Transverse polarization of Lambda hyperons in hadronic collisions

Since our goal is to investigate the potential of extracting the flavor dependence of the polarized FF in hadronic collisions, we first present the differential cross section of the single inclusive jet production with a given flavor in $pp$ collisions. Although the flavor of a reconstructed jet in experiments is not well defined, we still employ this phrase since we have limited ourselves at the LO approximation. A jet is simply a parton at LO. The flavor dependent cross section is given by

where $f_{a/b}$ is the parton distribution function with $x_{i}$ the momentum fraction of the hadron carried by the parton and $\mu_{f}$ the factorization scale, $y$ and $\bm{k}_{T}$ denote the rapidity and transverse momentum of parton $j$ which is also identified as the jet measured in experiments, and $y_{2}$ denotes the rapidity of unmeasured final state parton $c$. The momentum fractions $x_{1}$ and $x_{2}$ can be calculated with $x_{1}={k_{T}}(e^{y}+e^{y_{2}})/\sqrt{s}$ and $x_{2}={k_{T}}(e^{-y}+e^{-y_{2}})/\sqrt{s}$ for massless partons. $d\hat{\sigma}_{a+b\to j+c}/d\hat{t}$ denotes the partonic cross section. To compare with the single inclusive jet cross section measured in the experiments, we have to sum over all possible “jet flavors” since current experimental instruments cannot distinguish jet flavors.

To quantify the flavor component of final state jets, we define the production ratio of a given jet flavor as

The cross section of hadron within jet production at the LO can be obtained by convoluting the above jet cross section with the corresponding jet FF Yuan:2007nd ; Kang:2017btw . We obtain

where $z$ is the momentum fraction of the parton carried by the final state hadron and $\bm{p}_{\Lambda T}$ is the transverse momentum of $\Lambda$ with respect to the jet momentum. Here $M_{\Lambda}$ is the Lambda mass and $S_{\Lambda T}$ is the transverse spin vector of $\Lambda$. $D_{1j}^{\Lambda}$ is the unpolarized FF, and $D_{1T,j}^{\perp\Lambda}$ is the polarized FF.

The transverse polarization of final state $\Lambda$ hyperons along the $\bm{n}\equiv\frac{\bm{k}_{j}\times\bm{p}_{\Lambda}}{|\bm{k}_{j}\times\bm{p}_{\Lambda}|}$ direction is given by

Here $\Lambda^{\uparrow}$ indicates $\bm{S}_{\Lambda T}=\bm{n}$, and $\Lambda^{\downarrow}$ means $\bm{S}_{\Lambda T}=-\bm{n}$.

Inserting Eq. (3) into Eq. (4), it is straightforward to obtain

Since this paper mainly focuses on the flavor dependence of the polarized FF $D_{1T,j}^{\perp\Lambda}$, we neglect the QCD evolution effect for simplicity and take Gaussian ansatz for $\bm{p}_{\Lambda T}$ distribution, namely

Here, $\Delta_{\Lambda}$ is the Gaussian width which, as an approximation, is chosen the same between the unpolarized and polarized FFs. With the Gaussian ansatz and taking average over the magnitude of $k_{T}$, the final expression for the Lambda polarization can be simplified to

In the numerical calculation, we employ the DSV parametrization deFlorian:1997zj to provide the unpolarized Lambda FF $D_{1j}^{\Lambda}$ and mainly employ the isospin symmetric CLPSW parametrization Chen:2021hdn for the polarized one $D_{1T,j}^{\perp}$. For comparison, we have also used the isospin violated DMZ DAlesio:2020wjq and CKT Callos:2020qtu parametrizations in some calculations.

Furthermore, in the hybrid factorization framework, the factorization scale of the initial state collinear parton distribution functions (PDFs) is usually chosen to be $k_{T}$ with $k_{T}$ being the transverse momentum of the jet in the lab frame. However, that of the final state TMD jet FFs is usually chosen to be $k_{T}R$ with $R$ being the jet cone size. In the numerical evaluation, the DGLAP evolution of collinear PDFs has been taken care of by the CT18 PDF Hou:2019efy . The QCD evolution of the final state $D_{1T}^{\perp}$ has been extensively investigated in Gamberg:2021iat ; Kang:2010xv . As discussed in Appendix C, this effect also does not change our conclusion. Therefore, we take the Gaussian ansatz for simplicity.

To demonstrate the potential of revealing the flavor dependence, we first calculate the production ratios of different flavor jets in $pp$ collisions at the RHIC energy. The numerical results at the LO accuracy are presented as a function of jet $k_{T}$ in the upper panel of Fig. 1 and as a function of jet rapidity in the lower panel, where we employ CT18 PDFs Hou:2019efy . The qualitative feature is well-known: the valance quark contribution increases with increasing $k_{T}$ and increasing rapidity, while the gluon contribution decreases. The same feature is also exhibited at the LHC energies, as shown in Figs. 12 and 13 in the appendix. Therefore, by comparing the Lambda transverse polarization at different rapidities and jet transverse momenta, we obtain the power to extract the polarized gluon FFs which has been poorly constrained in previous studies DAlesio:2020wjq ; Callos:2020qtu ; Chen:2021hdn .

For instance, the CLPSW parametrization Chen:2021hdn is proposed to describe the Lambda transverse polarization in $e^{+}e^{-}$ annihilation at LO. Therefore, the gluon fragmentation function is absent in the previous analysis. However, it is important in making predictions for the hadronic collisions, since they amount to a sizable contribution. In this study, we propose two extreme prescriptions. In the first prescription, we simply assume that the gluon does not contribute to the Lambda polarization by setting $D_{1T,g}^{\perp}=0$. In the second prescription, we assume that $D_{1T,g}^{\perp}=D_{1T,u}^{\perp}=D_{1T,d}^{\perp}$. The numerical results are shown in Fig. 2. The upper panel shows the results using the first prescription, while the lower panel shows those in the second prescription. The real case should deviate from these two simple prescriptions and might be somewhere in between. However, the difference between two panels demonstrates this process is indeed sensitive to the gluon FF. Therefore, it is plausible to extract the gluon FF. More numerical predictions for the LHC energies are presented in the appendix.

The second process we investigate is the single inclusive jet production in $p\bar{p}$ collisions. Unlike the case in $pp$ collisions, the forward and backward directions are no longer symmetric in $p\bar{p}$ collisions. We show the production rate of different flavor jets as a function of rapidity in Fig. 3. It is apparent that there are more $u$ quark jets in the forward rapidity (proton-going direction) and there are more $\bar{u}$ quark jets in the backward rapidity (antiproton-going direction). This forward-backward asymmetric enables us to examine the difference between quark and antiquark fragmentation functions. This point can be fully appreciated from the numerical results shown in Fig. 4. The CLPSW, DMZ and CKT parametrizations have parametrized the $u$ and $\bar{u}$ fragmentation differently, so that the magnitude of forward-backward asymmetry also varies. Therefore, including the Tevatron experiment in the global analysis can cast more light on the flavor dependence of $D_{1T}^{\perp}$.

The last process we study in this subsection is the single inclusive jet production in $pA$ collisions. Similar to the $p\bar{p}$ collisions, there is also a forward-backward asymmetry. However, in this process, we can study the difference between $u$ and $d$ quark fragmentation function. We present the production ratios of different flavor jets as a function of jet rapidity in Fig. 5, which clearly shows that there are more $u$ quark jets in the proton-going direction and more $d$ quark jets in the nucleus-going direction. For nuclear modified PDFs we take EPPS21 sets Eskola:2021nhw .

As underlined in the previous work Chen:2021hdn , the Lambda hyperon is an isospin singlet. The isospin symmetry requires $D_{1T,u}^{\perp}=D_{1T,d}^{\perp}$. This requirement has been respected in the CLPSW parametrization, while it has been severely violated in the DMZ and CKT parametrizations. The answer to whether or not the isospin symmetry is violated in the hadronization process is vitally important for understanding strong interaction at the non-perturbative regime. We thus present the numerical results for the Lambda polarization at different rapidities in $pA$ collisions in Fig. 6. Apparently, there is almost no difference between forward and backward rapidities in the isospin symmetric CLPSW parametrization. However, one can observe distinguish difference in the DMZ and CKT parametrizations. Therefore, future experimental measurements on this aspect can eventually clarify this issue.

To summarize, a global analysis of the Lambda transverse polarization in different kinematic regions in $pp$, $p\bar{p}$, and $pA$ collisions has great potential in pinning down the flavor dependence of the polarized fragmentation function $D_{1T}^{\perp}$.

Besides the single-inclusive jet production, another important channel deserving study is the $\gamma/Z^{0}$-boson associated jet production. Unlike the single-inclusive jet production dominated by strong interaction, this process is an electroweak process. The electroweak vertex modifies the production ratios of jets with different flavors. Therefore, the phenomenological study of this channel provides a complementary platform to investigate flavor dependence.

To be more specific, the process we consider is $p+p\to{\rm jet}(\to\Lambda)+\gamma/Z^{0}+X$. Here, we only consider the prompt production for the final state photon or $Z^{0}$-boson. At LO, there are two partonic channels that contribute, namely $q+g\to q+\gamma/Z^{0}$ and $q+\bar{q}\to g+\gamma/Z^{0}$. Therefore, the gluon contribution is strongly suppressed at low jet $k_{T}$ and in the central rapidity.

Furthermore, for the photon associated jet production, the partonic cross sections are given by

Due to the $e_{q}^{2}$ factor, the $u$ quark contribution dominates in almost the whole kinematic region. However, as shown in Fig. 7, the gluon contribution is flat with increasing rapidity. This is a feature not expected in the single inclusive jet production where the gluon contribution is strongly suppressed in the forward rapidity. Therefore, this process provides with a fresh angle in the global analysis.

On the other hand, for the $Z^{0}$ associated jet production, the partonic cross sections are given by

Here $M_{Z}$ is the mass of $Z^{0}$-boson, $\theta_{W}$ is the Weinberg angle and $c_{1}^{q}\equiv(c_{V}^{q})^{2}+(c_{A}^{q})^{2}$. In this case, the contribution from $u$ quark is still important. However, it does not play the dominant role any more. As shown in Fig. 8, $u$, $d$ and gluon contributions are, roughly speaking, equally important.

We present our numerical predictions for $\Lambda$ transverse polarization in the vecor boson associated jet production processes in $pp$ collisions in Fig. 9. The left plot is the result for photon-jet production process and the right plot is that for $Z^{0}$-jet production. Both show that polarization can reach as large as $10\sim 20\%$.

In the relativistic heavy-ion collisions, most of the hard scatterings come from the QCD events where both nuclei break up. However, since the large nucleus is accompanied by an enormous number of quasireal photons, it is also quite plausible that only one of the quasireal photons interact with the other nucleus in the peripheral nucleus-nucleus collisions. In this case, the projectile nucleus (the photon-providing one) does not break up and appears in the final state. The target nucleus (photon-striking one) breaks. In this work, we refer the photon-going direction as the forward direction.

According to the Equivalent Photon Approximation (EPA), we have the following collision

where the photon distribution of the projectile nucleus is given by Baltz:2007kq

Here, $\zeta=x_{\gamma}M_{{p}}(R_{\rm{A}}+R_{\rm{B}})$ with $x_{\gamma}$ being the light-cone momentum fraction carried by the quasireal photon. $Z$ is the nuclear charge number, $M_{{p}}$ is the mass of the proton. $R_{\rm{A}}$ and $R_{\rm B}$ are the nucleus radiuses of the projectile and target nuclei. This process resembles SIDIS quite a lot. However, the photon in this process is almost onshell which is different from the case in SIDIS.

The partonic hard scatterings at LO are $\gamma q\to gq$ and $\gamma g\to q\bar{q}$ with partonic cross sections given by

As shown in Fig. 10, the gluon contribution is very important in the backward rapidity. However, it becomes negligible in the forward rapidity. Therefore, this process also offers the optimal platform to investigate the gluon FFs.

We show our numerical predictions for Lambda transverse polarization in photon-nucleus collisions in Fig. 11. The results without gluon contribution are shown in the left panel, and those with gluon contribution are shown in the right panel. As expected, the difference between these two becomes sizable in the backward rapidity, registering the potential to probe the gluon FFs.