Probing the Spatial Coherence of Chiral-Imbalance Fluctuations with Hyperon Spin Correlations

Introduction

Topological transitions of non-Abelian gauge fields can generate fluctuating axial charge through the chiral anomaly, giving rise to local chiral imbalance in the quark-gluon plasma (QGP) [1, 2, 3, 4]. An effective axial chemical potential, $\mu_5$, characterizes this imbalance in a near-equilibrium description. In the strong magnetic fields generated in relativistic heavy-ion collisions, chiral imbalance can drive the chiral magnetic effect (CME) [5, 6]. Experimental searches for the CME have extensively employed charge-dependent azimuthal correlations, particularly the $\Delta\gamma$ observable [7, 8, 9, 10]. However, substantial backgrounds from local charge conservation and resonance decays complicate the extraction of the CME signal [11, 12, 13, 14].

Beyond constraining the overall strength of axial-charge fluctuations, an important open question is how chiral-imbalance fluctuations are correlated in space [3, 6, 7]. An event-wise uniform chiral imbalance and spatially fluctuating chiral domains can have the same fluctuation strength but distinctly different spatial correlations. Conventional CME-sensitive observables, which generally integrate over broad regions of the evolving collision system, do not directly resolve these spatial correlations. This motivates the search for complementary observables sensitive to the spatial organization of axial-charge fluctuations.

Hyperon-pair spin correlations offer a promising probe of this structure. Local chiral imbalance can bias quark helicity populations. Although sign-symmetric chiral fluctuations can yield vanishing event-averaged spin polarization, two-particle spin correlations can remain finite. If this partonic response is transferred to final-state $\Lambda$ and $\bar{\Lambda}$ spins, their parity-violating weak decays make the resulting spin correlations experimentally accessible through joint decay-angular distributions [15, 16, 17]. Their dependence on the momentum-space pair separation $\Delta R=\sqrt{(\Delta\eta)^2+(\Delta\phi)^2}$ can retain information about the spatial organization of chiral fluctuations, because transport and collective expansion correlate final-state momenta with emission positions. However, $\Delta R$ is not a direct measure of the spatial separation between chiral domains.

In this Letter, we investigate differential hyperon-pair spin correlations using the AMPT model supplemented with chiral anomaly transport [18, 19, 20]. In the studied kinematic region, the $\mu_5$-induced correlations substantially exceed the thermal-vorticity-plus-shear baseline, and the $\Lambda\bar{\Lambda}$ correlation closely follows the average of the $\Lambda\Lambda$ and $\bar{\Lambda}\bar{\Lambda}$ correlations. This motivates a common-mode observable combining all three pair channels. We further find that the ratio of short- to long-range correlations depends strongly on the effective spatial coherence scale but only weakly on the imposed $\mu_5$ magnitude within the parameter range explored. Together with the absolute correlation strength, this ratio provides complementary, experimentally testable constraints on the spatial organization of chiral-imbalance fluctuations.

Physical mechanism and measurable spin correlations

In a parity-symmetric event ensemble, configurations with opposite topological charges have no preferred sign. Consequently, the event-averaged chiral imbalance vanishes, while its local fluctuations and spatial correlations can remain finite. In the massless-quark and weak-magnetic-field limits, the axial-charge density $n_5=n_R-n_L$ is related to $\mu_5$ by

where $T$ and $\mu$ denote the temperature and quark chemical potential, respectively, with the normalization convention of the cited calculation [5, 21]. This relation illustrates how a local $\mu_5$ generates a local axial-charge imbalance.

For a quark or antiquark species $a$, we define the local helicity asymmetry as

where $f_{a,\pm}$ are the phase-space distributions for helicity $h=\pm1$. In the massless limit, axial-charge imbalance is directly related to the imbalance of helicity populations, with the appropriate particle and antiparticle conventions. For the finite-mass strange quarks considered here, $H_a$ describes the helicity response to the local chiral field and is not assumed to be linearly proportional to $\mu_5$.

For a relativistic quark in a helicity eigenstate $h = \boldsymbol{\sigma} \cdot\hat{\mathbf{p}} = \pm1$, the spin is parallel or antiparallel to its momentum, $\mathbf{s} = (h/2)\hat{\mathbf{p}}$. For two such particles,

where $\theta_{ab}$ is the three-dimensional opening angle between their momenta.

For two spin-1/2 particles, we define the scalar spin correlation as [17, 22],

where $C_{ij}^{ab} = \langle\sigma_i^{(a)} \otimes\sigma_j^{(b)}\rangle$ is the spin-correlation tensor, with consistently defined spin axes.

For a fixed chiral-field configuration and fixed phase-space coordinates, we assume that the helicities of the two particles are independently sampled. Their conditional expectation value then factorizes as $\langle h_a h_b\rangle_{\mu_5} = \langle h_a\rangle_{\mu_5}\langle h_b\rangle_{\mu_5} = H_a H_b$, where subscript $\mu_5$ indicates finite chiral-field configuration. Averaging over field configurations gives

Thus, conditional independence does not imply the absence of an ensemble-averaged correlation: both particles can respond to the same fluctuating chiral field. The event-averaged helicity polarization can vanish for sign-symmetric fluctuations, while $\langle H_aH_b\rangle$ remains finite.

Using Eq. (2), the contribution induced by chiral-imbalance fluctuations, denoted by the superscript $(\chi)$, takes the schematic form

where subscript $\theta_{ab}$ indicates the averages are taken over events and particle pairs at a specified opening angle $\theta_{ab}$. The angular factor reflects helicity–spin locking, while $\langle H_aH_b\rangle_{\theta_{ab}}$ encodes the correlated helicity response to the fluctuating chiral field. This mechanism does not require direct pairwise spin interactions or quantum entanglement between the particles. The physical picture is illustrated in Figure 1.

To connect the partonic mechanism to experimentally accessible hyperons, we adopt the conventional SU(6) constituent-quark picture, in which the $ud$ pair in a directly produced $\Lambda$ carries predominantly zero spin and the hyperon spin is carried by its constituent $s$ quark; the corresponding assumption applies to $\bar{\Lambda}$ and $\bar{s}$ [23, 24]. Accordingly, the spin response of a directly produced $\Lambda$

$(\bar{\Lambda})$ is taken to follow that of its constituent $\bar{s}$ quark immediately before coalescence, with the spin vectors expressed in a consistent basis. This is a spin-transfer prescription in the numerical realization, rather than an exact identity for all hyperons; subsequent hadronic evolution can modify the resulting correlations.

We study the final-state hyperon-pair correlation $P_{AB}(\Delta R)$, where $A,B \in\{\Lambda,\bar{\Lambda}\}$ and $\Delta R = \sqrt{(\Delta\eta)^2 + (\Delta\phi)^2}$, $-\pi\leq\Delta\phi\leq\pi$. Here, $\eta$ is pseudorapidity and $\Delta\phi$ is the wrapped relative azimuthal angle. The dependence on $\Delta R$ is motivated by two complementary considerations.

First, $\Delta R$ is kinematically related to the opening angle $\Theta_{AB}$ between the final-state hyperon momenta. For small angular separations, $\Theta_{AB} \simeq\Delta R/\cosh\bar{\eta}$, where $\bar{\eta} = (\eta_A+\eta_B)/2$. Consequently, differential measurements in $\Delta R$ retain information about the angular dependence in Eq. (5), although the final-state hyperon opening angle is not identical to the partonic angle $\theta_{ab}$.

Second, transport and collective expansion establish statistical correlations between emission positions and final-state momenta [25]. Pairs selected in different $\Delta R$ intervals can therefore sample different distributions of source separations and, consequently, different correlations of the chiral field. If larger $\Delta R$ preferentially selects particles from more widely separated source regions, finite spatial coherence can suppress their helicity-pair response relative to that of small-$\Delta R$ pairs. This space–momentum mapping is dynamical and model dependent: $\Delta R$ is not a direct measure of the coordinate-space separation between chiral domains. The interplay of angular response and spatial coherence motivates studying differential rather than only pair-integrated spin correlations.

Experimentally, the scalar hyperon-pair correlation can be extracted from the acceptance-corrected joint weak-decay angular distribution, $\frac{1}{2}\frac{dN}{N\,d\cos\theta^*} = \frac{1}{2}[1+\alpha_A\alpha_B P_{AB}(\Delta R)\cos\theta^*]$, where $\alpha_A$ and $\alpha_B$ are the signed weak-decay parameters, and $\theta^*$ is the angle between the daughter (anti)proton momentum directions in their respective hyperon rest frames [17]. The measured $P_{AB}$ is the total spin correlation from all sources; isolating the contribution from chiral fluctuations requires controlling other sources of pair-spin correlations.

Experimentally benchmarked numerical realization.— To study the proposed spin-correlation response, we employ the string-melting AMPT model [18] supplemented with the Chiral Anomaly Transport (CAT) module [19, 20] for Au+Au collisions at $\sqrt{s_{NN}} = 200$ GeV. AMPT describes partonic evolution, quark coalescence, and hadronic rescattering, while CAT incorporates chiral transport in the presence of electromagnetic fields and axial-charge imbalance [26, 27, 28]. Following the spin-transfer prescription described above, we assign each $\Lambda$ ($\bar{\Lambda}$) the polarization of its constituent $s$ ($\bar{s}$) quark during quark coalescence [23]. Thermal-vorticity and shear [29, 30, 31] contributions are calculated separately from the local flow and temperature fields, providing a conventional spin-polarization baseline.

Rather than dynamically generating individual topological transitions, we impose controlled $\mu_5$ configurations to study how their spatial correlations are reflected in final-state hyperon spins. We consider two limiting cases: an event-wise uniform field, $\mu_5(x)=s\mu_{5,0}$, where $s=\pm1$ is sampled once per event; and a spatially fluctuating domain field, $\mu_5(x_i)=s_i\mu_{5,0}$, where $s_i=\pm1$ is independently sampled for each spatial cell, both with equal probabilities. The two configurations share the same ensemble mean and local variance, $\langle\mu_5\rangle=0$ and $\langle\mu_5^2\rangle=\mu_{5,0}^2$, but differ in their spatial correlations.

We characterize the spatial correlations of the imposed field through the two-point covariance $C_5(x_a,x_b)=\langle\mu_5(x_a)\mu_5(x_b)\rangle$, where the average is taken over field configurations and $\langle\mu_5(x)\rangle=0$. Here, $x_a$ and $x_b$ denote the locations at which the two particles sample the chiral field. For the event-wise uniform field, $C_5(x_a,x_b)=\mu_{5,0}^2$ throughout the system. For finite-size domains with independently sampled signs, $C_5=\mu_{5,0}^2$ within the same cell and vanishes between different cells. Although the field product between different cells is $\pm\mu_{5,0}^2$ in an individual event, its ensemble average is zero. The cell size $\ell_{\mathrm{cell}}$ therefore controls the effective spatial coherence of the imposed field, while the uniform configuration represents the long-coherence limit. Here, $\ell_{\mathrm{cell}}$ is a phenomenological coarse-graining scale rather than a direct measure of the microscopic sphaleron size.

Figure 2 benchmarks complementary ingredients of the numerical framework against existing STAR measurements. The global and longitudinal polarization observables [33, 35, 36, 37] constrain the conventional spin response, including contributions from thermal vorticity and shear [38, 39]. For the CME-sensitive $\Delta\gamma^{112}$ observable, including finite $\mu_5$ improves the description of the measured correlation in the present calculation. However, the substantial conventional backgrounds contributing to $\Delta\gamma^{112}$ prevent this comparison from uniquely determining the axial-charge fluctuation strength. Importantly, these observables do not independently resolve the spatial correlations of the axial-charge field. Differential hyperon-pair spin correlations can therefore provide complementary information on the spatial structure of chiral-imbalance fluctuations.

A common-mode response across pair channels

Figure 3(a) shows the differential hyperon-pair spin correlations in 20–60% Au+Au collisions at $\sqrt{s_{NN}} = 200$ GeV, for $|\eta| < 1$ and $0.5 < p_{\mathrm{T}} < 6$ GeV/$c$. At $\mu_5 = 0$, the baseline spin correlations generated by thermal vorticity and shear are substantially smaller than the finite-$\mu_5$ results over most of the investigated range, consistent with previous findings [40]. We therefore neglect the thermal-vorticity and shear contributions in the following analysis and focus on the correlations induced by chiral-imbalance fluctuations. With finite $\mu_5$, the correlations change from positive at small $\Delta R$ to negative at large $\Delta R$. This demonstrates the strong sensitivity of the pair observable to chiral imbalance within the present transport framework.

An additional feature is the approximate channel independence of this response: $P_{\Lambda\bar{\Lambda}}$ closely follows the average of $P_{\Lambda\Lambda}$ and $P_{\bar{\Lambda}\bar{\Lambda}}$. We quantify this consistency through

which is consistent with zero within statistical uncertainties [Fig. 3(b)]. This agreement motivates combining the three channels into a common-mode observable.

We therefore define the common-mode spin correlation as

The fixed $1 : 2 : 1$ combination preserves a channel-independent signal, $P^{(\alpha)}_{ab}(\Delta R) \simeq S(\Delta R)$, while canceling a background with the specific sign structure. $P^{(B)}_{ab}(\Delta R) = q_aq_bB(\Delta R)$, where $q_{\Lambda} = +1$ and $q_{\bar{\Lambda}} = -1$. For this background, $P^{(B)}_{\mathrm{common}} = 0$, whereas $\Delta P^{(B)} = -2B$. The combination thus retains the common chiral response while providing selective suppression of particle–antiparticle sign-dependent backgrounds. Such as spin correlation induced by the strong-force fields [41]. It does not, however, remove backgrounds shared by all three channels, including possible hadronization-induced correlations. The coefficients define a fixed linear combination of the channel-level correlations, rather than a weighting by the observed pair yields.

Complementary sensitivity to fluctuation strength and spatial coherence

Having established the common response across hyperon-pair channels, we investigate how its magnitude and differential structure encode the properties of chiral-imbalance fluctuations. We select two fixed, nonoverlapping intervals, $0 \leq\Delta R < 1.0$ and $2.5 \leq\Delta R < 3.5$, to characterize short- and long-range correlations. For a given pair channel $ab$ and interval $\Omega$, we define the pair-weighted correlation as

where $dN_{ab}/d\Delta R$ is the differential pair yield. The common-mode correlation in each interval is obtained by applying Eq. (7) to the three channel-averaged correlations. We denote the resulting observables by $P_{\mathrm{short}} = P_{\mathrm{common}}^{0\leq\Delta R<1.0}$, $P_{\mathrm{long}} = P_{\mathrm{common}}^{2.5\leq\Delta R<3.5}$, and define their ratio as

The angular factor in Eq. (5) changes sign at $\theta_{ab} = \pi/2$ when the helicity-pair response remains finite. However, the partonic opening angle $\theta_{ab}$ does not map uniquely onto the final-state hyperon separation $\Delta R$, owing to the momentum distributions, transport, and hadronization. We therefore use fixed intervals that sample the positive and negative regions of the calculated correlation, avoiding the need to define a model-dependent zero-crossing point. These intervals provide a consistent basis for comparing different configurations of the chiral field.

of the chiral field.

Figure 4(a) shows the dependence of $P_{\mathrm{short}}$ and $P_{\mathrm{long}}$ on the imposed magnitude $\mu_{5,0}$ for different spatial coherence scales. The short-range correlation is positive, whereas the long-range correlation is negative, consistent with the angular dependence of the proposed mechanism. For finite domains, both correlations increase in absolute magnitude with $\mu_{5,0}$ and generally with the cell size $l_{\mathrm{cell}}$. The event-wise uniform configuration produces substantially larger correlations, illustrating the dilution of the net spin response by spatially fluctuating domain signs. For visibility, the uniform results in panel (a) are multiplied by 0.2; this factor is not applied to the ratio.

The ratio in Figure 4(b) exhibits a different sensitivity. Over the investigated range $\mu_{5,0} = 0.55$–$0.80$ GeV, $R_{SL}$ depends only weakly on the imposed fluctuation magnitude but shows a clear dependence on the spatial coherence scale. It decreases from approximately 3.5–3.6 for $l_{\mathrm{cell}} = 0.5$ fm to about 2.8 for 2.0 fm, while the uniform configuration yields approximately 1.2. Thus, more localized chiral fluctuations suppress the long-range anticorrelation more strongly relative to the short-range positive correlation. The pronounced contrast between the uniform and finite-domain configurations further demonstrates the sensitivity of $R_{SL}$ to the spatial organization of the chiral field.

The weak dependence of $R_{SL}$ on $\mu_{5,0}$ suggests an approximate separation between the fluctuation amplitude and spatial response. Phenomenologically, the numerical trends can be expressed as

where $A$ characterizes the overall amplitude response and $K_{\mathrm{short,long}}$ describe the effective spatial response after transport and hadronization. Consequently,

with much of the common amplitude dependence canceling in the ratio. This factorization is a phenomenological interpretation of the observed trends, not a universal relation.

These results suggest a complementary strategy for constraining chiral-imbalance fluctuations: $R_{SL}$ provides a relatively amplitude-insensitive probe of the effective spatial coherence, while the absolute correlation magnitudes retain sensitivity to the fluctuation strength. Their combined use can therefore help constrain both properties within a specified dynamical framework. Since $R_{SL}$ is constructed from $P_{\mathrm{short}}$ and $P_{\mathrm{long}}$, a quantitative extraction must account for the statistical and systematic covariance between them.

Summary.— We propose differential hyperon-pair spin correlations as a probe of the strength and spatial coherence of chiral-imbalance fluctuations in the QGP. In the present transport framework, chiral-imbalance fluctuations generate a finite two-particle spin correlation even when the event-averaged helicity polarization vanishes. The induced correlation exceeds the thermal-vorticity and shear contributions in the investigated kinematic range and exhibits an approximately common response across the $\Lambda\Lambda$, $\bar{\Lambda}\bar{\Lambda}$, and $\Lambda\bar{\Lambda}$ channels. This motivates the common-mode observable $P_{\mathrm{common}}$ and the channel-difference observable $\Delta P$. We further find that the short-to-long-range ratio $R_{SL} = -P_{\mathrm{short}}/P_{\mathrm{long}}$ is strongly sensitive to the effective spatial coherence scale but only weakly dependent on the imposed $\mu_5$ magnitude over the explored parameter range, while the absolute correlations retain sensitivity to both. These observables can be extracted from joint weak-decay angular distributions and provide complementary, experimentally testable constraints on chiral-imbalance fluctuations. Their interpretation will require controlling other sources of pair-spin correlations, including those associated with hadronization and possible quantum correlations. The inferred spatial coherence scale is model dependent and should not be identified directly with the microscopic sphaleron size.

Acknowledgments.— This work is supported in part by the National Natural Science Foundation of China (NSFC) Grant Nos. 12235016, 12221005, 12305146 and by the CAS Project for Young Scientists in Basic Research under Grant No. YSBR-088.