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

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.


Introduction
Quark polarization and spin correlations are properties that have been extensively researched for top quarks at the LHC, both theoretically (e.g., refs.[1][2][3][4][5][6][7][8][9][10][11]) and experimentally (e.g., refs.[12][13][14][15][16][17]), 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 → t t, in which b quarks are available from the t → W + b decay, and c and s quarks from the subsequent decay W + → sc [18][19][20], and using samples of pp → W − c for c quarks [21].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 → q 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 [22][23][24].Such measurements were subsequently performed for the b quark using Λ b baryons [25][26][27], confirming the expectation of a sizable polarization transfer from the b quark to the Λ b .Analogous measurements have shown that the s-quark longitudinal polarization is preserved in Λ baryons that carry a significant fraction of the jet momentum [28][29][30].This method to access quark spin information has been analyzed in the context of the LHC in refs.[18][19][20][21], 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 Λ b in inclusive QCD samples, which is expected to be small, have been reported by LHCb [31,32] and CMS [33].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.[18,19]) as well as transverse polarization.This will provide certain cross-grained information about the spin-dependent fragmentation functions [34,35] of the b and c quarks hadronizing to the Λ b and Λ c baryons, respectively.Polarization and spin correlation measurements can also be sensitive to BSM contributions to b b or cc production.Similar ideas apply to ss, 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 [36,37].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 b and cc expected in QCD after validating our simulation on t 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.

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/Λ QCD ) [22][23][24].This happens since m q ≫ Λ QCD implies that the effect on the heavy quark spin via the chromomagnetic dipole moment, which scales as µ q ∝ 1/m q , is negligible.
If the heavy quark q (= b or c) ends up in a Λ q baryon, the baryon polarization is approximately equal to the quark polarization.It is so because the Λ 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 Σ q or Σ * q baryon, which are analogous to the Λ q but with the light quarks forming a spin and isospin triplet, the Λ q baryons produced in Σ ( * ) q → Λ q π decays will not carry the same polarization as the original quark [24].The Λ q from these decay channels are hard to distinguish from the direct Λ q production and thus they lower the polarization retention.Due to similar reasons, it is essentially impossible to extract polarization information from meson decays [24,38].
The polarization loss effect due to the contamination of the Λ q sample with Σ ( * ) q → Λ q π decays has been analyzed in refs.[18,24] and it was found that the inclusive Λ 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 r P = P(Λ q ) P(q) , ( where P denotes polarization and 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 t samples [18], and for charm quarks possibly also in W +c samples [21].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 Λ baryons from s quarks still preserve a large fraction of the polarization [28][29][30].
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, and is described by the so-called spin-dependent (or polarized ) fragmentation functions [34,35,39,40].These functions vary slowly as a function of the energy scale of the process due to the renormalization group evolution [39,40].Characterizing these effects and taking them into account will require high-statistics measurements that could follow up the measurements motivated here and in refs.[18,19,21].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 Λ baryons with high values of z.The dependence of the Λ polarization on z has been measured in Z decays at LEP [28][29][30] confirming the expectation [41] that the polarization of the initial strange quark is preserved primarily in high-z Λ baryons.For example, as has been estimated in ref. [19] based on the LEP measurements, roughly 60% of the strange-quark polarization is preserved in Λ baryons with z > 0.3.Additional information can be obtained using t t samples at the LHC [19].

Baryon Production and Relevant Decay Channels
The fragmentation fractions for producing the baryons of interest at high energies are shown in table 1.While the numbers for b → Λ b and c → Λ c are available in the literature, for the Λ we obtained the number from a Pythia [47] simulation.The simulation results are shown in figure 1, which presents the integrated fragmentation functions Fragmentation Fraction Decay Scheme BR Spin analyzing power b → Λ b 7.0% [18] Λ b → X c µ − νµ 11% [42] α µ − ≈ −0.26, α νµ ≈ 1 [18,43] with Λ → pπ − 2.7% [42] with Λ + c reco.2.0% [42] c → Λ c 6.4% [44] Λ + c → pK − π + 6.3% [42] α eff ≈ 0.662 [45] Λ + c → Λµ + ν µ 3.5% [42] α µ + ≈ 1 [46] with Λ → pπ − 2.2% [42] s → Λ 2.8% [47,48] Λ → pπ − 64% [42] α p ≈ 0.75 [42] (z > 0.3) Table 1: Baryon fragmentation fractions, relevant decay schemes, branching ratios (BR) and spin analyzing powers (asymmetry parameters) of the decays.and similarly for s → Λ.This gives the following fragmentation fractions for z 0 = 0.3 (a cut motivated at the end of section 2): We also compute the sum of the Λ and Λ numbers for a comparison with the AKKII databased results [48].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 Λ b decays, we consider three types of selections (similar to ref. [18]): • Inclusive Selection, where X c represents any collection of particles containing a charmed hadron, which is usually a Λ + c . 1 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 Λ → pπ − decay, to reduce backgrounds from the semileptonic b-meson decays.
• Exclusive Selection, where the X c is required to contain a fully reconstructible Λ + c decay (i.e., with charged products only), to reduce backgrounds from the semileptonic meson decays and facilitate the reconstruction of the Λ b decay kinematics for the polarization and spin correlation measurements.The full list of the Λ + c decays we include here is provided in a later section, in table 15.In the semileptonic Λ + c decay case, only the selection with the Λ → pπ − decay will be considered.Decays with electrons instead of muons, for both the Λ b and Λ + c , can be used The fragmentation fraction for z > z 0 , for s → Λ, s → Λ, and s → Λ/ Λ, where Λ/ Λ denotes that either a Λ or Λ is produced in the s-quark hadronization.The Pythia simulation was run at √ s = 14 TeV with p T > 400 GeV and |η| < 2.5 cuts on the jets, which were clustered with the anti-k t jet algorithm with radius R = 0.4 [52,53] and matched to the parton-level s quarks.We also show the data-based AKKII results [48].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 [49][50][51].
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) α is defined by writing the angular distribution of the decay as where θ is the angle between the momentum of the decay product and the direction of the baryon polarization ⃗ P, in the baryon rest frame. 2 This provides a handle for measuring the baryon and antibaryon polarizations and spin correlations.
The Λ + c → pK − π + 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, α eff , that corresponds to the sensitivity that can be obtained with a full amplitude analysis [45].
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. [18], 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 [54][55][56][57][58][59][60] and 50% for the c quark [61,62]. 3To 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.[64][65][66][67].Unfolding will be required to account for the approximations made.

Polarization and Spin Correlations
We will now review the mathematical description of the polarization and spin correlations of q q pairs and describe how these properties are expected to manifest themselves in angular distributions of the baryon decays.

Quark-Antiquark Pair Spin State Description
The spin state of a quark and antiquark is described by a density matrix of the form [68] ρ = 1 4 Here 1 is a 2 × 2 unit matrix, σ i are the Pauli matrices, the indices (summation over which is implied) represent the coordinate axes, B± are three-dimensional vectors characterizing the polarization of the quark and antiquark, 4 and C is a 3 × 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 t literature [8,13,15], we will use the orthogonal set of axes { k, n, r}.The axis 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 p the momentum direction of one of the incoming partons, and by Θ the scattering angle of the outgoing quark, such that cos Θ = k • p. Then the axes n and r are defined as The sign(cos Θ) factor is included in eq.(4.2) 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 Θ < 0 and cos Θ > 0, as can also be seen explicitly in refs.[69,70].It is of note that the inclusion of the sign(cos Θ) factor leads to partial cancellation for events originating from q q [69,70].It can be useful to also do a measurement without this factor to be more sensitive to q q-initiated contributions.CMS measured the Λ b polarization along the n axis without the sign factor, using an amplitude analysis of the decay5 Λ b → J/ψ(→ µ + µ − )Λ(→ pπ − ) at CM energies √ s = 7 and 8 TeV, finding P = 0.00 ± 0.06 (stat.)± 0.06 (syst.)[33].LHCb conducted Λ b polarization measurements using the same decay channel and found the polarization along 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 √ s = 7, 8 and 13 TeV, respectively [32].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 { k, n, r}, we can write the polarization vectors and spin correlation matrix as or equivalently In this way of writing, the symmetric part of the matrix C is described by the components and the antisymmetric part by The range of possible values for b ± i , c ij , and c ℓ is [−1, 1].These quantities in general depend on the production process, the partonic CM energy squared ŝ, and the quark's scattering angle Θ in relation to the proton going in the positive direction of the ẑ 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 θ 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 [69,70].Small contributions involving the electroweak interactions are expected in many of the components (see, e.g., ref. [8], where they were computed for the top quark), but we will not consider them in this paper.

Decay Angular Distributions
Similar to the t t case [8], if we denote the momentum vector of one of the baryon decay products in the baryon rest frame by p 1 , and the momentum vector of one of the antibaryon decay products in the antibaryon rest frame by p 2 , their angular distributions are given by where Here B± and C are the quark and antiquark polarization vectors and their spin correlation matrix from eqs. (4.3)-(4.4).The factors α + and α − , referring to the baryon and antibaryon, respectively, are the spin analyzing powers of their decays, as defined in eq.(3.3).The factors r i are the polarization retention factors from eq. (2.1): r L for the k axis and r T for the n and r axes. 6The factor f is the sample purity, namely the fraction of signal events out of the total number of selected events: The multiplication by f in eq.(4.9) is only correct if we assume that the effect of the background is only to dilute the B ± 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 and rewrite eq.(4.9) as ) where we used eqs.(4.6)-(4.7).In the expression for C − ij , the axes corresponding to the indices i, j and ℓ are related via î × ĵ = l.
Defining sets of spherical coordinates around the axes of interest for the baryon and antibaryon and integrating eq.(4.8) over the azimuthal angles gives where θ + i (θ − j ) is the angle between the direction of the decay product and the chosen axis, in the baryon (antibaryon) rest frame. 7Integrating over one of the θs in eq.(4.13), we get the distributions through which one can measure B ± i to obtain the quark and antiquark polarization components b ± i .Converting the double differential distribution in eq.(4.13) to a distribution differential in the product cos θ + i cos θ − j , we obtain 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 ℓ ) parts because of their different CP properties (recall table 2), one can derive from eq. (4.8) the distribution where The components c ij (with i ̸ = j) and c ℓ can be computed from C ± ij via eq.(4.12).

Statistical Uncertainty Evaluation
The statistical uncertainty of the coefficients B ± i , C ij , and C ± ij , when they are extracted from fits of data to eqs.(4.14), (4.15), and (4.16), respectively, is given approximately by where N is the number of events.These results are derived in appendix A. Using eq.(4.12) we then obtain for the statistical uncertainties of the polarization and spin correlation components where N sig is the number of signal events, and the notation c ij(ℓ) means to refer at once to c ij from eq. (4.6) and c ℓ from eq. (4.7).The uncertainty in eq.(4.18) 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.

Quark Polarization and Spin Correlations in QCD
We used MadGraph [71] to obtain the leading-order (LO) QCD expectations for the polarization and spin correlations in pp → b b and pp → cc.
We first validated our procedure on the process pp → t t, results for which are available in the literature [11].As a technical tool, we decayed the top quark as t → bℓ + v ℓ with Mad-Spin [72] (and similarly for t) to obtain the polarization and spin correlation information of the t and 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. [11].As a check, we have also run the full matrix element simulation in MadGraph, namely pp → bℓ + ν ℓ bℓ − νℓ 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 [8].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. [11].
To obtain the polarizations and spin correlations for pp → b b and pp → cc, we cannot follow a procedure exactly analogous to what we did for pp → t 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 → t 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 → bℓ + v ℓ , 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. [73], that the spin analyzing power of the lepton remains the same.
Unlike in the t t case, where an inclusive measurement without any cuts on the t and t is possible (with the use of unfolding), a measurement in b b or cc 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 → c transitions in the b b case is equivalent in terms of the cross section to applying the cuts |η| < 2.4 and p T > 79 GeV on the quarks ("jets").In the cc case the same procedure leads to a p T > 115 GeV cut on the quarks. 8We 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 t case, as a check.For the inclusive b b and cc 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 t, b b, and cc.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.Table 4 presents the analogous results for the HL-LHC with √ s = 14 TeV, where our effective cuts on the quarks are |η| < 2.5 and p T > 59 GeV for b b and p T > 80 GeV for cc.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.

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 → q 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 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 [74] and CMS [75].We will be using the ATLAS cuts. 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 [71] and/or Pythia [47] simulations and reliance on elements of existing ATLAS and CMS analyses.For jet clustering, the Pythia simulations are interfaced with FastJet [53], where we use the anti-k t algorithm with radius R = 0.4 [52].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.

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 [74] and the online ones for CMS [75].We will be using the ATLAS cuts.
For the jet based triggers, which are relevant for the hadronic channels c → Λ + c → pK − π + and s → Λ → pπ − , for Run 2 we added the requirement |η| < 2.4 (even though the trigger functions up to |η| = 2.8 [76]) so that the jets will be within the tracker.For the HL-LHC we require |η| < 3.8.
For the semileptonic channels b → Λ b → X c µ − νµ and c → Λ + c → Λµ + ν µ 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 Λ + 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 [75,77].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 [77].
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 → Λ b → X c µ − νµ channel, we also looked at b-tagging triggers.In ATLAS, the single b-jet trigger [78] requires E T > 225 GeV (which is expected to become p T > 180 GeV at the HL-LHC [74]), 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 [75].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 [36,37].The muon p T threshold varied between 7 and 12 GeV depending on the luminosity, its track was required to be within |η| < 1.5, and satisfy a requirement on impact parameter significance.Despite the lower integrated luminosity of this dataset (∼ 50 fb −1 ) and the η restriction, the soft p T threshold allowed collecting more statistics than the standard Run 2 muon triggers.

Analyses of pp → ss
For measuring the polarization and spin correlations in pp → ss, we consider events with s → Λ and Λ → pπ − .

Λ Reconstruction, Efficiency and Signal Yield
The decay Λ → pπ − has a very distinct signature of a highly displaced vertex with a pair of oppositely charged tracks that reconstruct the Λ mass if they are assigned the proton and charged pion masses.The other similar decay, K S → π + π − , will usually fail the Λ 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 Λ decays were reconstructed in multiple analyses in ATLAS (e.g., [79][80][81][82]) and CMS (e.g., [33, 83-86]).
There is, however, an important obstacle to reconstructing the decays of energetic Λ 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 [87-89].It has been used in multiple searches for long-lived BSM particles [90-97].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 max T ≈ 300 mm, with the mean reconstruction efficiency up to this decay radius being roughly ϵ track ≈ 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 [89].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 ϵ track ≈ 80% up to d max T ≈ 400 mm [98].We will be using the efficiency numbers with the LRT algorithm to estimate the Λ reconstruction efficiency, after accounting for the probability for the Λ to decay within the ranges mentioned above.
The average Λ decay radius in pp → ss events can be estimated as where cτ ≈ 7.9 cm is the Λ lifetime and z is the Λ momentum fraction from eq. (2.2).Since the cross section decreases fast as a function of p T,jet and z, we can obtain rough estimates for ⟨d T ⟩ by taking p T,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 Λ baryons are not correlated with the original strange-quark polarization, as was discussed in section 2. This gives ⟨d T ⟩ ∼ 9.8 m for Run 2 and 8.5 m for the HL-LHC.Since ⟨d T ⟩ ≫ d max T , the probability for the Λ to decay sufficiently early is ϵ d T ≃ d max T /⟨d T ⟩ ≈ 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 ϵ d T ≈ 2.2% for Run 2 and 3.4% for the HL-LHC.For background jets (which are a mixture of all pp → jj processes apart from pp → ss) the numbers are 2.4% and 3.7%, respectively.Run 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 ss ).
The full reconstruction efficiency for the Λ → pπ − decay in signal jets is ϵ Λ = ϵ d T ϵ 2 track ≈ 0.93% for Run 2 and 2.2% for the HL-LHC.For spin correlation measurements we need the efficiency for reconstructing both a Λ and a Λ.Despite the correlation between the p T of the two jets, we can still obtain a rough estimate by simply squaring the efficiency, ϵ Λ Λ ≃ ϵ 2 Λ , because the jet p T values are distributed mainly near the threshold.This gives ϵ Λ Λ ≈ 9 × 10 −5 for Run 2 and 5 × 10 −4 for the HL-LHC.
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 ss pairs available for the spin correlation measurement (N ss ) computed as where the cross sections σ ss (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.

Background
The dominant background is due to Λ baryons produced in dijet processes other than pp → ss.While the number of Λ 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 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 ss events, where we matched the jets to partons to count only Λ baryons from s (but not s) as signal.The numbers are presented as a function of z 0 , the cut on the momentum fraction z carried by the Λ.From these plots we can calculate the sample purity.For z 0 = 0.3, the purity is f ≈ 0.9% for both Run 2 and the HL-LHC.

Measurement prospects
With the signal event counts and purities, we can use eq. the polarization components b ± i at the HL-LHC to be given by r i ∆b ± i ≈ 0.27.We show the result for the product r i ∆b ± 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 ∼ 0.6, as mentioned in section 2. The physical range of b ± 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, ss 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.

Analyses of pp → cc
For cc polarization and spin correlation measurements, we will consider in turn three possible analysis channels in terms of the Λ + c decays: the hadronic channel where Λ + c → pK − π + , the semileptonic channel where Λ + c → Λµ + ν µ , and the mixed channel with the hadronic decay in one jet and the semileptonic decay in the other.

Hadronic Channel
The signature of the Λ + c → pK − π + decay is one negatively and two positively charged tracks coming from a common vertex.They should also reconstruct the Λ + c mass for an assignment of the proton and π + masses to the positively charged tracks and the K − mass to the negatively charged one.Such decays were reconstructed by CMS in refs.[99,100].
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. [18].Extrinsic backgrounds from pp → b b should be considered as well.They include Λ + cc, hadronic Run 2 sample purity (when a single jet is considered) to be f ≈ 6.9%.CMS also report the signal efficiency to be roughly ϵ Λ + c ≈ 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 [101][102][103][104][105] 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 − π + 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.
To calculate the expected number of signal events, we computed the pp → cc cross sections σ cc 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 Λ + c → pK − π + decay.For the spin correlations we require this Λ + c decay in one jet and the analogous Λ− c decay in the other.The expected numbers of events for measurements of polarization (N c ) and spin correlations (N cc ), calculated as ) 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 ∆b ± i based on eq.(4.18).The prospects are borderline for Run 2 but seem good for the HL-LHC.

Semileptonic Channel
A potentially more promising avenue is the semileptonic decay Λ + c → Λµ + ν µ with Λ → pπ − .While the branching ratio of this decay chain (3.5% for Λ + c → Λµ + ν µ , 64% for Λ → pπ − [42]) 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 Λ decay reconstructed in the tracker because it will be difficult to extract the Λ + c polarization information if the muon will be the only product associated with the Λ + c decay.Besides that, the Λ requirement strongly suppresses the intrinsic background due to semileptonic D-meson decays since they are kinematically forbidden from producing a Λ (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 Λ decays, and the shortness of the Λ + c lifetime (cτ ≈ 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 [50,51,[106][107][108]) as well as a reconstructed Λ → pπ − decay.To ensure that the Λ originates from a Λ + c decay, one should demand the Λ trajectory (inferred from the momenta of its decay products) to form a displaced vertex with the muon.Events in which the Λ trajectory is consistent with both a common vertex with the muon and the primary vertex, can still be accepted if the Λ carries a significant fraction (e.g., above 20%) of the jet momentum, since Λ 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 b production.We can estimate the size of this background by starting with the inclusive branching ratio for b jets to contain a Λ or Λ baryon, which was measured to be [42] BR b → (b-hadron) → Λ/ Λ + X ≈ (5.9 ± 0.6)% .( Accounting for us requiring a Λ (not a Λ) while on the other hand collecting background from both the b and b jet, we are left with the same number.We will further assume that the probability of having a µ + in association with the Λ is roughly 10%. 9The corresponding numbers are compared with the signal in table 9.
There are a number of ways to reduce the b 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 Λ + c and b-hadron lifetimes (see Table 9: Properties of the semileptonic Λ + c signal and the b b background: fragmentation fractions (FF) [44], branching ratios (BR) [42] and lifetimes [42].The cc and b 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 ΛX sample contain a µ + .The numbers shown are before any discriminating cuts.table 9) and apply an upper bound on the transverse displacement d T of the secondary vertex.For example, requiring d T < 0.77 mm×(p jet T /115 GeV) has an efficiency of 40% for the signal and 10% for the background, giving 60% purity. 10Various 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 [109] and CMS [50], which would lead to 71% purity in our case.The performance of charm tagging should be even better when applied to the Λ + 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 Λ 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 ϵ c ≈ 40% and neglect the remaining background.
Table 10 shows the Run 2 cross sections and numbers of events available for measurements of polarization (N c ) and spin correlations (N cc ).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 → cc, 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 ) where ϵ µ ≈ 50% is the efficiency for the muon to pass the isolation requirement of the single muon trigger [77], ϵ c,2 ≡ 2ϵ c − ϵ 2 c is the efficiency for any of the two jets to pass charm tagging, and ϵ Λ includes the decay radius and reconstruction efficiencies of the Λ, where cc, semileptonic polarization ≈ 300 mm, the average reconstruction efficiency for each track is expected to be ϵ track ≈ 65%, and for the HL-LHC, in the range of up to d max T ≈ 400 mm, it will be ϵ track ≈ 80%.Averaging over the whole range is justified as the mean decay radius is ⟨d T ⟩ ≫ d max T .We estimate ⟨d T ⟩ with the assumption that p Λ T ∼ (0.5/3) p jet T , where the factor of 0.5 is the typical c-jet momentum fraction carried by the Λ + c and the division by 3 roughly accounts for the Λ being one out of three decay products.For the jet p T , we used the rough estimate p jet T ∼ max(p jet T , m cut jj /3) where pjet T ≈ 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.

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.
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 Table 12: The fragmentation fractions [44] and branching ratios [42] of inclusive semimuonic decays of the common charmed hadrons.
events in the sample is then where we assume an efficiency of ϵ Λ + c →pK − π + ≈ 25% for the hadronic decay selection [100] and ϵ µ ≈ 50% for passing the isolation requirement of the single muon trigger [77].The resulting numbers of events are given in table 13.
The requirement of the muon in the second jet is also expected to remove part of the background that is observed under the Λ + c → pK − π + peak in the CMS measurement [100].While it will not eliminate background coming from cc or b 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 cc, mixed channel, polarization  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 cc.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 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 where ϵ Λ + c →pK − π + ≈ 25% is the hadronic decay efficiency [100], ϵ Λ describes the Λ decay acceptance and reconstruction efficiency (estimated as in section 6.3.2),ϵ µ ≈ 50% is the efficiency loss due to the isolation requirement of the single muon trigger [77], and ϵ 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 µ + from the c jet or a µ − from the c jet can trigger the event.This also affects the equivalent p jet T values we use to estimate the Λ displacement acceptance, which become pjet T = 97 GeV for Run 2 and pjet T = 72 GeV for the HL-LHC.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 Λ + c peak in the inclusive CMS sample [100] is primarily intrinsic (i.e., from cc), 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 → gg, where the jets rarely contain a muon and a Λ, 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 cc, mixed channel, spin correlations 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.

Analyses of pp → b b
We propose using the b → Λ b → X c µ − νµ process, with the muon-based triggers from table 5, to measure the polarization and spin correlations in b b events.As introduced in section 3, we will consider three types of selection: Inclusive, Semiinclusive, 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 Λ baryon (via its decay Λ → pπ − ) 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 Λ + 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 Λ + c decay is that the kinematics of the Λ 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.11

Inclusive Selection
The main requirement of this selection is the presence a muon in the jet, similar to soft muon b tagging [49,78,106,107,115,116].
To suppress extrinsic backgrounds with prompt or mildly displaced (in particular, cc) muons, we assume applying a b tagging algorithm that may use the muon impact parameter significance, or p rel T (the component of muon momentum transverse to the jet axis) [106], or Selection Decay Modes Branching Ratio Table 15: Decay modes relevant to the three Λ b selections and their branching ratios [42].
The Semi-inclusive and Exclusive selections are done on top of the Inclusive selection.
other variables.For our estimates, we will assume the b tagging efficiency to be ϵ b ≈ 80% and allow ourselves to neglect the above backgrounds.Another source of extrinsic background is pp → t t → b bX.This background will be small since the t t cross section is smaller than the b b cross section for any fixed b 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 → cµ − νµ , 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 → cf f ′ , c → sµ + ν µ .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 b.There is a different way to reduce this background significantly.The muons emitted in b → c transitions (as in our signal) are usually more energetic than those emitted in the c → s transitions in the b → c → s chain (which are background), so it can be useful to look into the ratio between the muon and jet p T , z = p µ T /p jet T .Figure 4 shows the z distributions for muons from both b and c hadrons in pp → b b events with √ 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, (The plot normalizations do not include the branching ratios.)this z cut results in decent efficiency and purity (ϵ z ≈ 77%, f z ≈ 84%), while the impact on the no-cut case is small (ϵ z ≈ 99%, f z ≈ 94%).From these results, we find that the c hadrons background is small and we will neglect it.
The numbers of events available for measurements of polarization and spin correlations, respectively, are given by where ϵ µ ≈ 50% is the efficiency for the muon to pass the isolation requirement of the single muon trigger [77], ϵ b,2 ≡ 2ϵ b − ϵ 2 b is the efficiency for any of the two jets to pass the b tagging condition, and ϵ z,2 ≡ 2ϵ z − ϵ 2 z is the efficiency for any of the two muons to pass the momentum fraction cut. 12The cross sections were obtained using a Pythia simulation of pp → b 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.
By accounting for the fragmentation fractions and semi-muonic decay branching ratios of the B0 , B − , and B0 s mesons (see table 18), we estimate the sample purity to be f ≈ 7.4% for a single jet.For spin correlation measurements, which involve two jets, this number is squared, giving the purity of f ≈ 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.

Exclusive Selection
The exclusive selection starts with the inclusive selection and requires, in addition, a fully reconstructed Λ + c decay, to eliminate most of the background due to the semileptonic Bmeson decays, which usually produce charmed mesons rather than baryons.
no cut 39000 6.7 × 10   Table 18: The fragmentation fractions (FF) [18,117] and branching ratios (BR) [42] of the signal (Λ b ) and the intrinsic background in the Inclusive Selection. Process
Some of the Λ + c decay channels listed in table 15 come with an efficiency price: K S and Λ can decay too far to be reconstructed, Σ ± 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 ϵ reco ≈ 50% for the Λ + c decays.The numbers of events available for measurements of polarization and spin correlations, respectively, using the exclusive channel are given by where BR(Λ + c → reco.)≈ 18% is the branching fraction of the reconstructible decay modes of the Λ + c from table 15 and ϵ µ ≈ 50% is the efficiency for the muon to pass the isolation requirement of the single muon trigger [77].The resulting numbers of events are included in tables 16 and 17.
A remaining background in this channel is the semileptonic decays of B mesons to final states with a Λ + 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 Λ b , as shown in table 19.The branching ratios for B-meson decays involving a µ − in addition to the Λ + 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 → (c → Λ + c ) transition, while in the case of the b initial state is it produced following the b → c (c → Λ + c ) s chain, in the subsequent c → 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 − , B0 and B0 s decays from table 19 actually contribute to the background.Summing up their contributions (while assigning B0 s the average of the B − and B0 BRs) and assuming the probability of having a muon to be similar to the Λ b case, where the muon is also produced in a b → 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.

Semi-inclusive Selection
The semi-inclusive selection starts with the inclusive selection and requires, in addition, the presence of a reconstructed Λ baryon.This requirement eliminates most of the contributions from B-meson decays while accepting a large fraction of Λ b decays since Λ b decays usually proceed through a Λ + c , which in turn produces a Λ in about 38% of the cases [42].The numbers of events in this channel, for measurements of polarization and spin correlations, respectively, are given by (6.16) where ϵ µ ≈ 50% is the efficiency for the muon to pass the isolation requirement of the single muon trigger [77] and ϵ Λ is the Λ 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 ∼ 65% (∼ 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 pjet T ≈ 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 with13 p Λ T ∼ (0.7/3 2 ) p jet T , 54% of the Λ satisfy the decay radius condition for the Run 2 (with p jet T = 79 GeV) and 76% satisfy it for the HL-LHC (with p jet T = 59 GeV).This efficiency will be lowered significantly when we apply high invariant mass cuts.We account for that by taking p jet T ∼ max(p jet T , m cut jj /3) in our estimate of the typical p Λ T .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 Λ baryon production in b-hadron decays, after subtracting the signal contribution.We have already discussed this measurement, eq.(6.6), in section 6.3.2 in the context of the background for the semileptonic cc channel.Following similar arguments, including an assumption of a 10% probability of having a µ − alongside the Λ (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 Λ + c that are listed in table 19. 14 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 Table 20: Properties of the b 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 ΛX sample contain a µ − .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 Λ + c shown in table 19.
on the muon p T fraction z.After subtracting that contribution we obtain a sample purity of f ≈ 57%.

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 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.

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 [36,37]. 15The dataset contains about N 0 = 10 10 pp → b 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 b polarization measurements.We 15 Similar data parking was done during part of Run 3 [118].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.
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 where we accounted for the fact that only half of the triggering decays come from a b (while the other half are from a 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 → c → µ chain relative to the contribution of the direct b → µ 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: ) For the semi-inclusive selection, our rough estimate for the Λ reconstruction efficiency, ϵ Λ , will be as follows.Since the muon that triggers the event, with a p T threshold that varied between 7 and 12 GeV [36,37], is produced in the Λ b decay together with the Λ + c , which subsequently decays to the Λ and additional particles, we estimate p Λ T to be around a few GeV.As a result, the Λ baryons will usually decay sufficiently early to be reconstructed by an analogue of the LRT algorithm [87-89], but not at the shortest distances where the efficiency is maximal.We will therefore take ϵ Λ ≃ ϵ 2 track with ϵ track ≈ 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 where we took a ballpark figure of Aϵ ∼ 0.2 for the acceptance times efficiency on the nontriggering side of the event (including the acceptance in η).The contribution of b → c → µ 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 .28)

Expected Precision
The expected precision of the b b polarization and spin correlation measurements for either a muon or neutrino spin analyzer can be computed using eqs.(4.18)-(4.20).As indicated in table 1, the neutrino in the Λ 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 |α µ − | and the spin correlation uncertainties by α 2 µ − , where α µ − ≈ −0.26.
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.

Summary and Discussion
In this work, we analyzed the feasibility of measuring quark polarization and spin correlations in high-energy pp → q 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 Λ b decays, with dijet invariant mass cuts up to ∼ 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 Λ + 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 ss 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.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 channel → inclusive inclusive/inclusive inclusive/exclusive 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 that will be measured in the spin correlation analyses depend on the polarization retention factors r L and r T via eq.(4.12), 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 ), 16 one can extract r L and r T using the (partly redundant) relations 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. [35]).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 t [18] and W +c [21] samples.The polarization and spin correlation measurements can in principle also be utilized to probe for BSM contributions to pp → q 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.The analytical results are in the limit of an infinite number of bins, while the numerical (Monte Carlo) results are for 10 bins.
where the function Φ is the Lerch transcendent, Li 2 is the dilogarithm, and c in eq.(A.14) stands for C ± ij /2.In figure 5 we plot the uncertainties of the coefficients as a function of the coefficient values based on the analytic formulas in eqs.(A.12)-(A.14),which correspond to the limit of a large number of bins.We also show the results of a Monte Carlo simulation of a χ 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, ∆X, which prevents bins much smaller than ∆X from being useful.
While the polarization and spin correlation components (b ± i , c ij , c ℓ ) 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 α ± in eq.(4.12) will limit the angular distribution coefficients B ± i , C ii , and C ± ij /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

Figure 1 :
Figure1: The fragmentation fraction for z > z 0 , for s → Λ, s → Λ, and s → Λ/ Λ, where Λ/ Λ denotes that either a Λ or Λ is produced in the s-quark hadronization.The Pythia simulation was run at √ s = 14 TeV with p T > 400 GeV and |η| < 2.5 cuts on the jets, which were clustered with the anti-k t jet algorithm with radius R = 0.4[52,53] and matched to the parton-level s quarks.We also show the data-based AKKII results[48].

Figure 2 :
Figure 2: Distributions of X = cos θ + i cos θ − i (left) and X = cos θ + i cos θ − j ± cos θ + j cos θ − i (right), from which the spin correlations can be extracted.The distributions are based on eqs.(4.15)-(4.16),and are shown for three different values of the coefficient C ii (left) and C ± ij /2 (right).

Figure 3 :
Figure3: Expected number of events with reconstructed Λ baryons satisfying z > z 0 for the signal originating from ss events and backgrounds with the final state partons specified in the legend, for Run 2 (left) and the HL-LHC (right).

Figure 4 :
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).(Theplot normalizations do not include the branching ratios.)

Figure 5 :
Figure 5: Statistical uncertainty (multiplied by √ N , where N is the number of events) of the angular distribution coefficients B ± i (black), C ii (blue), and C ± ij /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.

Table 2 :
P and CP properties of the polarization and spin correlation components.

Table 3 :
Spin correlations in b b and cc events with and without cuts for Run 2 ( √ s = 13 TeV).For comparison, we also show the same quantities for the t t case.The uncertainties shown are the statistical uncertainties of our simulation.

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 cc ) 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.

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.
we again rely on the LRT algorithm [87-89].As discussed in section 6.2.1, for Run 2, in the range of Λ decay radii d T up to d max T

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 ∆c ij(ℓ) denotes the uncertainties in c ij and c ℓ , which are of the same size.

Table 14 :
Cross sections, expected numbers of signal events and expected statistical uncertainties for the cc 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).

Table 17 :
HL-LHC cross sections (with trigger-motivated cuts) and signal event counts

Table 21 :
∆c ii r i r j ∆c ij(ℓ) r 2 i ∆c ii r i r j ∆c ij(ℓ) ∆c ii r i r j ∆c ij(ℓ) r 2 i ∆c ii r i r j ∆c ij(ℓ) Expected statistical uncertainties for the polarization and spin correlation measurements in b b in Run 2 with a neutrino spin analyzer for different selection channels.The shorthand ∆c ij(ℓ) denotes the uncertainties in c ij and c ℓ , which are of the same size.channel→inclusiveinclusive/inclusive inclusive/exclusivem jj cut [GeV] r i ∆b ± i r 2 i ∆c ii r i r j ∆c ij(ℓ) r 2 i ∆c ii r i r j ∆c ij(ℓ) semi-inclusive semi-inclusive/semi-inclusive semi-inclusive/inclusive m jj cut [GeV] r i ∆b ± i r 2 i ∆c ii r i r j ∆c ij(ℓ) r2i ∆c ii r i r j ∆c ij(ℓ) ∆c ii r i r j ∆c ij(ℓ) r 2 i ∆c ii r i r j ∆c ij(ℓ)

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