Transverse Momentum Conservation in Collisions

• Transverse Momentum Conservation is a strict kinematic constraint requiring the net transverse momentum to equal zero, fundamentally shaping event dynamics.
• It induces measurable multiparticle azimuthal correlations and nonflow effects that alter cumulant signatures and challenge conventional flow models.
• TMC modifies single-particle spectra and alignment patterns, offering essential insights for distinguishing kinematic baselines from collective behavior.

Transverse Momentum Conservation (TMC) is the exact kinematic constraint that the total transverse momentum of all produced particles in a high-energy collision sums to zero, $\sum_i \vec{p}_{T,i} = 0$. This global restriction, required by fundamental conservation laws, is not merely a trivial background: it profoundly influences multiparticle correlations, the structure of single-particle spectra in small systems, and the interpretation of signatures of collective dynamics such as flow, especially in $pp$, $p$+Pb, and peripheral heavy-ion collisions.

1. Fundamental Formalism of TMC in Multiparticle Systems

The rigorous theoretical implementation of TMC introduces a two-dimensional delta function into the $N$-particle distribution,

$$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$

where $f(\vec{p}_i)$ is the factorized single-particle transverse momentum spectrum and $A$ a normalization constant (Bzdak et al., 2017). For observables involving only $k \ll N$ particles, TMC is incorporated via the marginalization over the remaining $N-k$ momenta and approximating their sum by a bivariate Gaussian (central limit) for large $N$. This results in a $pp$0-particle subensemble distribution,

$pp$1

where $pp$2 is the second moment of the spectrum over full phase space (Bzdak et al., 2017, Bzdak et al., 2010, Pei et al., 2024, Pei et al., 17 Mar 2025).

2. TMC-Induced Multiparticle Azimuthal Correlations

TMC induces nontrivial "nonflow" correlations among all particles, generating nonzero values of multiparticle azimuthal cumulants $pp$3 and moment-based observables such as symmetric and asymmetric cumulants. For $pp$4-particle cumulants at harmonic $pp$5, TMC yields

$pp$6

where $pp$7 is the total multiplicity (Xie et al., 2022). This sign structure and strong suppression with $pp$8 (or, more precisely, $pp$9) are central: for example, $p$0, $p$1, $p$2, $p$3 in the pure TMC limit, in contrast to the usual flow-dominated expectations (Bzdak et al., 2017, Pei et al., 2024, Xie et al., 2022).

Explicitly, for two particles,

$p$4

and for $p$5, the cumulants scale as

$p$6

There is an explicit dependence on the shape of the $p$7 spectrum—harder spectra amplify TMC-induced correlations (Bzdak et al., 2017).

3. TMC and Correlation Factorization Breaking

TMC leads to the breakdown of factorization of two-particle azimuthal correlations, expressed through the ratios

$p$8

where $p$9 are Fourier coefficients of the pair distribution (Pei et al., 27 Jan 2026). Under TMC,

$N$0

producing $N$1 and $N$2 in small systems, in agreement with CMS $N$3+Pb data—a direct challenge to hydrodynamic models predicting $N$4 for all $N$5. The magnitude of $N$6 is enhanced at high $N$7 and low multiplicity and grows with increasing difference $N$8. This sign-alternating rule and the quantitative reproduction of $N$9 and $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$0 trends in multiplicity and $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$1 bins by the TMC+flow analytic framework resolve longstanding data puzzles (Pei et al., 27 Jan 2026).

4. Interplay with Collective Flow and Mixed Systems

While TMC sets a strict "nonflow" baseline, collective flow—arising from hydrodynamics or other mechanisms—superposes as genuine multi-particle correlations. In cumulant measures, flow yields contributions with different sign structure:

• $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$2 (negative)
• $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$3 (positive)
• $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$4 (negative)

When both TMC and flow are present, their competition produces sign-flip transitions as multiplicity increases: At low $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$5, TMC dominates, with its characteristic sign and $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$6 scaling, leading for instance to $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$7. As $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$8 grows, the flow term overtakes, and the sign of the observable changes (e.g., $$f_N(\vec{p}_1, ..., \vec{p}_N) = \frac{1}{A}\, \delta^2\Big(\sum_{i=1}^{N} \vec{p}_i\Big) \prod_{i=1}^{N} f(\vec{p}_i),$$9 for large $f(\vec{p}_i)$0), a phenomenon now numerically and analytically established as a clean probe of collectivity onset and possible hot-spot substructure in the proton (Xie et al., 2022, Pei et al., 2024).

Symmetric cumulants such as $f(\vec{p}_i)$1 and higher $f(\vec{p}_i)$2-particle cumulants have their sign and magnitude shaped by the relative strength of TMC and flow, with TMC dominating at small $f(\vec{p}_i)$3 and flow-driven mode-coupling appearing at larger $f(\vec{p}_i)$4 (Pei et al., 17 Mar 2025). Experimental comparisons (ATLAS, CMS) in $f(\vec{p}_i)$5 and $f(\vec{p}_i)$6+Pb collision systems confirm that accurate modeling of correlations requires combining both contributions (Pei et al., 2024, Pei et al., 17 Mar 2025).

5. Impact on Single-Particle Spectra

TMC modifies single-particle spectra when exact momentum conservation is imposed. In fully microcanonical statistical models—where total system energy $f(\vec{p}_i)$7 and momentum $f(\vec{p}_i)$8 are fixed—the one-particle $f(\vec{p}_i)$9 spectrum for $A$0 massless particles is

$A$1

which features a sharp suppression at large $A$2: $A$3 vanishes for $A$4 (Begun et al., 2012). For typical $A$5-like parameters ($A$6, $A$7 MeV, $A$8 GeV), the effect of energy-momentum conservation reduces the apparent spectrum by $A$9–$k \ll N$0 at $k \ll N$1 GeV compared to the naive Boltzmann exponential. This constraint must be taken into account when interpreting the high-$k \ll N$2 behavior and extracting physical parameters.

6. TMC and Event Shape / Alignment Phenomena

The enforcement of TMC among the most energetic particles in small systems can produce coplanarity or alignment patterns, as observed in cosmic-ray "family" events and modeled in contemporary heavy-ion codes such as HYDJET++. By requiring that the sum of the top $k \ll N$3 particles' $k \ll N$4 falls below a small threshold $k \ll N$5, the leading particles are forced into nearly collinear or back-to-back configurations, maximizing the alignment variable $k \ll N$6. Simulations show that the probability $k \ll N$7 for events to exhibit extreme alignment sharply increases as $k \ll N$8 decreases, reproducing key features of experimental alignment data in cosmic-ray and collider experiments. This effect emphasizes that selection plus TMC suffices to generate strong coplanarity without the need for exotic dynamics (Lokhtin et al., 2024).

7. Experimental and Theoretical Implications

TMC produces multiparticle correlations of definite sign and strength that persist at the percent level for low multiplicity, high $k \ll N$9 cuts, and even for moderate $N-k$0 in small-$N-k$1 systems (Bzdak et al., 2017, Xie et al., 2022). Its signatures are present in all azimuthal cumulant and symmetric cumulant observables, and its role is enhanced as the spectrum hardens. Consequently, any claim of collective behavior or new physics in small systems requires explicit subtraction or modeling of the TMC baseline. This necessity extends to the interpretation of STAR, CMS, and ATLAS measurements of azimuthal correlations, alignment, and flow observables. Furthermore, analytic and numerical frameworks now enable separation of pure TMC, pure flow, and their interplay, allowing the determination of thresholds where collectivity overtakes kinematic constraints. The systematic mapping of sign changes, $N-k$2-scaling, and $N-k$3-dependence in data thus provides sensitive probes of both kinematic and dynamical aspects of multiparticle production.

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References: (Bzdak et al., 2017, Pei et al., 27 Jan 2026, Pei et al., 2024, Xie et al., 2022, Lokhtin et al., 2024, Pei et al., 17 Mar 2025, Bzdak et al., 2010, Begun et al., 2012).