Center-of-Mass Imaging in Heavy-Ion Collisions

• Center-of-Mass Imaging is a femtoscopic approach that reconstructs pion emission sources directly in the emitter’s global frame, providing clearer spatial morphology.
• The method employs a small-ΔE selection to suppress temporal phase effects, thereby reducing distortions from pair-dependent Lorentz frames.
• Model tests and AMPT simulations confirm that CMFS imaging yields a source image closely matching the true distribution while minimizing non-Gaussian tail artifacts.

Center-of-Mass Imaging is a femtoscopic source-imaging strategy for identical-pion correlations in heavy-ion collisions that reconstructs the source directly in the source center-of-mass frame (CMFS) rather than in the conventional pair center-of-mass frame (CMFP). Its motivation is that standard imaging in the CMFP is mathematically clean because the pair energy difference vanishes there, but each pion pair has its own CMFP, so the reconstructed source is assembled from many pair-dependent Lorentz frames. The CMFS formulation instead aims to recover a source image in a single fixed frame, more directly tied to the actual spatial scale and morphology of the emitter, by suppressing the residual temporal phase term through a small-$\Delta E$ pair selection (Ren et al., 2018, Feng et al., 3 Nov 2025).

1. Frame choice and physical meaning

In two-pion femtoscopy, the object of interest is the source function, namely the distribution of relative separations between emitted pions. Historically, Brown–Danielewicz-type imaging reconstructs this source from measured correlation functions without imposing a Gaussian ansatz. In the conventional formulation, the reconstruction is carried out in the CMFP, the frame in which the total three-momentum of a given pion pair vanishes. That choice is exact for the standard inversion formula, but it is also geometrically awkward because different pairs have different CMFP boosts; the resulting image therefore mixes many different pair-dependent Lorentz frames rather than representing a single unified source frame (Ren et al., 2018).

Center-of-Mass Imaging replaces that pair-wise frame assignment by the CMFS, the global frame associated with the emitting source itself. In that frame the source geometry is fixed, so the reconstructed image is intended to reflect the emitter more directly. Later work sharpened this point by arguing that part of the long non-Gaussian tail seen in RHIC pion imaging is methodological rather than purely physical: CMFP imaging can itself broaden the source because the reconstructed separations are evaluated after different Lorentz transformations for different pairs (Feng et al., 3 Nov 2025).

The frame distinction is not merely terminological. For a spherical source of radius $10$ fm, the 2018 study found that model source functions in the CMFS are nearly identical for two different momentum distributions, whereas in the CMFP they differ visibly; it also emphasized that the CMFS source vanishes beyond roughly the source diameter, while the CMFP source can extend to much larger separations, even beyond $25$ fm (Ren et al., 2018).

2. Correlation formalism and the CMFS obstruction

Neglecting final-state interactions, the two-pion correlation function is written as

$$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$

where $d(x)$ is the relative space-time separation distribution, $\mathbf q$ is the relative momentum, $\mathbf r$ is the relative spatial separation, and $\Delta t$ is the emission-time difference. In the CMFP one has $\Delta E=0$, so the correlation reduces to

$$C(\mathbf q)=1+\int \cos(\mathbf q\cdot \mathbf r)\,S(\mathbf r)\,d\mathbf r, \qquad S(\mathbf r)=\int d(x)\,dt.$$

After angle averaging, the one-dimensional imaging relation used in both papers is

$10$0

with inversion

$10$1

The 2025 paper describes this as the one-dimensional Koonin-Pratt–type relation in the simplified noninteracting, spherically symmetric case, while the 2018 paper situates it within standard Brown–Danielewicz imaging (Ren et al., 2018, Feng et al., 3 Nov 2025).

The CMFS difficulty is immediate: in general $10$2, so the phase $10$3 cannot be dropped exactly. Physically, the time extent of emission contaminates the apparent spatial separation. The standard CMFP treatment avoids this because the kernel depends only on space there; the CMFS treatment does not. A plausible implication is that, in a moving pair frame, unequal emission times shift the effective separation by a velocity-times-time term, but the CMFS program avoids reconstructing that pair-dependent quantity directly and instead tries to suppress the temporal phase itself (Ren et al., 2018).

3. Operational definition of the method

The central approximation is to select identical pion pairs with very small energy difference

$10$4

so that

$10$5

and hence

$10$6

Under that condition, the CMFS correlation function approximately obeys the same inversion logic as in the CMFP: $10$7 This small-$10$8 condition is the formal basis of Center-of-Mass Imaging in the pion-source context (Ren et al., 2018).

Operationally, one works in the CMFS, selects identical pion pairs, computes $10$9, and imposes a cut such as

$25$0

Using only those pairs, one constructs $25$1 in the CMFS, angle-averages to obtain $25$2, and reconstructs the one-dimensional source image $25$3 by Fourier inversion. As in earlier imaging work, the benchmark “model source” for comparison is obtained from generated pion pairs with relative momentum less than $25$4 MeV/$25$5 (Ren et al., 2018).

The 2025 refinement places the conceptual emphasis on staying in one global source frame throughout. Its practical prescription remains the same in substance—construct the two-pion correlation function in the CMFS and retain only pairs satisfying

$25$6

so that the neglected $25$7 term is small—but it interprets the gain explicitly as elimination of frame mixing and reduction of kinematic correlations generated by varying pair boosts (Feng et al., 3 Nov 2025).

4. Model tests and AMPT validation

The 2018 paper first verified that standard imaging behaves as expected in the conventional frame. Using the string-melting AMPT model for central Au+Au collisions at $25$8 GeV with impact parameter $25$9 fm and $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$0 events, it found good agreement between the directly computed source function and the reconstructed one in the CMFP (Ren et al., 2018).

It then tested CMFS imaging in a controlled toy model: a homogeneous sphere of radius $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$1 fm, pion momenta drawn from a Boltzmann distribution with $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$2 MeV, and a one-sided Gaussian emission-time distribution

$$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$3

Using $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$4 pion pairs, the paper showed that CMFS imaging without any $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$5 cut gives a clear discrepancy from the true source. Once the cut is applied, the reconstructed source approaches the true source; for $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$6 MeV, the difference is described as “almost ignored.” Replacing the Boltzmann momentum distribution by a uniform one from $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$7 to $$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$8 GeV/$$C(\mathbf q)=1+\int d^4x\,\cos(\mathbf q\cdot \mathbf r-\Delta E\,\Delta t)\, d(x),$$9 produced the same trend, indicating that the effectiveness of the cut is only weakly influenced by the underlying momentum spectrum (Ren et al., 2018).

The same study then examined bias from the cut itself in AMPT. Comparing the full CMFS source function with the source restricted to pairs satisfying $d(x)$0 MeV, it found that the energy cut alters the source only slightly. It further compared directly computed model sources with reconstructed CMFS images for three impact parameters—$d(x)$1 fm with $d(x)$2 events, $d(x)$3 fm with $d(x)$4 events, and $d(x)$5 fm with $d(x)$6 events—and reported generally good agreement. The only noticeable mismatch occurred around $d(x)$7 fm in the $d(x)$8 fm case, attributed to the discreteness of the measured correlation function rather than to failure of the method. The paper also reported that using an even smaller $d(x)$9 cut did not materially change the result, indicating that $\mathbf q$0 MeV was already sufficient in that study (Ren et al., 2018).

5. Non-Gaussian tails and the 2025 reformulation

The 2025 paper reinterprets a long-standing RHIC observation: source images reconstructed from two-pion correlations often show a pronounced non-Gaussian tail. Earlier analyses had usually associated that tail with an extended halo, for example from long-lived resonance decays. The 2025 result does not reject physical non-Gaussianity, but it separates two contributions: genuine source structure and frame-dependent distortion produced by CMFP imaging itself (Feng et al., 3 Nov 2025).

To expose the issue, the paper used a simulated Gaussian source

$\mathbf q$1

with $\mathbf q$2 fm, $\mathbf q$3 fm/$\mathbf q$4, Boltzmann momenta corresponding to $\mathbf q$5 MeV, and a pair cut

$\mathbf q$6

In the CMFP, the imaged source agreed with the model source in that frame, but the result displayed a stronger tail than a Gaussian source in the source frame would suggest. In the CMFS, restricting to pairs with $\mathbf q$7 MeV produced an imaged source that agreed very well with the model source evaluated in the CMFS and was much closer to Gaussian (Feng et al., 3 Nov 2025).

The paper then turned to AMPT string-melting events for Au+Au collisions at $\mathbf q$8 GeV, $\mathbf q$9 fm, and identical $\mathbf r$0 pairs. It compared CMFP results obtained with $\mathbf r$1 simulated events to CMFS results obtained with $\mathbf r$2 events, explicitly attributing the much larger CMFS event count to the restrictive small-$\mathbf r$3 selection. The reconstructed CMFS source had a much smaller long-range tail than the CMFP source. The interpretation advanced there is that the CMFP tail contains both physical and frame-artifact contributions, whereas the CMFS tail reflects only the physical non-Gaussianity of the pion source. The evidence is presented visually rather than through goodness-of-fit measures, moments, or tail exponents (Feng et al., 3 Nov 2025).

6. Scope, limitations, and status

The method is explicitly limited to one-dimensional imaging. Both papers use a noninteracting correlation formalism, so in practice the construction applies to correlation functions built from Coulomb-corrected data. The 2018 paper describes the method as preliminary, and the 2025 paper states that the small-$\mathbf r$4 approximation is validated empirically rather than by a more rigorous analytical error estimate (Ren et al., 2018, Feng et al., 3 Nov 2025).

Statistics are a central practical constraint. The cut

$\mathbf r$5

strongly reduces the available pair sample, which is why large event counts appear in the AMPT tests, especially in the 2025 CMFS study. The 2025 paper also notes the absence of a systematic optimization study for the $\mathbf r$6 threshold, and it emphasizes that its comparisons are primarily morphological. The 2018 paper adds that resolution and discreteness of the measured $\mathbf r$7 can affect the inversion, particularly at large $\mathbf r$8 (Ren et al., 2018, Feng et al., 3 Nov 2025).

Within those limits, Center-of-Mass Imaging establishes a clear methodological alternative to pair-frame imaging. Its defining claim is that, once the $\mathbf r$9 phase is sufficiently suppressed, the reconstructed source in the CMFS is tied to a single physical frame, is less sensitive to the pair momentum distribution, and yields a more direct representation of source geometry than the conventional CMFP image. The later reformulation further argues that this single-frame construction substantially reduces the artificial non-Gaussian tail induced by CMFP frame mixing, while leaving the genuinely physical tail available for interpretation (Ren et al., 2018, Feng et al., 3 Nov 2025).