Sensitivity studies on the CKM angle $\gamma$ in $\Lambda_{b}^{0}\rightarrow
D\Lambda$ decays

Shunan<EMAIL_ADDRESS>Yi<EMAIL_ADDRESS>Zewen<EMAIL_ADDRESS>Wenbin
<EMAIL_ADDRESS>
1State Key Laboratory of Nuclear Physics and Technology & School of Physics,
Peking University, Beijing 100871, China
2School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China

The sensitivity of the CKM angle $\gamma$ in $\Lambda_{b}^{0}\rightarrow
D\Lambda$ decays has been studied using the decay parameter $\alpha$ as an
observable in addition to the decay rate asymmetry. Feasibility studies show
that adding this observable improves the sensitivity on $\gamma$ by up to 60%
and makes the decays one of the most promising places to measure angle
$\gamma$ and to search for $C\\!P$ violation in $b$-baryon decays.

One of the primary goals of flavor physics is to search for extra sources of
$C\\!P$ violation by over-constraining the CKM matrix [1, 2]. The CKM angle,
$\gamma\equiv\arg[-V_{ud}V_{ub}^{*}/V_{cd}V_{cb}^{*}]$, is one of the least
known CKM parameters and limits the power of the global fit [3, 4]. Many
methods have been proposed to measure the angle $\gamma$ using tree-level
transitions of $b\rightarrow c$ and $b\rightarrow u$ with negligible
theoretical uncertainties [5]. They are categorized according to the decays of
$D$ meson. Throughout the paper, $D$ is used to indicate a quantum mixture of
$D^{0}$ and $\overline{D}^{0}$. The GLW or ADS method [6, 7, 8] refers to $D$
decays into $C\\!P$ eigenstates or non-charge-conjugated final states, where
the favored (suppressed) $b$-decay is followed by a suppressed (favored) $D$
decay. The BPGGSZ method is used in case of multi-body $D$ decays, i.e.
$K_{\mathrm{S}}^{0}\pi^{+}\pi^{-}$ or $K_{\mathrm{S}}^{0}K^{+}K^{-}$ [9, 10,
11]. The current world average for $\gamma$ from direct measurements is
$(66.2^{+3.4}_{-3.6})^{\circ}$, dominated by the results from $B^{+}$ decays
[12]. Meanwhile the $\gamma$ values determined from global fits are
$(65.6^{+0.9}_{-2.7})^{\circ}$ from CKMfitter [3] and $(65.8\pm 2.2)^{\circ}$
from UTfit [4]. Further improvement on direct measurements is eagerly desired.
In addition, no $\gamma$ measurement has ever been performed in $b$-baryon
decays. New measurements will not only add further precision on the global
$\gamma$ determination from tree-level processes, but also help check if the
CKM mechanism works in the baryon sector by comparing the $\gamma$ determined
from the decays of $b$-baryons to that from $b$-mesons.

Actually, $C\\!P$ violation has only been found in meson decays, many searches
in hyperon, $c$\- or $b$-baryon decays have been extensively performed [13,
14, 15, 16, 17, 18, 19, 20, 21, 22] in the past ten years, no $C\\!P$
violation is observed yet. Finding $C\\!P$ violation in baryon decays will be
an important step towards understanding the large matter-antimatter asymmetry
observed in Universe [23]. The current focus on search for $C\\!P$ violation
in $b$-baryon decays are in charmless final states, however, when the $\gamma$
determined in $b$-baryon decays into charmed particles is statistically
inconsistent with zero, it will also be an unambiguous sign of $C\\!P$
violation. The ADS decay of $\Lambda_{b}^{0}\rightarrow D(K^{+}\pi^{-})pK^{-}$
has recently been observed and $C\\!P$ violation parameters are measured [24].
Throughout the paper, charge-conjugation is applied if not specified.
Extracting the angle $\gamma$ from $\Lambda_{b}^{0}\rightarrow DpK^{-}$ decays
are complicated due to many resonances involved. The spin-1/2 nature of
$\Lambda_{b}^{0}$ and $p$ baryons makes it even more difficult than the
partner decays of $B^{+}\rightarrow DK^{+}$. Additional strong parameters have
to be added to take into account different angular momentum contributions to
the decay amplitudes and thus dilute the sensitivity to $\gamma$.

In this paper, we perform the sensitivity studies on the angle $\gamma$ in
$\Lambda_{b}^{0}\rightarrow D\Lambda$ decays, which have similar Feynman
diagrams as $\Lambda_{b}^{0}\rightarrow DpK^{-}$ in the $\Lambda^{*}$ region.
Unlike $\Lambda_{b}^{0}\rightarrow DpK^{-}$, an additional observable can be
measured adding further constraints to $\gamma$ [25].

The angular momentum between $D$ and $\Lambda$ could be either $S$-wave or
$P$-wave and one has to take both into account when measuring the angle
$\gamma$. The $S$-wave decay amplitudes are written as

$\displaystyle A_{S}(\Lambda_{b}^{0}\rightarrow D^{0}\Lambda)$
$\displaystyle=$ $\displaystyle A_{c,S},$ (1) $\displaystyle
A_{S}(\Lambda_{b}^{0}\rightarrow\overline{D}^{0}\Lambda)$ $\displaystyle=$
$\displaystyle A_{c,S}r_{B,S}e^{i(\delta_{B,S}-\gamma)},$ (2)

for the $b\rightarrow c$ and $b\rightarrow u$ processes, respectively. The
parameters $r_{B,S}$ and $\delta_{B,S}$ describe the strong interaction parts
of the amplitude ratio between the two processes. The corresponding $P$-wave
amplitudes are denoted by replacing the subscript of $S$ into $P$. For the
charge-conjugated decay amplitudes, the weak phase $\gamma$ changes the sign
while other parts remain the same.

The $D^{0}$ and $\overline{D}^{0}$ decay amplitudes into a final state $f$ are
given by $A_{D}(f)$ and $\bar{A}_{D}(f)$, respectively. Studies in
$B^{+}\rightarrow DK^{+}$ have shown that bias on $\gamma$ caused by ignoring
$D^{0}$-$\overline{D}^{0}$ mixing effect and $C\\!P$ violation in $D$ decays
is less than 1∘ [26, 27, 28]. In the recent LHCb $\gamma$ combination [29],
they find that by fitting the $\gamma$ and $D$ mixing and $C\\!P$ violation
observables together, additional constraints on the $D$ mixing parameters are
obtained. In this study, we neglect these contributions since their effects on
$\gamma$ are marginal for the current small statistics of
$\Lambda_{b}^{0}\rightarrow D\Lambda$ decays and focus more on methodology
itself. In the assumption, we have $\bar{A}_{D}(\bar{f})=A_{D}(f)$ and
$\bar{A}_{D}(f)=A_{D}(\bar{f})$. The full decay amplitude of $S$-wave is

$A_{S}=A_{c,S}[A_{D}(f)+r_{B,S}e^{i(\delta_{B,S}-\gamma)}\bar{A}_{D}(f)],$ (3)

and the decay rate of $\Lambda_{b}^{0}\rightarrow D(f)\Lambda$ is given by

$\displaystyle\Gamma(\Lambda_{b}^{0}\rightarrow D(f)\Lambda)$ $\displaystyle=$
$\displaystyle|A_{S}|^{2}+|A_{P}|^{2}$ $\displaystyle\propto$
$\displaystyle(1+r^{2})|A_{D}(f)|^{2}+(r_{B,S}^{2}+r^{2}r_{B,P}^{2})|\bar{A}_{D}(f)|^{2}$
$\displaystyle+$ $\displaystyle
2[r_{B,S}\Re(e^{i(\delta_{B,S}-\gamma)}\bar{A}_{D}A_{D}^{*})+r_{B,P}r^{2}\Re(e^{i(\delta_{B,P}-\gamma)}\bar{A}_{D}A_{D}^{*})].$

The factor $|A_{c,S}|^{2}$ is omitted for simplicity and $A_{c,P}$ is replaced
by $re^{i\delta}A_{c,S}$. The parameters $r$ and $\delta$ are to be determined
from data. Comparing Eq. S0.Ex2 with that in $B^{+}\rightarrow DK^{+}$ decays
[29], one can immediately find the decay rate in $\Lambda_{b}^{0}$ decays are
more complicated and three additional strong parameters, $r$, $r_{B,P}$ and
$\delta_{B,P}$, have to be introduced to take into account contributions from
$P$-wave. The subsequent decay of $\Lambda\rightarrow p\pi^{-}$ only
contributes an overall factor of $|A_{S}^{\Lambda}|^{2}+|A_{P}^{\Lambda}|^{2}$
to the decay rate, where $A_{S}^{\Lambda}$ and $A_{P}^{\Lambda}$ are the $S$\-
and $P$-wave amplitudes of $\Lambda\rightarrow p\pi^{-}$, respectively, and
are not listed.

There are other observables proposed to study $C\\!P$ violation in hyperon
decays [30, 31], the decay parameters. They have been proposed previously to
measure the angle $\gamma$ in $\Lambda_{b}^{0}\rightarrow D\Lambda$ decays
[25]. The decay parameters contain interference information between $S$\- and
$P$-waves. As $\Lambda_{b}^{0}$ produced in $pp$ collisions at LHC is found to
be unpolarized [32], the only relevant decay parameter for
$\Lambda_{b}^{0}\rightarrow D\Lambda$ is

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{2\Re(A_{S}^{*}A_{P})}{|A_{S}|^{2}+|A_{P}|^{2}}.$ (5)

In the paper $\alpha_{-(+)}$ is used to denote $\alpha$ obtained from
$\Lambda_{b}^{0}$ ($\overline{\Lambda}_{b}^{0}$) decays. The other two decay
parameters $\beta=2\Im(A_{S}^{*}A_{P})/(|A_{S}|^{2}+|A_{P}|^{2})$ and
$\gamma^{\prime}=(|A_{S}|^{2}-|A_{P}|^{2})/(|A_{S}|^{2}+|A_{P}|^{2})$ can be
measured if $\Lambda_{b}^{0}$ was polarized, which can also add constraints to
the measurements of the angle $\gamma$ [25]. We use $\gamma^{\prime}$ for the
decay parameter to distinguish it from the angle $\gamma$.

The decay parameter $\alpha$ can be measured using angular distributions

$\frac{\textrm{d}\Gamma(\Lambda_{b}^{0}\rightarrow
D\Lambda(p\pi^{-}))}{\textrm{d}\Phi}\propto
1+\alpha_{-}^{\Lambda_{b}}\alpha_{-}^{\Lambda}\cos\theta.$ (6)

Here $\theta$ is the helicity angle of the $\Lambda\rightarrow p\pi^{-}$
decay, defined as the direction of $\pi^{-}$ with respect to the direction of
$D$ in the rest frame of $\Lambda$. The $\alpha_{-}^{\Lambda_{b}}$ and
$\alpha_{-}^{\Lambda}$ are the decay parameters of the $\Lambda_{b}^{0}$ and
$\Lambda$ decays. The $\alpha_{-}^{\Lambda}$ has been measured very precisely
[33] and $\alpha_{-}^{\Lambda_{b}}$ is determined uniquely. This offers us
additional observables to constrain the angle $\gamma$ as

$\displaystyle\Re(A_{S}^{*}A_{P})$ $\displaystyle\propto$ $\displaystyle
r[|A_{D}|^{2}\cos\delta+r_{B,S}r_{B,P}\cos(\delta+\delta_{B,P}-\delta_{B,S})|\bar{A}_{D}|^{2}$
$\displaystyle+$
$\displaystyle\Re(r_{B,S}e^{-i(\delta_{B,S}-\gamma-\delta)}A_{D}\bar{A}_{D}^{*}+r_{B,P}e^{i(\delta_{B,P}-\gamma+\delta)}A_{D}^{*}\bar{A}_{D})].$

Combing Eq. S0.Ex3 and Eq. S0.Ex2, the formula for $\alpha_{-}^{\Lambda_{b}}$
is obtained. In the above formula, despite the additional parameter $\delta$,
there are more observables when considering the many decays in this study and
their charge-conjugated ones. When $\alpha$ in different $D$ decays are the
same, one obtains $r_{B,S}=r_{B,P}$ and $\delta_{B,S}=\delta_{B,P}$, the
complications of $\Lambda_{b}^{0}$ decays degenerate and the sensitivity on
$\gamma$ from $\Lambda_{b}^{0}\rightarrow D\Lambda$ can be considered in a
similar way as in $B^{+}\rightarrow DK^{+}$. However, $r_{B}$ ($\sim 0.4$) in
$\Lambda_{b}^{0}\rightarrow DpK$ is larger than that in $B^{+}$ ($\sim 0.1$)
[29] and results in larger $C\\!P$ violation effects.

Sensitivities on $\gamma$ are studied using the decay rates, with and without
the decay parameter $\alpha$. The decay channels of $D$ mesons considered are
$D\rightarrow K^{+}K^{-}$, $D\rightarrow\pi^{+}\pi^{-}$, $D\rightarrow
K^{\pm}\pi^{\mp}$, $D\rightarrow K_{\mathrm{S}}^{0}\pi^{+}\pi^{-}$ and
$D\rightarrow K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$. For multi-body final states,
amplitude variations over the phase space needs to be taken into account. It
could be done either by using an amplitude model to describe the $D$ decays or
by a model-independent method where the $D$ decay information is obtained from
charm factories using quantum correlated $D$ samples from
$\psi(3770)\rightarrow D^{0}\overline{D}^{0}$. The model-independent method
has the advantage of small systematic uncertainties and is more preferred for
precise measurements. In this paper, we take the model-independent strategy
where the phase space of $D$ decays is divided into different bins and only
the integrated information in these bins

$c_{i}+is_{i}\equiv R_{D}^{i}e^{i\delta_{D}^{i}}\equiv\frac{\int_{\textrm{bin
i}}A_{D}(f)\overline{A}^{*}_{D}(f)\textrm{d}\Phi}{\sqrt{\int_{\textrm{bin
i}}|A_{D}(f)|^{2}\textrm{d}\Phi}\sqrt{\int_{\textrm{bin
i}}|\overline{A}_{D}(f)|^{2}\textrm{d}\Phi}}$ (8)

are used. The strong phase parameters $c_{i}$ and $s_{i}$ of $D\rightarrow
K_{\mathrm{S}}^{0}\pi^{+}\pi^{-}$ decays are measured by CLEO-c and BESIII
experiments [34, 35, 36] and the coherent factors $R_{D}$ and effective strong
phase difference $\delta_{D}^{i}$ for $D\rightarrow
K^{\mp}\pi^{\pm}\pi^{+}\pi^{-}$ are measured using the CLEO-c and BESIII data
[37, 38]. The combinations of the results from the two experiments have also
been performed [35, 36, 38].

The $\Lambda_{b}^{0}\rightarrow D\Lambda$ has not been observed yet.
Predictions based on factorization approach [25, 39, 40, 41] indicates that
the branching fraction of $\Lambda_{b}^{0}\rightarrow D\Lambda$ is about 200
times smaller than that of $\Lambda_{b}^{0}\rightarrow J/\psi\Lambda$. Around
300 $\Lambda_{b}^{0}\rightarrow D(K^{-}\pi^{+})\Lambda$ events are expected to
be reconstructed based on the data collected by LHCb in the years 2011-2018
[32], and is used for the sensitivity estimation. The numbers for the other
channels are scaled accordingly based on the formalism described previously.
However, the yields of $D\rightarrow K_{\mathrm{S}}^{0}\pi^{+}\pi^{-}$ and
$D\rightarrow K^{\mp}\pi^{\pm}\pi^{+}\pi^{-}$ are further scaled by a factor
of 0.16 and 0.18 [42, 43, 44] to take into account their efficiency
differences with respect to the two-body decays.

In the study, $r_{B,S}$ and $r_{B,P}$ are set to 0.4 and 0.3, respectively, as
both $\Lambda_{b}^{0}\rightarrow D^{0}\Lambda$ and
$\Lambda_{b}^{0}\rightarrow\overline{D}^{0}\Lambda$ decays are color
suppressed. The $r$ value describes the amplitude ratio between $S$\- and
$P$-waves and three sets of values (0.5, 1 or 2) are chosen. The three phases,
$\delta$, $\delta_{B,S}$ and $\delta_{B,P}$, are also set to three different
values, $0^{\circ}$, $60^{\circ}$ and $150^{\circ}$. The $\gamma$ input value
is chosen to be $65^{\circ}$ and the charm inputs are taken from their
averaged values [33] or from those measured with CLEO-c or BESIII data [34,
37, 35, 36, 38].

Pseudo data samples are generated according to the yields and $\alpha$
obtained from a set of parameters. A $\chi^{2}$ fit is then performed, where
yields and $\alpha$ are considered to be independent as is shown in toys. In
addition to the physical parameters, normalization factors are also considered
as free parameters. The two-body $D$ decays share the same normalization
factor following what has been done in real measurements, while the
$K_{\mathrm{S}}^{0}\pi^{+}\pi^{-}$ and $K^{\mp}\pi^{\pm}\pi^{+}\pi^{-}$ have
standalone normalization factors. This procedure is performed 1000 times for
each set of inputs, and the uncertainty of $\gamma$ is obtained by a Gaussian
fit to its distribution from the toy samples. No bias is found in all the
parameter settings. The procedures have been validated using parameters found
in $B^{+}\rightarrow DK^{+}$ decays [29] with $r=0$. An uncertainty of 2.4∘
for the angle $\gamma$ is achieved. The slightly better precision is due to
ignoring background contributions.

Based on the above settings, the sensitivity of $\gamma$ in
$\Lambda_{b}^{0}\rightarrow D\Lambda$ decays is found to be around
$12-36^{\circ}$ as shown in Fig. 1. Dependence on the input values of
$r,\delta_{B,S}$ and $\delta_{B,P}$ are found, while little on $\delta$. The
sensitivity on $\gamma$ is improved by up to 60% when including the observable
$\alpha$. Using the current data collected by the LHCb experiment, it may be
possible to observe $C\\!P$ violation in $\Lambda_{b}^{0}\rightarrow D\Lambda$
decays, and a sensitivity around $4-11^{\circ}$ will be achievable with the
data collected in the LHCb upgrade I period [45]. We strongly suggest the LHCb
experiment performing the measurements proposed in this paper.

Figure 1: Statistical uncertainties of $\gamma$ from feasibility studies
(left) with and (right) without the decay parameter $\alpha$. The parameters
$r_{B,S}$, $r_{B,P}$ and $\delta$ are set to be 0.4, 0.3 and 60∘,
respectively, for all toy samples.

Note added:

After the first submission to arXiv, we are informed by Prof. Guy Wilkinson
and Anton Poluektov that the idea to use the decay parameters in
$\Lambda_{b}^{0}\rightarrow D\Lambda$ decays has been proposed previously in
hep-ph/0112220 and published in Phys. Rev. D65 (2002) 073029. The paper
described the possibility of extracting the weak phase $\gamma$ using the
decay parameters $\alpha^{\prime}$, $\beta^{\prime}$ and $\gamma^{\prime}$
together with the decay rate. In the updated version of our paper, we modify
our text (and title) to reflect this fact.

In addition to the proposal of using the decay parametes as additional
observables for $\gamma$ measurement, we point out that only $\alpha$ can be
measured as $\Lambda_{b}^{0}$ from $pp$ collisions are unpolarized and explain
the experimental method to extract $\alpha$. We perform the first sensitivity
study with and without the decay parameter $\alpha$, proving that $\gamma$ can
be extracted in the $\Lambda_{b}^{0}\rightarrow D\Lambda$ decays. Including
the observable $\alpha$ can improve the sensitivity by up to 60%, depending on
the strong parameters. The sensitivities for the current LHCb data and also
the data collected during its upgrade are also given. In the sensitivity
study, additional channels of $D\rightarrow K_{S}\pi^{+}\pi^{-}$ and
$D\rightarrow K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$ are considered.

## Acknowledgements

This work is partially supported by National Natural Science Foundation of
China (NSFC) under grant No. 11975015, the Fundamental Research Funds for the
Central Universities. We thank Prof. Haibo Li, Yuehong Xie and Yuming Wang for
useful discussions on the topic. We thank Prof. Guy Wilkinson and Anton
Poluektov for pointing out the previous work and for their insight suggestions
to improve the manuscripts.

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