⚛️ phenomenology

Higgs pseudo-observables in $e^+e^-\to h\gamma$

This paper analyzes the process $e^+e^-\to h\gamma$ using Higgs pseudo-observables to demonstrate that deviations from the Standard Model due to heavy new physics are encoded in on-shell $h\gamma\gamma$ and $hZ\gamma$ couplings, which can be fully characterized by measuring the process at different FCC-ee energies to resolve sign ambiguities left by decay constraints.

Original authors: Fausto Boniotti, Gino Isidori, Christiane Mayer, Emanuelle Pinsard

Published 2026-09-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fausto Boniotti, Gino Isidori, Christiane Mayer, Emanuelle Pinsard

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

The Higgs boson is the final piece of the puzzle that explains why particles have mass, a discovery that completed the Standard Model of particle physics. While scientists have confirmed its existence, the deeper work of understanding exactly how it behaves is just beginning. The Higgs does not exist in isolation; it constantly interacts with other particles, and the strength of these interactions holds the key to whether our current understanding of the universe is complete or if there are hidden forces waiting to be found. One of the most promising ways to probe these interactions is by watching the Higgs boson interact with light. In the known laws of physics, a Higgs boson and a photon (a particle of light) should not be able to appear together from a collision of electrons and positrons unless they do so through a complex, indirect process involving heavy particles that exist only for a fleeting moment. This makes the event rare, but it also makes it a highly sensitive detector for new physics, as any unseen heavy particles could subtly alter the outcome.

A team of researchers has now mapped out this rare process with unprecedented clarity, offering a new way to interpret future experiments at the Future Circular Collider for electron-positron collisions. They focused on the specific reaction where an electron and a positron collide to produce a Higgs boson and a photon. Because this process happens only through quantum loops—where virtual particles briefly pop in and out of existence—it is theoretically clean and highly sensitive to new, heavy particles that might be too massive to create directly. The researchers broke down the mathematical description of this event into three distinct, independent parts: one part driven by the exchange of a virtual photon, another by a virtual Z boson (a heavy cousin of the photon), and a third part that involves no such exchange at all. This separation is crucial because it allows scientists to isolate the specific "fingerprints" of the Higgs boson's interactions with light and the Z boson, separating them from the messy background of other quantum effects.

The core finding of this work is that if new physics exists in the form of very heavy particles, its primary effect on this process will be to change the strength of the Higgs boson's connection to light and to the Z boson. These connections are known as effective couplings, and they act like knobs that determine how strongly the Higgs talks to these particles. The researchers showed that for heavy new physics, these are the only knobs that need to be turned to explain any deviations from the Standard Model. They demonstrated that more complicated, direct interactions between the colliding particles and the final products are so heavily suppressed that they would only appear if the new physics were much lighter or if the energy scales were vastly different. This means that by measuring the rate of these events, scientists can effectively measure the values of these two knobs with high precision.

However, there is a catch that this paper elegantly resolves. Measurements of the Higgs boson decaying into light or a Z boson and a photon can tell us the size of these knobs, but they cannot tell us their direction. In the language of physics, these interactions can be positive or negative, and the decay rates alone cannot distinguish between a positive value and a negative one of the same magnitude. This creates an ambiguity, like knowing the volume of a sound but not whether the wave is pushing or pulling. The researchers found that the process of creating a Higgs and a photon together is sensitive to the interference between the photon and Z boson contributions. Because the way these two contributions mix changes depending on the energy of the collision, measuring the event rate at different energy levels allows scientists to determine the relative sign of these interactions. By running the collider at two different energies, such as 160 and 240 gigaelectronvolts, the experiment can act as a switch that reveals whether the new physics is reinforcing or canceling out the known effects.

The study also provides a diagnostic tool for when things do not fit the standard picture. If future measurements show a large deviation that cannot be explained by simply turning the two known knobs, it would signal that the assumption of heavy new physics is wrong. In such a case, the new particles would have to be light enough to be produced directly or to create a distinct energy pattern that cannot be mimicked by simple adjustments. To illustrate this, the authors considered a hypothetical scenario involving a new, heavy particle that couples to electrons and the Higgs. They showed that for this particle to create a noticeable effect, it would need to be very close in mass to the collision energy and have a very specific, unnatural set of connections. If such a signal were found, it would point to a new type of particle that is light and strongly coupled, rather than the heavy, hidden particles that are the usual suspects in these searches.

Ultimately, this work establishes a robust framework for interpreting the next generation of data. It confirms that the process of creating a Higgs boson and a photon is a powerful, model-independent probe of the Higgs sector. By decomposing the event into its fundamental, gauge-invariant parts, the researchers have provided a clear path to distinguishing between different types of new physics. The ability to resolve the sign ambiguity of the Higgs couplings is a unique advantage of this method, offering information that decay measurements alone cannot provide. If the data from future colliders aligns with the predictions of this framework, it will confirm that the Higgs boson behaves exactly as the Standard Model expects, perhaps with slight adjustments to its interaction strengths. If the data deviates in a way that breaks this framework, it will be a direct signal of new, lighter particles entering the scene, fundamentally changing our understanding of the forces that shape the universe.

Technical Summary: Higgs pseudo-observables in $e^+e^- \to h\gamma$

Problem and Motivation
Precise measurements of Higgs-boson couplings are central to the physics program of future $e^+e^-$ colliders like FCC-ee. While current LHC data are consistent with the Standard Model (SM), significant room remains for non-standard effects, particularly in loop-induced couplings such as $hZ\gamma$ . The associated production process $e^+e^- \to h\gamma$ is a unique probe: it is loop-suppressed in the SM (occurring only beyond tree level due to the negligible electron Yukawa coupling), resulting in a small cross-section ( $\sim 0.1$ fb) but a clean experimental signature. This suppression naturally enhances sensitivity to new physics (NP). However, interpreting potential deviations requires a framework that distinguishes between modifications of on-shell effective couplings and genuine contact interactions, while addressing the inherent sign ambiguity of Higgs decay rates.

Methodology
The authors employ the framework of Higgs pseudo-observables, which characterize interactions based on the pole structure of physical amplitudes, independent of field redefinitions or operator bases in Effective Field Theories (EFT).

Amplitude Decomposition: The paper revisits the complete SM one-loop calculation for $e^+e^- \to h\gamma$ . The amplitude is decomposed into three separately gauge-invariant components:
- Photon-pole contribution: Mediated by an $s$ -channel virtual photon.
- $Z$ -pole contribution: Mediated by an $s$ -channel virtual $Z$ boson.
- Non-pole remainder: Comprising box diagrams ( $W$ - and $Z$ -boxes) and regular parts of vertex functions.
  The authors explicitly verify the gauge independence of these three components in a generic $R_\xi$ gauge.
EFT Power Counting: The analysis of heavy NP effects is grounded in SMEFT and HEFT power counting. The authors establish a hierarchy where modifications to the $h\gamma\gamma$ and $hZ\gamma$ pole residues arise at lower operator dimensions (dimension 6 in SMEFT, dimension 5 in HEFT) compared to genuine $e^+e^-h\gamma$ contact interactions (dimension 8 in SMEFT, dimension 7 in HEFT). Consequently, for heavy new physics, deviations are dominated by shifts in the pole residues.
Pseudo-observable Parametrization: The leading NP effects are parametrized by two real pseudo-observables, $\kappa_{\gamma\gamma}$ and $\kappa_{Z\gamma}$ , defined as the ratios of the NP-modified on-shell amplitudes to the SM amplitudes for $h \to \gamma\gamma$ and $h \to Z\gamma$ . The analysis focuses on CP-even effects, noting that decay rates constrain the magnitudes $|\kappa|$ but leave the relative signs undetermined.

Key Results

SM Baseline: The authors confirm the known SM cross-section, which exhibits a broad maximum around $\sqrt{s} \approx 240\text{--}250$ GeV and a destructive interference region near the $t\bar{t}$ threshold. The process is dominated by the left-handed $q$ -type amplitude.
Sign Ambiguity Resolution: A primary finding is that measurements of $e^+e^- \to h\gamma$ $e^{+} e^{-} \to hγ$ at multiple center-of-mass energies can resolve the discrete sign ambiguity of the pseudo-observables. While Higgs decay rates ( $h \to \gamma\gamma, Z\gamma$ $h \to γ γ, Z γ$ ) constrain $|\kappa_{\gamma\gamma}|$ $∣ κ_{γ γ} ∣$ and $|\kappa_{Z\gamma}|$ $∣ κ_{Z γ} ∣$ , they cannot determine the signs. The interference between the photon-pole and $Z$ $Z$ -pole amplitudes in $e^+e^- \to h\gamma$ $e^{+} e^{-} \to hγ$ depends on the relative signs of $\kappa_{\gamma\gamma}$ $κ_{γ γ}$ and $\kappa_{Z\gamma}$ $κ_{Z γ}$ .
- The paper demonstrates that the energy dependence of the cross-section ratio $R[\sqrt{s}](h\gamma) = \sigma_{NP}/\sigma_{SM}$ differs significantly for the four possible sign assignments $(\pm \kappa_{\gamma\gamma}, \pm \kappa_{Z\gamma})$ .
- Specifically, measurements at FCC-ee energies (e.g., 160 GeV and 240 GeV) allow for the selection of the correct sign branch and provide an over-constrained test of the hypothesis that NP affects only the pole residues.
Heavy vs. Light New Physics:
- Heavy NP: If new degrees of freedom are heavy ( $M_{NP} \gg \sqrt{s}$ ), deviations are fully captured by shifts in $\kappa_{\gamma\gamma}$ and $\kappa_{Z\gamma}$ . The paper notes that achieving a large enhancement in the cross-section via heavy NP is difficult unless there is a tuned cancellation or a sign flip that alters the interference pattern.
- Light/Intermediate NP: The authors illustrate that if an observed enhancement cannot be described by the pole pseudo-observables (consistent with decay data), it implies the presence of new degrees of freedom not far above the accessible energy scale. Using a simplified $Z'$ model, they show that such states generate non-local energy dependencies (via propagator effects) that cannot be absorbed into the two pole residues. This scenario requires specific, non-generic coupling hierarchies (e.g., strong coupling to $h\gamma$ relative to fermions).

Significance and Claims
The paper claims that the process $e^+e^- \to h\gamma$ offers a complementary and largely model-independent perspective on Higgs physics. Its significance lies in three main areas:

Gauge-Invariant Framework: The decomposition into pole and non-pole terms provides a robust, gauge-invariant basis for analyzing NP, applicable to both linear (SMEFT) and non-linear (HEFT) realizations of electroweak symmetry breaking.
Resolving Ambiguities: The process provides unique information inaccessible via decay rates alone: the ability to determine the relative signs of the effective $h\gamma\gamma$ and $hZ\gamma$ couplings through energy-dependent interference effects.
Diagnostic Tool: The energy dependence of the cross-section serves as a diagnostic for the nature of new physics. Consistency with the two-pseudo-observable description validates the "heavy NP" assumption. Inconsistency across different energies signals the presence of lighter, non-local new degrees of freedom, distinguishing between local EFT expansions and resonant or near-threshold new physics.

The authors conclude that while direct $h\gamma$ measurements at FCC-ee may not significantly improve the precision on individual coupling magnitudes beyond what is achievable via Higgs decays (given projected uncertainties), their primary value is the consistency test of the pseudo-observable description and the resolution of the sign ambiguity.

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