††institutetext: Department of Physics, Ben-Gurion University, Beer-Sheva
8410501, Israel

# Prospects for measuring quark polarization and
spin correlations in $b\bar{b}$ and $c\bar{c}$ samples at the LHC

Yevgeny Kats  and David Uzan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Polarization and spin correlations have been studied in detail for top quarks
at the LHC, but have been explored very little for the other flavors of
quarks. In this paper we consider the processes $pp\to q\bar{q}$ with $q=b$,
$c$ or $s$. Utilizing the partial preservation of the quark’s spin information
in baryons in the jet produced by the quark, we examine possible analysis
strategies for ATLAS and CMS to measure the quark polarization and spin
correlations. We find polarization measurements for the $b$ and $c$ quarks to
be feasible, even with the currently available datasets. Spin correlation
measurements for $b\bar{b}$ are possible using the CMS Run 2 parked data,
while such measurements for $c\bar{c}$ will become possible with higher
integrated luminosity. For the $s$ quark, we find the measurements to be
challenging with the standard triggers. We also provide leading-order QCD
predictions for the polarization and spin correlations expected in the
$b\bar{b}$ and $c\bar{c}$ samples with the cuts envisioned for the above
analyses. Apart from establishing experimentally the existence of spin
correlations in $b\bar{b}$ and $c\bar{c}$ systems produced in $pp$ collisions,
the proposed measurements can provide new information on the polarization
transfer from quarks to baryons and might even be sensitive to physics beyond
the Standard Model.

##  1 Introduction

Quark polarization and spin correlations are properties that have been
extensively researched for top quarks at the LHC, both theoretically (e.g.,
refs. Bernreuther:2001rq ; Bernreuther:2004jv ; Mahlon:2010gw ;
Baumgart:2011wk ; Baumgart:2012ay ; Baumgart:2013yra ; Bernreuther:2013aga ;
Bernreuther:2015yna ; Behring:2019iiv ; Afik:2020onf ; Severi:2022qjy ) and
experimentally (e.g., refs. ATLAS:2014abv ; ATLAS:2016bac ; CMS:2016piu ;
CMS:2019nrx ; ATLAS:2019zrq ; ATLAS:ttbar-entanglement ), but have not yet
been explored much for the $b$, $c$, or $s$ quarks. Measurements of these
quantities can provide interesting information on both Standard Model (SM) and
Beyond the Standard Model (BSM) interactions. There exist proposals for
methods to measure quark polarizations in samples of $pp\to t\bar{t}$, in
which $b$ quarks are available from the $t\to W^{+}b$ decay, and $c$ and $s$
quarks from the subsequent decay $W^{+}\to\bar{s}c$ Galanti:2015pqa ;
Kats:2015cna ; Kats:2023gul , and using samples of $pp\to W^{-}c$ for $c$
quarks Kats:2015zth . The $b$, $c$, and $s$ quarks in these processes are
expected to be highly polarized due to the electroweak interaction. On the
other hand, in the current paper we want to examine quark-antiquark pair
production, $pp\to q\bar{q}$, where $q$ can be either $b$, $c$ or $s$. These
processes are dominated by QCD interactions, which produce the quarks
unpolarized at the leading order. However, sizable spin correlations are
expected, as we will quantify, similar to the top-quark case.

Unlike the top quark, whose spin information can be obtained from the angular
distributions of its decay products, the $b$, $c$ and $s$ quarks are only
observed as jets of hadrons, making it more challenging to obtain the spin
information of the original quarks. It can nevertheless be done by measuring
the polarization and spin correlations of baryons produced from these quarks.
This approach was originally proposed for measuring the longitudinal
polarization of the heavy quarks ($b$ and $c$) produced in $Z$ decays at LEP
Mannel:1991bs ; Ball:1992fw ; Falk:1993rf . Such measurements were
subsequently performed for the $b$ quark using $\Lambda_{b}$ baryons
ALEPH:1995aqx ; OPAL:1998wmk ; DELPHI:1999hkl , confirming the expectation of
a sizable polarization transfer from the $b$ quark to the $\Lambda_{b}$.
Analogous measurements have shown that the $s$-quark longitudinal polarization
is preserved in $\Lambda$ baryons that carry a significant fraction of the jet
momentum ALEPH:1996oew ; ALEPH:1997an ; OPAL:1997oem . This method to access
quark spin information has been analyzed in the context of the LHC in refs.
Galanti:2015pqa ; Kats:2015cna ; Kats:2023gul ; Kats:2015zth , where it was
shown that a number of interesting measurements of longitudinal polarization
of quarks produced in top-quark decays are possible even with the statistics
of Run 2. In addition, attempts to measure the transverse polarization of
$\Lambda_{b}$ in inclusive QCD samples, which is expected to be small, have
been reported by LHCb LHCb:2013hzx ; LHCb:2020iux  and CMS CMS:2018wjk . When
moving to spin correlations, the cost in statistics increases significantly
since the prices for the fragmentation to baryons, the branching ratios of the
useful decays, and the reconstruction efficiency, are squared. It is therefore
an example of analyses that will benefit from the increase in statistics
offered by the high-luminosity phase of the LHC.

The spin correlation measurements will allow quantifying the effect of the
polarization transfer from quarks to baryons for longitudinal polarization
(cross-checking the information that would presumably be obtained even earlier
in the analyses proposed in refs. Galanti:2015pqa ; Kats:2015cna ) as well as
transverse polarization. This will provide certain cross-grained information
about the spin-dependent fragmentation functions Chen:1994ar ; Adamov:2000is
of the $b$ and $c$ quarks hadronizing to the $\Lambda_{b}$ and $\Lambda_{c}$
baryons, respectively. Polarization and spin correlation measurements can also
be sensitive to BSM contributions to $b\bar{b}$ or $c\bar{c}$ production.
Similar ideas apply to $s\bar{s}$, but we will find the corresponding
measurements to be challenging.

We will consider both the Run 2 and the High Luminosity LHC (HL-LHC) datasets
of ATLAS and CMS, including the CMS parked $b$-hadron dataset CMS-DP-2019-043
; Bainbridge:2020pgi . The goal of the current paper is to do a broad survey
of the possible analyses, considering a variety of baryon decay modes and
selection schemes. We will therefore restrict ourselves to rough estimates of
the expected sensitivity in each case, leaving more detailed simulations of
individual analyses and the consideration of systematic uncertainties to
future work. We also leave to future work the exploration of similar
opportunities in LHCb. While limited in the integrated luminosity and
acceptance, the LHCb detectors offer superior tracking and particle
identification capabilities. It is therefore plausible that a complementary
set of analyses, for low-$p_{T}$ quarks, will be possible in LHCb. It should
be noted, however, that the assumption of the factorization between the quark
production and its hadronization can break down at low $p_{T}$, making the
result interpretation difficult.

The rest of the paper is organized as follows. Section 2 reviews the
essentials of quark polarization retention in baryons. Section 3 provides
details on baryon production and discusses the baryon decay modes that will be
relevant to us. Section 4 reviews the formalism for the description of
polarization and spin correlations and presents the angular distributions
through which these quantities can be measured. In section 5 we simulate the
polarization and spin correlations for $b\bar{b}$ and $c\bar{c}$ expected in
QCD after validating our simulation on $t\bar{t}$. In section 6, we describe
the various possible analysis channels in detail, discuss the most important
backgrounds in each case, and estimate the expected sample purity and
measurement precision. We summarize the conclusions in section 7. Appendix A
presents the derivation of formulas for the statistical uncertainties of the
polarization and spin correlation measurements.

##  2 Polarization Retention in Baryons

For the heavy quarks, namely $b$ and (to some extent) $c$, the polarization is
expected to be preserved through the hadronization (on timescales of order
$1/\Lambda_{\rm QCD}$) Mannel:1991bs ; Ball:1992fw ; Falk:1993rf . This
happens since $m_{q}\gg\Lambda_{\rm QCD}$ implies that the effect on the heavy
quark spin via the chromomagnetic dipole moment, which scales as
$\mu_{q}\propto 1/m_{q}$, is negligible.

If the heavy quark $q$ ($=b$ or $c$) ends up in a $\Lambda_{q}$ baryon, the
baryon polarization is approximately equal to the quark polarization. It is so
because the $\Lambda_{q}$ structure in the framework of the quark model is
$qud$ with the $u$ and $d$ forming a spin singlet, so all the spin is on the
$q$. If, on the other hand, the $q$ ends up in a $\Sigma_{q}$ or
$\Sigma_{q}^{\ast}$ baryon, which are analogous to the $\Lambda_{q}$ but with
the light quarks forming a spin and isospin triplet, the $\Lambda_{q}$ baryons
produced in $\Sigma_{q}^{(\ast)}\to\Lambda_{q}\pi$ decays will not carry the
same polarization as the original quark Falk:1993rf . The $\Lambda_{q}$ from
these decay channels are hard to distinguish from the direct $\Lambda_{q}$
production and thus they lower the polarization retention. Due to similar
reasons, it is essentially impossible to extract polarization information from
meson decays Falk:1993rf ; Alonso:2021boj .

The polarization loss effect due to the contamination of the $\Lambda_{q}$
sample with $\Sigma_{q}^{(\ast)}\to\Lambda_{q}\pi$ decays has been analyzed in
refs. Falk:1993rf ; Galanti:2015pqa  and it was found that the inclusive
$\Lambda_{q}$ samples end up carrying between roughly $50\%$ and $80\%$ of the
original quark polarization, and this number may differ between the cases of
longitudinal and transverse polarization (with respect to the fragmentation
axis). In the current paper we will not repeat the discussions on the various
approaches to estimating these effects but only parameterize them in terms of
the longitudinal and transverse _polarization retention factors_ , $r_{L}$ and
$r_{T}$, defined as

$\displaystyle
r_{\hat{\mathcal{P}}}=\frac{\mathcal{P}(\Lambda_{q})}{\mathcal{P}(q)}\;,$ (1)

where $\mathcal{P}$ denotes polarization and $\hat{\mathcal{P}}=L$ or $T$
denotes whether its direction is longitudinal or transverse with respect to
the fragmentation axis. We also note that $r_{L}$ for bottom and charm quarks
can be measured by ATLAS and CMS in their Run 2 $t\bar{t}$ samples
Galanti:2015pqa , and for charm quarks possibly also in $W$+$c$ samples
Kats:2015zth . Measurements of both $r_{L}$ and $r_{T}$ for the different
quark flavors using spin correlations would be one of the goals of the
analyses proposed in the current paper.

The above heavy-quark argument does not apply to the $s$ quark, but LEP
experiments have shown that $\Lambda$ baryons from $s$ quarks still preserve a
large fraction of the polarization ALEPH:1996oew ; ALEPH:1997an ; OPAL:1997oem
.

Describing the polarization transfer from a quark to a baryon in terms of two
numbers, $r_{L}$ and $r_{T}$, is an approximation. More generally, the
polarization transfer will depend on the fraction of the jet momentum carried
by the baryon,

$z=\frac{p_{T}^{\Lambda_{q}}}{p_{T}^{\rm jet}}\;,$ (2)

and is described by the so-called _spin-dependent_ (or _polarized_)
_fragmentation functions_ Chen:1994ar ; Stratmann:1996hn ; deFlorian:1997zj ;
Adamov:2000is . These functions vary slowly as a function of the energy scale
of the process due to the renormalization group evolution Stratmann:1996hn ;
deFlorian:1997zj . Characterizing these effects and taking them into account
will require high-statistics measurements that could follow up the
measurements motivated here and in refs. Galanti:2015pqa ; Kats:2015cna ;
Kats:2015zth . The only case where it is absolutely essential to take the $z$
dependence into account is that of the strange quark. Due to their low mass,
soft strange quarks are copiously produced in parton showering. To reduce
these contributions, it is essential to focus on $\Lambda$ baryons with high
values of $z$. The dependence of the $\Lambda$ polarization on $z$ has been
measured in $Z$ decays at LEP ALEPH:1996oew ; ALEPH:1997an ; OPAL:1997oem
confirming the expectation Gustafson:1992iq  that the polarization of the
initial strange quark is preserved primarily in high-$z$ $\Lambda$ baryons.
For example, as has been estimated in ref. Kats:2015cna  based on the LEP
measurements, roughly $60\%$ of the strange-quark polarization is preserved in
$\Lambda$ baryons with $z>0.3$. Additional information can be obtained using
$t\bar{t}$ samples at the LHC Kats:2015cna .

##  3 Baryon Production and Relevant Decay Channels

Fragmentation Fraction | Decay Scheme | BR | Spin analyzing power
---|---|---|---
$b\to\Lambda_{b}$ | $7.0\%$ Galanti:2015pqa  | $\Lambda_{b}\to X_{c}\mu^{-}\bar{\nu}_{\mu}$ |  $11\%$ Workman:2022ynf  |  $\alpha_{\mu^{-}}\approx-0.26,\,\alpha_{\bar{\nu}_{\mu}}\approx 1$ Manohar:1993qn ; Galanti:2015pqa
with $\Lambda\to p\pi^{-}$ | 2.7% Workman:2022ynf  |
with $\Lambda_{c}^{+}$ reco. | 2.0% Workman:2022ynf  |
$c\to\Lambda_{c}$ | $6.4\%$ Lisovyi:2015uqa  | $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ |  $6.3\%$ Workman:2022ynf  |  $\alpha_{\rm eff}\approx 0.662$ LHCb:2022sck
$\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$ |  $3.5\%$ Workman:2022ynf  |  $\alpha_{\mu^{+}}\approx 1$ Czarnecki:1994pu
with $\Lambda\to p\pi^{-}$ |  $2.2\%$ Workman:2022ynf  |
$s\to\Lambda$ | $2.8\%$ Sjostrand:2014zea ; Albino  | $\Lambda\to p\pi^{-}$ | $64\%$ Workman:2022ynf  | $\alpha_{p}\approx 0.75$ Workman:2022ynf
$(z>0.3)$
Table 1: Baryon fragmentation fractions, relevant decay schemes, branching
ratios (BR) and spin analyzing powers (asymmetry parameters) of the decays.

The fragmentation fractions for producing the baryons of interest at high
energies are shown in table 1. While the numbers for $b\to\Lambda_{b}$ and
$c\to\Lambda_{c}$ are available in the literature, for the $\Lambda$ we
obtained the number from a Pythia Sjostrand:2014zea  simulation. The
simulation results are shown in figure 1, which presents the integrated
fragmentation functions

$f(s\to\Lambda,\,z>z_{0})=\int_{z_{0}}^{1}f_{s\to\Lambda}(z)\,dz$ (3)

and similarly for $s\to\bar{\Lambda}$. This gives the following fragmentation
fractions for $z_{0}=0.3$ (a cut motivated at the end of section 2):

$\displaystyle f(s\to\Lambda,~{}z>0.3)=2.8\%\,,\qquad
f(s\to\bar{\Lambda},~{}z>0.3)=0.3\%\,.$ (4)

We also compute the sum of the $\Lambda$ and $\bar{\Lambda}$ numbers for a
comparison with the AKKII data-based results Albino . A reasonable rough
agreement is seen, except at the high-$z$ tail where neither of the methods
can be trusted.

Table 1 also shows the decay channels we consider, selected based on their
branching ratio (BR), spin analyzing power, and feasibility of identification
and reconstruction. For the semileptonic $\Lambda_{b}$ decays, we consider
three types of selections (similar to ref. Galanti:2015pqa ):

  * •

_Inclusive Selection_ , where $X_{c}$ represents any collection of particles
containing a charmed hadron, which is usually a $\Lambda_{c}^{+}$.111The
branching ratios for $\Lambda_{b}$ decays to charmed mesons are very small,
e.g. $\operatorname{BR}(\Lambda_{b}\to D^{0}p\pi^{-})\approx 6\times 10^{-4}$,
$\operatorname{BR}(\Lambda_{b}\to D^{(\ast)+}p\pi^{-}\pi^{-})\approx 8\times
10^{-4}$ Workman:2022ynf . We will neglect them in the following. This
selection, which will not apply any conditions to the particles produced along
with the muon, will have high signal efficiency, but also unsuppressed
backgrounds from semileptonic decays of $b$ mesons.

  * •

_Semi-inclusive Selection_ , where the $X_{c}$ is required to contain a
$\Lambda\to p\pi^{-}$ decay, to reduce backgrounds from the semileptonic
$b$-meson decays.

  * •

_Exclusive Selection_ , where the $X_{c}$ is required to contain a fully
reconstructible $\Lambda_{c}^{+}$ decay (i.e., with charged products only), to
reduce backgrounds from the semileptonic meson decays and facilitate the
reconstruction of the $\Lambda_{b}$ decay kinematics for the polarization and
spin correlation measurements. The full list of the $\Lambda_{c}^{+}$ decays
we include here is provided in a later section, in table 15.

In the semileptonic $\Lambda_{c}^{+}$ decay case, only the selection with the
$\Lambda\to p\pi^{-}$ decay will be considered. Decays with electrons instead
of muons, for both the $\Lambda_{b}$ and $\Lambda_{c}^{+}$, can be used as
well. Their branching ratios and spin analyzing powers are approximately the
same as for the decays with muons, and the trigger thresholds for electrons
and muons are comparable. However, we will conservatively not take decays with
electrons into account since reconstruction of electrons inside jets usually
has low efficiency or high background ATLAS:2016lxn ; CMS:2021scf ;
CMS:2023aim .

Figure 1: The fragmentation fraction for $z>z_{0}$, for $s\to\Lambda$,
$s\to\bar{\Lambda}$, and $s\to\Lambda/\bar{\Lambda}$, where
$\Lambda/\bar{\Lambda}$ denotes that either a $\Lambda$ or $\bar{\Lambda}$ is
produced in the $s$-quark hadronization. The Pythia simulation was run at
$\sqrt{s}=14$ TeV with $p_{T}>400$ GeV and $|\eta|<2.5$ cuts on the jets,
which were clustered with the anti-$k_{t}$ jet algorithm with radius $R=0.4$
Cacciari:2008gp ; Cacciari  and matched to the parton-level $s$ quarks. We
also show the data-based AKKII results Albino .

The last column in table 1 indicates the decay products whose angular
distributions are intended to be used for the polarization and spin
correlation measurements. The spin analyzing power (or the asymmetry
parameter) $\alpha$ is defined by writing the angular distribution of the
decay as

$\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta}=\frac{1}{2}\left(1+\alpha\mathcal{P}\cos\theta\right)\,,$
(5)

where $\theta$ is the angle between the momentum of the decay product and the
direction of the baryon polarization $\mathcal{\vec{P}}$, in the baryon rest
frame.222To define $\alpha$ for the corresponding antibaryon decay, we will
follow the common convention (as in refs. Bernreuther:2001rq ;
Bernreuther:2004jv ; Bernreuther:2015yna , for example, but unlike in ref.
CMS:2019nrx ) of adding a minus sign in front of the $\alpha$ in eq. (5) for
the antibaryon, such that for $CP$ conserving decays $\alpha$ has the same
value for the baryon and antibaryon decay. This provides a handle for
measuring the baryon and antibaryon polarizations and spin correlations.

The $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ decay can proceed via various
intermediate resonances, with different angular distributions in each case,
and we quote the effective value of the spin analyzing power, $\alpha_{\rm
eff}$, that corresponds to the sensitivity that can be obtained with a full
amplitude analysis LHCb:2022sck .

Reconstructing the kinematics of the semileptonic decays, which is needed for
the polarization and spin correlation measurements, is not straightforward
because neutral particles cannot be assigned to a vertex and neutrinos are not
observed at all. However, it can be done with certain approximations using the
method described in detail in sections 4.2.2 and 4.2.3 of ref. Galanti:2015pqa
, for example. It uses the fact that the energy fraction carried by a heavy-
flavored hadron (relative to the original quark) has a relatively peaked
distribution, with an average value around $70\%$ for the $b$ quark
ALEPH:2001pfo ; DELPHI:2011aa ; OPAL:2002plk ; SLD:2002poq ; ATLAS:2022miz ;
ATLAS:2021agf ; CMS-PAS-TOP-18-012  and $50\%$ for the $c$ quark OPAL:1997edj
; ALEPH:1999syy .333These numbers are appropriate for quark $p_{T}$ values of
tens of GeV. They decrease slowly as a function of the energy scale due to
renormalization group evolution Cacciari:2005uk . To reconstruct the neutrino
momentum, the vector pointing from the primary vertex to the baryon decay
vertex is taken as the baryon flight direction. See also refs. Dambach:2006ha
; LHCb:2015eia ; LHCb:2020ist ; Ciezarek:2016lqu . Unfolding will be required
to account for the approximations made.

##  4 Polarization and Spin Correlations

We will now review the mathematical description of the polarization and spin
correlations of $q\bar{q}$ pairs and describe how these properties are
expected to manifest themselves in angular distributions of the baryon decays.

###  4.1 Quark-Antiquark Pair Spin State Description

The spin state of a quark and antiquark is described by a density matrix of
the form Fano:1983zz

$\rho=\frac{1}{4}\left(\mathbbm{1}\otimes\mathbbm{1}+\tilde{B}^{+}_{i}\,\sigma^{i}\otimes\mathbbm{1}+\tilde{B}^{-}_{i}\,\mathbbm{1}\otimes\sigma^{i}+\tilde{C}_{ij}\,\sigma^{i}\otimes\sigma^{j}\right).$
(6)

Here $\mathbbm{1}$ is a $2\times 2$ unit matrix, $\sigma^{i}$ are the Pauli
matrices, the indices (summation over which is implied) represent the
coordinate axes, $\mathbf{\tilde{B}}^{\pm}$ are three-dimensional vectors
characterizing the polarization of the quark and antiquark,444The quark
polarization is $\mathbf{\tilde{B}}^{+}$, and the antiquark polarization is
$-\mathbf{\tilde{B}}^{-}$. and $\mathbf{\tilde{C}}$ is a $3\times 3$ matrix
that characterizes the spin correlations between them. The tilde symbol is
used here to distinguish between these properties of the quark and antiquark
and the measurable coefficients of the related angular distributions that will
be defined in the next subsection.

As common in the $t\bar{t}$ literature Bernreuther:2015yna ; ATLAS:2016bac ;
CMS:2019nrx , we will use the orthogonal set of axes
$\\{\hat{k},\hat{n},\hat{r}\\}$. The axis $\hat{k}$ is defined as the
direction of the outgoing quark’s momentum in the partonic center-of-mass (CM)
frame. To define the other axes, we denote by $\hat{p}$ the momentum direction
of one of the incoming partons, and by $\Theta$ the scattering angle of the
outgoing quark, such that $\cos\Theta=\hat{k}\cdot\hat{p}$. Then the axes
$\hat{n}$ and $\hat{r}$ are defined as

$\hat{n}=\operatorname{sign}(\cos\Theta)\,\frac{\hat{p}\times\hat{k}}{\sin\Theta}\;,\qquad\hat{r}=\operatorname{sign}(\cos\Theta)\,\frac{\hat{p}-\hat{k}\cos\Theta}{\sin\Theta}\;.$
(7)

The $\operatorname{sign}(\cos\Theta)$ factor is included in eq. (7) to account
for the Bose symmetry of the $gg$ initial state, meaning that without this
sign factor the $gg$-initiated contributions to the polarizations and spin
correlations of the sample as a whole will cancel between events with
$\cos\Theta<0$ and $\cos\Theta>0$, as can also be seen explicitly in refs.
Dharmaratna:1996xd ; Bernreuther:1995cx . It is of note that the inclusion of
the $\operatorname{sign}(\cos\Theta)$ factor leads to partial cancellation for
events originating from $q\bar{q}$ Dharmaratna:1996xd ; Bernreuther:1995cx .
It can be useful to also do a measurement without this factor to be more
sensitive to $q\bar{q}$-initiated contributions. CMS measured the
$\Lambda_{b}$ polarization along the $\hat{n}$ axis without the sign factor,
using an amplitude analysis of the decay555The advantage of this decay chain
is that every product is charged and therefore has a track, so one can
reconstruct it exactly, leading to low backgrounds and precise kinematics. The
downside of this channel is its tiny branching ratio ($\sim 3\times 10^{-5}$).
$\Lambda_{b}\to J/\psi(\to\mu^{+}\mu^{-})\Lambda(\to p\pi^{-})$ at CM energies
$\sqrt{s}=7$ and $8$ TeV, finding $\mathcal{P}=0.00\pm 0.06\,(\text{stat.})\pm
0.06\,(\text{syst.})$ CMS:2018wjk . LHCb conducted $\Lambda_{b}$ polarization
measurements using the same decay channel and found the polarization along
$\hat{n}$ (without the sign factor) to be within $68\%$ credibility level
intervals of $[-0.06,0.05]$, $[-0.04,0.05]$ and $[-0.01,0.07]$ at
$\sqrt{s}=7$, $8$ and $13$ TeV, respectively LHCb:2020iux . The absence of the
sign factor in the LHCb measurement does not lead to a cancellation of the
$gg$-initiated contributions since the LHCb detectors have coverage only in
the forward direction.

Given the orthonormal basis $\\{\hat{k},\hat{n},\hat{r}\\}$, we can write the
polarization vectors and spin correlation matrix as

$\mathbf{\tilde{B}}^{\pm}=b^{\pm}_{k}\hat{k}+b^{\pm}_{n}\hat{n}+b^{\pm}_{r}\hat{r}$
(8)

and

$\displaystyle\mathbf{\tilde{C}}=\;$ $\displaystyle
c_{kk}\hat{k}\hat{k}+c_{nn}\hat{n}\hat{n}+c_{rr}\hat{r}\hat{r}$
$\displaystyle+c_{rk}(\hat{r}\hat{k}+\hat{k}\hat{r})+c_{nr}(\hat{n}\hat{r}+\hat{r}\hat{n})+c_{kn}(\hat{k}\hat{n}+\hat{n}\hat{k})$
(9)
$\displaystyle+c_{n}(\hat{r}\hat{k}-\hat{k}\hat{r})+c_{k}(\hat{n}\hat{r}-\hat{r}\hat{n})+c_{r}(\hat{k}\hat{n}-\hat{n}\hat{k})$

or equivalently

$\mathbf{\tilde{C}}=\begin{pmatrix}c_{kk}&~{}~{}&c_{kn}+c_{r}&~{}~{}&c_{rk}-c_{n}\\\
c_{kn}-c_{r}&~{}~{}&c_{nn}&~{}~{}&c_{nr}+c_{k}\\\
c_{rk}+c_{n}&~{}~{}&c_{nr}-c_{k}&~{}~{}&c_{rr}\end{pmatrix}.$ (10)

In this way of writing, the symmetric part of the matrix $\tilde{\textbf{C}}$
is described by the components

$c_{ij}=\frac{1}{2}\left(\tilde{C}_{ij}+\tilde{C}_{ji}\right)$ (11)

and the antisymmetric part by

$c_{\ell}=\frac{1}{2}\epsilon_{ij\ell}\tilde{C}_{ij}\,.$ (12)

The range of possible values for $b_{i}^{\pm}$, $c_{ij}$, and $c_{\ell}$ is
$[-1,1]$. These quantities in general depend on the production process, the
partonic CM energy squared $\hat{s}$, and the quark’s scattering angle
$\Theta$ in relation to the proton going in the positive direction of the
$\hat{z}$ axis.

Table 2 classifies the polarization and spin correlation components according
to their $P$ and $CP$ properties. The $P$ and $CP$ invariance of QCD
(neglecting $\theta_{\rm QCD}$) allows spin correlations only in $c_{kk}$,
$c_{rr}$, $c_{nn}$, and $c_{rk}$. For the polarizations, nonzero
$b_{n}^{+}=b_{n}^{-}$ are allowed, but expected to be small as these
polarizations only appear at NLO QCD and are proportional to the quark mass
Dharmaratna:1996xd ; Bernreuther:1995cx . Small contributions involving the
electroweak interactions are expected in many of the components (see, e.g.,
ref. Bernreuther:2015yna , where they were computed for the top quark), but we
will not consider them in this paper.

Component | P | CP
---|---|---
$b_{k}^{+}+b_{k}^{-}$ | Odd | Even
$b_{k}^{+}-b_{k}^{-}$ | Odd | Odd
$b_{r}^{+}+b_{r}^{-}$ | Odd | Even
$b_{r}^{+}+b_{r}^{-}$ | Odd | Odd
$b_{n}^{+}+b_{n}^{-}$ | Even | Even
$b_{n}^{+}-b_{n}^{-}$ | Even | Odd
$c_{kk}$ | Even | Even
$c_{rr}$ | Even | Even
$c_{nn}$ | Even | Even
$c_{rk}$ | Even | Even
$c_{n}$ | Even | Odd
$c_{nr}$ | Odd | Even
$c_{k}$ | Odd | Odd
$c_{kn}$ | Odd | Even
$c_{r}$ | Odd | Odd
Table 2: $P$ and $CP$ properties of the polarization and spin correlation
components.

###  4.2 Decay Angular Distributions

Similar to the $t\bar{t}$ case Bernreuther:2015yna , if we denote the momentum
vector of one of the baryon decay products in the baryon rest frame by
$\mathbf{p}_{1}$, and the momentum vector of one of the antibaryon decay
products in the antibaryon rest frame by $\mathbf{p}_{2}$, their angular
distributions are given by

$\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{1}d\Omega_{2}}=\frac{1}{(4\pi)^{2}}\left(1+\mathbf{B}^{+}\cdot\mathbf{\hat{p}}_{1}+\mathbf{B}^{-}\cdot\mathbf{\hat{p}}_{2}-\mathbf{\hat{p}}_{1}\cdot\mathbf{C}\cdot\mathbf{\hat{p}}_{2}\right),$
(13)

where

$B^{\pm}_{i}=\alpha_{\pm}\,r_{i}\,f\,\tilde{B}^{\pm}_{i}\,,\qquad
C_{ij}=\alpha_{+}\alpha_{-}\,r_{i}\,r_{j}\,f\,\tilde{C}_{ij}\,.$ (14)

Here $\mathbf{\tilde{B}}^{\pm}$ and $\mathbf{\tilde{C}}$ are the quark and
antiquark polarization vectors and their spin correlation matrix from eqs.
(8)–(9). The factors $\alpha_{+}$ and $\alpha_{-}$, referring to the baryon
and antibaryon, respectively, are the spin analyzing powers of their decays,
as defined in eq. (5). The factors $r_{i}$ are the polarization retention
factors from eq. (1): $r_{L}$ for the $\hat{k}$ axis and $r_{T}$ for the
$\hat{n}$ and $\hat{r}$ axes.666We work here in the approximation that the
baryon momentum is parallel to the quark momentum and the only effect is the
scaling of the polarization by the corresponding factor. The factor $f$ is the
sample purity, namely the fraction of signal events out of the total number of
selected events:

$f=\frac{N_{\rm sig}}{N_{\rm bg}+N_{\rm sig}}\;.$ (15)

The multiplication by $f$ in eq. (14) is only correct if we assume that the
effect of the background is only to dilute the $B^{\pm}$ and $C$ coefficients.
This would be the case, for example, for a background consisting of
unpolarized and uncorrelated baryon-antibaryon pairs. If the background has a
more general angular dependence, it will add a bias that will need to be
subtracted. In some cases it will be possible to measure the bias using
sidebands; in other cases it can be estimated through simulation.

It will also be useful for us to define

$C_{ij}^{\pm}=C_{ij}\pm C_{ji}$ (16)

and rewrite eq. (14) as

$B^{\pm}_{i}=\alpha_{\pm}r_{i}fb^{\pm}_{i}\,,\quad
C_{ii}=\alpha_{+}\alpha_{-}r_{i}^{2}fc_{ii}\,,\quad
C_{ij}^{+}=2\alpha_{+}\alpha_{-}r_{i}r_{j}fc_{ij}\,,\quad
C_{ij}^{-}=2\alpha_{+}\alpha_{-}r_{i}r_{j}fc_{\ell}\,,$ (17)

where we used eqs. (11)–(12). In the expression for $C_{ij}^{-}$, the axes
corresponding to the indices $i$, $j$ and $\ell$ are related via
$\hat{i}\times\hat{j}=\hat{\ell}$.

Defining sets of spherical coordinates around the axes of interest for the
baryon and antibaryon and integrating eq. (13) over the azimuthal angles gives

$\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta^{+}_{i}d\cos\theta^{-}_{j}}=\frac{1}{4}\left(1+B^{+}_{i}\cos\theta^{+}_{i}+B^{-}_{j}\cos\theta^{-}_{j}-C_{ij}\cos\theta^{+}_{i}\cos\theta^{-}_{j}\right),$
(18)

where $\theta^{+}_{i}$ ($\theta^{-}_{j}$) is the angle between the direction
of the decay product and the chosen axis, in the baryon (antibaryon) rest
frame.777For the antibaryon, we take the reference axes to be opposite to the
$\\{\hat{k},\hat{n},\hat{r}\\}$ axes defined for the baryon, following the
convention of ref. Bernreuther:2015yna . In combination with the convention
described in footnote 2, this makes our signs for $B^{\pm}_{i}$ and $C_{ij}$
agree with both ref. Bernreuther:2015yna  and ref. CMS:2019nrx . Integrating
over one of the $\theta$s in eq. (18), we get the distributions

$\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta^{\pm}_{i}}=\frac{1}{2}\left(1+B^{\pm}_{i}\cos\theta^{\pm}_{i}\right),$
(19)

through which one can measure $B^{\pm}_{i}$ to obtain the quark and antiquark
polarization components $b^{\pm}_{i}$. Converting the double differential
distribution in eq. (18) to a distribution differential in the product
$\cos\theta^{+}_{i}\cos\theta^{-}_{j}$, we obtain

$\frac{1}{\sigma}\frac{d\sigma}{d(\cos\theta^{+}_{i}\cos\theta^{-}_{j})}=\frac{1}{2}\left(1-C_{ij}\cos\theta^{+}_{i}\cos\theta^{-}_{j}\right)\ln\left(\frac{1}{|\cos\theta^{+}_{i}\cos\theta^{-}_{j}|}\right),$
(20)

through which one can measure the $C_{ij}$ prefactors related to the spin
correlations. We imagine using them to extract the diagonal ($c_{ii}$)
components of the spin correlation matrix. For the off-diagonal components,
which are useful to divide into symmetric ($c_{ij}$) and antisymmetric
($c_{\ell}$) parts because of their different $CP$ properties (recall table
2), one can derive from eq. (13) the distribution

$\frac{1}{\sigma}\frac{d\sigma}{dX_{\pm}}=\frac{1}{2}\left(1-\frac{C_{ij}^{\pm}}{2}X_{\pm}\right)\cos^{-1}(|X_{\pm}|)\,,$
(21)

where
$X_{\pm}=\cos\theta^{+}_{i}\cos\theta^{-}_{j}\pm\cos\theta^{+}_{j}\cos\theta^{-}_{i}$.
The components $c_{ij}$ (with $i\neq j$) and $c_{\ell}$ can be computed from
$C_{ij}^{\pm}$ via eq. (17).

Figure 2: Distributions of $X=\cos\theta^{+}_{i}\cos\theta^{-}_{i}$ (left) and
$X=\cos\theta^{+}_{i}\cos\theta^{-}_{j}\pm\cos\theta^{+}_{j}\cos\theta^{-}_{i}$
(right), from which the spin correlations can be extracted. The distributions
are based on eqs. (20)–(21), and are shown for three different values of the
coefficient $C_{ii}$ (left) and $C_{ij}^{\pm}/2$ (right).

The distributions in eq. (19) are straight lines, while the shapes of the
distributions in eqs. (20) and (21) are shown in figure 2. They are plotted
for $C_{ii}$ and $C_{ij}^{\pm}/2$ values of $\pm 1$ (which are the upper and
lower bounds) and $0$.

###  4.3 Statistical Uncertainty Evaluation

The statistical uncertainty of the coefficients $B^{\pm}_{i}$, $C_{ij}$, and
$C_{ij}^{\pm}$, when they are extracted from fits of data to eqs. (19), (20),
and (21), respectively, is given approximately by

$\Delta B^{\pm}_{i}\simeq\frac{\sqrt{3}}{\sqrt{N}}\;,\quad\Delta
C_{ij}\simeq\frac{3}{\sqrt{N}}\;,\quad\Delta
C_{ij}^{\pm}\simeq\frac{3\sqrt{2}}{\sqrt{N}}\;,$ (22)

where $N$ is the number of events. These results are derived in appendix A.
Using eq. (17) we then obtain for the statistical uncertainties of the
polarization and spin correlation components

$\displaystyle\Delta
b^{\pm}_{i}\simeq\frac{\sqrt{3}}{|r_{i}\alpha_{\pm}|\sqrt{fN_{\rm sig}}}\;,$
(23) $\displaystyle\Delta
c_{ii}\simeq\frac{3}{r_{i}^{2}|\alpha_{+}\alpha_{-}|\sqrt{fN_{\rm sig}}}\;,$
(24) $\displaystyle\Delta
c_{ij(\ell)}\simeq\frac{3}{\sqrt{2}\,|r_{i}r_{j}\alpha_{+}\alpha_{-}|\sqrt{fN_{\rm
sig}}}\;,$ (25)

where $N_{\rm sig}$ is the number of signal events, and the notation
$c_{ij(\ell)}$ means to refer at once to $c_{ij}$ from eq. (11) and $c_{\ell}$
from eq. (12). The uncertainty in eq. (23) applies to $b_{i}^{+}$ and
$b_{i}^{-}$ separately. The quantities with definite $P$ and $CP$ properties
formed from the sums and differences of $b_{i}^{+}$ and $b_{i}^{-}$ (recall
table 2) will have lower relative statistical uncertainties since both the
quark and antiquark measurements will contribute.

We note that these formulas only provide rough estimates of the expected
statistical uncertainties, mainly because they do not take into account
effects of unfolding or nontrivial angular distributions of the background.

##  5 Quark Polarization and Spin Correlations in QCD

We used MadGraph Alwall:2014hca  to obtain the leading-order (LO) QCD
expectations for the polarization and spin correlations in $pp\to b\bar{b}$
and $pp\to c\bar{c}$.

We first validated our procedure on the process $pp\to t\bar{t}$, results for
which are available in the literature Severi:2022qjy . As a technical tool, we
decayed the top quark as $t\to b\ell^{+}v_{\ell}$ with MadSpin
Artoisenet:2012st  (and similarly for $\bar{t}$) to obtain the polarization
and spin correlation information of the $t$ and $\bar{t}$ from the angular
distributions of the leptons. We used the NN23LO1 parton distribution
functions and the default dynamical factorization and renormalization scales
defined in MadGraph. The simulation was inclusive, with no cuts applied, as
relevant for the comparison with the numbers in ref. Severi:2022qjy . As a
check, we have also run the full matrix element simulation in MadGraph, namely
$pp\to b\ell^{+}\nu_{\ell}\bar{b}\ell^{-}\bar{\nu}_{\ell}$ without separating
the process into production and decay, and obtained consistent results.

The symmetries of LO QCD dictate that only the components $c_{kk}$, $c_{rr}$,
$c_{nn}$, and $c_{rk}$ can be nonzero Bernreuther:2015yna . We indeed find all
the other components to be consistent with zero. For the non-vanishing
components, we found good agreement with the LO results of ref. Severi:2022qjy
.

To obtain the polarizations and spin correlations for $pp\to b\bar{b}$ and
$pp\to c\bar{c}$, we cannot follow a procedure exactly analogous to what we
did for $pp\to t\bar{t}$ since MadGraph does not allow decaying $b$ and $c$
quarks (which we need as a technical tool to extract the spin information of
the quarks). Instead, relying on the flavor blindness of QCD, we simulated
$pp\to t\bar{t}$ with the top-quark mass set to $m_{b}$ or $m_{c}$ (and its
width set to a negligible value). We have also lowered the masses of the
particles that participate in the top decay $t\to b\ell^{+}v_{\ell}$, so that
the decay will still happen. We have verified, by simulating the decay of a
polarized top quark in this model using the method of ref.
BuarqueFranzosi:2019boy , that the spin analyzing power of the lepton remains
the same.

Unlike in the $t\bar{t}$ case, where an inclusive measurement without any cuts
on the $t$ and $\bar{t}$ is possible (with the use of unfolding), a
measurement in $b\bar{b}$ or $c\bar{c}$ will typically be limited by triggers.
(Triggers will be discussed in detail in section 6.1.) As we will see in later
sections, the muon-based triggers are the best for our purposes. Since we work
in MadGraph, at the level of parton-level quarks, instead of applying cuts on
the muons produced in the hadron decays, we will apply roughly equivalent cuts
on the quarks. Using a Pythia simulation, we found that applying the Run 2
ATLAS dimuon trigger threshold to muons from $b\to c$ transitions in the
$b\bar{b}$ case is equivalent in terms of the cross section to applying the
cuts $|\eta|<2.4$ and $p_{T}>79$ GeV on the quarks (“jets”). In the $c\bar{c}$
case the same procedure leads to a $p_{T}>115$ GeV cut on the quarks.888The
difference between $b\bar{b}$ and $c\bar{c}$ is mostly related to the fact
that a $b$ hadron typically carries around $70\%$ of the $b$-jet momentum,
while a $c$ hadron carries around $50\%$ of the $c$-jet momentum, so $c$
quarks need to be more energetic for their muons to pass the trigger
threshold. We will use these cuts here as an example.

The results for the non-vanishing spin correlation components are shown in
table 3. We also present the inclusive values for a more meaningful comparison
with the $t\bar{t}$ case, as a check. For the inclusive $b\bar{b}$ and
$c\bar{c}$ simulations we fixed the factorization and renormalization scales
to $10$ GeV, to avoid large artifacts from low energies. As can be seen in the
table, the inclusive values are rather similar between $t\bar{t}$, $b\bar{b}$,
and $c\bar{c}$. The cuts, on the other hand, take us to a completely different
regime. This is understandable since the inclusive contributions are dominated
by a region near the production threshold while the cuts select regions away
from the threshold.

|  $t\bar{t}$, no cuts |  $b\bar{b}$, no cuts |  $c\bar{c}$, no cuts |  $b\bar{b}$ with cuts |  $c\bar{c}$ with cuts
---|---|---|---|---|---
$c_{kk}$ |  $0.324\pm 0.006$ |  $0.296\pm 0.004$ |  $0.284\pm 0.004$ | $-0.987\pm 0.004$ | $-0.984\pm 0.006$
$c_{rr}$ |  $0.009\pm 0.006$ |  $0.004\pm 0.004$ | $-0.006\pm 0.004$ | $-0.603\pm 0.004$ | $-0.609\pm 0.006$
$c_{nn}$ |  $0.333\pm 0.006$ |  $0.299\pm 0.004$ |  $0.298\pm 0.004$ |  $0.591\pm 0.004$ |  $0.603\pm 0.006$
$2c_{rk}$ | $-0.211\pm 0.008$ | $-0.197\pm 0.006$ | $-0.188\pm 0.006$ | $-0.038\pm 0.006$ | $-0.008\pm 0.009$
Table 3: Spin correlations in $b\bar{b}$ and $c\bar{c}$ events with and
without cuts for Run 2 ($\sqrt{s}=13$ TeV). For comparison, we also show the
same quantities for the $t\bar{t}$ case. The uncertainties shown are the
statistical uncertainties of our simulation.

Table 4 presents the analogous results for the HL-LHC with $\sqrt{s}=14$ TeV,
where our effective cuts on the quarks are $|\eta|<2.5$ and $p_{T}>59$ GeV for
$b\bar{b}$ and $p_{T}>80$ GeV for $c\bar{c}$. We can see similar effects from
the cuts as in the Run 2 case.

To assess systematic uncertainties, we examined the effects of varying the
renormalization and factorization scales up and down by a factor of $2$. While
there was a significant effect on the cross sections, there were no
significant effects on the spin correlation coefficients relative to the
statistical uncertainties of our simulations, which are shown in tables 3 and
4.

|  $b\bar{b}$, no cuts |  $c\bar{c}$, no cuts |  $b\bar{b}$ with cuts |  $c\bar{c}$ with cuts
---|---|---|---|---
$c_{kk}$ |  $0.298\pm 0.004$ |  $0.276\pm 0.004$ | $-0.972\pm 0.004$ | $-0.995\pm 0.006$
$c_{rr}$ | $-0.001\pm 0.004$ | $-0.009\pm 0.004$ | $-0.587\pm 0.004$ | $-0.610\pm 0.006$
$c_{nn}$ |  $0.302\pm 0.004$ |  $0.284\pm 0.004$ |  $0.577\pm 0.004$ |  $0.596\pm 0.006$
$2c_{rk}$ | $-0.187\pm 0.006$ | $-0.188\pm 0.006$ | $-0.054\pm 0.006$ | $-0.015\pm 0.009$
Table 4: Spin correlations in $b\bar{b}$ and $c\bar{c}$ events with and
without cuts for the HL-LHC with $\sqrt{s}=14$ TeV. The uncertainties shown
are the statistical uncertainties of our simulation.

##  6 Proposed Analyses and Their Prospects

In this section we will consider a variety of analysis channels for measuring
the polarizations or spin correlations in $pp\to q\bar{q}$ processes with
$q=s$, $c$, or $b$, using the baryon decays that were listed in table 1.

We will need to address a variety of backgrounds. There are _intrinsic
backgrounds_ , which arise from the same parton-level process as the signal
but with a different hadron decay passing the selection. There are also
_extrinsic backgrounds_ , which arise due to other parton-level processes that
may involve the same baryon decay as the signal or another similar hadron
decay. Lastly, there are _combinatorial backgrounds_ (which may be of an
intrinsic or extrinsic origin), which are a result of random tracks forming by
chance a vertex similar to that of the baryon decay of interest. While the
probability of this happening will usually be low, such a background can still
be significant if the total cross section of the corresponding process is
large.

We will assess the feasibility of each channel in terms of the precision that
can be achieved and the sample purity. We will do it with the help of MadGraph
Alwall:2014hca  and/or Pythia Sjostrand:2014zea  simulations and reliance on
elements of existing ATLAS and CMS analyses. For jet clustering, the Pythia
simulations are interfaced with FastJet Cacciari , where we use the
anti-$k_{t}$ algorithm with radius $R=0.4$ Cacciari:2008gp . Apart from
trigger-motivated cuts and background reduction cuts relevant in each case, we
will present our results for several values of dijet invariant mass ($m_{jj}$)
cuts (in cases where statistics allows that) since such a selection can
enhance the sensitivity to BSM effects that become sizable only at high
energies.

We will start by presenting our assumed datasets, based on the current and
future planned LHC parameters and triggers, and then proceed to discussing the
individual analysis channels, each with its own backgrounds and selection
strategy.

###  6.1 Benchmark Datasets

We will consider the currently available full Run 2 dataset as well as the HL-
LHC dataset. Table 5 presents the main parameters defining these datasets,
including the standard trigger-motivated cuts that we will be assuming. The
numbers shown in the table are for the offline cuts from ATLAS ATL-PHYS-
PUB-2019-005  and the online ones for CMS Collaboration:2759072 . We will be
using the ATLAS cuts.

| ATLAS | CMS
---|---|---
| Run 2 | HL-LHC | Run 2 | HL-LHC
Collider energy $\sqrt{s}$ [TeV] | 13 | 14 | 13 | 14
Integrated luminosity $\mathcal{L}$ [fb-1] | 140 | 3000 | 140 | 3000
Jet $p_{T}$ cut [GeV] | 460 | 400 | 500 | 520
Double muon $p_{T}$ cut (without isolation) [GeV] | 15 | 10 | 37, 27 | 37, 27
Single muon $p_{T}$ cut (with isolation) [GeV] | 27 | 20 | 24 | 24
Double electron $p_{T}$ cut (without isolation) [GeV] | 18 | 10 | 25 | 25
Single electron $p_{T}$ cut (with isolation) [GeV] | 27 | 22 | 28 | 32 or 26
Jet $|\eta|$ cut | 2.4 | 3.8 | 2.4 | 4.0
Muon $|\eta|$ cut | 2.4 | 2.5 | 2.4 | 2.4
Electron $|\eta|$ cut | 2.4 | 2.5 | 2.4 | 2.4
Table 5: The collider energy, luminosity, and trigger-motivated cuts for Run 2
of the LHC and those that are planned for the HL-LHC in ATLAS ATL-PHYS-
PUB-2019-005  and CMS Collaboration:2759072 . We will be using the ATLAS cuts.

For the jet based triggers, which are relevant for the hadronic channels
$c\to\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ and $s\to\Lambda\to p\pi^{-}$, for Run
2 we added the requirement $|\eta|<2.4$ (even though the trigger functions up
to $|\eta|=2.8$ Owen:2302730 ) so that the jets will be within the tracker.
For the HL-LHC we require $|\eta|<3.8$.

For the semileptonic channels $b\to\Lambda_{b}\to X_{c}\mu^{-}\bar{\nu}_{\mu}$
and $c\to\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$ we can use the muon
triggers, whose thresholds are much softer than those of the jet triggers.
Even though the muon carries only a fraction of the jet energy, the muon
triggers will still provide higher statistics. Since our muons are inside
jets, the triggers of interest are primarily those that do not require the
muons to be isolated. That is not a problem for events with two muons since
double muon triggers without isolation requirements have sufficiently low
thresholds. However, in some of the analyses that we will describe
(semileptonic $\Lambda_{c}^{+}$ decay on one side and hadronic on the other
side, or polarization measurements without any requirement on the second jet)
just a single muon will be present. Single muon triggers without isolation
have the relatively high thresholds of $50$ GeV ATLAS:2020gty ;
Collaboration:2759072 . We will instead be relying on the ATLAS single muon
trigger (included in table 5) with a loose isolation criterion, which has
around $50\%$ efficiency for muons in heavy-flavor jets ATLAS:2020gty .

As was mentioned in section 3, decays with electrons instead of muons can be
considered as well (even though reconstruction of electrons inside jets
usually has low efficiency or high background), and table 5 shows the
corresponding triggers.

In the context of the $b\to\Lambda_{b}\to X_{c}\mu^{-}\bar{\nu}_{\mu}$
channel, we also looked at $b$-tagging triggers. In ATLAS, the single $b$-jet
trigger ATLAS:2021piz  requires $E_{T}>225$ GeV (which is expected to become
$p_{T}>180$ GeV at the HL-LHC ATL-PHYS-PUB-2019-005 ), and the double $b$-jet
trigger demands $E_{T}>150$ GeV for the leading jet $E_{T}>50$ GeV for the
subleading one. Similarly, CMS is expected to have a double $b$-jet trigger
with a $p_{T}>128$ GeV cut at the HL-LHC Collaboration:2759072 . Even though
these thresholds are significantly lower than those of the generic jet
triggers, we have checked that the much lower thresholds of the muon-based
triggers still result in more statistics. This happens in our particular case
because the $b$ jets we are interested in always contain a muon. The $b$-jet
triggers can however be useful at high invariant masses, where they can
recover much of the efficiency loss due to the loose isolation requirement of
the low-$p_{T}$ single-muon triggers.

In addition to the standard trigger paths listed in table 5, we will consider
the CMS “parked” $b$-hadron dataset that was collected during part of Run 2
using the _data parking_ strategy with a single displaced muon trigger CMS-
DP-2019-043 ; Bainbridge:2020pgi . The muon $p_{T}$ threshold varied between 7
and 12 GeV depending on the luminosity, its track was required to be within
$|\eta|<1.5$, and satisfy a requirement on impact parameter significance.
Despite the lower integrated luminosity of this dataset ($\sim 50$ fb-1) and
the $\eta$ restriction, the soft $p_{T}$ threshold allowed collecting more
statistics than the standard Run 2 muon triggers.

###  6.2 Analyses of $pp\to s\bar{s}$

For measuring the polarization and spin correlations in $pp\to s\bar{s}$, we
consider events with $s\to\Lambda$ and $\Lambda\to p\pi^{-}$.

####  6.2.1 $\Lambda$ Reconstruction, Efficiency and Signal Yield

The decay $\Lambda\to p\pi^{-}$ has a very distinct signature of a highly
displaced vertex with a pair of oppositely charged tracks that reconstruct the
$\Lambda$ mass if they are assigned the proton and charged pion masses. The
other similar decay, $K_{S}\to\pi^{+}\pi^{-}$, will usually fail the $\Lambda$
mass reconstruction, and moreover can be vetoed without significant loss of
signal efficiency by requiring that the two tracks do not reconstruct the
$K_{S}$ mass when assigned the charged pion masses. These $\Lambda$ decays
were reconstructed in multiple analyses in ATLAS (e.g., ATLAS:2011xhu ;
ATLAS:2012cvl ; ATLAS:2014swk ; ATLAS:2019pqg ) and CMS (e.g., CMS:2011jlm ;
CMS:2018wjk ; CMS:2019isl ; CMS:2020zzv ; CMS:2021rvl ).

There is, however, an important obstacle to reconstructing the decays of
energetic $\Lambda$ baryons. With increasing $p_{T}$ they quickly become too
displaced to be successfully reconstructed within the volume of the tracker.
This can result in a very significant efficiency loss. To optimize the
reconstruction efficiency of highly displaced tracks, ATLAS have developed the
Large Radius Tracking (LRT) algorithm ATL-PHYS-PUB-2017-014 ; ATL-PHYS-
PUB-2019-013 ; ATLAS:2023nze . It has been used in multiple searches for long-
lived BSM particles ATLAS:2017tny ; ATLAS:2019kpx ; ATLAS:2019fwx ;
ATLAS:2019jcm ; ATLAS:2020xyo ; ATLAS:2020wjh ; ATLAS:2021jig ; ATLAS:2023oti
. The LRT algorithm looks at hits remaining after the standard reconstruction,
and tries to reconstruct the remaining tracks with looser conditions on the
transverse and longitudinal impact parameters. The addition of this algorithm
allows keeping decent track reconstruction efficiency up to decay radii
$d_{T}$ of about $d_{T}^{\rm max}\approx 300$ mm, with the mean reconstruction
efficiency up to this decay radius being roughly $\epsilon_{\rm track}\approx
65\%$. It is of note that LRT was applied to only about $10\%$ of the events
in Run 2, but is going to be used regularly in Run 3 and at the HL-LHC after
the LRT processing time was significantly improved ATLAS:2023nze . Moreover,
with the ATLAS tracker upgrade planned for the HL-LHC, it will be possible to
address even larger decay radii, with average reconstruction efficiency of
roughly $\epsilon_{\rm track}\approx 80\%$ up to $d_{T}^{\rm max}\approx 400$
mm Strebler:2022nkh . We will be using the efficiency numbers with the LRT
algorithm to estimate the $\Lambda$ reconstruction efficiency, after
accounting for the probability for the $\Lambda$ to decay within the ranges
mentioned above.

The average $\Lambda$ decay radius in $pp\to s\bar{s}$ events can be estimated
as

$\langle d_{T}\rangle=\langle\gamma\beta_{T}\rangle\,c\tau=\frac{\langle
p_{T,\Lambda}\rangle\,c\tau}{m_{\Lambda}}=\frac{\langle z\,p_{T,{\rm
jet}}\rangle\,c\tau}{m_{\Lambda}}\;,$ (26)

where $c\tau\approx 7.9$ cm is the $\Lambda$ lifetime and $z$ is the $\Lambda$
momentum fraction from eq. (2). Since the cross section decreases fast as a
function of $p_{T,{\rm jet}}$ and $z$, we can obtain rough estimates for
$\langle d_{T}\rangle$ by taking $p_{T,{\rm jet}}$ to be the trigger-motivated
jet $p_{T}$ threshold from table 5, and $z=0.3$, which is the lowest value of
$z$ we will be willing to use since much softer $\Lambda$ baryons are not
correlated with the original strange-quark polarization, as was discussed in
section 2. This gives $\langle d_{T}\rangle\sim 9.8$ m for Run 2 and $8.5$ m
for the HL-LHC. Since $\langle d_{T}\rangle\gg d_{T}^{\rm max}$, the
probability for the $\Lambda$ to decay sufficiently early is
$\epsilon_{d_{T}}\simeq d_{T}^{\rm max}/\langle d_{T}\rangle\approx 3.1\%$ for
Run 2 and $4.7\%$ for the HL-LHC. More accurate numbers that we obtained from
a Pythia simulation (which accounts for the full jet $p_{T}$ and $z$
distributions above their corresponding thresholds) are
$\epsilon_{d_{T}}\approx 2.2\%$ for Run 2 and 3.4% for the HL-LHC. For
background jets (which are a mixture of all $pp\to jj$ processes apart from
$pp\to s\bar{s}$) the numbers are $2.4\%$ and $3.7\%$, respectively.

The full reconstruction efficiency for the $\Lambda\to p\pi^{-}$ decay in
signal jets is $\epsilon_{\Lambda}=\epsilon_{d_{T}}\,\epsilon_{\rm
track}^{2}\approx 0.93\%$ for Run 2 and $2.2\%$ for the HL-LHC. For spin
correlation measurements we need the efficiency for reconstructing both a
$\Lambda$ and a $\bar{\Lambda}$. Despite the correlation between the $p_{T}$
of the two jets, we can still obtain a rough estimate by simply squaring the
efficiency, $\epsilon_{\Lambda\bar{\Lambda}}\simeq\epsilon_{\Lambda}^{2}$,
because the jet $p_{T}$ values are distributed mainly near the threshold. This
gives $\epsilon_{\Lambda\bar{\Lambda}}\approx 9\times 10^{-5}$ for Run 2 and
$5\times 10^{-4}$ for the HL-LHC.

Run 2
---
$\sigma_{s\bar{s}}$ [pb] | $N_{s}$ | $N_{s\bar{s}}$
3.1 | 54 | $<1$

HL-LHC
---
$\sigma_{s\bar{s}}$ [pb] | $N_{s}$ | $N_{s\bar{s}}$
8.3 | $8.1\times 10^{3}$ | 2

Table 6: Cross sections (with trigger-motivated cuts) and signal event counts
(after the full selection) for measurements of $s$-quark polarization
($N_{s}$) and spin correlations ($N_{s\bar{s}}$).

Table 6 shows the numbers of expected signal events in Run 2 and at the HL-
LHC. We show both the number of $s$ jets available for the polarization
measurement ($N_{s}$) and the number of $s\bar{s}$ pairs available for the
spin correlation measurement ($N_{s\bar{s}}$) computed as

$\displaystyle N_{s}$
$\displaystyle=\mathcal{L}\,\sigma_{s\bar{s}}\,f(s\to\Lambda,~{}z>0.3)\operatorname{BR}(\Lambda\to
p\pi^{-})\,\epsilon_{\Lambda}\,,$ (27) $\displaystyle N_{s\bar{s}}$
$\displaystyle=\mathcal{L}\,\sigma_{s\bar{s}}\,f^{2}(s\to\Lambda,~{}z>0.3)\operatorname{BR}^{2}(\Lambda\to
p\pi^{-})\,\epsilon_{\Lambda\bar{\Lambda}}\,,$ (28)

where the cross sections $\sigma_{s\bar{s}}$ (with the trigger-motivated cut
on the jets) were computed in MadGraph. We see from table 6 that the number of
events available for spin correlation measurements is going to be too low even
at the HL-LHC. We will therefore proceed with investigating the prospects of
polarization measurements only.

####  6.2.2 Background

The dominant background is due to $\Lambda$ baryons produced in dijet
processes other than $pp\to s\bar{s}$. While the number of $\Lambda$ baryons
produced in most of these processes falls with $z$ faster than in the signal,
their total cross section is large. As a result, their contribution ends up
being significant.

Figure 3: Expected number of events with reconstructed $\Lambda$ baryons
satisfying $z>z_{0}$ for the signal originating from $s\bar{s}$ events and
backgrounds with the final state partons specified in the legend, for Run 2
(left) and the HL-LHC (right).

Figure 3 shows the results of a Pythia simulation for the signal and
backgrounds, where the backgrounds are split into three categories according
to the produced partons: $gg$, $qg$, and $qq$ (where $q$ represents all
flavors of quarks and antiquarks). The two leading jets in each event are
considered as $s$ jet candidates, except for $s\bar{s}$ events, where we
matched the jets to partons to count only $\Lambda$ baryons from $s$ (but not
$\bar{s}$) as signal. The numbers are presented as a function of $z_{0}$, the
cut on the momentum fraction $z$ carried by the $\Lambda$. From these plots we
can calculate the sample purity. For $z_{0}=0.3$, the purity is $f\approx
0.9\%$ for both Run 2 and the HL-LHC.

####  6.2.3 Measurement prospects

With the signal event counts and purities, we can use eq. (23) to compute the
expected precision of the polarization measurements. We find the expected
statistical uncertainty on the polarization components $b^{\pm}_{i}$ at the
HL-LHC to be given by $r_{i}\Delta b^{\pm}_{i}\approx 0.27$. We show the
result for the product $r_{i}\Delta b^{\pm}_{i}$ to provide numbers that are
independent of the systematic uncertainty of the polarization retention
factors $r_{i}$. We remind the reader, however, that $r_{L}\sim 0.6$, as
mentioned in section 2. The physical range of $b^{\pm}_{i}$ values is
$[-1,1]$. Only the HL-LHC results were discussed here since the Run 2 numbers
are far from being promising.

We conclude that with the standard triggers we assumed, $s\bar{s}$ spin
correlations cannot be measured, while polarization measurements might be
possible at the HL-LHC, although the statistical uncertainty of the result is
expected to be high, and the low purity of the sample will make the
measurements difficult.

###  6.3 Analyses of $pp\to c\bar{c}$

For $c\bar{c}$ polarization and spin correlation measurements, we will
consider in turn three possible analysis channels in terms of the
$\Lambda_{c}^{+}$ decays: the _hadronic channel_ where $\Lambda_{c}^{+}\to
pK^{-}\pi^{+}$, the _semileptonic channel_ where
$\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$, and the _mixed channel_ with the
hadronic decay in one jet and the semileptonic decay in the other.

####  6.3.1 Hadronic Channel

The signature of the $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ decay is one
negatively and two positively charged tracks coming from a common vertex. They
should also reconstruct the $\Lambda_{c}^{+}$ mass for an assignment of the
proton and $\pi^{+}$ masses to the positively charged tracks and the $K^{-}$
mass to the negatively charged one. Such decays were reconstructed by CMS in
refs. CMS:2019uws ; CMS:2023frs .

There are intrinsic backgrounds from various other decays of charmed hadrons,
with examples shown in table 7. Some approaches for reducing them were
discussed in ref. Galanti:2015pqa . Extrinsic backgrounds from $pp\to
b\bar{b}$ should be considered as well. They include $\Lambda_{c}^{+}$ baryons
produced in $\Lambda_{b}$ or $B$-meson decays as well as other decays that
produce vertices that pass the selection. These contributions can be
suppressed significantly using the large displacement of the $b$-hadron decays
($c\tau\approx 0.45$ mm) relative to the $\Lambda_{c}^{+}$ ($c\tau\approx
0.06$ mm). There is also the combinatorial background of three general tracks
forming by chance a fake $\Lambda_{c}^{+}$-like vertex.

Fragmentation Fraction [%] | Decay Scheme | Branching Ratio [%]
---|---|---
$c\to\Lambda_{c}^{+}$ | 6.4 |  $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ (signal) | 6.3
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}\pi^{0}$ | 4.4
$\Lambda_{c}^{+}\to\Sigma^{+}\pi^{-}\pi^{+}$ | 4.5
$\Lambda_{c}^{+}\to\pi^{+}\pi^{-}\pi^{+}\Lambda$ | 3.6
$c\to D^{+}$ | 22.7 | $D^{+}\to\pi^{+}K^{-}\pi^{+}$ | 9.4
$D^{+}\to\pi^{+}K^{-}\pi^{+}\pi^{0}$ | 6.2
$D^{+}\to\pi^{+}\pi^{-}\pi^{+}K_{S}^{0}$ | 3.1
$D^{+}\to\pi^{+}\pi^{-}\pi^{+}\pi^{0}$ | 1.2
$c\to D^{0}$ | 61.8 | $D^{0}\to\pi^{+}K^{-}\pi^{+}\pi^{-}$ | 8.2
$D^{0}\to\pi^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$ | 4.3
$c\to D_{s}^{+}$ | 8.2 | $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$ | 5.4
$D_{s}^{+}\to K^{+}K^{-}\pi^{+}\pi^{0}$ | 5.5
$D_{s}^{+}\to\pi^{+}K^{-}\pi^{+}K^{0}_{S}$ | 1.5
Table 7: The $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ signal and some of its
intrinsic backgrounds. The numbers are from refs. Lisovyi:2015uqa ;
Workman:2022ynf . The background decays will usually fail to reconstruct the
$\Lambda_{c}^{+}$ mass—a requirement that will strongly suppress them.

Simulating the backgrounds for the $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ decay
with publicly available tools is nontrivial for us due to several reasons.
First, the displacement of the $\Lambda_{c}^{+}$ vertex is small, so a careful
simulation of the tracking resolution, as well as the vertexing algorithm, are
needed to assess the impact of any displacement-related cuts. Second,
simulating combinatorial background requires good control of tails of
distributions of high cross section processes.

We can however get an idea about the size of the expected background and the
signal efficiency from the CMS analysis in ref. CMS:2023frs . CMS used several
variables to select $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ candidates: the
$\chi^{2}$ of the vertex fit to the three tracks, the angle between the
$\Lambda_{c}^{+}$ candidate momentum and the vector connecting the production
and decay vertices in the transverse plane, the transverse separation
significance between the two vertices, and the $\Lambda_{c}^{+}\,$ $p_{T}$
fractions carried by the kaon and by the proton. A distribution of the
$pK^{-}\pi^{+}$ invariant mass was then constructed, with the
$\Lambda_{c}^{+}$ contribution appearing as a narrow peak. The number of
events in the peak is about $7.9\%$ of the total number of events in the peak
region for $\Lambda_{c}^{+}$ candidates with $20<p_{T}<30$ GeV. CMS estimate
the prompt $\Lambda_{c}^{+}$ fraction in the peak to be between $0.79$ to
$0.88$. We will therefore take our rough estimate for the sample purity (when
a single jet is considered) to be $f\approx 6.9\%$. CMS also report the signal
efficiency to be roughly $\epsilon_{\Lambda_{c}^{+}}\approx 25\%$, which we
will also assume. It should be noted that the invariant mass resolution will
be improved by $20$–$50\%$ at the HL-LHC ATL-PHYS-PUB-2018-041 ; ATL-PHYS-
PUB-2018-032 ; ATL-PHYS-PUB-2018-035 ; CMS-PAS-FTR-18-013 ; CMS-PAS-FTR-18-033
so the purity will improve accordingly. We shall be optimistic and reduce the
background under the peak by a factor of two (i.e., approximately double the
purity) in our estimates for the HL-LHC. We note that the upgraded tracking
detectors will likely improve the efficacy of the other cuts used in the
selection as well.

The sidebands of the $pK^{-}\pi^{+}$ invariant mass distribution can be used
to measure the angular distributions of the background. They will need to be
subtracted to obtain the polarization information of the signal from the
events in the peak region.

$c\bar{c}$, hadronic | Run 2
---|---
$m_{jj}$ cut [GeV] |  $\sigma_{c\bar{c}}$ [pb] | $N_{c}$ | $r_{i}\Delta b^{\pm}_{i}$ | $N_{c\bar{c}}$
no cut | 3.0 | 420 | 0.48 | $<1$
1000 | 2.6 | 360 |  |
1500 | 0.57 | 80 |  |
2000 | 0.13 | 18 |  |

HL-LHC
---
$\sigma_{c\bar{c}}$ [pb] | $N_{c}$ | $r_{i}\Delta b^{\pm}_{i}$ | $N_{c\bar{c}}$
8.0 | $2.4\times 10^{4}$ | 0.045 | 24
4.7 | $1.4\times 10^{4}$ | 0.059 | 14
0.93 | 2800 | 0.13 | 2
0.21 | 650 | 0.28 |

Table 8: Cross sections (with trigger-motivated cuts) and signal event counts
(after the full selection) for measurements of $c$-quark polarization
($N_{c}$) and spin correlations ($N_{c\bar{c}}$) in the hadronic channel. The
expected statistical uncertainties for the polarization measurements are shown
as well. The sample purity is $6.9\%$ for Run 2, and double that for the HL-
LHC.

To calculate the expected number of signal events, we computed the $pp\to
c\bar{c}$ cross sections $\sigma_{c\bar{c}}$ with the jet trigger cuts from
table 5 using a MadGraph simulation. In addition, for the $c$ polarization
measurements we require the $c$ jet to contain a $\Lambda_{c}^{+}\to
pK^{-}\pi^{+}$ decay. For the spin correlations we require this
$\Lambda_{c}^{+}$ decay in one jet and the analogous $\bar{\Lambda}_{c}^{-}$
decay in the other. The expected numbers of events for measurements of
polarization ($N_{c}$) and spin correlations ($N_{c\bar{c}}$), calculated as

$\displaystyle N_{c}$
$\displaystyle=\mathcal{L}\,\sigma_{c\bar{c}}\,f(c\to\Lambda_{c}^{+})\operatorname{BR}(\Lambda_{c}^{+}\to
pK^{-}\pi^{+})\,\epsilon_{\Lambda_{c}^{+}}\,,$ (29) $\displaystyle
N_{c\bar{c}}$
$\displaystyle=\mathcal{L}\,\sigma_{c\bar{c}}\,f^{2}(c\to\Lambda_{c}^{+})\operatorname{BR}^{2}(\Lambda_{c}^{+}\to
pK^{-}\pi^{+})\,\epsilon_{\Lambda_{c}^{+}}^{2}\,,$ (30)

are shown in table 8 for Run 2 and the HL-LHC. We also provide results that
correspond to different cuts on the dijet invariant mass, $m_{jj}$, as such
cuts can enhance the sensitivity to new physics contributions. As can be seen
in the table, the number of events available for the spin correlation
measurements in this channel is small even for the HL-LHC (considering also
the relatively low purity), so we proceed with the analysis of polarization
measurements only. The table shows the expected statistical uncertainties for
the polarization measurements (multiplied by the polarization retention
factors) $r_{i}\Delta b^{\pm}_{i}$ based on eq. (23). The prospects are
borderline for Run 2 but seem good for the HL-LHC.

####  6.3.2 Semileptonic Channel

A potentially more promising avenue is the semileptonic decay
$\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$ with $\Lambda\to p\pi^{-}$. While
the branching ratio of this decay chain ($3.5\%$ for
$\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$, $64\%$ for $\Lambda\to p\pi^{-}$
Workman:2022ynf ) is lower than that of the hadronic decay, the muon triggers
have very low $p_{T}$ thresholds (see table 5) so there is a potential for
getting more data. We do not consider a selection without a $\Lambda$ decay
reconstructed in the tracker because it will be difficult to extract the
$\Lambda_{c}^{+}$ polarization information if the muon will be the only
product associated with the $\Lambda_{c}^{+}$ decay. Besides that, the
$\Lambda$ requirement strongly suppresses the intrinsic background due to
semileptonic $D$-meson decays since they are kinematically forbidden from
producing a $\Lambda$ (which would have to be produced together with an
antibaryon, for baryon number conservation, while no sufficiently light
antibaryons exist).

This channel also has disadvantages, such as the low reconstruction efficiency
of $\Lambda$ decays, and the shortness of the $\Lambda_{c}^{+}$ lifetime
($c\tau\approx 0.06$ mm) which leads to large uncertainties in its flight
direction reconstruction, which is needed for the neutrino reconstruction.
Although these challenges exist we will analyze the potential of this channel.

Our selection in this channel requires the jet to contain a muon (as done
sometimes for charm jet tagging ATLAS:2015thz ; CMS:2017wtu ; ATLAS-
CONF-2018-055 ; CMS:2021scf ; CMS:2023aim ) as well as a reconstructed
$\Lambda\to p\pi^{-}$ decay. To ensure that the $\Lambda$ originates from a
$\Lambda_{c}^{+}$ decay, one should demand the $\Lambda$ trajectory (inferred
from the momenta of its decay products) to form a displaced vertex with the
muon. Events in which the $\Lambda$ trajectory is consistent with both a
common vertex with the muon and the primary vertex, can still be accepted if
the $\Lambda$ carries a significant fraction (e.g., above $20\%$) of the jet
momentum, since $\Lambda$ baryons produced in parton showering and
hadronization will usually be soft. We expect this requirement to have high
signal efficiency and significant background suppression, but estimating the
efficiency and purity quantitatively requires a detailed simulation which is
beyond the scope of the current work.

There is an extrinsic background from $b\bar{b}$ production. We can estimate
the size of this background by starting with the inclusive branching ratio for
$b$ jets to contain a $\Lambda$ or $\bar{\Lambda}$ baryon, which was measured
to be Workman:2022ynf

$\operatorname{BR}\left(b\to\mbox{($b$-hadron)}\to\Lambda/\bar{\Lambda}+X\right)\approx(5.9\pm
0.6)\%\,.$ (31)

Accounting for us requiring a $\Lambda$ (not a $\bar{\Lambda}$) while on the
other hand collecting background from both the $b$ and $\bar{b}$ jet, we are
left with the same number. We will further assume that the probability of
having a $\mu^{+}$ in association with the $\Lambda$ is roughly $10\%$.999This
assumption is based on the fact that the off-shell $W$ boson that produces the
muon can also produce other flavors of leptons and quarks, to the extent
allowed by phase space. While the most common decay chain $b\to c\to s$ (for
example) involves two $W$ bosons, they produce muons of opposite charges, only
one of which is relevant to our signal. The corresponding numbers are compared
with the signal in table 9.

Fragmentation and Decay | FF [%] | BR [%] | FF$\times$BR [%] | Lifetime [s]
---|---|---|---|---
Signal: $c\to\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$ | 6.4 | 3.5 | 0.22 | $2.0\cdot 10^{-13}$
Background: |  |  |  |
$b$+$\bar{b}\to\mbox{($b$ and $\bar{b}$ hadrons)}\to\Lambda\mu^{+}\nu_{\mu}X$ |  |  | 0.59 | $1.6\cdot 10^{-12}$
_based on:_ $b\to\mbox{($b$-hadron)}\to\Lambda/\bar{\Lambda}+X$ |  |  | _5.9_ |
Table 9: Properties of the semileptonic $\Lambda_{c}^{+}$ signal and the
$b\bar{b}$ background: fragmentation fractions (FF) Lisovyi:2015uqa ,
branching ratios (BR) Workman:2022ynf  and lifetimes Workman:2022ynf . The
$c\bar{c}$ and $b\bar{b}$ production cross sections are similar for given cuts
on the jets. The number shown for the background is already summed over the
two jets in the event and involves the assumption that roughly $10\%$ of the
events in the $\Lambda X$ sample contain a $\mu^{+}$. The numbers shown are
before any discriminating cuts.

There are a number of ways to reduce the $b\bar{b}$ background significantly.
One can immediately veto events in which both the secondary vertex from the
$b$-hadron decay and the tertiary vertex from the subsequent charmed hadron
decay can be distinguished. Moreover, one can use the order-of-magnitude
difference between the $\Lambda_{c}^{+}$ and $b$-hadron lifetimes (see table
9) and apply an upper bound on the transverse displacement $d_{T}$ of the
secondary vertex. For example, requiring
$d_{T}<0.77~{}\mbox{mm}\times(p_{T}^{\rm jet}/115~{}\mbox{GeV})$ has an
efficiency of $40\%$ for the signal and $10\%$ for the background, giving
$60\%$ purity.101010To obtain these numbers we assumed that the jet $p_{T}$
fraction carried by the hadron is $0.5$ for the $\Lambda_{c}^{+}$ and $0.7$
for the $b$ hadrons. Various additional discriminants between $b$ and $c$ jets
exist and are used in $c$ tagging algorithms. For a $c$-jet efficiency of
$40\%$, a $b$-jet efficiency as low as $6\%$ is achieved in both ATLAS
ATLAS:2022qxm  and CMS CMS:2021scf , which would lead to $71\%$ purity in our
case. The performance of charm tagging should be even better when applied to
the $\Lambda_{c}^{+}$ sample than to an inclusive sample of charmed hadrons
because the lifetimes of the more common charmed hadrons are closer to the
$b$-hadron lifetimes. In addition, the properties of our decay of interest can
suppress the background further. In particular, one can require that the
displaced vertex formed by the muon track and the inferred $\Lambda$
trajectory (if it is distinct from the primary vertex) should not contain any
additional tracks. Based on these arguments, we will assume a charm tagging
efficiency of $\epsilon_{c}\approx 40\%$ and neglect the remaining background.

$\bm{c\bar{c}}$, semileptonic | polarization
---|---
$m_{jj}$ cut [GeV] |  $\sigma_{c\bar{c}}$ [pb] | $N_{c}$ | $r_{i}\Delta b^{\pm}_{i}$
no cut | 2500 | $8.3\times 10^{3}$ | 0.019
100 | 2200 | $7.2\times 10^{3}$ | 0.020
300 | 350 | $1.2\times 10^{3}$ | 0.051
500 | 66 | 160 | 0.14
750 | 14 | 22 | 0.37

spin correlations
---
$\sigma_{c\bar{c}}$ [pb] | $N_{c\bar{c}}$
2300 | 3
2000 | 2
250 |
48 |
9.3 |

Table 10: Run 2 cross sections (with trigger-motivated cuts) and signal event
counts (after the full selection) for measurements of $c$-quark polarization
and spin correlations in the semileptonic channel. The expected statistical
uncertainties for the polarization measurements are shown as well.

Table 10 shows the Run 2 cross sections and numbers of events available for
measurements of polarization ($N_{c}$) and spin correlations ($N_{c\bar{c}}$).
The cross sections are based on the single muon trigger acceptance for the
polarization and the double muon trigger for spin correlations. They were
obtained using a Pythia simulation of $pp\to c\bar{c}$, where we allowed the
charmed hadrons to decay only to final states with a muon and applied the
trigger cuts from table 5 to muons from charmed hadrons inside the two leading
jets. These cross sections do not include branching ratios. The numbers of
events were calculated as

$\displaystyle N_{c}$
$\displaystyle=\mathcal{L}\,\sigma_{c\bar{c}}\,f(c\to\Lambda_{c}^{+})\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})\operatorname{BR}(\Lambda\to
p\pi^{-})\,\epsilon_{\mu}\,\epsilon_{c}\,\epsilon_{\Lambda}\,,$ (32)
$\displaystyle N_{c\bar{c}}$
$\displaystyle=\mathcal{L}\,\sigma_{c\bar{c}}\,f^{2}(c\to\Lambda_{c}^{+})\operatorname{BR}^{2}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})\operatorname{BR}^{2}(\Lambda\to
p\pi^{-})\,\epsilon_{c,2}\,\epsilon^{2}_{\Lambda}\,,$ (33)

where $\epsilon_{\mu}\approx 50\%$ is the efficiency for the muon to pass the
isolation requirement of the single muon trigger ATLAS:2020gty ,
$\epsilon_{c,2}\equiv 2\epsilon_{c}-\epsilon_{c}^{2}$ is the efficiency for
any of the two jets to pass charm tagging, and $\epsilon_{\Lambda}$ includes
the decay radius and reconstruction efficiencies of the $\Lambda$, where we
again rely on the LRT algorithm ATL-PHYS-PUB-2017-014 ; ATL-PHYS-PUB-2019-013
; ATLAS:2023nze . As discussed in section 6.2.1, for Run 2, in the range of
$\Lambda$ decay radii $d_{T}$ up to $d_{T}^{\rm max}\approx 300$ mm, the
average reconstruction efficiency for each track is expected to be
$\epsilon_{\rm track}\approx 65\%$, and for the HL-LHC, in the range of up to
$d_{T}^{\rm max}\approx 400$ mm, it will be $\epsilon_{\rm track}\approx
80\%$. Averaging over the whole range is justified as the mean decay radius is
$\langle d_{T}\rangle\gg d_{T}^{\rm max}$. We estimate $\langle d_{T}\rangle$
with the assumption that $p_{T}^{\Lambda}\sim(0.5/3)\,p_{T}^{\rm jet}$, where
the factor of $0.5$ is the typical $c$-jet momentum fraction carried by the
$\Lambda_{c}^{+}$ and the division by $3$ roughly accounts for the $\Lambda$
being one out of three decay products. For the jet $p_{T}$, we used the rough
estimate $p_{T}^{\rm jet}\sim\max(\bar{p}_{T}^{\rm jet},m_{jj}^{\rm cut}/3)$
where $\bar{p}_{T}^{\rm jet}\approx 115$ GeV is the cut on jets that is
equivalent to the cuts of both the single and double muon triggers of Run 2,
like we already discussed in a slightly different context in section 5. For
the HL-LHC, the equivalent jet cuts are $85$ GeV and $80$ GeV for the single
and double muon triggers, respectively.

Table 10 also shows the expected statistical uncertainty of the polarization
measurements, while the number of events for spin correlation measurements in
Run 2 is too small to be useful. Table 11 presents the analogous results for
the HL-LHC, where spin correlation measurements become feasible. We see that
the semileptonic channel is superior to the hadronic channel, expect for
polarization measurements at high $m_{jj}$ at the HL-LHC, where the two
channels are comparable.

$\bm{c\bar{c}}$, semilep. | polarization
---|---
$m_{jj}$ cut [GeV] |  $\sigma_{c\bar{c}}$ [pb] | $N_{c}$ | $r_{i}\Delta b^{\pm}_{i}$
no cut | 9400 | $1.7\times 10^{6}$ | 0.001
100 | 6700 | $1.2\times 10^{6}$ | 0.002
300 | 620 | $9.8\times 10^{4}$ | 0.005
500 | 110 | $1.1\times 10^{4}$ | 0.017
750 | 23 | $1.6\times 10^{3}$ | 0.043
1000 | 5.4 | 280 | 0.10
1500 | 0.62 | 22 | 0.37

spin correlations
---
$\sigma_{c\bar{c}}$ [pb] | $N_{c\bar{c}}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
13000 | $2.4\times 10^{3}$ | 0.060 | 0.042
7700 | $1.5\times 10^{3}$ | 0.078 | 0.054
540 | 70 | 0.35 | 0.25
89 | 4 |  |
16 |  |  |
4.1 |  |  |
0.47 |  |  |

Table 11: HL-LHC cross sections (with trigger-motivated cuts) and signal event
counts (after the full selection) for measurements of $c$-quark polarization
and spin correlations in the semileptonic channel. The expected statistical
uncertainties are shown as well. The shorthand $\Delta c_{ij(\ell)}$ denotes
the uncertainties in $c_{ij}$ and $c_{\ell}$, which are of the same size.

####  6.3.3 Mixed Channel

We can also look at a mixed channel, where the decay in one of the jets is
semileptonic and in the other hadronic. This channel combines the ability to
trigger on a muon (with a low $p_{T}$ threshold) on one side with the higher
BR and a cleaner decay (without a neutrino) on the other side.

$h_{c}$ |  $f(c\to h_{c})$ [%] |  $\operatorname{BR}(h_{c}\to\mu^{+}X)$ [%]
---|---|---
$\Lambda_{c}^{+}$ | $6.4$ | $3.5$
$D^{+}$ | $22.7$ | $17.6$
$D^{0}$ | $61.8$ | $6.8$
$D_{s}^{+}$ | $8.2$ | $\approx 6$
Table 12: The fragmentation fractions Lisovyi:2015uqa  and branching ratios
Workman:2022ynf  of inclusive semi-muonic decays of the common charmed
hadrons.

We propose using the hadronic side of the event for polarization measurements
due to the ability to fully reconstruct the decay kinematics, without the need
to account for a neutrino. Moreover, one can then enjoy the _inclusive_
semimuonic decays of _all_ charmed hadrons (see table 12) in the second jet to
trigger the event. The expected number of signal events in the sample is then

$\displaystyle N_{c}$
$\displaystyle=\mathcal{L}\,\sigma_{c\bar{c}}\,f(c\to\Lambda_{c}^{+})\operatorname{BR}(\Lambda_{c}^{+}\to
pK^{-}\pi^{+})\,\epsilon_{\Lambda_{c}^{+}\to pK^{-}\pi^{+}}$
$\displaystyle\quad\times\sum_{h_{c}=\Lambda_{c}^{+},D^{+},D^{0},D_{s}^{+}}f(c\to
h_{c})\operatorname{BR}(h_{c}\to\mu^{+}X)\,\epsilon_{\mu}\,,$ (34)

where we assume an efficiency of $\epsilon_{\Lambda_{c}^{+}\to
pK^{-}\pi^{+}}\approx 25\%$ for the hadronic decay selection CMS:2023frs  and
$\epsilon_{\mu}\approx 50\%$ for passing the isolation requirement of the
single muon trigger ATLAS:2020gty . The resulting numbers of events are given
in table 13.

$\bm{c\bar{c}}$, mixed channel, polarization
---
$m_{jj}$ cut | Run 2 | HL-LHC
[GeV] |  $\sigma_{c\bar{c}}$ [pb] | $N_{c}$ | $r_{i}\Delta b^{\pm}_{i}$ |  $\sigma_{c\bar{c}}$ [pb] | $N_{c}$ | $r_{i}\Delta b^{\pm}_{i}$
no cut | 2500 | $1.6\times 10^{4}$ | $0.021-0.080$ | 9400 | $1.3\times 10^{6}$ | $0.002-0.006$
100 | 2200 | $1.3\times 10^{4}$ | $0.022-0.085$ | 6700 | $9.0\times 10^{5}$ | $0.003-0.007$
300 | 350 | 2600 | $0.055-0.21$ | 620 | $8.4\times 10^{4}$ | $0.009-0.024$
500 | 66 | 520 | $0.13-0.49$ | 110 | $1.4\times 10^{4}$ | $0.021-0.059$
750 | 14 | 86 | $\gtrsim 0.28$ | 23 | $3.1\times 10^{3}$ | $0.047-0.13$
1000 | 3.5 | 22 |  | 5.4 | 730 | $0.097-0.26$
1500 | 0.40 | 2 |  | 0.62 | 84 | $\gtrsim 0.29$
Table 13: Cross sections, expected numbers of signal events and expected
statistical uncertainties for the $c$-quark polarization measurements using
the hadronic decay in the mixed channel of $c\bar{c}$. We show a range of
values for the statistical uncertainties, corresponding to purities between
$100\%$ and $6.9\%$ for Run 2 and $13.8\%$ for the HL-LHC (see text).

The requirement of the muon in the second jet is also expected to remove part
of the background that is observed under the $\Lambda_{c}^{+}\to
pK^{-}\pi^{+}$ peak in the CMS measurement CMS:2023frs . While it will not
eliminate background coming from $c\bar{c}$ or $b\bar{b}$, the muon
requirement will strongly suppress the combinatorial background from high
cross section dijet final states without heavy flavors (e.g., $gg$). Since we
do not know the composition of the background observed by CMS, we present in
table 13 a range of values for the statistical uncertainties, for purities
varying between $100\%$ and the hadronic decay purity of $6.9\%$ for Run 2 (as
in the CMS measurement) and $13.8\%$ for the HL-LHC (recall the discussion in
section 6.3.1). The results in table 13 are comparable to those of the
semileptonic channel.

For spin correlation measurements, the expected number of signal events is

$\displaystyle N_{c\bar{c}}$
$\displaystyle=\mathcal{L}\,\sigma_{c\bar{c}}\,f^{2}(c\to\Lambda_{c}^{+})\operatorname{BR}(\Lambda_{c}^{+}\to
pK^{-}\pi^{+})\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})\operatorname{BR}(\Lambda\to
p\pi^{-})$ $\displaystyle\quad\times\epsilon_{\Lambda_{c}^{+}\to
pK^{-}\pi^{+}}\,\epsilon_{\Lambda}\,\epsilon_{\mu}\,\epsilon_{c}\,,$ (35)

where $\epsilon_{\Lambda_{c}^{+}\to pK^{-}\pi^{+}}\approx 25\%$ is the
hadronic decay efficiency CMS:2023frs , $\epsilon_{\Lambda}$ describes the
$\Lambda$ decay acceptance and reconstruction efficiency (estimated as in
section 6.3.2), $\epsilon_{\mu}\approx 50\%$ is the efficiency loss due to the
isolation requirement of the single muon trigger ATLAS:2020gty , and
$\epsilon_{c}$ is the charm tagging efficiency discussed in section 6.3.2.
Table 14 shows the expected numbers of events for Run 2 and the HL-LHC. While
the same single-muon trigger is used here as in the polarization measurements,
the cross sections are higher by a factor of $2$ because either a $\mu^{+}$
from the $c$ jet or a $\mu^{-}$ from the $\bar{c}$ jet can trigger the event.
This also affects the equivalent $p_{T}^{\rm jet}$ values we use to estimate
the $\Lambda$ displacement acceptance, which become $\bar{p}_{T}^{\rm jet}=97$
GeV for Run 2 and $\bar{p}_{T}^{\rm jet}=72$ GeV for the HL-LHC.

$\bm{c\bar{c}}$, mixed channel, spin correlations
---
$m_{jj}$ cut | Run 2 | HL-LHC
[GeV] |  $\sigma_{c\bar{c}}$ [pb] | $N_{c\bar{c}}$ |  $\sigma_{c\bar{c}}$ [pb] | $N_{c\bar{c}}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
no cut | 5000 | 19 | 19000 | $3.9\times 10^{3}$ | $0.072-0.19$ | $0.050-0.13$
100 | 4300 | 16 | 13000 | $2.7\times 10^{3}$ | $0.085-0.23$ | $0.060-0.16$
300 | 710 | 2 | 1200 | 200 | $\gtrsim 0.32$ | $\gtrsim 0.22$
Table 14: Cross sections, expected numbers of signal events and expected
statistical uncertainties for the $c\bar{c}$ spin correlation measurements in
the mixed channel. We show a range of values for the statistical
uncertainties, corresponding to purities between $100\%$ and $13.8\%$ (see
text).

While the semileptonic decay selection is almost background-free (as discussed
in section 6.3.2), various processes can mimic the hadronic decay. If the
background observed under the hadronic $\Lambda_{c}^{+}$ peak in the inclusive
CMS sample CMS:2023frs  is primarily intrinsic (i.e., from $c\bar{c}$), it
will not be affected by the semileptonic selection on the other side of the
event and our rough purity estimates will be $6.9\%$ in Run 2 and $13.8\%$ at
the HL-LHC. If, on the other hand, the background is mostly extrinsic and
comes from a process such as $pp\to gg$, where the jets rarely contain a muon
and a $\Lambda$, the semileptonic selection will eliminate most of it and the
sample will be almost background-free. For Run 2, the number of events is too
small for a meaningful measurement regardless of the background. For the HL-
LHC, we show in table 14 a range of values for the expected precision
corresponding to the above range of possible purity values. The expected
precision is comparable to that of the semileptonic channel.

###  6.4 Analyses of $pp\to b\bar{b}$

We propose using the $b\to\Lambda_{b}\to X_{c}\mu^{-}\bar{\nu}_{\mu}$ process,
with the muon-based triggers from table 5, to measure the polarization and
spin correlations in $b\bar{b}$ events.

As introduced in section 3, we will consider three types of selection:
Inclusive, Semi-inclusive, and Exclusive. In the Inclusive Selection, no
attempt is made to reduce the intrinsic background due to semileptonic
$B$-meson decays. Such a selection can still be competitive because of its
high signal efficiency. In the Semi-Inclusive selection, we require in
addition to the muon the reconstruction of a $\Lambda$ baryon (via its decay
$\Lambda\to p\pi^{-}$) originating from the vicinity of the displaced vertex.
This reduces the $B$-meson background. The last selection type is the
Exclusive Selection. In this selection, in addition to the muon, we require a
full reconstruction of one of $\Lambda_{c}^{+}$ decays by a set of tracks
consistent with originating from a common vertex. This significantly
suppresses the $B$-meson background too. Another advantage of the complete
reconstruction of a $\Lambda_{c}^{+}$ decay is that the kinematics of the
$\Lambda_{b}$ decay as a whole can then be reconstructed more accurately. In
table 15 we list all the decay channels relevant to the signal in each
selection type.111111In the Exclusive Selection, we include decays with the
$\Sigma^{\pm}$ baryons in the final state. While these particles will often
decay before passing through the entire tracker, reconstruction of such short
tracks is possible ATLAS:2017oal ; ATLAS:2022rme ; CMS:2019ybf ; CMS:2020atg ;
CMS:2023mny .

Selection | Decay Modes | Branching Ratio
---|---|---
Inclusive | $\Lambda_{b}\to X_{c}\mu^{-}\bar{\nu}_{\mu}$ | 11%
Semi-inclusive | $\Lambda_{c}^{+}\to\Lambda X$ | 38%
$\Lambda\to p\pi^{-}$ | 64%
Exclusive | $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ | 6.3%
$\Lambda_{c}^{+}\to\Lambda\pi^{+}\to p\pi^{-}\pi^{+}$ | 0.8%
$\Lambda_{c}^{+}\to pK_{S}\to p\pi^{-}\pi^{+}$ | 1.1%
$\Lambda_{c}^{+}\to\Lambda\pi^{+}\pi^{+}\pi^{-}\to p\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ | 2.3%
$\Lambda_{c}^{+}\to pK_{S}\pi^{+}\pi^{-}\to p\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ | 1.1%
$\Lambda_{c}^{+}\to\Sigma^{+}\pi^{+}\pi^{-}$ | 4.5%
$\Lambda_{c}^{+}\to\Sigma^{-}\pi^{+}\pi^{+}$ | 1.9%
total | 18%
Table 15: Decay modes relevant to the three $\Lambda_{b}$ selections and their
branching ratios Workman:2022ynf . The Semi-inclusive and Exclusive selections
are done on top of the Inclusive selection.

####  6.4.1 Inclusive Selection

The main requirement of this selection is the presence a muon in the jet,
similar to _soft muon $b$ tagging_ CMS:2012feb ; ATLAS:2015thz ; ATLAS:2016lxn
; CMS:2017wtu ; ATLAS:2021piz ; ATLAS:2022jbw .

To suppress extrinsic backgrounds with prompt or mildly displaced (in
particular, $c\bar{c}$) muons, we assume applying a $b$ tagging algorithm that
may use the muon impact parameter significance, or $p_{T}^{\rm rel}$ (the
component of muon momentum transverse to the jet axis) ATLAS:2015thz , or
other variables. For our estimates, we will assume the $b$ tagging efficiency
to be $\epsilon_{b}\approx 80\%$ and allow ourselves to neglect the above
backgrounds.

Another source of extrinsic background is $pp\to t\bar{t}\to b\bar{b}X$. This
background will be small since the $t\bar{t}$ cross section is smaller than
the $b\bar{b}$ cross section for any fixed $b\bar{b}$ invariant mass. It can
be suppressed further by vetoing the presence of additional objects such as
isolated leptons or multiple energetic jets.

The intrinsic background is the more prominent one. An important contribution
is made by the semileptonic $B$-meson decays via the same underlying process
as in the signal, $b\to c\mu^{-}\bar{\nu}_{\mu}$, but we do not attempt to
reduce it in the inclusive selection as we want to keep the signal efficiency
high. Another source of intrinsic background is the decay chain $b\to
cf\bar{f}^{\prime}$, $c\to s\mu^{+}\nu_{\mu}$. The fact that the muon in this
case has the wrong charge cannot be used to eliminate this background because
in the inclusive selection we do not know which jet is coming from the $b$ and
which from the $\bar{b}$. There is a different way to reduce this background
significantly. The muons emitted in $b\to c$ transitions (as in our signal)
are usually more energetic than those emitted in the $c\to s$ transitions in
the $b\to c\to s$ chain (which are background), so it can be useful to look
into the ratio between the muon and jet $p_{T}$, $z=p_{T}^{\mu}/p_{T}^{\rm
jet}$. Figure 4 shows the $z$ distributions for muons from both $b$ and $c$
hadrons in $pp\to b\bar{b}$ events with $\sqrt{s}=13$ TeV simulated in Pythia
with the single-muon trigger cuts. In a sample without any further cuts, the
contribution of muons from $c$-hadron decays is small to begin with, due to
their lower efficiency to pass the trigger $p_{T}$ cut. For a dijet invariant
mass cut of $m_{jj}>1000$ GeV, their relative contribution is significant, but
concentrated at lower values of $z$ than the muons from $b$ hadrons. Based on
these results, we will apply a cut of $z>0.2$ in all cases. In the example
with the $m_{jj}>1000$ GeV cut, this $z$ cut results in decent efficiency and
purity ($\epsilon_{z}\approx 77\%$, $f_{z}\approx 84\%$), while the impact on
the no-cut case is small ($\epsilon_{z}\approx 99\%$, $f_{z}\approx 94\%$).
From these results, we find that the $c$ hadrons background is small and we
will neglect it.

Figure 4: The momentum fraction $z$ of the muons relative to their respective
jets. The graphs compare the distributions for muons originating directly from
$b$ hadrons with those from $c$ hadrons in $b$ jets, with no invariant mass
cut (left) and with $m_{jj}>1000$ GeV (right). (The plot normalizations do not
include the branching ratios.)

$m_{jj}$ cut [GeV] |  $\sigma_{b\bar{b}}$ [pb] | $N^{i}_{b}$ | $N^{s}_{b}$ | $N^{e}_{b}$
---|---|---|---|---
no cut | 9400 | $4.0\times 10^{6}$ | $2.0\times 10^{5}$ | $3.6\times 10^{5}$
100 | 7200 | $3.1\times 10^{6}$ | $1.5\times 10^{5}$ | $2.8\times 10^{5}$
300 | 560 | $2.3\times 10^{5}$ | $1.0\times 10^{4}$ | $2.1\times 10^{4}$
500 | 82 | $3.1\times 10^{4}$ | $1.0\times 10^{3}$ | $2.8\times 10^{3}$
750 | 14 | $4.8\times 10^{3}$ | 120 | 430
1000 | 3.4 | $1.1\times 10^{3}$ | 22 | 100
1500 | 0.34 | 100 |  | 9
parked data |  | $3.5\times 10^{8}$ | $3.6\times 10^{7}$ | $3.1\times 10^{7}$
purity $f$ [%] |  | $7.4$ | $57$ | $66$
(a) Polarization

$m_{jj}$ cut | $\sigma_{b\bar{b}}$ | $N^{ii}_{b\bar{b}}$ | $N^{ss}_{b\bar{b}}$ | $N^{ee}_{b\bar{b}}$ | $N^{is}_{b\bar{b}}$ | $N^{ie}_{b\bar{b}}$ | $N^{se}_{b\bar{b}}$
---|---|---|---|---|---|---|---
[GeV] |  [pb]
no cut | 10000 | $8.0\times 10^{4}$ | 200 | 640 | $8.1\times 10^{3}$ | $1.4\times 10^{4}$ | 730
100 | 5900 | $4.7\times 10^{4}$ | 121 | 380 | $4.8\times 10^{3}$ | $8.5\times 10^{3}$ | 430
300 | 340 | $2.7\times 10^{3}$ | 4 | 21 | 230 | 490 | 20
500 | 46 | 360 |  | 2 | 20 | 65 | 1
parked data |  | $1.1\times 10^{6}$ | $1.1\times 10^{4}$ | 8700 | $2.2\times 10^{5}$ | $1.9\times 10^{5}$ | $2.0\times 10^{4}$
purity $f$ [%] |  | $0.55$ | $32$ | $44$ | $4.2$ | $4.9$ | $38$
(b) Spin correlations

Table 16: Run 2 cross sections (with trigger-motivated cuts) and signal event
counts (after the full selection) for measurements of (a) $b$-quark
polarization, and (b) $b\bar{b}$ spin correlations in the inclusive, semi-
inclusive, and exclusive selection channels. The last row shows the expected
sample purity.

The numbers of events available for measurements of polarization and spin
correlations, respectively, are given by

$\displaystyle N_{b}^{i}$
$\displaystyle=\mathcal{L}\,\sigma_{b\bar{b}}\,f(b\to\Lambda_{b})\operatorname{BR}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\,\epsilon_{b}\,\epsilon_{z}\,\epsilon_{\mu}\,,$
(36) $\displaystyle N_{b\bar{b}}^{ii}$
$\displaystyle=\mathcal{L}\,\sigma_{b\bar{b}}\,f^{2}(b\to\Lambda_{b})\operatorname{BR}^{2}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\,\epsilon_{b,2}\,\epsilon_{z,2}\,,$ (37)

where $\epsilon_{\mu}\approx 50\%$ is the efficiency for the muon to pass the
isolation requirement of the single muon trigger ATLAS:2020gty ,
$\epsilon_{b,2}\equiv 2\epsilon_{b}-\epsilon_{b}^{2}$ is the efficiency for
any of the two jets to pass the $b$ tagging condition, and
$\epsilon_{z,2}\equiv 2\epsilon_{z}-\epsilon_{z}^{2}$ is the efficiency for
any of the two muons to pass the momentum fraction cut.121212Once one of the
muons is classified as signal, the second muon can be classified as signal or
background based on its charge. The cross sections were obtained using a
Pythia simulation of $pp\to b\bar{b}$, where we allowed the bottom hadrons to
decay only to final states with a muon (not including cases in which the muon
comes from a charmed hadron) and applied the single or double muon trigger
cuts from table 5 to muons from bottom hadrons inside the two leading jets.
These cross sections do not include branching ratios. The resulting numbers of
events are shown in tables 16 and 17 for Run 2 and the HL-LHC, respectively,
along with the numbers obtained for the other selections. While here we
considered the standard muon-based triggers, table 16 also presents numbers
corresponding to the CMS parked dataset. It will be discussed separately in
section 6.4.5.

$m_{jj}$ cut [GeV] |  $\sigma_{b\bar{b}}$ [pb] | $N^{i}_{b}$ | $N^{s}_{b}$ | $N^{e}_{b}$
---|---|---|---|---
no cut | 33000 | $3.0\times 10^{8}$ | $3.3\times 10^{7}$ | $2.7\times 10^{7}$
100 | 18300 | $1.7\times 10^{8}$ | $1.8\times 10^{7}$ | $1.5\times 10^{7}$
300 | 990 | $8.2\times 10^{6}$ | $7.3\times 10^{5}$ | $7.4\times 10^{5}$
500 | 126 | $9.3\times 10^{5}$ | $6.3\times 10^{4}$ | $8.4\times 10^{4}$
750 | 21 | $1.5\times 10^{5}$ | $7.6\times 10^{3}$ | $1.4\times 10^{4}$
1000 | 5.6 | $3.7\times 10^{4}$ | $1.6\times 10^{3}$ | $3.3\times 10^{3}$
1500 | 0.49 | $2.9\times 10^{3}$ | 94 | 260
2000 | 0.088 | 490 | 12 | 44
purity $f$ [%] |  | $7.4$ | $57$ | $66$
(a) Polarization

$m_{jj}$ cut | $\sigma_{b\bar{b}}$ | $N^{ii}_{b\bar{b}}$ | $N^{ss}_{b\bar{b}}$ | $N^{ee}_{b\bar{b}}$ | $N^{is}_{b\bar{b}}$ | $N^{ie}_{b\bar{b}}$ | $N^{se}_{b\bar{b}}$
---|---|---|---|---|---|---|---
[GeV] |  [pb]
no cut | 39000 | $6.7\times 10^{6}$ | $8.1\times 10^{4}$ | $5.4\times 10^{4}$ | $1.5\times 10^{6}$ | $1.2\times 10^{6}$ | $1.3\times 10^{5}$
100 | 15000 | $2.6\times 10^{6}$ | $3.1\times 10^{4}$ | $2.1\times 10^{4}$ | $5.7\times 10^{5}$ | $4.7\times 10^{5}$ | $5.1\times 10^{4}$
300 | 570 | $9.6\times 10^{4}$ | 610 | 780 | $1.5\times 10^{4}$ | $1.7\times 10^{4}$ | $1.4\times 10^{3}$
500 | 74 | $1.2\times 10^{4}$ | 35 | 98 | $1.3\times 10^{3}$ | $2.2\times 10^{3}$ | 120
750 | 12 | $2.0\times 10^{3}$ | 3 | 16 | 150 | 360 | 13
1000 | 2.9 | 460 |  | 3 | 27 | 82 | 2
purity $f$ [%] |  | $0.55$ | $32$ | $44$ | $4.2$ | $4.9$ | $38$
(b) Spin correlations

Table 17: HL-LHC cross sections (with trigger-motivated cuts) and signal event
counts (after the full selection) for measurements of (a) $b$-quark
polarization, and (b) $b\bar{b}$ spin correlations in the inclusive, semi-
inclusive, and exclusive selection channels. The last row shows the expected
sample purity.

By accounting for the fragmentation fractions and semi-muonic decay branching
ratios of the $\bar{B}^{0}$, $B^{-}$, and $\bar{B}_{s}^{0}$ mesons (see table
18), we estimate the sample purity to be $f\approx 7.4\%$ for a single jet.
For spin correlation measurements, which involve two jets, this number is
squared, giving the purity of $f\approx 0.55\%$. Such a low purity can be
problematic as it means that even a small mismodelling of the background can
ruin the measurement. The next two selection strategies that we discuss offer
much higher purities, at the expense of statistics.

Process | FF [%] | BR [%]
---|---|---
$b\to\Lambda_{b}\to\mu^{-}\bar{\nu}_{\mu}X$ | 7.0 | 11
$b\to\bar{B}^{0}\to\mu^{-}\bar{\nu}_{\mu}X$ | 40.1 | 10.3
$b\to B^{-}\to\mu^{-}\bar{\nu}_{\mu}X$ | 40.1 | 11.0
$b\to\bar{B}^{0}_{s}\to\mu^{-}\bar{\nu}_{\mu}X$ | 10.4 | 10.2
Table 18: The fragmentation fractions (FF) Galanti:2015pqa ; HFLAV:2014fzu
and branching ratios (BR) Workman:2022ynf  of the signal ($\Lambda_{b}$) and
the intrinsic background in the Inclusive Selection.

####  6.4.2 Exclusive Selection

The exclusive selection starts with the inclusive selection and requires, in
addition, a fully reconstructed $\Lambda_{c}^{+}$ decay, to eliminate most of
the background due to the semileptonic $B$-meson decays, which usually produce
charmed mesons rather than baryons.

Some of the $\Lambda_{c}^{+}$ decay channels listed in table 15 come with an
efficiency price: $K_{S}$ and $\Lambda$ can decay too far to be reconstructed,
$\Sigma^{\pm}$ can produce a kinked track, and the channels with five tracks
need each track to be successfully reconstructed. To account for this, as a
rough estimate, we will assume a reconstruction efficiency of $\epsilon_{\rm
reco}\approx 50\%$ for the $\Lambda_{c}^{+}$ decays.

The numbers of events available for measurements of polarization and spin
correlations, respectively, using the exclusive channel are given by

$\displaystyle N_{b}^{e}$
$\displaystyle=\mathcal{L}\,\sigma_{b\bar{b}}\,f(b\to\Lambda_{b})\operatorname{BR}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\operatorname{BR}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{b}\,\epsilon_{z}\,\epsilon_{\rm reco}\,\epsilon_{\mu}\,,$
(38) $\displaystyle N_{b\bar{b}}^{ee}$
$\displaystyle=\mathcal{L}\,\sigma_{b\bar{b}}\,f^{2}(b\to\Lambda_{b})\operatorname{BR}^{2}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\operatorname{BR}^{2}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{b,2}\,\epsilon_{z,2}\,\epsilon_{\rm reco}^{2}\,,$ (39)

where $\operatorname{BR}(\Lambda_{c}^{+}\to{\rm reco.})\approx 18\%$ is the
branching fraction of the reconstructible decay modes of the $\Lambda_{c}^{+}$
from table 15 and $\epsilon_{\mu}\approx 50\%$ is the efficiency for the muon
to pass the isolation requirement of the single muon trigger ATLAS:2020gty .
The resulting numbers of events are included in tables 16 and 17.

Process | FF [%] | BR [%]
---|---|---
$b\to\Lambda_{b}\to\Lambda_{c}^{+}X$ | 7.0 | $\approx 100$
$b\to B^{-}\to\Lambda_{c}^{+}X$ | 40.1 | $2.8^{+1.1}_{-0.9}$
$b\to\bar{B}^{0}\to\Lambda_{c}^{+}X$ | 40.1 | $5.0^{+2.1}_{-1.5}$
$b\to\bar{B}_{s}^{0}\to\Lambda_{c}^{+}X$ | 10.4 | unknown
$\bar{b}\to B^{+}\to\Lambda_{c}^{+}X$ | 40.1 | $2.1^{+0.9}_{-0.6}$
$\bar{b}\to B^{0}\to\Lambda_{c}^{+}X$ | 40.1 | $<3.1$
$\bar{b}\to B_{s}^{0}\to\Lambda_{c}^{+}X$ | 10.4 | unknown
Table 19: The fragmentation fractions (FF) Galanti:2015pqa ; HFLAV:2014fzu
and branching ratios (BR) Workman:2022ynf  for $\Lambda_{c}^{+}$ production
from the $\Lambda_{b}$ (signal) and $B$ mesons (background).

A remaining background in this channel is the semileptonic decays of $B$
mesons to final states with a $\Lambda_{c}^{+}$ baryon. While the branching
ratios of the meson decays to baryons are small, their contribution is
enhanced by their large fragmentation fractions compared with that of the
$\Lambda_{b}$, as shown in table 19. The branching ratios for $B$-meson decays
involving a $\mu^{-}$ in addition to the $\Lambda_{c}^{+}$ have not been
measured, but such decays definitely exist. In the case of the $b$ initial
state, the muon can be produced in the $b\to(c\to\Lambda_{c}^{+})$ transition,
while in the case of the $\bar{b}$ initial state is it produced following the
$\bar{b}\to\bar{c}\,(c\to\Lambda_{c}^{+})\,\bar{s}$ chain, in the subsequent
$\bar{c}\to\bar{s}$ transition. Events with muons from the latter source will
largely fail the cut on the muon momentum fraction $z$ discussed in section
6.4.1, so only the $B^{-}$, $\bar{B}^{0}$ and $\bar{B}_{s}^{0}$ decays from
table 19 actually contribute to the background. Summing up their contributions
(while assigning $\bar{B}_{s}^{0}$ the average of the $B^{-}$ and
$\bar{B}^{0}$ BRs) and assuming the probability of having a muon to be similar
to the $\Lambda_{b}$ case, where the muon is also produced in a $b\to c$
transition, we obtain a sample purity of $f=66\%$. Since the muons and
neutrinos in these background processes are produced in the decays of spinless
mesons, their impact on the angular distributions will be trivial.

####  6.4.3 Semi-inclusive Selection

The semi-inclusive selection starts with the inclusive selection and requires,
in addition, the presence of a reconstructed $\Lambda$ baryon. This
requirement eliminates most of the contributions from $B$-meson decays while
accepting a large fraction of $\Lambda_{b}$ decays since $\Lambda_{b}$ decays
usually proceed through a $\Lambda_{c}^{+}$, which in turn produces a
$\Lambda$ in about $38\%$ of the cases Workman:2022ynf .

The numbers of events in this channel, for measurements of polarization and
spin correlations, respectively, are given by

$\displaystyle N_{b}^{s}$
$\displaystyle=\mathcal{L}\,\sigma_{b\bar{b}}\,f(b\to\Lambda_{b})\operatorname{BR}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}(\Lambda\to
p\pi^{-})\,\epsilon_{b}\,\epsilon_{z}\,\epsilon_{\Lambda}\,\epsilon_{\mu}\,,$
(40) $\displaystyle N_{b\bar{b}}^{ss}$
$\displaystyle=\mathcal{L}\,\sigma_{b\bar{b}}\,f^{2}(b\to\Lambda_{b})\operatorname{BR}^{2}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\operatorname{BR}^{2}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}^{2}(\Lambda\to
p\pi^{-})\,\epsilon_{b,2}\,\epsilon_{z,2}\,\epsilon_{\Lambda}^{2}\,,$ (41)

where $\epsilon_{\mu}\approx 50\%$ is the efficiency for the muon to pass the
isolation requirement of the single muon trigger ATLAS:2020gty  and
$\epsilon_{\Lambda}$ is the $\Lambda$ decay reconstruction efficiency, where
we base our estimates on the ATLAS reconstruction with the LRT algorithm
discussed in section 6.2.1. The mean track reconstruction efficiency is $\sim
65\%$ ($\sim 80\%$) in the range of decay radii up to 300 mm (400 mm) for Run
2 (HL-LHC). The typical $b$-jet $p_{T}$ corresponding to the muons in the jets
passing the double-muon trigger is $\bar{p}_{T}^{\rm jet}\approx 79$ GeV ($59$
GeV) for Run 2 (HL-LHC). For the single-muon trigger it is $81$ GeV ($62$ GeV)
for Run 2 (HL-LHC). By an analytical calculation we find that with131313The
factor of $0.7$ is the typical $b$-jet momentum fraction carried by a $b$
hadron, and the $1/3^{2}$ factor is a rough estimate of the momentum splitting
in a typical semileptonic $\Lambda_{b}$ decay and the subsequent
$\Lambda_{c}^{+}$ decay. $p_{T}^{\Lambda}\sim(0.7/3^{2})\,p_{T}^{\rm jet}$,
54% of the $\Lambda$ satisfy the decay radius condition for the Run 2 (with
$p_{T}^{\rm jet}=79$ GeV) and 76% satisfy it for the HL-LHC (with $p_{T}^{\rm
jet}=59$ GeV). This efficiency will be lowered significantly when we apply
high invariant mass cuts. We account for that by taking $p_{T}^{\rm
jet}\sim\max(\bar{p}_{T}^{\rm jet},m_{jj}^{\rm cut}/3)$ in our estimate of the
typical $p_{T}^{\Lambda}$. The resulting numbers of events are included in
tables 16 and 17.

The background in this channel can be estimated using the inclusive
measurement of $\Lambda$ baryon production in $b$-hadron decays, after
subtracting the signal contribution. We have already discussed this
measurement, eq. (31), in section 6.3.2 in the context of the background for
the semileptonic $c\bar{c}$ channel. Following similar arguments, including an
assumption of a $10\%$ probability of having a $\mu^{-}$ alongside the
$\Lambda$ (as a rough estimate), we obtain the number _“Total”_ in the second
row in table 20. Part of it is the signal contribution, which is shown in the
first row of the table. After subtracting it, we are left with the background,
which is shown in the third row. The last row shows an estimate for the part
of the background coming from processes involving the $\Lambda_{c}^{+}$ that
are listed in table 19.141414For the $B^{0}$ BR, we took its upper bound. For
the $B_{s}^{0}$ and $\bar{B}_{s}^{0}$, we took the BR to be the average of the
other BRs. This contribution seems to account for the majority of the
background (although one should keep in mind the large uncertainties in the
input data and the assumptions made). As we discussed in section 6.4.2, only
the $b$-initiated processes from table 19, which account for roughly $60\%$ of
the background contributions in that table, will pass the cut on the muon
$p_{T}$ fraction $z$. After subtracting that contribution we obtain a sample
purity of $f\approx 57\%$.

| Process | FF [%] | BR [%] | FF$\times$BR [%]
---|---|---|---|---
_Signal_ | $b\to\Lambda_{b}\to\Lambda\mu^{-}\bar{\nu}_{\mu}X$ | 7.0 | 3.8 | 0.29
_Total_ |  $b$+$\bar{b}\to\mbox{(hadrons)}\to\Lambda\mu^{-}\bar{\nu}_{\mu}X$ |  |  | 0.59
_Background_ |  |  |  | 0.30
via $\Lambda_{c}^{+}$ |  $b$+$\bar{b}\to\mbox{(mesons)}\to(\Lambda_{c}^{+}\to\Lambda X)\mu^{-}\bar{\nu}_{\mu}X$ |  |  | 0.20
Table 20: Properties of the $b\bar{b}$ signal and background in the semi-
inclusive channel: fragmentation fractions (FF) and branching ratios (BR). The
number shown for _Total_ is already summed over the two jets in the event and
involves the assumption that roughly $10\%$ of the events in the $\Lambda X$
sample contain a $\mu^{-}$. The _Background_ number is obtained as the
difference between _Total_ and _Signal_. The last row shows the part of the
background that comes from the processes involving the $\Lambda_{c}^{+}$ shown
in table 19.

####  6.4.4 Mixed Selections

For spin correlation measurements, one can also consider one type of selection
(inclusive, semi-inclusive, or exclusive) for one of the jets and another type
for another. The numbers of events for these mixed selections are given by

$\displaystyle N_{b\bar{b}}^{is}$
$\displaystyle=2\,\mathcal{L}\,\sigma_{b\bar{b}}\,f^{2}(b\to\Lambda_{b})\operatorname{BR}^{2}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}(\Lambda\to
p\pi^{-})\,\epsilon_{b,2}\,\epsilon_{z,2}\,\epsilon_{\Lambda}\;,$ (42)
$\displaystyle N_{b\bar{b}}^{ie}$
$\displaystyle=2\,\mathcal{L}\,\sigma_{b\bar{b}}\,f^{2}(b\to\Lambda_{b})\operatorname{BR}^{2}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\operatorname{BR}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{b,2}\,\epsilon_{z,2}\,\epsilon_{\rm reco}\;,$ (43)
$\displaystyle N_{b\bar{b}}^{se}$
$\displaystyle=2\,\mathcal{L}\,\sigma_{b\bar{b}}\,f^{2}(b\to\Lambda_{b})\operatorname{BR}^{2}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}(\Lambda\to p\pi^{-})$
$\displaystyle\quad\times\operatorname{BR}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{b,2}\,\epsilon_{z,2}\,\epsilon_{\Lambda}\,\epsilon_{\rm
reco}$ (44)

for the inclusive/semi-inclusive, inclusive/exclusive and semi-
inclusive/exclusive selections, respectively. We included a factor of two
since different selections can be applied to each of the jets interchangeably
and contribute the same. The resulting numbers are included in tables 16 and
17.

####  6.4.5 CMS Parked Data

It is also interesting to consider the CMS _parked_ $b$ dataset, which was
collected during part of Run 2 using a special data acquisition strategy CMS-
DP-2019-043 ; Bainbridge:2020pgi .151515Similar data parking was done during
part of Run 3 CMS:2023gfb . We are unaware of the specifics of data parking
that might be planned for the HL-LHC, so we will not address that case. We
hope that the analyses we propose in this paper, for the $b$, $c$, and $s$
quarks, will add to the motivation for such data streams. The dataset contains
about $N_{0}=10^{10}\,$ $pp\to b\bar{b}$ events, in which one side of the
event includes a displaced muon that triggered the event, with a $p_{T}$
threshold that varied between $7$ and $12$ GeV. We will consider using these
semileptonic decays with muons for the $b$ and $\bar{b}$ polarization
measurements. We will also consider spin correlation measurements, in which we
will demand a semileptonic decay with a muon also on the other side of the
event.

The number of events available for the polarization measurements with the
inclusive selection is

$N_{b}^{i}\simeq\frac{1}{2}\,f(b\to\Lambda_{b})\,N_{0}\approx 3.5\times
10^{8}\,,$ (45)

where we accounted for the fact that only half of the triggering decays come
from a $b$ (while the other half are from a $\bar{b}$) and took the
approximation that the inclusive semileptonic branching ratios and muon
kinematics are approximately the same for the different $b$ hadrons. We
neglect the contribution of muons from the $b\to c\to\mu$ chain relative to
the contribution of the direct $b\to\mu$ process since the more energetic
muons from the direct decay pass the trigger threshold much more easily,
similar to what we have seen in figure 4 (left). For the semi-inclusive and
exclusive selections, we compute the number of events as follows:

$\displaystyle N_{b}^{s}$
$\displaystyle=N_{b}^{i}\,\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}(\Lambda\to p\pi^{-})\,\epsilon_{\Lambda}\;,$ (46)
$\displaystyle N_{b}^{e}$
$\displaystyle=N_{b}^{i}\,\operatorname{BR}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{\rm reco}\;.$ (47)

For the semi-inclusive selection, our rough estimate for the $\Lambda$
reconstruction efficiency, $\epsilon_{\Lambda}$, will be as follows. Since the
muon that triggers the event, with a $p_{T}$ threshold that varied between $7$
and $12$ GeV CMS-DP-2019-043 ; Bainbridge:2020pgi , is produced in the
$\Lambda_{b}$ decay together with the $\Lambda_{c}^{+}$, which subsequently
decays to the $\Lambda$ and additional particles, we estimate
$p_{T}^{\Lambda}$ to be around a few GeV. As a result, the $\Lambda$ baryons
will usually decay sufficiently early to be reconstructed by an analogue of
the LRT algorithm ATL-PHYS-PUB-2017-014 ; ATL-PHYS-PUB-2019-013 ;
ATLAS:2023nze , but not at the shortest distances where the efficiency is
maximal. We will therefore take $\epsilon_{\Lambda}\simeq\epsilon_{\rm
track}^{2}$ with $\epsilon_{\rm track}\approx 65\%$ as a ballpark figure.

Our rough estimate for the number of events for the spin correlation
measurement with the inclusive selection on both sides will be

$N_{b\bar{b}}^{ii}\simeq
f^{2}(b\to\Lambda_{b})\operatorname{BR}(\Lambda_{b}\to
X_{c}\mu^{-}\bar{\nu}_{\mu})\,{\cal A}\epsilon\,N_{0}\approx 1.1\times
10^{6}\,,$ (48)

where we took a ballpark figure of ${\cal A}\epsilon\sim 0.2$ for the
acceptance times efficiency on the non-triggering side of the event (including
the acceptance in $\eta$). The contribution of $b\to c\to\mu$ decays from that
side of the event can be eliminated based on the muon charge (compared with
the triggering muon). For the other selections, we have

$\displaystyle N_{b\bar{b}}^{ss}$
$\displaystyle=N_{b\bar{b}}^{ii}\,\operatorname{BR}^{2}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}^{2}(\Lambda\to p\pi^{-})\,\epsilon_{\Lambda}^{2}\;,$ (49)
$\displaystyle N_{b\bar{b}}^{ee}$
$\displaystyle=N_{b\bar{b}}^{ii}\,\operatorname{BR}^{2}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{\rm reco}^{2}\;,$ (50) $\displaystyle N_{b\bar{b}}^{is}$
$\displaystyle=2N_{b\bar{b}}^{ii}\,\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}(\Lambda\to p\pi^{-})\,\epsilon_{\Lambda}\;,$ (51)
$\displaystyle N_{b\bar{b}}^{ie}$
$\displaystyle=2N_{b\bar{b}}^{ii}\,\operatorname{BR}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{\rm reco}\;,$ (52) $\displaystyle N_{b\bar{b}}^{se}$
$\displaystyle=2N_{b\bar{b}}^{ii}\,\operatorname{BR}(\Lambda_{c}^{+}\to\Lambda
X)\operatorname{BR}(\Lambda\to
p\pi^{-})\operatorname{BR}(\Lambda_{c}^{+}\to{\rm
reco.})\,\epsilon_{\Lambda}\,\epsilon_{\rm reco}\;.$ (53)

####  6.4.6 Expected Precision

The expected precision of the $b\bar{b}$ polarization and spin correlation
measurements for either a muon or neutrino spin analyzer can be computed using
eqs. (23)–(25). As indicated in table 1, the neutrino in the $\Lambda_{b}$
decay has a larger spin analyzing power than the muon and therefore has the
potential to provide higher precision. Since the neutrino is not observed
directly, a reconstruction is required, and it requires certain
approximations, as discussed at the end of section 3. However, the decay
reconstruction is needed anyway, even if the muon is used as the spin
analyzer. We will therefore present the statistical uncertainties that can be
obtained with the neutrino. The analogous results for the muon can be obtained
by dividing the polarization uncertainties by $|\alpha_{\mu^{-}}|$ and the
spin correlation uncertainties by $\alpha_{\mu^{-}}^{2}$, where
$\alpha_{\mu^{-}}\approx-0.26$.

channel $\to$ | inclusive | inclusive/inclusive | inclusive/exclusive
---|---|---|---
$m_{jj}$ cut [GeV] | $r_{i}\Delta b^{\pm}_{i}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
no cut | 0.003 | 0.14 | 0.10 | 0.11 | 0.079
100 | 0.004 | 0.18 | 0.13 | 0.15 | 0.10
300 | 0.013 |  |  |  |
500 | 0.036 |  |  |  |
750 | 0.093 |  |  |  |
1000 | 0.19 |  |  |  |
parked data | 0.0003 | 0.039 | 0.027 | 0.031 | 0.022

channel $\to$ | semi-inclusive | semi-inclusive/semi-inclusive | semi-inclusive/inclusive
---|---|---|---
$m_{jj}$ cut [GeV] | $r_{i}\Delta b^{\pm}_{i}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
no cut | 0.005 | 0.36 | 0.25 | 0.16 | 0.11
100 | 0.006 | 0.47 | 0.33 | 0.21 | 0.15
300 | 0.022 |  |  |  |
500 | 0.072 |  |  |  |
750 | 0.21 |  |  |  |
parked data | 0.0004 | 0.050 | 0.035 | 0.031 | 0.022

channel $\to$ | exclusive | exclusive/exclusive | exclusive/semi-inclusive
---|---|---|---
$m_{jj}$ cut [GeV] | $r_{i}\Delta b^{\pm}_{i}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
no cut | 0.003 | 0.18 | 0.11 | 0.18 | 0.13
100 | 0.004 | 0.23 | 0.16 | 0.23 | 0.16
300 | 0.015 |  |  |  |
500 | 0.040 |  |  |  |
750 | 0.10 |  |  |  |
1000 | 0.21 |  |  |  |
parked data | 0.0004 | 0.049 | 0.034 | 0.034 | 0.024

Table 21: Expected statistical uncertainties for the polarization and spin
correlation measurements in $b\bar{b}$ in Run 2 with a neutrino spin analyzer
for different selection channels. The shorthand $\Delta c_{ij(\ell)}$ denotes
the uncertainties in $c_{ij}$ and $c_{\ell}$, which are of the same size.

Table 21 shows the expected uncertainties for Run 2, for both the standard
datasets and the CMS parked data. The results are promising for the
polarization, and with the parked data also for the spin correlations.
Analogous results for the HL-LHC are presented in table 22.

channel $\to$ | inclusive | inclusive/inclusive | inclusive/exclusive
---|---|---|---
$m_{jj}$ cut [GeV] | $r_{i}\Delta b^{\pm}_{i}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
no cut | 0.0004 | 0.015 | 0.011 | 0.012 | 0.0086
100 | 0.0005 | 0.025 | 0.017 | 0.020 | 0.014
300 | 0.0022 | 0.13 | 0.091 | 0.10 | 0.071
500 | 0.0063 | 0.36 | 0.26 | 0.29 | 0.20
750 | 0.016 |  |  |  |
1000 | 0.032 |  |  |  |
1500 | 0.11 |  |  |  |
2000 | 0.27 |  |  |  |

channel $\to$ | semi-inclusive | semi-inclusive/semi-inclusive | semi-inclusive/inclusive
---|---|---|---
$m_{jj}$ cut [GeV] | $r_{i}\Delta b^{\pm}_{i}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
no cut | 0.0004 | 0.018 | 0.013 | 0.012 | 0.0084
100 | 0.0005 | 0.029 | 0.021 | 0.019 | 0.013
300 | 0.0027 | 0.21 | 0.15 | 0.12 | 0.082
500 | 0.0091 |  |  |  |
750 | 0.026 |  |  |  |
1000 | 0.058 |  |  |  |
1500 | 0.24 |  |  |  |

channel $\to$ | exclusive | exclusive/exclusive | exclusive/semi-inclusive
---|---|---|---
$m_{jj}$ cut [GeV] | $r_{i}\Delta b^{\pm}_{i}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$ | $r_{i}^{2}\Delta c_{ii}$ | $r_{i}r_{j}\Delta c_{ij(\ell)}$
no cut | 0.0004 | 0.019 | 0.013 | 0.013 | 0.0093
100 | 0.0005 | 0.031 | 0.022 | 0.021 | 0.015
300 | 0.0025 | 0.16 | 0.11 | 0.13 | 0.091
500 | 0.0070 |  |  |  |
750 | 0.018 |  |  |  |
1000 | 0.037 |  |  |  |
1500 | 0.13 |  |  |  |
2000 | 0.32 |  |  |  |

Table 22: Expected statistical uncertainties for the polarization and spin
correlation measurements in $b\bar{b}$ at the HL-LHC with a neutrino spin
analyzer for different selection channels.

##  7 Summary and Discussion

In this work, we analyzed the feasibility of measuring quark polarization and
spin correlations in high-energy $pp\to q\bar{q}$ events in ATLAS or CMS,
where the quark $q$ is $b$, $c$, or $s$. Such measurements can be done using
the decay angular distributions of baryons produced from these quarks. Our
main results are as follows:

  * ✦

In the $b$-quark case, Run 2 data allows measuring the polarization in a
number of channels of semileptonic $\Lambda_{b}$ decays, with dijet invariant
mass cuts up to $\sim 1$ TeV. Particularly precise measurements are possible
with the CMS parked data. Spin correlation measurements are also feasible in
multiple channels, although the statistical uncertainties will be sizable with
the standard Run 2 datasets. An opportunity to do more precise measurements is
again offered by the Run 2 CMS parked dataset.

  * ✦

For the $c$ quark, polarization measurements with $\Lambda_{c}^{+}$ decays
using either the semileptonic or mixed channel are possible with the Run 2
dataset, and they can be extended to dijet invariant masses above 1 TeV at the
HL-LHC using the same channels or the hadronic channel. Spin correlation
measurements will be feasible at the HL-LHC in the semileptonic and mixed
channels.

  * ✦

The $s$-quark case is challenging in terms of the statistics that can be
collected with the standard triggers. Polarization measurements become only
borderline feasible with the statistics of the HL-LHC, and systematic
uncertainties may pose challenges as well due to the low purity of the sample.
There will not be enough statistics for $s\bar{s}$ spin correlation
measurements even at the HL-LHC.

In table 23, we provide a more detailed summary of the measurement channels we
have considered, showing in which of them a measurement is expected to be
possible.

| Polarization
---|---
Quark | Channel | Run 2 | HL-LHC
standard | parked
$s$ |  |  |  | ( )
$c$ | hadronic | ( ) |  |
semileptonic |  |  |
mixed | ( ) |  |
$b$ | inclusive | ( ) | ( ) | ( )
semi-inclusive |  |  |
exclusive |  |  |

| Spin Correlations
---|---
Quark | Channel | Run 2 | HL-LHC
standard | parked
$s$ |  |  |  |
$c$ | hadronic |  |  |
semileptonic |  |  |
mixed |  |  |
$b$ | inclusive/inclusive | ( ) | ( ) | ( )
semi-inclusive/semi-inclusive |  |  |
exclusive/exclusive |  |  |
inclusive/exclusive | ( ) | ( ) | ( )
inclusive/semi-inclusive | ( ) | ( ) | ( )
exclusive/semi-inclusive |  |  |

Table 23: The prospects in Run 2 and HL-LHC datasets for the different
analysis channels examined. The top table is for polarization and the bottom
is for spin correlations. Check marks indicate that a measurement is expected
to be possible, and crossed check marks indicate borderline cases. Parentheses
around a check mark indicate low sample purity, under 10%. The “parked” column
refers to the CMS parked $b$-hadron dataset CMS-DP-2019-043 ;
Bainbridge:2020pgi .

We would like to note that doing this broad survey of the different possible
analysis channels required us to use various assumptions and approximations,
as we described along the way. Therefore, our predicted sensitivities should
be viewed as rough estimates. The actual performance of the corresponding
analyses may be worse because of neglected backgrounds or effects of
unfolding. On the other hand, improvements in the reconstruction of electrons
within jets to make them comparable to muons would enhance the performance.
Advanced machine learning techniques that have been entering the field
recently can also boost the performance by improving object identification and
reconstruction and/or background rejection. Finally, new trigger paths that
might be deployed in future runs of the LHC have the potential to improve the
statistics of all of the proposed analyses.

Since the angular distribution coefficients $C_{ii}$ and $C_{ij}^{\pm}$ that
will be measured in the spin correlation analyses depend on the polarization
retention factors $r_{L}$ and $r_{T}$ via eq. (17), the analyses proposed in
this paper can be used to determine these factors for $c$ and $b$ quarks. By
relying on spin correlation components allowed to be large by the symmetries
of QCD (specifically $c_{kk}$, $c_{nn}$, $c_{rr}$, and $c_{rk}$),161616In
practice, $c_{rk}$ happens to be small when $p_{T}$ cuts are applied, as can
be seen in tables 3 and 4. one can extract $r_{L}$ and $r_{T}$ using the
(partly redundant) relations

$r_{L}^{2}=\frac{C_{kk}}{c_{kk}\alpha_{+}\alpha_{-}f}\;,\quad
r_{T}^{2}=\frac{C_{nn}}{c_{nn}\alpha_{+}\alpha_{-}f}\;,\quad
r_{T}^{2}=\frac{C_{rr}}{c_{rr}\alpha_{+}\alpha_{-}f}\;,\quad
r_{L}r_{T}=\frac{C_{rk}^{+}}{2c_{rk}\alpha_{+}\alpha_{-}f}\;.$ (54)

The measurements of $r_{L}$ and $r_{T}$ can lead to further understanding of
the polarization transfer in QCD fragmentation and act as a test for different
models of low-energy QCD phenomenology (e.g., ref. Adamov:2000is ). It will
also be interesting to compare the values of $r_{L}$ that will be obtained in
these spin correlation measurements with those that will hopefully be obtained
in polarization measurements of $b$ and $c$ quarks in $t\bar{t}$
Galanti:2015pqa  and $W$+$c$ Kats:2015zth  samples.

The polarization and spin correlation measurements can in principle also be
utilized to probe for BSM contributions to $pp\to q\bar{q}$ processes,
especially using the numerous components in which the SM contribution is
expected to be small. Whether such searches can be competitive with other
probes of the same BSM scenarios is a subject for another study.

###### Acknowledgements.

This research was supported in part by the Israel Science Foundation (grants
no. 780/17 and 1666/22) and the United States - Israel Binational Science
Foundation (grant no. 2018257).

##  Appendix A Statistical Uncertainty Formulas

The angular distributions in eqs. (19), (20) and (21) are all of the form

$\frac{1}{\sigma}\frac{d\sigma}{dX}=\frac{1}{2}\left(1+cX\right)f(|X|)\,,$
(55)

where

$X\;=\;\cos\theta^{\pm}_{i}\,,\quad\cos\theta^{+}_{i}\cos\theta^{-}_{j}\,,\quad\cos\theta^{+}_{i}\cos\theta^{-}_{j}\pm\cos\theta^{+}_{j}\cos\theta^{-}_{i}$
(56)

and

$f(|X|)\;=\;1\,,\quad\;\;\ln\left(\frac{1}{|X|}\right),\quad\;\;\cos^{-1}(|X|)\,,\qquad\qquad\qquad$
(57)

respectively. We imagine extracting the value of $c$ from the data with a
$\chi^{2}$ minimization procedure for a binned distribution of $X$. The
$\chi^{2}$ function is

$\chi^{2}(c)=\sum_{i}\frac{\left[n_{i}-\frac{1}{2}\left(1+cX_{i}\right)f(|X_{i}|)N\Delta
X\right]^{2}}{\sigma_{i}^{2}}\;,$ (58)

where the summation is over bins of size $\Delta X$, $n_{i}$ is the number of
events observed in bin $i$, $X_{i}$ is the value of $X$ in the middle of that
bin, $N$ is the total number of events, and

$\sigma_{i}^{2}\simeq
n_{i}\simeq\frac{1}{2}\left(1+cX_{i}\right)f(|X_{i}|)N\Delta X$ (59)

is the squared uncertainty of the expected number of events in bin $i$.
Solving the minimization condition $d\chi^{2}/dc=0$ for $c$ and then
propagating the statistical uncertainties of $n_{i}$, we find for the
uncertainty of $c$:

$\Delta c=\frac{1}{S(c)\sqrt{N}}\;,$ (60)

with

$S^{2}(c)\simeq\frac{\Delta
X}{2}\sum_{i}\frac{X_{i}^{2}}{1+cX_{i}}\,f(|X_{i}|)\,,$ (61)

where we assumed $n_{i}\gg 1$. In the limit of a large number of bins we can
approximate the sum with an integral and obtain

$S^{2}(c)\simeq\frac{1}{2}\int_{-1}^{1}\frac{X^{2}}{1+cX}\,f(|X|)\,dX=\int_{0}^{1}\frac{X^{2}}{1-c^{2}X^{2}}\,f(X)\,dX\,.$
(62)

The uncertainties on the parameters $B^{\pm}_{i}$ ($\equiv B$), $C_{ii}$, and
$C_{ij}^{\pm}/2$ are then given by

$\displaystyle\Delta B$ $\displaystyle=\frac{1}{S_{B}(B)\sqrt{N}}\;,$ (63)
$\displaystyle\Delta C_{ii}$
$\displaystyle=\frac{1}{S_{C_{ii}}(C_{ii})\sqrt{N}}\;,$ (64)
$\displaystyle\Delta(C_{ij}^{\pm}/2)$
$\displaystyle=\frac{1}{S_{C_{ij}^{\pm}/2}(C_{ij}^{\pm}/2)\sqrt{N}}\;,$ (65)

where

$\displaystyle S_{B}^{2}(B)$
$\displaystyle\simeq\int_{0}^{1}\frac{X^{2}}{1-B^{2}X^{2}}\,dX=\frac{\operatorname{arctanh}B-B}{B^{3}}\;,$
(66) $\displaystyle S_{C_{ii}}^{2}(C_{ii})$
$\displaystyle\simeq\int^{0}_{1}\frac{X^{2}}{1-C_{ii}^{2}X^{2}}\ln\left(\frac{1}{X}\right)dX=\frac{1}{9}+\frac{1}{4}\,C_{ii}^{2}\,\Phi(C_{ii}^{2},2,\tfrac{5}{2})\;,$
(67) $\displaystyle S_{C_{ij}^{\pm}/2}^{2}(c)$
$\displaystyle\simeq\int^{0}_{1}\frac{X^{2}}{1-c^{2}X^{2}}\cos^{-1}(X)\,dX$
$\displaystyle=\frac{1}{4ic^{3}}\left[2\,\mbox{Li}_{2}\left(\frac{1}{\left(ic+\sqrt{1-c^{2}}\right)^{2}}\right)-8\,\mbox{Li}_{2}\left(\sqrt{1-c^{2}}-ic\right)\right.$
(68)
$\displaystyle\qquad\qquad\left.+\,2\left(\pi+2i\ln\left(\frac{\sqrt{1-c^{2}}-1}{c}\right)\right)\arcsin
c-4ic+\pi^{2}\right]\;,$

where the function $\Phi$ is the Lerch transcendent, $\mbox{Li}_{2}$ is the
dilogarithm, and $c$ in eq. (68) stands for $C_{ij}^{\pm}/2$.

Figure 5: Statistical uncertainty (multiplied by $\sqrt{N}$, where $N$ is the
number of events) of the angular distribution coefficients $B^{\pm}_{i}$
(black), $C_{ii}$ (blue), and $C_{ij}^{\pm}/2$ (red) as a function of the
coefficient values, calculated both analytically (solid curves) and
numerically (dots) The analytical results are in the limit of an infinite
number of bins, while the numerical (Monte Carlo) results are for 10 bins.

In figure 5 we plot the uncertainties of the coefficients as a function of the
coefficient values based on the analytic formulas in eqs. (66)–(68), which
correspond to the limit of a large number of bins. We also show the results of
a Monte Carlo simulation of a $\chi^{2}$ minimization procedure with 10 bins.
There is a good agreement between the two sets of results, except very close
to $B=-1$ or $1$, where the analytical formula, which assumes infinitesimally
small (but at the same time highly populated) bins, gives vanishing
uncertainty. This happens because bins near $X=1$ or $-1$ in these cases
contain a vanishingly small fraction of events, making the uncertainty in
those bins vanishingly small too, which in turn leads to a vanishingly small
uncertainty in $B$. This situation is unphysical because measurements of $X$
always have some uncertainty, $\Delta X$, which prevents bins much smaller
than $\Delta X$ from being useful.

While the polarization and spin correlation components ($b^{\pm}_{i}$,
$c_{ij}$, $c_{\ell}$) can in principle be anywhere in the range $[-1,1]$, the
polarization retention factors $r_{i}$ and in many cases also the purity $f$
or the spin analyzing powers $\alpha_{\pm}$ in eq. (17) will limit the angular
distribution coefficients $B^{\pm}_{i}$, $C_{ii}$, and $C_{ij}^{\pm}/2$ to a
smaller range of values. This, together with the flatness of the curves in
figure 5 in the vicinity of zero, implies that the uncertainties can usually
be approximated by their values at zero. We have

$S_{B}(0)=\frac{1}{\sqrt{3}}\;,\qquad S_{C_{ii}}(0)=\frac{1}{3}\;,\qquad
S_{C_{ij}^{\pm}/2}(0)=\frac{\sqrt{2}}{3}\;,$ (69)

which leads to eq. (22).

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