New physics contributions to $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$ decays

Abstract

The decay modes $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$ are dominated by electroweak penguins that are small in the standard model. In this work we investigate the contributions to these penguins from a model with an additional $U(1)'$ gauge symmetry and show there effects on the branching ratios of $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$. In a scenario of the model, where $Z^\prime$ couplings to the left-handed quarks vanish, we show that the maximum enhancement occurs in the branching ratio of $\bar{B}^0_s\rightarrow \,\pi ^0\,\eta '$ where it can reach 6 times the SM prediction. On the other hand, in a scenario of the model where $Z^\prime$ couplings to both left-handed and right-handed quarks do not vanish, we find that $Z^\prime$ contributions can enhance the branching ratio of $B^0_s\rightarrow \,\rho ^0\,\eta$ up to one order of magnitude comparing to the SM prediction for several sets of the parameter space where both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints are satisfied. This kind of enhancement occurs for a rather fine-tuned point where the $\Delta M_{B_s}$ constraint on $\mid S_\mathrm{SM} (B_s) + S_{Z'} (B_s)\mid$ is fulfilled by overcompensating the SM via $S_{Z'} (B_s) \simeq -2 S_\mathrm{SM} (B_s)$.

1 Introduction

The purely isospin-violating decays $\bar{B}_s \rightarrow \phi \,\pi \,(\rho ^0)$, $\bar{B}_s \rightarrow \pi \, \eta (\eta ')$ and $\bar{B}_s \rightarrow \rho ^0\, \eta (\eta ')$ are dominated by the electroweak penguins [1,2,3,4]. These penguins are small in the standard model and can serve as a probe of new physics beyond the standard model. The decay modes $\bar{B}_s \rightarrow \phi \,\pi \,(\rho ^0 )$ have been studied within SM in different frameworks such as QCD factorization as in Refs. [5, 6], in PQCD as in Ref. [7] and using Soft Collinear Effective Theory (SCET) as in Refs. [8, 9]. The study has been extended to include NP models namely, a modified $Z^0$ penguin, a model with an additional $U(1)'$ gauge symmetry and the MSSM using QCDF [6]. In addition, the investigation of NP in these decay modes has been recently extended to include supersymmetric models with non-universal A-term [9] and two Higgs doublet models (2HDMs) [10] using SCET. The results of these studies showed that the additional $Z\,'$ boson of the $U(1)'$ gauge symmetry with couplings to leptons switched off can lead to an enhancement in their branching ratios (BRs) up to an order of magnitude making these decays are interesting for LHCb and future B factories searches [6].

Recently tension between the SM and data related to $b\rightarrow s\ell ^+ \ell ^-$ channels has become apparent. In particular LHCb has reported deviations from the Standard Model predictions in the processes $B\rightarrow K^*\mu ^+\mu ^-$ and $B_s\rightarrow \phi \mu ^+\mu ^-$ that are mediated by the $b\rightarrow s\mu ^+ \mu ^-$ transition. Moreover, the deviations include the process $B\rightarrow K\mu ^+\mu ^-$ through the ratio $R_K$ defined as

$$\begin{aligned} R_K = \frac{Br (B\rightarrow K\mu ^+\mu ^-)}{Br(B\rightarrow K e^+e^-)}. \end{aligned}$$

(1)

These anomalies can be naturally accommodated in $Z'$ models as have been found in Refs. [11,12,13,14,15,16]. These findings serve as a general phenomenological motivation for $Z'$ models and for the search of analogous tensions in hadronic B decays. In this work we investigate the phenomenological implications of a leptophobic $Z'$ model on the decay modes $\bar{B}_s \rightarrow \pi \, \eta (\eta ')$ and $\bar{B}_s \rightarrow \rho ^0\, \eta (\eta ')$. In this model $Z'$ couplings to quarks are not related to their couplings to leptons and thus can avoid the tight constraints from semileptonic decays [6].

The decay modes $\bar{B}_s \rightarrow \pi \, \eta (\eta ')$ and $\bar{B}_s \rightarrow \rho ^0\, \eta (\eta ')$ have been studied within SM using different frameworks such as Naive Factorization (NF) [2], generalized factorization [3], POCD [7, 17] and QCDF [4, 18]. On the other hand, using SCET, an investigation of $\bar{B}_s \rightarrow \pi \, \eta (\eta ')$ has been carried out in Ref. [19], while the decay modes $\bar{B}_s \rightarrow \rho ^0\, \eta (\eta ')$ have been studied in Ref. [8]. NP effects namely 2HDMs have been investigated in these decay modes in Ref. [2] using NF and using generalized factorization in Ref. [20]. In our study we will adopt SCET as a framework for the calculation of the amplitudes [21,22,23,24].

SCET provides a systematic and rigorous way to deals with the processes in which energetic quarks and gluons have different momenta modes such as hard, soft and collinear modes. The power counting in SCET reduces the complexity of the calculations. In addition, the factorization formula given by SCET is perturbative to all powers in an $\alpha _s$ expansion.

This paper is organized as follows. In Sect. 2, we review the decay amplitude for $B \rightarrow M_1M_2$ within SCET framework. Accordingly, we present the SM predictions of the branching ratios of the decay modes under the study in Sect. 3. Then we proceed to analyze NP contributions namely the non-universal $Z^{\prime }$ model in Sect. 4. Finally, we give our conclusion in Sect. 5.

2 $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$ decays in SCET

The decay amplitude of B meson into two light final states mesons $M_1 , M_2$ at LO in $\alpha _s(m_b)$ expansion in SCET can be written as [8]:

(2)

here $M_1 M_2$ can be PP or PV where P stands for pseudoscalar meson and V stands for vector meson and $\phi _{M}(u)$ is the light-cone distribution amplitude (LCDA) of the meson M. $A^{M_1 M_2}_{cc}$ represents the non-perturbative long distance charm contribution to the amplitude. The hadronic parameters $\zeta ^{{B} M}$, $\zeta ^{ {B} M}_g$, $\zeta _J^{{B} M}$ and $\zeta _{Jg}^{{B} M}$, in the framework of SCET, are treated as non-perturbative parameters that can be fitted using the experimental data of the branching fractions and CP asymmetries of the non-leptonic B and $B_s$ decays [8, 19, 25, 26]. The hard kernels $T_{i}$, $T_{ig}$, $T_{iJ}(u)$ and $T_{iJg}(u)$ for $i=1,2$ are functions of the Wilson coefficients of the weak effective Hamiltonian. The expressions of these kernels for a certain $B\rightarrow M_1 M_2$ decay mode in the case of SM can be obtained using the formulas given in the appendix of Ref. [8].

Table 1 Hard kernels of $\bar{B}^0_s\rightarrow \eta _{s,q}\, \pi ^0 (\rho ^0)$ decays. The hard kernels $T_{iJ}$, $T_{iJg}$, for $i=1,2$, can be obtained through the replacement $c_i^s\rightarrow b_i^s$

In many extensions of the SM the weak effective Hamiltonian can have a new set of operators $\tilde{Q}_i$ that are obtained by flipping the chirality of the SM four-quark operators from left to right. Following a similar treatment to that in Ref. [8] we find that the effect of these new operators can be incorporated in the expressions of the hard kernels $T_{i}$, $T_{ig}$, $T_{iJ}(u)$ and $T_{iJg}(u)$. In Table 1 we present the explicit expressions of the hard kernels $T_{i}$, $T_{ig}$, $T_{iJ}(u)$ relevant to the decay channels $\bar{B}_s\rightarrow \eta _s\, \pi ^0$, $\bar{B}_s\rightarrow \eta _{q}\, \pi ^0$, $\bar{B}_s\rightarrow \eta _{s}\, \rho ^0$ and $\bar{B}_s\rightarrow \eta _{q}\, \rho ^0$. The coefficients $c^s_i$ and $b^s_i$ are functions of Wilson coefficients of the weak effective Hamiltonian. After extending the SM weak effective Hamiltonian to include right-handed operators $\tilde{Q}_i$ generated by NP we find that

$$\begin{aligned} c_{2}^{(s)}= & {} \lambda _u^{(s)}\Big [C_{2}-\tilde{C}_2+\frac{1}{N_c}(C_{1}-\tilde{C}_1)\Big ] \nonumber \\&-\frac{3}{2} \lambda _t^{(s)}\Big [C_{9}-\tilde{C}_9+\frac{1}{N_c}(C_{10}-\tilde{C}_{10})\Big ], \nonumber \\ c_3^{(s)}= & {} -\frac{3}{2}\lambda _t^{(s)}\Big [C_7-\tilde{C}_7+\frac{1}{N_c}(C_8-\tilde{C}_8)\Big ], \end{aligned}$$

(3)

and

$$\begin{aligned} b_{2}^{(s)}= & {} \lambda _u^{(s)}\Big [C_{2}+\tilde{C}_2+\frac{1}{N_c}\Big (1-\frac{m_b}{\omega _3}\Big )(C_{1}+\tilde{C}_1)\Big ]\nonumber \\&-\frac{3}{2}\lambda _t^{(s)}\Big [C_{9}+\tilde{C}_9+\frac{1}{N_c}\Big (1-\frac{m_b}{\omega _3}\Big )(C_{10}+\tilde{C}_{10})\Big ], \nonumber \\ b_3^{(s)}= & {} - \frac{3}{2} \lambda _t^{(s)}\Big [C_7-\tilde{C}_7+ \frac{1}{N_c} \Big (1-\frac{m_b}{\omega _2}\Big )(C_8-\tilde{C}_8)\Big ] , \end{aligned}$$

(4)

where ${C}_i = C^\mathrm{SM}_i + C^{NP}_i$ and $\tilde{C}_i = \tilde{C}^\mathrm{SM}_i +\tilde{C}^{NP}_i$. The Wilson coefficients $\tilde{C}_i$ correspond to the four-quark operators in the weak effective Hamiltonian that have right chirality. In the SM such operators are absent and hence $\tilde{C}^\mathrm{SM}_i=0$. In Eq. (4) we have $\omega _2=u m_{\bar{B}_s}$ and $\omega _3=-\bar{u} m_{\bar{B}_s}$ with u is the momentum fraction of the positive quark in the emitted meson and $N_c=3$. From charge conjugation and isospin we have $\phi _{\pi (\rho )}(u)= \phi _{\pi (\rho )}(1-u)$ [26]. Thus we can write

$$\begin{aligned} \int ^1_0 \mathrm{d}u \frac{\phi _{M}(u)}{\bar{u}}= & {} \int ^1_0 \mathrm{d}u \frac{\phi _{M}(1-u)}{1-u} = \int ^1_0 \mathrm{d}u \frac{\phi _{M}(u)}{u} \nonumber \\= & {} \langle \chi ^{-1}\rangle _{M} \end{aligned}$$

(5)

for $M=\pi$ and $M=\rho$. This relation together with the relation

$$\begin{aligned} \int ^1_0 \mathrm{d}u \phi _{M}(u)=1 \end{aligned}$$

(6)

allow us to perform the integrals in Eq. (2) and express the results in terms of the hadronic parameter $\langle \chi ^{-1}\rangle _{M}$. For the physical states $\eta$ and $\eta ^{\prime }$ they are related to the flavor basis $\eta _s$ and $\eta _q$ through [19]:

$$\begin{aligned} \left( \begin{array}{c} \eta \\ \eta ^{\prime } \end{array} \right) = \left( \begin{array}{cc} \cos \phi &{} - \sin \phi \\ \sin \phi &{} \cos \phi \end{array} \right) ~ \left( \begin{array}{c} \eta _q \\ \eta _s \end{array} \right) . \end{aligned}$$

(7)

Here the mixing angle is measured as $\phi = 46^\circ$ [27]. Upon using this relation we can easily calculate the decay amplitudes of $\bar{B}_s \rightarrow \eta M$ and $\bar{B}_s \rightarrow \eta ^{\prime } M$ for $M=\pi$ and $M=\rho$.

3 $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$ decays in the standard model

In this section we give our predictions for the branching ratios of $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$ decays in the standard model. In our analysis, we use the different set of values given in Ref. [8] for the hadronic parameters $\zeta ^{{B} M}$, $\zeta ^{ {B} M}_g$, $\zeta _J^{{B} M}$ and $\zeta _{Jg}^{{B} M}$ corresponding to the two solutions obtained from the $\chi ^2$ fit and assuming a $20\%$ error in their values due to the SU(3) symmetry breaking. It should be noted that the values that enter the SCET predictions are obtained from a fit to data assuming SM Wilson coefficients. They are valid for NP analysis provided the NP contribution to those channels which dominate the fit is small compared to the SM contribution. This is the case for the considered $Z'$ scenarios. We use for the inverse moment of the $\rho$ meson light-cone distribution amplitude $\langle \chi ^{-1}\rangle _{\rho }= 3.45$ [28] and $\langle \chi ^{-1}\rangle _{\pi }= 2.9\pm 0.4$ [29].

The amplitudes of $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$ decays in the SM can be obtained by setting $\tilde{C}^\mathrm{SM}_i = C^{Z'}_i = \tilde{C}^{Z'}_i = 0$. For $\bar{B}_s \rightarrow \pi ^0\,\eta ^{(')}$ decays we obtain

$$\begin{aligned}&\mathcal{A}_1(\bar{B}^0_s\rightarrow \eta \, \pi ^0)\times 10^6\simeq - 5.6\, C^\mathrm{SM}_9\,\lambda ^s_{c} \nonumber \\&\quad -(2.3\, C^\mathrm{SM}_1 + 3.7\, C^\mathrm{SM}_2 )\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta \, \pi ^0)\times 10^6\simeq - 5.1\, C^\mathrm{SM}_9\,\lambda ^s_{c}\nonumber \\&\quad - (1.6\, C^\mathrm{SM}_1 + 3.4\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_1(\bar{B}^0_s\rightarrow \eta ' \, \pi ^0)\times 10^6\simeq 0.4\, C^\mathrm{SM}_9\,\lambda ^s_{c}\nonumber \\&\quad + (0.1\, C^\mathrm{SM}_1+0.3\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta ' \, \pi ^0)\times 10^6\simeq 1.8\, C^\mathrm{SM}_9\,\lambda ^s_{c} \nonumber \\&\quad + (2.7\, C^\mathrm{SM}_1+1.2\, C^\mathrm{SM}_2 )\lambda ^s_{u}, \end{aligned}$$

(8)

while for $\bar{B}_s \rightarrow \rho ^0 \,\eta ^{(')}$ decays we obtain

$$\begin{aligned}&\mathcal{A}_1(\bar{B}_s\rightarrow \eta \,\rho ^0 )\times 10^6\simeq - 6.3\, C^\mathrm{SM}_9 \,\lambda ^s_{c} \nonumber \\&\quad - (3.4\, C^\mathrm{SM}_1 +4.2\,C^\mathrm{SM}_2 )\lambda ^s_{u} ,\nonumber \\&\mathcal{A}_2(\bar{B}_s\rightarrow \eta \,\rho ^0 )\times 10^6\simeq - 3.0\, C^\mathrm{SM}_9 \,\lambda ^s_{c} \nonumber \\&\quad - (1.6\, C^\mathrm{SM}_1+2.0\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_1(\bar{B}_s\rightarrow \eta '\,\rho ^0)\times 10^6\simeq 3.3\,C^\mathrm{SM}_9 \,\lambda ^s_{c} \nonumber \\&\quad + (0.8\, C^\mathrm{SM}_1+2.2\, C^\mathrm{SM}_2 )\lambda ^s_{u} ,\nonumber \\&\mathcal{A}_2(\bar{B}_s\rightarrow \eta '\,\rho ^0 )\times 10^6\simeq 8.2\, C^\mathrm{SM}_9\,\lambda ^s_{c} \nonumber \\&\quad + (6.3\, C^\mathrm{SM}_1+5.4\, C^\mathrm{SM}_2 )\lambda ^s_{u} , \end{aligned}$$

(9)

where we have used the unitarity of the CKM matrix to write $\lambda ^s_{t}= - \lambda ^s_{u}-\lambda ^s_{c}$ and also the hierarchy of the SM Wilson coefficients $C^\mathrm{SM}_1\gg C^\mathrm{SM}_{i}$ for $i=2,3,4,5,6,7,8,9,10$ and $C^\mathrm{SM}_9\gg C^\mathrm{SM}_{i}$ for $i=7,8,10$. The amplitudes $\mathcal{A}_1$ and $\mathcal{A}_2$ refers to solutions 1 and 2 of the SCET parameters, respectively. From the CKM matrix we can write to a good approximation $\lambda ^s_{c} \simeq \mathrm{Re} (\lambda ^s_{c}) \simeq 0.04$ and $|\lambda ^s_{u}| \simeq 2\times 10^{-2} \lambda ^s_{c}$. At leading order we have

$$\begin{aligned} C^\mathrm{SM}_1= 1.1, \quad C^\mathrm{SM}_2= -0.253, \quad C^\mathrm{SM}_9= -10.3\times 10^{-3} \end{aligned}$$

(10)

Clearly the real parts of the amplitudes in Eqs. (8) and (9) are dominant by the terms proportional to $C^\mathrm{SM}_9\,\lambda ^s_{c} \simeq -4\times 10^{-4}$. This can be attributed to several reasons. First, the cancellation that takes place in the $\lambda ^s_{u}$ terms of the amplitudes due to the sign difference between $C^\mathrm{SM}_1$ and $C^\mathrm{SM}_2$. Second, the sign difference between the terms proportional to $\lambda ^s_{c}$ and $\lambda ^s_{u}$ after taking into account the minus sign of the Wilson coefficient $C^\mathrm{SM}_9$. Finally, due to the hierarchy $|\lambda ^s_{u}| \simeq 2\times 10^{-2} \lambda ^s_{c}$. Another remark, the imaginary parts of the amplitudes in Eqs. (8) and (9) are suppressed as they are proportional to $|\lambda ^s_{u}| \simeq 2\times 10^{-2} \lambda ^s_{c} \simeq 10 ^{-4}$. As a consequence the predicted branching ratios for these decay modes are small as shown in Table 2. The last two columns give the predictions corresponding to the two solutions of the SCET parameters obtained from the $\chi ^2$ fit. The errors on the SCET predictions are due to SU(3) breaking effects and errors due to SCET parameters, respectively.

Table 2 Branching ratios of $\bar{B}^0_s\rightarrow \eta ^{(')}\, \pi ^0$ and $\bar{B}_s\rightarrow \eta ^{(')}\, \rho ^0$ decays in $10^{-6}$ units. The last two columns give the predictions corresponding to the two solutions of the SCET parameters obtained from the $\chi ^2$ fit. On the SCET predictions the errors are due to SU(3) breaking effects and errors due to SCET parameters respectively. For a comparison with previous studies in the literature, we list the results evaluated in QCDF [5], PQCD [7]

4 $Z'$ model contributions to $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\, \eta ^{(')}$ decays

One of the possible extension of the SM is to enlarge the SM gauge group to include additional $U(1)^{\prime }$ gauge group. This possibility is well motivated in several beyond SM theories such as theories with large extra dimensions [30] and grand unified theories [31]. As a consequence of the $U(1)^{\prime }$ gauge symmetry a new gauge boson, $Z^{\prime }$, arises. Basically $Z^{\prime }$ can have either family universal couplings or family non-universal couplings to the SM fermions. In the case that $Z^{\prime }$ gauge couplings are family universal they remain diagonal even in the presence of fermion flavor mixing by the GIM mechanism [36]. On the other hand and in some models like string models it is possible to have family-non-universal $Z^{\prime }$ couplings, due to the different constructions of the different families [36,37,38,39]. This scenario with family-non-universal couplings has theoretical and phenomenological motivations. For instance, possible anomalies in the Z-pole $b \bar{b}$ asymmetries suggest that the data are better fitted with a non-universal $Z^{\prime }$ [40]. Recent studies of the phenomenology of $Z^{\prime }$ have been performed in Refs. [13, 41,42,43,44,45,46,47]. For a detailed review of the physics of $Z^{\prime }$ gauge bosons we refer to Ref. [48].

In our analysis we will follow Refs. [6, 13, 40,41,42,43,44,45,46,47, 49,50,51,52,53,54,55] and consider non-universal $Z^{\prime }$ couplings in a way independent to a specific $Z^{\prime }$ model. Neglecting Z–$Z'$ mixing and assuming the absence of exotic fermions that can mix with the SM fermions through the $Z'$ couplings, the quark–antiquark–$Z^{\prime }$ interaction Lagrangian can be written as [6, 49, 53]

$$\begin{aligned} \mathcal{L}^{\mathrm{eff}}_{Z^{\prime }} = -\frac{g_{U(1)'}}{2\sqrt{2}}\sum _{ij} \bar{q}_i \left[ \zeta _L^{ij}\gamma ^{\mu }(1-\gamma _5) +\zeta _R^{ij}\gamma ^{\mu }(1+\gamma _5)\right] q_j Z^{\prime }_{\mu }, \end{aligned}$$

(11)

where i and j denote different quark flavors of the same type quarks. In order to simplify our analysis we introduce the parameters

$$\begin{aligned} i \Delta ^{ij}_{L,R} \equiv -\frac{g_{U(1)'}}{\sqrt{2}}\zeta _{L,R}^{ij}. \end{aligned}$$

(12)

In terms of these parameters we find that $Z^{\prime }$ contributions to the electroweak penguins relevant to our decay processes, at the electroweak scale, are given as

$$\begin{aligned} C^{Z^{\prime }}_{7}\,= & {} \,\,\frac{4}{3} \frac{ M_W^2 \Delta _L^{sb}}{g^2 M_{Z'}^2\lambda _t^{(s)}}\, \left( \Delta _R^{uu} -\Delta _R^{dd} \right) , \nonumber \\ \tilde{C}^{Z^{\prime }}_7\,= & {} \,\,\frac{4}{3} \frac{ M_W^2 \Delta _R^{sb}}{g^2 M_{Z'}^2\lambda _t^{(s)}}\, \left( \Delta _L^{uu} -\Delta _L^{dd} \right) , \nonumber \\ C^{Z^{\prime }}_{9}\,= & {} \, \,\frac{4}{3}\frac{ M_W^2 \Delta _L^{sb}}{g^2 M_{Z'}^2\lambda _t^{(s)}}\, \left( \Delta _L^{uu}-\Delta _L^{dd}\right) ,\nonumber \\ \tilde{C}^{Z^{\prime }}_9\,= & {} \, \,\frac{4}{3}\frac{ M_W^2 \Delta _R^{sb}}{g^2 M_{Z'}^2\lambda _t^{(s)}}\, \left( \Delta _R^{uu}-\Delta _R^{dd}\right) . \end{aligned}$$

(13)

The $SU(2)_L$ invariance implies that $\zeta _{L}^{uu}=\zeta _{L}^{dd}$ [6]. As a consequence $\Delta _{L}^{uu}=\Delta _{L}^{dd}$ and thus we are left with only two non-vanishing coefficients:

$$\begin{aligned} C^{Z^{\prime }}_{7}\,= & {} \,\,\frac{4}{3} \frac{ M_W^2 \Delta _L^{sb}}{g^2 M_{Z'}^2\lambda _t^{(s)}}\, \left( \Delta _R^{uu} -\Delta _R^{dd} \right) , \nonumber \\ \tilde{C}^{Z^{\prime }}_9\,= & {} \, \,\frac{4}{3}\frac{ M_W^2 \Delta _R^{sb}}{g^2 M_{Z'}^2\lambda _t^{(s)}}\, \left( \Delta _R^{uu}-\Delta _R^{dd}\right) . \end{aligned}$$

(14)

The last equation indicates that $Z'$ contributions to the Electroweak Wilson coefficients vanish in the case of $\Delta ^{uu}_{R} = \Delta ^{dd}_{R} = 0$ or $\Delta ^{uu}_{R} = \Delta ^{dd}_{R}$.

We discuss now the constraints imposed on the $\Delta ^{ij}_{L,R}$ parameters. To avoid tight constraints from semileptonic decays we consider $Z'$ model with vanishing couplings to leptons. In this model $Z'$ mass is much less constrained [6]. This can be explained as leptophobic $Z'$ bosons can avoid detection via traditional Drell–Yan processes. This choice can be adopted as the couplings of the $Z'$ boson to quarks are not related to their couplings to leptons. This leptophobic $Z'$ boson can appear in models with an $E_6$ gauge symmetry [56].

The most stringent constraints on the couplings $\Delta _L^{sb}$ and $\Delta _R^{sb}$ stem from $B_s$–$\overline{B}_s$ mixing. The effective Hamiltonian governs $B_s$–$\overline{B}_s$ mixing can be written as [42, 57, 58]

$$\begin{aligned} \mathcal{H}^{\Delta f=2}_{\mathrm{eff}}= C^{\mathrm{VLL}}_1 Q^{\mathrm{VLL}}_1+ C^{\mathrm{VRR}}_1 Q^{\mathrm{VRR}}_1+C^{\mathrm{LR}}_1 Q^{\mathrm{LR}}_1+ C^{\mathrm{LR}}_2 Q^{\mathrm{LR}}_2 \end{aligned}$$

(15)

where the four-quark operators are given by

$$\begin{aligned} Q^{\mathrm{VLL}}_1= & {} \left[ \bar{b}_\alpha \gamma ^\mu P_L s_\alpha \right] \left[ \bar{b}_\beta \gamma ^\mu P_L s_\beta \right] , \nonumber \\ Q^{\mathrm{VRR}}_1= & {} \left[ \bar{b}_\alpha \gamma ^\mu P_R s_\alpha \right] \left[ \bar{b}_\beta \gamma ^\mu P_R s_\beta \right] , \nonumber \\ Q^{\mathrm{LR}}_1= & {} \left[ \bar{b}_\alpha \gamma ^\mu P_L s_\alpha \right] \left[ \bar{b}_\beta \gamma ^\mu P_R s_\beta \right] , \nonumber \\ Q^{\mathrm{LR}}_2= & {} \left[ \bar{b}_\alpha P_L s_\alpha \right] \left[ \bar{b}_\beta P_R s_\beta \right] . \end{aligned}$$

(16)

The $\Delta B_s = 2$ mass difference is given by [42]

$$\begin{aligned} \Delta M_{B_s}=\frac{G^2_F}{6\pi ^2}M^2_W m_{B_s}|\lambda ^s_t|^2 F^2_{B_s}\hat{B}_{B_s}\eta _B |S(B_s)|. \end{aligned}$$

(17)

The expression of $S(B_s)$ can be written [42]

$$\begin{aligned} S(B_s)= & {} S_0(x_t)+[\Delta S(B_s)]_{\mathrm{VLL}}+[\Delta S(B_s)]_{\mathrm{VRR}} \nonumber \\&+\;[\Delta S(B_s)]_{\mathrm{LR}} \equiv |S(B_s)| e^{i \theta ^{B_s}_S} \end{aligned}$$

(18)

where the loop function $S_0(x_t)$ stems from the SM contribution to the $\Delta B_s = 2$ mass difference,

$$\begin{aligned} S_0(x_t)=\frac{4 x_t-11 x^2_t+x^3_t}{4(1-x_t)^2}-\frac{3 x^2_t\log x_t}{2(1-x_t)^3}, \end{aligned}$$

(19)

with $x_t= \frac{m^2_t}{m^2_W}$. The rest of the quantities in $S(B_s)$ account for the $Z^{\prime }$ contribution to the $\Delta B_s = 2$ mass difference. The expressions for $[\Delta S(B_s)]_{\mathrm{VLL}(\mathrm{VRR})}$ are given as [42]

$$\begin{aligned}{}[\Delta S(B_s)]_{\mathrm{VLL}(\mathrm{VRR})}= & {} \left[ \frac{\Delta _{L(R)}^{bs}}{\lambda ^s_t}\right] ^2\frac{4\tilde{r}}{M_{Z^\prime }^2 g^2_\mathrm{SM}}. \end{aligned}$$

(20)

Here $\tilde{r}$ is a factor that accounts for QCD renormalization group effects. Explicit expressions for $\tilde{r}$ and $g^2_\mathrm{SM}$ can be found in Ref. [43]. Turning now to the expression of $[\Delta S(B_s)]_{\mathrm{LR}}$ one finds that [42]

$$\begin{aligned}{}[\Delta S(B_s)]_{\mathrm{LR}}= & {} \frac{\Delta _L^{bs} \Delta _R^{bs}}{M_{Z^\prime }^2 T(B_s)}\big [C^{\mathrm{LR}}_1(\mu _{Z'})\langle Q^{\mathrm{LR}}_1(\mu _{Z'},B_s)\rangle \nonumber \\&+C^{\mathrm{LR}}_2(\mu _{Z'})\langle Q^{\mathrm{LR}}_2(\mu _{Z'},B_s)\rangle \big ] \end{aligned}$$

(21)

where

$$\begin{aligned} T(B_s)= & {} \frac{G^2_F}{12\pi ^2}M^2_W m_{B_s}|\lambda ^s_t|^2 F^2_{B_s}\hat{B}_{B_s}\eta _B,\nonumber \\ C^{\mathrm{LR}}_1(\mu _{Z'})= & {} 1+\frac{\alpha _s}{4\pi }\left( -\log \frac{M_{Z^\prime }^2}{\mu ^2_{Z'}}-\frac{1}{6}\right) ,\nonumber \\ C^{\mathrm{LR}}_2(\mu _{Z'})= & {} \frac{\alpha _s}{4\pi }\left( -6\log \frac{M_{Z^\prime }^2}{\mu ^2_{Z'}}-1\right) . \end{aligned}$$

(22)

The central values of the matrix elements $\langle Q^{\mathrm{LR}}_{1,2}(\mu _{Z'},B_s)\rangle$ can be found in Table 1 in Ref. [42]. In order to take the experimental and the hadronic uncertainties into account we follow Ref. [42] and require that the theory reproduce the data for $\Delta M_{B_s}$ within $\pm 5\%$. Thus for $\Delta M^{Exp}_{B_s}=17.761(22)\, ps^{-1}$ [59] the allowed range reads

$$\begin{aligned} 16.9/ ps \le \Delta M_{B_s} \le 18.6 /ps . \end{aligned}$$

(23)

The previous relation can be used to set constraints on the parameters $\Delta _{L,R}^{bs}$ once the values of $M_{Z'}$ are given. The most stringent constraints on $M_{Z'}$ are provided by CMS experiment [60]. For the sequential $Z'$ model the lower bound for $M_{Z'}$ is 2.59 TeV, while in other models values as low as 1 TeV are still possible.

In addition to the constraint from $\Delta M_{B_s}$ we need to take into account the constraint from $S_{\psi \phi }$ which can be defined as [42]

$$\begin{aligned} S_{\psi \phi }= \sin (2|\beta _s|-2\phi _{B_s}) \end{aligned}$$

(24)

where the phases $\beta _s$ and $\phi _{B_s}$ are defined by

$$\begin{aligned} V_{ts}= - |V_{ts}|e^{-i\beta _s}, \quad 2\phi _{B_s} = -\theta ^{B_s}_S \end{aligned}$$

(25)

with $\beta _s \simeq -1^\circ$ and $\theta ^{B_s}_S$ being the phase of $S(B_s)$ given in Eq. (18). The LHCb measurement of $S_{\psi \phi }$ reads [61]

$$\begin{aligned} S_{\psi \phi }= 0.002\pm 0.087. \end{aligned}$$

(26)

To use $S_{\psi \phi }$ as a constraint on the parameter space, we follow Ref. [42] and require $S_{\psi \phi }$ to vary in the range

$$\begin{aligned} - 0.18 \le S_{\psi \phi } \le 0.18 . \end{aligned}$$

(27)

In our analysis we will consider two scenarios. In the first scenario, the right-handed scenario (RHS), we assume $\Delta _{R}^{ij} \ne 0$ and $\Delta _{L}^{ij}= 0$. Note that the scenario with $\Delta _{L}^{ij} \ne 0$ and $\Delta _{R}^{ij}= 0$ is not interesting for our decay modes as this scenario leads to the vanishing of the Wilson coefficients. In the second scenario, the left–right scenario (LRS), we assume $\Delta _{R}^{ij} \ne 0$ and $\Delta _{L}^{ij} \ne 0$. In Ref. [42] a scenario with a left–right symmetry in the $Z'$-couplings to quarks i.e. $\Delta _{L}^{ij}= \Delta _{R}^{ij}$ has been adopted. This scenario is not relevant to our decay modes, as it leads to vanishing $Z'$ contributions to the amplitudes. This can be explained thus: the $SU(2)_L$ invariance implies that $\Delta _{L}^{uu}= \Delta _{L}^{dd}$ and hence $\Delta _{R}^{uu}- \Delta _{R}^{dd} = \Delta _{L}^{uu}- \Delta _{L}^{dd} = 0$ leading to vanishing $C^{Z'}_7$ and $\tilde{C}^{Z'}_9$. Thus in our LRS scenario we take $\Delta _{L}^{ij} \ne \Delta _{R}^{ij}$.

Fig. 1

Left allowed regions in the $\mathrm{Re} (\Delta _{R}^{bs})$–$\mathrm{Im} (\Delta _{R}^{bs})$ plane in RHS where red (blue) color corresponds to the bounds on $\Delta M_{B_s}$ ($S_{\psi \phi }$). Middle allowed regions in the $\mathrm{Re} (\Delta _{R}^{bs})$–$\mathrm{Re} (\Delta _{L}^{bs})$ plane in LRS from the bounds on $\Delta M_{B_s}$ corresponding to the case $\mathrm{Im} (\Delta _{L}^{bs})=\mathrm{Im} (\Delta _{R}^{bs})=0$. Right allowed regions in the $\mathrm{Im} (\Delta _{R}^{bs})$–$\mathrm{Im} (\Delta _{L}^{bs})$ plane in LRS from the bounds on $\Delta M_{B_s}$ corresponding to the case $\mathrm{Re}(\Delta _{L}^{bs})=\mathrm{Re} (\Delta _{R}^{bs})=0$. In all plots we take $M_{Z^\prime }=1$ TeV

We start our analysis by investigating the parameter space in the two scenarios. In the RHS the parameter space consists of the points $\big (M_{Z^\prime }, \mathrm{Re} (\Delta _{R}^{bs}), \mathrm{Im} (\Delta _{R}^{bs}) \big )$. For a given value of $M_{Z^\prime }$ we can use the constraints from $\Delta M_{B_s}$ and $S_{\psi \phi }$ to show the allowed regions in the $\mathrm{Re} (\Delta _{R}^{bs})$–$\mathrm{Im} (\Delta _{R}^{bs})$ plane. In Fig. 1 left, we plot the allowed regions in the $\mathrm{Re} (\Delta _{R}^{bs})$–$\mathrm{Im} (\Delta _{R}^{bs})$ plane for a value of $M_{Z^\prime }=1$ TeV. The red (blue) color corresponds to the allowed regions from the bounds on $\Delta M_{B_s}$ ($S_{\psi \phi }$). Clearly, from the figure, combining both constraints reduces the allowed regions in the $\mathrm{Re} (\Delta _{R}^{bs})$–$\mathrm{Im} (\Delta _{R}^{bs})$ plane.

We turn now to the LRS. The parameter space in this case consists of the points $\big (M_{Z^\prime }, \mathrm{Re} (\Delta _{L}^{bs}), \mathrm{Im} (\Delta _{L}^{bs}) \big ),\mathrm{Re} (\Delta _{R}^{bs}), \mathrm{Im} (\Delta _{R}^{bs}) \big )$. For a given value of $M_{Z^\prime }$ we can use the constraints from $\Delta M_{B_s}$ and $S_{\psi \phi }$ to find the allowed regions in the $\big (\mathrm{Re} (\Delta _{L}^{bs}), \mathrm{Im} (\Delta _{L}^{bs}),\mathrm{Re} (\Delta _{R}^{bs}), \mathrm{Im} (\Delta _{R}^{bs}) \big )$ space. In Fig. 1, middle, we show the allowed regions in the $\mathrm{Re} (\Delta _{R}^{bs})$–$\mathrm{Re} (\Delta _{L}^{bs})$ plane from the bounds on $\Delta M_{B_s}$ at $M_{Z^\prime }=1$ TeV corresponding to case $\mathrm{Im} (\Delta _{L}^{bs})=\mathrm{Im} (\Delta _{R}^{bs})=0$. Their is no bound from $S_{\psi \phi }$ in this case. In the same figure, right, we show the allowed regions in the $\mathrm{Im} (\Delta _{R}^{bs})$–$\mathrm{Im} (\Delta _{L}^{bs})$ plane in LRS from the bounds on $\Delta M_{B_s}$ corresponding to the case $\mathrm{Re}(\Delta _{L}^{bs})=\mathrm{Re} (\Delta _{R}^{bs})=0$. Regarding the bounds from $S_{\psi \phi }$, for this case, we find that they are so loose. This can be explained by $Z'$ contribution to $\Delta M_{B_s}$ being dominated by the new LR operators which become real in this case and hence the phase $\theta ^{B_s}_S \simeq 0$. In addition to the previous two cases in the LRS there is a general case where none of the real or imaginary parts of $\Delta _{L}^{bs}$ and $\Delta _{R}^{bs}$ are equal to zero. In Table 3 we list some sample sets that satisfy both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints at $M_{Z'} = 1$ TeV corresponding to this general case. In obtaining these sets we run each of the real and imaginary parts of $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ over the interval $[-0.01,0.01]$ and require both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints to be satisfied. Having discussed the parameter space we proceed to an estimate of the predictions for the $Z'$ Wilson coefficients and accordingly the branching ratios.

We see from Eq. (14) that the Wilson coefficients $C^{Z^{\prime }}_{7}$ and $\tilde{C}^{Z^{\prime }}_9$ depend on the difference $\Delta _R^{uu}-\Delta _R^{dd}$. Clearly $C^{Z^{\prime }}_{7}$ and $\tilde{C}^{Z^{\prime }}_9$ will vanish if the couplings $\zeta _{R}^{qq}$, and hence $\Delta _{R}^{qq}$, are universal. On the other hand the maximum values of the Wilson coefficients $C^{Z^{\prime }}_{7}$ and $\tilde{C}^{Z^{\prime }}_9$ correspond to the maximum value of the coupling difference $\Delta _R^{uu}-\Delta _R^{dd}$. In our analysis we assume that the difference $\Delta _R^{uu}-\Delta _R^{dd}$ is real and $\Delta _R^{uu}-\Delta _R^{dd} = 1$ to get an estimation of the upper values of the branching ratios of $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$.

In the RHS scenario $\Delta ^{sb}_L = 0$ and $\Delta ^{sb}_R\ne 0$. As a result $C^{Z^{\prime }}_{7} =0$ and $\tilde{C}^{Z^{\prime }}_9 \ne 0$. This means that $Z^{\prime }$ contributes to the amplitude of the given decay process only through the non-vanishing $\tilde{C}^{Z^{\prime }}_9$.

Table 3 Sample sets of the parameter space at $M_{Z^\prime }=1$ TeV that satisfy both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints

Fig. 2

Left (right): contours of the real (imaginary) part of $\tilde{C}^{Z^{\prime }}_9$ normalized by the SM Wilson coefficient ${C}^\mathrm{SM}_9$ in the RHS. The shaded red (blue) region is allowed from the bounds on $\Delta M_{B_s}$ ($S_{\psi \phi }$) for $M_{Z^\prime }=1$ TeV

The amplitudes of $\bar{B}_s \rightarrow \pi ^0 \,\eta ^{(')}$ including $Z^{\prime }$ contributions then become

$$\begin{aligned}&\mathcal{A}_1(\bar{B}^0_s\rightarrow \eta \, \pi ^0)\times 10^6 \simeq - \left( 5.6 -1.8\, \mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. -1.8\, \mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \,I\right) C^\mathrm{SM}_9\,\lambda ^s_{c} -(2.3\, C^\mathrm{SM}_1 + 3.7\, C^\mathrm{SM}_2 )\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta \, \pi ^0)\times 10^6\simeq - \left( 5.1-3.3\,\mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. -3.3\,\mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \,I\right) C^\mathrm{SM}_9\,\lambda ^s_{c} - (1.6\, C^\mathrm{SM}_1 + 3.4\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_1(\bar{B}^0_s\rightarrow \eta ' \, \pi ^0)\times 10^6\simeq \left( 0.4- 0.3\,\mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. -0.3\,\mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) I \right) C^\mathrm{SM}_9\,\lambda ^s_{c} + (0.1\, C^\mathrm{SM}_1+0.3\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta ' \, \pi ^0)\times 10^6 \simeq \left( 1.8 +6.6\,\mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. +6.6\,\mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) I\right) C^\mathrm{SM}_9\,\lambda ^s_{c} + (2.7\, C^\mathrm{SM}_1+1.2\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\ \end{aligned}$$

(28)

while for $\bar{B}_s \rightarrow \rho ^0 \,\eta ^{(')}$ decays we obtain

$$\begin{aligned}&\mathcal{A}_1(\bar{B}_s\rightarrow \eta \,\rho ^0 )\times 10^6\simeq -\left( 6.3- 0.3 \mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. - 0.3 \mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) I\right) C^\mathrm{SM}_9 \,\lambda ^s_{c} - (3.4\, C^\mathrm{SM}_1 +4.2\,C^\mathrm{SM}_2 )\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}_s\rightarrow \eta \,\rho ^0 )\times 10^6\simeq -\left( 3.0 -0.1 \mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. -0.1 \mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) I\right) C^\mathrm{SM}_9 \,\lambda ^s_{c} - (1.6\, C^\mathrm{SM}_1+2.0\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_1(\bar{B}_s\rightarrow \eta '\,\rho ^0 )\times 10^6 \simeq \left( 3.3 - 3.1 \mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. - 3.1 \mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) I \right) C^\mathrm{SM}_9 \,\lambda ^s_{c} + (0.8\, C^\mathrm{SM}_1+2.2\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}_s\rightarrow \eta '\,\rho ^0 )\times 10^6 \simeq \left( 8.2 \,+5.2 \mathrm{Re}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) \right. \nonumber \\&\quad \left. +5.2 \mathrm{Im}\left( \frac{\tilde{C}^{Z^{\prime }}_9}{C^\mathrm{SM}_9}\right) I\right) C^\mathrm{SM}_9\,\lambda ^s_{c} + (6.3\, C^\mathrm{SM}_1+5.4\, C^\mathrm{SM}_2 )\lambda ^s_{u}.\nonumber \\ \end{aligned}$$

(29)

Fig. 3

Left (right): contours of ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}({\mathcal R}^{\,\rho ^0\,\eta '}_{2})$ in the RHS. The shaded red (blue) region is allowed from the bounds on $\Delta M_{B_s}$ ($S_{\psi \phi }$) for $M_{Z^\prime }=1$ TeV

We discuss now the predictions of $\tilde{C}^{Z^{\prime }}_9$. In Fig. 2 we show the contours of the real and imaginary parts of $\tilde{C}^{Z^{\prime }}_9$ normalized by the SM Wilson coefficient ${C}^\mathrm{SM}_9$. The shaded red (blue) region is allowed from the bounds on $\Delta M_{B_s}$ ($S_{\psi \phi }$) for a value of $M_{Z^\prime }=1$ TeV. As can be seen from the figure, the real part of $\tilde{C}^{Z^{\prime }}_9$ can reach a maximum value of about $25\%$ of the SM Wilson coefficient ${C}^\mathrm{SM}_9$. On the other hand the imaginary part of $\tilde{C}^{Z^{\prime }}_9$ can reach a maximum value equal to ${C}^\mathrm{SM}_9$ at the point $\big (\mathrm{Re} (\Delta _{R}^{bs}) = 0,\mathrm{Im} (\Delta _{R}^{bs})= \pm 0.018 \big )$ in the same figure. At this point we find that $S_{\psi \phi }\simeq 0.035$, satisfying the $S_{\psi \phi }$ bound in Eq. (27). Moreover, at the same point, we find that $[\Delta S(B_s)]_{\mathrm{VRR}}= -4.4\simeq -2\, S_0(x_t)$ i.e. $S_{Z'} (B_s) \simeq -2 S_\mathrm{SM} (B_s)$. Thus $\mid S_\mathrm{SM} (B_s) + S_{Z'} (B_s)\mid \simeq \mid - S_\mathrm{SM} (B_s)\mid$ and thus the point $\big (\mathrm{Re} (\Delta _{R}^{bs}) = 0,\mathrm{Im} (\Delta _{R}^{bs}) = \pm 0.018\big )$ satisfies $\Delta M_{B_s}$ constraint.

As can be seen from Eqs. (28) and (29) the decay amplitudes $\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta ' \, \pi ^0)$ and $\mathcal{A}_2(\bar{B}_s\rightarrow \eta '\,\rho ^0 )$ have the largest coefficients of the real and imaginary parts of $\tilde{C}^{Z^{\prime }}_9$ compared to the other amplitudes. Thus we expect that these amplitudes receive the largest enhancements due to $\tilde{C}^{Z^{\prime }}_9$ and consequently their branching ratios. We define the ratio ${\mathcal R}^{\,M_1M_2}_{i}=\big (BR^{SM+Z'}_{i}(\bar{B}_s\rightarrow M_1M_2)-BR^\mathrm{SM}_{i}(\bar{B}_s\rightarrow M_1M_2)\big )/BR^\mathrm{SM}_{i}(\bar{B}_s\rightarrow M_1M_2)$ where $i= 1,2$ refers to solutions 1, 2 for the SCET parameter space and BR refers to the branching ratio. The numerical value of ${\mathcal R}^{\,M_1M_2}_{i}$ gives an estimation of the size of the enhancement or the suppression in the branching ratios due to the contributions of $Z'$ to the amplitude of the given decay process. In Fig. 3, left (right), we show the contours of ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}({\mathcal R}^{\,\rho ^0\,\eta '}_{2})$ over the allowed regions in the $\mathrm{Re} (\Delta _{R}^{bs})$–$\mathrm{Im} (\Delta _{R}^{bs})$ plane, satisfying both $S_{\psi \phi }$ and $\Delta M_{B_s}$ constraints, for a value of $M_{Z^\prime }=1$ TeV. We see from Fig. 3, left, that ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ can reach a maximum value of about 6 at the point $\big (\mathrm{Re} (\Delta _{R}^{bs}) = 0,\mathrm{Im} (\Delta _{R}^{bs}) = - 0.018\big )$. This means that at this point $Z^\prime$ contributions can enhance the total branching ratio of $\bar{B}^0_s\rightarrow \,\pi ^0\,\eta '$ to six times the SM prediction. Recall that at this point $S_{\psi \phi }\simeq 0.035$ satisfying the $S_{\psi \phi }$ bound in Eq. (27) and $S_{Z'} (B_s) \simeq -2 S_\mathrm{SM} (B_s)$ resulting in $\mid S_\mathrm{SM} (B_s) + S_{Z'} (B_s)\mid \simeq \mid - S_\mathrm{SM} (B_s)\mid$ and thus $\Delta M_{B_s}$ constraint is also satisfied. On the other hand from Fig. 3, right, we see that ${\mathcal R}^{\,\rho ^0\,\eta '}_{2}$ can reach a maximum value of only about 2.5 also at the point $\big (\mathrm{Re} (\Delta _{R}^{bs}) = 0,\mathrm{Im} (\Delta _{R}^{bs}) = - 0.018\big )$. Thus the enhancement in the total branching ratio of the decay mode $B^0_s\rightarrow \,\rho ^0\,\eta '$ is not much compared to the enhancement in decay mode $\bar{B}^0_s\rightarrow \,\pi ^0\,\eta '$.

We consider now LRS scenario in which $\Delta ^{sb}_L\ne 0$ and $\Delta ^{sb}_R\ne 0$. As a consequence ${C}^{Z^{\prime }}_7 \ne 0$ and $\tilde{C}^{Z^{\prime }}_9\ne 0$ and hence the amplitudes of $\bar{B}_s \rightarrow \pi ^0 \,\eta ^{(')}$ can be written as

$$\begin{aligned}&\mathcal{A}_1(\bar{B}^0_s\rightarrow \eta \, \pi ^0)\times 10^6\simeq (5.6\, C^{Z^{\prime }}_7- 5.6\, C^\mathrm{SM}_9+ 1.8\, \tilde{C}^{Z^{\prime }}_9 )\lambda ^s_{c} \nonumber \\&\quad -(2.3\, C^\mathrm{SM}_1 + 3.7\, C^\mathrm{SM}_2 )\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta \, \pi ^0)\times 10^6\simeq ( 5.1\, C^{Z^{\prime }}_7 - 5.1\, C^\mathrm{SM}_9+3.3\, \tilde{C}^{Z^{\prime }}_9)\lambda ^s_{c}\nonumber \\&\quad - (1.6\, C^\mathrm{SM}_1 + 3.4\, C^\mathrm{SM}_2)\lambda ^s_{u},\nonumber \\&\mathcal{A}_1(\bar{B}^0_s\rightarrow \eta ' \, \pi ^0)\times 10^6 \simeq (-0.4\, C^{Z^{\prime }}_7+ 0.4\, C^\mathrm{SM}_9- 0.3 \,\tilde{C}^{Z^{\prime }}_9)\lambda ^s_{c}\nonumber \\&\quad + (0.1\, C^\mathrm{SM}_1+0.3\, C^\mathrm{SM}_2)\lambda ^s_{u}\nonumber \\&\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta ' \, \pi ^0)\times 10^6 \simeq ( -1.8 \, C^{Z^{\prime }}_7 +1.8\, C^\mathrm{SM}_9+ 6.6\, \tilde{C}^{Z^{\prime }}_9)\lambda ^s_{c}\nonumber \\&\quad + (2.7\, C^\mathrm{SM}_1+1.2 C^\mathrm{SM}_2 )\lambda ^s_{u}, \end{aligned}$$

(30)

while for $\bar{B}_s \rightarrow \rho ^0 \,\eta ^{(')}$ decays we obtain

$$\begin{aligned}&\mathcal{A}_1(\bar{B}_s\rightarrow \eta \,\rho ^0 )\times 10^6 \simeq (-11.4 \,C^{Z^{\prime }}_7- 6.3\, C^\mathrm{SM}_9 + 0.3\, \tilde{C}^{Z^{\prime }}_9 )\lambda ^s_{c} \nonumber \\&\quad - (3.4\, C^\mathrm{SM}_1 +4.2\,C^\mathrm{SM}_2 )\lambda ^s_{u} ,\nonumber \\&\mathcal{A}_2(\bar{B}_s\rightarrow \eta \,\rho ^0 )\times 10^6 \simeq (-13.3\, C^{Z^{\prime }}_7- 3.0\, C^\mathrm{SM}_9 +0.1\, \tilde{C}^{Z^{\prime }}_9)\lambda ^s_{c} \nonumber \\&\quad - (1.6\, C^\mathrm{SM}_1+2.0\, C^\mathrm{SM}_2 )\lambda ^s_{u},\nonumber \\&\mathcal{A}_1(\bar{B}_s \rightarrow \eta '\,\rho ^0 )\times 10^6 \simeq (-1.9\, C^{Z^{\prime }}_7+3.3\,C^\mathrm{SM}_9-3.1\,\tilde{C}^{Z^{\prime }}_9 )\lambda ^s_{c}\nonumber \\&\quad + (0.8\, C^\mathrm{SM}_1+2.2\, C^\mathrm{SM}_2 )\lambda ^s_{u},\nonumber \\&\mathcal{A}_2(\bar{B}_s\rightarrow \eta '\,\rho ^0 )\times 10^6 \simeq (-2.6 \, C^{Z^{\prime }}_7+8.2\, C^\mathrm{SM}_9 +5.2\, \tilde{C}^{Z^{\prime }}_9)\lambda ^s_{c} \nonumber \\&\quad + (6.3\, C^\mathrm{SM}_1+5.4\, C^\mathrm{SM}_2 )\lambda ^s_{u}. \end{aligned}$$

(31)

Fig. 4

Left contours of $\mathrm{Re}({C}^{Z^{\prime }}_7)$ ($\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)$) in green (blue) color normalized by the SM Wilson coefficient ${C}^\mathrm{SM}_9$ in the LRS corresponding to the case $\mathrm{Im}(\Delta ^{sb}_R)= \mathrm{Im}(\Delta ^{sb}_L) =0$. Right contours of $\mathrm{Im}({C}^{Z^{\prime }}_7)$ ($\mathrm{Im}(\tilde{C}^{Z^{\prime }}_9)$) in orange (magenta) color normalized by the SM Wilson coefficient ${C}^\mathrm{SM}_9$ in the LRS corresponding to the case $\mathrm{Re}(\Delta ^{sb}_R)= \mathrm{Re}(\Delta ^{sb}_L) =0$. In both plots the shaded colored regions are allowed from the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV

We discuss now the predictions of ${C}^{Z^{\prime }}_7$ and $\tilde{C}^{Z^{\prime }}_9$ in the LRS. We consider the case for which $\mathrm{Im}(\Delta ^{sb}_R)= \mathrm{Im}(\Delta ^{sb}_L) =0$. In this case ${C}^{Z^{\prime }}_7\equiv \mathrm{Re}({C}^{Z^{\prime }}_7)$ and $\tilde{C}^{Z^{\prime }}_9 \equiv \mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)$. In Fig. 4, left, we show the contours of $\mathrm{Re}({C}^{Z^{\prime }}_7)$ ($\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)$) in green (blue) color normalized by the SM Wilson coefficient ${C}^\mathrm{SM}_9$. The shaded red regions are allowed from the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV as discussed before. The contours of $\mathrm{Re}({C}^{Z^{\prime }}_7)$ ($\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)$) are straight lines as ${C}^{Z^{\prime }}_7$ ($\tilde{C}^{Z^{\prime }}_9$) is a function of $\mathrm{Re}(\Delta ^{sb}_L)$ ($\mathrm{Re}(\Delta ^{sb}_R)$) only. Clearly from the figure $\mathrm{Re}({C}^{Z^{\prime }}_7)$ can reach a maximum value around $0.4\, {C}^\mathrm{SM}_9$ while $\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)$ can reach a maximum value around $0.6\, {C}^\mathrm{SM}_9$. However, $\big (\mathrm{Re}({C}^{Z^{\prime }}_7), \mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)\big ) = \big ( 0.4, 0.6\big )$ are excluded by the bounds on $\Delta M_{B_s}$, which require $\mathrm{Re}(\Delta ^{sb}_R)$ and $\mathrm{Re}(\Delta ^{sb}_L)$, and hence $\mathrm{Re}({C}^{Z^{\prime }}_7)$ and $\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)$, which should not be large simultaneously.

We consider another case where $\mathrm{Re}(\Delta ^{sb}_R)= \mathrm{Re}(\Delta ^{sb}_L) =0$. In this case ${C}^{Z^{\prime }}_7\equiv \mathrm{Im}({C}^{Z^{\prime }}_7)$ and $\tilde{C}^{Z^{\prime }}_9 \equiv \mathrm{Im}(\tilde{C}^{Z^{\prime }}_9)$. In Fig. 4, right, we show the contours of $\mathrm{Im}({C}^{Z^{\prime }}_7)$ ($\mathrm{Im}(\tilde{C}^{Z^{\prime }}_9)$) in orange (magenta) color normalized by the SM Wilson coefficient ${C}^\mathrm{SM}_9$. The shaded blue regions are allowed from the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV. Recall that the constraints from $S_{\psi \phi }$ are so loose as discussed before. The conclusion for this case is the same as the previous case with just doing the replacements $\mathrm{Re}({C}^{Z^{\prime }}_7)\rightarrow \mathrm{Im}({C}^{Z^{\prime }}_7)$ and $\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)\rightarrow \mathrm{Im}(\tilde{C}^{Z^{\prime }}_9)$.

We finally consider the general case where none of the real or imaginary parts of $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ is equal to zero. In Table 4 we list the predictions of ${C}^{Z^{\prime }}_7$ and $\tilde{C}^{Z^{\prime }}_9$ corresponding to some sample sets of the parameter space allowed by both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints for $M_{Z'}= 1$ TeV. As before, in obtaining these results we run each of the real and imaginary parts of $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ over the interval $[-0.01,0.01]$ requiring that both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints be satisfied. From the table we note that $\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)= \mathrm{Im}(\tilde{C}^{Z^{\prime }}_9) \simeq 0.6\,C^\mathrm{SM}_9$ corresponding to set III of the allowed parameter space. This is the maximum value of $\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)\big (\mathrm{Im}(\tilde{C}^{Z^{\prime }}_9)\big )$ obtained in our scan for all points in the parameter space that satisfy $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints. Regarding ${C}^{Z^{\prime }}_7$ we find that $\mathrm{Re}({C}^{Z^{\prime }}_7)\big (\mathrm{Im}({C}^{Z^{\prime }}_7)\big )$ can reach a maximum value around $0.4\, {C}^\mathrm{SM}_9$.

Table 4 Predictions of ${C}^{Z^{\prime }}_7$ and $\tilde{C}^{Z^{\prime }}_9$ corresponding to some sample sets of the parameter space for $M_{Z^\prime }=1$ TeV allowed by both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints

Finally, we turn to the predictions of the branching ratios of the processes under consideration. We start with $\bar{B}^0_s\rightarrow \eta (\eta ')\, \pi ^0$ decays. Their related amplitudes are given in Eq. (30). Clearly the amplitude $\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta '\, \pi ^0)$ has the largest coefficient of $\tilde{C}^{Z^{\prime }}_9$ compared to the coefficients of both the ${C}^{Z^{\prime }}_7$ and the ${C}^\mathrm{SM}_9$ in the other amplitudes. As a result this amplitude receives the largest enhancement due to $Z^{\prime }$ contributions. In Fig. 5 left, we show the contours of ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ in the LRS for the case $\mathrm{Im}(\Delta ^{sb}_L)=\mathrm{Im}(\Delta ^{sb}_R)=0$ where the shaded red regions satisfy the bounds on $\Delta M_{B_s}$. In the same figure right, we show the contours of ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ for the case $\mathrm{Re}(\Delta ^{sb}_L)=\mathrm{Re}(\Delta ^{sb}_R)=0$ where the shaded blue regions are allowed by the bounds on $\Delta M_{B_s}$. In both plots we take $M_{Z^\prime }=1$ TeV. We see from the figure that ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ can reach a maximum value of about 2.5 in both cases.

In Table 5 we list the predictions of ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ corresponding to the general case where non of the real or imaginary parts of $\Delta _{L}^{bs}$ and $\Delta _{R}^{bs}$ is equal to zero and are allowed by both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints. As before, in obtaining these results we run each of the real and imaginary parts of $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ over the interval $[-0.01,0.01]$ requiring that both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints be satisfied for $M_{Z^\prime }=1$ TeV. From the table we note that ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}\simeq 4.6$ corresponding to set I of the allowed parameter space. This means that $Z^\prime$ contributions can enhance the total branching ratio of $B^0_s\rightarrow \,\pi ^0\,\eta '$ 4.6 times the SM prediction. This is the maximum value we obtained in our scan for all sets of the parameter space that satisfy both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints.

Fig. 5

Left contours of ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ in the LRS for the case $\mathrm{Im}(\Delta ^{sb}_L)=\mathrm{Im}(\Delta ^{sb}_R)=0$. Right contours of ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ in the LRS for the case $\mathrm{Re}(\Delta ^{sb}_L)=\mathrm{Re}(\Delta ^{sb}_R)=0$. In both plots the shaded colored regions satisfy the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV

We turn now to the decay modes $\bar{B}^0_s\rightarrow \eta (\eta ')\, \rho ^0$. Their related amplitudes are given in Eq. (31). We note that the amplitude $\mathcal{A}_2(\bar{B}^0_s\rightarrow \eta \, \rho ^0)$ has the largest coefficient of ${C}^{Z^{\prime }}_7$ compared to the coefficients of both the $\tilde{C}^{Z^{\prime }}_9$ and the ${C}^\mathrm{SM}_9$ in the other amplitudes. As a result this amplitude receives the largest enhancement due to $Z^{\prime }$ contributions. In Fig. 6 left, we show the contours of ${\mathcal R}^{\,\rho ^0\,\eta }_{2}$ in the LRS for the case $\mathrm{Im}(\Delta ^{sb}_L)=\mathrm{Im}(\Delta ^{sb}_R)=0$. In the same figure right, we show the contours of ${\mathcal R}^{\,\rho ^0\,\eta }_{2}$ in the LRS for the case $\mathrm{Re}(\Delta ^{sb}_L)=\mathrm{Re}(\Delta ^{sb}_R)=0$. In the figure the shaded colored regions satisfy the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV. We see from the figure that $Z^\prime$ contributions can enhance the total branching ratio of $B^0_s\rightarrow \,\rho ^0\,\eta$ by about one order of magnitude compared to the SM prediction. In fact this is the conclusion also for the general case where none of the real or imaginary parts of $\Delta _{L}^{bs}$ and $\Delta _{R}^{bs}$ is equal to zero as shown in Table 6. In that table we list the predictions of ${\mathcal R}^{\,\rho ^0\,\eta }_{2}$ corresponding to some sample sets of the parameter space allowed by both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints. As before, in obtaining these results we run each of the real and imaginary parts of $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ over the interval $[-0.01,0.01]$ requiring that both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints be satisfied at a value $M_{Z^\prime }=1$ TeV. From the table we see that $Z^\prime$ contributions can enhance the total branching ratio of $B^0_s\rightarrow \,\rho ^0\,\eta$ up to one order of magnitude comparing to the SM prediction.

Table 5 Predictions for ${\mathcal R}^{\,\pi ^0\,\eta '}_{2}$ corresponding to some sample sets of the parameter space for $M_{Z^\prime }=1$ TeV allowed by both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints

Fig. 6

Left contours of ${\mathcal R}^{\,\eta \,\rho ^0}_{2}$ in the LRS for the case $\mathrm{Im}(\Delta ^{sb}_L)=\mathrm{Im}(\Delta ^{sb}_R)=0$. Right contours of ${\mathcal R}^{\,\rho ^0\,\eta }_{2}$ in the LRS for the case $\mathrm{Re}(\Delta ^{sb}_L)=\mathrm{Re}(\Delta ^{sb}_R)=0$. In both plots the shaded colored regions satisfy the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV

Finally, from Fig. 6 we note that the enhancement of the branching ratios by an order of magnitude occurs in the regions in the parameter space corresponding to the thin branches in the figure. In these regions the $[\Delta S(B_s)]_{\mathrm{VLL}(\mathrm{VRR})}$ and $[\Delta S(B_s)]_{\mathrm{LR}}$ contributions to $B_s$–$\bar{B}_s$ mixing given in Eqs. (20) and (21) cancel each other to a large extent. For a proper interpretation of the given upper limit for the enhancement it is thus necessary to know the degree of fine-tuning between $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ for the corresponding points in the parameter space. For the case of real $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$, the fine-tuning can be quantified by the measure $X_{B_s}$ introduced in Eq. (26) in Ref. [62]. The corresponding expression reads

$$\begin{aligned} X_{B_s}= & {} \frac{(\Delta ^{sb}_L)^2+(\Delta ^{sb}_R)^2 - b_{B_s}\Delta ^{sb}_L\Delta ^{sb}_R}{(\Delta ^{sb}_L)^2+(\Delta ^{sb}_R)^2 + b_{B_s} \Delta ^{sb}_L\Delta ^{sb}_R} \end{aligned}$$

(32)

where

$$\begin{aligned} b_{B_s}= & {} \frac{(\lambda ^s_t)^2 g^2_\mathrm{SM}}{4\tilde{r} T(B_s)}\big [C^{\mathrm{LR}}_1(\mu _{Z'})\langle Q^{\mathrm{LR}}_1(\mu _{Z'},B_s)\rangle \nonumber \\&+C^{\mathrm{LR}}_2(\mu _{Z'})\langle Q^{\mathrm{LR}}_2(\mu _{Z'},B_s)\rangle \big ]. \end{aligned}$$

(33)

At $M_{Z^\prime } = 1$ TeV we find that $b_{B_s} \simeq -10.8$. In Fig. 7 we show the points in the $\Delta ^{sb}_L-\Delta ^{sb}_R$ satisfying $X_{B_s} < 10$, $10< X_{B_s} < 100$ and $100 < X_{B_s}$ in green, orange and blue colors, respectively. All colored points satisfy the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV. Clearly from Figs. 6, 7 if we exclude points that lead to $100 < X_{B_s}$ from the parameter space and exclude their corresponding predictions of the branching ratios we still can have an enhancement by an order of magnitude. As before this enhancement occurs for rather fine-tuned points where $\Delta M_{B_s}$ constraint on $\mid S_\mathrm{SM} (B_s) + S_{Z'} (B_s)\mid$ is fulfilled by overcompensating the SM via $S_{Z'} (B_s) \simeq -2 S_\mathrm{SM} (B_s)$. For instance, this enhancement is still allowed for the points lying on the thin red curved branch in Fig. 6 for which $\Delta ^{sb}_R < -0.04$ and $\Delta ^{sb}_L < -0.06$. In fact this conclusion agrees with the findings of Ref. [62] where one found that sizable enhancements in the branching ratios of the process under the considerations are still possible for a fine-tuning of $X_{B_s} \le 100$.

Table 6 Predictions for ${\mathcal R}^{\,\rho ^0\,\eta }_{2}$ corresponding to some sample sets of the parameter space for $M_{Z^\prime }=1$ TeV allowed by both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints

Fig. 7

Points satisfying $X_{B_s} < 10$, $10< X_{B_s} < 100$ and $100 < X_{B_s}$ in green, orange and blue colors, respectively. All colored points satisfy the bounds on $\Delta M_{B_s}$ for $M_{Z^\prime }=1$ TeV

5 Conclusion

In this work we have studied the decay modes $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$ within a model with an additional $U(1)'$ gauge symmetry and adopting SCET as a framework to calculate the amplitudes. We have derived the contributions to the amplitudes, within Soft Collinear Effective Theory, arising from new physics contributions to the weak effective Hamiltonian.

In the study we have considered a leptophobic $Z'$ boson where its couplings to leptons vanish. Such a model can appear in models with an $E_6$ gauge symmetry. In this model $Z'$ mass is much less constrained and the strongest constraints on the parameter space can be obtained by considering $B_s$–$\overline{B}_s$ mixing. We considered two scenarios, the Right-Handed Scenario, RHS, and Left–Right Scenario, LRS, based on the couplings of $Z'$ to quarks that appear in the Wilson coefficients. In these scenarios we discussed the constraints on the parameter space from considering $B_s$–$\overline{B}_s$ mixing. As a consequence, we presented the predictions of the Wilson coefficients and accordingly the branching ratios of $\bar{B}_s \rightarrow \pi ^0(\rho ^0 )\,\eta ^{(')}$.

In the RHS we found that the real part of $\tilde{C}^{Z^{\prime }}_9$ can reach a maximum value of about $25\%$ of ${C}^\mathrm{SM}_9$. On the other hand the imaginary part of $\tilde{C}^{Z^{\prime }}_9$ can reach a maximum value equal to ${C}^\mathrm{SM}_9$. As a result we found that the decay amplitudes $\mathcal{A}_2(\bar{B}^0_s\rightarrow \pi ^0 \eta ')$ and $\mathcal{A}_2(\bar{B}_s\rightarrow \rho ^0\,\eta ' )$ receive the largest enhancements due to the contributions of $\tilde{C}^{Z^{\prime }}_9$ as they have the largest coefficients of the real and imaginary parts of $\tilde{C}^{Z^{\prime }}_9$ compared to the other amplitudes. Accordingly we found that $Z^\prime$ contributions can enhance the total branching ratio of $\bar{B}^0_s\rightarrow \,\pi ^0\,\eta '$ to six times the SM prediction while for $B^0_s\rightarrow \,\rho ^0\,\eta '$ it is just 2.5 times the SM prediction. This kind of enhancement occurs for a rather fine-tuned point where $\Delta M_{B_s}$ constraint on $\mid S_\mathrm{SM} (B_s) + S_{Z'} (B_s)\mid$ is fulfilled by overcompensating the SM via $S_{Z'} (B_s) \simeq -2 S_\mathrm{SM} (B_s)$. Moreover, the constraint from $S_{\psi \phi }$ is also satisfied as the $Z^\prime$ coupling to the b and s quarks is real at this point.

In the LRS the parameter space consists of the real and imaginary parts of $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ in addition to $M_{Z'}$. At a value of $M_{Z'}= 1$ TeV we scanned the real and imaginary parts of $\Delta ^{sb}_L$ and $\Delta ^{sb}_R$ over the interval $[-0.01,0.01]$ requiring that both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints be satisfied. As a consequence we found that the maximum enhancements in the $Z'$ Wilson coefficients correspond to $\mathrm{Re}(\tilde{C}^{Z^{\prime }}_9)= \mathrm{Im}(\tilde{C}^{Z^{\prime }}_9) \simeq 0.6\,C^\mathrm{SM}_9$ and $\mathrm{Re}( {C}^{Z^{\prime }}_7)= \mathrm{Im}({C}^{Z^{\prime }}_7) \simeq 0.4\,C^\mathrm{SM}_9$.

Regarding the branching ratios we found that $Z^\prime$ contributions can enhance the branching ratio of $B^0_s\rightarrow \,\pi ^0\,\eta '$ by about 4.6 times the SM prediction. Moreover, we found that the branching ratio of $B^0_s\rightarrow \,\rho ^0\,\eta$ can be enhanced up to one order of magnitude compared to the SM prediction for several sets of the parameter space satisfying both $\Delta M_{B_s}$ and $S_{\psi \phi }$ constraints and for a fine-tuning of $X_{B_s} \le 100$. For these points $\Delta M_{B_s}$ constraint on $\mid S_\mathrm{SM} (B_s) + S_{Z'} (B_s)\mid$ is fulfilled by overcompensating the SM via $S_{Z'} (B_s) \simeq -2 S_\mathrm{SM} (B_s)$.