Robustness of double-logarithmic small-$x$ saturation of the proton spin sum rule: exact amplitudes, uncertainty budget, and orbital angular momentum

Ermolaev (Eur. Phys. J. C 85 (2025) 360) argued that quark and gluon helicity distributions with all double logarithms of $1/x$ resummed, combined with data from the Relativistic Heavy Ion Collider (RHIC), reproduce the proton spin sum rule $S_P=1/2$ without large orbital angular momentum (OAM). The claim, however, rests on a Bessel-function interpolation of the helicity amplitudes, tentative numerics with fixed inputs ($\omega_0=0.86$, constant initial distributions, RHIC $x$-cutoffs $10^{-3}$ and $5\times10^{-2}$), and neglected OAM. The problem is to settle this claim with controlled uncertainties: evaluate $S_q$ and $S_g$ with the exact double-logarithmic Mellin-space amplitudes instead of the interpolation and quantify the approximation error; establish robustness of the saturation conclusion under variation of the inputs and inclusion of subleading logarithms; and compute or bound the small-$x$ OAM contributions so that the statement that large OAM is unnecessary becomes definitive. These three items remain unaddressed in the DLA framework.

Background

The proton spin puzzle originated with the European Muon Collaboration (EMC) measurement (Ashman et al., Phys. Lett. B 206 (1988) 364) that the quark and gluon spin contributions alone fell short of the angular-momentum conservation requirement $S_q + S_g = 1/2$, where $S_q = \frac{1}{2}\int_0^1 dx\, \Delta\Sigma(x)$ and $S_g = \int_0^1 dx\, \Delta G(x)$ are built from the helicity distributions $\Delta\Sigma$ (quark spin alignment) and $\Delta G$ (gluon spin alignment). The modern sum rule $S_P = S_q + S_g + L_q + L_g$ (Jaffe–Manohar; Ji) adds quark and gluon orbital angular momenta (OAM), the angular momentum carried by partons' transverse motion.

The helicity distributions evolve with the resolution scale $Q^2$ via DGLAP evolution equations, but at small Bjorken-$x$ (the fractional parton momentum) each order of perturbation theory generates powers of $\alpha_s \ln^2(1/x)$, which must be resummed. The double-logarithmic approximation (DLA) resums the leading series $(\alpha_s\ln^2(1/x))^n$. Two frameworks do this for helicity: the small-$x$ helicity evolution equations of Kovchegov, Pitonyak, and Sievert, as revised by Cougoulic, Tarasov, and Tawabutr (KPS-CTT), and the older Ermolaev–Greco–Troyan (EGT) approach based on infrared evolution equations (IREEs — a technique that extracts the leading logarithmic behavior of scattering amplitudes by comparing them at successive transverse-momentum cutoffs). In the EGT framework the perturbative components of the helicities are Mellin amplitudes $f_{ij}(\omega)$ (the Mellin transform rewrites an $x$-distribution as an integral over the conjugate variable $\omega$, where evolution equations become algebraic), whose rightmost singularity, the intercept $\omega_0$, controls the small-$x$ growth $h(x)\sim x^{-\omega_0}$. With running-coupling effects included, $\omega_0 = 0.86$, fixed by the Principle of Minimal Sensitivity (choosing the renormalization scale where predictions are stationary).

Ermolaev's paper "Proton spin in double-logarithmic approximation" (the source of this problem) estimates the proton spin as follows. The Relativistic Heavy Ion Collider (RHIC), a polarized-proton collider, constrains the partial helicity integrals over the measured $x$-ranges (denoted with bars) to $0.15\le\bar S_q\le0.20$ for $x\ge10^{-3}$ and $0.13\le\bar S_g\le0.26$ for $x\ge5\times10^{-2}$. Because the exact DLA amplitudes $f_{ij}(\omega)$ of the EGT papers are "quite complicated" in Mellin space, the paper replaces them by an interpolation with the correct intercept: $\widetilde h_{q,g}(x) = C'_{q,g}\, I_2(\xi\sqrt{\omega_0})/\xi$ with $\xi = \ln(1/x)$ and $I_2$ a modified Bessel function. Matching the normalization to the RHIC intervals gives enhancement factors $S_q = 1.18\, \bar S_q$ and $S_g = 1.85\, \bar S_g$, hence $0.42 \le S_q + S_g \le 0.72$, which contains $1/2$: the sum rule appears saturated by helicities alone, suggesting large OAM is not needed. The paper itself flags three weaknesses, which are exactly the open problem: the exact amplitudes were never used; the numerics are tentative (fixed $\omega_0$, constant initial distributions at small $x$, fixed cutoffs, no subleading logarithms); and OAM was dropped entirely. This matters because in the rival KPS-CTT framework the small-$x$ OAM evolution derived by Kovchegov and Manley gives OAM distributions the same leading intercept as the helicities — OAM is not parametrically suppressed at small $x$ — and the first phenomenological application of small-$x$ helicity evolution to polarized $pp$ data (Adamiak et al., Phys. Rev. D 112 (2025) 094032) still finds a total helicity between 0.02 and 0.51 for $x > 10^{-7}$, so whether helicities alone saturate the proton spin remains unsettled on every side.

Problem Statement

Within the double-logarithmic (DLA) framework of Ermolaev (Eur. Phys. J. C 85 (2025) 360), establish — with quantified theoretical uncertainty — whether the DL-resummed quark and gluon helicity contributions, matched to the RHIC constraints ($0.15\le\bar S_q\le0.20$ for $x\ge10^{-3}$; $0.13\le\bar S_g\le0.26$ for $x\ge5\times10^{-2}$), saturate the proton spin sum rule $S_P = S_q + S_g + L_q + L_g = 1/2$ without large orbital angular momentum. The objective is a definitive, uncertainty-quantified verdict on helicity-only saturation, which depends on three linked results, one for each loophole of the source paper's claim:

1. Exact amplitudes. Evaluate $S_q$ and $S_g$ using the exact DLA Mellin-space amplitudes $f_{ij}(\omega)$ of the Ermolaev–Greco–Troyan IREE solutions (in place of the interpolation $\widetilde h_{q,g}(x) = C'_{q,g} I_2(\xi\sqrt{\omega_0})/\xi$), and quantify the approximation error in the enhancement factors (the shortcut gives $S_q = 1.18\,\bar S_q$, $S_g = 1.85\,\bar S_g$) and in the resulting interval for $S_P$.

2. Robustness. Establish whether the conclusion that the helicity-only sum satisfies $S_q + S_g \in [0.42, 0.72] \ni 1/2$ survives variation of the inputs the source paper fixed by hand: the intercept $\omega_0 = 0.86$, the treatment of initial small-$x$ parton distributions as constants (vs. realistic nonperturbative parameterizations constrained without double-counting the RHIC data), the cutoffs $x_1$, $x_2$, the fixed resolution scale $Q^2\approx10\ \mathrm{GeV}^2$, and the effect of subleading (single-logarithmic and beyond-DL) corrections.

3. OAM. Either compute the small-$x$ OAM contributions $L_q$, $L_g$ in a controlled treatment consistent with the factorization framework used (the source author argues OAM requires $k_T$-factorization rather than collinear factorization), or bound them by an estimate of the small-$x$ OAM that does not merely restate the sum-rule identity $L_q + L_g = 1/2 - S_q - S_g$, so that the statement 'large OAM is not necessary' becomes definitive at the level of the full sum rule.

The objective also includes identifying, with an uncertainty budget, which input or correction could overturn that verdict.

References

1. B. I. Ermolaev, 'Proton spin in double-logarithmic approximation', Eur. Phys. J. C 85 (2025) 360. DOI: 10.1140/epjc/s10052-025-14067-3; arXiv:2402.16385, https://arxiv.org/abs/2402.16385
2. B. I. Ermolaev, 'Parton helicities at arbitrary x and Q^2 in double-logarithmic approximation', Eur. Phys. J. C 86 (2026) 48. DOI: 10.1140/epjc/s10052-025-15247-x; arXiv:2505.07359, https://arxiv.org/abs/2505.07359
3. Daniel Adamiak, Nicholas Baldonado, Yuri V. Kovchegov, Ming Li, W. Melnitchouk, Daniel Pitonyak, Nobuo Sato, Matthew D. Sievert, Andrey Tarasov, Yossathorn Tawabutr, 'First study of polarized proton-proton scattering with small-x helicity evolution', Phys. Rev. D 112 (2025) 094032. DOI: 10.1103/9gnx-ycs4; arXiv:2503.21006, https://arxiv.org/abs/2503.21006
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5. Brandon Manley, 'Orbital angular momentum small-x evolution: exact results in the large-Nc limit', JHEP 04 (2024) 055. DOI: 10.1007/JHEP04(2024)055; arXiv:2401.05508, https://arxiv.org/abs/2401.05508
6. Ignacio Borsa, Marco Stratmann, Werner Vogelsang, Daniel de Florian, Rodolfo Sassot, 'Next-to-Next-to-Leading Order Global Analysis of Polarized Parton Distribution Functions', Phys. Rev. Lett. 133 (2024) 151901. DOI: 10.1103/PhysRevLett.133.151901; arXiv:2407.11635 (preprint titled 'NNLO Global Analysis of Polarized Parton Distribution Functions'), https://arxiv.org/abs/2407.11635
7. B. I. Ermolaev, M. Greco, S. I. Troyan, 'Overview of the spin structure function g_1 at arbitrary x and Q^2', Riv. Nuovo Cim. 33 (2010) 57. DOI: 10.1393/ncr/i2010-10052-3; arXiv:0905.2841, https://arxiv.org/abs/0905.2841
8. Florian Cougoulic, Yuri V. Kovchegov, Andrey Tarasov, Yossathorn Tawabutr, 'Quark and gluon helicity evolution at small x: revised and updated', JHEP 07 (2022) 095. DOI: 10.1007/JHEP07(2022)095; arXiv:2204.11898, https://arxiv.org/abs/2204.11898