Probing new physics in $\bar{B}^0 \to D^{(\ast)} \tau^- \bar\nu_{\tau}$ using the longitudinal, transverse, and normal polarization components of the tau lepton

TL;DR

This paper investigates tau lepton polarization in B meson decays to identify potential new physics effects, showing that polarization measurements are highly sensitive to beyond Standard Model interactions.

Contribution

It provides a model-independent analysis of tau polarization in B decays, constraining new physics scenarios and exploring their impact on polarization observables.

Findings

Tau polarization differs significantly from SM predictions without new physics.

Polarization components are highly sensitive to scalar and tensor new physics operators.

Normal polarization could be sizable if complex Wilson coefficients are present.

Abstract

We study the longitudinal, transverse, and normal polarization components of the tau lepton in the decays $\bar{B}^0 \to D^{(\ast)} \tau^- \bar\nu_{\tau}$ and discuss their role in searching for new physics (NP) beyond the standard model (SM). Starting with a model-independent effective Hamiltonian including non-SM four-Fermi operators, we obtain experimental constraints on different NP scenarios and investigate their effects on the polarization observables. In the SM the longitudinal and transverse polarizations of the tau lepton differ substantially from the corresponding zero lepton mass values of $P_L=-1$ and $P_T=0$. In addition, $P_L$ and $P_T$ are very sensitive to NP effects. For the transverse polarization this holds true, in particular, for the effective tensor operator in the case of $\bar{B}^0 \to D^\ast$ and for the scalar operator in the case of $\bar{B}^0 \to D$. The $T$-odd normal polarization $P_N$, which is predicted to be negligibly small in the SM, can be very sizable assuming NP complex Wilson coefficients. We also discuss in some detail how the three polarization components of the tau lepton can be measured with the help of its subsequent leptonic and semihadronic decays.

Tables

Table 1. Table 1: Parameters of the dipole approximation in Eq. ( 42 ) for B ¯ 0 → D ( ∗ ) → superscript ¯ 𝐵 0 superscript 𝐷 ∗ \bar{B}^{0}\to D^{(\ast)} form factors. Zero-recoil values of the form factors are also listed for comparison with the HQET.

	${\overline{B}}^0\to D^{\ast }$									${\overline{B}}^0\to D$
	$A_0$	$A_{+}$	$A_{-}$	$V$	$G^S$	$G_0^T$	$G_1^T$	$G_2^T$		$F_{+}$	$F_{-}$	$F^S$	$F^T$
$F\left(0\right)$	1.62	0.67	$-0.77$	0.77	$-0.50$	$-0.073$	0.73	$-0.37$		0.79	$-0.36$	0.80	0.77
$a$	0.34	0.87	0.89	0.90	0.87	1.23	0.90	0.88		0.75	0.77	0.22	0.76
$b$	$-0.16$	0.057	0.070	0.075	0.060	0.33	0.074	0.065		0.039	0.046	$-0.098$	0.043
$F\left(q_{\max }^2\right)$	1.91	0.99	$-1.15$	1.15	$-0.74$	$-0.13$	1.10	$-0.55$		1.14	$-0.53$	0.89	1.11
$F^{HQET}\left(q_{\max }^2\right)$	1.99	1.12	$-1.12$	1.12	$-0.62$	0	1.12	$-0.50$		1.14	$-0.54$	0.88	1.14

Table 2. Table 2: q 2 superscript 𝑞 2 q^{2} averages of the polarization components and the total polarization. The two rows labeled by SM (CCQM) contain our predictions using SM effective operators with transition form factors calculated in the CCQM. The predicted intervals for the observables in the presence of NP are given in correspondence with the 2 ​ σ 2 𝜎 2\sigma allowed regions of the NP couplings depicted in Fig.  5 .

${\overline{B}}^0\to D$
SM (CCQM)	$0.33$	$0.84$	$0$	$0.91$
$S_L$	$(0.36,0.67)$	$(-0.68,0.33)$	$(-0.76,0.76)$	$(0.89,0.96)$
$T_L$	$(0.13,0.31)$	$(0.78,0.83)$	$(-0.17,0.17)$	$(0.79,0.90)$
${\overline{B}}^0\to D^{\ast }$
SM (CCQM)	-0.50	0.46	0	0.71
$S_L$	$(-0.40,-0.14)$	$(0.47,0.62)$	$(-0.20,0.20)$	$(0.69,0.70)$
$T_L$	$(-0.36,0.24)$	$(-0.61,0.26)$	$(-0.17,0.17)$	$(0.23,0.69)$
$V_R$	$-0.50$	$(0.32,0.43)$	$0$	$(0.48,0.67)$

Equations

$${\vec{e}}_L=\frac{{\vec{p}}_{\tau }}{|{\vec{p}}_{\tau }|},{\vec{e}}_N=\frac{{\vec{p}}_{\tau }\times {\vec{p}}_{D^{(\ast )}}}{|{\vec{p}}_{\tau }\times {\vec{p}}_{D^{(\ast )}}|},{\vec{e}}_T={\vec{e}}_N\times {\vec{e}}_L,$$

$$s_L^{\mu }=(0,{\vec{e}}_L),s_N^{\mu }=(0,{\vec{e}}_N),s_T^{\mu }=(0,{\vec{e}}_T).$$

$$s^{\mu}_{L}=\Big{(}\frac{|\vec{p}_{\tau}|}{m_{\tau}},\frac{E_{\tau}}{m_{\tau}}\frac{\vec{p}_{\tau}}{|\vec{p}_{\tau}|}\Big{)},$$

$$P_i(q^2)=\frac{d\Gamma (s_i^{\mu })/d{q}^2-d\Gamma (-s_i^{\mu })/d{q}^2}{d\Gamma (s_i^{\mu })/d{q}^2+d\Gamma (-s_i^{\mu })/d{q}^2},i=L,N,T,$$

$${\tau }^{-}$$

$$p_{\tau }^{\mu }=E_{\tau }(1,{\beta }_{\tau }\sin {\theta }_{\tau },0,{\beta }_{\tau }\cos {\theta }_{\tau }),$$

$$\frac{d\Gamma}{dq^{2}d\cos\theta_{dP}}={\cal B}_{d}\frac{d\Gamma}{dq^{2}}\frac{1}{2}\,\Big{(}1+A_{d}\cdot|\vec{P}(q^{2})|\cos\theta_{dP}\Big{)},$$

$${\vec{p}}_d=|{\vec{p}}_d|(\sin {\theta }_d\cos \chi ,\sin {\theta }_d\sin \chi ,\cos {\theta }_d).$$

$$\frac{d\Gamma}{dq^{2}d\cos\theta_{d}d\chi/2\pi}={\cal B}_{d}\frac{d\Gamma}{dq^{2}}\frac{1}{2}\big{[}1+A_{d}(P_{T}(q^{2})\sin\theta_{d}\cos\chi+P_{N}(q^{2})\sin\theta_{d}\sin\chi+P_{L}(q^{2})\cos\theta_{d})\big{]}.$$

$$\frac{d\Gamma}{dq^{2}d\cos\theta_{d}}={\cal B}_{d}\frac{d\Gamma}{dq^{2}}\frac{1}{2}\Big{(}1+A_{d}P_{L}(q^{2})\cos\theta_{d}\Big{)}$$

$$A_{FB}^P=\frac{d\Gamma (F)-d\Gamma (B)}{d\Gamma (F)+d\Gamma (B)}=A_d{P}_L(q^2).$$

$$\frac{d\Gamma}{dq^{2}d\chi/2\pi}={\cal B}_{d}\,\frac{d\Gamma}{dq^{2}}\Big{(}1+A_{d}\frac{\pi}{4}\big{(}P_{T}(q^{2})\cos\chi+P_{N}(q^{2})\sin\chi\big{)}\Big{)}$$

$$\frac{d\Gamma_{\pi}}{dq^{2}d\cos\theta_{\pi}d\chi/2\pi}={\cal B}_{\pi}\frac{d\Gamma}{dq^{2}}\frac{1}{2}\big{[}1+P_{T}(q^{2})\sin\theta_{\pi}\cos\chi+P_{N}(q^{2})\sin\theta_{\pi}\sin\chi+P_{L}(q^{2})\cos\theta_{\pi}\big{]},$$

$$\frac{d\Gamma_{\pi}}{dq^{2}d\chi/2\pi}={\cal B}_{\pi}\frac{d\Gamma}{dq^{2}}\Big{(}1+\frac{\pi}{4}P_{T}(q^{2})\cos\chi\Big{)}.$$

$$\frac{d\Gamma_{\rho}}{dq^{2}d\cos\theta_{\rho}d\chi/2\pi}={\cal B}_{\rho}\frac{d\Gamma}{dq^{2}}\frac{1}{2}\Big{[}1+\frac{m_{\tau}^{2}-2m_{\rho}^{2}}{m_{\tau}^{2}+2m_{\rho}^{2}}(P_{T}(q^{2})\sin\theta_{\rho}\cos\chi+P_{L}(q^{2})\cos\theta_{\rho})\Big{]},$$

$$\frac{d{\Gamma }_{\rho }}{d\cos \theta }=\frac{3}{8}(1+{\cos }^2\theta ){\Gamma }^T+\frac{3}{4}{\sin }^2\theta {\Gamma }^L,$$

$${\epsilon }^{\alpha }(0)=\frac{1}{m_{\rho }{m}_{\tau }p}(m_{\rho }^2{p}_{\tau }^{\alpha }-p_{\rho }{p}_{\tau }{p}_{\rho }^{\alpha }).$$

$$\frac{d{\Gamma }_{\rho }^L}{d{q}^{2}d\cos {\theta }_{\rho }d\chi /2\pi }$$

$$\frac{d{\Gamma }_{\rho }^T}{d{q}^{2}d\cos {\theta }_{\rho }d\chi /2\pi }$$

$$\frac{d\Gamma_{\ell}}{dq^{2}dxd\cos\theta_{\ell}d\chi/2\pi}=\frac{d\Gamma}{dq^{2}}\frac{\Gamma_{0}}{\Gamma_{\tau}}\beta x\big{[}G_{1}(x)+G_{2}(x)\big{(}P_{T}(q^{2})\sin\theta_{\ell}\cos\chi+P_{L}(q^{2})\cos\theta_{\ell}\big{)}\big{]},$$

$$G_1=x(3-2x)-(4-3x)y^2,G_2=\beta x(1-2x+3y^2),$$

$$\frac{<\beta x{G}_2(x)>}{<\beta x{G}_1(x)>}=-\frac{1}{12}(1+8y^2-32y^3+\dots ).$$

$$\frac{d{\Gamma }_{\ell }}{d{q}^{2}d\chi /2\pi }=\frac{d\Gamma }{d{q}^2}{\mathcal{B}}_{\ell }^0(1+P_T{A}_{\chi }\cos \chi ),\mathrm{where}{A}_{\chi }=-\frac{\pi }{12}(1+8y^2-32y^3+\dots ).$$

$$z=\frac{E_{\ell }}{E_{\tau }}=\frac{{\beta }_{\tau }p\cos {\theta }_{\ell }+E}{m_{\tau }}=\frac{x}{2}({\beta }_{\tau }\beta \cos {\theta }_{\ell }+1).$$

$$\frac{d\Gamma_{\ell}}{dq^{2}dxdzd\chi/2\pi}={\cal B}_{\ell}^{0}\frac{d\Gamma}{dq^{2}}2\big{(}G_{1}(x)+G_{2}(x)(P_{L}\cos\theta_{\ell}+P_{T}\sin\theta_{\ell}\cos\chi)\big{)}.$$

$$\frac{d{\Gamma }_{\ell }}{d{q}^{2}dzd\chi /2\pi }$$

$$\frac{d\Gamma_{\ell}}{dq^{2}d\chi/2\pi}={\cal B}_{\ell}^{0}\frac{d\Gamma}{dq^{2}}\Big{(}1-\frac{\pi}{12}P_{T}(q^{2})\cos\chi\Big{)}.$$

$$\frac{d\Gamma_{\ell}}{dq^{2}dzd\chi/2\pi}={\cal B}_{\ell}\frac{d\Gamma}{dq^{2}}\big{[}f(q^{2},z)+g(q^{2},z)P_{L}(q^{2})+h(q^{2},z)P_{T}(q^{2})\cos\chi],$$

$$f(q^2,z)$$

$$g(q^2,z)$$

$$h(q^2,z)$$

Full text

I   Introduction

The exclusive semileptonic decays $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ have been measured by the BABAR Lees:2012xj , Belle Huschle:2015rga ; Sato:2016svk , and LHCb Aaij:2015yra collaborations in an effort to unravel the well-known $R_{D^{(*)}}$ puzzle which has persisted for several years Ivanov:2016qtw ; Crivellin:2012ye ; Becirevic:2012jf ; DeVito:2004zs ; Nierste:2008qe ; Faustov:2012nk ; Celis:2012dk ; Biancofiore:2013ki ; Duraisamy:2013kcw ; Sakaki:2013bfa ; Duraisamy:2014sna ; Calibbi:2015kma ; Greljo:2015mma ; Bauer:2015knc ; Fajfer:2015ycq ; Barbieri:2015yvd ; Barbieri:2016las ; Boucenna:2016wpr ; Li:2016vvp ; Nandi:2016wlp ; Alok:2016qyh ; Deshpand:2016cpw ; Kim:2016yth ; Bardhan:2016uhr ; Dutta:2016eml ; Wang:2016ggf ; Celis:2016azn ; Bhattacharya:2016zcw ; Becirevic:2016yqi ; Becirevic:2016oho . Recently, the Belle collaboration reported a new measurement of the decay $\bar{B}^{0}\to D^{\ast}\tau^{-}\bar{\nu}_{\tau}$ using the hadronic $\tau^{-}$ decay modes $\tau^{-}\to\pi^{-}\nu_{\tau}$ and $\tau^{-}\to\rho^{-}\nu_{\tau}$, in which they found $R_{D^{\ast}}=0.270\pm 0.035(\text{stat.})^{+0.028}_{-0.025}(\text{syst.})$ Hirose:2016wfn . Taking this new result into account, the current world averages of the ratios are $R_{D}=0.406\pm 0.050$ and $R_{D^{\ast}}=0.311\pm 0.016$, which exceed the SM predictions of $R_{D}=0.300\pm 0.008$ Na:2015kha ; Lattice:2015rga ; Aoki:2016frl and $R_{D^{\ast}}=0.252\pm 0.003$ Fajfer:2012vx by $2.1\sigma$ and $3.6\sigma$, respectively.

In Ref. Hirose:2016wfn the Belle collaboration also reported on the first measurement of the longitudinal polarization of the tau lepton in the decay $\bar{B}^{0}\to D^{\ast}\tau^{-}\bar{\nu}_{\tau}$ with the result $P_{L}^{\tau}=-0.38\pm 0.51(\text{stat.})^{+0.21}_{-0.16}(\text{syst.})$. The errors of this measurement are quite large but this pioneering measurement has opened a completely new window on the analysis of the dynamics of the semileptonic $B\to D$ and $B\to D^{*}$ transitions. The hope is that, with the Belle II super-B factory nearing completion, more precise values of the polarization can be achieved in the future, which would shed more light on the search for possible NP in these decays.

In this paper we shall study the longitudinal, transverse, and normal polarization components of the $\tau^{-}$ in the semileptonic decays $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$. In order to set up our notation we define three orthogonal unit vectors as follows:

[TABLE]

where $\vec{p}_{\tau}$ and $\vec{p}_{D^{(*)}}$ are the three-momenta of the $\tau^{-}$ and the $D^{(*)}$ meson in the $W^{-}_{off-shell}$ rest frame. In the following we shall loosely refer to this frame as the $W^{-}$ rest frame. The three unit vectors $\vec{e}_{T}$, $\vec{e}_{N}$, and $\vec{e}_{L}$ form a right-handed coordinate system. The longitudinal $(L)$, normal $(N)$, and transverse $(T)$ polarization four-vectors of the $\tau^{-}$ in its rest frame are given by

[TABLE]

A Lorentz boost from the $\tau^{-}$ rest frame to the $W^{-}$ rest frame transforms only the longitudinal polarization four-vector according to

[TABLE]

leaving the normal $(s^{\mu}_{N})$ and transverse $(s^{\mu}_{T})$ polarization four-vectors unchanged. The longitudinal, normal, and transverse polarization components of the tau are given by

[TABLE]

where $q^{\mu}=p_{B}^{\mu}-p_{D^{(*)}}^{\mu}$ is the momentum transfer. We note that the terms longitudinal polarization and longitudinal polarization component are often used interchangeably, as in this paper. The same convention is used for the normal and transverse polarizations.

The normal polarization component $P_{N}$ is a $T$-odd observable and is predicted to be zero in the SM in the absence of final state interactions which are known to be negligibly small. However, in some extended versions of the SM such as the two-Higgs-doublet models, the minimal supersymmetric standard model, and the leptoquark model, large values of $P_{N}$ are possible through the introduction of $CP$-violating phases Garisto:1994vz ; Tsai:1996ps ; Wu:1997uaa ; Lee:2001nw .

The longitudinal polarization $P_{L}$ has also been used as a promising observable in order to probe NP in $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ Tanaka:2010se ; Fajfer:2012vx ; Datta:2012qk ; Tanaka:2012nw ; Bhattacharya:2015ida ; Becirevic:2016hea . $P_{L}$ has been found to be very sensitive to the scalar and tensor operators. It has been shown in Ref. Tanaka:2010se ; Tanaka:2012nw that some correlations between $P_{L}$ and the decay rate are very useful for NP prediction. In addition, the NP couplings can be extracted from $P_{L}$ with much less uncertainties as compared to those from other observables Bhattacharya:2015ida .

In Ref. Ivanov:2015tru we have calculated the SM values of the longitudinal and transverse polarization of the charged lepton in the decays $\bar{B}^{0}\to D^{(\ast)}\ell^{-}\bar{\nu}_{\ell}$. The polarization components have been calculated in the so-called helicity basis where the polarization components are given in terms of bilinear forms of the helicity amplitudes of the current-induced $B\to D^{(\ast)}$ transitions. Depending on the phase space region the transverse tau polarization can become quite large. On average one has $<P^{\tau}_{T}>=0.84$ ($B\to D$) and $<P^{\tau}_{T}>=0.46$ ($B\to D^{\ast}$) Ivanov:2015tru compared to $<P^{\ell}_{T}>=0$ for $m_{\ell}=0$ in both cases. For the longitudinal polarization one has $<P^{\tau}_{L}>=0.33$ ($B\to D$) and $<P^{\tau}_{L}>=-0.50$ ($B\to D^{\ast}$) Ivanov:2015tru which one has to compare with the zero lepton mass result $<P^{\ell}_{L}>=-1$, again in both cases 111One of the authors of the present paper (J.G.K.) records his contentment that the values $<P^{\tau}_{L}>=0.33$ ($B\to D$) and $<P^{\tau}_{L}>=-0.53$ ($B\to D^{\ast})$ calculated in the early paper Korner:1989qb are quite close to those calculated in  Ivanov:2015tru .. For the averages of the total polarization $|\vec{P}^{\tau}|$ one obtains $<|\vec{P}^{\tau}|>=0.91$ ($B\to D$) and $<|\vec{P}^{\tau}|>=0.71$ ($B\to D^{\ast})$. In this paper we also consider the transverse polarization in the presence of NP and compare its NP sensitivity with that of $<P^{\tau}_{L}>$ and $<P^{\tau}_{N}>$. The discussion of NP contributions to the transverse and normal polarization components of the $\tau^{-}$ are new.

Since the $\tau^{-}$ lepton decays weakly, its polarization is revealed through its ensuing decay distributions, i.e. it is self-analyzing. As analyzing modes for the $\tau^{-}$ polarization we will consider the four dominant $\tau^{-}$ decay modes

[TABLE]

where we have added the respective branching fractions in brackets. In the next section, we will show how the three polarization components of the tau can be measured by using its decays as polarization analyzers and how well each mode can serve as polarization analyzer. The remaining parts of the paper are organized as follows: in Sec. III we introduce some formalism concerning the semileptonic transitions, including the derivation of the polarization formulae in the presence of NP. An analysis of NP effects on the polarizations is given in Sec. IV. Finally, we summarize the main results in Sec. V.

II   Analyzing the polarization of the tau through its decays

The polarization components of the $\tau^{-}$ in $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ can be measured by using the decay products of the $\tau^{-}$ as polarization analyzers. The kinematics of the decay $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ followed by a $\tau^{-}$ decay is depicted in Fig. 1, where $d^{-}=\pi^{-},\,\rho^{-},\,e^{-},\,\mu^{-}$.

In the $W^{-}$ rest frame, $\theta_{\tau}$ is the angle between the $\tau^{-}$ three-momentum and the direction opposite to the direction of the $D^{(*)}$ meson. In the $\tau^{-}$ rest frame, $\theta_{d}$ is the angle between the three-momentum of the final tau daughter $d^{-}$ and the longitudinal polarization axis which is chosen to coincide with the direction of the $\tau^{-}$ in the $W^{-}$ rest frame (helicity basis). The production plane defined by the decay $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ is spanned by the three-momenta of the $\tau^{-}$ and the $D^{(\ast)}$ while the $\tau^{-}\to d^{-}+X$ decay plane is spanned by the three-momentum of the $d^{-}$ and the longitudinal polarization axis. The angle $\chi$ is the azimuthal angle between the two planes. We choose a right-handed $xyz$ coordinate system in the $W^{-}$ rest frame such that the $z$ axis is opposite to the direction of the mesons $\bar{B}^{0}$ and $D^{(\ast)}$, and the three-momentum of the $\tau^{-}$ lies in the $(xz)$ plane. In this system the $\tau^{-}$ momentum is given by

[TABLE]

where $E_{\tau}=(q^{2}+m_{\tau}^{2})/2\sqrt{q^{2}}$ is the energy and $\beta_{\tau}=|\vec{p}_{\tau}|/E_{\tau}=\sqrt{1-m_{\tau}^{2}/E_{\tau}^{2}}$ the velocity of the $\tau^{-}$ in the $W^{-}$ rest frame.

Let us discuss the spin-momentum correlations in the $\tau^{-}$ rest frame. Since we are dealing with two-body decays ($\tau^{-}\to\pi^{-}(\rho^{-})+\nu_{\tau}$) or quasi-two-body decays ($\tau^{-}\to\ell^{-}+X$) there is only one independent spin-momentum scalar product which can be taken to be $(\vec{p}_{d}\cdot\vec{P})$, where $\vec{p}_{d}$ is the three-momentum of the $d^{-}$ and $\vec{P}$ is the polarization vector of the $\tau^{-}$. Note that in a three-body decay as e.g. in $t\to b+\ell^{+}+\nu_{\ell}$ there are two possible spin-momentum scalars which provide for a richer spin-momentum correlation structure (see e.g. Korner:1998nc ; Groote:2006kq ). Returning to the two-body decays treated here the differential polar angle distribution is given by

[TABLE]

where $\theta_{dP}$ is the polar angle between the momentum $\vec{p}_{d}$ and the polarization vector $\vec{P}$ of the $\tau^{-}$, and ${\cal B}_{d}$ and $A_{d}$ are the branching fraction and the analyzing power of the decay $\tau^{-}\to d^{-}+X$, respectively. Note that the magnitude of the analyzing power has to satisfy $|A_{d}|\leq 1$ to guarantee the positivity of rates for $|\vec{P}|=1$.

The polar angle $\theta_{dP}$ appearing in Eq. (7) is experimentally not accessible since the direction of the polarization vector $\vec{P}$ of the $\tau^{-}$ is a priori unknown. However, one can define experimentally accessible angles $\theta_{d}$ and $\chi$ through the representation of the momentum vector $\vec{p}_{d}$ in the production plane (see Fig. 1) via

[TABLE]

In terms of the angles $\theta_{d}$ and $\chi$, the decay distribution reads

[TABLE]

Through an analysis of the decay distribution (9) one can determine the three components of the $q^{2}$-dependent polarization vector $\vec{P}(q^{2})=(P_{T}(q^{2}),P_{N}(q^{2}),P_{L}(q^{2}))$.

Upon $\chi$ integration, one obtains

[TABLE]

such that the forward-backward polarization asymmetry is given by

[TABLE]

Upon $\cos\theta_{d}$ integration one has

[TABLE]

with an effective azimuthal analyzing power of $A_{d}\pi/4$.

II.1 The semihadronic modes $\tau^{-}\to\pi^{-}\nu_{\tau}$

and $\tau^{-}\to\rho^{-}\nu_{\tau}$

The differential decay rate of $\bar{B}^{0}\to D^{(\ast)}\tau^{-}(\to\pi^{-}\nu_{\tau})\bar{\nu}_{\tau}$ reads

[TABLE]

where ${\cal B}_{\pi}$ is the branching fraction of $\tau^{-}\to\pi^{-}\nu_{\tau}$ and $\Gamma$ is the decay rate of $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$. Note that the analyzing power of the decay $\tau^{-}\to\pi^{-}\nu_{\tau}$ is $100\%$. In the following we shall drop explicit reference to the component $P_{N}$ in the angular decay distribution. After $\cos\theta_{\pi}$ integration, one obtains

[TABLE]

The effective azimuthal analyzing power is quite large with $\pi/4=78.54\%$.

For the decay $\bar{B}^{0}\to D^{(\ast)}\tau^{-}(\to\rho^{-}\nu_{\tau})\bar{\nu}_{\tau}$ one has

[TABLE]

where ${\cal B}_{\rho}$ is the branching fraction of $\tau^{-}\to\rho^{-}\nu_{\tau}$. It is apparent that one looses analyzing power compared to the case $\tau^{-}\to\pi^{-}\nu_{\tau}$ since $(m_{\tau}^{2}-2m_{\rho}^{2})/(m_{\tau}^{2}+2m_{\rho}^{2})=0.4485<1$.

However, one can retain the full analyzing power if one projects out the longitudinal and transverse components of the $\rho^{-}$, which can be achieved by an angular analysis of the decay $\rho^{-}\to\pi^{-}+\pi^{0}$ in the rest frame of the $\rho^{-}$. The polar angle distribution of the decay $\rho^{-}\to\pi^{-}+\pi^{0}$ reads

[TABLE]

where $\theta$ is the polar angle of the $\pi^{-}$ with respect to the original flight direction of the $\rho^{-}$. Technically, one can project out the longitudinal piece of the $\rho^{-}$ with the help of the normalized longitudinal polarization four-vector of the $\rho^{-}$ which reads

[TABLE]

One can check that $p_{\rho}\cdot\varepsilon(0)=0$ and that the polarization four-vector is correctly normalized: $\varepsilon^{\ast}(0)\cdot\varepsilon(0)=-1$. In the rest frame of the $\rho^{-}$ one has $\varepsilon^{\alpha}(0)=(0;0,0,1)$. The transverse contribution can be obtained from the difference $\Gamma^{T}=\Gamma-\Gamma^{L}$.

The longitudinal and transverse differential decay distributions of the $\rho^{-}$ are finally given by

[TABLE]

By separating the two distributions on has regained the full analyzing power of $100\%$ in both cases. This can e.g. be done by projection: ${\cal P}^{L}=2(1-5/2\cos^{2}\theta)$ will project out the longitudinal and ${\cal P}^{T}=-(1-5\cos^{2}\theta)$ the transverse component. It is evident that the sum of the two distributions (II.1) gives the result Eq. (15).

II.2 The leptonic modes

$\tau^{-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\tau}$ $(\ell=e,\mu)$

Using the results of e.g. Ref. Fischer:2002hn one finds

[TABLE]

where, as usual, we have defined a scaled energy variable $x=2E/m_{\tau}$ where $E=(|\vec{p_{\ell}}|^{2}+m_{\ell}^{2})^{1/2}$ is the energy of the final charged lepton $\ell^{-}$ in the $\tau^{-}$ rest frame. Here, $\Gamma_{0}=G_{F}^{2}m_{\tau}^{5}/192\pi^{3}$ is the reference rate for the leptonic decay of final-state massless leptons $m_{\ell}=0$, and $\Gamma_{\tau}$ is the total decay width of the $\tau^{-}$. Note that the expression to the right of $\Gamma_{0}/\Gamma_{\tau}$ integrates to 1 for $m_{\ell}=0$ as it should be. For later purposes we define a reference branching ratio ${\cal B}_{\ell}^{0}=\Gamma_{0}/\Gamma_{\tau}$.

The coefficient functions in Eq. (19) are given by Fischer:2002hn

[TABLE]

where $y=m_{\ell}/m_{\tau}$ and $\beta=\sqrt{1-4y^{2}/x^{2}}=\sqrt{1-m_{\ell}^{2}/E^{2}}=p/E$. We mention that the next-to-leading order QED radiative corrections to the leptonic polarized $\tau^{-}$ decays can also be found in Ref. Fischer:2002hn .

The polar and azimuthal analyzing power is determined by the ratio $G_{2}(x)/G_{1}(x)$. By averaging over $x$ $(2y\leq x\leq 1+y^{2})$, one obtains

[TABLE]

The azimuthal analyzing power is given by

[TABLE]

For $m_{\ell}=0$ one finds $A_{\chi}=-0.262$ which increases by $3.2\%$ for $m_{\ell}=m_{\mu}$ [see Eq. (22)].

Another possibility to analyze the polarization of the $\tau^{-}$ is to describe the leptonic decay of the polarized $\tau^{-}$ in terms of the variables $(x,\,z)$, where $z=E_{\ell}/E_{\tau}$ is the fractional energy $E_{\ell}$ of the daughter lepton and the energy $E_{\tau}$ of the $\tau^{-}$ both in the $W^{-}$ rest frame Bullock:1992yt . For the dependence $z=z(x,\,\cos\theta_{\ell})$, one finds

[TABLE]

It is important to realize that $E$ (energy of the daughter lepton in the $\tau^{-}$ rest frame) is no longer fixed but becomes a variable to be integrated over.

Let us first discuss the so-called collinear approximation $\beta_{\tau}=1$ and the zero lepton mass limit $\beta=1$ introduced in Ref. Bullock:1992yt to analyze the longitudinal polarization of the $\tau^{-}$. The approximation $\beta_{\tau}=1$ is good for the small recoil (i.e. large $q^{2}$) region. The approximation $\beta=1$ holds for the limiting case when one can neglect the lepton mass in the final state. With these approximations the twice differential rate reads

[TABLE]

By integrating Eq. (24) over $x$ in the region $z\leq x\leq 1$, one obtains

[TABLE]

The differential rate and the contribution proportional to $P_{L}$ agree with Eq. (2.8) of Ref. Bullock:1992yt . Upon $z$ integration $(0\leq z\leq 1)$, one obtains the azimuthal distribution

[TABLE]

The analyzing power is $\pi/12=26.18\%$, which is in agreement with the corresponding result in Eq. (22).

The calculation for $\beta_{\tau}\neq 1$ and $\beta=1$ is slightly more difficult and has been done by Tanaka and Watanabe Tanaka:2010se for the differential rate and the longitudinal contribution proportional to $P_{L}$. The decay distribution in terms of $dz$ is written as

[TABLE]

where ${\cal B}_{\ell}$ is the branching fraction of $\tau^{-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\tau}$. Neglecting the lepton mass $m_{\ell}$, i.e. setting $\beta=1$, the functions $f$, $g$, and $h$ are given by

[TABLE]

for $0\leq z\leq(1-\beta_{\tau})/2$, and

[TABLE]

for $(1-\beta_{\tau})/2\leq z\leq(1+\beta_{\tau})/2$. Equations. (II.2) and (II.2) are obtained by integrating Eq. (24) over $x$ in the regions $2z/(1+\beta_{\tau})\leq x\leq 2z/(1-\beta_{\tau})$ and $2z/(1+\beta_{\tau})\leq x\leq 1$, respectively.

In the collinear approximation $\beta_{\tau}=1$, the first region $0\leq z\leq(1-\beta_{\tau})/2$ shrinks to zero, while the second region $(1-\beta_{\tau})/2\leq z\leq(1+\beta_{\tau})/2$ simplifies to $0\leq z\leq 1$. The collinear forms of the functions $f(q^{2},z)$ and $g(q^{2},z)$ in Eq. (25) can be obtained by simply substituting $\beta_{\tau}=1$ in Eq. (II.2). However, it is quite subtle to recover the collinear form of $h(q^{2},z)$ in Eq. (25) from Eq. (II.2) since the treatments of the integral in two cases are different, depending on whether $\beta_{\tau}=1$ or $\beta_{\tau}\neq 1$.

Yet another method to analyze the longitudinal polarization of the $\tau^{-}$ has been suggested in Ref. Alonso:2016gym where a forward-backward asymmetry is defined with respect to $\cos\theta^{\ast}$, where $\theta^{\ast}$ is the angle between the final charged lepton and the recoiling $D^{(*)}$ in the $W^{-}$ rest system. At the end of this section we shall also discuss a different basis, the so-called off-diagonal basis, where the $z$ axis is chosen to point in the direction of the polarization vector of the $\tau^{-}$.

II.3 The off-diagonal basis

In their papers Mahlon:1995zn ; Parke:1996pr ; Mahlon:1997uc Mahlon, Parke, and Shadmi introduced the so-called off-diagonal (OD) basis to maximize spin-spin correlation effects in top quark pair production in $e^{+}e^{-}$ and hadronic interactions. As shown in Ref. Groote:2010zf ; Kaldamae:2016gws the off-diagonal basis amounts to choosing the $z$ axis to point in the direction of the polarization vector of the top quark, or, in this application, of the polarization vector of the $\tau^{-}$. For the sake of simplicity, we shall only discuss the off-diagonal basis for the SM case where $P_{N}=0$.

The relevant rotation to the off-diagonal basis is achieved by a rotation in the $(\vec{e}_{L},\vec{e}_{T})$ plane by an angle $\theta_{OD}$ where $\theta_{OD}$ is the polar angle of the tau polarization relative to the tau three-momentum, measured anticlockwise from the direction of the tau. One has

[TABLE]

In the off-diagonal basis (denoted by a prime), the transverse component of the polarization vector is zero $P^{\prime}_{T}=0$ and the azimuthal contribution proportional to $\cos\chi$ in the angular decay distributions vanishes. Therefore, the sensitivity of the polar angle measurement proportional to $\cos\theta^{\prime}_{d}$ is enhanced since $|\vec{P}|=\sqrt{P_{T}^{2}+P_{L}^{2}}\geq|P_{L}|$. Here, $\theta^{\prime}_{d}$ is the polar angle between the three-momentum of the $d^{-}$ and the $z$ direction in the off-diagonal basis.

This discussion suggests a possible search strategy to experimentally determine the polarization vector of the $\tau^{-}$ from a set of polar measurements alone. Take a set of directions $z$ in the $(\vec{e}_{L},\vec{e}_{T})$ plane and maximize the forward-backward polarization asymmetry $A_{FB}^{P}=A_{d}P_{L}(q^{2})$ for this set. The $z$ direction corresponding to this maximal value gives the direction of the $\tau^{-}$ polarization vector $\vec{P}$, and the corresponding value of $P_{L}(q^{2})$ obtained from $A_{FB}^{P}=A_{d}P_{L}(q^{2})$ determines its magnitude $|\vec{P}|$.

In Fig. 2 we display the $q^{2}$ dependence of the angle $\theta_{OD}$ for the $B\to D$ and $B\to D^{\ast}$ transitions. In the case of the $B\to D$ transition the angle $\theta_{OD}$ slightly changes in the range $(50\degree,70\degree)$ for almost the whole $q^{2}$ region and quickly decreases from $50\degree$ to $0\degree$ for $q^{2}\gtrsim 10~{}\text{GeV}^{2}$. In the case of the $B\to D^{\ast}$ transition the angle $\theta_{OD}$ monotonically increases with $q^{2}$ from about $80\degree$ to $180\degree$.

The $q^{2}$ dependence of the angle $\theta_{OD}$ is obviously related to the correlation between the longitudinal and transverse polarization components, or in other words, the orientation and the length of the polarization vector. In Fig. 3 we show how the apex of the polarization vector moves in the $(P_{L},P_{T})$ plane when $q^{2}$ increases from threshold $q^{2}=m_{\tau}^{2}$ to the zero-recoil points $q^{2}=(m_{\bar{B}^{0}}-m_{D^{(*)}})^{2}$. The apexes move within the unit circle since $|\vec{P}|\leq 1$. Both trajectories start off at threshold and end up at the zero-recoil points. As $q^{2}$ increases, the polarization vector of the $\tau^{-}$ turns into the direction of its three-momentum (for the $B\to D$ transition) or opposite to it (for the $B\to D^{\ast}$ transition). Both transverse polarization components vanish at zero recoil as follows from the helicity analysis in Sec. III. It is interesting to note that in the case of the $B\to D^{\ast}$ transition the dots are approximately equally spaced on the trajectory, which indicates a moderate rotation of the polarization vector when $q^{2}$ increases. In contrast, the polarization vector in the case of the $B\to D$ transition rotates quite fast for $q^{2}\gtrsim 10~{}\text{GeV}^{2}$. These behaviors are also reflected in the $q^{2}$ dependence of the angle $\theta_{OD}$ shown in Fig. 2. The average values of the polar angle $\theta_{OD}$ read $<\theta_{OD}>=130\degree$ for $B\to D^{\ast}$ and $<\theta_{OD}>=64\degree$ for $B\to D$.

III   Effective operators and helicity amplitudes

Assuming that all neutrinos are left-handed and that NP effects only influence leptons of the third generation, the effective Hamiltonian for the quark-level transition $b\to c\tau^{-}\bar{\nu}_{\tau}$ is given by

[TABLE]

where the four-Fermi operators $\mathcal{O}_{X}$ are defined as

[TABLE]

and $X$’s are the NP complex Wilson coefficients which are equal to zero in the SM.

The invariant form factors describing the hadronic transitions $\bar{B}^{0}\to D$ and $\bar{B}^{0}\to D^{\ast}$ are defined as follows:

[TABLE]

where $P=p_{1}+p_{2}$, $q=p_{1}-p_{2}$, and $\epsilon_{2}$ is the polarization vector of the $D^{\ast}$ meson which satisfies the condition $\epsilon_{2}^{\dagger}\cdot p_{2}=0$. The particles are on their mass shells: $p_{1}^{2}=m_{1}^{2}=m_{\bar{B}^{0}}^{2}$ and $p_{2}^{2}=m_{2}^{2}=m_{D^{(\ast)}}^{2}$.

Using the helicity technique first described in Refs. Korner:1987kd ; Korner:1989ve ; Korner:1989qb and further discussed in our recent papers Ivanov:2015tru ; Ivanov:2016qtw one obtains the ratio of branching fractions $R_{D^{(*)}}(q^{2})$ as follows:

[TABLE]

where

[TABLE]

[TABLE]

Here, $\delta_{\ell}=m^{2}_{\ell}/2q^{2}$ is the helicity flip factor, $g_{V}\equiv V_{L}+V_{R}$, $g_{S}\equiv S_{L}+S_{R}$, $g_{P}\equiv S_{L}-S_{R}$, and the index $n$ runs through $(0,+,-)$. The definition of the hadronic helicity amplitudes in terms of the invariant form factors are presented in the Appendix. The expressions for ${\cal H}_{tot}^{D^{(\ast)}}$ in Eqs. (35) and (36) agree with the results of Ref. Sakaki:2013bfa . Note that in this paper we do not consider interference terms between different NP operators since we assume the dominance of only one NP operator besides the SM contribution.

In the remaining part of this section we provide the formulae for the polarization components of the $\tau^{-}$ including NP contributions. Starting from the definition given in Eq. (4) one easily obtains the differential decay rate for a given spin projection in a given direction by using the Dirac projection operators, which results in the replacement of

[TABLE]

in the relevant traces. The $W^{-}$ rest frame polarization vectors $s_{i}^{\mu}$ are given by Faessler:2002ut ; Gutsche:2013oea ; Gutsche:2015mxa

[TABLE]

The longitudinal polarization reads

[TABLE]

We emphasize that the longitudinal polarization of the $\tau^{-}$ is defined in the $W^{-}$ rest frame with $\vec{p}_{\tau}$ defining the longitudinal direction, and not in the rest frame of the parent $\bar{B}^{0}$ meson.

Similarly, the transverse polarization is given by

[TABLE]

As can be seen directly from Eq. (III), the transverse polarization vanishes in the zero lepton mass limit $m_{\ell}=0$ due to the overall factor $\sqrt{\delta_{\ell}}=m_{\ell}/\sqrt{2q^{2}}$. Physically this comes about since the lepton is $100\%$ longitudinally polarized for $m_{\ell}=0$ and thus there is no room for a transverse polarization. It is the lepton mass that brings in the transverse polarization which, in fact, is quite large in the case of the $\tau^{-}$. In the SM the transverse polarization can be seen to vanish at zero recoil as a result of the zero-recoil relations $H_{t}=0$ and $H_{\pm}=H_{0}$ (see the Appendix).

The normal polarization is zero in the SM because we take the form factors and thereby the helicity amplitudes to be real. In the presence of NP CP-violating complex Wilson coefficients, they obtain nonzero contributions from the imaginary part of the coefficients as can be seen in Eq. (III). Both $P_{N}^{D}$ and $P_{N}^{D^{\ast}}$ are sensitive to the tensor and scalar operators. The normal polarization reads

[TABLE]

IV   Numerical analysis

It is important to note that all the discussions and expressions that we have provided so far are model independent. Now, in order to make numerical predictions we use the form factors calculated in the covariant confined quark model (CCQM) Ivanov:2016qtw which has been developed in several previous papers by our group (see Refs. Branz:2009cd ; Ivanov:2011aa ; Ivanov:2015woa and references therein). One can also employ the form factors obtained from the heavy quark effective theory (HQET) with better controlled errors. However, in this section, we only aim at clarifying the role of the tau polarization in searching for NP; therefore, the use of our form factors is well suited. For example, the longitudinal polarizations calculated in our model assuming only the SM operator are $<P^{D}_{L}>=0.33$ and $<P^{D^{\ast}}_{L}>=-0.50$, which are in very good agreement with other results in the literature $<P^{D}_{L}>=0.325\pm 0.009$ Tanaka:2010se and $<P^{D^{\ast}}_{L}>=-0.497\pm 0.013$ Tanaka:2012nw ; Hirose:2016wfn .

IV.1 Form factors in the CCQM

As has been discussed in detail in Ref. Ivanov:2016qtw we calculate the current-induced $B\to D^{(\ast)}$ transitions from their one-loop quark diagrams. As a result the various form factors in our model are represented by three-fold integrals which are calculated by using fortran codes in the full kinematical momentum transfer region $0\leq q^{2}\leq q^{2}_{max}=(m_{\bar{B}^{0}}-m_{D^{(\ast)}})^{2}$. Our numerical results for the form factors are well represented by a double-pole parametrization

[TABLE]

The parameters of the form factors for the $\bar{B}^{0}\to D$ and $\bar{B}^{0}\to D^{\ast}$ transitions are listed in Table 1.

We also list the zero-recoil values of the form factors for comparison with the corresponding HQET results which can e.g. be found in Ref. Ivanov:2016qtw . The agreement between the two sets of zero-recoil values is within $10\%$. It is worth mentioning that we obtain a nonzero result for the form factor $G_{0}^{T}$ at zero recoil, which is predicted to vanish in the HQET.

In Fig. 4, we compare our form factors with the Alonso-Kobach-Camalich (AKC) form factors calculated in Ref. Alonso:2016gym where they have used theoretical input from the HQET, lattice calculations, and equation of motion (EOM) relations. We rewrite the AKC form factors in our notation using the relations between the two sets of form factors. The form factor $F_{0}(q^{2})$ in Fig. 4 is given by

[TABLE]

It is seen that our form factors share quite similar shapes with the corresponding AKC ones. The first plot in Fig. 4 shows that our form factors $F_{+}(q^{2})$ and $F_{0}(q^{2})$ (solid lines) satisfy the relation $F_{0}(0)=F_{+}(0)$ while the corresponding AKC form factors (dashed lines) are slightly different at $q^{2}=0$. This is due to the fact that in their paper Alonso:2016gym , the authors used different parametrizations for $F_{+}(q^{2})$ and $F_{0}(q^{2})$. More specifically, they used the Caprini-Lellouch-Neubert parametrization for $F_{+}(q^{2})$ Caprini:1997mu ; Tanaka:2012nw , but the Bourrely-Caprini-Lellouch parametrization for $F_{0}(q^{2})$ Bourrely:2008za ; Na:2015kha . However, the difference $F_{0}(0)-F_{+}(0)\approx 0.03$ lies within the uncertainty of $F_{+}(q^{2})$ at $q^{2}=0$, which reads $F_{+}(0)=0.664(34)$ Na:2015kha .

We note that in Ref. Ivanov:2015tru the heavy quark limit (HQL) in our approach was explored in great detail for the heavy-to-heavy $\bar{B}^{0}\to D^{(\ast)}$ transitions. In Ref. Ivanov:2015tru we also calculated the Isgur-Wise function and considered the near-recoil behavior of the form factors. A brief discussion of the subleading corrections to the HQL arising from finite quark masses can be found in Appendix B of our paper Ivanov:2016qtw . Note that our form factors do not satisfy the EOM relations since the $b$ and $c$ quarks in the relevant propagators in the quark loop are off their mass shells.

Finally, we briefly discuss some error estimates within our model. We fix our model parameters (the constituent quark masses, the infrared cutoff, and the hadron size parameters) by minimizing the functional $\chi^{2}=\sum\limits_{i}\frac{(y_{i}^{\rm expt}-y_{i}^{\rm theor})^{2}}{\sigma^{2}_{i}}$ where $\sigma_{i}$ is the experimental uncertainty. If $\sigma$ is too small then we take its value of 10$\%$. Moreover, we observed that the errors of the fitted parameters are of the order of 10$\%$. Thus we estimate the model uncertainties to lie within 10$\%$.

IV.2 Experimental constraints

Within the SM (without any NP operators) our model calculation yields $R(D)=0.267$ and $R(D^{\ast})=0.238$, which are consistent with other SM predictions given in Refs. Na:2015kha ; Lattice:2015rga ; Aoki:2016frl ; Fajfer:2012vx within $10\%$. Assuming the dominance of only one NP operator in Eq. (31) at a time (besides the SM contribution), we compare the calculated ratios $R_{D^{(\ast)}}$ with the current experimental data $R_{D}=0.406\pm 0.050$ and $R_{D^{\ast}}=0.311\pm 0.016$ given in Sec. I and obtain the allowed regions for the NP couplings as shown in Fig. 5.

It is important to note that while determining these regions, we also take into account a theoretical error of $10\%$ for the ratios $R(D^{(\ast)})$. The operator $\mathcal{O}_{S_{R}}$ is excluded at $2\sigma$ and is not presented here. The operator $\mathcal{O}_{V_{L}}$ is not excluded, but it does not affect the polarizations in general and, therefore, will not be considered in what follows. In other words, only three NP operators $\mathcal{O}_{V_{R}}$, $\mathcal{O}_{S_{L}}$, and $\mathcal{O}_{T_{L}}$ can modify the polarizations. In each allowed region at $2\sigma$ we find the best-fit value for each NP coupling. The best-fit couplings read

[TABLE]

and are marked with an asterisk.

IV.3 Theoretical predictions

The $\tau^{-}$ polarization components in $\bar{B}^{0}\to D^{\ast}\tau^{-}\bar{\nu}_{\tau}$ are shown in Fig. 6. In each column we present one component in the presence of different NP couplings $S_{L}$, $V_{R}$, and $T_{L}$, one by one. In each row one can see how one NP coupling affects the three components at the same time. All the plots are in one scale so that one can quickly compare the sensitivity of different polarization components to different NP couplings.

Let us begin with the longitudinal polarization (left column in Fig. 6). The longitudinal polarization $P_{L}^{D^{\ast}}$ is not affected by $V_{R}$ but is very sensitive to $S_{L}$ and $T_{L}$. Both $S_{L}$ and $T_{L}$ tend to increase $P_{L}^{D^{\ast}}$ and shift the zero-crossing point from that in the SM. In the presence of $S_{L}$, $P_{L}^{D^{\ast}}$ starts at a higher value but converges to its SM value at high $q^{2}$ and its shape is similar to the SM one. In contrast to $S_{L}$, $T_{L}$ changes $P_{L}^{D^{\ast}}$ thoroughly: $P_{L}^{D^{\ast}}$ now starts at a lower position but can be positive for the most part of the whole $q^{2}$ region and maximally diverts from its SM prediction at high $q^{2}$.

The transverse polarization $P_{T}^{D^{\ast}}$ (center column in Fig. 6) has the same sensitivity to $S_{L}$ and $V_{R}$ but $S_{L}$ tends to increase $P_{T}^{D^{\ast}}$ while $V_{R}$ tends to decrease $P_{T}^{D^{\ast}}$. The transverse polarization is extremely sensitive to $T_{L}$ and its sign can be changed in the presence of $T_{L}$. It is interesting to note that $S_{L}$ increases both $P_{L}^{D^{\ast}}$ and $P_{T}^{D^{\ast}}$, while $T_{L}$ amplifies $P_{L}^{D^{\ast}}$ but lowers $P_{T}^{D^{\ast}}$. When $T_{L}$ is present, largest deviations of $P_{T}^{D^{\ast}}$ from its SM prediction happen at low $q^{2}$, which is opposite to the case of $P_{L}^{D^{\ast}}$.

Regarding the normal polarization $P_{N}^{D^{\ast}}$ (right column in Fig. 6), it is sensitive to both $S_{L}$ and $T_{L}$ but slightly more to $S_{L}$. $P_{N}^{D^{\ast}}$ can be both positive or negative and its absolute value can reach about $0.2$. It is worth noting that $P_{N}^{D^{\ast}}$ is much less sensitive to $T_{L}$ in comparison with $P_{L}^{D^{\ast}}$ and $P_{T}^{D^{\ast}}$.

Next we turn to the $\tau^{-}$ polarizations in $\bar{B}^{0}\to D\tau^{-}\bar{\nu}_{\tau}$, which are shown in Fig. 7.

It is readily seen that all three polarization components in this case are much more sensitive to $S_{L}$ than to $T_{L}$. In the presence of $T_{L}$, the polarizations $P_{L}^{D}$ and $P_{T}^{D}$ can be positively or negatively enhanced but their shapes over the whole $q^{2}$ range are similar to those in the SM. In contrast, the scalar coupling $S_{L}$ changes the shapes of $P_{L}^{D}$ and $P_{T}^{D}$ dramatically and can even imply a zero-crossing point, which is impossible in the SM. This distinct effect of $S_{L}$ may give some hints for experimental study. The normal polarization $P_{N}^{D}$ can reach about $\pm 0.2$ under the effect of $T_{L}$ while it can even reach about $\pm 0.8$ when $S_{L}$ is present.

The $q^{2}$ dependence of the polarizations bears powerful information for discriminating between different NP scenarios. One possible approach is to make use of it to perform a bin-by-bin analysis in order to probe NP in different $q^{2}$ regions. One can also calculate the average polarizations over the whole $q^{2}$ region. When calculating the $q^{2}$ averages one has to multiply the numerator and denominator of (III), (III), and (III) by the $q^{2}$-dependent piece of the phase-space factor given by $C(q^{2})=|\mathbf{p_{2}}|(q^{2}-m_{\tau}^{2})^{2}/q^{2},$ where $|{\bf p_{2}}|=\lambda^{1/2}(m_{1}^{2},m_{2}^{2},q^{2})/2m_{1}$ is the momentum of the daughter meson. For example, the average longitudinal polarization $\langle P_{L}^{D}\rangle$ can then be calculated according to

[TABLE]

The predictions for the mean polarizations are summarized in Table 2.

Again, one sees that the $\tau^{-}$ polarization components in $\bar{B}^{0}\to D\tau^{-}\bar{\nu}_{\tau}$ are extremely sensitive to $S_{L}$. When $S_{L}$ is present, $\langle P_{L}^{D}\rangle$ can be as large as $0.67$, $\langle P_{T}^{D}\rangle$ can reach $-0.68$, and $\langle P_{N}^{D}\rangle$ can even reach $\pm 0.76$. It is interesting to note that if one measures $\langle P_{L}^{D}\rangle$ and finds any excess over the SM value, it would be a clear sign of $S_{L}$. Meanwhile, the $\tau^{-}$ longitudinal and transverse polarization components in $\bar{B}^{0}\to D^{\ast}\tau^{-}\bar{\nu}_{\tau}$ are more sensitive to $T_{L}$. The coupling $T_{L}$ can enhance $\langle P_{L}^{D^{\ast}}\rangle$ from the SM value of $-0.50$ up to $0.24$, or lower $\langle P_{T}^{D^{\ast}}\rangle$ from $0.46$ down to $-0.61$. Notably, the average transverse polarization $\langle P_{T}^{D}\rangle$ is almost insensitive to $T_{L}$ in comparison with $S_{L}$. When $T_{L}$ is present, one finds $0.78\leq\langle P_{T}^{D}\rangle\leq 0.83$, which is almost the same as the SM value $\langle P_{T}^{D}\rangle=0.84$. In contrast, if $S_{L}$ is present, one has $-0.68\leq\langle P_{T}^{D}\rangle\leq 0.33$, which is much lower than the SM prediction. This unique property of $\langle P_{T}^{D}\rangle$ may play a very important role in probing the scalar coupling $S_{L}$. It is also interesting to note that the average total polarization $<|\vec{P}^{D^{\ast}}|>$ is almost insensitive to $S_{L}$.

V   Summary and conclusions

We have studied the longitudinal, transverse, and normal polarization components of the $\tau^{-}$ in the semileptonic decays $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ in the presence of NP scalar, vector, and tensor interactions based on an SM-extended effective Hamiltonian. Constraints on the space of NP couplings have been obtained from experiments at $B$ factories and LHCb including the most recent result of the Belle collaboration Hirose:2016wfn . We have also briefly discussed how to extract the polarization of the $\tau^{-}$ from the distribution of its most prominent subsequent decay modes.

All the polarization components are sensitive to the scalar coupling $S_{L}$ and the tensor coupling $T_{L}$. Besides, the transverse polarization $P_{T}^{D^{\ast}}$ is also sensitive to the vector coupling $V_{R}$. The longitudinal and transverse polarizations are more sensitive to $T_{L}$ in the case of $\bar{B}^{0}\to D^{\ast}\tau^{-}\bar{\nu}_{\tau}$, but more to $S_{L}$ in the case of $\bar{B}^{0}\to D\tau^{-}\bar{\nu}_{\tau}$. $P_{N}^{D^{\ast}}$ is equally sensitive to $T_{L}$ and $S_{L}$, while $P_{N}^{D}$ is much more sensitive to $S_{L}$ than to $T_{L}$. The normal polarization can reach about $\pm 0.8$ in $\bar{B}^{0}\to D\tau^{-}\bar{\nu}_{\tau}$ if $S_{L}$ is present, and about $\pm 0.2$ in other cases. These observations may provide some insights to look for NP in the decays $\bar{B}^{0}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$.

Acknowledgements.

The authors thank the Heisenberg-Landau Grant for providing support for their collaboration. M.A.I. acknowledges the financial support of the PRISMA Cluster of Excellence at the University of Mainz. M.A.I. and C.T.T. greatly appreciate the warm hospitality of the Mainz Institute for Theoretical Physics (MITP) at the University of Mainz.

Appendix A Helicity amplitudes

In this appendix, we express the helicity amplitudes used in the main text in terms of the invariant form factors defined in Eq. (III). A detailed derivation of these relations can be found in our recent paper Ivanov:2016qtw .

For the $\bar{B}^{0}\to D$ transition:

[TABLE]

where $|{\bf p_{2}}|=\lambda^{1/2}(m_{1}^{2},m_{2}^{2},q^{2})/2m_{1}$ is the momentum of the daughter meson.

For the $\bar{B}^{0}\to D^{\ast}$ transition:

[TABLE]

The dependence of the helicity amplitudes and the invariant form factors on $q^{2}$ have been omitted for simplicity.