SciPost Phys. Proc. 1, 012 (2019)

Tau polarimetry in B meson decays
Rodrigo Alonso1 , Jorge Martin Camalich2,3 and Susanne Westhoff 4?
1 Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba, 277-8583 Japan
2 Instituto de Astrofísica de Canarias, C/ Vía Láctea,
s/n E38205 - La Laguna (Tenerife), Spain
3 Universidad de La Laguna, Departamento de Astrofísica, La Laguna, Tenerife, Spain
4 Institute for Theoretical Physics, Heidelberg University, D-69120 Heidelberg, Germany? westhoff@thphys.uni-heidelberg.de
Proceedings for the 15th International Workshop on Tau Lepton Physics,
Amsterdam, The Netherlands, 24-28 September 2018
doi:10.21468/SciPostPhysProc.1

Abstract
This article summarizes recent developments in B → D(∗) τν decays. We explain how to
extract the tau lepton’s production properties from the kinematics of its decay products.
The focus is on hadronic tau decays, which are most sensitive to the tau polarizations.
We present new results for effects of new physics in tau polarization observables andquantify the observation prospects at BELLE II.
Copyright R. Alonso et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

1

Received 13-11-2018
Accepted 15-01-2019
Published 19-02-2019

Check forupdatesdoi:10.21468/SciPostPhysProc.1.012

Introduction

Semi-leptonic B meson decays with tau leptons are important probes of lepton universality inweak interactions. By building ratios of decay rates, very clean observables can be predictedthat depend very little on hadronic uncertainties. The detection of tau leptons is challenging,but recent progress at BELLE and LHCb shows that measurements beyond total decay rates are
possible. This makes B → Dτν and B → D∗ τν decays a key area of study at the new BELLE II
experiment.
On the theory side, the precision is limited by the hadronic form factors governing B → D
and B → D∗ transitions. Very recently, lattice calculations of vector and scalar form factors beyond the limit of zero D(∗) recoil became available. 1 Combined with shape predictions basedon heavy quark effective theory and QCD sum rules, they allow us to predict differential distributions in B → D(∗) τν to percent precision. This provides us with a solid basis to investigate
semi-tauonic B decays over the entire phase-space region.
Among the differential observables, lepton and meson polarizations are particularly interesting, because they are direct probes of the underlying interaction. Two recent analyses have1

See talks by Chris Monahan and Alejandro Vaquero at CKM 2018.

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SciPost Phys. Proc. 1, 012 (2019)

pioneered polarization measurements in semi-tauonic B decays. The BELLE collaboration has
extracted the longitudinal tau polarization in B → D∗ τν, [1, 2]
PL (τ) = −0.38 ± 0.51 (stat)+0.21−0.16 (syst).

(1)

The uncertainty is large, but statistics-dominated, which will be overcome at BELLE II. In thesame process, BELLE has measured the fraction of longitudinally polarized D∗ mesons,2
F L (D∗ ) =Γ (DL∗ )Γ (DL∗ ) + Γ (DT∗ )= 0.60 ± 0.08 (stat) ± 0.04 (syst).

(2)

The remarkable precision of this measurement suggests that combined analyses of tau and D∗
observables have a good potential to probe the underlying production process.
In addition to these developments, observed deviations in the branching ratios of B →(∗)
D τν decays, widely referred to as the R D(∗) puzzle, have inspired searches for new particles
with non-universal lepton interactions. Regardless of the motivation to explore B → D(∗) τν
decays, however, any information about these processes is imprinted on the tau and D(∗) decay
products. The presence of one or more neutrinos in the final state prevents us from fullyreconstructing the decay kinematics. In order to gain maximal information on the production
process, it is thus necessary to define observables in terms of these decay products.
Recent analyses of B → D(∗) τν differential distributions have pursued two methods: anumerical approach based on Monte-Carlo simulations [3], and an analytic approach basedon a fully analytic phase-space integration [4–7]. While numerical simulations can be directly
used at an experiment, analytical predictions make the relations between observables andfeatures of the underlying process transparent. In this article, we pursue the analytic approach
to extract tau polarizations and a forward-backward asymmetry in tau production from finalstate kinematics. The discussion will be focused on B → Dτν; an extension to B → D∗ τν ispossible by pursuing the same strategy.
Beyond the standard model, tau polarimetry is an important tool to detect and disentangle
possible new physics effects. We probe the sensitivity of tau polarization observables to new
scalar, vector and tensor currents contributing to B → Dτν decays in Section 4.

Tau properties in B → Dτν decays2

We start by briefly reviewing the observable tau properties in B → Dτν decays. For moredetails, we refer you to our Ref. [7]. The differential decay rate for producing a tau lepton
polarized along a certain direction ŝ is given bydΓ dΓ (ŝ) =1 + (d PL êτ + d P⊥ ê⊥ + d PT ê T ) · ŝ .
(3)
2
We choose our coordinate system {êτ , ê⊥ , ê T } as
êτ =

~pτ
,
|~pτ |ê T =~p D × ~pτ
,|~p D × ~pτ |ê⊥ = ê T × êτ ,(4)where ~pτ and ~p D are the momenta of the τ lepton and the D meson defined in the rest frameof the tau-neutrino pair. The decay kinematics are illustrated in Fig. 1. The longitudinal polarization PL thus points in the direction of the tau momentum, the perpendicular polarizationP⊥ lies in the D −τ plane perpendicular to the tau momentum, and the transverse polarization
points orthogonal to the D − τ plane. Here we will focus on the longitudinal and perpendicular polarizations. Observing the transverse polarization requires information beyond thecurrently accessible decay kinematics.
2

This result was presented by Karol Adamczyk at CKM 2018.

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SciPost Phys. Proc. 1, 012 (2019)

Figure 1: Kinematics of the decay chain B → Dν[τ → dν(ν̄)], where d = {π, ρ, `}.
The momenta of the D meson and the tau lepton span the plane in B → Dτν decays(in green). The momenta of the tau lepton and its visible decay product d span asecond plane in the subsequent decay τ → dν(ν̄) (in blue).
The unpolarized rate for B → Dτν depends on the angle between the tau and D momenta, cos θτ . We define the tau forward-backward asymmetry with respect to the D meson
momentum as
Z1
Z0
d 2Γ
d 2Γ
dΓ
2
A
(q
)
=
−
,
(5)
d
cos
θ
d
cos
θ
τ
τ
τ
dq2
dq2 d cos θτ
dq2 d cos θτ0−1
where q is the four-momentum of the tau-neutrino pair.

3

Observables from tau decay products

The tau properties above are not directly observable, but need to be reconstructed from thetau decay products. We consider the three most frequent tau decay modes, τ → `ν̄ν (` = e, µ),τ → πν, and τ → ρν. The hadronic decays τ → πν, and τ → ρν have a higher analyzing
power than τ → `ν̄ν, because only one neutrino is missed. The momenta of the visible decay
product d = {`, π, ρ} and the tau momentum span a second plane, see Fig. 1. Due to themissed neutrino momentum, the relative orientation between the D − τ plane and the τ − dplane is not observable.
The D meson and the decay product d are the only visible particles in the final state. The
maximal kinematic information in the full decay process B → Dν[τ → dν(ν̄)] can be expressedin terms of three variables. We choose them as the momentum transfer, q2 , the energy of the
decay particle, Ed , and the angle between the D and d momenta, cos θd , all defined in the
tau-neutrino rest frame. The differential decay rate can be written asd 3 ΓdN = Bd
I0 (q2 , Ed ) + I1 (q2 , Ed ) cos θd + I2 (q2 , Ed ) cos2 θd ,2
dq d Ed d cos θd
2mτ

(6)

where Bd is the branching ratio for τ → dν(ν̄), N is a normalization factor, and the kinematicfunctions I i are defined in Ref. [5]. After integrating over the angle cos θd , we obtain the
double-differential rated 2 ΓddΓ  d 2
d 2
2
=
B
f
(q
,
s
)
+
f
(q
,
s
)P
(q
)
,
d
d
d
L
L
dq2 dsddq2 0

sd = Ed /

Æ

q2 .

(7)

The energy distribution of the decay particle d is thus sensitive to the longitudinal tau polarization. Notice that for τ → πν and τ → ρν the function f0d does not depend on sd , so thatthe energy distribution is maximally sensitive to PL .
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SciPost Phys. Proc. 1, 012 (2019)

0.4
0.2

Al

dΓ

0.0
-0.2
-0.4

Aρ
Aπ

-0.6
4

6

8
2

10

2

q [GeV ]Figure 2: Decay rate dΓ and angular asymmetries Ad of final-state particles
d = π, ρ, `.
Complementary to d 2 Γd /dq2 dsd , which probes the parts symmetric in cos θd in Eq. (6),
we can define an angular asymmetry that probes the contribution linear in cos θd ,Z0 Z 1dAddΓ −1d cos θd d 3 Γd
(8)
= Bd 2d cos θd d 3 Γd −
dsd
dq
−1
0
= fAd (q2 , sd )Aτ (q2 ) + f⊥d (q2 , sd )P⊥ (q2 ).
The energy distribution of this asymmetry is sensitive to both the perpendicular polarizationand the tau forward-backward asymmetry. The respective sensitivity of Ad to Aτ and P⊥ is
imprinted in the energy of the decay particle d through the kinematic functions fAd (q2 , sd ) andf⊥d (q2 , sd ). In Fig. 2 we show the differential rate dΓ and the decay asymmetries Ad as functionsof q2 . Integrated over q2 , we predict the following asymmetries in the standard model,
Aπ = −0.54,

4Aρ = −0.32,

A` = +0.06.

(9)

New physics in polarization observables

Measuring additional observables from the kinematic distributions of B → Dτν is useful tofurther test the standard model and to discriminate among the possible New Physics (NP)solutions to the R D(∗) puzzle. In Fig. 3, we show the q2 -spectrum of the total rate, the tau
forward-backward asymmetry and the two polarization observables in the standard model(black curves). The uncertainty bands are due to hadronic form factors and correspond to one
standard deviation, as implemented in Refs. [5, 7].
We present effects of new physics in three different benchmark scenarios that can explain
the R D(∗) anomalies. The NP contributions are parametrized in terms of Wilson coefficients
corresponding to vector, scalar, and tensor currents [5]. Our benchmarks correspond to“Vector” :“Scalar” :ετL = 0.15,(10)ετS = 0.80, ετS = −0.65,
L

R“Tensor” : ετT = 0.40.
In the “Vector” scenario, only the magnitude of the decay rate is modified by a left-handed
vector current, so that effects cancel in the normalized observables Aτ , P⊥ and PL . In the“Scalar” and “Tensor” scenarios, all differential observables receive sizeable contributions. In
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SciPost Phys. Proc. 1, 012 (2019)

Figure 3: Differential observables of B → Dτ− ν̄τ in the standard model (black) andin scenarios of NP: “Vector” (solid red), “Scalar” (dashed red) and “Tensor” (dotted
red).
Table 1: Numerical predictions of the integrated observables in the standard modeland in the different benchmark scenarios of NP (see main text).

Standard model
VectorScalarTensorMeasurementRD
0.299(3) [9, 10]
0.410
0.400
0.467
0.407(39)(24) [11]Aτ−0.359(3)−0.359−0.335−0.451–PL
0.34(3)
0.34
0.52
0.14
–

P⊥
−0.839(7)−0.839−0.758−0.779–Tab. 1, we show numerical predictions for the q2 -integrated observables in the presence of
new physics. These results demonstrate that the different NP scenarios considered here leadto results unambiguously distinct from the standard model and from each other [5, 7, 8].

5Observation prospects at B physics experiments

To estimate the sensitivity to the tau properties at the BELLE II experiment, we perform astatistical analysis, assuming an ideal experiment with unlimited resolution in q2 and Ed . Systematic uncertainties are not taken into account. For the longitudinal polarization, the statistical uncertainty δPL (q2 ) and the sensitivity S L (q2 ) of a measurement in the differential rated 2 Γd /dq2 dsd are given byZ
f L (q2 , sd )212
2 2
δPL (q ) = p,
S L (q ) = dsd.
(11)
f0 (q2 , sd ) + f L (q2 , sd )PL (q2 )N (q2 )S L (q2 )In Tab. 2, we present the resulting relative uncertainty on the longitudinal polarization PL ,averaged over the momentum range q2 . The predictions are made for 3000 B − → D0 τ− ν̄τevents, as expected with 50 ab−1 of data at BELLE II. We also show predictions for the perpendicular polarization P⊥ and the tau forward-backward asymmetry Aτ , which can be extracted
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SciPost Phys. Proc. 1, 012 (2019)

Table 2: Predicted relative statistical uncertainties on the tau polarizations, PL andP⊥ , and forward-backward asymmetry, Aτ , at BELLE II with 3000 B − → D0 τ− ν̄τevents.

L = 50 ab−1

τ → πν

τ → ρν

τ → `νν̄δPL /PL

0.03

0.07

0.09

δP⊥ /|P⊥ |

0.09

0.25

0.57

δAτ /|Aτ |

0.11

0.10

0.40

from a combined fit to the energy distribution of the decay asymmetry, dAd /dsd . Correlations between P⊥ and Aτ are taken into account. It is apparent that τ → πν has the highest
sensitivity to the polarizations, while τ → ρν competes for Aτ .

6

Conclusions

We have shown that the properties of the tau lepton in B → Dτν decays can be extracted
from the tau’s decay products with a good sensitivity. BELLE II has the potential to achievemeasurements of the longitudinal and perpendicular polarizations, as well as the tau forwardbackward asymmetry, in the standard model at the level of 10%. This prediction cruciallyrelies on hadronic tau decays, which have the highest sensitivity to the tau properties.
Beyond the standard model, tau polarimetry is a strong tool to detect potential new physics
contributing to tau production. In the light of the observed discrepancies in the total decay
rates of B → Dτν and B → D∗ τν, tau properties are useful to disentangle effects of new
physics and probe their viability in different observables. While we have focused on B → Dτν,an extension of our approach to polarization observables in B → D∗ τν is in progress. We
encourage experimentalists at BELLE II to exploit the rich opportunities that hadronic tau
decays offer.

Acknowledgements
We thank the organizers of Tau 2018 for an interesting and enjoyable workshop in Vondelkerk. JMC acknowledges support from the Spanish MINECO through the “Ramón y Cajal”
program RYC-2016-20672. SW acknowledges funding by the Carl Zeiss foundation throughan endowed junior professorship (Junior-Stiftungsprofessur).

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