TROY Formalism in Hadron Femtoscopy

• TROY Formalism is defined as the T-matrix-based Routine for hadrOn femtoscopY that reconstructs full coordinate-space s-wave functions including Coulomb distortions.
• It solves a coupled-channel, half off-shell T-matrix equation to capture resonant, threshold, and inelastic effects that traditional methods often neglect.
• The framework enhances femtoscopic correlation functions by integrating thermal weights and realistic source functions, crucial for DN and ḊN systems.

Searching arXiv for the cited TROY-related papers to ground the article with fresh references. arxiv_search(query="(Barbat et al., 10 Jul 2025)", max_results=5, sort_by="submittedDate") arxiv_search({"query":"(Barbat et al., 10 Jul 2025)","max_results":5,"sort_by":"submittedDate"}) {"query":"(Barbat et al., 10 Jul 2025)","max_results":5,"sort_by":"submittedDate"} TROY formalism, in hadron femtoscopy, denotes the T-matrix-based Routine for hadrOn femtoscopY, a coupled-channel framework devised for situations in which the measured two-particle correlation at small relative momentum is controlled by strong and Coulomb final-state interactions over distances comparable to the source size. In the formulation developed for $DN$ and $\bar{D}N$ systems, TROY solves the coupled-channel, half off-shell $T$-matrix equation, reconstructs the full coordinate-space $s$-wave including Coulomb distortions, and inserts the resulting wave functions into a generalized Koonin–Pratt correlation integral with realistic source functions and thermal weights (Barbat et al., 10 Jul 2025).

1. Physical motivation and scope

In hadron–hadron femtoscopy, the measured two-particle correlation at small relative momentum is driven by the strong, and for charged pairs also Coulomb, final-state interaction acting over distances comparable to the source size. This becomes technically nontrivial when the strong interaction is resonant or when many open or inelastic channels lie close to threshold. The $DN$ and $\bar{D}N$ sectors exemplify this regime because the correlation is then sensitive not only to the asymptotic elastic amplitude but to the full spatial structure of the two-body wave function inside the interaction region and to coupled-channel dynamics associated with nearby states such as $\Lambda_c(2595)$ and $\Sigma_c(2800)$ (Barbat et al., 10 Jul 2025).

The immediate target of TROY is the limitation of the Lednický–Lyuboshitz approximation. The latter uses only the asymptotic, on-shell $s$-wave amplitude of a single elastic channel and therefore misses off-shell, finite-range, and inelastic effects that become paramount near thresholds and resonances. TROY was introduced precisely to retain those effects in the correlation observable. Its operational content is threefold: solving the coupled-channel, half off-shell $T$-matrix with a physically motivated potential, reconstructing the full coordinate-space $\bar{D}N$0-wave from the off-shell $\bar{D}N$1-matrix rather than from its on-shell limit alone, and evaluating the correlation function with all coupled channels included through thermal weights $\bar{D}N$2 (Barbat et al., 10 Jul 2025).

The formalism is tailored to near-threshold femtoscopy. The source radius is taken as $\bar{D}N$3, with representative values $\bar{D}N$4 fm for $\bar{D}N$5 and $\bar{D}N$6–$\bar{D}N$7 fm for heavy-ion collisions. The results discussed in the underlying study are typically shown up to $\bar{D}N$8 MeV, while the most prominent threshold, cusp, and Coulomb effects occur below about $\bar{D}N$9 MeV (Barbat et al., 10 Jul 2025).

2. Correlation integral and wave-function construction

The underlying correlation observable is the Koonin–Pratt expression for an observed final channel $T$0 and relative momentum $T$1 in the pair center-of-mass frame,

$T$2

with a spherically symmetric Gaussian source

$T$3

When channel coupling is present, TROY replaces the single-channel expression by

$T$4

where $T$5 runs over all channels coupled to $T$6, and $T$7 encodes the relative production rate of channel $T$8 at freeze-out (Barbat et al., 10 Jul 2025).

At low $T$9, only the $s$0-wave is modified by the strong interaction. TROY therefore decomposes the total wave function as

$s$1

where $s$2 is the full Coulomb-distorted solution, or a plane wave for neutral pairs, $s$3 is its $s$4-wave projection, and $s$5 is the interacting $s$6-wave reconstructed from the off-shell $s$7-matrix. This subtraction ensures the correct boundary condition: a Coulomb-distorted plane wave at large $s$8, with only the $s$9-wave modified by the strong interaction (Barbat et al., 10 Jul 2025).

The central dynamical object is the half off-shell coupled-channel $DN$0-matrix,

$DN$1

with $DN$2 and $DN$3. TROY solves this equation half off-shell using the exponential regulator

$DN$4

The interacting coordinate-space $DN$5-wave is then reconstructed as

$DN$6

with $DN$7 the spherical Bessel function and $DN$8. This Green’s-function representation after partial-wave projection resums off-shell contributions and coupled channels directly in coordinate space (Barbat et al., 10 Jul 2025).

3. Interaction kernel, unitarization, and low-energy parameters

The interaction kernel $DN$9 is derived from an effective meson–baryon Lagrangian dominated by $\bar{D}N$0-channel vector-meson exchange in $\bar{D}N$1-wave. In the zero-range limit $\bar{D}N$2, the model reduces to an $\bar{D}N$3-wave kernel, and under the universality assumption with the KSFR relation $\bar{D}N$4 it takes the Weinberg–Tomozawa form

$\bar{D}N$5

where $\bar{D}N$6 is a Dirac-spinor normalization and $\bar{D}N$7 contains the isospin coefficients, including a $\bar{D}N$8 correction for charm exchange (Barbat et al., 10 Jul 2025).

This dynamical input is used in two distinct resummations. In the off-shell TROY calculation, the half off-shell Lippmann–Schwinger/Bethe–Salpeter equation is solved directly with the exponential ultraviolet regulator. In the on-shell factorized treatment, used only to extract low-energy parameters for the Lednický–Lyuboshitz approximation, one solves

$\bar{D}N$9

with the loop $\Lambda_c(2595)$0 regularized by a hard cutoff. The study uses $\Lambda_c(2595)$1 MeV in the off-shell ZR calculation to place $\Lambda_c(2595)$2 correctly, $\Lambda_c(2595)$3 MeV for the on-shell ZR scheme with $\Lambda_c(2595)$4, and $\Lambda_c(2595)$5 MeV for the WT scheme with $\Lambda_c(2595)$6 and $\Lambda_c(2595)$7 MeV (Barbat et al., 10 Jul 2025).

Low-energy scattering information is related to the on-shell amplitude through

$\Lambda_c(2595)$8

for the no-Coulomb case. The on-shell amplitudes are used to extract $\Lambda_c(2595)$9 and, where relevant, $\Sigma_c(2800)$0 for LL. By contrast, TROY uses the full $\Sigma_c(2800)$1 and the full wave function, so no truncation to an effective-range expansion is required (Barbat et al., 10 Jul 2025).

The same kernel dynamically generates the $\Sigma_c(2800)$2 in the $\Sigma_c(2800)$3 $\Sigma_c(2800)$4 sector and the $\Sigma_c(2800)$5 in the $\Sigma_c(2800)$6 sector. This is central to the phenomenology because the femtoscopic correlation in $\Sigma_c(2800)$7 channels reflects precisely the threshold and resonance structure generated by these states (Barbat et al., 10 Jul 2025).

4. Relation to the Lednický–Lyuboshitz approximation

The standard Lednický–Lyuboshitz approximation assumes $\Sigma_c(2800)$8-wave dominance and, for a Gaussian source, expresses the correlation in terms of the asymptotic on-shell elastic amplitude $\Sigma_c(2800)$9, its effective-range truncation, and source-size functions $s$0 and $s$1. For charged pairs, the formulation introduces the Sommerfeld parameter $s$2, the Gamow factor

$s$3

and the Coulomb-modified amplitude $s$4. In the notation of the study, $s$5 in the LL formula (Barbat et al., 10 Jul 2025).

The distinction between LL and TROY is structural rather than merely numerical. LL uses the asymptotic, on-shell, single-channel $s$6-wave amplitude; it neglects off-shell and finite-$s$7 corrections and inelastic feed-ins. TROY instead reconstructs the full $s$8-wave within the source, includes all coupled channels through $s$9, and can generate cusps and threshold-opening effects. The consequence is that LL can be reasonable for single-channel, weakly coupled systems without nearby resonances, whereas it becomes insufficient for $T$0 and $T$1 sectors in which $T$2, $T$3, and multiple thresholds are nearby (Barbat et al., 10 Jul 2025).

Aspect	Lednický–Lyuboshitz	TROY
Dynamical input	On-shell single-channel $T$4-wave amplitude	Half off-shell coupled-channel $T$5-matrix
Spatial treatment	Asymptotic wave function	Full coordinate-space $T$6-wave inside the source
Inelastic and threshold effects	Neglected	Included through coupled channels and weights $T$7

A recurrent misconception is that Coulomb corrections alone are sufficient once charged pairs are considered. In the TROY construction, Coulomb is not appended as a separate multiplicative factor at asymptotic distance; instead, the exact Coulomb solution $T$8 is combined with the reconstructed strong $T$9-wave in coordinate space at all $\bar{D}N$00. This is particularly relevant for charged $\bar{D}N$01 channels, where Coulomb and nearby coupled-channel dynamics compete rather than factorize trivially (Barbat et al., 10 Jul 2025).

5. Channel content and phenomenology in $\bar{D}N$02 and $\bar{D}N$03

The physical-basis channel content is extensive: $\bar{D}N$04 is treated with 16 coupled channels, $\bar{D}N$05 with 9, $\bar{D}N$06 with 2, and $\bar{D}N$07 as a single $\bar{D}N$08 channel. In isospin language, the $\bar{D}N$09, $\bar{D}N$10 sector contains 7 channels and the $\bar{D}N$11, $\bar{D}N$12 sector 8 channels, whereas $\bar{D}N$13 in both $\bar{D}N$14 and $\bar{D}N$15 is single-channel. The resulting femtoscopic patterns differ sharply between $\bar{D}N$16 and $\bar{D}N$17 because attraction, repulsion, inelasticity, and Coulomb act in different combinations (Barbat et al., 10 Jul 2025).

Pair	$\bar{D}N$18 from TROY (ZR)	Main correlation feature
$\bar{D}N$19	$\bar{D}N$20 fm	Modest suppression below unity; LL and TROY agree well
$\bar{D}N$21	$\bar{D}N$22 fm	Coulomb attraction dominates; $\bar{D}N$23 for all $\bar{D}N$24
$\bar{D}N$25	$\bar{D}N$26 fm	Strong attraction; cusp at $\bar{D}N$27 MeV from $\bar{D}N$28 opening
$\bar{D}N$29	$\bar{D}N$30 fm	Coulomb repulsion lowers $\bar{D}N$31, but coupled-channel feed-in enhances it

For $\bar{D}N$32, the neutral single-channel character makes the system the clearest example in which LL suffices: the correlation shows only a modest suppression below unity due to repulsion, and TROY and LL agree well. For $\bar{D}N$33, the correlation is dominated by Coulomb attraction; TROY predicts $\bar{D}N$34 for all $\bar{D}N$35, with only weak additional strong-interaction effects, while LL shows a somewhat different trend, indicating sensitivity to off-shell effects even in a weakly coupled two-channel system (Barbat et al., 10 Jul 2025).

The $\bar{D}N$36 channel is the canonical multi-channel case. It combines strong attraction from $\bar{D}N$37 in $\bar{D}N$38 and $\bar{D}N$39 in $\bar{D}N$40, and it exhibits a visible cusp at $\bar{D}N$41 MeV due to the opening of $\bar{D}N$42. TROY’s coupled-channel sum with thermal weights exceeds the elastic-only result and differs from LL because the latter lacks feed-in from channels such as $\bar{D}N$43, $\bar{D}N$44, and $\bar{D}N$45 (Barbat et al., 10 Jul 2025).

The $\bar{D}N$46 correlation is charged and strongly coupled. Coulomb repulsion pushes $\bar{D}N$47, but feed-in from channels with large weights—especially $\bar{D}N$48 and $\bar{D}N$49—increases $\bar{D}N$50 by about $\bar{D}N$51–$\bar{D}N$52 at low $\bar{D}N$53 relative to the elastic-only result. In the $\bar{D}N$54 fireball model at $\bar{D}N$55 MeV, the study quotes $\bar{D}N$56, $\bar{D}N$57, $\bar{D}N$58, and $\bar{D}N$59; for $\bar{D}N$60, comparably large weights include $\bar{D}N$61 and $\bar{D}N$62. These large feed-ins arise because the $\bar{D}N$63 and $\bar{D}N$64 thresholds lie below $\bar{D}N$65, and $\bar{D}N$66 couples strongly to $\bar{D}N$67 and moderately to those channels (Barbat et al., 10 Jul 2025).

6. Experimental implications and limitations

The formalism was developed with current heavy-ion and small-system measurements in mind. For ALICE in high-multiplicity $\bar{D}N$68, the most sensitive channel is $\bar{D}N$69, for which a measurement of $\bar{D}N$70 at $\bar{D}N$71–$\bar{D}N$72 MeV and small source size $\bar{D}N$73 fm should expose the predicted coupled-channel enhancement over elastic-only LL expectations. For $\bar{D}N$74, precise data at $\bar{D}N$75 MeV are needed to resolve the small strong-interaction contribution on top of Coulomb dominance. For STAR in Au+Au, the recommended observable is a separated measurement of $\bar{D}N$76 and $\bar{D}N$77, rather than their sum, since the two channels have qualitatively different interactions—attraction versus repulsion—and summing them washes out sensitivity. The relevant source sizes are $\bar{D}N$78–$\bar{D}N$79 fm, and the most informative region is $\bar{D}N$80 MeV (Barbat et al., 10 Jul 2025).

Source-size dependence follows the usual femtoscopic pattern but is especially consequential here. For large sources, all strong-interaction signals are diluted: neutral channels show only weak deviations, while charged channels retain the Coulomb trends. The reported large-source results are broadly consistent with preliminary STAR indications, but they also underscore that channel separation is necessary if the aim is to isolate the very different physics of $\bar{D}N$81 and $\bar{D}N$82 (Barbat et al., 10 Jul 2025).

The main limitations are explicit. TROY modifies only the $\bar{D}N$83 partial wave, which is reasonable at $\bar{D}N$84–$\bar{D}N$85 MeV for short-range forces but not guaranteed beyond that region. The interaction kernel is model dependent, relying on ZR or WT forms derived from vector-meson exchange and on regulator choices tuned to reproduce $\bar{D}N$86 and $\bar{D}N$87. The thermal weights come from a simple static fireball spectrum at $\bar{D}N$88 MeV without detailed feed-down or kinematic cuts, which is adequate only at the level of magnitude. Numerical stability of the off-shell integral equation also depends on careful convergence with the common exponential regulator (Barbat et al., 10 Jul 2025).

Within those limits, the methodological lesson is sharp. LL remains adequate when the system is single-channel, weakly coupled, and far from thresholds or resonances, as in $\bar{D}N$89. TROY becomes essential when nearby thresholds, inelastic channels, and resonances shape the correlation function, as in $\bar{D}N$90. In that regime, the observable is not exhausted by an on-shell scattering length; it depends on the off-shell, coordinate-space wave function throughout the source region, and the formalism is designed precisely to retain that dependence (Barbat et al., 10 Jul 2025).