REVIEW
Bounds on transverse momentum dependent distribution and fragmentation functions
High Energy Physics - Phenomenology · Submitted 23 Dec 1999 · 26 references
Pith's verdict: The TMD positivity bounds are correct and became standard; the fragmentation half is asserted by analogy rather than derived, which is a presentation gap, not an error.
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Referee report
The paper
The paper claims
The paper derives upper bounds on the transverse-momentum-dependent quark distribution and fragmentation functions by requiring that the lightcone quark correlator, viewed as a matrix in combined Dirac and target-spin space, be positive sem
The reading
What carries the argument
The load-bearing object is the matrix $M = (Φ(x,p_T)γ^+)^T$ written on the basis of good lightcone quark spinors and then tensored with the target spin density matrix, giving a $4×4$ Hermitian matrix whose entries are the twelve leading-order transverse-momentum-dependent functions. Its positive semidefiniteness is the entire engine: taking one-dimensional subspaces reproduces the trivial bounds $f_1 ≥ 0$ and $|g_{1L}| ≤ f_1$, two-dimensional subspaces give the transversity and $h_{1T}^{⊥(1)}$ bounds, and the full eigenvalue conditions give the quadratic bounds that tie the T-odd Sivers-type function to the T-even $g_{1T}$ and the Collins-type function to $h_{1L}^{⊥}$.
Extended reading notes
Core claim
The authors take the lightcone quark correlator $Φ(x,p_T)$, transpose it in Dirac space, and extend it to a matrix in the tensor product of Dirac space and the target's $2×2$ spin space. Positivity of that matrix for any vector in the combined space — which follows from its representation as a sum of squares of lightcone wave-function amplitudes — forces the eigenvalues to be non-negative, and those eigenvalues translate into inequalities among the leading-order transverse-momentum-dependent functions. The results are $|h_1| ≤ (f_1+g_{1L})/2$, the Soffer bound recovered as a special case, together with the new $|h_{1T}^{⊥(1)}| ≤ (f_1-g_{1L})/2$, and the paired quadratic bounds $(g_{1T}^{(1)})^2+(f_{1T}^{⊥(1)})^2 ≤ (p_T^2/4M^2)(f_1+g_{1L})(f_1-g_{1L})$ and $(h_{1L}^{⊥(1)})^2+(h_{1}^{⊥(1)})^2 ≤ (p_T^2/4M^2)(f_1+g_{1L})(f_1-g_{1L})$. Writing the functions in terms of two positive combinations $F$ and $G$ with $f_1 = F+G$ and $g_1 = F-G$, and four parameters $α, β, γ, δ$ bounded in magnitude by one, the authors convert the eigenvalue conditions into the compact pair $|γ+δ|^2 ≤ (1-α)(1+β)$ and $|γ-δ|^2 ≤ (1+α)(1-β)$, which is the geometric region plotted in the paper's figure.
Load-bearing premise
The bounds on fragmentation functions, including the ceiling on the Collins function, are asserted by analogy with the distribution case rather than derived from the fragmentation correlator, which is a sum over final states with a different Dirac and trace structure; the premise is that this correlator has the same positive-semidefinite form.
Editorial extensions
If this is right
- Every azimuthal or single-spin asymmetry measured at HERMES, COMPASS, RHIC or LEP has a computable upper ceiling built from $f_1$ and $g_1$, so an anomalously large reported asymmetry would signal either a breakdown of the leading-order description or an error.
- The Collins function $H_1^{⊥(1)}$, which governs the fragmentation of a transversely polarized quark into a pion, is bounded by $(|P_{π⊥}|/2zM_π)D_1(z,P_{π⊥}^2)$, so its magnitude is tied to the unpolarized fragmentation function at the same kinematics.
- The T-odd Sivers-type distribution $f_{1T}^{⊥}$ and the T-even $g_{1T}$ are not independent: their squares must fit inside a common envelope set by $f_1$ and $g_1$, which constrains fits that treat them separately.
- Functions such as $g_{1T}^{(1)}$ and $h_{1L}^{⊥(1)}$ are forced to vanish proportionally to $|p_T|$ as $p_T → 0$, which fixes the small-$p_T$ shape of the asymmetries they generate.
Reading between the lines
- Because the bounds are inequalities among functions that all evolve with the hard scale, comparing a measured asymmetry against its ceiling at two different $Q$ values would test whether the correlator's positivity structure survives evolution, a question the paper flags as open.
- The same eigenvalue argument, applied to the gluon correlator rather than the quark one, would produce ceilings on the gluon Sivers and Boer-Mulders functions; the paper does not attempt this.
- If the Collins function is found to sit close to its ceiling, then the $e^+e^-$ dihadron asymmetry and the SIDIS Collins asymmetry should be correlated through the shared $D_1$ normalization, which is a checkable pattern rather than a single number.
- Model builders can use the allowed region in the $(α,β)$ plane as a filter: a spectator-model calculation whose transversity and $h_{1T}^{⊥(1)}$ fall outside the shaded region is internally inconsistent, not merely unusual.
Original abstract
We give bounds on the distribution and fragmentation functions that appear at leading order in deep inelastic 1-particle inclusive leptoproduction or in Drell-Yan processes. These bounds simply follow from positivity of the defining matrix elements and are an important guidance in estimating the magnitude of the azimuthal and spin asymmetries in these processes.
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.
The abstract, sentence by sentence
We give bounds on the distribution and fragmentation functions that appear at leading order in deep inelastic 1-particle inclusive leptoproduction or in Drell-Yan processes. These bounds simply follow from positivity of the defining matrix elements and are an important guidance in estimating the magnitude of the azimuthal and spin asymmetries in these processes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We give bounds on the distribution and fragmentation functions that appear at leading order in deep inelastic 1-particle inclusive leptoproduction or in Drell-Yan processes. | contribution 0.92 | 0.93 | 0.61 | asserted 1.00 |
| s1 | These bounds simply follow from positivity of the defining matrix elements and are an important guidance in estimating the magnitude of the azimuthal and spin asymmetries in these processes. | interpretation 0.98 | 0.54 | 0.39 | asserted 1.00 |
What was examined, and what was not
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Figures
Showing 2 of the paper's 3 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9912490 by the authors.
References
How the literature uses it
32 arXiv papers cite this in 51 sentences
Papers citing it · 37 sentences without a confident label
Uses its method or tool 1 paper
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To demonstrate the violations of the positivity bound by mixed-order calculations for the elementary T-even and T-odd functions we employ the positivity relations derived in Ref. [ref] to impose constraints on the FFs, which should also hold for model SFs.
Monte Carlo Implementation of Polarized Hadronization · Positivity bounds for splitting functions · 0.92
Cites it as background 13 papers
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Theoretical framework: There have been extensive studies of high-$p_T$ hadron production in $p$-$p$ collisions, where the intrinsic transverse momentum of partons within the incoming proton [ref] , and the transverse momentum of the final hadron with respect to the high-$p_T$ parton [ref] have been included.
Azimuthal Anisotropy at high transverse momentum in $p$-$p$ and $p$-$A$ collisions · Introduction · 0.98
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Before proceeding with twist-3 let us pause and revisit positivity bounds [ref] .
Semi-Inclusive Deep Inelastic Scattering in Wandzura-Wilczek-type approximation · Inequalities and a cross check · 0.92
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Corresponding bounds have been formulated for transverse-momentum dependent distributions (TMDs) in [ref] , for impact parameter distributions in [ref] , and for double parton distributions (DPDs) in [ref] .
Double parton distributions out of bounds in colour space · Introduction · 0.80
Discussion
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