Hadron Form Factors and Local-Current Structure

Hadron form factors are the scalar functions in an on-shell matrix element of a specified local current. Lorentz symmetry fixes the allowed tensor structures; current conservation and discrete symmetries reduce them; normalization at zero momentum transfer ties them to charges and moments. They describe current structure of stable external states, not parton probability densities and not, without analytic continuation, unstable-resonance structure.

Required background. Hadron quantum numbers and the QCD spectrum supplies the external-state labels; form factors and local operator insertions supplies LSZ reduction and matrix-element normalization. Helpful background. Renormalized composite-operator insertions supplies current renormalization and mixing.

Current matrix elements and invariant amplitudes

Adopt relativistically normalized one-hadron states,

$$\langle h(p',s')|h(p,s)\rangle =2E_{\mathbf p}(2\pi)^3\delta_{s's}\delta^{(3)}(\mathbf p'-\mathbf p),$$

and define $q=p'-p$, $t=q^2$, and $Q^2=-t\ge0$ in the spacelike region. A form factor is dimensionless when it multiplies a tensor structure with the same mass dimension as the current matrix element; alternative conventions may extract powers of the hadron mass.

For a conserved vector current between identical spin-zero states, Lorentz covariance and current conservation give

$$\langle h(p')|J^\mu(0)|h(p)\rangle=(p'+p)^\mu F(t).$$

Indeed, the other possible vector $q^\mu$ is removed by $q_\mu J^\mu=0$, while $q\cdot(p'+p)=p'^2-p^2=0$. For a parity-even vector current between identical spin-$\tfrac12$ states,

$$\langle h(p',s')|J^\mu(0)|h(p,s)\rangle =\bar u(p',s') \left[ \gamma^\mu F_1(t) +\frac{i\sigma^{\mu\nu}q_\nu}{2M}F_2(t) \right]u(p,s),$$

where $\sigma^{\mu\nu}=\tfrac{i}{2}[\gamma^\mu,\gamma^\nu]$. The Dirac equations and Gordon identity reduce any equivalent decomposition built from $\gamma^\mu$, $(p'+p)^\mu$, and $q^\mu$ to these two structures. Contracting with $q_\mu$ gives zero because

$$\bar u(p')q\!\!\!/u(p)=\bar u(p')(p'\!\!\!/-p\!\!\!/)u(p)=0, \qquad q_\mu\sigma^{\mu\nu}q_\nu=0.$$

The first equality uses equal on-shell masses; a transition between different masses needs its own complete decomposition. The electromagnetic spin-$\tfrac12$ parameterization and its normalization are developed explicitly in Schwartz 2014, § 32.1.2, pp. 669–672.

Charge, magnetization, and radii

It is useful in spacelike kinematics to introduce

$$\tau=\frac{Q^2}{4M^2}, \qquad G_E(Q^2)=F_1(-Q^2)-\tau F_2(-Q^2), \qquad G_M(Q^2)=F_1(-Q^2)+F_2(-Q^2).$$

In the Breit frame, where $q^0=0$ and $\mathbf p'=-\mathbf p=\mathbf q/2$, these combinations multiply the non-spin-flip charge and spin-flip magnetic structures:

$$\langle J^0\rangle=2M\,\chi_{s'}^\dagger\chi_s\,G_E(Q^2), \qquad \langle\mathbf J\rangle =\chi_{s'}^\dagger i\boldsymbol\sigma\times\mathbf q\,\chi_s\,G_M(Q^2).$$

For a conserved electromagnetic current expressed in units of the elementary charge,

$$F_1(0)=Q_h, \qquad F_2(0)=\kappa_h, \qquad \mu_h=\frac{e}{2M}G_M(0)=\frac{e}{2M}(Q_h+\kappa_h).$$

For a charged state, the electric mean-square radius convention is

$$\langle r_E^2\rangle =-\frac{6}{G_E(0)} \left.\frac{dG_E(Q^2)}{dQ^2}\right|_{Q^2=0}.$$

For a neutral state $G_E(0)=0$, so one instead defines $\langle r_E^2\rangle=-6G_E'(0)$ in the stated charge units. These slopes are invariant form-factor parameters. Calling them literal three-dimensional rest-frame density moments requires additional nonrelativistic or frame-dependent assumptions.

Analytic structure and continuation

As a function of timelike $t$, a form factor is analytic apart from poles and cuts allowed by states carrying the current’s quantum numbers. When one subtraction suffices, a representative dispersion relation is

$$F(t)=F(0)+\frac{t}{\pi} \int_{t_0}^{\infty}dt', \frac{\operatorname{Im}F(t')}{t'(t'-t-i0)}.$$

The threshold $t_0$, number of subtractions, and asymptotic assumptions are part of the statement. Timelike resonant structure enters through the spectral function; fitting a pole-shaped term on the real axis does not remove the need to specify cuts, channels, and continuation. For an unstable external hadron, a process-independent resonance form factor is associated with a pole residue of a higher-point amplitude, not an ordinary matrix element between asymptotic one-resonance states. That construction begins with coupled-channel resonance poles.

Local currents are not parton correlators

Object	Operator and kinematics	Variables	Typical exact relation	Interpretation limit
Elastic form factor	Local current, same incoming and outgoing hadron	$t$	$F_1(0)$ is the conserved charge	Not a frame-independent three-dimensional density
Transition form factor	Local current, different external states	invariant momentum transfers and helicities	Ward identities relate allowed structures	Unstable states require pole-residue definitions
PDF	Forward light-ray operator with a Wilson line	$x$, renormalization scale	local twist-two moments where defined	Not localized in transverse position
GPD	Off-forward light-ray operator	$x$, skewness $\xi$, $t$, scale	first moments give local-current form factors	Density interpretation is restricted, especially at $\xi\ne0$
TMD	Transverse-separated light-ray operator, staple link, and soft subtraction	$x$, $\mathbf k_T$, renormalization and rapidity scales	weighted or matched limits under stated schemes	Wilson-line geometry and process class matter

The first-moment link between GPDs and form factors follows because integrating over the parton fraction collapses the light-ray separation to a local current Diehl 2003, §§ 3.2–3.3. It does not make the full $x$-dependent GPD a form factor. The operator definitions and interpretation boundaries continue on Partonic Structure, Spin, and Hadron Tomography.

From measurement or correlation function to a result

Stage	Required input	Output	Dominant checks and limitations
Lorentz analysis	state spins, parities, masses, current transformation law	complete independent form-factor basis	count helicity amplitudes; impose Ward identities without overconstraining transitions
Current definition	flavor structure, renormalization scheme and scale, improvement terms	renormalized operator	conserved-current normalization; mixing with operators allowed by symmetries
Experimental extraction	cross sections or polarization observables, radiative corrections, reaction model	form factors over measured kinematics	acceptance, two-boson exchange, correlated normalization, model dependence
Euclidean extraction	two- and three-point correlators, covariance, current matching	finite-volume matrix elements	excited states, disconnected contractions, finite volume, lattice spacing, quark masses
Parametrization or continuation	kinematic data and analytic assumptions	radii, moments, timelike continuation	truncation, threshold placement, covariance propagation, asymptotic constraints

The detailed Euclidean method is handled by three-point functions, matrix elements, and disconnected contributions. A usable handoff includes state and spin conventions, current normalization, complete covariance, fit windows, excited-state model, and continuum/volume information.

Independent checks and limitations

• Ward identity: verify $q_\mu\langle J^\mu\rangle=0$ for a conserved current. For unequal-mass transitions, do not discard longitudinal structures until the correct Ward identity is applied.
• Zero-transfer normalization: compare $F_1(0)$ or the spin-zero $F(0)$ with the generator charge. A lattice local current may require a finite normalization even when the continuum current is conserved.
• Dimensions and basis changes: all $F_i$, $G_E$, and $G_M$ above are dimensionless. Transforming to helicity or multipole form factors must preserve the number of independent amplitudes and introduce explicit mass factors.
• Kinematic singularities: choose a basis whose apparent poles do not create unphysical singularities at $Q^2=0$ or pseudothresholds.
• Analyticity: parameterizations should place the first branch point at the correct crossed-channel threshold and respect any stated unitarity bounds.
• Domain: finite-temperature, nuclear-medium, and genuinely inclusive responses need different matrix elements. Mutable numerical averages are not supplied here.

Common pitfalls

Using the slope sign in $t$ and $Q^2$ interchangeably. Since $Q^2=-t$, $d/dQ^2=-d/dt$. State the variable before defining a radius.

Calling Sachs form factors frame-independent densities. $G_E$ and $G_M$ are invariant combinations, but their simple density-like reading comes from Breit-frame and nonrelativistic reasoning.

Forgetting current renormalization. A form-factor decomposition is purely kinematic; the inserted composite operator still needs a scheme, scale when applicable, and matching to the desired continuum current.

Informal self-check

Starting from the spin-$\tfrac12$ decomposition, verify current conservation and determine the charge and magnetic moment at $Q^2=0$.

Answer

Contracting with $q_\mu$ kills the Pauli term because a symmetric product $q_\mu q_\nu$ contracts an antisymmetric tensor. The Dirac term becomes $\bar u(p')(p'\!\!\!/-p\!\!\!/)u(p)=M\bar uu-M\bar uu=0$. Charge normalization gives $F_1(0)=Q_h$. Since $G_M(0)=F_1(0)+F_2(0)$, the magnetic moment is $e[Q_h+\kappa_h]/(2M)$ when $F_2(0)=\kappa_h$.

Handoffs

• Send a local-current matrix element with stable external states, normalization, and covariance to a phenomenological or Euclidean form-factor analysis.
• Send an $x$-dependent nonlocal correlator, its Wilson-line path, and both renormalization scales to the PDF/GPD/TMD route.
• Send unstable-channel quantum numbers and the production/scattering amplitude to the coupled-channel pole route before defining a resonance form factor.

References

• Diehl, Markus. “Generalized Parton Distributions.” Physics Reports 388 (2003): 41–277. DOI · Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, § 32.1.2. DOI