same category, the global-fit constraints are identical, and so are the neutral meson mixings constraints. However collider constraints are different for sub-scenarios (like A1 and A2) which have different $X$-charge assignments for quarks. We can clearly see that, on one hand, the allowed $2\sigma$ bands from global-fit have started to become narrower, while on the other hand, the constraints from LHC are becoming considerably more stringent. The current data with 140 fb^-1 total integrated luminosity cms-dilepton has essentially ruled out all the parameter space for these MFV-like models.

The freedom of choice of $V_{dL}$ allows us to find scenarios that survive the stringent collider and meson-mixing constraints above. This will be shown in the next subsection.

Transition from MFV-like mixing, i.e. $V_{dL}=V_{\rm CKM}$, to non-MFV mixing with $V_{dL}\neq V_{\rm CKM}$ would be severely constrained by measurements in the $K-\overline{K}$ sector, where the value of $\epsilon_{K}$ as given in eq. (21) is very well measured. However, these constraints can be evaded if $V_{dL}$ is chosen to be real. In the rest of the paper, we shall continue with real $V_{dL}$.

As seen in section 2.5, the resolution of $R_{K^{(*)}}$ anomalies needs $X_{S}\mathcal{R}_{\text{mix}}<0$. Writing the real $V_{dL}$ in terms of three mixing angles $\theta_{12}$, $\theta_{23}$, $\theta_{13}$ (similar to the CKM parameterization), $\mathcal{R}_{\text{mix}}$ in eq. (17) can be written as

Note that, since $\mathcal{R}_{\rm mix}$ can have either sign, the sign of $X_{S}$ can now be positive as well as negative. This allows the categories AA, BB and CC from table 2 to be viable candidates, in addition to the categories A, B and C considered earlier. Moreover, if the magnitude of $\mathcal{R}_{\rm mix}$ is large, the required values of $C_{9\mu}^{\rm NP}$ may become possible even with lower values of $g_{Z^{\prime}}/M_{Z^{\prime}}$, as can be seen from eq. (15). However, the parameter $\mathcal{R}_{\rm mix}$ cannot be too large, otherwise the simultaneous explanation of $b\to s\ell\ell$ anomalies along with neutral meson mixing constraints from $B_{d/s}-\overline{B}_{d/s}$ mixing would be difficult. Thus, a modest enhancement of $\mathcal{R}_{\rm mix}$ is required to make these scenarios compatible with the global fits, neutral meson mixing data, and collider constraints.

Since $[\cos\theta_{12}\cos\theta_{13}]_{\rm CKM}\approx 1$, one would need $[\sin 2\theta_{23}]_{dL}\gtrsim[\sin 2\theta_{23}]_{\rm CKM}$ for the enhancement in $\mathcal{R}_{\rm mix}$. In a simplified scenario, we can take $\theta_{12,dL}\approx 0$ and $\theta_{13,dL}\approx 0$, which leads to