Partonic Structure, Spin, and Hadron Tomography

PDFs, GPDs, and TMDs are different renormalized hadron matrix elements, not different plots of one universal three-dimensional density. A PDF is forward and collinear, a GPD is off-forward and collinear, and a TMD retains transverse parton momentum together with a process-class-dependent staple Wilson line and rapidity scale. Their moments and limits connect to charges, form factors, and spin sum rules only under the hypotheses stated below.

Required background. Hadron form factors and current structure supplies local-current matrix elements; collinear factorization and operator PDFs supplies the separation of short-distance coefficients from light-ray operators. Helpful background. TMD factorization and rapidity evolution supplies soft subtraction and Collins–Soper evolution.

Light-ray correlators select different information

Let $a^\pm=(a^0\pm a^3)/\sqrt2$ and take the hadron to carry large $P^+$. A representative renormalized quark correlator is

$$\Phi_q^{[\Gamma]}(x;\mu) =\frac12\int\frac{dz^-}{2\pi} e^{ixP^+z^-} \langle P,S| \bar q(-z^-/2)\,\Gamma\, W[-z^-/2,z^-/2]q(z^-/2) |P,S\rangle_\mu,$$

with $z^+=0$, $\mathbf z_T=0$, and a straight Wilson line $W$ along the lightlike separation. The Wilson line is required for gauge invariance. The leading-twist projections define, up to the displayed normalization convention,

$$\Gamma=\gamma^+ \ longrightarrow\ f_1^q(x,\mu), \qquad \Gamma=\gamma^+\gamma^5 \ longrightarrow\ g_1^q(x,\mu), \qquad \Gamma=i\sigma^{j+}\gamma^5 \ longrightarrow\ h_1^q(x,\mu).$$

They describe unpolarized, helicity, and transversity structure, respectively. Their scale dependence is operator renormalization and is compensated by short-distance coefficients in a factorized observable. The derivation of collinear factorization and DGLAP evolution belongs to the required background route; the present task is to identify which hadron matrix element a claim uses. A standard operator-to-cross-section construction is given in Schwartz 2014, §§ 32.2–32.3.

For local moments it is convenient to extend the quark distribution to $-1<x<1$ with $q(-x)=-\bar q(x)$ for $x>0$. Taylor expansion of the light-ray operator then gives symmetric traceless twist-two operators,

$$\int_{-1}^{1}dx\,x^{n-1}q(x,\mu) \ longleftrightarrow\ \langle P| \bar q\gamma^{\{\mu_1}iD^{\mu_2}\cdots iD^{\mu_n\}}q -\text{traces} |P\rangle_\mu.$$

The arrow denotes equality after applying the chosen normalization and contracting with the lightlike direction; it is not a pointwise reconstruction of $q(x)$. Low moments yield useful checks:

$$\int_0^1dx\,[q(x)-\bar q(x)]=N_q,$$ $$\sum_q\int_0^1dx\,x[q(x,\mu)+\bar q(x,\mu)] +\int_0^1dx\,xg(x,\mu)=1.$$

The first is a conserved flavor-number sum rule. The second is the momentum sum rule at a common scheme and scale. Individual quark and gluon momentum fractions evolve even though their sum does not.

GPDs: off-forward collinear structure

Set $\bar P=(p'+p)/2$, $\Delta=p'-p$, $t=\Delta^2$, and $\xi=-\Delta^+/(2\bar P^+)$. The spin-$\tfrac12$ unpolarized quark GPDs are defined by

$$\begin{aligned} &\frac12\int\frac{dz^-}{2\pi}e^{ix\bar P^+z^-} \langle p',s'| \bar q(-z^-/2)\gamma^+Wq(z^-/2) |p,s\rangle_\mu \\[2pt] &\qquad=\frac{1}{2\bar P^+}\bar u(p',s') \left[ \gamma^+H^q(x,\xi,t;\mu) +\frac{i\sigma^{+\alpha}\Delta_\alpha}{2M} E^q(x,\xi,t;\mu) \right]u(p,s). \end{aligned}$$

This differs from a PDF because the initial and final hadron momenta differ. It differs from a form factor because the light-ray separation, and therefore the parton fraction $x$, has not been integrated out. The exact reductions make the distinction testable:

$$H^q(x,0,0;\mu)=q(x,\mu),$$ $$\int_{-1}^{1}dx\,H^q(x,\xi,t;\mu)=F_1^q(t), \qquad \int_{-1}^{1}dx\,E^q(x,\xi,t;\mu)=F_2^q(t).$$

Lorentz covariance makes higher Mellin moments polynomials in $\xi$ of bounded degree. The apparent $\xi$ independence of the first moments is the lowest polynomiality check Diehl 2003, §§ 3.2–3.3.

For the gauge-invariant kinetic angular-momentum decomposition, the quark total angular momentum obeys Ji’s relation

$$J_q(\mu)=\frac12\int_{-1}^{1}dx\,x \bigl[H^q(x,0,0;\mu)+E^q(x,0,0;\mu)\bigr].$$

The quark and gluon pieces are scale- and scheme-dependent, while their conserved total is not. This relation determines total quark angular momentum, not orbital angular momentum alone; extracting the latter also requires the quark spin contribution in the same convention Ji 1997, pp. 610–613.

At zero skewness, a transverse Fourier transform has a controlled impact-parameter interpretation for a transversely localized fast hadron:

$$q(x,\mathbf b_T)= \int\frac{d^2\boldsymbol\Delta_T}{(2\pi)^2} e^{-i\mathbf b_T\cdot\boldsymbol\Delta_T} H^q(x,0,-\boldsymbol\Delta_T^2).$$

It correlates longitudinal fraction $x$ with transverse position $\mathbf b_T$ relative to the light-front transverse center. It is not a rest-frame three-dimensional density. At $\xi\ne0$, the initial and final states carry different longitudinal momenta, so even this probability interpretation is lost Diehl 2003, § 3.10.

TMDs: transverse separation and rapidity structure

A TMD retains $\mathbf z_T$ before Fourier transformation:

$$\Phi_q^{[\Gamma]}(x,\mathbf k_T;\mu,\zeta) \sim \int dz^-d^2\mathbf z_T\, e^{ixP^+z^- -i\mathbf k_T\cdot\mathbf z_T} \frac{\langle P,S|\bar q(-z/2)\Gamma W_{\rm staple}q(z/2)|P,S\rangle} {\sqrt{S_{\rm soft}}}.$$

The proportionality sign suppresses convention-dependent factors and the explicit rapidity regulator. Unlike the straight collinear link, $W_{\rm staple}$ runs toward future or past light-cone infinity according to the color flow of the factorized process. The soft factor removes double counting, and the rapidity scale $\zeta$ accompanies the ordinary renormalization scale $\mu$. A TMD result without the link direction, soft-subtraction scheme, $\mu$, and $\zeta$ is incomplete Collins 2011, ch. 13, pp. 479–539.

The link direction explains why naively time-reversal-odd functions such as the Sivers function change sign between future-link and past-link process classes. It does not mean that every TMD is arbitrary or process dependent: factorization predicts a controlled universality pattern. The leading-twist spin decomposition and its angular modulations are tabulated in Bacchetta et al. 2007, §§ 2–3.

Naively integrating a renormalized TMD over all $\mathbf k_T$ is ultraviolet sensitive. The small-$b_T$ operator product expansion matches a TMD onto collinear PDFs with perturbative coefficients; equality to a collinear PDF cannot be asserted by simply setting $b_T=0$ inside an arbitrary rapidity scheme.

A classification that prevents false tomography

Correlator	Momentum transfer	Separation retained	Gauge link and scales	Valid structural reading
Collinear PDF	$\Delta=0$	light-ray $z^-$; $\mathbf z_T=0$	straight collinear link; $\mu$	distribution in longitudinal fraction within collinear factorization
GPD	$\Delta\ne0$	light-ray $z^-$; $\mathbf z_T=0$	straight link; $\mu$	correlation of longitudinal fraction with momentum transfer; impact-parameter density only at $\xi=0$ under light-front hypotheses
TMD	$\Delta=0$	$z^-$ and $\mathbf z_T$	staple link, soft subtraction; $\mu,\zeta$	transverse-momentum dependence for a factorization class, with controlled link dependence

For each extracted quantity, the method chain should remain visible:

Input observable	Factorized object	Essential theory inputs	Validation and uncertainty sources
Inclusive hard cross section	collinear PDF	coefficient functions, DGLAP evolution, heavy-flavor scheme	scale variation, sum rules, correlated data systematics, parameterization bias
Exclusive hard amplitude	GPD convolution	hard kernel, evolution, real/imaginary dispersion structure	polynomiality, form-factor moments, kinematic coverage, deconvolution assumptions
Small-transverse-momentum spectrum or spin asymmetry	TMD convolution	hard and soft factors, rapidity evolution, link class	matching region, power corrections, nonperturbative large-$b_T$ model, sign tests
Euclidean matrix elements or moments	renormalized nonlocal/local operator	matching, mixing, finite-momentum formalism	lattice spacing, volume, excited states, large-momentum and truncation limits

No row turns an extracted function into an observable by itself. The measurable cross section or amplitude, factorization theorem, operator definition, fitting assumptions, covariance, and source must be reported separately.

Independent checks and limitations

• Gauge invariance: include the correct Wilson line and, for TMDs, its staple direction and soft subtraction. Changing the path changes the operator.
• Support and charges: collinear distributions should respect the declared $x$ support and conserved flavor-number sum rules. Conventions for negative $x$ must be stated before taking moments.
• Momentum and spin: test the momentum sum rule at one common scale. For spin decompositions, distinguish helicity, transversity, total angular momentum, and orbital angular momentum; they are matrix elements of different operators.
• Polynomiality: Mellin moments of GPDs must have the allowed polynomial dependence on $\xi$. A flexible fit that violates this is not a GPD parameterization consistent with Lorentz covariance.
• Matching limits: $H(x,0,0)=q(x)$ and the first GPD moments reproduce form factors in consistent conventions. TMD-to-PDF matching requires the regulated small-$b_T$ limit, not an unqualified integral.
• Scales and schemes: PDFs and GPDs require $\mu$; TMDs require both $\mu$ and a rapidity scale. Quark/gluon spin and momentum partitions are not scale-free percentages.
• Domain: factorization can fail through insufficient hard-scale separation, Glauber exchange, endpoint regions, or power corrections. The applicable theorem is process and kinematics dependent.

Common pitfalls

Combining $x$, $\mathbf b_T$, and $\mathbf k_T$ into a classical picture. Quantum correlators do not generally admit simultaneous probability distributions for conjugate variables. State the projection and its positivity conditions.

Using “tomography” as a three-dimensional density claim. The clean impact-parameter interpretation uses $\xi=0$ and a fast, transversely localized state. It does not reconstruct a frame-independent rest-frame image.

Suppressing the Wilson line. Link geometry carries factorization information, especially for TMDs. Omitting it can hide a sign change or even change which theorem applies.

Informal self-check

A claim presents a function of $(x,\mathbf k_T)$ but gives only a renormalization scale $\mu$. What information is missing before it can be identified as a TMD?

Answer

One needs the bilocal operator and spin projection, staple direction or process class, rapidity regulator and soft-subtraction convention, rapidity scale $\zeta$, and the factorization theorem and kinematic region that connect the function to an observable. The result also needs a statement of evolution and matching to the collinear regime. Dependence on $(x,\mathbf k_T)$ alone is not an operator definition.

Handoffs

• Send the collinear operator, scheme, scale, and hard process to the operator-PDF factorization route.
• Send the staple geometry, $(\mu,\zeta)$, soft scheme, and measured transverse-momentum region to the TMD factorization route.
• Send $x$-integrated local-current moments and stable external-state conventions back to the form-factor route.
• Treat mutable global fits, phenomenological averages, and comparative status statements as source-dated research evidence rather than evergreen definitions.

References

• Bacchetta, Alessandro, Markus Diehl, Klaus Goeke, Andreas Metz, Piet J. Mulders, and Marc Schlegel. “Semi-Inclusive Deep Inelastic Scattering at Small Transverse Momentum.” Journal of High Energy Physics 02 (2007): 093. DOI · Open PDF
• Collins, John C. Foundations of Perturbative QCD. Cambridge: Cambridge University Press, 2011, ch. 13. DOI
• Diehl, Markus. “Generalized Parton Distributions.” Physics Reports 388 (2003): 41–277. DOI · Open PDF
• Ji, Xiangdong. “Gauge-Invariant Decomposition of Nucleon Spin.” Physical Review Letters 78 (1997): 610–613. DOI · Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, ch. 32. DOI