# CP Violation in Charmed and Bottom Baryon Physics

John M. Yelton  University of Florida, Gainesville, FL, USA, 32611

###### Abstract

This is an experimentalist’s view of the recent results on, and prospects for,
CP Violation in charmed and bottom baryons.

##  I Introduction

The phenomenon of CP violation has been discussed in several contributions to
this conference already. However, these studies have focussed on mesons.
Studies of baryon decays are usually one or decades behind those of mesons,
but in CP violation the situation is even more difficult. Whereas in neutral
mesons there can be mixing between particle and anti-particle, and two
distinct interfering amplitudes which can produce time-dependent CP violation,
baryon number is a very strong constraint and the window provided by mixing is
not available. This means that any CP violation in baryons needs to be
$direct$. However, though the difficulties are many, the prize is great. If we
want to understand why we are here today (and made out of matter), we need to
understand CP violation in baryons.

This is an experimentalist’s review of recent results on, and prospects for,
CP violation studies in charmed and bottom baryons.

##  II The Experiments

It is no surprise to find that the two featured current experiments are LHCb
(representing the “energy frontier” of $pp$ collisions), and Belle II
(representing the “intensity frontier” of $e^{+}e^{-}$ annihilation).

Table 1 shows the comparison between the two. The first and last rows show
huge benefits of LHCb. These are partly compensated by the other entries in
the table, which means that there will be a complementarity of approaches in
the next few years.

Table 1: Comparison of LHCb and Belle II for CP violation studies in baryons. $\sigma(b+\overline{b}){\rm nb}$ |  $\sim 150,000$ $\sim 1$ |
---|---|---
$\int Ldt({\rm fb}^{-1})$ | $\sim 50$ |  $\sim 50,000$ (eventually…)
Background Level | Higher | Lower
Typical Efficiency | Lower | Higher
Neutral Efficiency | Lower | Higher
Decay time resolution | Great | Good
Production Ratios | Baryons produced | Baryons/Antibaryons
| preferentially | produced equally
Collision spot size | Bigger | Tiny
Particles produced | Includes b baryons | Only $c$ and $s$

##  III LHCb Search for CP Violation in $\Xi_{b}\to pK^{-}K^{-}$

Why look at this mode? Interference between two amplitudes with different weak
and strong phases leads to CP violation in decay. Weak phases are associated
with the complex elements of the CKM matrix Strong phases associated with
hadronic final-state effects. In general, the common decays of $b$ hadrons to
charm quarks have one amplitude that dominates and so there is no useful
interference between amplitudes to use for CP violation studies. This is
similar to the situation in meson decays.

This analysis is presented in XIB . The signal is around 500 events. As CP
violation is expected to be enhanced in the baryon resonance region, the
analysis performs a fit to the entire Dalitz plot taking into account the six
excited $\Lambda$ and $\Sigma$ resonances tabulated in the Particle Data Book
PDG  in the region 1.4-2.0 ${\rm GeV}/c^{2}$, searching for a significant
difference between particle and anti-particles. The results are that there is
no significant CP violation in any of the six quasi-two-body decays. The
statistical uncertainties are large (not surprising when several resonances
are barely significant), and so will improve with more data. The systematic
uncertainties will also improve.

##  IV LHCb Search for CP Violation in $\Lambda_{b}\to DpK^{-}$

This is suppressed decay mode and comes in different varieties. The one that
is interesting is $\Lambda_{c}^{0}\to[K^{+}\pi^{-}]_{D}pK^{-}$ where the kaons
have different charges. Note that we do not specify whether the $K\pi$ is a
$D$ or a $\bar{D}$ as both can contribute. This decay has contributions both
from $b\to c$ and $b\to u$ which are of the same order of magnitude, and an
interference that depends on the CKM angle $\gamma$.

What is measured is the asymmetry:

$A=\frac{(B_{particle}-B_{antiparticle})}{(B_{particle}+B_{antiparticle})}$ ,

where $B_{particle}=B(\Lambda_{b}^{0}\to[K^{+}\pi^{-}]_{D}pK^{-})$ and
$B_{antiparticle}=B({\bar{\Lambda}}_{b}^{0}\to[K^{-}\pi^{+}]_{D}{\bar{p}}K^{+})$.
Note that the interference is anticipated to be larger in regions of phase
space that involve excited $\Lambda$ states.

The analysis is presented in DpK . The measured CP asymmetries are:

$A_{CP}=0.12\pm 0.09{\rm(stat)}^{+0.02}_{-0.03}({\rm syst.})$

and

$A_{CP}=0.01\pm 0.16{\rm(stat)}^{+0.02}_{-0.03}({\rm syst.})$

where this first is integrated over all phase-space and the second in the low-
mass (resonance) $pK$ region. The conclusion is that there is no evidence of
CP violation.

##  V LHCb Search for CP Violation in $\Lambda_{b}^{0}\to
D\pi^{-}\pi^{+}\pi^{-}$

There are two methods now in use to analyze this decay mode, which has the
benefit of relatively high statistics p3pi . The first uses “scalar triple
products”, defined in GD , which involve the momentum correlation between the
proton, the positive pion and the “faster” of the negative pions, and the
analogs for antiparticles.

$C={\vec{p}_{p}}\cdot({\vec{p}}_{\pi^{-}}\times{\vec{p}}_{\pi^{+}})$ and
${\bar{C}}={\vec{p}_{\bar{p}}}\cdot({\vec{p}}_{\pi^{+}}\times{\vec{p}}_{\pi^{-}})$

The data is then divided into four pieces, each with yields N (i.e. divide by
two by particle/antiparticle, and again by 2 according to the sign of the
scalar triple product.

The Triple Product Asymmetry is then defined for particles and antiparticle:

$A=\frac{N(C_{T}>0)-N(C_{T}<0)}{N(C_{T}>0)+N(C_{T}<0)}$ and
${\bar{A}}=\frac{{\vec{N}}({\vec{C}_{T}}>0)-{\bar{N}}({\bar{C}_{T}}<0)}{{\bar{N}}({\bar{C}_{T}}>0)+{\bar{N}}({\bar{C}_{T}}<0)}$,

where these two would be the same if there is no CP violation. Then, the CP
violating piece looked for is defined as:

$a_{CP}=\frac{1}{2}(A_{T}-{\bar{A}_{T}})$

Integrated over all phase-space they find $a_{CP}=(-0.7\pm 0.7\pm 0.2)\%$
which is clearly consistent with zero.

Once again there is the expectation that any CP Violation will be in any area
of phase-space dominated by resonances, and here there are two sorts of
resonances - excited nucleons ($N^{*}$), and excited mesons (e.g. $a_{1}$).
Each of these preferred regions are then divided into bins dependent on the
relative angles of the final state particles.

The biggest discrepency from the non-CPV expectation is found in the $N^{*}$
enhanced sample when plotted in terms of the angle between the decay planes of
the $(\pi^{+}\pi^{-})_{slow}$ and $(p\pi^{-})_{fast}$. This discrepency is of
2.9 $\sigma$ significance. A previous version of this analysis provided some
excitement as it found 3.3 $\sigma$ significance on 25% of the dataset, but
the extra data has not enhanced the effect.

LHCb have also analyzed the data in a less model dependent manner using
independent unbinned tests MW . Again, no significant evidence of CP violation
has been found.

##  VI CP Violation Searches using Charmed Baryons

The charm sector has some experimental advantages over the B sector. One is
the complementary datasets available in $e^{+}e^{-}$ annihalations, and the
other is the comparatively low multiplicies in the final states.

The basic research thrusts comprise choosing a suitable decay mode and
measuring its branching fraction, measuring the asymmetry parameter, $\alpha$
defined below, and then measuring the difference in asymmetry paramaters in
particles and anti-particles. Once again, no measurable CPV is expected in
2-body Cabibbo-favored decays, and singly-Cabibbo-suppressed decays are a more
promising laboratory, with effects of order $10^{-3}$ to be expected.

Typical decay modes that we can hope to produce interesting results are
$\Lambda_{c}^{+}\to\Lambda K^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{0}K^{+}$
which have measured branching fractions of $(6.1\pm 1.2)\times 10^{-4}$ and
$(5.2\pm 0.8)\times 10^{-4}$ respectively. In $e^{+}e^{-}$ machines these
(particularly the former) can be detected with good efficiency and purity.

In decays of $\Lambda_{c}^{+}$ to a baryon and pseudo-scalar meson, the
$\alpha$ parameter is defined to be: $\alpha=\frac{2Re(s\cdot
p}{|s|^{2}+|p|^{2}}$ where $s$ and $p$ are the parity-violating s-wave and the
parity-conserving p-wave amplitudes. Operationally, for a decay such as
$\Lambda_{c}^{+}\to\Lambda K^{+}$, we define an angle $\theta_{\Lambda}$,
which is the angle between the proton momentum in the $\Lambda$ frame and the
$\Lambda$ in the $\Lambda_{c}^{+}$ frame. The data is divided into bins of
$cos(\Theta_{\Lambda})$, and then use the equation:

$dN/dcos\theta_{\Lambda}\propto 1+\alpha(\Lambda_{c}^{+}\to\Lambda
K^{+})\alpha(\Lambda\to p\pi^{-})cos\theta_{\Lambda}$

In other words, split the data into bins of $cos\theta_{\Lambda}$, fit the
mass peak in each plot, find the slope and extract the product of the $\alpha$
parameters ($\alpha$ for a $\Lambda$ is well-known).

The $\alpha$ parameters are found separately for particles and antiparticles,
and the then the CP-violating parameter defined as:

$A_{CP}^{\alpha}=\frac{\alpha(\Lambda_{c}^{+})+\alpha(\Lambda_{c}^{+})}{\alpha(\Lambda_{c}^{+})-\alpha(\Lambda_{c}^{+})}$

The $\alpha$ parameters are interesting in their own right, as they are
parameters, along with the partial width, that can in principle be predicted
theoretically in different models of weak decays. However, until recently,
neither the experimental measurements or the theoretical models were of
sufficient quality for such comparisons to be meaningful. However, the
situation is changing. For example, in 2001 CLEO measured $\alpha$ for
$\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ to be $-0.56\pm 0.39^{+0.10}_{-0.09}$, but that
paper did not receive a single citation for the first eight years! Recently,
BELLE BXic  measured $\alpha$ for particles and anti-particles and found
$A_{CP}^{\alpha}=0.024\pm 0.052\pm 0.012$. This, of course is a Cabibbo-
allowed decay and finding CP violation in such a mode would be a major
surprise. However, in the Cabibbo-suppressed modes $\Lambda_{c}^{+}\to\Lambda
K^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{0}K^{+}$ precisions of the order of 0.1
and 0.3 would seem to be attainable for each $1\ ab^{-1}$ of data (i.e. the
already taken Belle dataset). Although this may not be sufficient, the
prospect of $50\ {\rm ab}^{-1}$ Belle II data opens realistic possibilities
for these and other similar modes.

##  VII  LHCb Measurements in $\Lambda_{c}^{+}\to ph^{+}h^{-}$

These, with $h=\pi$ or $K$, are two Cabibbo-suppressed decays of the
$\Lambda_{c}^{+}$, and the big challenge is to understand the detection
efficiency of protons and anti-protons. One way of avoiding this LHCphh  is to
use two related decays and compare them. That is, instead of

$A_{CP}(f)=\frac{\Gamma(f)-\Gamma(\bar{f})}{\Gamma(f)+\Gamma(\bar{f})}$

use

$A_{Raw}=\frac{N(f)-N(\bar{f})}{N(f)+N(\bar{f})}$

where $f$ refers to a specific final state and $N$ is the number detected.

Then we can say that the difference in the $A_{Raw}$ parameters will be a good
approximation to the the difference of the $A_{CP}$ parameters.

Although the raw asymmetries of the two modes are significantly non-zero, the
difference is $\Delta_{CP}=(0.30\pm 0.91\pm 0.61)\%$ and therefore there is no
evidence of CP violation.

##  VIII  LHCb Measurements in $\Xi_{c}^{+}\to K^{+}\pi^{-}$

This Cabibbo-suppressed decay is a very useful one for LHCb as the statistics
are very high and the background suppression good pKpi . The Cabibbo-allowed
decay of $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ is used as a control. The
substructure of these decays are very rich, and CP violation, if it exists in
these modes, is likely to depend on where in the Dalitz plot the decay lies.
Two different analysis techniques are used. One is binned into areas of the
Dalitz plot, searching for localized differences in the plots for baryons and
anti-baryons. The second is unbinned. However, no significant differences are
found between baryons and anti-baryons, and the results are consistent with
the hypothesis of no CP violation.

##  IX  Conclusions

I have reviewed a number of searches for CP violation in charmed baryon
baryons. No significant violation has been observed. However, the sensitivity
is not at the level that this is a surprise.

It is clear that more predictions for specific decay modes for CPV would be
very useful. I expect in the next few years to have more measurements of the
asymmetry parameter, $\alpha$, whch are interesting in their own right and can
lead to measurements of CP violation. I would like to point out that many of
the searches involve knowledge of the excited hyperons in the substructure of
the heavy baryon decays. Research in the area of excited hyperons is under-
appreciated. However, clearly the biggest challenge in the next decade is to
collect big enough datasets that comparatively rare decays can be studied with
good statistics.

###### Acknowledgements.

This work is supported in part be DOE contract DE-SC0009824.

## References

  * (1) R. Aaij et al. (LHCb Collaboration), D. 104, 052010, (2021).
  * (2) R. L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083CO1 (2022).
  * (3) R. Aaij et al. (LHCb Collaboration), D. 104, 112008, (2021).
  * (4) R. Aaij et al. (LHCb Collaboration), D. 102, 051101, (2020).
  * (5) G. Durieuz, JHEP 10, 005 (2016).
  * (6) M. Williams, Phys. Rev. D. 84, 054015, (2011).
  * (7) Y. .B. Li et al. (Belle Collaboration), Phys. Rev. Lett. 127, 121803, (2021).
  * (8) S. Chan et al. (CLEO Collaboration), Phys. Rev. D. 63 111102 (2001).
  * (9) R. Aaij et al. (LHCb Collaboration), JHEP 03, 182 (2018).
  * (10) R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C80 (2020).