LHCb anomaly in $\boldsymbol{B\to K^*\mu^+ \mu^-}$ optimised observables and potential of $\boldsymbol{Z^\prime}$ Model

TL;DR

This paper investigates how a non-universal Z' model can explain anomalies observed in B to K* mu+ mu- decay angular observables, suggesting potential new physics beyond the Standard Model.

Contribution

It explores the impact of a non-universal Z' model on angular observables in B decays, showing it can accommodate existing anomalies and proposing new observables for future tests.

Findings

Z' model parameters influence angular observables

The P'_5 anomaly can be explained within the Z' model

Additional data needed to confirm the model's validity

Abstract

Over the last few years LHCb with present energies found some discrepancies in $b\to s\ell^+\ell^-$ FCNC transitions including anomalies in the angular observables of $B\to K^*\mu^+\mu^-$, particularly in $P_5^\prime$, in low dimuon mass region. Recently, these anomalies are confirmed by Belle, CMS and ATLAS. As the direct evidence of physics beyond-the-SM is absent so far, therefore, these anomalies are being interpreted as indirect hint of new physics. In this context, we study the implication of non universal family of $Z^\prime$ model to the angular observables $P_{1,2,3}$, $P^{\prime}_{4,5,6}$ and newly proposed lepton flavor universality violation observables, $Q_{4,5}$, in $B\to K^*(\to K\pi)\mu^+ \mu^-$ decay channel in the low dimuon mass region. To see variation in the values of these observables from their standard model values, we have chosen the different scenarios of the $Z^\prime$ model. It is found that these angular observables are sensitive to the values of the parameters of $Z^\prime$ model. We have also found that with the present parametric space of $Z^\prime$ model, the $P_5^\prime$-anomaly could be accommodated. However, more statistics on the anomalies in the angular observables are helpful to reveal the status of the considered model and, in general, the nature of new physics.

Figures

Tables

Table 1. Table 1: The values of the fit parameters involved in the calculations of the form factors given in Eq. ( 8 ) zwicky05 .

	$r_1$	$r_2$	$m_R^2\left({\text{GeV}}^2\right)$	$m_{fit}^2\left({\text{GeV}}^2\right)$
$V\left(s\right)$	0.923	-0.511	28.30	49.40
$A_1\left(s\right)$		0.290		40.38
$A_2\left(s\right)$	-0.084	0.342		52.00

Table 2. Table 2: Results for ⟨ P ⟩ delimited-⟨⟩ 𝑃 \langle P\rangle -observables and their comparison with maximum likelihood fit results of ref. 2015lhcb in different bin size.

Obs.	SM Prediction	${𝒮}_1$	${𝒮}_2$	${𝒮}_3$	Measurement 2015lhcb
GeV2
$⟨P_1⟩$	$-0.002$	$-0.002\leftrightarrow -0.008$	$-0.002\leftrightarrow -0.002$	$-0.002\leftrightarrow -0.009$	$-{0.099}_{-0.163}^{+0.168}\pm 0.014$
$⟨P_2⟩$	$-0.106$	$-0.134\leftrightarrow -0.113$	$-0.116\leftrightarrow -0.102$	$0.042\leftrightarrow -0.059$	$-{0.003}_{-0.052}^{+0.051}\pm 0.007$
$⟨P_3⟩$	$-0.0001$	$0.000\leftrightarrow -0.0002$	$-0.000\leftrightarrow -0.001$	$-0.000\leftrightarrow -0.0001$	${0.113}_{-0.079}^{+0.079}\pm 0.006$
$⟨P_4^{\prime }⟩$	$0.267$	$0.175\leftrightarrow 0.155$	$0.230\leftrightarrow 0.171$	$0.405\leftrightarrow 0.380$	${0.185}_{-0.154}^{+0.158}\pm 0.023$
$⟨P_5^{\prime }⟩$	$0.740$	$0.747\leftrightarrow 0.473$	$0.712\leftrightarrow 0.497$	$-0.209\leftrightarrow 0.424$	${0.387}_{-0.133}^{+0.132}\pm 0.052$
$⟨P_6^{\prime }⟩$	$-0.158$	$-0.447\leftrightarrow -0.585$	$-0.384\leftrightarrow -0.566$	$0.466\leftrightarrow 0.400$	${0.034}_{-0.135}^{+0.134}\pm 0.015$
GeV2
$⟨P_1⟩$	$-0.007$	$-0.008\leftrightarrow -0.008$	$-0.007\leftrightarrow -0.008$	$-0.006\leftrightarrow -0.006$	$-{0.415}_{-0.636}^{+0.519}\pm 0.038$
$⟨P_2⟩$	$-0.433$	$-0.417\leftrightarrow -0.161$	$-0.406\leftrightarrow -0.187$	$0.097\leftrightarrow -0.347$	$-{0.373}_{-0.199}^{+0.146}\pm 0.027$
$⟨P_3⟩$	$0.0001$	$-0.000\leftrightarrow 0.001$	$0.000\leftrightarrow 0.001$	$0.001\leftrightarrow 0.001$	${0.350}_{-0.254}^{+0.330}\pm 0.015$
$⟨P_4^{\prime }⟩$	$0.023$	$-0.113\leftrightarrow -0.173$	$-0.040\leftrightarrow -0.13$	$0.170\leftrightarrow 0.200$	$-{0.163}_{-0.240}^{+0.232}\pm 0.021$
$⟨P_5^{\prime }⟩$	$0.225$	$0.275\leftrightarrow -0.208$	$0.211\leftrightarrow -0.141$	$0.249\leftrightarrow 0.160$	${0.289}_{-0.202}^{+0.220}\pm 0.023$
$⟨P_6^{\prime }⟩$	$-0.078$	$-0.432\leftrightarrow -0.533$	$-0.400\leftrightarrow -0.536$	$0.689\leftrightarrow 0.520$	$-{0.463}_{-0.221}^{+0.202}\pm 0.012$
GeV2
$⟨P_1⟩$	-0.023	$-0.025\leftrightarrow -0.026$	$-0.024\leftrightarrow -0.025$	$-0.032\leftrightarrow -0.024$	${0.571}_{-1.714}^{+2.404}\pm 0.045$
$⟨P_2⟩$	$-0.228$	$-0.215\leftrightarrow 0.154$	$-0.188\leftrightarrow 0.110$	$-0.341\leftrightarrow -0.280$	$-{0.636}_{-1.735}^{+0.444}\pm 0.015$
$⟨P_3⟩$	0.001	$-0.000\leftrightarrow 0.002$	$0.001\leftrightarrow 0.002$	$0.004\leftrightarrow 0.004$	${0.745}_{-0.861}^{+2.587}\pm 0.030$
$⟨P_4^{\prime }⟩$	$-0.282$	$-0.355\leftrightarrow -0.394$	$-0.320\leftrightarrow -0.371$	$-0.314\leftrightarrow -0.205$	$-{0.713}_{-1.305}^{+0.410}\pm 0.024$
$⟨P_5^{\prime }⟩$	$-0.400$	$-0.204\leftrightarrow -0.667$	$-0.339\leftrightarrow -0.628$	$0.722\leftrightarrow -0.294$	$-{0.066}_{-0.364}^{+0.343}\pm 0.023$
$⟨P_6^{\prime }⟩$	$-0.066$	$-0.313\leftrightarrow -0.350$	$-0.309\leftrightarrow -0.372$	$0.568\leftrightarrow 0.508$	${0.205}_{-0.341}^{+0.962}\pm 0.013$
GeV2
$⟨P_1⟩$	-0.055	$-0.053\leftrightarrow -0.053$	$-0.054\leftrightarrow -0.053$	$-0.064\leftrightarrow -0.062$	${0.180}_{-0.348}^{+0.364}\pm \mathrm{0.0.027}$
$⟨P_2⟩$	$0.206$	$0.088\leftrightarrow 0.357$	$0.170\leftrightarrow 0.341$	$-0.407\leftrightarrow 0.146$	${0.042}_{-0.087}^{+0.088}\pm 0.011$
$⟨P_3⟩$	$0.001$	$-0.000\leftrightarrow 0.003$	$0.000\leftrightarrow 0.003$	$0.003\leftrightarrow 0.004$	${0.083}_{-0.184}^{+0.187}\pm 0.023$
$⟨P_4^{\prime }⟩$	$-0.443$	$-0.460\leftrightarrow -0.472$	$-0.452\leftrightarrow -0.465$	$-0.477\leftrightarrow -0.446$	$-{0.448}_{-0.172}^{+0.169}\pm 0.020$
$⟨P_5^{\prime }⟩$	$-0.761$	$-0.492\leftrightarrow -0.837$	$-0.653\leftrightarrow -0.829$	$0.682\leftrightarrow -0.514$	$-{0.300}_{-0.159}^{+0.158}\pm 0.023$
$⟨P_6^{\prime }⟩$	$-0.036$	$-0.182\leftrightarrow -0.198$	$-0.178\leftrightarrow -0.214$	$0.249\leftrightarrow 0.268$	$-{0.032}_{-0.166}^{+0.167}\pm 0.007$

Table 3. Table 3: Results for ⟨ P ⟩ delimited-⟨⟩ 𝑃 \langle P\rangle -observables for s ∈ [ 1.0 , 6.0 ] 𝑠 1.0 6.0 s\in[1.0,6.0] GeV 2 and their comparison with LHCb maximum likelihood fit results of ref. 2015lhcb in different bin size, Belle results belle2016 ; Wehle:2016yoi .

Obs.	SM Prediction	${𝒮}_1$	${𝒮}_2$	${𝒮}_3$	Measurement
$⟨P_1⟩$	-0.033$\pm$0.001	$-0.032\leftrightarrow -0.034$	$-0.033\leftrightarrow -0.033$	$-0.039\leftrightarrow -0.036$	${0.080}_{-0.245}^{+0.248}\pm 0.044$ 2015lhcb
$⟨P_2⟩$	$0.091\pm 0.033$	$0.133\leftrightarrow -0.162$	$0.087\leftrightarrow -0.135$	$0.254\leftrightarrow 0.106$	$-{0.162}_{-0.073}^{+0.072}\pm 0.010$ 2015lhcb
$⟨P_3⟩$	$0.001\pm 0.000$	$-0.000\leftrightarrow 0.002$	$0.000\leftrightarrow 0.002$	$0.003\leftrightarrow 0.003$	${0.205}_{-0.134}^{+0.135}\pm 0.017$ 2015lhcb
$⟨P_4^{\prime }⟩$	$-0.264\pm 0.014$	$-0.333\leftrightarrow -0.368$	$-0.298\leftrightarrow -0.347$	$-0.195\leftrightarrow -0.256$	$-{0.336}_{-0.122}^{+0.124}\pm 0.12$ 2015lhcb $-{0.095}_{-0.309}^{+0.302}\pm 0.174$ belle2016 $-{0.22}_{-0.34}^{+0.35}\pm 0.15$ Wehle:2016yoi
$⟨P_5^{\prime }⟩$	$-0.378\pm 0.051$	$-0.197\leftrightarrow -0.617$	$-0.322\leftrightarrow -0.583$	$0.572\leftrightarrow -0.260$	$-{0.049}_{-0.108}^{+0.107}\pm 0.014$ 2015lhcb ${0.385}_{-0.285}^{+0.276}\pm 0.099$ belle2016 ${0.43}_{-0.28}^{+0.26}\pm 0.10$ Wehle:2016yoi
$⟨P_6^{\prime }⟩$	$-0.056\pm -0.000$	$-0.287\leftrightarrow -0.330$	$-0.276\leftrightarrow -0.345$	$0.452\leftrightarrow 0.403$	$-{0.166}_{-0.108}^{+0.108}\pm 0.021$ 2015lhcb $-{0.202}_{-0.270}^{+0.278}\pm 0.172$  belle2016

Table 4. Table 4: Values of Wilson coefficients at μ b = 4 ⋅ 8 subscript 𝜇 𝑏 ⋅ 4 8 \mu_{b}=4\cdot 8 .

$C_1\left({\mu }_b\right)$	$C_2\left({\mu }_b\right)$	$C_3\left({\mu }_b\right)$	$C_4\left({\mu }_b\right)$	$C_5\left({\mu }_b\right)$	$C_6\left({\mu }_b\right)$	$C_7^{\mathrm{eff}}\left({\mu }_b\right)$	$C_8^{\mathrm{eff}}\left({\mu }_b\right)$	$C_9\left({\mu }_b\right)$	$C_{10}\left({\mu }_b\right)$
-0.2632	1.0111	-0.0055	-0.0806	0.0004	0.0009	-0.2923	-0.1663	4.0749	-4.3085

Table 5. Table 5: The numerical values of the Z ′ superscript 𝑍 ′ Z^{\prime} parameters ConstrainedZPC1 ; UTfit .

	$\left|{ℬ}_{sb}\right|\times {10}^{-3}$	${\varphi }_{sb}\left(\text{Degree}\right)$	$S_{\ell \ell }^{LR}\times {10}^{-2}$	$D_{\ell \ell }^{LR}\times {10}^{-2}$
$𝒮1$	$1.09\pm 0.22$	$-72\pm 7$	$-2.8\pm 3.9$	$-6.7\pm 2.6$
$𝒮2$	$2.20\pm 0.15$	$-82\pm 4$	$-1.2\pm 1.4$	$-2.5\pm 0.9$
$𝒮3$	$4.0\pm 1.5$	$150\pm 10$ or $\left(-150\pm 10\right)$	$0.8$	$-2.6$

Table 6. Table 6: Values of input parameters.

${\alpha }_{em}\left(M_Z\right)=1/128.940$	Misiak:2006zs	${\alpha }_s\left(M_Z\right)=0.1184\pm 0.0007$	pdg
$m_e=0.51099\times {10}^{-3}\mathrm{GeV}$	pdg	$m_{\mu }=0.10565837\mathrm{GeV}$	pdg
$m_B=5.27950\mathrm{GeV}$	pdg	$m_{K^{\ast }}=0.89594\mathrm{GeV}$	pdg
$m_b^{1S}=4.68\pm 0.03\mathrm{GeV}$	Bauer:2004ve	$m_s=0.095\pm 0.005\mathrm{GeV}$	pdg
$m_c^{\overline{MS}}\left(m_c\right)=1.27\pm 0.09\mathrm{GeV}$	pdg
$\left|V_{tb}\right|=0.999139\pm 0.000045$	pdg	$\left|V_{ts}\right|=\left(40.5\pm 0.11\right)\cdot {10}^{-3}$	pdg
$f_B=194\pm 10\mathrm{MeV}$	Mahmoudi:2012un	${\lambda }_B=460\pm 110\mathrm{MeV}$	Ball:2006nr
$f_{K^{\ast },\left|\right|}=220\pm 5$ MeV	Ball:2006eu	$f_{K^{\ast },⟂}=185\pm 9\mathrm{MeV}$	Ball:2006eu
$a_{1,\left|\right|}=0.03\pm 0.03$	zwicky05	$a_{2,\left|\right|}=0.08\pm 0.06$	zwicky05
$a_{1,⟂}=0.03\pm 0.03$	zwicky05	$a_{2,⟂}=0.08\pm 0.06$	zwicky05

Equations

$${\mathcal{M}}^{\text{SM}}(b\to s{\ell }^{+}{\ell }^{-})=-\frac{\alpha G_F}{2\sqrt{2}\pi }{V}_{tb}{V}_{ts}^{\ast }$$

$$\displaystyle\times\bigg{\{}\langle K^{*}(p_{K^{*}},\epsilon)|\bar{s}\gamma^{\mu}Lb|B(p_{B})\rangle(C_{9}^{\rm eff}\bar{\ell}\gamma^{\mu}\ell+C_{10}^{\rm SM}\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)$$

$$\displaystyle-2m_{b}C_{7}^{\rm eff}\langle K^{*}(p_{K^{*}},\epsilon)|\bar{s}i\sigma_{\mu\nu}\frac{q^{\nu}}{q^{2}}Rb|B(p_{B})\rangle\,\bar{\ell}\gamma^{\mu}\ell\bigg{\}},$$

$${\mathcal{H}}_{\mathrm{eff}}^{Z^{\prime }}$$

$$\text{where,}{\Lambda }_{sb}$$

$$\text{and }{C}_{10}^{Z^{\prime }}$$

$$\text{vspace}10.6cm{\mathcal{S}}_{\ell \ell }^{LR}$$

$${\mathcal{M}}^{tot}(B\to K^{\ast }{\ell }^{+}{\ell }^{-})=-\frac{\alpha G_F}{2\sqrt{2}\pi }{V}_{tb}{V}_{ts}^{\ast }$$

$$\displaystyle\times\bigg{\{}\langle K^{*}(p_{K^{*}},\epsilon)|\bar{s}\gamma^{\mu}Lb|B(p_{B})\rangle(C_{9}^{\text{tot}}\bar{\ell}\gamma^{\mu}\ell+C_{10}^{\text{tot}}\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)$$

$$\displaystyle-2m_{b}C_{7}^{\rm eff}\langle K^{*}(p_{K^{*}},\epsilon)|\bar{s}i\sigma_{\mu\nu}\frac{q^{\nu}}{q^{2}}Rb|B(p_{B})\rangle\bar{\ell}\gamma^{\mu}\ell\bigg{\}},$$

$$⟨K^{\ast }(p_{K^{\ast }},ϵ)|\bar{s}{\gamma }^{\mu }Lb|B(p_B)⟩=-i{q}_{\mu }\frac{2m_{K^{\ast }}}{s}{ϵ}^{\ast }\cdot q$$

$$\displaystyle\times\bigg{[}A_{3}(s)-A_{0}(s)\bigg{]}-\epsilon_{\mu\nu\lambda\sigma}\epsilon^{*\nu}p^{\lambda}_{K^{*}}q^{\sigma}\frac{2V(s)}{(m_{B}+m_{K^{*}})}$$

$$+i{ϵ}_{\mu }^{\ast }(m_B+m_{K^{\ast }})A_1(s)$$

$$\mp i(p_B+p_{K^{\ast }}{)}_{\mu }{ϵ}^{\ast }\cdot q\frac{A_2(s)}{(m_B+m_{K^{\ast }})},$$

$$⟨K^{\ast }(p_{K^{\ast }},ϵ)|\bar{s}i{\sigma }_{\mu \nu }{q}^{\nu }Rb|B(p_B)⟩=2{ϵ}_{\mu \nu \lambda \sigma }{ϵ}^{\ast \nu }{p}_{K^{\ast }}^{\lambda }{q}^{\sigma }{T}_1(s)$$

$$\displaystyle+~{}i\epsilon^{*}\cdot q\bigg{\{}q_{\mu}-\frac{(p_{B}+p_{K^{*}})_{\mu}s}{(m_{B}^{2}-m_{K^{*}}^{2})}\bigg{\}}T_{3}(s)$$

$$\displaystyle+~{}i\bigg{\{}\epsilon^{*}_{\mu}(m_{B}^{2}-m_{K^{*}}^{2})-(p_{B}+p_{K^{*}})_{\mu}\epsilon^{*}\cdot q\bigg{\}}~{}T_{2}(s),$$

$$A_3(s)$$

$${\xi }_{\perp }$$

$${\xi }_{\parallel }$$

$$V(s)$$

$$A_2(s)$$

$$T_1(s)$$

$$\frac{d^4\Gamma }{dsd\cos {\theta }_{\ell }d\cos {\theta }_{K^{\ast }}d\varphi }$$

$$\tilde{\Gamma }\left(s,{\theta }_{\ell },{\theta }_{K^{\ast }},\varphi \right)=$$

$$J_1^s{\sin }^2{\theta }_{K^{\ast }}+J_1^c{\cos }^2{\theta }_{K^{\ast }}+\left(J_2^s{\sin }^2{\theta }_{K^{\ast }}+J_2^c{\cos }^2{\theta }_{K^{\ast }}\right)\cos 2{\theta }_{\ell }$$

$$+J_3{\sin }^2{\theta }_{K^{\ast }}{\sin }^2{\theta }_{\ell }\cos 2\varphi +J_4\sin 2{\theta }_{K^{\ast }}\sin 2{\theta }_{\ell }\cos \varphi$$

$$+J_5\sin 2{\theta }_{K^{\ast }}\sin {\theta }_{\ell }\cos \varphi +(J_6^s{\sin }^2{\theta }_{K^{\ast }}+J_6^c{\cos }^2{\theta }_{K^{\ast }})\cos {\theta }_{\ell }$$

$$+J_7\sin 2{\theta }_{K^{\ast }}\sin {\theta }_{\ell }\sin \varphi +J_8\sin 2{\theta }_{K^{\ast }}\sin 2{\theta }_{\ell }\sin \varphi$$

$$+J_9{\sin }^2{\theta }_{K^{\ast }}{\sin }^2{\theta }_{\ell }\sin 2\varphi .$$

$$4m_{\ell }^2⩽s⩽{\left(m_B-m_{K^{\ast }}\right)}^2,0⩽{\theta }_{\ell }⩽\pi ,$$

$$0⩽{\theta }_{K^{\ast }}⩽\pi ,0⩽\varphi ⩽2\pi ,$$

$$P_1(s)$$

$$P_4(s)$$

$$P_6(s)$$

$$P_4^{\prime }$$

$$P_5^{\prime }$$

$$P_6^{\prime }$$

$$P_2^{\text{exp}}$$

$$P_6^{\prime \text{exp}}$$

$$J_1^s$$

$$J_1^c$$

$$J_2^s$$

$$J_2^c$$

Full text

I Introduction

In flavor physics, the study of rare $B$ meson decays provide us a powerful tool, not only to test the standard model (SM) at loop level but also to search the possible new physics (NP). searching of NP in rare decays of B-meson demands to focus on those observables which contain minimum hadronic uncertainties such that they can be predicted precisely in the SM and are available at current colliders. In exclusive rare B meson decays, the main source of hadronic uncertainties come from the form factors which are non-perturbative quantities and are difficult to compute. In addition, these uncertainties may preclude the signature of any possible NP. From this point of view, among all rare decays, the four body decay channel, $B\to K^{*}(\to K\pi)\mu^{+}\mu^{-}$, have a special interest in literature due to the fact that it gives a large variety of angular observables, namely, $P_{i}$ $(i=1,2,3)$ and $P_{i}^{\prime}$ $(i=4,5,6)$ DescotesGenon:2012zf which are free from hadronic uncertainties Descotes-Genon:2013vna . The comparison between the theoretical predictions of these kind of observables in the SM with the experimental data could be helpful to clear some smog on the physics beyond the SM.

From experimental point of view, few years back, LHCb measured the values of these angular observables for the decay channel $B\to K^{*}(\to K\pi)\mu^{+}\mu^{-}$. These measurements found a 3.7$\sigma$ deviation in the value of $P_{5}^{\prime}$, with 1 fb*-1* luminosity in the $s\in[4.30,8.68]$ GeV2 bin 2013lhcb . Recently, this discrepancy again seen at LHCb with a 3$\sigma$ deviation with 3 fb*-1* luminosity in comparatively two shorter adjacent bins $s\in[4,6]$ GeV2 2015lhcb and $s\in[6,8]$ GeV2 which is also confirmed by Belle in the larger bin $s\in[4,8]$ GeV2 belle2016 ; Wehle:2016yoi . The very recent results from ATLAS ATLAS-CONF-2017-023 and CMS CMS-PAS-BPH-15-008 ; Khachatryan:2015isa collaborations, presented in Moriond 2017, are also confirmed this discrepancy. Furthermore, LHCb also found 2.6$\sigma$ deviation in the value of $R_{K}=$ Br$(B\to K\mu^{+}\mu^{-})/$Br$(B\to Ke^{+}e^{-})$ rk , and \mathrel{\raise 1.29167pt\hbox{>\kern-7.5pt\lower 4.30554pt\hbox{\sim}}}2\sigma in the Br$(B_{s}\to\phi\mu^{+}\mu^{-})$ lhcbphi . Interestingly, all these deviations belong to the flavor changing neutral current (FCNC) transitions, $b\to s\ell^{+}\ell^{-}$, where $\ell^{-}$ denotes the final state leptons.

The anomalies, mentioned above, are slowly piled up and received a considerable attention in the literature (see for instance Matias:2012xw ; Altmannshofer:2017fio ). It is also important to mention here that even the angular observables are form factor independent (FFI) but for precise theoretical predictions, one needs to incorporate the factorizable and non-factorizable QCD corrections. The factorizable corrections absorb in the hadronic form factors while the non-factorizable corrections arise from hard scattering of the process and do not belong to the form factors. In this respect, there are some studies which focus to the question whether these anomalies emerge from unknown factorizable power corrections or from NP Capdevila:2017ert ; Chobanova:2017ghn . However, global fit analysis with present data, strongly pointed out that the interpretation of mentioned anomalies through the NP is a valid option Altmannshofer:2017fio . In the present study, to determine the values of angular observables, we have included both type of corrections up to next-to-leading order (NLO) and their expressions are given in Appendix B.

From NP point of view, several extensions of the SM have been put forwarded models ; Egede:2010zc ; Altmannshofer14 ; bobeth13 ; thurth14 ; jager14 ; Allanach:2015gkd ; Crivellin:2015era ; Jager:2017gal . Among these, the $Z^{\prime}$ model is economical due to the fact that besides the SM gauge group, it requires only one extra $U(1)^{\prime}$ gauge symmetry associated with a neutral gauge boson, called $Z^{\prime}$. The nature of couplings of the $Z^{\prime}$ boson with the quarks and leptons leads the FCNC transitions to the tree level. In this model, the NP effects comes only through the short distance Wilson coefficients which are encapsulated in the new coefficients $C_{9}^{\text{tot}}=C_{9}^{\text{SM}}+C_{9}^{Z^{\prime}}$, $C_{10}^{\text{tot}}=C_{10}^{\text{SM}}+C_{10}^{Z^{\prime}}$, while operator basis remained unchange.

Several previous studies shown a possible interpretation to alleviate the mismatch between the experimental data of different observables for the decay $B\to K^{*}\mu^{+}\mu^{-}$ and their SM predictions in terms of $Z^{\prime}$ model Descotes-Genon:2013wba ; Altmannshofer:2013foa ; Gauld:2013qba ; Gauld:2013qja ; Buras:2013dea ; Buras:2013qja without any conflict. Therefore, it is natural to ask whether the $Z^{\prime}$ model could explain the recently observed anomalies in the angular observables of the decay channel $B\to K^{*}(\to K\pi)\mu^{+}\mu^{-}$. With this motivation, in the current study, we have analyzed the optimal observables $P_{1,2,3}$ and $P^{\prime}_{4,5,6}$, in the low dimuon mass region, for the $B\to K^{*}(\to K\pi)\mu^{+}\mu^{-}$ in the SM and in the $Z^{\prime}$ model. Besides these observables, we have also calculated the violation of lepton flavor universality (LFU) observables namely, $Q_{4(5)}=P^{\prime\mu}_{4(5)}-P^{\prime e}_{4(5)}$ Capdevila:2016ivx . For numerical calculations of these observables, we have used the LCSR values of the hadronic form factors zwicky05 and for $Z^{\prime}$ parameters, we have used the Utfit collaboration values, called as $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ and another different scenario, called $\mathcal{S}_{3}$ which numerical values are listed in Tab. (5).

We would like to mention here that the considered scenarios labeled as $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$ have same coupling structure of the $Z^{\prime}$ boson with the quarks and the leptons. However, the underlying difference between these scenarios is related to the different fit values of parameters such as new weak phase and couplings of $Z^{\prime}$ model, for considered decay process, available in the literature. For example, by using the all available experimental data on $B_{s}-\bar{B}_{s}$ mixing, Utfit collaboration has found two solutions of new weak phase, $\phi_{sb}$, that arises due to the measurement ambiguities in the data and referred as $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$. Similarly, another possible constraint on parameters of $Z^{\prime}$ model is discussed in newcon4 that, hereafter, we label as $\mathcal{S}_{3}$.

This paper is organized as follows: Section II.1, contains the effective Hamiltonian for the $b\to s\ell^{+}\ell^{-}$ transition in the SM and in the $Z^{\prime}$ model. The $B\to K^{*}$ matrix elements in terms of form factors and the expression of differential decay distributions are also given in this section. Formulae for the angular observables in section II.2. In section III, we have plotted the angular observables and their average values against dimuon mass $s$ and we have given phenomenological analysis of these observables. In the last section we conclude our work. Appendix A contains the analytical expressions of the angular observables and the values of input parameters. The contributions of factorizable and non-factorizable corrections at NLO are summarized in Appendix B.

II Formulation for the Analysis

II.1 Matrix Elements and Form Factors

In the standard model, FCNC transition, $b\to s\ell^{+}\ell^{-}$, occurs at loop level which amplitude can be written as,

[TABLE]

where $L,R=(1\mp\gamma^{5})$, $p_{K^{*}}$ and $\epsilon$ are momentum and polarization of $K^{*}$ meson, respectively, while $p_{B}$ is the momentum of $B$ meson.

In the presence of $Z^{\prime}$ the FCNC transitions could occur at tree level and the Hamiltion can be written in the following form (see detail in the refs. PL1 ; QCh ; KCh1 ; VB3 )

[TABLE]

The $\mathcal{B}_{sb}$ is the coupling of $Z^{\prime}$ with quarks and $\mathcal{B}^{L}_{\ell\ell}$, $\mathcal{B}^{R}_{\ell\ell}$ are left and right-handed couplings fo $Z^{\prime}$ with leptons. One can notice from Eq. (II.1) that in the $Z^{\prime}$ model, operator basis remains the same as in the SM while Wilson coefficients, $C_{9}$ and $C_{10}$, get modified. The total amplitude for the decay $B\to K^{*}\ell^{+}\ell^{-}$ is the sum of SM and $Z^{\prime}$ contributions, and can be written as follows,

[TABLE]

where $C_{9}^{\rm tot}=C_{9}^{\rm eff}+\Lambda_{sb}C_{9}^{Z^{\prime}}$ and $C_{10}^{\rm tot}=C_{10}^{\rm SM}+\Lambda_{sb}C_{10}^{Z^{\prime}}$.

The matrix elements for $B\to K^{*}$ transition, appears in Eq. (4), can be written in terms of form factors as follows

[TABLE]

[TABLE]

where,

[TABLE]

Here $A_{0,1,2}(s)$, $V(s)$, $T_{1,2,3}(s)$ are the form factors and contain hadronic uncertainties. At leading order by using the heavy quark limit, the QCD form factors follow the symmetry relations and can be expressed in terms of two universal form factors $\xi_{\perp}$ and $\xi_{\parallel}$ hiller2 ; Beneke:2000wa .

[TABLE]

It is also important to mention here that the angular observables are soft form factor independent at LO in $\alpha_{s}$ (i.e., not totally dependent of FF). There is residual dependence has been discussed, computed systematically and included in the predictions of the main papers of the field and even if, as expeted, does not play an important role induce certain mild dependence on FF. In addition, for the $s$ dependence of the universal form factors there are different parametrization Straub:2015ica , however, we have analyzed that the choice of parametrization is not so important at low $s$. In the current study, we use the following parametrization of LCSR approach zwicky05 .

[TABLE]

where the parameters $r_{1,2}$, $m^{2}_{R}$ and $m^{2}_{fit}$ are listed in Tab. (1). The uncertainty in the universal form factors $\xi_{\perp}$ and $\xi_{\parallel}$ arises from the uncertainty in the different parameters using in LCSR approach which is about $11\%$ and $14\%$, respectively, as discussed in hiller2 .

At NLO, the relations between the $T_{i}(s)$ where ($i=1,2,3$) and the invariant amplitudes $\mathcal{T}_{\perp,\parallel}(s)$, where $\mathcal{T}_{\perp,\parallel}=\mathcal{T}^{-}_{\perp,\parallel}$, read as Beneke:2001at .

[TABLE]

where $\mathcal{E}_{K^{*}}=(m_{B}^{2}+m_{K^{*}}^{2}-s)/2m_{B}$ is the energy of kaon in the rest frame of $B$-meson and $\mathcal{T}_{\perp,\parallel}(s)$ are defined in Eq. (23) of Appendix B.

The four-fold differential decay distribution for the cascade decay $B\to K^{*}(\to K\pi)\ell^{+}\ell^{-}$ is completely described by the four independent kinematical variables: the three angles; $\theta_{K^{*}}$ is the angle between the $K$ and $B$ mesons in the rest frame of $K^{*}$, $\theta_{\ell}$ is the angle between lepton and $B$ meson in the dilepton rest frame while $\phi$ is the azimuthal angle between the dilepton rest frame and $K^{*}$ rest frame and the fourth variable is dilepton invariant squared mass $s$. The explicit dependence of differential decay distribution on these kinematical variables can be expressed as follows

[TABLE]

where

[TABLE]

The full physical region phase space of kinematical variables is given by

[TABLE]

where $m_{B}$, $m_{K^{*}}$, $m_{\ell}$ are the masses of $B$ meson, $K^{*}$ meson and lepton, respectively.

The expressions of coefficients $J_{i}^{(a)}=J_{i}^{(a)}\left(s\right)$ for $i=1,....,9$ and $a=s,c$ as a function of the dilepton mass $s$, are given in Appendix A in Eq. (16). As we do not take the scalar contribution in this study, therefore, $J_{6}^{c}=0$.

II.2 Expressions of the Angular Observables

The definitions of FFI angular observables (optimal observables) are given in ref. Matias:2012xw ,

[TABLE]

The primed observables (related to the $P_{i}$ ($i=4,5,6$)) which are simpler and more efficient to fit experimentally are defined as,

[TABLE]

III Results and Discussion

In this section, we will present the numerical analysis of the angular observables. The authors would like to mention here that all of the numerical results are taken from the self-written Mathematica code. Before the analysis, we would like to write the different definitions of angular observables that are opted by LHCb 2015lhcb and theoretically used in the literature,

[TABLE]

For the numerical analysis the values of LCSR form factors, and relevant fit parameters are listed in Tab. (1). The values of Wilson coefficients and other input parameters are listed in Appendix A in Tabs. (4) and (6), respectively. Regarding the coupling parameters of $Z^{\prime}$ with quarks and leptons, there are some severe constraints come from different inclusive and exclusive $B-$ meson channels ConstrainedZPC1 . Particularly, coming from the two different fitting values for $B_{s}-\bar{B}_{s}$ mixing data by the UTFit collaborationUTfit . In this study, we call these two fitting values as $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ and their numerical values are listed in Tab.(5). We have considered another scenario which denoted by $\mathcal{S}_{3}$ in the present study that are obtained from the analysis of $B\to X_{s}\mu^{+}\mu^{-}$ newcon1a , $B\to K^{*}\mu^{+}\mu^{-}$ newcon1 ; newcon2 and $B\to\mu^{+}\mu^{-}$ newcon3 . The numerical values of scenario $\mathcal{S}3$ are chosen from cpv5 ; newcon4 and also listed in the Tab. (5). The purpose of the following analysis is to check that these constrained of $Z^{\prime}$ parameters could accomodate the anomalies in the angular observables, particularly, in $P_{5}^{\prime}$.

III.1 $P$-observables in different bin size

The numerical values of angular observables in different low $s$ bins in SM and in $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$ are given in Tab. (2). For comparison with experimental measurements, the maximum likelihood fit results of LHCb 2015lhcb are also given in the table. The ranges in the values of angular observables in $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$ is found by setting the upper and lower values of parametric space of these scenarios. These results are also shown graphically in Figs. (1) and (2) where black crosses are the data points taken from the last column of Tab. (2) and black dashed line correspond to the SM while green, red and blue bands correspond to the $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$ scenarios of the $Z^{\prime}$ model, respectively. The upper curve of the band corresponds the upper values of parametric space while the lower curve of the band corresponds the lower values of parametric space of the scenario. In our different bin sized analysis, we have not included the preliminary results from Belle belle2016 ; Wehle:2016yoi , ATLAS ATLAS-CONF-2017-023 and CMS333see figure 6 of Altmannshofer:2017fio for the recent analysis with these new results.CMS-PAS-BPH-15-008 ; Khachatryan:2015isa because their bin intervals are different from LHCb 2015lhcb that we have discussed in this section. In Fig. (1), the gray shaded region corresponds to the uncertainty in the SM values due to the uncertainty in different input parameters. One can see from the left panel of Figs. (1) and (2) that the uncertainty band in SM not preclude the effects of $Z^{\prime}$ model. Therefore, we have not provided the SM uncertainty in Tab. (2) and hence in the right panel of Figs. (1) and (2).

The plots in first and third rows of Fig. (1), represent the variation in the values of $P_{1,3}$ and their average values $\langle P_{1,3}\rangle$ as a function of $s$ in the SM and in the different scenarios of $Z^{\prime}$ model. From these graphs one can see that the values of these observables are quite small in the SM and not much enhanced when we incorporate the $Z^{\prime}$ effects. One can also see from Fig. (1) that the SM values of $\langle P_{1}\rangle$ lie inside the measured values. As the error in the measurement is huge, therefore, no potent result can be drawn from this observable with the current data. On the other hand the values of $\langle P_{3}\rangle$ in last two bins are within the measured values while in first two bins the SM values are out of the measured bars. However, to say something about any discrepancy in these observables, reduction in the experimental uncertainties are required.

Plots in second row of Fig. (1), show the variation in the values of $P_{2}$ and its average $\langle P_{2}\rangle$ against dilepton mass $s$. It could be seen from these figures that the values of these observables are significantly influenced in the presence of $Z^{\prime}$ effects. The right plot in the second row of Fig. (1) shows that the SM values of $\langle P_{2}\rangle$ in the bins $s\in[1.1,2.5]$ and $s\in[2.5,4.0]$ lie within the measurements and also in the bin $s\in[4.0,6,0]$ when the theoretical uncertainties of the input parameters are taken into account. However, in the first bin $s\in[0.1,0.98]$, the SM value of $\langle P_{2}\rangle$ looks mismatch from the experimental value. But it is worthy to mention here that the measurement performed by LHCb in this bin is without including the $m_{\ell}-$ suppressed terms which are important at very low $s$ region and it was found in Descotes-Genon:2015uva , that the impact of these terms is about $23\%$ reduction in the value of $\langle P_{2}\rangle$. Regarding this, it is mentioned in Capdevila:2017ert that in the first bin, LHCb actually measured $\langle\hat{P}_{2}\rangle$ instead of $\langle P_{2}\rangle$. Therefore, in principle, one could say that, up-till now, there is no mismatch between the SM predicted values of $\langle P_{2}\rangle$ with the experimental values.

In the first row of Fig. (2), we have displayed $P_{4}^{\prime}$ and it’s average value, $\langle P_{4}^{\prime}\rangle$, in the SM and in the different scenarios of $Z^{\prime}$ model as a function of $s$. One can see from these plots that the $Z^{\prime}$ effects are quite significant in the $P_{4}^{\prime}$ values at low $s$ region but mild at larger values of $s$. However, the SM values of $\langle\mathcal{P}_{4}^{\prime}\rangle$ in all four bins lie inside the measured values.

The results of $P_{5}^{\prime}$ and it’s average value $\langle P_{5}^{\prime}\rangle$ in the SM and in the $Z^{\prime}$ models are presented in the second row of Figs. (2). The values are significantly changed from the SM values when we incorporate the $Z^{\prime}$ effects. It can be noticed in the bin $s=4\text{ to }6$ GeV2, the SM average value $\langle P_{5}^{\prime}\rangle$ mismatch with the experimental values and as mentioned in the introduction that LHCb found 3$\sigma$ deviation in this bin. It could be seen from the figure that this discrepancy can be alleviated by $\mathcal{S}_{3}$ (red band) of $Z^{\prime}$ model. On the other hand for the Utfit scenarios, namely, $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ it can be noticed that when we take the upper and lower limit values of the current parametric space of these scenarios (green and blue bands), the $P_{5}^{\prime}$ anomaly in the bin $s\in[4,6]$GeV2 can not be accommodated. However, if the values of different parameters are chosen randomly within the allowed range then one could accommodate the $P_{5}^{\prime}$ anomaly in this bin by $\mathcal{S}_{1}$ but not with $\mathcal{S}_{2}$. Therefore, it looks that the $\mathcal{S}_{2}$ of Utfit is not consistent with the present data while the parametric space of $\mathcal{S}_{1}$, the left (right) couplings, ($B_{\ell\ell}^{L}$, $B_{\ell\ell}^{R}$), of $Z^{\prime}$ with leptons is severely constraint as shown in Fig. (3).

In the third row of Fig. (2), we have shown the variation of $P_{6}^{\prime}$ and $\langle P_{6}^{\prime}\rangle$ as a function of $s$. Similar to $P_{1,3}$, the SM value of this observable is also suppressed. As seen from the graph that SM value of $P_{6}^{\prime}$ consistent with the data with large error bars, however there is 2$\sigma$ deviation in one bin $s\in[1.1,2.5]$ which probably will be disappear when data will increase. One can also notice that in contrast to the $P_{1,3}$, the value of $P_{6}^{\prime}$ significantly enhanced in the $Z^{\prime}$ model. It is also noticed that in the $Z^{\prime}$ model the value of $P_{6}^{\prime}$ is positive in scenarios $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ while becomes negative in $\mathcal{S}_{3}$. As for the present analysis in $\mathcal{S}_{3}$, we set the value of $\phi_{sb}=150\pm 10$, in contrast to this, if we choose $\phi_{sb}=-150\pm 10$ which is also allowed (see Tab. (V)), then this negative value becomes positive.

III.2 $P$-observables in $s\in[1.0,6.0]$ GeV2

Besides the analysis of angular observables in shorter bins at low $s$ region (discussed in previous section), we have also analyzed these observables in the full $s\in[1.0,6.0]$ GeV2 region. The results for $P$-observables in $s\in[1.0,6.0]$ GeV2 are summarized in Tab. (3) and corresponding plots are shown in Fig. (4). In this figure, black error bar corresponds to LHCb result 2015lhcb while, magenta and yellow error bars correspond to Belle measurements for some of these observables belle2016 ; Wehle:2016yoi . However, it is good to mention here that LHCb results are in the bin $s\in[1.1,6.0]$ GeV2 while Belle measurements belle2016 ; Wehle:2016yoi are in the $s\in[1.0,6.0]$ GeV2. In addition, recently, the ATLAS collaboration announced its results for $s\in[0.04,6.0]$ GeV2 ATLAS-CONF-2017-023 which is not included in the current analysis. The empty red boxes in the plots of $\langle P_{2}\rangle$ and $\langle P^{\prime}_{6}\rangle$ represent the $\mathcal{S}_{3}$ scenario when we choose $\phi_{sb}=-150\pm 10$, while, other legends are same as in Figs. (1) and (2).

From Fig. (4), one can immediately notice that the values of $\langle P_{1}\rangle$ and $\langle P_{4}^{\prime}\rangle$ in the SM and in all the three scenarios of $Z^{\prime}$ lie within the current measurements, however, the error bars are huge. Therefore, to extract any information about the NP requires the precise measurement of these observables. It is also noticed that the values of $\langle P_{1}\rangle$ in the SM and in the $Z^{\prime}$ scenarios are very close, consequently, this observable even after the reduction of error bars not a good candidate to constrained the $Z^{\prime}$ parametric space. On the other hand $\langle P_{4}^{\prime}\rangle$ could be helpful to constraint the $Z^{\prime}$ parametric space, if any mismatch will appear in future in the bin [1,6] GeV2. The SM value of $\langle P_{3}\rangle$ is small and not enhanced in $Z^{\prime}$ model. However, the measured value is well above the SM prediction with huge error bars and need precision to draw any conclusion from this observable as well. From graph of $\langle P_{2}\rangle$ in Fig. (4), one can deduced that the SM value of $\langle P_{2}\rangle$ not lie within the measured value of LHCb. However, the values of $\langle P_{2}\rangle$ in $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are within the measurements while in $\mathcal{S}_{3}$, the value is out side the measured error bars. For $\langle P^{\prime}_{6}\rangle$, we have two different measurements as shown in the plot and contrast to the $\langle P_{2}\rangle$, the value of $\langle P^{\prime}_{6}\rangle$ lie within these measurements. However, similar to $\langle P_{2}\rangle$ the values of $\langle P^{\prime}_{6}\rangle$ in $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ lie within the measurements while the value in $\mathcal{S}_{3}$ lies outside the measured values (see red bands in both plots). Regarding $\mathcal{S}_{3}$, it is interesting to check whether the values of $\langle P_{2}\rangle$ and $\langle P^{\prime}_{6}\rangle$ could be reduced to current measurements. For this purpose, we choose the weak phase with opposite sign i.e., $\phi_{sb}=-150\pm 10$ (see Tab. 5 in Appendix A) and represent them in plots by empty red boxes. In Fig. 4, by looking the empty red box in $\langle P_{2}\rangle$ plot, the value is reduced but still well above the current measurement. In contrast, the value of $\langle P^{\prime}_{6}\rangle$ reduce to the Belle measurements belle2016 . However, more statistics on the observables $\langle P_{2}\rangle$ and $\langle P^{\prime}_{6}\rangle$ are helpful to constrained the $Z^{\prime}$ parameters, particularly, the sign and the magnitude of new weak phase $\phi_{sb}$.

For $\langle P^{\prime}_{5}\rangle$ plots of Fig. (3), the values in the SM and in $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ lie out side the error bars of experimental data points while the values in the $\mathcal{S}_{3}$ well inside the all data points shown in figure. In general, from the plots of Fig. (3), one concludes that the considered model do have potential to remove mismatch between theory and experiment but it is not so conclusive at present. We hope more precise measurements will clear the situation.

III.3 $Q_{4,5}$ for $s\in[1.0,6.0]$ GeV2

In Fig. (5), we have plotted the lepton flavor universality violation (LFUV) observables $\langle Q_{4(5)}\rangle$ against $s$. The values are quite small in the SM approximately $\langle Q_{4(5)}\rangle=8.8\pm 2.1\times 10^{-3}(7.5\pm 3.6\times 10^{-3})$ in the bin $s\in[1,6]$GeV2. We have also found that the effects of $Z^{\prime}$ are negligible. This fact is trivial, since Eq. (3) implies $C_{9,10}^{Z^{\prime},\mu}=C_{9,10}^{Z^{\prime},e}$. However, error bars are quite large and need more experimental data to find the accurate values of these observables.

IV Conclusion

In the present study, we have calculated the angular observables $P_{i}$ and their average values $\langle P_{i}\rangle$ in the SM and in the noun-universal family of $Z^{\prime}$ model for the decay channel $B\to K^{*}(\to K\pi)\ell^{+}\ell^{-}$. The expressions of the angular observables are given in the form of coefficient $J_{i}(s)$ which are written in terms of auxiliary functions $g_{i}(h_{i})$ in Eq. (16). As in the literature, these $J_{i}(s)$ coefficients, in general, expressed via transversity amplitudes, $A_{\perp}$, $A_{\parallel}$ and $A_{0}$, so the relations of these transversity amplitudes with auxiliary function $g_{i}(h_{i})$ are also given in Eq. (LABEL:TARelation). To see the $Z^{\prime}$ effects on these observables, we have used the UtFit collaboration constraints for the $Z^{\prime}$ parameters, called as scenarios $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$. Besides, we also consider another scenario, called $\mathcal{S}_{3}$, in the present study. From the present analysis, in all three scenarios of $Z^{\prime}$ for small values of $s$ i.e the large recoil region, the values of angular observables are significantly changed from their SM values. The current analysis shown that except the sceanario $\mathcal{S}_{1}$, the scenarios $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$ of $Z^{\prime}$ model has potential to accommodate the mismatch between the recent experimental measurements and the SM values of some of the angular observables in some bins of $s$. For instance, there is a discrepancy between experimentally measured value and SM value of $P_{5}^{\prime}$ in the region $s\in[4,6]$ GeV2 and in the current study it is found that scenario $\mathcal{S}_{3}$ of $Z^{\prime}$ could be adjusted this mismatch value with the measured value in this bin. On the other hand, this mismatch can not be accommodated on taking the maximum and minimum values of different parameters of scenarios $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ of UtFit collaborations. However, when we choose the random values of different parameters in the allowed region of these scenarios, one can accommodate the $P_{5}^{\prime}$ anomaly with scenario $\mathcal{S}_{2}$ but not with scenario $\mathcal{S}_{1}$. It is also noticed that the $P_{5}^{\prime}$ anomaly further constraint on the allowed parametric space of $\mathcal{S}_{2}$. Furthermore, we have also calculated the angular observables $\langle P_{i}\rangle$ and the LFUV observables $\langle Q_{4,5}\rangle$ in the large bin $s\in[1,6]$ and plotted with the measured data, however, the error bar is quite large in this bin and more static is needed to draw results. Here, we would like to comment that CMS and ATLAS collaborations recently announced preliminary results on angular observables in Moriond 2017 which still show the tension between experimental measurements and the SM predictions. Therefore, in general, one can say, as data will be enlarge and the statistical error will be reduced then these observables are quite promising to say something about the constraints on coupling of $Z^{\prime}$ boson with the quarks and leptons and consequently about the status of $Z^{\prime}$ model.

Appendix A

The expressions of $J_{i}$ appeared in Eqs. (13) and (14) are as follows,

[TABLE]

[TABLE]

[TABLE]

where $g_{i}$($h_{i}$), $i=1,\cdots,3$ are the auxiliary functions and given as follows,

[TABLE]

[TABLE]

and a${}_{0}=\frac{E_{K^{*}}}{m_{K^{*}}}\frac{\xi_{\parallel}}{\Delta_{\parallel}}$ where $\Delta_{\parallel}$ is given in Appendix B in Eq. (LABEL:delta).

Traditionally, the $J$’s are given in terms of transversity amplitudes but we have written in terms of $g_{i}(h_{i})$ functions given in Eq. (17) $A_{0,\parallel,\perp}$. The $A_{0,\parallel,\perp}$ are related with $g_{i}(h_{i})$ as follows

[TABLE]

where $\mathcal{N}=\alpha\,G_{F}|V_{tb}V_{ts}^{*}|\sqrt{\frac{s\,\beta_{\ell}\,p_{K^{*}}}{3\cdot 2^{10}\pi^{5}m_{B}^{3}}}$. We would like to mention here that our expressions of $J$’s are consistent with the literature for example given in refs. Matias:2012xw ; ball .

The values of Wilson coefficients at NNLO, $Z^{\prime}$ parameters and other input parameters are listed in Tabs. (4), (5) and (6), respectively.

Appendix B

The expression of $\Delta_{\parallel}$, appear in the definition of a0 below Eq. (18), written as follows

[TABLE]

and contributes only for massive leptons. The light-cone distribution amplitude (LCDA) $\Phi_{\bar{K}^{*},a}$ for transversely $\left(a=\perp\right)$ and longitudinally $\left(a=\parallel\right)$ polarized $K^{*}$ can be written as Beneke:2001at ; ball03

[TABLE]

where $L=-(m^{2}_{b}-s)/s\ln\left(1-s/m^{2}_{b}\right)$ and $a_{i}\left(\bar{K}^{*}\right)_{a}$ are the Gegenbauer coefficients. The moments are

[TABLE]

where $\Phi_{B,\pm}$ are the two $B$-meson light-cone distribution amplitudes Beneke:2001at . The $\lambda^{-1}_{B,-}\left(s\right)$ can be expressed as:

[TABLE]

where $\omega_{0}=2(m_{B}-m_{b})$. The $\xi_{a}$ are the universal form factors,

[TABLE]

The $B\rightarrow K^{*}$ matrix elements in heavy quark limit depend on four independent functions $\mathcal{T}_{a}^{\pm}$ $\left(a=\perp,\parallel\right)$. In the low $s$, ($1.0<s<6.0\,$GeV2), the invariant amplitudes $\mathcal{T}_{\perp,\parallel}$ at NLO within QCDf are given in hiller2 ; ball ; Beneke:2001at ,

[TABLE]

where $\Xi_{\perp}\equiv 1$, $\Xi_{\parallel}\equiv m_{K^{*}}/E_{K^{*}}$ and the factorization scale $\mu_{f}=\sqrt{m_{b}\Lambda_{QCD}}$. The coefficient functions $C_{a}$ and hard scattering functions $T_{a,\pm}$ are written as

[TABLE]

The form factor terms $C_{a}^{\left(0\right)}$ at LO are

[TABLE]

[TABLE]

where $h(s,m_{q})$ is well-known fermionic loop function.

The coefficients $C_{a}^{\left(1\right)}$ at NLO is divided into a factorizable and a non-factorizable part as

[TABLE]

At NLO the factorizable correction reads Beneke:2001at ; beneke05

[TABLE]

The non-factorizable corrections are,

[TABLE]

where $\Delta M$ depends on the mass renormalization convention for $m_{b}$. These corrections are obtained from the matrix elements of four-quark and chromomagnetic dipole operators Beneke:2001at that are embedded in $F^{(7,9)}_{1,2}$ and $F^{(7,9)}_{8}$ asatryan01 ; greub08 .

At LO the hard-spectator scattering term $T_{a,\pm}^{(0)}\left(u,\omega\right)$ from weak annihilation diagram is Beneke:2001at

[TABLE]

[TABLE]

The contributions to $T_{a}^{(1)}$ at NLO also contain a factorizable as well as non-factorizable part

[TABLE]

Including $\mathcal{O}\left(\alpha_{s}\right)$ corrections the factorizable term to $T_{a,\pm}^{(1)}$ are given by Beneke:2001at ; beneke05

[TABLE]

where $\bar{u}=1-u$. The non-factorizable correction comes through the matrix elements of four-quark operators and the chromomagnetic dipole operator

[TABLE]

The $t_{a}\left(u,m_{q}\right)$ functions are given by

[TABLE]

where $B_{0}$ and $I_{1}$ are

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]