Off-Shell Sudakov Form Factors in QFT

• Off-shell Sudakov form factors quantify double-logarithmic corrections by using external state virtualities as infrared regulators, resulting in a doubling of the logarithmic contributions compared to on-shell cases.
• In planar N=4 SYM, the exponentiation is governed by the octagon anomalous dimension instead of the conventional cusp anomalous dimension, highlighting a shift in analytic structure.
• Beyond one loop, the violation of standard soft–collinear factorization necessitates additional cross-talk factors, impacting resummation techniques and high-energy QCD phenomenology.

The off-shell Sudakov form factor quantifies the impact of infrared (IR) and collinear logarithmic enhancements in quantum field theory when external states carry nonzero virtuality. Unlike the on-shell case—where double-logarithmic terms are regulated by infinitesimal particle masses or photon mass regularization—off-shell form factors use the external state virtualities as IR cutoffs, fundamentally altering the analytic structure and resummation properties. These structures appear in perturbative QED, QCD, maximally supersymmetric Yang–Mills (𝒩=4 SYM), and play foundational roles in factorization, operator product expansion, and the paper of IR singularities. Recent developments have revealed that in planar 𝒩=4 SYM, the exponentiation of the off-shell Sudakov form factor is governed by the octagon (Γ_oct) anomalous dimension, supplanting the familiar cusp anomalous dimension, and that factorization between soft and collinear sectors is violated beyond one-loop.

1. Definition and Analytic Structure

In the standard Sudakov regime, consider a vertex function with two external fermion legs of momenta $p_1$, $p_2$ (with virtualities $p_1^2, p_2^2$) and an off-shell photon of momentum $q = p_1 + p_2$. The off-shell Sudakov form factor $\Gamma^{\mathrm{off}}(p_1^2, p_2^2, q^2)$ captures the leading double-logarithmic corrections as: $$\Gamma^{\mathrm{off}}(p_1^2, p_2^2, q^2) = \exp\left[-\,\frac{\alpha}{2\pi} \ln\frac{q^2}{p_1^2} \ln\frac{q^2}{p_2^2} \right]$$ In contrast, the on-shell limit ($p_i^2 = m^2$, photon regulated by mass $\mu$) yields: $$\Gamma^{\mathrm{on}}(m^2, m^2, q^2) = \exp\left[-\,\frac{\alpha}{4\pi} \ln^2\frac{q^2}{\mu^2} \right]$$ Thus, the coefficient of the double-logarithmic term is twice as large in the off-shell case compared to the on-shell case, reflecting a key qualitative difference in the phase-space structure of IR singularities (Forte, 2020).

In gauge theories such as 𝒩=4 SYM, the natural generalization involves form factors of composite operators evaluated between external off-shell states, often with all legs carrying mass $m$ (Coulomb-branch regularization) (Belitsky et al., 25 Nov 2024, Belitsky et al., 2023). The kinematic variable is $t = m^2 / Q^2$ ($Q^2$ being the momentum exchange scale).

2. Exponentiation, Double Logarithms, and the Factor of Two

The double-logarithmic structure in the off-shell regime arises from integrals over the mutually independent soft and collinear regions of the phase space. In this scenario, the phase-space limits for the energy $E \to 0$ ("soft") and for small angles $\theta \to 0$ or $\pi$ ("collinear") are independently regulated by $p_i^2$, allowing the two logarithmic integrations to factorize. Performing the real-emission calculation using the eikonal (soft) approximation, the one-loop off-shell form factor emerges as: $$\Gamma^{(1),\mathrm{off}}(p_1^2, p_2^2, q^2) = -\frac{\alpha}{2\pi} \ln\frac{q^2}{p_1^2} \ln\frac{q^2}{p_2^2}$$ whereas

$$\Gamma^{(1),\mathrm{on}}(m^2, m^2, q^2) = -\frac{\alpha}{4\pi} \ln^2\frac{q^2}{\mu^2}$$

This difference of a factor of two in the coefficients reflects the independent regulation by $p_i^2$ in the off-shell case, compared to overlapping IR and collinear regulating scales in the on-shell scenario (Forte, 2020).

To all orders, these double logarithms exponentiate due to the factorization of soft/collinear emissions and the universal structure of the multi-particle phase-space in the double-logarithmic region: $$\Gamma^{\mathrm{off}}(p_1^2, p_2^2, q^2) =\exp\left[-\,\frac{\alpha}{2\pi} \ln\frac{q^2}{p_1^2} \ln\frac{q^2}{p_2^2} \right]$$ Similar exponentiation holds in non-Abelian gauge theories, with the replacements for color factors and relevant anomalous dimension structures.

3. Octagon vs. Cusp Anomalous Dimensions in 𝑁=4 SYM

In planar 𝒩=4 SYM, a particularly striking property emerges: the coefficient of the $\ln^2 t$ term in the exponent is governed by the octagon anomalous dimension, $\Gamma_\mathrm{oct}$, not the usual cusp anomalous dimension $\Gamma_\mathrm{cusp}$ (Belitsky et al., 2022, Belitsky et al., 2023, Belitsky et al., 25 Nov 2024): $$\log F_\text{off-shell}(t, g) = - \frac{1}{2} \Gamma_\mathrm{oct}(g) \ln^2 t - D(g) + \mathcal{O}(m^2)$$ This is in contrast to the on-shell regime, where the double log is universally set by $\Gamma_\mathrm{cusp}$. For the off-shell Sudakov form factor,

$$\Gamma_\mathrm{oct}(g) = \frac{2}{\pi^2} \ln[\cosh(2\pi g)]$$

where $g^2 = g^2_{\rm YM} N_c / (4\pi)^2$, and $D(g)$ encodes the finite (hard) contributions.

At the multi-loop level, this distinction persists:

• The IR divergences exponentiate, with the $\ln^2 t$ coefficient given by $\Gamma_\mathrm{oct}(g)$ through three loops, conjectured to all orders.
• The anomaly arises because taking the on-shell limit before vs. after the dimensional continuation does not commute, yielding a different IR structure (Belitsky et al., 2022, Belitsky et al., 28 May 2025).

4. Minimal Off-Shell Form Factors and Remainder Functions

For higher-point and higher-loop minimal form factors of half-BPS operators in 𝒩=4 SYM, the exponentiation pattern persists: $$F_n\big(p_i^2\big) = \exp\left[ \mathcal G^{(1)}(p_i^2) + \mathcal G^{(2)}(p_i^2) + \cdots \right]$$ with

$$\mathcal G^{(L)}(p_i^2) = -\frac{\Gamma_{\rm oct}^{(L)}}{4}\sum_{i=1}^n L_i^2 + \frac{\Gamma_{\rm oct}^{(L)}}{2}\sum_{i=1}^n L_i \ell_i - \frac{\Gamma_{\rm oct}^{(L)}}{4}\sum_{i=1}^n \ell_i^2 - \frac{n}{2} D^{(L)}$$

where $L_i = \ln(-p_i^2/\mu^2)$, $\ell_i = \ln(s_{i, i+1}/\mu^2)$, $s_{i, i+1} = (p_i + p_{i+1})^2$, and $D^{(L)}$ are loop-level constants (Belitsky et al., 25 Nov 2024). Subtracting these exponentiated logs leaves a finite remainder function—the symbol of which for $n=3$ coincides with the corresponding on-shell (conformal) remainder, with different higher-weight terms. This structural correspondence highlights the universality of the symbol alphabet and the transcendentality properties within planar 𝒩=4 SYM.

5. Factorization Properties and their Violation

Factorization of the Sudakov form factor into hard, jet (collinear), and soft components underpins the modern approach to resummation and power counting in perturbative QFT. While the on-shell Sudakov form factor admits precise factorization into these sectors using Wilson lines and eikonal approximations, the off-shell scenario in planar 𝒩=4 SYM presents an explicit breakdown of this structure beyond one loop (Belitsky et al., 28 May 2025).

Explicit region analysis using the method of regions decomposes loop integrals into "hard," "collinear," and "ultrasoft" scaling:

• While the hard region cleanly separates, starting at two loops collinear and ultrasoft regions are intertwined via subleading denominators.
• For example, the two-loop ladder region $T_{21}^{\rm c-us}$ does not factorize as $$T_{21}^{\rm c-us}\neq T_{11}^{\rm c} T_{11}^{\rm us}$$, signaling non-multiplicative structure.
• At three loops, twist factors $z_{21}(\varepsilon)$, $z_{22}(\varepsilon)$ in mixed regions ($c\,c\,us$, $c\,us\,us$) further obstruct any attempt to restore multiplicative factorization (Belitsky et al., 28 May 2025).

As a result, the usual decomposition

$$\mathcal F_2 = h(\varepsilon)\; \times\; J(\varepsilon,\sqrt t)\; \times\; S(\varepsilon,t)$$

must be augmented by a nontrivial cross-talk factor $\mathcal R(\varepsilon, t)$ encapsulating the violation of standard soft–collinear separation: $$\mathcal F_2 = h(\varepsilon)\; \times\; J(\varepsilon,\sqrt t)\; \times\; S(\varepsilon,t) \; \times\; \mathcal R(\varepsilon,t)$$ The cross-talk factor $\mathcal R(\varepsilon,t)$ is non-vanishing at two and three loops, irreparably breaking conventional factorization.

6. Skewed Sudakov Regime and Asymmetric Off-Shellness

The "skewed Sudakov regime" refers to the configuration where one external leg is nearly on-shell and the other carries finite virtuality (Kim et al., 2018). In massless QED, this regime is governed by a unique linear evolution equation for the form factor $F(t, z)$, whose solution can be represented spectrally using harmonic numbers and eigenfunctions: $$F(t,z) = e^{\frac{3}{2}t} \sum_{n=0}^\infty e^{- H_{n+1} t} (1 - z)^n$$ where $t = (\alpha/4\pi) \ln (Q^2/\mu^2)$, $z = 1/(1 + y)$, $y = q^2/(2p \cdot p')$. The perturbative expansion organizes into multiple polylogarithms, and in the $z \rightarrow 1$ limit, the asymptotics match the standard exponentiation of doubly logarithmic corrections. In the limit of decreasing virtuality on one leg, the form factor grows with the opening of phase space for soft emission, distinct from the purely off-shell Sudakov suppression (Kim et al., 2018).

7. Off-Shell Sudakov Factors in High-Energy QCD Phenomenology

Sudakov form factors are essential theoretical ingredients in small-$x$ factorization, transverse-momentum-dependent (TMD) resummation, and saturation models. The inclusion of an off-shell Sudakov factor in dipole models modifies the “bare” dipole cross section to incorporate hard-scale dependence, as relevant for HERA data fits (Goda et al., 2022). The Sudakov factor $S(r,Q^2)$ in $r$-space is: $$S_{\mathrm{pert}}^{(1)}(r, Q^2) = \frac{C_A}{2\pi} \int_{\mu_b^2}^{Q^2} \frac{d\mu^2}{\mu^2} \alpha_s(\mu^2) \ln \left(\frac{Q^2}{\mu^2}\right)$$ where $\mu_b = C_S / r$, $C_S = 2 e^{-\gamma_E}$. The Sudakov-suppressed dipole cross section is then convoluted with the photon wave function for observables such as $F_2$ in DIS. The perturbative Sudakov effect substantially improves the agreement of GBW/BGK models with HERA $F_2$ at large $Q^2$ (Goda et al., 2022).

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• (Forte, 2020) On the Sudakov form factor, and a factor of two
• (Belitsky et al., 2022) Exact off-shell Sudakov form factor in N=4 SYM
• (Belitsky et al., 2023) Off-shell form factor in N=4 sYM at three loops
• (Belitsky et al., 25 Nov 2024) Off-shell minimal form factors
• (Belitsky et al., 28 May 2025) Off-shell form factor: factorization is violated
• (Kim et al., 2018) Skewed Sudakov Regime, Harmonic Numbers, and Multiple Polylogarithms
• (Goda et al., 2022) Sudakov effects and the dipole amplitude

These works collectively establish the defining properties, analytic structure, anomalous dimensions, factorization properties, and phenomenological relevance of off-shell Sudakov form factors across abelian and non-abelian gauge theories.