REVIEW
Time-reversal odd distribution functions in leptoproduction
High Energy Physics - Phenomenology · Submitted 26 Nov 1997 · 25 references
Pith's verdict: This is where h1⊥ and f1T⊥ enter as leading-twist objects and get matched to specific azimuthal asymmetries; the asymmetry tables are the real contribution, and the missing gauge link in Eq. (1) is the loose thread.
T0 · AI review
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Referee report
The paper
The paper claims
The paper catalogues every place in lepton-hadron scattering where a time-reversal-odd quark function could show up, and argues that most of them appear as distinctive azimuthal asymmetries rather than as shifts in the total rate.
The reading
What carries the argument
The machinery is the lightfront quark-quark correlation function $Phi(x,p_T)$ (and its fragmentation analogue $Delta(z,k_T)$), expanded in Dirac projections. Lorentz invariance, hermiticity and parity fix the expansion to a finite list of scalar functions of $x$ and $p_T^2$; dropping the time-reversal constraint is precisely what adds $f_{1T}^{perp}$ and $h_1^{perp}$ to that list. The hadron tensor at leading order is a convolution of $Phi$ and $Delta$ with a transverse-momentum delta function tying $p_T+q_T$ to $k_T$ (Eq. 7), and the observables are defined as cross sections weighted by powers of $Q_T$ and by trigonometric functions of the azimuthal angles $varphi_{ell h}$, $varphi_{ell S}$, $varphi_{ell S_h}$ (Eq. 11). Because the weight carries the angular dependence, the azimuthal asymmetry becomes an integral over $p_T$ and $k_T$ that picks out a specific moment of a specific pair of functions. The paper also supplies the bridge between the two languages: relations such as $f_T(x) = -d/dx, f_{1T}^{perp(1)}(x)$ and $h(x) = -d/dx, h_1^{perp(1)}(x)$ connect the $p_T$-integrated twist-three functions to moments of the leading-twist $p_T$-dependent ones, and the combinations collected in Eqs. (26)–(31) isolate the genuinely interaction-dependent parts of the twist-three fragmentation functions.
Extended reading notes
Core claim
The paper's central claim is that allowing time-reversal-odd functions in the transverse-momentum-dependent quark correlators produces a specific, enumerable set of leading-order azimuthal asymmetries in leptoproduction, and that these asymmetries are the practical handles for finding the odd functions. The most striking entry is the fully unpolarized case: weighting the cross section by $Q_T^2/(MM_h)$ and by $cos(2varphi_{ell h})$ projects out a term proportional to the product of the two odd functions, $h_1^{perp(1)}$ from the target and $H_1^{perp(1)}$ from the fragmentation, with the kinematic factor $(1-y)$ (Eq. 13). A second key entry is the single-spin asymmetry $sin(varphi_{ell h}-varphi_{ell S})$ for an unpolarized lepton beam on a transversely polarized target, proportional to $f_{1T}^{perp(1)}(x_B),D_1(z_h)$, which the authors single out as probably the easiest way to look for $f_{1T}^{perp}$ because it requires only a polarized target and no final-state polarimetry. The remaining tables pair the odd functions with T-even partners in single-spin asymmetries requiring polarimetry of the produced hadron, and identify exactly one odd-odd double-spin asymmetry at leading order. For the transverse-momentum-integrated case the authors push to order $1/Q$ and give the twist-three functions, noting that all leading-twist functions are T-even and that only $f_T$, $e_L$, $h$ and their fragmentation counterparts carry the odd behaviour at subleading order.
Load-bearing premise
The whole catalogue rests on the assumption that the cross section factorizes at leading order into a transverse-momentum-dependent distribution and fragmentation function convolved as in Eq. (7), and that T-odd distribution functions can exist at all; the authors themselves note that factorization for the transverse-momentum-integrated functions has not yet been proven and that the origin of the T-odd distributions is conjectural.
Editorial extensions
If this is right
- A measurement of the $cos(2varphi_{ell h})$ asymmetry in fully unpolarized lepton-hadron scattering, weighted by $Q_T^2/(MM_h)$, would test whether both $h_1^{perp}$ and $H_1^{perp}$ are non-zero, since the asymmetry is their product.
- A measurement of $sin(varphi_{ell h}-varphi_{ell S})$ in $e p^{uparrow} to e,pi X$ or $e p^{uparrow} to e K X$ would probe $f_{1T}^{perp}$ without any final-state polarimetry, requiring only a transversely polarized target.
- Polarimetry of a produced hyperon, as in $e p to e Lambda^{uparrow} X$ read out through the $p pi^-$ final state, would give access to $D_{1T}^{perp}$ and to $h_1^{perp}$ through the three T-odd single-spin asymmetries in Table III.
- The one leading-order odd-odd double-spin asymmetry, $cos(varphi_{ell S}-varphi_{ell S_h})$ weighted by $Q_T^2/(MM_h)$, would isolate the product $f_{1T}^{perp(1)} D_{1T}^{perp(1)}$ in the same $e p^{uparrow} to e Lambda^{uparrow} X$ process where the even-even transverse-spin transfer is measured.
- Because the asymmetries involving $D_1$ survive jet production with $D_1(z) = delta(1-z)$, the azimuthal asymmetry $cos(varphi_{ell,rm jet}-varphi_{ell S})$ offers a channel for $g_{1T}$ that needs no hadron identification.
Reading between the lines
- The unpolarized $cos(2varphi_{ell h})$ asymmetry is a product of two odd functions, so a null result cannot distinguish 'no T-odd distributions' from 'no T-odd fragmentation'; only the single-spin asymmetries, where one side is T-even, break that degeneracy experimentally.
- The derivative relations between twist-three and moment-of-leading-twist functions mean an experiment that measures the $1/Q$ azimuthal spin asymmetries and the leading-order $Q_T$-weighted ones is measuring the same underlying objects twice, which gives an internal consistency check rather than independent information.
- If the odd distributions arise from gluonic poles as the authors' companion work suggests, their scale dependence should differ from that of the T-even functions, so the asymmetries' $Q^2$ evolution is a second, independent handle beyond the azimuthal shape.
- The authors' own note that final-state interaction phases might average to zero for single-hadron production would, if correct, remove the fragmentation-side odd functions entirely and leave only the distribution-side ones — a sharp divide the tables make visible.
Original abstract
We consider the various asymmetries, notably single spin asymmetries, that appear in leptoproduction as a consequence of the presence of time-reversal odd distribution functions. This could facilitate experimental searches for time-reversal odd phenomena.
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 consider the various asymmetries, notably single spin asymmetries, that appear in leptoproduction as a consequence of the presence of time-reversal odd distribution functions. This could facilitate experimental searches for time-reversal odd phenomena.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the various asymmetries, notably single spin asymmetries, that appear in leptoproduction as a consequence of the presence of time-reversal odd distribution functions. | contribution 0.95 | 0.51 | 0.19 | asserted 0.88 |
| s1 | This could facilitate experimental searches for time-reversal odd phenomena. | interpretation 0.86 | 0.07 | 0.24 | hedged possibility 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
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- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-ph/9711485 by the authors.
References
How later papers used it
153 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 of 3 papers with this label
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Assumed a factorized Gaussian dependence on parton and hadron transverse momenta from this paper and defined a transverse-momentum-weighted distribution.
Sivers and Collins Single Spin Asymmetries (2008) · 0.93 · paraphrased: the licence does not allow quoting
How the literature uses it
136 arXiv papers cite this in 232 sentences
Papers citing it · 102 sentences without a confident label
Uses its method or tool 3 papers
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Still to the present day, the state-of-the-art approach to subleading-twist TMD observables is the one of Refs. [ref] , based on a TMD tree-level formalism, which we adopt here.
Semi-Inclusive Deep Inelastic Scattering in Wandzura-Wilczek-type approximation · TMDs, FFs and structure functions · 0.97
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In SIDIS, it can be avoided using the TSAs weighted with powers of the outgoing hadron transverse momentum $$ [ref] .
Measurement of $q_T$-weighted TSAs in 2015 COMPASS Drell-Yan data · Introduction · 0.81
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We assume a factorized Gaussian dependence on parton and hadron transverse momenta [ref] with $B_G(z)=(1+z^2 p_h_1^2/K^2_H_1)^-1/2$ and define $H_1^(z) ≡H_1^⊥(1/2) a(z)=$ $∫^2K_T|K_T|/2zm_π H_1^⊥a(z,K_T)$.
Sivers and Collins Single Spin Asymmetries · Transversity and Collins effect · 0.93
Uses its data or values 1 paper
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The derived relations are verified numerically using unpolarized TMDs from the recent N4LL extraction ART23 [ref] , demonstrating the consistency of the method. * The first TMM provides information about the distribution of the average transverse momentum shift of partons [ref] , and it is expressed via collinear distributions of twist three.
Transverse momentum moments · Introduction · 0.81
Compares with it 2 papers
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Here we present a comparison of the eight leading-twist quark TMD-FFs as given in the helicity formalism with those defined in the notation of the Amsterdam group [ref] .
General helicity formalism for two-hadron production in $e^+e^-$ annihilation within a TMD approach · Comparison with other notations · 1.00
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Appendix C of Ref. [ref] gives a detailed comparison of the above notation with the one of the Amsterdam group [ref] , often adopted in the literature (see also Eq. ([ref])).
Azimuthal and Single Spin Asymmetries in Hard Scattering Processes · Spin and TMD parton distribution functions at leading twist · 0.86
Cites it as background 95 papers
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The distribution of linearly polarized gluons in nucleons is given by the Boer-Mulders' function $h_1^(x,k_^2)$ [ref] .
Azimuthal Anisotropy at high transverse momentum in $p$-$p$ and $p$-$A$ collisions · Introduction · 0.88
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The key ingredient for these TSAs is the Boer-Mulders function $h_1^⊥$ [ref] , which is a naively time-reversal odd transverse-momentum-dependent distribution function and provides the necessary phase required for TSAs.
Boer-Mulders function of the pion and pretzelosity distribution of the proton in the polarized pion-proton Drell-Yan process at COMPASS · Introduction · 0.97
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Several relations between (forward) twist-3 and (moments of) $_T$-dependent PDFs have been proposed in the literature [ref] .
Lorentz invariance relations between parton distributions and the Wandzura-Wilczek approximation · Introduction · 0.92
Discussion
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