Probing the Tau Anomalous Magnetic Moment at Colliders: From Ultra-Peripheral Collisions to the Precision Frontier

Abstract: The anomalous magnetic moment of the tau lepton, $a_τ$, represents a fundamental test of the Standard Model (SM) and a high-sensitivity probe for New Physics in the third generation of leptons. Due to the tau's extremely short lifetime, traditional spin-precession measurements remain inaccessible, necessitating innovative experimental strategies at high-energy colliders. This review provides a comprehensive overview of the current experimental landscape, highlighting the recent paradigm shift from LEP-era constraints to the unprecedented precision reached at the LHC. We emphasize the importance of Ultra-Peripheral Heavy-Ion Collisions (UPCs), which act as a photon-photon collider'' of extreme intensity. By leveraging the $Z^4$ enhancement of the coherent photon flux in Lead-Lead ($PbPb$) interactions, these collisions provide a theoretically robustquasi-static'' environment. These results are critically compared with the latest measurements from proton-proton collisions, including the recent CMS observation of the $γγ\to ττ$ process and the ATLAS constraints from the high-mass Drell-Yan tail. We evaluate their complementarity and the challenges related to Effective Field Theory validity at the TeV scale. Finally, we outline the future prospects for $a_τ$ at Belle II and the Future Circular Collider (FCC) stages. While FCC-hh in $PbPb$ mode provides a theoretically clean environment, its sensitivity remains limited to $\mathcal{O}(10<sup>{-2})$. Conversely, the next generation of lepton facilities, specifically Belle II and FCC-ee, aims for the $\mathcal{O}(10<sup>{-5})$ level, required to probe SM electroweak loop corrections. Long-term projections for a high-energy Muon Collider suggest a potential reach of $\mathcal{O}(10<sup>{-6})$.

• The paper demonstrates that ultra-peripheral PbPb collisions provide a precise, on-shell photon probe to measure the tau’s anomalous magnetic moment, bypassing spin-precession limitations.
• It leverages effective field theory and photon flux convolution methods in both pp and UPC regimes to constrain dipole operator contributions from potential new physics.
• The study highlights complementary approaches across colliders and outlines future prospects at Belle II, FCC, and muon colliders for reaching unprecedented a_tau sensitivity.

Probing the Tau Anomalous Magnetic Moment at Colliders: A Comprehensive Assessment

Theoretical Foundations of $\boldsymbol{a_\tau}$: Operator Structure and New Physics Sensitivity

The tau anomalous magnetic moment, $a_\tau = (g_\tau-2)/2$, is a fundamental, quantum-loop-induced observable that provides a stringent test of the Standard Model's (SM) electromagnetic sector for third-generation leptons. While the electron and muon anomalous magnetic moments ($a_e$, $a_\mu$) are determined with extraordinary precision, $a_\tau$ remains significantly less constrained due to the tau's short lifetime ($\approx 290\,\mathrm{fs}$), rendering spin-precession-based storage ring methods inapplicable. However, the tau's large mass amplifies chirality-flipping new physics (NP) contributions, typically scaling as $m_\tau^2/\Lambda_{NP}^2$, resulting in an intrinsic sensitivity to high-scale phenomena by a factor of $\sim 280$ over the muon. Many extensions of the SM, including SUSY, leptoquark, and $Z'$ models, predict enhancements to $a_\tau$ via dipole operators.

The general electromagnetic interaction of the tau is encoded in the vertex function $a_\tau = (g_\tau-2)/2$0, parameterized by four dimensionless form factors $a_\tau = (g_\tau-2)/2$1. The Pauli form factor $a_\tau = (g_\tau-2)/2$2 defines the static anomalous magnetic moment, accumulated from SM loop corrections—QED (dominant), electroweak, and hadronic—along with any NP contributions. Current SM calculations converge to $a_\tau = (g_\tau-2)/2$3, with irreducible uncertainty dominated by hadronic vacuum polarization and light-by-light contributions. The EFT approach, particularly SMEFT, enables model-independent mapping between experimental bounds on $a_\tau = (g_\tau-2)/2$4 and the Wilson coefficients of the dipole operators, establishing correlations with $a_\tau = (g_\tau-2)/2$5 and $a_\tau = (g_\tau-2)/2$6 couplings and two-boson contact interactions.

Experimental Methodology: From LEP to LHC—$a_\tau = (g_\tau-2)/2$7 as a High-Precision Probe

Traditional direct measurement of $a_\tau = (g_\tau-2)/2$8 is inaccessible; all constraints derive from its effective modification of $a_\tau = (g_\tau-2)/2$9 and $a_e$0 vertices in scattering and pair-production processes. At LEP, the DELPHI measurement using $a_e$1 (photon-fusion) yielded a $a_e$2 CL exclusion of $a_e$3. At the LHC, photon-photon fusion is dissected across two principal collision regimes: $a_e$4 (proton-proton) and heavy-ion ultra-peripheral ($a_e$5, UPC).

In $a_e$6 processes, exclusive $a_e$7 production occurs with both protons remaining intact (elastic), one dissociating (semi-elastic), or both breaking up (inelastic). The latter two introduce PDF and hadronic uncertainties, but the statistics are higher. The UPC regime, operational in $a_e$8 collisions, leverages the coherent $a_e$9-enhanced photon flux from highly Lorentz-boosted nuclei in the impact parameter domain $a_\mu$0, ensuring the observables probe nearly on-shell ($a_\mu$1) photons and suppress strong interaction backgrounds.

The experimental cross section is given by a convolution of equivalent photon fluxes with the elementary $a_\mu$2 cross section. For heavy ions, robust theoretical modeling of the photon spectrum is possible, enabling precise extraction of $a_\mu$3 from the measured yields.

Impact of LHC Heavy-Ion Collisions: The "Photon Collider" Paradigm

LHC $a_\mu$4 UPCs now set the leading constraints in the near-static limit on $a_\mu$5. The amplified luminosity due to coherent photon emission ($a_\mu$6 scaling) enables sensitivity at orders of magnitude surpassing LEP, while systematic uncertainties are strongly suppressed relative to $a_\mu$7 or $a_\mu$8 environments. Recent ATLAS and CMS analyses yield $a_\mu$9 and $a_\tau$0 respectively at the $a_\tau$1 CL.

(Figure 1)

Figure 1: Effective photon-photon luminosity as a function of invariant mass $a_\tau$2, illustrating the $a_\tau$3 enhancement in $a_\tau$4 and the extended energy reach of $a_\tau$5 collisions.

UPCs uniquely access the region $a_\tau$6 GeV (where $a_\tau$7 is the $a_\tau$8 CM energy), allowing direct constraints on the static $a_\tau$9 with minimal EFT translation. In contrast, $\approx 290\,\mathrm{fs}$0 data cover higher energy scales, necessitating careful SMEFT interpretation.

Complementarity of $\approx 290\,\mathrm{fs}$1 and UPC Channels: Kinematics, Systematics, and EFT Considerations

There exists intrinsic complementarity between UPC and $\approx 290\,\mathrm{fs}$2 approaches. UPC measurements, close to $\approx 290\,\mathrm{fs}$3, minimize theoretical ambiguities and directly probe $\approx 290\,\mathrm{fs}$4, while $\approx 290\,\mathrm{fs}$5 collisions access much larger $\approx 290\,\mathrm{fs}$6 (up to several TeV), improving raw statistical precision but introducing dependence on the running of the dipole form factor and potential breakdown of the EFT expansion at high energies.

In the latest CMS and ATLAS $\approx 290\,\mathrm{fs}$7 analyses, the quoted bounds, $\approx 290\,\mathrm{fs}$8 (CMS, (2604.19665)), and $\approx 290\,\mathrm{fs}$9 (ATLAS, (2604.19665)), reach $m_\tau^2/\Lambda_{NP}^2$0–level sensitivity but probe largely virtual photons and the high-mass Drell-Yan region, so model assumptions about operator truncation and energy suppression are critical. The effective limits are thus on combinations of SMEFT Wilson coefficients, and their translation to $m_\tau^2/\Lambda_{NP}^2$1 depends on the hierarchy $m_\tau^2/\Lambda_{NP}^2$2.

Future Prospects: Projected Sensitivities at Belle II, FCC, and High-Energy Muon Colliders

The next-generation lepton colliders—Belle II and especially FCC-ee, operating at the $m_\tau^2/\Lambda_{NP}^2$3 pole with $m_\tau^2/\Lambda_{NP}^2$4 $m_\tau^2/\Lambda_{NP}^2$5 pairs—are expected to reach $m_\tau^2/\Lambda_{NP}^2$6 sensitivities in the $m_\tau^2/\Lambda_{NP}^2$7 range via precision measurements of $m_\tau^2/\Lambda_{NP}^2$8 polarization, spin correlations, and radiative processes (2604.19665). These facilities have inherently clean experimental environments and minimal systematics, allowing model-independent extractions that can start probing the SM value of $m_\tau^2/\Lambda_{NP}^2$9 and potential NP signals.

In the energy frontier, the FCC-hh in $\sim 280$0 UPC mode, despite enormous beam energy, is projected to achieve only $\sim 280$1 due to the limitation of accessible photon energies by the nuclear form factor. In contrast, a multi-TeV muon collider—by exploiting Drell-Yan and vector-boson fusion channels, as well as rare decays $\sim 280$2—could, in principle, attain sensitivity down to $\sim 280$3, but only with sizable luminosity and advanced event selection strategies.

Crucially, at all future energy-frontier facilities, inclusion of the SMEFT-induced two-boson and higher-dimension interactions, as well as a global analysis incorporating both neutral and charged processes, is necessary for complete and consistent interpretation of $\sim 280$4-related observables.

Implications and Outlook

Current and future experimental programs are driving $\sim 280$5 from a less-constrained, theoretically motivated observable into a precision collider test of the SM and a direct probe of lepton-universality-violating new physics scenarios. The distinctive combination of heavy-ion UPCs (providing clean, robust constraints in the static regime) and high-luminosity $\sim 280$6 running (enabling high-energy sensitivity but requiring stringent EFT control) constitutes an optimal, mutually reinforcing approach.

As sensitivity approaches the $\sim 280$7 level at Belle II and FCC-ee, genuine SM loop-induced contributions and a wealth of NP scenarios (SUSY, leptoquarks, $\sim 280$8) can be discriminated in third-generation leptons. Meanwhile, energy-frontier colliders like a muon collider could potentially resolve tiny deviations indicative of ultra-heavy new states via enhanced dipole amplitudes, provided the limitations of the EFT at high $\sim 280$9 are comprehensively addressed.

Conclusion

The collider-based program for probing $Z'$0 has undergone a significant transformation, with the UPC channel now delivering SM-scale probes in robust kinematics. Ongoing and planned experiments are poised to explore the full electroweak dipole structure of the tau and its BSM sensitivity. The interplay between UPC ($Z'$1), high-energy $Z'$2 collisions, and future lepton and muon machines will determine whether evidence for new physics emerges in the tau sector or if the SM remains unchallenged, now extended to the precision frontier for third-generation charged leptons.