⚛️ phenomenology

T-odd Effect from Two-Photon Exchange in SIDIS at Low Transverse Momentum

This paper investigates T-odd effects arising from two-photon exchange in semi-inclusive deeply inelastic scattering at low transverse momentum, demonstrating that while the absorptive part of the scattering amplitude is divergent, the resulting physical differential cross-section remains finite and exhibits distinct angular dependencies that offer new avenues for detecting effects beyond the one-photon approximation and searching for new physics.

Original authors: J. P. Ma, G. P. Zhang

Published 2026-09-18

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: J. P. Ma, G. P. Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep inside the matter that makes up our world, protons are not solid, indivisible spheres but rather bustling cities of smaller particles called quarks and gluons. To map the three-dimensional layout of these cities, physicists fire high-speed electrons at protons and watch how the electrons scatter. When an electron hits a proton, it often knocks out a quark, which then transforms into a spray of new particles, including a single identifiable hadron. By studying the angles and speeds of these outgoing particles, scientists can reconstruct the internal structure of the proton. For decades, the standard way to interpret these collisions has relied on a simplified picture: the electron and the quark exchange a single particle of light, known as a photon, to interact. This "one-photon" approximation has been incredibly successful, allowing researchers to extract detailed maps of how quarks move and spin inside the proton. However, nature is rarely as simple as our first models suggest, and there are subtle effects that this single-particle exchange misses.

A team of researchers has now turned their attention to one of those missing pieces: the possibility that the electron and the quark might exchange two photons at once. While this sounds like a minor correction, it introduces a specific type of behavior that is invisible in the simpler model. In physics, certain interactions are "time-reversal odd," meaning they behave differently if you were to run the clock backward. In the standard one-photon picture, these effects are absent. But when two photons are exchanged, the interaction acquires a complex, "absorptive" quality that breaks this symmetry. The researchers set out to calculate exactly how this two-photon exchange changes the outcome of the collision, specifically when the resulting hadron flies off with a low sideways speed. They wanted to know if this subtle effect could be seen in the data and how it would alter the patterns scientists observe.

The team performed a rigorous calculation of the scattering process, focusing on the mathematical "imaginary" part of the interaction, which corresponds to this absorptive quality. They found that while the raw calculation contained a mathematical infinity—a common issue when dealing with massless particles like photons—the final physical result was perfectly finite and well-behaved. This is a crucial finding because it means the effect is real and calculable, not just a theoretical artifact. When they applied this result to the full collision process, they discovered that the two-photon exchange introduces new contributions to the scattering pattern that depend on the spin of the incoming electron. These new contributions change the way the final particles are distributed in space, creating a distinct angular signature that differs from the standard one-photon predictions.

The researchers showed that these new effects would manifest as specific shifts in the angles at which the outgoing particles appear. If the incoming electron is polarized, meaning its spin is aligned in a particular direction, the two-photon exchange causes the final particles to cluster in ways that the standard model does not predict. The team derived a complete formula for these angular distributions, identifying six different ways to measure the effect through asymmetries. These asymmetries are essentially comparisons of how often particles appear in one direction versus another, and they would be zero if the two-photon effect did not exist. The presence of these asymmetries would serve as a clear signal that the interaction is more complex than the simple single-photon exchange.

This work is significant for two main reasons. First, it provides a precise prediction for a known physical effect that goes beyond the standard approximation, allowing future experiments to test the limits of our current understanding of quantum electrodynamics in these collisions. Second, and perhaps more importantly, it clears the path for searching for entirely new physics. Because the two-photon effect creates a specific "background" of time-reversal odd behavior, scientists must understand it perfectly before they can claim to have discovered something new, such as a violation of fundamental symmetries that would point to physics beyond the standard model. By mapping out exactly what the standard theory predicts in this subtle regime, the researchers have given experimentalists the tools they need to distinguish between a known correction and a truly revolutionary discovery.

Technical Summary: T-odd Effects from Two-Photon Exchange in SIDIS at Low Transverse Momentum

Problem Statement
The paper addresses the theoretical description of Semi-Inclusive Deeply Inelastic Scattering (SIDIS) in the kinematic region where the produced hadron possesses low transverse momentum ( $P_{h\perp} \ll Q$ ). Standard theoretical predictions for SIDIS rely on the approximation of one-photon exchange between the lepton and the quark (tree-level QED). While QED radiative corrections (vertex, propagator, and real photon radiation) have been studied to improve precision, the contribution from two-photon exchange has often been excluded. The authors investigate specific "T-odd" effects arising from this two-photon exchange mechanism. These effects are absent in the one-photon approximation but emerge at the next-to-leading order due to the absorptive (imaginary) part of the scattering amplitude. Understanding these effects is crucial for two reasons: first, to identify new physical effects beyond the standard one-photon approximation in SIDIS; and second, to establish a baseline for distinguishing Standard Model T-odd effects from potential new CP-violating interactions in future lepton-proton collider searches (such as the EicC or EIC).

Methodology
The study proceeds through a systematic calculation and factorization approach:

Lepton-Quark Scattering Amplitude: The authors first calculate the scattering amplitude for the process $\ell(k_1) + q(p_1) \to \ell(k_2) + q(p_2)$ involving the exchange of two photons. They compute the absorptive part of this amplitude, which arises from the cut diagram where the intermediate states go on-shell. The calculation is performed in $d = 4 - \epsilon$ dimensions to handle potential divergences.
Analysis of Divergences: The calculation reveals that the absorptive part contains an Infrared (I.R.) divergence associated with the term $A_1$ in the amplitude decomposition. However, the authors demonstrate that this divergent term is proportional to the tree-level amplitude. Since physical T-odd effects arise from the interference between the absorptive part and the tree-level amplitude, the divergent contribution cancels out, rendering the physical observables finite.
TMD Factorization: The authors apply Transverse-Momentum-Dependent (TMD) factorization to the SIDIS process. They express the differential cross-section as a convolution of perturbative coefficient functions (derived from the lepton-quark scattering) with TMD quark distributions and TMD quark fragmentation functions.
- The calculation is performed at the leading power in the expansion parameter $\lambda \sim P_{h\perp}/Q$ .
- Gauge invariance is addressed by noting that collinear gluon contributions are absorbed into the gauge links of the TMD definitions, while soft gluon contributions form a soft factor.
Angular Distribution Derivation: Using the factorized cross-section, the authors derive the angular dependence of the T-odd effects. They consider a frame where the virtual photon moves in the $-z$ direction and the initial hadron in the $+z$ direction. They analyze the azimuthal angles of the produced hadron ( $\phi_h$ ) and the transverse spin of the initial hadron ( $\phi_s$ ).

Key Contributions and Results

Finite Physical Observables: The paper explicitly demonstrates that although the absorptive part of the two-photon exchange amplitude is I.R. divergent, the resulting physical differential cross-section is finite. The divergence is confined to the $A_1$ term, which does not contribute to the T-odd interference terms.
New Angular Dependences: The inclusion of T-odd effects from two-photon exchange introduces new angular dependencies in the differential cross-section that differ from those obtained in the one-photon exchange approximation. Specifically, the cross-section receives contributions proportional to the lepton helicity ( $\lambda_\ell$ ) and involves chirality-odd TMD distributions (such as $h_{1T}^\perp$ , $h_{1L}^\perp$ , $H_{1T}^\perp$ , $H_{1L}^\perp$ ) and their convolutions. The results show that when the initial lepton is unpolarized, there is no contribution to the differential cross-section; when the lepton is polarized, the effects depend on the transverse spin of the initial hadron ( $|\vec{S}_\perp|$ ).
Complete Angular Distribution: Unlike previous studies that provided results for only a single asymmetry, this work derives the complete angular distribution for the T-odd effect in SIDIS with an unpolarized final hadron.
Six New Asymmetries: The authors define six specific asymmetries ( $A_L[\cos 2\phi_h]$ , $A_L[\sin 2\phi_h]$ , $A_L[\cos 3\phi_h]$ , $A_L[\sin 3\phi_h]$ , $A_L[\cos \phi_h]$ , $A_L[\sin \phi_h]$ ) that are sensitive to these T-odd effects. These asymmetries vanish if the absorptive part from two-photon exchange is neglected. The results show that these asymmetries are proportional to the lepton helicity and the transverse spin magnitude of the initial hadron ( $|\vec{S}_\perp|$ ).
Consistency Check: The authors note that their result for one specific asymmetry (corresponding to the $\sin 2\phi_h$ term) agrees with a previous result found in the literature [17].

Significance
The paper claims that these results are significant for two primary motivations:

Beyond One-Photon Approximation: The study provides a theoretical framework for detecting physical effects in SIDIS that go beyond the standard one-photon exchange approximation. It highlights that T-odd effects, driven by the absorptive part of two-photon exchange, are a necessary component of the theoretical description in the low transverse momentum region.
Search for New Physics: The work is motivated by the need to search for new physics, specifically new CP-violating interactions, at future lepton-proton colliders. Since the initial state in lepton-proton scattering is not a CP-eigenstate, effects from new CP-violating interactions can mix with Standard Model T-odd effects. Therefore, a precise understanding of the Standard Model T-odd effects (as calculated here) is a prerequisite for isolating and identifying true new physics signals.

The authors maintain a modest tone, presenting their work as a derivation of the complete angular distribution and a set of asymmetries to detect these specific QED-induced effects, rather than claiming the immediate discovery of new physics.

Technical Summary: T-odd Effects from Two-Photon Exchange in SIDIS at Low Transverse Momentum

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