Un-binned Angular Analysis of 𝑩 → 𝑫∗ ℓ𝝂
and the Right-handed Current
Zhuoran Huang
APCTP
In collaboration with Emi Kou, Cai-Dian Lü and Ru-Ying Tang
Flavor Physics Workshop 2021
1

Motivation• Semileptonic 𝐵 → 𝐷 (∗) ℓ𝜈 decay• 𝑅(𝐷 (∗) ) anomalies• 𝑉𝑐𝑏 𝑝𝑢𝑧𝑧𝑙𝑒
inclusive decay 𝐵 → 𝑋𝑐 ℓ𝜈 (𝑋𝑐 = 𝐷, 𝐷∗ , 𝐷0∗ …)
HQE, Optical theorem, OPE

exclusive decay 𝐵 → 𝐷 (∗) ℓ𝜈
form factor calcualtion: lattice, LCSR

in. 42.16(50) vs ex. 39.70(60) parametrization: CLN(-like)/BGL~ 3𝜎 deviation[M. Bordone et al. ′21] [S. Iguro et al. ′20]

Relation between the R.H. vector current and the 𝑽𝒄𝒃 puzzle

𝐶𝑉𝐿 = 1 and 𝐶𝑉𝑅 = 0 in the SM
𝐶𝑉𝑅 ≠ 0 in the Left-Right symmetric modelfrom 𝑊𝐿 − 𝑊𝑅 mixing [E. Kou et al. ′13]
[A. Crivellin et al. ′15]

𝐵 → 𝐷∗ ℓ𝜈 vs 𝐵 → 𝑋𝑐 ℓ𝜈: 𝐶𝑉𝑅 ~ − 5%
𝐵 → 𝐷ℓ𝜈 vs 𝐵 → 𝑋𝑐 ℓ𝜈: 𝐶𝑉𝑅 ~ 5%

Considerable ex. uncertainties.
Theo. uncertainty from lattice QCD input.
More measurements needed!

Theoretical Framework
Differential decay rate (𝒎𝝁,𝒆 → 𝟎):𝐽𝑖 experimentally measurable, includes 𝐻+ ,𝐻− , 𝐻0 , 𝐶𝑉𝐿 and 𝐶𝑉𝑅 (SM and BSM).

𝑱𝒊 functions:Kinematic variables in 𝑩 → 𝑫∗ → 𝑫𝝅 ℓ𝝂𝜃ℓ the angle between the lepton andthe direction opposite the B-mesonin the virtual W-boson rest frame;

𝜃𝑣 the angle between the D meson andthe direction opposite the B meson inthe D∗ rest frame;

𝜒 the tilting angle between the two
decay planes spanned by the W and D
systems in the B meson rest frame;

𝑤 the dimensionless four-momentum
transfer.
[A. Abdesselam et al, Belle Collaboration ′17]

Helicity amplitudes in CLN and BGL parametrizations

BGL parametrization (analyticity based)

CLN parametrization (HQE based)

Blaschke factors:
𝑷𝒈 , 𝑷𝒇 , 𝑷𝑭𝟏
outer functions:𝝓𝒈 , 𝝓𝒇 , 𝝓𝑭𝟏

Un-binned Angular Analysis
Normalised PDF:

Exsiting binned analysis(projected 𝝌𝟐 fit):
Belle ’17 ’18; BaBar ‘19

The experimental determination of < 𝒈𝒊 >can be pursued by the maximum
likelihood method :Angular observables allow todetermine 𝐶𝑉𝑅 without theintervention of the 𝑉𝑐𝑏 puzzle!

Pseudo data generation
Pseudo data generated using CLN parameters
fitted by Belle [E. Waheed et al, ′18]
𝑵𝒆𝒗𝒆𝒏𝒕 = (𝟓𝟑𝟎𝟔, 𝟖𝟗𝟑𝟒, 𝟏𝟎𝟓𝟐𝟓, 𝟏𝟏𝟐𝟒𝟏, 11392,11132,10555,9726,8693,7497)
Pseudo data generated using BGL parameters
fitted by Belle [E. Waheed et al, ′18]

𝑵𝒆𝒗𝒆𝒏𝒕 = (𝟓𝟑𝟎𝟔, 𝟖𝟗𝟑𝟒, 𝟏𝟎𝟓𝟐𝟓, 𝟏𝟏𝟐𝟒𝟏, 11392,11132,10555,9726,8693,7497)< 𝒈𝒊 > generated in 10 bins with covariancematrices by toy Monte-Carlo method
Total event number: 95k as in Belle analysis
Using pseudo data we fit theoretical formula
including 𝑪𝑽𝑹 (on top of form factors). Note
𝑽𝒄𝒃 is not possible to fit any more because itcancels in 𝒈𝒊 !

< 𝒈𝒊 > generated in 10 w-bins

𝝌𝟐 utilized in the CLN/BGL fit
Notes:1.) 𝑪𝑽𝑹 and 𝑽𝒄𝒃 are dependent
using decay rates in the fit as thechanges in both parameters
directly impact 𝑩𝒓(𝑩 → 𝑫∗ ℓ𝝂)

We include the lattice input by introducing

with 𝒉𝑨𝟏 𝟏 = 𝟎. 𝟗𝟎𝟔 ± 𝟎. 𝟎𝟏𝟑
by Fermilab/MILC

2.) the angular fit does notconverge as 𝑪𝑽𝑹 is not independentof the vector form factor
Lattice input of the vector form factoris crucial for determining 𝐶𝑉𝑅 !
𝑹𝟏 (𝟏) ~𝟒% error 𝒉𝑽 (𝟏) ~7% error[T. Kaneko et al, ′19]

Fit of 𝑪𝑽𝑹
CLN fit:𝒉𝑽 (𝟏)
𝑹𝟏 𝟏 =𝒉𝑨𝟏 (𝟏)

𝐶𝑉𝑅 can be determined to a
precision of ~2 (4)% in CLN(BGL) parametrization.

BGL fit:Central value of 𝐶𝑉𝑅 is ~0 because of Belle measurement.

𝐶𝑉𝑅 and the vector form factor are highly correlated!

Contour Plots

If new lattice results turn out to be different from theexperimental fitted value (assuming SM), non-zero
𝐶𝑉𝑅 can be hinted.

Fit of 𝑪𝑽𝑹 using forward-backward asymmetry (FBA) only
𝑭𝑩𝑨~ < 𝐠 𝟔𝐬 >Advantage: one angle measurement

𝐶𝑉𝑅 can be determined at a precision of
2.2% using FBA alone! Almost as goodas the full set of < 𝑔𝑖 >!
𝑰𝒎 𝑪𝑽𝑹 can also be determined at
precision of 𝟎. 𝟕% for both CLN andBGL!

Summary & Conclusions•The normalized angular observables < 𝒈𝒊 > for 𝑩 → 𝑫∗ 𝑫𝝅 ℓ𝝂 determined in
the un-binned analysis are useful for the precision measurement of 𝑪𝑽𝑹 bycircumventing the 𝑽𝒄𝒃 puzzle.

•𝑪𝑽𝑹 is highly dependent on the vector form factor, thus it can only bedetermined with the vector form factor calculated by lattice.

•The real (imaginary) part of 𝑪𝑽𝑹 can be determined at precision of 2-4 (1) %using the full set of < 𝒈𝒊 >.

•FBA (< 𝒈𝟔𝒔 > ) can determine 𝑪𝑽𝑹 at almost equally good precision, thus it ishighly proposed to be measured in the near future.

Thank you!

Backup
SM fit including 𝑽𝒄𝒃

SM fit results in CLN parametrization

w dependence in 𝝌𝟐 :SM fit results in BGL parametrization

The factor 𝜶 is a constant, which relates the numberof events and the decay rate:ഥ pairs produced from 𝚼(𝟒𝐒)
number of 𝑩𝑩
𝑩𝟎 lifetime
𝑩+ /𝑩𝟎 production ratio at Belle𝜶 = 𝟔. 𝟔𝟏𝟔 𝟔. 𝟔𝟏𝟑 × 𝟏𝟎𝟏𝟖 in CLN (BGL)
parametrization
Experimental efficiency: 𝝐 = ~ 𝟒. 𝟖 × 𝟏𝟎−𝟐15