How large is the longitudinal unpolarized structure function in semi-inclusive deep inelastic scattering?

R_SIDIS compares two structure functions, not two cross sections

For an unpolarized beam and an unpolarized target, the azimuthally integrated semi-inclusive deep inelastic scattering (SIDIS) cross section is described by two structure functions, F_UU,T and F_UU,L, which generalize the transverse and longitudinal structure functions of inclusive deep inelastic scattering. The paper studies the ratio R_SIDIS = F_UU,L / F_UU,T and its dependence on transverse momentum.

This ratio should not be confused with the inclusive longitudinal-to-transverse cross-section ratio R = σ_L/σ_T, which has been measured with moderate precision and is broadly quoted as R ≃ 0.25. R_SIDIS is a ratio of SIDIS structure functions, and the paper's estimate of about 0.3 at low transverse momentum is a statement about that different quantity, in a specific kinematic setting rather than a global constant.

Power counting made F_UU,L look negligible at low q_T

In the low-transverse-momentum region q_T ≪ Q, transverse-momentum-dependent (TMD) factorization applies, and at leading power in the hard scale Q the transverse structure function F_UU,T can be written as a convolution of unpolarized TMD parton distribution and fragmentation functions f_1 and D_1. In that same regime F_UU,L is explicitly power suppressed relative to F_UU,T, receiving next-to-next-to-leading-power (NNLP) contributions that scale as q_T^2 / Q^2 times F_UU,T, plus contributions from higher-twist TMD distributions.

Because F_UU,T is predicted to dominate by power counting, most recent analyses of transverse-momentum-differential SIDIS data have focused on it, while F_UU,L has received little attention. The paper's starting point is that this suppression does not by itself make the longitudinal term numerically small, especially where q_T is not extremely small compared with Q.

The estimate stitches TMD and collinear descriptions together

The two ends of the transverse-momentum spectrum are described by different mechanisms. At small q_T, intrinsic and perturbatively evolved transverse-momentum dependence is captured by TMD factorization; at large q_T, where q_T ∼ Q, the transverse momentum is generated by hard partonic radiation and collinear factorization applies as a fixed-order expansion in the strong coupling α_s.

For F_UU,T, the two descriptions are connected by matching the TMD contribution at small q_T to the fixed-order collinear result at large q_T. For F_UU,L, that matching is not established: the twist-4 TMD terms at small q_T should in principle match onto twist-2 collinear terms at large q_T, but known problems with the standard approach remain unresolved. The authors therefore do not attempt to solve the matching problem; they interpolate between the two regions with a transition function w(q_T), so that F = w F_W + (1 − w) F_FO, and treat the result as an exploratory estimate.

The low-q_T model for F_UU,L follows the expected power counting, using a weight function ω(p_T^2) = (4M^2/Q^2)(p_T^2/M^2) applied to the unpolarized TMD PDF and FF, where M is the target hadron mass. Boer–Mulders and Collins contributions are omitted because their effect was found to be small relative to the kinematic contribution, and because no compatible phenomenological extraction of those functions was available for this framework.

TMD grids and collinear sets set the numerical scale

The numerical estimate uses TMD PDF and TMD FF grids from the MAP22 extraction to compute the W-term contribution to both F_UU,T and F_UU,L. The fixed-order calculation is taken at O(α_s) for both structure functions, with the MMHT2014 and DSS14 sets at NLO used for collinear PDFs and FFs respectively. To match the W and fixed-order terms consistently, the TMD grids are used at NNLL accuracy.

The authors note that O(α_s^2) fixed-order calculations and N^3LL TMD grids exist but were not implemented here, since doing so is not straightforward and lies beyond the scope of this work. That choice is one reason the resulting numbers are presented as estimates with quantified uncertainty rather than as precision predictions.

R_SIDIS reaches about 0.3 at low q_T and low Q

The central quantitative result is that the longitudinal contribution can be sizable: R_SIDIS ≃ 0.3 at low transverse momentum and hard scale. The paper reports that this produces a correction of the order of 10% to transverse-momentum-integrated observables, and that after integration over transverse momentum the difference between the estimated longitudinal structure function and the fixed-order result reaches approximately 5–10% at Q ≃ 2 GeV in the kinematics studied, becoming negligible at larger Q.

The transverse-momentum dependence of the ratio changes with the hard scale. In the COMPASS kinematics considered, at larger Q the ratio starts from small values at low q_T, rises rapidly, and then decreases following the fixed-order calculation. At low Q it starts from values on the order of 30% and decreases with increasing q_T, approaching the fixed-order description. The uncertainty on the ratio is sizable, particularly at low Q, where the input TMD and missing-higher-order uncertainties dominate.

Neglecting F_UU,L can bias TMD PDF and FF extractions

The relevance of F_UU,L extends beyond the structure function itself. Transverse-momentum-dependent SIDIS measurements are commonly presented as hadron multiplicities, ratios of semi-inclusive to inclusive DIS cross sections. In global TMD analyses, the inclusive DIS denominator contains both the transverse and longitudinal structure functions F_T and F_L, while the contribution of F_UU,L to the numerator is usually neglected.

The paper argues that including F_UU,L can affect the predicted shape and normalization of multiplicities differential in transverse momentum, and consequently the determination of unpolarized TMD PDFs and FFs. The same issue applies to angular modulations and spin asymmetries, whose experimental extraction involves ratios to the unpolarized SIDIS cross section. The authors also note that experimental information on the transverse-momentum dependence of F_UU,L remains very limited: the only available SIDIS measurements are integrated in transverse momentum and carry large experimental errors.

The largest effects sit at the lowest hard scales

The authors compared their small-transverse-momentum results with alternative approaches in the literature and found qualitative agreement. They also tested the impact of NNLP corrections to F_UU,T on their results, using the Madrid and BP expressions as well as their own model for that correction at x = 0.1, z = 0.5, and Q = 2 GeV. The impact on R_SIDIS is small and does not significantly change its size; the Madrid and BP expressions give a slightly smaller R_SIDIS at q_T/Q ≃ 0.6, and all results are compatible within the baseline uncertainty.

The estimate is explicitly subject to theoretical uncertainties from the interpolation, the input distributions, and missing higher-order corrections, and it lacks a complete theoretical description of F_UU,L and of the matching. Within those limits, the paper's practical priority is clear: the effects are largest at the relatively low values of Q encountered in fixed-target experiments, exactly where the estimates yield the largest R_SIDIS. That is the regime where dedicated measurements of R_SIDIS differential in transverse momentum would most directly establish the size of the longitudinal contribution and constrain its behavior.

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