Intrinsic transverse momentum and the polarized Drell-Yan process

P.J. Mulders; R.D. Tangerman

REVIEW

Intrinsic transverse momentum and the polarized Drell-Yan process

R.D. Tangerman, P.J. Mulders

High Energy Physics - Phenomenology · Submitted 4 Mar 1994 · 25 references

Pith's verdict: The foundational six-function TMD paper, and the derivation holds up — but the naive time-reversal constraint quietly excludes the two T-odd distributions that would have explained the NA10 asymmetry the authors puzzle over.

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arXiv IDhep-ph/9403227 (version 2)

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CitationP.J. Mulders, R.D. Tangerman. "Intrinsic transverse momentum and the polarized Drell-Yan process." arXiv:hep-ph/9403227 (1994). https://pith.science/paper/JIWZILPV

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Editor's note

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This is one of the two or three papers the whole TMD program rests on. If you work on Drell-Yan at measured QT or on SIDIS with transverse momentum, you already know the result; what's worth checking is whether the derivation holds, and it does. I'd take the reader's ACCEPT and add one point they and the stress test both missed. What's new: the Dirac expansion in Eq. (3.3), constrained by hermiticity, parity, and time reversal, gives nine amplitudes; power counting in Eqs. (3.7)-(3.8) keeps exactly three projections (gamma+, gamma+gamma5, i sigma^{i+} gamma5), and those yield six real distributions per flavor in Eq. (3.9). The sixth, h1T-perp, is genuinely absent from Ralston-Soper: the authors pin it to the A9 amplitude RS dropped from their Eq. (3.4), and the free-quark ensemble in Sec. IV confirms it is nonzero for kT not zero. That is clean, checkable work, and the cross sections in Eqs. (5.13)-(5.16) follow from the Fierz decomposition plus the appendix projection machinery, which is spelled out far enough to redo by hand. Soft spots, in proportion. The one that matters is the time-reversal constraint in Eq. (3.2). Imposing naive T on the correlator kills the two T-odd distributions, Sivers and Boer-Mulders, which is exactly why Sec. VI.C ends puzzled by the NA10 cos2phi asymmetry and reaches for higher twist or vacuum structure. Boer-Mulders is the answer, and it is excluded by construction here. The authors do flag the gauge-link question in Sec. III.A, so they are not blind to it, but the link between that open question and the missing T-odd functions is never drawn, and it is where the paper's framing dates fastest. Second, the explicit 1/QT powers in Eq. (5.12) are handled by the kinematical-zero argument of Sec. VI.B, which rests on a smoothness assumption stated but not tested; the Gaussian model of Sec. VI.C illustrates the rest rather than proving it. Minor, but know it if you use those formulas literally near QT to zero. Third, Ref. [18] is in preparation. Minor. Who it's for: anyone setting up TMD factorization, and anyone who wants to see where the modern eight-function counting comes from and why this paper has six. Recommendation: accept without hesitation, and cite it.

Referee report

Major objections (1)

[Sec. V, Eq. (5.12); Sec. VI.B] Several structure functions in Eq. (5.12) carry explicit inverse powers of Q_T, up to 1/Q_T^3 in V^TL_{2,2} (5.12e) and 1/Q_T^4 in V^2TT_{2,2} (5.12i). The manuscript argues that these are cancelled by kinematical zeros of the accompanying tensors, but the explicit demonstration is given only for V^LL_{2,2} in Eq. (6.11); the remaining cases are checked only within the Gaussian model of Sec. VI.C, and the general step from Eq. (6.9) to Eq. (6.10) invokes a Taylor expansion whose validity is asserted in one clause ('assuming the distribution functions to be sufficiently well-behaved'). Since the central claim is that all six distributions are observable at leading order in 1/Q, the finiteness of these structure functions at small Q_T is load-bearing. Please state the smoothness assumption precisely and give the order of the kinematical zero for the highest-order cases (5.12e) and (5.12i) explicitly, so that the cancellation does not rest on the Gaussian ansatz.

Minor objections (7)

[Sec. III.B, Eq. (3.3)] The statement that the nine amplitudes A1-A9 exhaust the Dirac structures allowed by the symmetries of Eq. (3.2) is asserted rather than shown. A one-line completeness argument (counting independent covariants built from P, k, and S subject to P·S=0) would let the reader verify that no leading projection is missed; the six-function count itself follows from enumerating the Dirac matrices with n_+ - n_- = 1, which is independent of Eq. (3.3).
[Eqs. (5.12b), (5.19), (6.16)] The relative sign of the first term of Eq. (6.16) appears inconsistent with the combination of Eq. (5.12b) and Eq. (5.19): inserting V^LL_T = -4I[g1L g1L-bar] into (5.19) gives -lambda_A lambda_B I[g1L g1L-bar]/I[f1 f1-bar], whereas (6.16) carries a plus sign. Please check which of the three equations needs correction.
[Ref. [18]; Sec. II.B and Sec. IV] Ref. [18] is listed as 'in preparation', yet it is the sole support for two claims: the statement in Sec. II.B that the T-versus-perp distinction matters at higher order, and the statement in Sec. IV that nonvanishing quark-quark-gluon correlations cause deviations from the free-field results. Either supply the argument or soften the claims until [18] is available.
[Sec. V.A and Eqs. (5.12)] The symbol T is used both for transverse momentum (k_T, q_T) and for the structure-function combination (0,0) - (1/3)(2,0) ('The subindex T stands for the linear combination...', Sec. V.A). Combined with the statement that T and perp subscripts may be interchanged at leading order, this makes Eqs. (5.12) hard to parse. A short glossary or table of the six functions with their names, tensor structures, and free-quark interpretations would help.
[Eq. (5.7)] The sign prescription for antiquarks ('either by d-bar1 (d2) if it concerns a vector or axial-tensor distribution, or by -d-bar1 (-d2) if it concerns an axial-vector distribution') is correct but compact. A small table listing, for each of the six functions, the antiquark replacement rule would make the convolution integrals of Sec. V.B reproducible without repeated reference to Eqs. (3.15)-(3.17).
[Sec. VI.C] The text says the transverse radius r 'in principle depends on both the particular distribution function, and on x', but the Gaussian model then uses a single r_A and r_B per hadron. Please state explicitly whether the model assumes a common r for all distributions in a given hadron, since the numerical coefficients in Eqs. (6.15) depend on that choice.
[Sec. II.A; Refs. [13], [14], [24]] Typographical: 'Jackob-Wick' (Sec. II.A) should read 'Jacob-Wick'; 'Pertubative Quantum Chromomdynamics' (Ref. [13]) should read 'Perturbative Quantum Chromodynamics'; and the author name in Ref. [24] appears as 'M¨ uller'.

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The paper

The paper claims

A spin-1/2 hadron's quark content at measured transverse momentum needs six distribution functions, not five, and all six are observable in polarized Drell-Yan scattering.

The reading

Deep inelastic scattering sees only the longitudinal momentum and helicity of quarks, because the quark transverse momentum is integrated over. The Drell-Yan process at measured lepton-pair transverse momentum $Q_T$ does not integrate it away, so the angular pattern of the lepton pair carries information about how quarks move sideways inside the hadron. The paper asks how many independent quark distributions a spin-1/2 hadron can have once both $x$ and $k_T^2$ are kept, and answers with six per flavor: one momentum density, two helicity functions, and three transverse-polarization functions. It then computes the leading-order cross sections for longitudinal and transverse beam polarizations as convolutions over exactly these six objects, and illustrates them with a free-quark ensemble. The payoff, if the bookkeeping holds, is a complete dictionary between measurable lepton-pair angular distributions and the intrinsic transverse motion of quarks.

What carries the argument

The engine is the Dirac decomposition of the quark correlation function $Phi_{ij}(PS;k)$, constrained by hermiticity, parity, and time reversal, then truncated to leading twist by counting light-cone components. Projecting with $gamma^+$, $gamma^+gamma_5$, and $i sigma^{i+} gamma_5$ yields the six distributions, and a Fierz rearrangement factorizes the Born diagram into a product of one correlation function from each hadron. The convolution integrals over the two quark transverse momenta are then re-expressed in the Collins-Soper $hat{x}, hat{y}, hat{z}$ basis orthogonal to the virtual photon, so that each Lorentz tensor multiplying a structure function is built from external vectors rather than from convolution variables. A Gaussian ansatz for the $k_T$ dependence is used to exhibit the kinematic zeros of each structure function as $Q_T to 0$.

Extended reading notes

Core claim

The paper's claim is that at leading order in $1/Q$, the quark-quark correlation function integrated over $k^-$ reduces to three Dirac projections — vector, axial-vector, and axial-tensor — and that these are parametrized by six real functions of $x$ and $k_T^2$ for each flavor, written in Eq. (3.9): $f_1$, $g_{1L}$, $g_{1T}$, $h_{1T}$, $h^{1}_{1L}$, and $h^{1}_{1T}$ in the paper's notation. This extends the earlier five-function classification of Ralston and Soper by one transverse-momentum-dependent function needed to describe quark transverse polarization when $k_T$ is nonzero; the extra function also appears in the free-quark ensemble, where the densities reduce to probability, helicity, and transverse-polarization densities. With this basis the authors compute the leading-order hadron tensor and give cross sections for unpolarized, longitudinally polarized, transversely polarized, and mixed cases, Eqs. (5.13)–(5.16), each term being a convolution over the six distributions. They also record what survives when $Q_T$ is integrated out (three functions, matching the familiar $f_1$, $g_1$, $h_1$) and which structure functions survive at $Q_T = 0$.

Load-bearing premise

The cross-section predictions assume that the diagrammatic expansion truncated to a single quark-quark correlation function, with an approximated momentum-conserving delta function, correctly captures the physics at lepton-pair transverse momentum of order the hadronic scale, even though the authors state that factorization in that regime has not been proven.

Editorial extensions

If this is right

Doubly polarized Drell-Yan experiments at measured $Q_T$ can in principle extract all six distributions per flavor, including the two that vanish from inclusive deep inelastic scattering.
The longitudinal-longitudinal asymmetry acquires an angle-dependent term proportional to $Q_T^2$ and to the average transverse radius of the $h^{1}_{1L}$ distributions, so a measured $φ$ dependence there would size the transverse spread of those functions.
Integrating over $Q_T$ collapses the description back to the three familiar functions $f_1$, $g_1$, and $h_1$, with $h_1$ receiving a specific $k_T^2/(2M^2)$ correction from $h^{1}_{1T}$.
The transverse-longitudinal asymmetry is suppressed by $1/Q$ after $Q_T$ integration, so any observed asymmetry of that type at low $Q_T$ signals intrinsic transverse momentum rather than a leading-twist longitudinal effect.
The unpolarized cross section at this order has no $φ$-dependent $θ$ term, so a measured $φ$ asymmetry in unpolarized scattering must come from polarization of the beams, higher twist, or physics outside this leading-order treatment.

Reading between the lines

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

The six-function basis is a statement about leading twist only; the authors note that additional quark-quark-gluon correlations enter at subleading order and spoil the free-quark relations among the distributions, so the clean counting should be read as a leading-order scaffold rather than a complete classification.
Because the cross-section predictions rely on the diagrammatic expansion truncated to a single correlator with an approximated momentum-conserving delta function, the practical test is whether the extracted distributions are process-independent between Drell-Yan and semi-inclusive deep inelastic scattering at comparable $Q_T$.
The observation that any beam polarization can generate a $φ$-dependent asymmetry in the unpolarized channel suggests that polarization control in the beam and target is a prerequisite for interpreting the reported unpolarized asymmetries as evidence of new QCD structure.
The Gaussian illustration makes the kinematic zeros explicit, which offers a clean experimental handle: the $Q_T$ scaling of each angular coefficient distinguishes the transverse-spin distributions from the helicity ones without needing absolute normalization.

Original abstract

In this paper we study the cross section at leading order in $1/Q$ for polarized Drell-Yan scattering at measured lepton-pair transverse momentum $Q_T$. We find that for a hadron with spin $1/2$ the quark content at leading order is described by six distribution functions for each flavor, which depend on both the lightcone momentum fraction $x$, and the quark transverse momentum $\bbox{k}_T^2$. These functions are illustrated for a free-quark ensemble. The cross sections for both longitudinal and transverse polarizations are expressed in terms of convolution integrals over the distribution functions.

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

In this paper we study the cross section at leading order in $1/Q$ for polarized Drell-Yan scattering at measured lepton-pair transverse momentum $Q_T$. We find that for a hadron with spin $1/2$ the quark content at leading order is described by six distribution functions for each flavor, which depend on both the lightcone momentum fraction $x$, and the quark transverse momentum $\bbox{k}_T^2$. These functions are illustrated for a free-quark ensemble. The cross sections for both longitudinal and transverse polarizations are expressed in terms of convolution integrals over the distribution functions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we study the cross section at leading order in $1/Q$ for polarized Drell-Yan scattering at measured lepton-pair transverse momentum $Q_T$.	contribution 1.00	0.86	0.07	asserted 0.87
s1	We find that for a hadron with spin $1/2$ the quark content at leading order is described by six distribution functions for each flavor, which depend on both the lightcone momentum fraction $x$, and the quark transverse momentum $\bbox{k}_T^2$.	result 0.94	0.91	0.36	asserted 1.00
s2	These functions are illustrated for a free-quark ensemble.	contribution 0.41	0.61	0.04	asserted 0.98
s3	The cross sections for both longitudinal and transverse polarizations are expressed in terms of convolution integrals over the distribution functions.	result 0.42	0.88	0.15	asserted 1.00

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

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not examined

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Figures

**Figure 1.** FIG. 1. Quark and antiquark handbag diagrams for inclusive D

Figures reproduced from arXiv:hep-ph/9403227 by the authors.

References

Works cited (25)

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R. D. Tangerman and P. J. Mulders, in preparation

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J. A. M. Vermaseren, Symbolic manipulation with FORM, version 2 (CAN, Kruislaan 413, 1098 SJ Amsterdam, 1991). 25 FIGURES FIG. 1. Quark and antiquark handbag diagrams for inclusive D IS. FIG. 2. The quark and antiquark Born diagrams for the Drell-Y an process. FIG. 3. The blob representing the quark correlation functio n. 26 This figure "fig1-1.png" is av...

View on Pith Pith review arXiv 1991

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Wrote down all Lorentz-covariant structures compatible with the symmetry properties of the correlator, analogous to the procedure outlined in this paper.

Exploring quark transverse momentum distributions with lattice QCD (2010) · 0.95 · paraphrased: the licence does not allow quoting

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Cites it as background 31 papers

Some of these relations such as, e.g, the quark-model Lorentz-invariance relations (qLIRs), are generic in the sense that they hold in quark models which respect Lorentz symmetry and contain no gauge field degrees of freedom [ref] , but are not valid in QCD [ref] .

Quark-model relations among TMDs in the parton model · Quark model relations among TMDs · 0.85
The so-called transverse momentum dependent (TMD) approach [ref] , allowing for the separation of the soft physics from the hard, perturbatively calculable parts, is the main and consolidated framework to deal with these observables.

Collins asymmetries for pion-in-jet production in polarized $\ell p$ collisions at the EIC · Introduction · 0.81
Early theoretical studies of TMDs in hadron production in proton-proton processes [ref] were followed by systematic investigations in semi-inclusive deep-inelastic scattering (SIDIS) [ref] and DY [ref] (also fragmentation functions [ref] enter the description of SIDIS).

The Drell-Yan process with pions and polarized nucleons · Introduction · 0.98

Discussion

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[1] [18]

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J. A. M. Vermaseren, Symbolic manipulation with FORM, version 2 (CAN, Kruislaan 413, 1098 SJ Amsterdam, 1991). 25 FIGURES FIG. 1. Quark and antiquark handbag diagrams for inclusive D IS. FIG. 2. The quark and antiquark Born diagrams for the Drell-Y an process. FIG. 3. The blob representing the quark correlation functio n. 26 This figure "fig1-1.png" is av...

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