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Next-to-leading Order Evolution of Transversity Distributions and Soffer's Inequality

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W. Vogelsang (CERN)

High Energy Physics - Phenomenology · Submitted 25 Jun 1997 · 35 references

Pith's verdict: A clean, standard NLO calculation that delivers the two-loop transversity kernels and a defensible valence-sector check of Soffer's inequality.

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arXiv IDhep-ph/9706511 (version 2)
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CitationW. Vogelsang (CERN). "Next-to-leading Order Evolution of Transversity Distributions and Soffer's Inequality." arXiv:hep-ph/9706511 (1997). https://pith.science/paper/ZXOJYGCN
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The paper

The paper claims

Vogelsang computes the two-loop evolution kernels for transversity quark densities and shows that Soffer's inequality for valence quarks survives next-to-leading-order QCD evolution.

The reading

Transversity densities describe how quark spins line up with a transversely polarized nucleon's spin, and they are the last piece of the twist-2 parton picture that has no measured values. Soffer's inequality, $|Δ_T f| ≤ ½(f + Δ_L f)$, is the one model-independent constraint tying them to the ordinary and helicity-weighted densities, and it is known to survive leading-order evolution. The paper asks whether it survives the next order, which requires the two-loop transversity splitting functions that nobody had computed. Vogelsang derives them in the $MS$ scheme and finds the relevant combination of splitting functions and Drell-Yan coefficient functions stays positive for all $x$, so an input set that satisfies the bound at $Q_0$ cannot break it at higher $Q^2$. That makes the inequality usable as a constraint on models and on future Drell-Yan fits rather than a leading-order accident.

What carries the argument

The calculation uses the Curci–Furmanski–Petronzio ladder method in light-like axial gauge: the squared amplitude for transversely polarized quark scattering is expanded into two-particle-irreducible kernels, which are finite before the connecting lines are integrated, and a projector onto transverse polarization, built from $u(p,s)ū(p,s) = -p̸ s̸ γ_5$, isolates the collinear poles. The coefficient of the $1/ε$ pole in the resulting factor $Δ_T Γ_{ii}$ is the splitting function. Because every Dirac trace carries exactly two $γ_5$'s, an anticommuting $γ_5$ suffices, and the pure-singlet splitting vanishes because its two separate quark lines each give a trace with one $γ_5$ and an odd number of other gamma matrices. The check of Soffer's inequality then rests on the observation that the $Q^2$-evolution exponent $-2(Δ_T P^{(0),n}_{qq} - P^{(0),n}_{qq})/β_0$ is positive, so the leading-order preservation is automatic and only the next-order remainder needs to be shown positive.

Extended reading notes

Core claim

The paper presents the next-to-leading-order (two-loop) quark-to-quark splitting functions for transversity, $Δ_T P^{(1)}_{qq}(x)$ and $Δ_T P^{(1)}_{q̄q}(x)$, in the $MS$ scheme, and shows that the pure-singlet piece $Δ_T P^{(1)}_{qq,PS}$ vanishes identically. With these in hand the author tests Soffer's inequality for valence densities. Choosing a factorization scheme in which the unpolarized, longitudinally polarized and transversely polarized non-singlet Drell-Yan cross sections each keep their leading-order form, the condition for the inequality to be preserved reduces to the positivity of $D̃(x) = (P^{(1)}_{qq}(x) - Δ_T P^{(1)}_{qq,-}(x)) - (β_0/4)(C^{DY}_q(x) - Δ_T C^{DY}_q(x))$. Evaluating it for three flavours, all $+$-prescriptions and $δ(1-x)$ terms cancel and $D̃(x)$ is positive everywhere, diverging as $x → 0$. The same holds for the $MS$ version without the coefficient-function terms, so the bound is maintained in the scheme in which unpolarized and helicity densities are normally quoted.

Load-bearing premise

The argument assumes the inequality already holds at the input scale, which the paper says cannot be proved or disproved in perturbative QCD, and it checks only valence (non-singlet) combinations, leaving the singlet sector to a future numerical study.

Editorial extensions

If this is right

  • Soffer's inequality can be imposed as a constraint on model inputs for transversity at a low scale, because NLO evolution will not violate it at higher scales for valence combinations.
  • A complete NLO treatment of transversely polarized Drell-Yan is now possible: the coefficient function and the evolution kernels are both available in a common scheme.
  • The first moment $Δ_T P^{(1),1}_{qq,-} = \tfrac{19}{8}C_F^2 - \tfrac{257}{72}C_F N_C + \tfrac{13}{18}C_F T_f$ supplies the NLO correction to the evolution of the nucleon's tensor charge.
  • The near-constant small-$x$ limit of the transversity splitting functions, $Δ_T P^{(1)}_{qq,-}(x) ≈ C_F(2C_F - N_C) + 2C_F^2 x\ln^2 x$, means transversity densities are suppressed relative to unpolarized and helicity densities as $Q^2$ rises at small $x$.
  • The results agree with the two-loop anomalous dimensions obtained independently from the operator product expansion, giving a cross-check between the ladder and OPE approaches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the bound is preserved in both the Drell-Yan scheme and $MS$, a practical recipe follows: fit transversity at a low scale subject to the inequality, then evolve without re-imposing it, and any violation found in data would point to a scheme mismatch or to non-perturbative input rather than to evolution.
  • The mild small-$x$ behaviour means the inequality's margin shrinks in the same region where transversity itself is most suppressed, so future small-$x$ Drell-Yan kinematics may be the least informative place to test the bound.
  • Because the check is confined to valence combinations, the natural next step is a numerical singlet study; the vanishing of the pure-singlet transversity kernel suggests the singlet mixing cannot by itself generate a violation, but that remains to be shown.
  • The disagreement recorded with an earlier NLO Drell-Yan calculation over the $d$-dimensional leading-order splitting function is testable: an independent recalculation of the collinear limit of $q̄q → μ^+μ^- g$ would settle which scheme that earlier result actually corresponds to.
Original abstract

We present a calculation of the two-loop splitting functions for the evolution of the twist-2 `transversity' parton densities of transversely polarized nucleons. We study the implications of our results for Soffer's inequality for the case of valence quark densities.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,267 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We present a calculation of the two-loop splitting functions for the evolution of the twist-2 `transversity' parton densities of transversely polarized nucleons. We study the implications of our results for Soffer's inequality for the case of valence quark densities.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a calculation of the two-loop splitting functions for the evolution of the twist-2 `transversity' parton densities of transversely polarized nucleons. contribution 0.97 0.96 0.18 asserted 0.99
s1 We study the implications of our results for Soffer's inequality for the case of valence quark densities. contribution 0.57 0.85 0.23 asserted 0.55

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.92.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The functions D˜ (x), D(x) (see Eqs. (46),(49)) for f = 3 active flavours. As discussed in Sec. 2 we have chosen a factorization scheme in which the NLO Drell￾Yan cross sections for the unpolarized, the longitudinally polarized and the transversely polarized cases all retain their respective LO forms. We have done this to deal with quan￾tities that are manifestly scheme-independent. From a practical point of view, this may not be very useful since the unpolarized and longitudinally polarized parton densities are usually defined in the MS scheme. In particular, when building models…
Figure 2. The Mellin-n moments of the NLO transversity splitting functions ∆TP (1) qq,− (solid line) and ∆TP (1) qq,+ (dashed line) versus real n ≥ 1. For comparison the dash-dotted curve shows the LO result ∆TP (0),n qq . of the full singlet evolution of the unpolarized and longitudinally polarized densities and thus has to be subject to a detailed numerical study. Combining our finding for the valence part with previous experience from LO [15] indicates that no surprises concerning Soffer’s inequality are expected in the singlet sector. Note added After completing this work we received

Figures reproduced from arXiv:hep-ph/9706511 by the authors.

References
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