The decay contribution to the parity-odd fragmentation functions

To be more specific, we first consider the $H\to h+X$ decay process with both $H$ and $h$ being spin-$1/2$ hadrons. We use ${\cal D}^{H}_{q}(\lambda_{q},\lambda_{H};z^{\prime},p_{T}^{\prime})$ to denote the helicity-dependent fragmentation function of the parent hadron $H$ and ${\cal D}^{h,H}_{q}(\lambda_{q},\lambda_{h};z,p_{T})$ to denote the decay contribution to that of the daughter hadron $h$ with $\lambda_{q}$ and $\lambda_{h/H}$ being the helicities of the quark and the corresponding hadron. $X$ is a collection of all the other unmeasured particles. Here, $z$ and $z^{\prime}$ are the longitudinal momentum fractions of daughter and parent hadrons respectively, while $p_{T}$ and $p_{T}^{\prime}$ are the corresponding transverse momenta with respect to the quark momentum.

For the parent hadron $H$ production, we neglect decay contributions and only consider the prompt production contribution, which is parity invariant. Therefore, we only have the following two possible structures as long as we only consider the longitudinal polarizations of the quark and the hadron,

where $D_{1,q}^{H,{\rm dir}}(z,p_{T})$ and $G_{1L,q}^{H,{\rm dir}}(z,p_{T})$ are the unpolarized fragmentation function and the longitudinal spin transfer for the prompt production respectively.

Once the transverse polarizations of the quark and/or the hadron are taken into account, the other six leading twist fragmentation functions can also be defined this way. These structures are beyond the scope of our current study. In this paper, we only study the longitudinal polarizations of both the quark and the hadron.

Since we are going to consider the general decay kinematics, we keep the transverse momentum dependence at the current stage. These transverse-momentum-dependent fragmentation functions are related to their collinear partners by

Thanks to the weak decay contribution, the total fragmentation function is allowed to break the parity symmetry. In general, we have

Here, $D_{1L,q}^{{\rm PV},h}$ and $G_{1,q}^{{\rm PV},h}$ represent the number densities of producing longitudinally polarized hadrons from unpolarized quarks and of producing unpolarized hadrons from longitudinally polarized quarks respectively. They are apparently P-odd. Notice that we have employed a different naming system with that in Ref. Kang and Kharzeev (2011), where $\tilde{D}$ in their paper is denoted as $G_{1,q}^{{\rm PV},h}$ in this paper. In this work, we adopt the following convention: (i) $D$ denotes a fragmentation function of an unpolarized quark and $G$ denotes that of a longitudinally polarized quark. (ii) The subscript $L$ is used to indicate the fragmentation function for the production of longitudinally polarized hadrons. (iii) The “PV” in the superscript of $D$ or $G$ is employed to represent the parity violation. We do not consider the $\theta$-vacuum term in the QCD Lagrangian so that the P-odd fragmentation functions solely arise from the weak decay.

Convoluting the parent hadron fragmentation function with the decay kernel function, we arrive at

where ${\cal D}^{h,H}$ denotes the decay contribution of the parent hadron $H$ to the fragmentation function of $h$ and ${dN(\lambda_{h},\lambda_{H})}/{dzd^{2}p_{T}}$ is the number density for a parent hadron $H$ with helicity $\lambda_{H}$ and four-momentum being specified by $z^{\prime}$ and $p_{T}^{\prime}$ to produce a daughter hadron $h$ with helicity $\lambda_{h}$ and four-momentum being labeled by $z$ and $p_{T}$. Notice that the helicity direction is frame-dependent for massive particles. Throughout this paper, the helicity is always defined in the lab frame. The decay contributions to spin-dependent fragmentation functions thus depend on the parton energy. Unlike the factorization scale dependence which obeys the QCD evolution equation, this dependence on the parton energy arises from the kinematics rather than from the factorization theorem.

For a two-body decay process, the differential distribution $dN/dzd^{2}p_{T}$ is given by

where $\alpha$, $\beta$ and $\gamma$ are constant parameters, $M_{H}$ is the mass of $H$, $\bm{p}_{h}^{*}$ is the three-momentum of $h$ in the $H$-rest frame, $p_{H}$ and $p_{h}$ are the four-momenta of $H$ and $h$ in the lab frame, $\bm{\omega}_{i}=\bm{p}_{H}/|\bm{p}_{H}|$ is the helicity direction of $H$ and $\bm{\omega}_{f}$ is the helicity direction of $h$ in the $H$-rest frame. Due to the Wick rotation, the helicity direction $\bm{\omega}_{f}$ is not along $\bm{p}_{h}^{*}$. It is given by

where, $E_{H/h}$ is the energy of $H/h$ in the lab frame, $\bm{p}_{H/h}$ is the three momentum of $H/h$ in the lab frame, $E_{h}^{*}$ is the energy of $h$ in the $H$-rest frame and $m_{h}$ is the mass of $h$. Despite the involved expression, the angle between $\bm{\omega}_{f}$ and $\bm{p}_{h}^{*}/|\bm{p}_{h}^{*}|$ is just the well-known Wick-angle $\theta_{\rm Wick}$. Taking the $m_{h}\to 0$ limit, we find that $\cos\theta_{\rm Wick}=\bm{\omega}_{f}\cdot\bm{p}_{h}^{*}/|\bm{p}_{h}^{*}|=1$. This means that the helicity of a massless particle does not change under Lorentz boosts.

We only consider the prompt production for the parent hadron $H$. Therefore, only parity-even terms survive. Benefiting from the number density interpretation of leading twist fragmentation functions, it is straightforward to obtain the decay contribution to the $h$ fragmentation function as

Furthermore, we can integrate over $d^{2}p_{T}$ and obtain

where the kernel functions are given by

Although the above expressions are complicated, the physical interpretations of these kernel functions are straightforward. For instance, $K_{U\to L}=\alpha\cos\theta_{\rm Wick}$ with $\theta_{\rm Wick}$ being the Wick angle. $K_{L\to U}=\alpha\cos\theta^{*}$ with $\theta^{*}$ the polar angle of $h$-momentum in the $H$-rest frame. $K_{L\to L}=\cos\theta^{*}\cos\theta_{\rm Wick}+\gamma\sin\theta^{*}\sin\theta_{\rm Wick}$. Furthermore, we have also run a simple Monte-Carlo simulation to perform the cross-check. Numerical evaluation with the above kernel functions can exactly reproduce the simulation results.

The polarization of a spin-$3/2$ hadron consists of a lot of modes. Each polarization mode is associated with multiple fragmentation functions Zhao et al. (2022) describing the production of spin-$3/2$ hadrons. Unfortunately, all of the spin-dependent fragmentation functions are poorly studied numerically so far. In this study, we average over the polarization of the parent hadron and only study the induced longitudinally polarization of the final state $\Lambda$-hyperon, and we arrive at

where $\alpha$ is the decay parameter of spin-$3/2$ hadron measuring the magnitudes of the parity violation in the weak decay. The rest of the calculation coincides with that for the spin-$1/2$ to spin-$1/2$ decay.

Besides the P-odd fragmentation functions of the $\Lambda$-hyperon, we also want to investigate those of pions. Most of the vector mesons decay through the strong interaction and therefore do not bring in any P-odd effect. Effectively, we only need to consider the weak decay of spin-$1/2$ hadrons. It is straightforward to obtain the decay kernel function for this case from Eq. (6) by setting $\lambda_{h}=0$ and $\bm{\omega}_{f}=0$. We obtain

When the formula is served like Eq. (21), the sign of the $\alpha$ parameter depends on which hadron is the one of interest. For instance, in the $\Lambda\to p+\pi^{-}$ decay process, $\alpha_{p}=-\alpha_{\pi}$.

As mentioned above, $D_{1L}^{\rm PV}(z)$ is responsible for generating longitudinally polarized hadrons from unpolarized partons. Therefore, we expect that there will be a signal of spontaneous longitudinal polarization for the $\Lambda$-hyperons produced in unpolarized high-energy collisions. The produced $\Lambda$-hyperons can further decay into the $p+\pi^{-}$ pair through the weak interaction. The polarization of the proton is usually averaged over since it is extremely difficult to measure with current experimental instruments. Eventually, the angular distribution of the final state proton follows the $dN/d\Omega\propto 1+\alpha{\cal P}_{L}\cos\theta^{*}$ in the $\Lambda$-rest frame. Here, ${\cal P}_{L}$ is the polarization of $\Lambda$, $\alpha$ is a constant, and $\theta^{*}$ is the polar angle of the proton momentum in the $\Lambda$-rest frame. Therefore, the $\Lambda$ polarization, ${\cal P}_{L}$ can be easily measured by extracting $\langle\cos\theta^{*}\rangle$ through its self-analyzing decay.

Quarks produced in $e^{+}e^{-}$ annihilation can acquire longitudinal polarization through the $Z^{0}$-boson exchange diagram and the interference term. The quark longitudinal polarization is significant at the $Z^{0}$-pole and it is negligible at relatively low energy. Eventually, this longitudinal polarization of quark can be inherited by the fragmented hadron through $G_{1L}$ which is a P-even fragmentation function and becomes the background of our study. The second part of contribution comes from P-odd fragmentation functions. At this point, we take a brief detour to the $pp$ collisions. In $pp$ collisions, the partonic collisions are dominated by the strong interaction. The fragmenting hadron is not polarized. Therefore, the unpolarized $pp$ collisions provide a better platform to probe the P-odd fragmentation functions. However, in this work, we still utilize the $e^{+}e^{-}$ annihilation process as an example to demonstrate this effect. Particularly, the ratio between these two terms in Eq. (31) offers information on the magnitude of P-odd effects. The final contribution to the longitudinal polarization of $\Lambda$ in $e^{+}e^{-}$ annihilation reads

Here, $\omega_{q}=e_{q}^{2}+c_{1}^{e}c_{1}^{q}\chi+\chi_{\rm int}^{q}c_{V}^{e}c_{V}^{q}$ is the weight function for the unpolarized quark production and $\Delta\omega_{q}=-c_{1}^{e}c_{3}^{q}\chi-c_{V}^{e}c_{A}^{q}\chi_{\rm int}^{q}$ with $c_{1}^{e}=(c_{V}^{e})^{2}+(c_{A}^{e})^{2}$, $c_{1}^{q}=(c_{V}^{q})^{2}+(c_{A}^{q})^{2}$, $c_{3}^{q}=2c_{V}^{q}c_{A}^{q}$, $\chi=Q^{4}/[(Q^{2}-M_{Z}^{2})^{2}+\Gamma_{Z}^{2}M_{Z}^{2}]\sin^{4}2\theta_{W}$, $\chi_{\text{int }}^{q}=-2e_{q}Q^{2}(Q^{2}-M_{Z}^{2})/[(Q^{2}-M_{Z}^{2})^{2}+\Gamma_{Z}^{2}M_{Z}^{2}]\sin^{2}2\theta_{W}$. $M_{Z}$ is the mass of $Z^{0}$-boson, $\Gamma_{Z}$ is the width of the $Z^{0}$-boson mass, $\theta_{W}$ is the Weinberg angle and $Q$ is the center-of-mass energy of the colliding leptons. For simplicity, we have only shown the contribution from quark to $\Lambda$. The contribution from antiquark is small but should be taken into account. However, notice that $\omega_{\bar{q}}=\omega_{q}$ and $\Delta\omega_{\bar{q}}=-\Delta\omega_{q}$. The first one comes from the simple fact that the cross section of quark production is the same with that of antiquark production. The second one indicates that quark and antiquark have the opposite helicities.

To proceed, a parameterization of the longitudinal spin transfer $G_{1L,q}(z)$ is required. We take $G_{1L,q}^{\Lambda}(z)=z^{0.8}D_{1,q}^{\Lambda}(z)$ and only consider contributions from light quarks. Such a simple parameterization can already provide a sufficiently good description of the $\Lambda$-hyperon polarization measured by the LEP experiment Buskulic et al. (1996); Ackerstaff et al. (1998). With these approximations at hand, we present our numerical result for the longitudinal polarization of $\Lambda$ hyperons at different collision energies in l.h.s. of Fig. 2. To illustrate the relative contribution from the P-odd fragmentation function, we define the ratio ${\cal R}(z)$ as

and show the corresponding numerical results in r.h.s. of Fig. 2. It is obvious that the dominant contribution to the hadron polarization arises from the P-odd effect when $Q\ll M_{Z}$. The relative contribution from $D_{1L}^{\rm PV}$ decreases as the collisional energy increases and reaches the minimum at the $Z^{0}$-pole. Thus, it is better to access this P-odd fragmentation function at relatively low energy collisions.

Furthermore, we show the longitudinal polarization of the $\Lambda$ hyperons produced in $e^{+}e^{-}$ annihilations at $\sqrt{s}=10$ GeV in the l.h.s. of Fig. 3 and the relative contribution from the P-odd fragmentation function as a function of collisional energy at different $z$ in the r.h.s. of Fig. 3. Notice that the ratio ${\cal R}$ even reaches unity at about $Q\sim 30$ GeV, which indicates that the contribution from the spin transfer of polarized quarks vanishes. The reason is straightforward. Both the $Z^{0}$-boson exchange term and the $\gamma Z^{0}$ interference term contribute to the quark polarization. However, the signs are different for $d$ and $s$ quarks. Namely, the interference term contributes to the positive polarization while the $Z^{0}$-boson exchange diagram contributes to the negative. At low-$Q^{2}$, the dominant contribution comes from the interference term, which results in positive polarization for the final state $\Lambda$ hyperon. However, at higher energy, $Z^{0}$-boson exchange diagram takes charge and eventually turns the polarization of $\Lambda$-hyperons into negative. Therefore, at a certain energy scale, there is a cross-over where the P-even contribution vanishes.

Although quarks produced through electromagnetic interaction in the $e^{+}e^{-}$-annihilation process are not polarized individually, the polarization correlation of quark and antiquark takes the maximum value Chen et al. (1995); Zhang and Wei (2023). The polarization correlation is thus proposed as a new probe for the $G_{1L}$ fragmentation function. In this work, we focus on the P-odd effect of this polarization correlation of the quark-antiquark pair.

Combining the P-even and P-odd contributions, the complete cross section of spin-0 dihadron production in $e^{+}e^{-}$ annihilation process is given by Boer et al. (1997)

where $\omega_{q}$ and $\Delta\omega_{q}$ have already defined in the previous subsection. Again, although we only lay out the contribution from $q\to h_{1}$ and $\bar{q}\to h_{2}$ here, the exchange between quark and antiquark is implicit.

The first term in Eq. (33) computes P-even contributions with the other terms evaluating P-odd ones. The emergence of P-odd terms clearly modifies the production rate of dihadron production. Depending on the hadron species, the modification could be a suppression or an enhancement. The magnitude of this modification can be quantified by ${\cal M}(z_{h_{1}},z_{h_{2}})$ which is defined as

The sign of $\cal M$ indicates if the modification is a suppression or an enhancement. If $\cal M$ is positive, we have a suppression. Otherwise, we have an enhancement.

We consider the $\pi^{+}\pi^{-}$-pair production and first show our numerical estimate of ${\cal M}(z_{\pi^{+}},z_{\pi^{-}})$ at $z_{\pi^{+}}=z_{\pi^{-}}=z$ as a function of $z$ in the left panel of Fig. 4. Effectively, the last two terms in Eq. (34) largely cancel each other due to the charge conjugation symmetry by choosing $z_{\pi^{+}}=z_{\pi^{-}}$. Therefore, we find ${\cal M}(z,z)\simeq\sum_{q}\omega_{q}[G_{1,q}^{\rm PV}(z)]^{2}/\sum_{q}\omega_{q}[D_{1,q}(z)]^{2}$ which is positive definite. The ratio $|G_{1,q}^{\rm PV}(z)/D_{1,q}(z)|$ is at the order of $10^{-2}$, which explains why the signal is tiny. As long as $z_{\pi^{+}}$ and $z_{\pi^{-}}$ are not the same, the polarized quark term contributes. This brings a significant parton energy dependence since the quark polarization is negligible at low $Q$ and takes the maximum at $Q=M_{Z}$. The negligible parton polarization at the low $Q$ region results in a tiny modification of dihadron pair production. However, the magnitude of the modification becomes 100 times larger at around the $Z^{0}$-pole. This feature is shown in the middle and right panels of Fig. 4.

In light of the above discussion, we recommend the following kinematic region to minimize the weak decay contribution and thus to better probe the strong-parity-violating effect. (1) Measure the $\pi^{+}\pi^{-}$-pair production in low-$Q$ $e^{+}e^{-}$ experiments; (2) Focus on the large $z$ region, which strongly suppresses the weak decay contribution; (3) Set $z_{\pi^{+}}$ and $z_{\pi^{-}}$ to be equal to minimize the contribution from polarized quarks.