# Strategy to measure tau $g-2$ via photon fusion in LHC proton collisions

Lydia Beresford<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron
DESY, Notkestr. 85, 22607 Hamburg, Germany  Savannah Clawson
<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,
22607 Hamburg, Germany  Jesse Liu<EMAIL_ADDRESS>Cavendish
Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

###### Abstract

Measuring the tau-lepton ($\tau$) anomalous magnetic moment
$a_{\tau}=(g_{\tau}-2)/2$ in photon fusion production
($\gamma\gamma\to\tau\tau$) tests foundational Standard Model principles.
However, $\gamma\gamma\to\tau\tau$ eludes observation in LHC proton collisions
(pp) despite enhanced new physics sensitivity from higher-mass reach than
existing probes. We propose a novel strategy to measure
$\text{pp}\to\text{p}(\gamma\gamma\to\tau\tau)\text{p}$ by introducing the
overlooked electron-muon signature with vertex isolation for signal
extraction. Applying the effective field theory of dipole moments, we estimate
95% CL sensitivity of $-0.0092<a_{\tau}<0.011$ assuming 300 fb-1 luminosity
and 5% systematics. This fourfold improvement beyond existing constraints
opens a crucial path to unveiling new physics imprinted in tau-lepton dipoles.

##  I  Introduction

Precise measurements of electromagnetic (EM) dipoles are fundamental tests of
the Standard Model (SM) that could reveal beyond-the-SM (BSM) physics. A
cornerstone SM principle is lepton universality, where all three generations
(electron $e$, muon $\mu$, tau-lepton $\tau$) couple equally to gauge bosons.
The leading SM loop correction from quantum fluctuations is also flavor
universal, shifting magnetic moments by the Schwinger term
$\alpha_{\text{EM}}/2\pi\simeq 0.0012$ Schwinger (1948); Kusch and Foley
(1948). The electron and muon anomalous magnetic moments
$a_{e,\mu}=(g_{e,\mu}-2)/2$ are now tested to 13 Odom  _et al._ (2006);
Hanneke  _et al._ (2011); Bouchendira  _et al._ (2011); Aoyama  _et al._
(2012a); Parker  _et al._ (2018); Aoyama  _et al._ (2012b); Keshavarzi  _et
al._ (2018); Davier  _et al._ (2020); Morel  _et al._ (2020); Fan  _et al._
(2023) and 10 decimal places Bennett  _et al._ (2006); Abi  _et al._ (2021);
Aguillard  _et al._ (2023); Aoyama  _et al._ (2020), respectively. However,
the tau-lepton counterpart $a_{\tau}$ is still compatible with zero to two
decimal places Tanabashi  _et al._ (2018) as its 0.3 ps proper lifetime ALEPH
Collaboration (1997); DELPHI Collaboration (2004a); L3 Collaboration (2000);
Belle Collaboration (2014) precludes storage-ring probes Aguillard  _et al._
(2023). The existence of tau-lepton loop interactions with photons in nature
thus remains strikingly untested.

The most precise single-experiment $a_{\tau}$ constraint is a
$-0.052<a_{\tau}^{\text{obs}}<0.013$ 95% CL limit by DELPHI DELPHI
Collaboration (2004b) at the Large Electron Positron Collider (LEP), with
similar precision by L3 and OPAL L3 Collaboration (1998); OPAL Collaboration
(1998). ATLAS and CMS recently pioneered Large Hadron Collider (LHC) probes of
$a_{\tau}$ using photon fusion production of tau-leptons
($\gamma\gamma\to\tau\tau$) in lead-lead (PbPb) data ATLAS Collaboration
(2023a); CMS Collaboration (2023); the ATLAS 95% CL limit is
$-0.057<a_{\tau}^{\text{obs}}<0.024$. Such large experimental uncertainties
relative to the SM prediction
$a_{\tau,\,\text{SM}}^{\text{pred}}=0.001\,177\,21\,(5)$ Eidelman and Passera
(2007) could conceal BSM dynamics motivated by lepton sector tensions Fukuyama
_et al._ (2008); Dutta and Mimura (2019); Davoudiasl and Marciano (2018);
Bauer  _et al._ (2020); Endo and Yin (2019); LHCb Collaboration (2015, 2023);
Belle Collaboration (2019); Allanach  _et al._ (2016); Di Chiara  _et al._
(2017); Biswas and Shaw (2019); Yin and Yokozaki (2016); Yamaguchi and Yin
(2018); Babu  _et al._ (2019); Iguro and Kitahara (2023); Allanach and Davighi
(2023); Allanach and Mullin (2023). Specific models predict quadratic scaling
$\delta a_{\ell}\propto m_{\ell}^{2}$ with lepton mass $m_{\ell}$ Moroi
(1996); Ibrahim and Nath (2000); Martin and Wells (2001), implying
$(m_{\tau}/m_{\mu})^{2}\simeq 280$ times larger effects for $a_{\tau}$ than
$a_{\mu}$. New physics can also violate charge-parity (CP) symmetry, inducing
an electric dipole $d_{\tau}$. Standard LHC proton-proton (pp) collisions
reach higher $\mathcal{O}\text{(TeV)}$ masses, enhancing BSM dipole
sensitivity over $\mathcal{O}\text{(100~{}GeV)}$ in PbPb del Aguila  _et al._
(1991); Beresford and Liu (2020); Dyndal  _et al._ (2020); Burmasov  _et al._
(2023); Goncalves  _et al._ (2021); Verducci  _et al._ (2023). Despite this
key benefit, cross-section yielding over 30 million events to date, and major
photon-fusion advances Piotrzkowski (2001); Baltz (2008); Albrow  _et al._
(2009); de Favereau de Jeneret  _et al._ (2009); d’Enterria and Lansberg
(2010); ATLAS Collaboration (2015a, 2018a, 2016a); CMS Collaboration (2012,
2013, 2016); ATLAS Collaboration (2021a, b, 2023b); d’Enterria and da Silveira
(2013); ATLAS Collaboration (2017); CMS Collaboration (2019a); ATLAS
Collaboration (2019a); Howarth (2020); Gonçalves  _et al._ (2020); Chapon  _et
al._ (2010); Fichet  _et al._ (2014); Ellis  _et al._ (2017); Knapen  _et al._
(2017); Baldenegro  _et al._ (2018); Ohnemus  _et al._ (1994); Schul and
Piotrzkowski (2008); Harland-Lang  _et al._ (2012); Beresford and Liu (2019);
Harland-Lang  _et al._ (2019a); Bruce  _et al._ (2020); d’Enterria  _et al._
(2023); ATLAS Collaboration (2023c); CMS and TOTEM Collaborations (2022); CMS
and TOTEM Collaborations (2023), $\gamma\gamma\to\tau\tau$ remarkably evades
observation in pp data.

Figure 1:  Tau-leptons produced from photon fusion in proton beams with
electron-muon $\tau\tau\to e\nu\nu\mu\nu\nu$ decays as a Feynman diagram
(left) and detector signature illustrating the vertex isolation technique for
the electron-muon vs pileup tracks (right). New physics can modify the
magnetic moment $\delta a_{\tau}$.

This paper proposes the strategy to measure $\gamma\gamma\to\tau\tau$ and tau-
lepton EM dipoles in LHC pp collisions (Fig. 1). We initiate the first Monte
Carlo (MC) simulation analysis of the
$\text{pp}\to\text{p}(\gamma\gamma\to\tau\tau)\text{p}$ signal that includes
important weak-boson backgrounds and detector effects neglected in earlier
work Atag and Billur (2010). Prevailing wisdom targets hadronic tau-lepton
decays for high signal rates, but inefficient triggers, formidable
backgrounds, and multiple pp interactions (pileup) obstruct detection. We
overcome these longstanding obstacles by leveraging recent progress ATLAS
Collaboration (2020a, 2021c, 2021d) to introduce the overlooked electron-muon
signature, track-vertex isolation techniques (Fig. 1, right), and kinematic
discriminants all unexplored for pp probes of $\gamma\gamma\to\tau\tau$. This
unlocks crucial access to high-momentum kinematics unique to pp events that
augment BSM dipole sensitivity. We also propose critical strategies for
controlling systematics. Our proposal complements other production modes
Samuel and Li (1994); Hayreter and Valencia (2013, 2013); Galon  _et al._
(2016); Fomin  _et al._ (2019); Fu  _et al._ (2019); Haisch  _et al._ (2024)
and future facilities Eidelman  _et al._ (2016); Crivellin  _et al._ (2022);
Koksal  _et al._ (2018); Billur and Köksal (2014); Rajaraman  _et al._ (2019);
Köksal (2019); Gutiérrez-Rodríguez  _et al._ (2019); Chwastowski  _et al._
(2022); Fael  _et al._ (2014); Köksal (2021); Chen and Wu (2019); Tran and
Kurihara (2021); Köksal  _et al._ (2019); Qian  _et al._ (2022), while
broadening the precision tau-lepton ALEPH Collaboration (2001); ATLAS
Collaboration (2018b); CMS Collaboration (2024); ATLAS Collaboration (2021e);
BaBar Collaboration (2009); BESIII Collaboration (2014); Belle-II
Collaboration (2023) and search programs Alves  _et al._ (2012); Abdallah  _et
al._ (2015); Barr and Liu (2017); ATLAS Collaboration (2016b, 2019b, 2020b,
2020c, 2020d); CMS Collaboration (2019b); CMS Collaboration (2020, 2022); LHCb
Collaboration (2020).

##  II  Model and simulation

Relativistic field theory generalizes the Schrödinger-Pauli Hamiltonian
$\mathcal{H}=-\bm{\mu}_{\tau}\cdot\mathbf{B}-\mathbf{d}_{\tau}\cdot\mathbf{E}$
describing EM dipoles into an effective Lagrangian coupling the Dirac spinor
tensor $\sigma^{\mu\nu}=\mathrm{i}[\gamma^{\mu},\gamma^{\nu}]/2$ to the photon
field $F_{\mu\nu}$

$\displaystyle\mathcal{L}_{\text{dipole}}=\tfrac{1}{2}\bar{\tau}_{\text{L}}\sigma^{\mu\nu}\left(a_{\tau}\tfrac{e}{2m_{\tau}}-\mathrm{i}d_{\tau}\gamma_{5}\right)\tau_{\text{R}}F_{\mu\nu},$
(1)

where $\tau_{\text{L\,(R)}}$ is the left- (right-) handed tau-lepton spinor
and $\gamma^{5}$ satisfies the $\\{\gamma^{5},\gamma^{\mu}\\}=0$
anticommutator. Quantum electrodynamics (QED) predicts a vanishing electric
dipole $d_{\tau}=0$ and vacuum fluctuations induce $a_{\tau}\neq 0$.

Figure 2: Cross-sections relative to the SM
$\sigma_{\gamma\gamma\to\tau\tau}/\sigma_{\gamma\gamma\to\tau\tau}^{\text{SM}}$
for elastic $\gamma\gamma\to\tau\tau$ production vs magnetic moment variations
$\delta a_{\tau}$ with proton-proton pp 14 TeV (solid), proton-lead pPb 8.8
TeV (dashed), lead-lead PbPb 5.52 TeV (dotted) beams given tau-lepton
transverse momentum $p_{\text{T}}^{\tau}>100$ GeV (squares, only pp), 30 GeV
(triangles), 3 GeV (circles).

New physics at high-mass scale $\Lambda$ can shift EM dipoles beyond SM values
at low momentum transfers $q$ satisfying $q^{2}\ll\Lambda^{2}$, parameterized
by SM Effective Field Theory (SMEFT) Escribano and Masso (1993). Following
Ref. Beresford and Liu (2020), we introduce the dimension-six SMEFT operator

$\displaystyle\mathcal{L}_{\text{SMEFT}}=\left(C_{\tau
B}/\Lambda^{2}\right)\bar{L}_{\tau}\sigma^{\mu\nu}\tau_{R}HB_{\mu\nu}$ (2)

that modifies $(a_{\tau},d_{\tau})$ at tree level Eidelman  _et al._ (2016),
where $L_{\tau}$ ($H$) is the tau-lepton (Higgs) doublet, $B_{\mu\nu}$ is the
hypercharge field, and $C_{\tau B}$ is a dimensionless, complex Wilson
coefficient in the Warsaw basis Grzadkowski  _et al._ (2010). We implement Eq.
(2) in FeynRules Alloul  _et al._ (2014) using the
SMEFTsim_general_alphaScheme_UFO model in SMEFTsim 3.0.2 Brivio  _et al._
(2017); Brivio (2021). The real (ceBRe33) and imaginary (ceBIm33) parts of
$C_{\tau B}$ in the restrict_SMlimit_massless parameter card map to dipole
shifts

$\displaystyle\delta a_{\tau}$
$\displaystyle=\frac{2m_{\tau}}{e}\frac{\mathrm{Re}\left[C_{\tau
B}\right]}{M},\quad\delta d_{\tau}=\frac{\mathrm{Im}\left[C_{\tau
B}\right]}{M},$ (3)

defining $M=\Lambda^{2}/(\sqrt{2}v\cos\theta_{W})$ in terms of the Weinberg
angle $\theta_{W}$ and $v=246$ GeV, $m_{\tau}=1.776$ GeV, and
$e=1/\sqrt{4\pi}$ in Heaviside-Lorentz units. We interface SMEFTsim with
MadGraph 3.5.0 Alwall  _et al._ (2011, 2014) for cross-section calculation and
MC event simulation.

Canonical calculations of LHC photon-fusion cross-sections
$\sigma_{\gamma\gamma\to\tau\tau}^{(\text{pp})}$ factorize into convolutions
of the elementary cross-section $\hat{\sigma}_{\gamma\gamma\to\tau\tau}$ with
the photon flux $n(x)$ Budnev  _et al._ (1975)

$\sigma_{\gamma\gamma\to\tau\tau}^{(\text{pp})}=\int\mathrm{d}x_{1}\mathrm{d}x_{2}\,n(x_{1})n(x_{2})\,\hat{\sigma}_{\gamma\gamma\to\tau\tau},$
(4)

where $x_{i}=E_{i}/E_{\text{beam}}$ is the photon energy $E_{i}$ emitted from
proton $i$ normalized to beam energy $E_{\text{beam}}$. We adopt the charge
form factor (ChFF) flux from gamma-UPC 1.0 Shao and d’Enterria (2022), which
includes nonfactorizable soft-survival corrections to Eq. (4). The cross-
section is proportional to the amplitude squared
$\hat{\sigma}_{\gamma\gamma\to\tau\tau}\propto|\mathcal{A}|^{2}$,

$\displaystyle|\mathcal{A}|^{2}=|\mathcal{A}_{\text{SM}}|^{2}+2\mathrm{Re}\left(\mathcal{A}_{\text{SM}}\mathcal{A}_{\text{BSM}}^{*}\right)+|\mathcal{A}_{\text{BSM}}|^{2}.$
(5)

Generating $\gamma\gamma\to\tau\tau$ events with up to two EFT vertices models
this SM-BSM quantum interference. The correspondence to static EM dipoles
formally applies in the real photon limit $q^{2}_{\gamma}\to 0$, satisfied for
elastic photons from LHC protons
$q^{2}_{\gamma}/m_{\tau}^{2}\simeq(280~{}\text{MeV}/1776~{}\text{MeV})^{2}\ll
1$.

We find the 14 TeV cross-section
$\sigma_{\gamma\gamma\to\tau\tau}^{\text{(pp)}}$ for elastic
$\gamma\gamma\to\tau\tau$ production is 150 pb, yielding $4.5\times 10^{7}$
events at 300 fb-1 luminosity. Imposing transverse momentum
$p_{\text{T}}^{\tau}>30~{}(100)$ GeV, the 100 (2.5) fb cross-section yields
favorable $3\times 10^{4}$ (750) events at 300 fb-1, whereas the 25 nb (5.1
pb) PbPb cross-section yields fewer (negligible) 100 (0.02) events at 4 nb-1.
Figure 2 shows the relative cross-section variations versus $\delta a_{\tau}$,
assuming $\delta d_{\tau}=0$ for different beams and minimum
$p_{\text{T}}^{\tau}$. Requiring $p_{\text{T}}^{\tau}>30~{}(100)$ GeV modifies
$\sigma_{\gamma\gamma\to\tau\tau}^{\text{(pp)}}$ by a measurable 5% for per-
mille $\delta a_{\tau}=0.005~{}(0.001)$ shifts, dramatically improving $\delta
a_{\tau}$ sensitivity when probing scales only accessible to pp beams. We
generate around 2 million elastic $\gamma\gamma\to\tau\tau$ events per point
for 27 coupling variations in the range $\delta a_{\tau}\in[-0.015,0.015]$
with $\delta d_{\tau}=0$, requiring $p_{\text{T}}^{\tau}>15$ GeV in MadGraph
and fully leptonic decays using Pythia 8.306 Sjostrand  _et al._ (2008);
Bierlich  _et al._ (2022) customized for photon fusion ipp (2023); ATLAS
Collaboration (2021c) to improve MC statistics. Dissociative photon-fusion
(indicated by p∗) and background simulation follows standard MadGraph+Pythia
procedures Alwall  _et al._ (2014); Ball  _et al._ (2013); Mangano  _et al._
(2007) detailed in Appendix .1, generating at least 1 million events per
process. All samples have Delphes 3.5.0 de Favereau  _et al._ (2014) detector
emulation applied. We assume 300 fb-1 for the 13–13.6 TeV dataset and 4000
fb-1 for the 14 TeV High-Luminosity LHC (HL-LHC) ATLAS Collaboration (2015b);
Appendix .3 shows minor differences between these energies so we generate only
14 TeV MC for simplicity.

##  III  Analysis proposal

Figure 3: Distributions normalized to 300 fb-1 for leading lepton
$p_{\text{T}}^{\ell_{1}}$ (left), dilepton mass $m_{\ell\ell}$ (center) and
opening angle $|\Delta\phi_{\ell\ell}|$ (right). Electron-muon preselection
$p_{\text{T}}^{e\,(\mu)}>18\,(15)$ GeV, $|\eta_{\ell}|<2.5$, and vertex
isolation $\Delta z_{\text{min}}>1$ mm via
$(\varepsilon_{\text{PU}},\varepsilon_{\text{UE}})=(50\%,0.4\%)$ are applied.
SM (BSM) elastic $\gamma\gamma\to\tau\tau$ is the solid (dashed) line; other
processes are stacked histograms.

We propose targeting the $\tau\tau\to e\nu\nu\mu\nu\nu$ decay with
$2\times\mathcal{B}(\tau\to e\nu\nu)\times\mathcal{B}(\tau\to\mu\nu\nu)\simeq
6\%$ Tanabashi  _et al._ (2018) branching ratio. We emulate standard dilepton
triggers selecting this signature by requiring the electron (muon) satisfy
current offline thresholds $p_{\text{T}}^{e\,(\mu)}>18\,(15)$ GeV ATLAS
Collaboration (2020e, f, 2022a, 2019c) and tracker acceptance
$|\eta_{e,\mu}|<2.5$. After this selection, several backgrounds mimic this
signal, which we denote _reducible_ (_irreducible_) if they vanish (persist)
in the limit of ideal detector efficiency and resolution. Quark-induced ditau
$qq\to\tau\tau\to e\nu\nu\mu\nu\nu$ (Drell-Yan) and diboson $qq\to WW\to
e\nu\mu\nu$ dominate reducible backgrounds; photon-induced diboson
$\gamma\gamma\to WW\to e\nu\mu\nu$ dominates irreducible backgrounds. We
design an (i) _observation analysis_ optimized for extracting the SM
$\gamma\gamma\to\tau\tau$ signal from backgrounds and (ii) _dipole analysis_
targeting BSM deviations amid backgrounds.

To reject reducible quark-induced backgrounds, we apply the track-vertex
isolation technique ATLAS Collaboration (2020a, 2021c). This requires the
electron-muon principal vertex (PV) and track nearest to the PV have
longitudinal separation $\Delta
z_{\text{min}}=\left|z_{\text{trk}}^{\text{min}}-z_{\text{PV}}\right|$ (Fig.
1, right). This isolation (track veto) condition selects photon-fusion
processes that exhibit no additional tracks associated to the PV, whereas
quark-induced backgrounds have many tracks from underlying event (UE) in
proton breakup. We follow Ref. ATLAS Collaboration (2021d) and emulate the
$\Delta z_{\text{min}}>1~{}\text{mm}$ condition assuming nominal
$p_{\text{T}}^{\text{trk}}>500$ MeV by applying a
$\varepsilon_{\text{UE}}=0.4\%$ efficiency to our quark-induced MC $(qq\to
WW,\tau\tau)$. Pileup tracks can also accidentally fall within the veto
window, so we further apply a $\varepsilon_{\text{PU}}=50\%$ efficiency based
on data ATLAS Collaboration (2021c, d) to all MC, which depends on the
process-independent track density governed by the beamspot width
$\sigma_{z}^{\text{beam}}\simeq 38$ mm in Run 2 ATLAS Collaboration (2021c).
We consider $\varepsilon_{\text{UE}}=0.4\%$ as conservative given measurements
show this is nearer 0.1% in $Z\to ee/\mu\mu$ data ATLAS Collaboration (2021c);
the difference arises from UE mismodelling in simulation ATLAS Collaboration
(2019d). We normalize our single (double) dissociative MadGraph MC to 0.5
(0.1) of the elastic cross-section to account for vertex isolation and soft
survival based on Ref. Harland-Lang  _et al._ (2020). Reconstructing UE tracks
outside nominal acceptance would improve $\varepsilon_{\text{UE}}$ by (i)
increasing tracker acceptance $|\eta^{\text{trk}}|<4$ at HL-LHC and (ii)
reducing track thresholds to $p_{\text{T}}^{\text{trk}}>200$ MeV. A
$p_{\text{T}}^{\text{trk}}>100$ MeV threshold is feasible ATLAS Collaboration
(2016c), but practically achieving this requires
$p_{\text{T}}^{\text{trk}}>200$ MeV to control fake tracks McCormack  _et al._
(2020). Low-$p_{\text{T}}$ tracking requires unconventional data processing
that reconstructs a subset of events passing kinematic selection offline
rather than all events online, akin to ATLAS _delayed stream_ and CMS _data
parking_ paradigms. We use a simple $\varepsilon_{\text{UE}}=0.04\%$ benchmark
to capture these possible improvements. Increased
$p_{\text{T}}^{\text{trk}}>1$ GeV ATLAS Collaboration (2021d) and pileup
nuance HL-LHC extrapolations beyond the scope of this work, so we capture this
simply with $\varepsilon_{\text{PU}}=10\%$. These considerations motivate the
$\varepsilon_{\text{UE}}\in[0.4\%,0.04\%]$ and
$\varepsilon_{\text{PU}}\in[50\%,10\%]$ benchmarks in our projections.

Turning to electron-muon kinematics, we henceforth denote $\ell\in[e,\mu]$.
The electron-muon pair from $\gamma\gamma\to\tau\tau$ is nearly back-to-back,
so requiring the azimuthal opening angle satisfy $|\Delta\phi_{\ell\ell}|>3.1$
provides 77% signal efficiency with little BSM dependence for 14%
$qq\to\tau\tau$ and 3% $WW$ efficiency. While the dilepton invariant mass
$m_{\ell\ell}$ from $qq\to\tau\tau$ peaks around 40 GeV from resonant
$Z$-boson production, a looser requirement $m_{\ell\ell}>20$ GeV retains
greater signal efficiency. Figure 3 shows $p_{\text{T}}^{\ell_{1}}$,
$m_{\ell\ell}$, $|\Delta\phi_{\ell\ell}|$ distributions motivating the
selection. We also require the stransverse mass defined in Refs. Barr  _et
al._ (2003); Lester and Nachman (2015); ATLAS Collaboration (2020b) satisfy
$m_{\text{T2}}^{100}<101$ GeV, which is bounded from above by a endpoint at
$m_{\tau}$ for $\tau\tau\to e\nu\nu\mu\nu\nu$. Similar to Ref. Beresford and
Liu (2019), we use the dilepton $p_{\text{T}}^{\ell\ell}$ as the missing
transverse momentum proxy in $m_{\text{T2}}^{100}$. To summarize the event
selection:

  * •

_Observation analysis_ considers both elastic and dissociative
$\gamma\gamma\to\tau\tau$ as signal following Refs. ATLAS Collaboration
(2021c, d), applying $p_{\text{T}}^{e\,(\mu)}>18\,(15)$ GeV,
$|\eta_{\ell}|<2.5$, $m_{\ell\ell}>20$ GeV, $\Delta z_{\text{min}}>1$ mm (via
$\varepsilon_{\text{UE}},\varepsilon_{\text{PU}}$),
$|\Delta\phi_{\ell\ell}|>3.1$, $m_{\text{T2}}^{100}<101$ GeV.

  * •

_Dipole analysis_ exploits the harder BSM $p_{\text{T}}^{\ell}$ spectrum by
binning in leading lepton
$p_{\text{T}}^{e_{1}\,(\mu_{1})}\in[18\,(15),40,\infty]$, introduced in Ref.
Beresford and Liu (2020), subtracting dissociative $\gamma\gamma\to\tau\tau$
to leave only elastic $\gamma\gamma\to\tau\tau$ as signal. We use two bins for
simplicity, but future work could improve sensitivity via finer binning.

For systematic uncertainties, we outline strategies for experimentalists to
control them with powerful data-driven techniques. The dominant background
systematic is UE track modelling, where control samples of Drell-Yan
$qq\to\ell\ell$ can constrain $qq\to\tau\tau$ to percent-level accuracy ATLAS
Collaboration (2020a, 2021c). The dominant $\gamma\gamma\to\tau\tau$ signal
uncertainties arise from initial-state dynamics that we could estimate to be
$\mathcal{O}(10\%)$ by comparing photon fluxes (Appendix .2) or alternative
generators Harland-Lang  _et al._ (2019b, 2020, c). Fortunately, we can
constrain these by measuring the high-statistics standard-candle
$\gamma\gamma\to\ell\ell$ process (24 pb with $p_{\text{T}}^{\ell}>5$ GeV,
$3.6\times 10^{6}$ events at 300 fb-1) to correct both absolute
$\gamma\gamma\to\tau\tau$ cross-sections and generator-level kinematics
important for dipole shape analyses. Deriving such corrections from
$\gamma\gamma\to ee/\mu\mu$ for $\gamma\gamma\to\tau\tau$ is well justified
given the flavor independence of both initial-state proton soft-survival
physics and QED final-state radiation. ATLAS has deployed these control
techniques in similar measurements ATLAS Collaboration (2023a, 2021c). We
emphasize measuring dilepton mass $m_{\ell\ell}$ and rapidity $y_{\ell\ell}$
in distinct azimuthal angle $|\Delta\phi_{\ell\ell}|$ regions (or the
correlated $p_{\text{T}}^{\ell\ell}$) would probe the composition of
dissociative $\text{pp}\to\text{p}(\gamma\gamma\to\ell\ell)\text{p}^{*}$
enhanced at low $|\Delta\phi_{\ell\ell}|<3.1$. This ensures simulation can
accurately model and subtract dissociative production so only the elastic
$\gamma\gamma\to\tau\tau$ signal remains for measuring EM dipoles in the
$q^{2}_{\gamma}\to 0$ limit. From recent $\gamma\gamma\to\ell\ell$
measurements ATLAS Collaboration (2018a, 2020a), we anticipate such data-
driven corrections to $\gamma\gamma\to\tau\tau$ predictions can reach percent-
level accuracy. With HL-LHC statistics, we anticipate experimental systematics
to be limiting: luminosity uncertainties already reach 0.8% precision ATLAS
Collaboration (2023d), while electron/muon momentum calibration will limit
kinematic corrections, where the $ee$ channel and upgraded trackers will
enable tuning in $2.5<|\eta_{e}|<4$. Together, this motivates the assumed [1%
5%] systematics for benchmarking our projections, similar to the approach of
Ref. ATLAS Collaboration (2021d).

##  IV  Sensitivity projections

Applying the event selection and assuming 300 fb-1 luminosity, we find our
_observation analysis_ yields $S=121$ signal and $B=443$ background counts
($S/B=0.27$) assuming
$(\varepsilon_{\text{UE}},\varepsilon_{\text{PU}})=(0.4\%,50\%)$. Quark-
induced processes $qq\to\tau\tau$ (96%) and $qq\to WW$ (3%) dominate the
background composition with $\gamma\gamma\to WW$ in remainder. Evaluating the
Poissonian asymptotic significance $Z_{A}(S,B,\zeta_{b})$ Cowan (2012); Cowan
_et al._ (2011) assuming $\zeta_{b}$ background systematics, this exceeds five
standard deviations $Z_{A}=5.4$ for $\zeta_{b}=1\%$. Assuming the more
accurate value of $\varepsilon_{\text{UE}}=0.1\%$, $Z_{A}=8.5$ is feasible for
$\zeta_{b}=5\%$. This establishes a procedure to observe
$\gamma\gamma\to\tau\tau$ using standard pp runs and tracking.

Figure 4: Summary of tau-lepton magnetic dipole $a_{\tau}=(g_{\tau}-2)/2$
sensitivity. Our projections (pink) assume 300 fb-1 and 4000 fb-1 luminosities
for various pileup $\varepsilon_{\text{PU}}$ and underlying-event
$\varepsilon_{\text{UE}}$ efficiencies with 5% (dark) and 1% (light)
systematics. Displayed are existing DELPHI DELPHI Collaboration (2004b) and
ATLAS ATLAS Collaboration (2023a) constraints (blue) alongside the SM
prediction $a_{\tau}^{\text{pred}}$ (orange) Eidelman and Passera (2007). The
thick (thin) lines indicate 68% CL (95% CL) limits.

We estimate our _dipole analysis_ sensitivity to BSM $\delta a_{\tau}$ using a
$\chi^{2}$ statistical test against the SM null hypothesis

$\displaystyle\chi^{2}_{r}=\frac{(S_{\text{SM+BSM}}-S_{\text{SM}})^{2}}{\sigma_{\text{stat}}^{2}+\sigma_{\text{syst}}^{2}}.$
(6)

We evaluate region $r$ with $B$ background yield, $S_{\text{SM}}$
($S_{\text{SM+BSM}}$) signal yield assuming SM (nonzero $\delta a_{\tau}$)
couplings, statistical $\sigma_{\text{stat}}^{2}=B+S_{\text{SM+BSM}}$ and
systematic
$\sigma_{\text{syst}}^{2}=(\zeta_{b}B)^{2}+(\zeta_{s}S_{\text{SM+BSM}})^{2}$
uncertainties paramterized by $\zeta_{b,s}$. We consider a simplified scheme
that assumes $\zeta=\zeta_{s}=\zeta_{b}$ for the benchmarks
$\zeta\in[1\%,5\%]$ and are uncorrelated to statistically combine using
$\chi^{2}=\sum_{\text{r}}\chi_{\text{r}}^{2}$. The inequality $\chi^{2}<1$
($\chi^{2}<3.84$) defines the 68% CL (95% CL) constraints on $\delta
a_{\tau}$.

Figure 4 summarizes our projected sensitivity for
$a_{\tau}=a_{\tau}^{\text{SM}}+\delta a_{\tau}$. With 300 fb-1 luminosity, we
find $-0.0092<a_{\tau}<0.011$ at 95% CL for the benchmarks
$(\zeta,\varepsilon_{\text{PU}},\varepsilon_{\text{UE}})=(5\%,50\%,0.4\%)$.
This is a fourfold improvement in magnitude over existing LEP DELPHI
Collaboration (2004b); L3 Collaboration (1998); OPAL Collaboration (1998) and
LHC ATLAS Collaboration (2023a); CMS Collaboration (2023) limits. Setting
$C_{\tau B}=-1$ as a benchmark in Eq. (3), this corresponds to new physics
scale sensitivity reaching $\Lambda^{95}_{C=-1}>430$ GeV at 95% CL (from
$\Lambda^{95}_{C=-1}>190$ GeV at LEP). For 4000 fb-1 HL-LHC extrapolation, we
also loosen $p_{\text{T}}^{\ell}>10$ GeV, $|\eta_{e}|<4$ GeV ATLAS
Collaboration (2022b) to find an eightfold improvement beyond LEP
$-0.0029<a_{\tau}<0.0046$ for the most optimistic scenario
$(\zeta,\varepsilon_{\text{PU}},\varepsilon_{\text{UE}})=(1\%,50\%,0.04\%)$,
corresponding to $\Lambda^{95}_{C=-1}>680$ GeV. This HL-LHC reach degrades
modestly to $-0.0067<a_{\tau}<0.0082$ with more realistic benchmarks
$(\zeta,\varepsilon_{\text{PU}},\varepsilon_{\text{UE}})=(1\%,10\%,0.4\%)$.
Our results establish an important strategy to overcome the longstanding
obstacles of measuring $\gamma\gamma\to\tau\tau$ and $a_{\tau}$ in standard
LHC runs. This introduces a novel avenue toward BSM physics via precision
competitive with the landmark one-loop QED prediction
$\alpha_{\text{EM}}/2\pi\simeq 0.0012$.

In summary, we proposed the strategy to measure $\gamma\gamma\to\tau\tau$ in
LHC proton beams, with $-0.0092<a_{\tau}<0.011$ (95% CL) dipole sensitivity
assuming 300 fb-1 luminosity and 5% systematics. This opens future work to
develop triggers and track-veto strategies for hadronic channels, machine
learning optimization, CP-sensitive observables for electric dipoles, and
combinations with ALICE, LHCb, and heavy ions. Experimental realization would
furnish a novel precision tau-lepton dipole program that could reveal new
physics in quantum fluctuations.

###### Acknowledgements.

We thank Will Barter, Markus Diehl, Aleksandra Dimitrievska, Mateusz Dyndal,
Zahra Ghorbanimoghaddam, Lucian Harland-Lang, Oldrich Kepka, Jakub Kremer,
Valerie Lang, Kristin Lohwasser, Maeve Madigan, Klaus Mönig, Anna Mullin,
Simone Pagan Griso, Andrew Pilkington, Philip Sommer, and Weronika Stanek-
Maslouska for helpful discussions. We are grateful to the hospitality of the
Institute for Particle Physics Phenomenology at Durham University hosting the
_Photon-induced Processes Workshop_ , and University of Hamburg hosting the
_European Physical Society Conference on High Energy Physics_ , which
facilitated in-person collaboration. LB and SC are supported by the Helmholtz
Association “Young Investigator Group” initiative. JL is supported by a Junior
Research Fellowship at Trinity College, University of Cambridge.

## References

  * Schwinger (1948) J. S. Schwinger, “On Quantum-Electrodynamics and the Magnetic Moment of the Electron,” Phys. Rev. 73, 416–417 (1948).
  * Kusch and Foley (1948) P. Kusch and H. M. Foley, “The Magnetic Moment of the Electron,” Phys. Rev. 74, 250 (1948).
  * Odom  _et al._ (2006) B. Odom, D. Hanneke, B. D’Urso, and G. Gabrielse, “New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron,” Phys. Rev. Lett. 97, 030801 (2006).
  * Hanneke  _et al._ (2011) D. Hanneke, S. Fogwell Hoogerheide, and G. Gabrielse, “Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment,” Phys. Rev. A 83, 052122 (2011), arXiv:1009.4831 [physics.atom-ph] .
  * Bouchendira  _et al._ (2011) R. Bouchendira, P. Cladé, S. Guellati-Khélifa, F. Nez, and F. Biraben, “New Determination of the Fine Structure Constant and Test of the Quantum Electrodynamics,” Phys. Rev. Lett. 106, 080801 (2011).
  * Aoyama  _et al._ (2012a) T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, “Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant,” Phys. Rev. Lett. 109, 111807 (2012a), arXiv:1205.5368 [hep-ph] .
  * Parker  _et al._ (2018) R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, “Measurement of the fine-structure constant as a test of the Standard Model,” Science 360, 191–195 (2018).
  * Aoyama  _et al._ (2012b) T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, “Complete Tenth-Order QED Contribution to the Muon g-2,” Phys. Rev. Lett. 109, 111808 (2012b), arXiv:1205.5370 [hep-ph] .
  * Keshavarzi  _et al._ (2018) A. Keshavarzi, D. Nomura, and T. Teubner, “Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis,” Phys. Rev. D 97, 114025 (2018), arXiv:1802.02995 [hep-ph] .
  * Davier  _et al._ (2020) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, “A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\mathbf{\bm{\alpha}(m_{Z}^{2})}$,” Eur. Phys. J. C 80, 241 (2020), arXiv:1908.00921 [hep-ph] .
  * Morel  _et al._ (2020) L. Morel, Z. Yao, P. Cladé, and S. Guellati-Khélifa, “Determination of the fine-structure constant with an accuracy of 81 parts per trillion,” Nature 588, 61–65 (2020).
  * Fan  _et al._ (2023) X. Fan, T. G. Myers, B. A. D. Sukra, and G. Gabrielse, “Measurement of the Electron Magnetic Moment,” Phys. Rev. Lett. 130, 071801 (2023), arXiv:2209.13084 [physics.atom-ph] .
  * Bennett  _et al._ (2006) G. W. Bennett _et al._ (Muon g-2), “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” Phys. Rev. D 73, 072003 (2006), arXiv:hep-ex/0602035 [hep-ex] .
  * Abi  _et al._ (2021) B. Abi _et al._ (Muon $g-2$), “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281 [hep-ex] .
  * Aguillard  _et al._ (2023) D. P. Aguillard _et al._ (Muon $g-2$), “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm,” Phys. Rev. Lett. 131, 161802 (2023), arXiv:2308.06230 [hep-ex] .
  * Aoyama  _et al._ (2020) T. Aoyama _et al._ , “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887, 1–166 (2020), arXiv:2006.04822 [hep-ph] .
  * Tanabashi  _et al._ (2018) M. Tanabashi _et al._ (Particle Data Group), “Review of Particle Physics,” Phys. Rev. D 98, 030001 (2018).
  * ALEPH Collaboration (1997) ALEPH Collaboration, “Updated measurement of the tau lepton lifetime,” Phys. Lett. B 414, 362–372 (1997), arXiv:hep-ex/9710026 .
  * DELPHI Collaboration (2004a) DELPHI Collaboration, “A Precise measurement of the tau lifetime,” Eur. Phys. J. C 36, 283–296 (2004a), arXiv:hep-ex/0410010 .
  * L3 Collaboration (2000) L3 Collaboration, “Measurement of the lifetime of the $\tau$ lepton,” Phys. Lett. B 479, 67–78 (2000), arXiv:hep-ex/0003023 .
  * Belle Collaboration (2014) Belle Collaboration, “Measurement of the $\tau$-lepton lifetime at Belle,” Phys. Rev. Lett. 112, 031801 (2014), arXiv:1310.8503 [hep-ex] .
  * DELPHI Collaboration (2004b) DELPHI Collaboration, “Study of tau-pair production in photon-photon collisions at LEP and limits on the anomalous electromagnetic moments of the tau lepton,” Eur. Phys. J. C 35, 159–170 (2004b), arXiv:hep-ex/0406010 [hep-ex] .
  * L3 Collaboration (1998) L3 Collaboration, “Measurement of the anomalous magnetic and electric dipole moments of the tau lepton,” Phys. Lett. B 434, 169–179 (1998).
  * OPAL Collaboration (1998) OPAL Collaboration, “An Upper limit on the anomalous magnetic moment of the tau lepton,” Phys. Lett. B 431, 188–198 (1998), arXiv:hep-ex/9803020 [hep-ex] .
  * ATLAS Collaboration (2023a) ATLAS Collaboration, “Observation of the $\gamma\gamma\rightarrow\tau\tau$ Process in $\mathrm{Pb}+\mathrm{Pb}$ Collisions and Constraints on the $\tau$-Lepton Anomalous Magnetic Moment with the ATLAS Detector,” Phys. Rev. Lett. 131, 151802 (2023a), arXiv:2204.13478 [hep-ex] .
  * CMS Collaboration (2023) CMS Collaboration, “Observation of $\tau$ Lepton Pair Production in Ultraperipheral Pb-Pb Collisions at $\sqrt{{s}_{NN}}=5.02\text{ }\text{ }\mathrm{TeV}$,” Phys. Rev. Lett. 131, 151803 (2023), arXiv:2206.05192 [nucl-ex] .
  * Eidelman and Passera (2007) S. Eidelman and M. Passera, “Theory of the tau lepton anomalous magnetic moment,” Mod. Phys. Lett. A 22, 159–179 (2007), arXiv:hep-ph/0701260 [hep-ph] .
  * Fukuyama  _et al._ (2008) T. Fukuyama, A. Ilakovac, and T. Kikuchi, “Lepton flavor violating leptonic/semileptonic decays of charged leptons in the minimal supersymmetric standard model,” Eur. Phys. J. C 56, 125–146 (2008), arXiv:hep-ph/0506295 [hep-ph] .
  * Dutta and Mimura (2019) B. Dutta and Y. Mimura, “Electron $g-2$ with flavor violation in MSSM,” Phys. Lett. B 790, 563–567 (2019), arXiv:1811.10209 [hep-ph] .
  * Davoudiasl and Marciano (2018) H. Davoudiasl and W. J. Marciano, “Tale of two anomalies,” Phys. Rev. D 98, 075011 (2018), arXiv:1806.10252 [hep-ph] .
  * Bauer  _et al._ (2020) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, “Axionlike Particles, Lepton-Flavor Violation, and a New Explanation of $a_{\mu}$ and $a_{e}$,” Phys. Rev. Lett. 124, 211803 (2020), arXiv:1908.00008 [hep-ph] .
  * Endo and Yin (2019) M. Endo and W. Yin, “Explaining electron and muon $g-2$ anomaly in SUSY without lepton-flavor mixings,” JHEP 08, 122 (2019), arXiv:1906.08768 [hep-ph] .
  * LHCb Collaboration (2015) LHCb Collaboration, “Measurement of the ratio of branching fractions $\mathcal{B}(\bar{B}^{0}\to D^{*+}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}^{0}\to D^{*+}\mu^{-}\bar{\nu}_{\mu})$,” Phys. Rev. Lett. 115, 111803 (2015), [Erratum: Phys. Rev. Lett. 115 (2015) 159901], arXiv:1506.08614 [hep-ex] .
  * LHCb Collaboration (2023) LHCb Collaboration, “Measurement of the ratios of branching fractions $\mathcal{R}(D^{*})$ and $\mathcal{R}(D^{0})$,” Phys. Rev. Lett. 131, 111802 (2023), arXiv:2302.02886 [hep-ex] .
  * Belle Collaboration (2019) Belle Collaboration, “Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^{\ast})$ with a semileptonic tagging method,” (2019), arXiv:1904.08794 [hep-ex] .
  * Allanach  _et al._ (2016) B. Allanach, F. S. Queiroz, A. Strumia, and S. Sun, “$Z^{\prime}$ models for the LHCb and $g-2$ muon anomalies,” Phys. Rev. D 93, 055045 (2016), [Erratum: Phys. Rev. D 95 (2017) 119902], arXiv:1511.07447 [hep-ph] .
  * Di Chiara  _et al._ (2017) S. Di Chiara, A. Fowlie, S. Fraser, C. Marzo, L. Marzola, M. Raidal, and C. Spethmann, “Minimal flavor-changing $Z^{\prime}$ models and muon $g-2$ after the $R_{K^{*}}$ measurement,” Nucl. Phys. B 923, 245–257 (2017), arXiv:1704.06200 [hep-ph] .
  * Biswas and Shaw (2019) A. Biswas and A. Shaw, “Reconciling dark matter, $R_{K^{(*)}}$ anomalies and $(g-2)_{\mu}$ in an ${L_{\mu}-L_{\tau}}$ scenario,” JHEP 05, 165 (2019), arXiv:1903.08745 [hep-ph] .
  * Yin and Yokozaki (2016) W. Yin and N. Yokozaki, “Splitting mass spectra and muon g - 2 in Higgs-anomaly mediation,” Phys. Lett. B 762, 72–79 (2016), arXiv:1607.05705 [hep-ph] .
  * Yamaguchi and Yin (2018) M. Yamaguchi and W. Yin, “A novel approach to finely tuned supersymmetric standard models: The case of the non-universal Higgs mass model,” PTEP 2018, 023B06 (2018), arXiv:1606.04953 [hep-ph] .
  * Babu  _et al._ (2019) K. S. Babu, T. Fukuyama, S. Khan, and S. Saad, “Peccei-Quinn Symmetry and Nucleon Decay in Renormalizable SUSY $SO$(10),” JHEP 06, 045 (2019), arXiv:1812.11695 [hep-ph] .
  * Iguro and Kitahara (2023) S. Iguro and T. Kitahara, “Electric Dipole Moments as Probes of $B$ Anomaly,” (2023), arXiv:2307.11751 [hep-ph] .
  * Allanach and Davighi (2023) B. Allanach and J. Davighi, “The Rumble in the Meson: a leptoquark versus a Z’ to fit b → s$\mu$+$\mu$- anomalies including 2022 LHCb ${R}_{K^{\left(\ast\right)}}$ measurements,” JHEP 04, 033 (2023), arXiv:2211.11766 [hep-ph] .
  * Allanach and Mullin (2023) B. Allanach and A. Mullin, “Plan B: New ${Z^{\prime}}$ models for $b\rightarrow sl^{+}l^{-}$ anomalies,” JHEP 09, 173 (2023), arXiv:2306.08669 [hep-ph] .
  * Moroi (1996) T. Moroi, “The Muon anomalous magnetic dipole moment in the minimal supersymmetric standard model,” Phys. Rev. D 53, 6565–6575 (1996), [Erratum: Phys.Rev.D 56, 4424 (1997)], arXiv:hep-ph/9512396 .
  * Ibrahim and Nath (2000) T. Ibrahim and P. Nath, “CP violation and the muon anomaly in N=1 supergravity,” Phys. Rev. D 61, 095008 (2000), arXiv:hep-ph/9907555 .
  * Martin and Wells (2001) S. P. Martin and J. D. Wells, “Muon Anomalous Magnetic Dipole Moment in Supersymmetric Theories,” Phys. Rev. D 64, 035003 (2001), arXiv:hep-ph/0103067 [hep-ph] .
  * del Aguila  _et al._ (1991) F. del Aguila, F. Cornet, and J. I. Illana, “The possibility of using a large heavy-ion collider for measuring the electromagnetic properties of the tau lepton,” Phys. Lett. B 271, 256–260 (1991).
  * Beresford and Liu (2020) L. Beresford and J. Liu, “New physics and tau $g-2$ using LHC heavy ion collisions,” Phys. Rev. D 102, 113008 (2020), [Erratum: Phys. Rev. D 106 (2022) 039902], arXiv:1908.05180 [hep-ph] .
  * Dyndal  _et al._ (2020) M. Dyndal, M. Klusek-Gawenda, M. Schott, and A. Szczurek, “Anomalous electromagnetic moments of $\tau$ lepton in $\gamma\gamma\to\tau^{+}\tau^{-}$ reaction in Pb+Pb collisions at the LHC,” Phys. Lett. B 809, 135682 (2020), arXiv:2002.05503 [hep-ph] .
  * Burmasov  _et al._ (2023) N. Burmasov, E. Kryshen, P. Bühler, and R. Lavicka, “Feasibility Studies of Tau-Lepton Anomalous Magnetic Moment Measurements in Ultraperipheral Collisions at the LHC,” Phys. Part. Nucl. 54, 590–594 (2023).
  * Goncalves  _et al._ (2021) V. P. Goncalves, D. E. Martins, and M. S. Rangel, “Exclusive dilepton production at forward rapidities in $PbPb$ collisions,” Eur. Phys. J. C 81, 220 (2021), arXiv:2012.08923 [hep-ph] .
  * Verducci  _et al._ (2023) M. Verducci, N. Vignaroli, C. Roda, and V. Cavasinni, “A study of the measurement of the $\tau$ lepton anomalous magnetic moment in high energy lead-lead collisions at LHC,” (2023), arXiv:2307.15160 [hep-ph] .
  * Piotrzkowski (2001) K. Piotrzkowski, “Tagging two photon production at the CERN LHC,” Phys. Rev. D 63, 071502 (2001), arXiv:hep-ex/0009065 [hep-ex] .
  * Baltz (2008) A. J. Baltz, “The Physics of Ultraperipheral Collisions at the LHC,” Phys. Rept. 458, 1–171 (2008), arXiv:0706.3356 [nucl-ex] .
  * Albrow  _et al._ (2009) M. G. Albrow _et al._ (FP420 R & D), “The FP420 R & D Project: Higgs and New Physics with forward protons at the LHC,” JINST 4, T10001 (2009), arXiv:0806.0302 [hep-ex] .
  * de Favereau de Jeneret  _et al._ (2009) J. de Favereau de Jeneret, V. Lemaitre, Y. Liu, S. Ovyn, T. Pierzchala, K. Piotrzkowski, X. Rouby, N. Schul, and M. Vander Donckt, “High energy photon interactions at the LHC,” (2009), arXiv:0908.2020 [hep-ph] .
  * d’Enterria and Lansberg (2010) D. d’Enterria and J.-P. Lansberg, “Study of Higgs boson production and its $b\overline{b}$ decay in $\gamma\mathrm{\text{$-$}}\gamma$ processes in proton-nucleus collisions at the LHC,” Phys. Rev. D 81, 014004 (2010), arXiv:0909.3047 [hep-ph] .
  * ATLAS Collaboration (2015a) ATLAS Collaboration, “Measurement of exclusive $\gamma\gamma\rightarrow\ell^{+}\ell^{-}$ production in proton-proton collisions at $\sqrt{s}=7$ TeV with the ATLAS detector,” Phys. Lett. B 749, 242–261 (2015a), arXiv:1506.07098 [hep-ex] .
  * ATLAS Collaboration (2018a) ATLAS Collaboration, “Measurement of the exclusive $\gamma\gamma\rightarrow\mu^{+}\mu^{-}$ process in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” Phys. Lett. B 777, 303–323 (2018a), arXiv:1708.04053 [hep-ex] .
  * ATLAS Collaboration (2016a) ATLAS Collaboration, “Measurement of exclusive $\gamma\gamma\rightarrow W^{+}W^{-}$ production and search for exclusive Higgs boson production in $pp$ collisions at $\sqrt{s}=8$ TeV using the ATLAS detector,” Phys. Rev. D 94, 032011 (2016a), arXiv:1607.03745 [hep-ex] .
  * CMS Collaboration (2012) CMS Collaboration, “Search for exclusive or semi-exclusive photon pair production and observation of exclusive and semi-exclusive electron pair production in $pp$ collisions at $\sqrt{s}=7$ TeV,” JHEP 11, 080 (2012), arXiv:1209.1666 [hep-ex] .
  * CMS Collaboration (2013) CMS Collaboration, “Study of Exclusive Two-Photon Production of $W^{+}W^{-}$ in $pp$ Collisions at $\sqrt{s}=7$ TeV and Constraints on Anomalous Quartic Gauge Couplings,” JHEP 07, 116 (2013), arXiv:1305.5596 [hep-ex] .
  * CMS Collaboration (2016) CMS Collaboration, “Evidence for exclusive $\gamma\gamma\to W^{+}W^{-}$ production and constraints on anomalous quartic gauge couplings in $pp$ collisions at $\sqrt{s}=7$ and 8 TeV,” JHEP 08, 119 (2016), arXiv:1604.04464 [hep-ex] .
  * ATLAS Collaboration (2021a) ATLAS Collaboration, “Measurement of light-by-light scattering and search for axion-like particles with 2.2 nb-1 of Pb+Pb data with the ATLAS detector,” JHEP 03, 243 (2021a), [Erratum: JHEP 11 (2021) 050], arXiv:2008.05355 [hep-ex] .
  * ATLAS Collaboration (2021b) ATLAS Collaboration, “Exclusive dimuon production in ultraperipheral Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV with ATLAS,” Phys. Rev. C 104, 024906 (2021b), arXiv:2011.12211 [nucl-ex] .
  * ATLAS Collaboration (2023b) ATLAS Collaboration, “Exclusive dielectron production in ultraperipheral Pb+Pb collisions at $\sqrt{s_{\textrm{NN}}}$ = 5.02 TeV with ATLAS,” JHEP 06, 182 (2023b), arXiv:2207.12781 [nucl-ex] .
  * d’Enterria and da Silveira (2013) D. d’Enterria and G. G. da Silveira, “Observing light-by-light scattering at the Large Hadron Collider,” Phys. Rev. Lett. 111, 080405 (2013), [Erratum: Phys. Rev. Lett. 116 (2016) 129901], arXiv:1305.7142 [hep-ph] .
  * ATLAS Collaboration (2017) ATLAS Collaboration, “Evidence for light-by-light scattering in heavy-ion collisions with the ATLAS detector at the LHC,” Nat. Phys. 13, 852–858 (2017), arXiv:1702.01625 [hep-ex] .
  * CMS Collaboration (2019a) CMS Collaboration, “Evidence for light-by-light scattering and searches for axion-like particles in ultraperipheral PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV,” Phys. Lett. B 797, 134826 (2019a), arXiv:1810.04602 [hep-ex] .
  * ATLAS Collaboration (2019a) ATLAS Collaboration, “Observation of light-by-light scattering in ultraperipheral Pb+Pb collisions with the ATLAS detector,” Phys. Rev. Lett. 123, 052001 (2019a), arXiv:1904.03536 [hep-ex] .
  * Howarth (2020) J. Howarth, “Elastic Potential: A proposal to discover elastic production of top quarks at the Large Hadron Collider,” (2020), arXiv:2008.04249 [hep-ph] .
  * Gonçalves  _et al._ (2020) V. P. Gonçalves, D. E. Martins, M. S. Rangel, and M. Tasevsky, “Top quark pair production in the exclusive processes at the LHC,” Phys. Rev. D 102, 074014 (2020), arXiv:2007.04565 [hep-ph] .
  * Chapon  _et al._ (2010) E. Chapon, C. Royon, and O. Kepka, “Anomalous quartic $WW\gamma\gamma$, $ZZ\gamma\gamma$, and trilinear $WW\gamma$ couplings in two-photon processes at high luminosity at the LHC,” Phys. Rev. D 81, 074003 (2010), arXiv:0912.5161 [hep-ph] .
  * Fichet  _et al._ (2014) S. Fichet, G. von Gersdorff, O. Kepka, B. Lenzi, C. Royon, and M. Saimpert, “Probing new physics in diphoton production with proton tagging at the Large Hadron Collider,” Phys. Rev. D 89, 114004 (2014), arXiv:1312.5153 [hep-ph] .
  * Ellis  _et al._ (2017) J. Ellis, N. E. Mavromatos, and T. You, “Light-by-Light Scattering Constraint on Born-Infeld Theory,” Phys. Rev. Lett. 118, 261802 (2017), arXiv:1703.08450 [hep-ph] .
  * Knapen  _et al._ (2017) S. Knapen, T. Lin, H. K. Lou, and T. Melia, “Searching for Axionlike Particles with Ultraperipheral Heavy-Ion Collisions,” Phys. Rev. Lett. 118, 171801 (2017), arXiv:1607.06083 [hep-ph] .
  * Baldenegro  _et al._ (2018) C. Baldenegro, S. Fichet, G. von Gersdorff, and C. Royon, “Searching for axion-like particles with proton tagging at the LHC,” JHEP 06, 131 (2018), arXiv:1803.10835 [hep-ph] .
  * Ohnemus  _et al._ (1994) J. Ohnemus, T. F. Walsh, and P. M. Zerwas, “$\gamma\gamma$ production of nonstrongly interacting SUSY particles at hadron colliders,” Phys. Lett. B 328, 369–373 (1994), arXiv:hep-ph/9402302 [hep-ph] .
  * Schul and Piotrzkowski (2008) N. Schul and K. Piotrzkowski, “Detection of two-photon exclusive production of supersymmetric pairs at the LHC,” Nucl. Phys. Proc. Suppl. 179-180, 289–297 (2008), arXiv:0806.1097 [hep-ph] .
  * Harland-Lang  _et al._ (2012) L. A. Harland-Lang, C. H. Kom, K. Sakurai, and W. J. Stirling, “Measuring the masses of a pair of semi-invisibly decaying particles in central exclusive production with forward proton tagging,” Eur. Phys. J. C 72, 1969 (2012), arXiv:1110.4320 [hep-ph] .
  * Beresford and Liu (2019) L. Beresford and J. Liu, “Search Strategy for Sleptons and Dark Matter Using the LHC as a Photon Collider,” Phys. Rev. Lett. 123, 141801 (2019), arXiv:1811.06465 [hep-ph] .
  * Harland-Lang  _et al._ (2019a) L. A. Harland-Lang, V. A. Khoze, M. G. Ryskin, and M. Tasevsky, “LHC Searches for Dark Matter in Compressed Mass Scenarios: Challenges in the Forward Proton Mode,” JHEP 04, 010 (2019a), arXiv:1812.04886 [hep-ph] .
  * Bruce  _et al._ (2020) R. Bruce _et al._ , “New physics searches with heavy-ion collisions at the CERN Large Hadron Collider,” J. Phys. G 47, 060501 (2020), arXiv:1812.07688 [hep-ph] .
  * d’Enterria  _et al._ (2023) D. d’Enterria, M. A. Tamlihat, L. Schoeffel, H.-S. Shao, and Y. Tayalati, “Collider constraints on massive gravitons coupling to photons,” Phys. Lett. B 846, 138237 (2023), arXiv:2306.15558 [hep-ph] .
  * ATLAS Collaboration (2023c) ATLAS Collaboration, “Search for an axion-like particle with forward proton scattering in association with photon pairs at ATLAS,” JHEP 07, 234 (2023c), arXiv:2304.10953 [hep-ex] .
  * CMS and TOTEM Collaborations (2022) CMS and TOTEM Collaborations, “First Search for Exclusive Diphoton Production at High Mass with Tagged Protons in Proton-Proton Collisions at $\sqrt{s}$ = 13 TeV,” Phys. Rev. Lett. 129, 011801 (2022), arXiv:2110.05916 [hep-ex] .
  * CMS and TOTEM Collaborations (2023) CMS and TOTEM Collaborations, “Search for high-mass exclusive $\gamma\gamma$$\to$ WW and $\gamma\gamma$$\to$ ZZ production in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” JHEP 07, 229 (2023), arXiv:2211.16320 [hep-ex] .
  * Atag and Billur (2010) S. Atag and A. A. Billur, “Possibility of Determining $\tau$ Lepton Electromagnetic Moments in ${\gamma\gamma\to\tau^{+}\tau^{-}}$ Process at the CERN-LHC,” JHEP 11, 060 (2010), arXiv:1005.2841 [hep-ph] .
  * ATLAS Collaboration (2020a) ATLAS Collaboration, “Observation and Measurement of Forward Proton Scattering in Association with Lepton Pairs Produced via the Photon Fusion Mechanism at ATLAS,” Phys. Rev. Lett. 125, 261801 (2020a), arXiv:2009.14537 [hep-ex] .
  * ATLAS Collaboration (2021c) ATLAS Collaboration, “Observation of photon-induced $W^{+}W^{-}$ production in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector,” Phys. Lett. B 816, 136190 (2021c), arXiv:2010.04019 [hep-ex] .
  * ATLAS Collaboration (2021d) ATLAS Collaboration, “Sensitivity to exclusive $WW$ production in photon scattering at the High Luminosity LHC,” ATL-PHYS-PUB-2021-026 (2021d).
  * Samuel and Li (1994) M. A. Samuel and G. Li, “How to measure the magnetic moment of the tau lepton,” Int. J. Theor. Phys. 33, 1471–1477 (1994).
  * Hayreter and Valencia (2013) A. Hayreter and G. Valencia, “Constraining $\tau$-lepton dipole moments and gluon couplings at the LHC,” Phys. Rev. D 88, 013015 (2013), [Erratum: Phys. Rev. D 91 (2015) 099902], arXiv:1305.6833 [hep-ph] .
  * Galon  _et al._ (2016) I. Galon, A. Rajaraman, and T. M. P. Tait, “$H\rightarrow\tau^{+}\tau^{-}\gamma$ as a probe of the $\tau$ magnetic dipole moment,” JHEP 12, 111 (2016), arXiv:1610.01601 [hep-ph] .
  * Fomin  _et al._ (2019) A. S. Fomin, A. Yu Korchin, A. Stocchi, S. Barsuk, and P. Robbe, “Feasibility of $\tau$ -lepton electromagnetic dipole moments measurement using bent crystal at the LHC,” JHEP 03, 156 (2019), arXiv:1810.06699 [hep-ph] .
  * Fu  _et al._ (2019) J. Fu, M. A. Giorgi, L. Henry, D. Marangotto, F. Martínez Vidal, A. Merli, N. Neri, and J. Ruiz Vidal, “Novel Method for the Direct Measurement of the $\tau$ Lepton Dipole Moments,” Phys. Rev. Lett. 123, 011801 (2019), arXiv:1901.04003 [hep-ex] .
  * Haisch  _et al._ (2024) U. Haisch, L. Schnell, and J. Weiss, “LHC tau-pair production constraints on $a_{\tau}$ and $d_{\tau}$,” SciPost Phys. 16, 048 (2024), arXiv:2307.14133 [hep-ph] .
  * Eidelman  _et al._ (2016) S. Eidelman, D. Epifanov, M. Fael, L. Mercolli, and M. Passera, “$\tau$ dipole moments via radiative leptonic $\tau$ decays,” JHEP 03, 140 (2016), arXiv:1601.07987 [hep-ph] .
  * Crivellin  _et al._ (2022) A. Crivellin, M. Hoferichter, and J. M. Roney, “Toward testing the magnetic moment of the tau at one part per million,” Phys. Rev. D 106, 093007 (2022), arXiv:2111.10378 [hep-ph] .
  * Koksal  _et al._ (2018) M. Koksal, A. A. Billur, A. Gutiérrez-Rodríguez, and M. A. Hernández-Ruíz, “Model-independent sensitivity estimates for the electromagnetic dipole moments of the $\tau$-lepton at the CLIC,” Phys. Rev. D 98, 015017 (2018), arXiv:1804.02373 [hep-ph] .
  * Billur and Köksal (2014) A. A. Billur and M. Köksal, “Probe of the electromagnetic moments of the tau lepton in gamma-gamma collisions at the CLIC,” Phys. Rev. D 89, 037301 (2014).
  * Rajaraman  _et al._ (2019) A. Rajaraman, J. N. Howard, R. Riley, and Tim M. P. Tait, “The $\tau$ Magnetic Dipole Moment at Future Lepton Colliders,” LHEP 2, 5 (2019), arXiv:1810.09570 [hep-ph] .
  * Köksal (2019) M. Köksal, “Search for the electromagnetic moments of the $\tau$ lepton in photon–photon collisions at the LHeC and the FCC-he,” J. Phys. G 46, 065003 (2019), arXiv:1809.01963 [hep-ph] .
  * Gutiérrez-Rodríguez  _et al._ (2019) A. Gutiérrez-Rodríguez, M. Köksal, A. A. Billur, and M. A. Hernández-Ruíz, “Feasibility at the LHC, FCC-he and CLIC for sensitivity estimates on anomalous $\tau$-lepton couplings,” (2019), arXiv:1903.04135 [hep-ph] .
  * Chwastowski  _et al._ (2022) J. J. Chwastowski, K. Piotrzkowski, and M. Przybycien, “Exclusive lepton pair production at the electron–ion collider,” Eur. Phys. J. C 82, 846 (2022), arXiv:2206.02466 [hep-ph] .
  * Fael  _et al._ (2014) M. Fael, L. Mercolli, and M. Passera, “Towards a determination of the tau lepton dipole moments,” Nucl. Phys. Proc. Suppl. 253-255, 103–106 (2014), arXiv:1301.5302 [hep-ph] .
  * Köksal (2021) M. Köksal, “Model-independent study on the anomalous $\tau\bar{\tau}\gamma$ couplings at the ILC,” Nucl. Phys. B 971, 115520 (2021), arXiv:2104.01003 [hep-ph] .
  * Chen and Wu (2019) X. Chen and Y. Wu, “Search for the Electric Dipole Moment and anomalous magnetic moment of the tau lepton at tau factories,” JHEP 10, 089 (2019), arXiv:1803.00501 [hep-ph] .
  * Tran and Kurihara (2021) H. M. Tran and Y. Kurihara, “Tau $g-2$ at $e^{-}e^{+}$ colliders with momentum dependent form factor,” Eur. Phys. J. C 81, 108 (2021), arXiv:2006.00660 [hep-ph] .
  * Köksal  _et al._ (2019) M. Köksal, A. A. Billur, A. Gutiérrez-Rodríguez, and M. A. Hernández-Ruíz, “The $\mu^{+}\mu^{-}$ collider to sensitivity estimates on the magnetic and electric dipole moments of the tau-lepton,” Int. J. Mod. Phys. A 34, 1950076 (2019), arXiv:1811.01188 [hep-ph] .
  * Qian  _et al._ (2022) S. Qian _et al._ , “Boosted Tau Lepton as a Microscope and Macroscope,” Adv. High Energy Phys. 2022, 4931241 (2022), arXiv:2201.07808 [hep-ph] .
  * ALEPH Collaboration (2001) ALEPH Collaboration, “Measurement of the tau polarization at LEP,” Eur. Phys. J. C 20, 401–430 (2001), arXiv:hep-ex/0104038 .
  * ATLAS Collaboration (2018b) ATLAS Collaboration, “Measurement of $\tau$ polarisation in $Z/\gamma^{*}\rightarrow\tau\tau$ decays in proton–proton collisions at $\sqrt{s}=8$ TeV with the ATLAS detector,” Eur. Phys. J. C 78, 163 (2018b), arXiv:1709.03490 [hep-ex] .
  * CMS Collaboration (2024) CMS Collaboration, “Measurement of the $\tau$ lepton polarization in Z boson decays in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” JHEP 01, 101 (2024), arXiv:2309.12408 [hep-ex] .
  * ATLAS Collaboration (2021e) ATLAS Collaboration, “Test of the universality of $\tau$ and $\mu$ lepton couplings in $W$-boson decays with the ATLAS detector,” Nature Phys. 17, 813–818 (2021e), arXiv:2007.14040 [hep-ex] .
  * BaBar Collaboration (2009) BaBar Collaboration, “Measurements of the $\tau$ mass and the mass difference of the ${\tau}^{+}$ and ${\tau}^{-}$ at BABAR,” Phys. Rev. D 80, 092005 (2009), arXiv:0909.3562 [hep-ex] .
  * BESIII Collaboration (2014) BESIII Collaboration, “Precision measurement of the mass of the $\tau$ lepton,” Phys. Rev. D 90, 012001 (2014), arXiv:1405.1076 [hep-ex] .
  * Belle-II Collaboration (2023) Belle-II Collaboration, “Measurement of the $\tau$-lepton mass with the Belle II experiment,” Phys. Rev. D 108, 032006 (2023), arXiv:2305.19116 [hep-ex] .
  * Alves  _et al._ (2012) D. Alves _et al._ (LHC New Physics Working Group), “Simplified Models for LHC New Physics Searches,” J. Phys. G 39, 105005 (2012), arXiv:1105.2838 [hep-ph] .
  * Abdallah  _et al._ (2015) J. Abdallah _et al._ , “Simplified Models for Dark Matter Searches at the LHC,” Phys. Dark Univ. 9-10, 8–23 (2015), arXiv:1506.03116 [hep-ph] .
  * Barr and Liu (2017) A. Barr and J. Liu, “Analysing parameter space correlations of recent 13 TeV gluino and squark searches in the pMSSM,” Eur. Phys. J. C 77, 202 (2017), arXiv:1608.05379 [hep-ph] .
  * ATLAS Collaboration (2016b) ATLAS Collaboration, “Search for new phenomena in dijet mass and angular distributions from $pp$ collisions at $\sqrt{s}=$ 13 TeV with the ATLAS detector,” Phys. Lett. B 754, 302–322 (2016b), arXiv:1512.01530 [hep-ex] .
  * ATLAS Collaboration (2019b) ATLAS Collaboration, “Search for low-mass resonances decaying into two jets and produced in association with a photon using $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” Phys. Lett. B 795, 56–75 (2019b), arXiv:1901.10917 [hep-ex] .
  * ATLAS Collaboration (2020b) ATLAS Collaboration, “Searches for electroweak production of supersymmetric particles with compressed mass spectra in $\sqrt{s}=$ 13 TeV $pp$ collisions with the ATLAS detector,” Phys. Rev. D 101, 052005 (2020b), arXiv:1911.12606 [hep-ex] .
  * ATLAS Collaboration (2020c) ATLAS Collaboration, “Search for direct stau production in events with two hadronic $\tau$-leptons in $\sqrt{s}=13$ TeV $pp$ collisions with the ATLAS detector,” Phys. Rev. D 101, 032009 (2020c), arXiv:1911.06660 [hep-ex] .
  * ATLAS Collaboration (2020d) ATLAS Collaboration, “Search for heavy Higgs bosons decaying into two tau leptons with the ATLAS detector using $pp$ collisions at $\sqrt{s}=13$ TeV,” Phys. Rev. Lett. 125, 051801 (2020d), arXiv:2002.12223 [hep-ex] .
  * CMS Collaboration (2019b) CMS Collaboration, “Search for supersymmetric partners of electrons and muons in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Phys. Lett. B 790, 140–166 (2019b), arXiv:1806.05264 [hep-ex] .
  * CMS Collaboration (2020) CMS Collaboration, “Search for Supersymmetry with a Compressed Mass Spectrum in Events with a Soft $\tau$ Lepton, a Highly Energetic Jet, and Large Missing Transverse Momentum in Proton-Proton Collisions at $\sqrt{s}=13$ TeV,” Phys. Rev. Lett. 124, 041803 (2020), arXiv:1910.01185 [hep-ex] .
  * CMS Collaboration (2022) CMS Collaboration, “Search for invisible decays of the Higgs boson produced via vector boson fusion in proton-proton collisions at $\sqrt{s}=13$ TeV,” Phys. Rev. D 105, 092007 (2022), arXiv:2201.11585 [hep-ex] .
  * LHCb Collaboration (2020) LHCb Collaboration, “Search for $A^{\prime}\to\mu^{+}\mu^{-}$ Decays,” Phys. Rev. Lett. 124, 041801 (2020), arXiv:1910.06926 [hep-ex] .
  * Escribano and Masso (1993) R. Escribano and E. Masso, “New bounds on the magnetic and electric moments of the tau lepton,” Phys. Lett. B 301, 419–422 (1993).
  * Grzadkowski  _et al._ (2010) B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, “Dimension-Six Terms in the Standard Model Lagrangian,” JHEP 10, 085 (2010), arXiv:1008.4884 [hep-ph] .
  * Alloul  _et al._ (2014) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, “FeynRules 2.0 – A complete toolbox for tree-level phenomenology,” Comput. Phys. Commun. 185, 2250–2300 (2014), arXiv:1310.1921 [hep-ph] .
  * Brivio  _et al._ (2017) I. Brivio, Y. Jiang, and M. Trott, “The SMEFTsim package, theory and tools,” JHEP 12, 070 (2017), arXiv:1709.06492 [hep-ph] .
  * Brivio (2021) I. Brivio, “SMEFTsim 3.0 — a practical guide,” JHEP 04, 073 (2021), arXiv:2012.11343 [hep-ph] .
  * Alwall  _et al._ (2011) J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, and T. Stelzer, “MadGraph 5: Going Beyond,” JHEP 06, 128 (2011), arXiv:1106.0522 [hep-ph] .
  * Alwall  _et al._ (2014) J. Alwall _et al._ , “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,” JHEP 07, 079 (2014), arXiv:1405.0301 [hep-ph] .
  * Budnev  _et al._ (1975) V. M. Budnev, I. F. Ginzburg, G. V. Meledin, and V. G. Serbo, “The two photon particle production mechanism. Physical problems. Applications. Equivalent photon approximation,” Phys. Rept. 15, 181–281 (1975).
  * Shao and d’Enterria (2022) H.-S. Shao and D. d’Enterria, “gamma-UPC: automated generation of exclusive photon-photon processes in ultraperipheral proton and nuclear collisions with varying form factors,” JHEP 09, 248 (2022), arXiv:2207.03012 [hep-ph] .
  * Sjostrand  _et al._ (2008) T. Sjostrand, S. Mrenna, and P. Z. Skands, “A Brief Introduction to PYTHIA 8.1,” Comput. Phys. Commun. 178, 852–867 (2008), arXiv:0710.3820 [hep-ph] .
  * Bierlich  _et al._ (2022) C. Bierlich _et al._ , “A comprehensive guide to the physics and usage of PYTHIA 8.3,” SciPost Phys. Codebases 8 (2022), 10.21468/SciPostPhysCodeb.8, arXiv:2203.11601 [hep-ph] .
  * ipp (2023) “IPPP Workshop on the modeling of photon-induced processes,” (2023), https://conference.ippp.dur.ac.uk/event/1193/.
  * Ball  _et al._ (2013) R. D. Ball _et al._ , “Parton distributions with LHC data,” Nucl. Phys. B 867, 244–289 (2013), arXiv:1207.1303 [hep-ph] .
  * Mangano  _et al._ (2007) M. L. Mangano, M. Moretti, F. Piccinini, and M. Treccani, “Matching matrix elements and shower evolution for top-quark production in hadronic collisions,” JHEP 01, 013 (2007), arXiv:hep-ph/0611129 [hep-ph] .
  * de Favereau  _et al._ (2014) J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi (DELPHES 3), “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” JHEP 02, 057 (2014), arXiv:1307.6346 [hep-ex] .
  * ATLAS Collaboration (2015b) ATLAS Collaboration, “ATLAS Phase-II Upgrade Scoping Document,”  (2015b), 10.17181/CERN.7CRX.AJHP.
  * ATLAS Collaboration (2020e) ATLAS Collaboration, “Performance of electron and photon triggers in ATLAS during LHC Run 2,” Eur. Phys. J. C 80, 47 (2020e), arXiv:1909.00761 [hep-ex] .
  * ATLAS Collaboration (2020f) ATLAS Collaboration, “Performance of the ATLAS muon triggers in Run 2,” JINST 15, P09015 (2020f), arXiv:2004.13447 [physics.ins-det] .
  * ATLAS Collaboration (2022a) ATLAS Collaboration, “Performance of the ATLAS Level-1 topological trigger in Run 2,” Eur. Phys. J. C 82, 7 (2022a), arXiv:2105.01416 [hep-ex] .
  * ATLAS Collaboration (2019c) ATLAS Collaboration, “Trigger menu in 2018,” ATL-DAQ-PUB-2019-001 (2019c).
  * ATLAS Collaboration (2019d) ATLAS Collaboration, “Measurement of distributions sensitive to the underlying event in inclusive $Z$-boson production in pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” Eur. Phys. J. C 79, 666 (2019d), arXiv:1905.09752 [hep-ex] .
  * Harland-Lang  _et al._ (2020) L. A. Harland-Lang, M. Tasevsky, V. A. Khoze, and M. G. Ryskin, “A new approach to modelling elastic and inelastic photon-initiated production at the LHC: SuperChic 4,” Eur. Phys. J. C 80, 925 (2020), arXiv:2007.12704 [hep-ph] .
  * ATLAS Collaboration (2016c) ATLAS Collaboration, “Charged-particle distributions at low transverse momentum in $\sqrt{s}=13$ TeV $pp$ interactions measured with the ATLAS detector at the LHC,” Eur. Phys. J. C 76, 502 (2016c), arXiv:1606.01133 [hep-ex] .
  * McCormack  _et al._ (2020) W. P. McCormack _et al._ (ATLAS), “CTD2020: Minimum Pt Track Reconstruction in ATLAS,” Proceedings of Connecting The Dots Workshop (2020), 10.5281/zenodo.4088478.
  * Barr  _et al._ (2003) A. Barr, C. Lester, and P. Stephens, “A variable for measuring masses at hadron colliders when missing energy is expected; $m_{\text{T2}}$: the truth behind the glamour,” J. Phys. G 29, 2343–2363 (2003), arXiv:hep-ph/0304226 [hep-ph] .
  * Lester and Nachman (2015) C. G. Lester and B. Nachman, “Bisection-based asymmetric MT2 computation: a higher precision calculator than existing symmetric methods,” JHEP 03, 100 (2015), arXiv:1411.4312 [hep-ph] .
  * Harland-Lang  _et al._ (2019b) L. A. Harland-Lang, V. A. Khoze, and M. G. Ryskin, “Exclusive LHC physics with heavy ions: SuperChic 3,” Eur. Phys. J. C 79, 39 (2019b), arXiv:1810.06567 [hep-ph] .
  * Harland-Lang  _et al._ (2019c) L. A. Harland-Lang, A. D. Martin, R. Nathvani, and R. S. Thorne, “Ad Lucem: QED Parton Distribution Functions in the MMHT Framework,” Eur. Phys. J. C 79, 811 (2019c), arXiv:1907.02750 [hep-ph] .
  * ATLAS Collaboration (2023d) ATLAS Collaboration, “Luminosity determination in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector at the LHC,” Eur. Phys. J. C 83, 982 (2023d), arXiv:2212.09379 [hep-ex] .
  * Cowan (2012) G. Cowan, “Discovery sensitivity for a counting experiment with background uncertainty,” Royal Holloway, London (2012), www.pp.rhul.ac.uk/~cowan/stat/medsig/medsigNote.pdf.
  * Cowan  _et al._ (2011) G. Cowan, K. Cranmer, E. Gross, and O. Vitells, “Asymptotic formulae for likelihood-based tests of new physics,” Eur. Phys. J. C71, 1554 (2011), [Erratum: Eur. Phys. J.C 73 (2013) 2501], arXiv:1007.1727 [physics.data-an] .
  * ATLAS Collaboration (2022b) ATLAS Collaboration, “Technical Design Report for the Phase-II Upgrade of the ATLAS TDAQ System,” CERN-LHCC-2017-020 (2022b), 10.17181/CERN.2LBB.4IAL.
  * Heinrich (2018) L. Heinrich, “lukasheinrich/pylhe v0.0.4,”  (2018), 10.5281/zenodo.1217032.

## Appendix

This Appendix provides supplementary details supporting the main text. Section
.1 provides further methodological details on the simulation of signal and
background processes. Section .2 expands the discussion on theoretical
systematic uncertainties. Section .3 shows validation studies for generator-
level distributions.

###  .1 Technical simulation details

For the elastic $\gamma\gamma\to\tau\tau$ process, we generate around 2
million MC events for each coupling variation and require
$p_{\text{T}}^{\tau}>15$ GeV in MadGraph by setting {15:15}=pt_min_pdg in the
run_card to improve generator statistics. The SM cross-section reduces from
150 pb to 0.72 pb after imposing $p_{\text{T}}^{\tau}>15$ GeV, corresponding
to an efficiency of 0.5%. We further improve generator statistics by requiring
tau-leptons decay fully leptonically in Pythia 8.306 Sjostrand  _et al._
(2008); Bierlich  _et al._ (2022) using:
15:onMode = off
15:onIfAny = 11 13.
We then account for the dileptonic branching fraction
$\mathcal{B}(\tau\tau\to\ell\nu\nu\ell\nu\nu)\simeq 12.4\%$ in cross-section
normalization. For elastic processes, the physical picture of the
$q_{\gamma}\to 0$ limit is that the EM fields surrounding the protons not only
comprise the photons pair creating tau-leptons but also provide the quasi-
static external EM field in which we measure the tau-lepton EM dipoles.

Single dissociation (elastic-inelastic photon fusion
$\text{pp}\to\text{p}(\gamma\gamma\to\tau\tau)\text{p}^{*}$ where one proton
dissociates) and double dissociation (inelastic-inelastic photon fusion
$\text{pp}\to\text{p}^{*}(\gamma\gamma\to\tau\tau)\text{p}^{*}$ where both
protons dissociate) processes are not implemented in the gammaUPC 1.0 package
inside MadGraph 3.5.0. Therefore, we use MadGraph 2.6.7 based on Ref. ATLAS
Collaboration (2021c) to implement the equivalent photon approximation Budnev
_et al._ (1975) prescription for the elastic photon $\gamma$, and the default
NNPDF23_lo_as_0130_qed parton distribution functions (PDF) Ball  _et al._
(2013) for the inelastic photon $\gamma^{*}$. We find the MadGraph cross-
section for single (double) dissociative photon-fusion production of tau pairs
to be 3.41 pb (2.58 pb) with a $p_{\text{T}}^{\tau}>15$ GeV generator
requirement. For the $\gamma\gamma\to WW$ process, no generator cuts are
imposed and we find the elastic (single dissociative) cross-section to be 74.1
fb (592 fb). To decay, shower and hadronize such processes, we use Pythia
configured specifically for photon-fusion processes generated in MadGraph. The
full Pythia settings follow Refs. ipp (2023); ATLAS Collaboration (2021c):

  * •

PartonLevel:ISR = off for elastic processes;

  * •

BeamRemnants:unresolvedHadron set to 0, 1/2, 3 for elastic, forward/backward
single dissociation and double dissociation, respectively;

  * •

All dissociation samples apply these settings:
PartonLevel:ISR = on
BeamRemnants:primordialKThard = 1.5
BeamRemnants:primordialKTremnant = 0.1
SpaceShower:dipoleRecoil = on
SpaceShower:pTmaxMatch = 0
SpaceShower:pTmaxFudge = 1.0
SpaceShower:pTdampMatch = 1
SpaceShower:pTdampFudge = 1.5

For other non-photon-induced background processes, we generate the Drell-Yan
$qq\to Z^{(*)}/\gamma^{*}\to\tau\tau$ (which we also denote $qq\to\tau\tau$ in
the main text) and diboson $qq\to WW\to\ell\nu\ell\nu$ processes to leading
order using MadGraph 3.5.0. We employ the default NN23LO PDFs Ball  _et al._
(2013) with up to one parton in the matrix element interfaced to Pythia for
parton shower and hadronization using MLM jet-parton matching Mangano  _et
al._ (2007) with merging scale xqcut set to 25 GeV. To improve generator
statistics, we decay the tau-leptons leptonically $\tau\to\ell\nu\nu$ for
$\ell\in[e,\mu]$ and impose a generator-level $p_{\text{T}}^{\ell}>5$ GeV cut
in MadGraph (5=ptl in run_card). After such requirements, the cross-section
for these Drell-Yan (diboson) samples is 154 pb (12.6 pb). Top quark pair
production is observed to be negligible in $e\mu$ selections after track-
vertex isolation ATLAS Collaboration (2020a, 2021c) given the high track
multiplicities of the two heavy-flavor jets, so we avoid simulating this.

###  .2 Theoretical systematic uncertainties

For the $\gamma\gamma\to\tau\tau$ signal, the elementary cross-section is
crucially well predicted by quantum electrodynamics and we expect the dominant
modeling uncertainties arise from the photon flux and rescattering effects
involving the outgoing protons called soft survival. Uncertainties in the
gamma-UPC photon flux can be theoretically estimated using alternative form
factors such as the Electric Dipole (EDFF) model studied in Ref. Shao and
d’Enterria (2022), which shows our baseline (ChFF) more accurately models the
data but has a slightly harder $p_{\text{T}}^{\tau}$ and $m_{\tau\tau}$
spectrum than the other photon-flux models. For
$\text{pp}\to\text{p}(\gamma\gamma\to\tau\tau)\text{p}$ with
$p_{\text{T}}^{\tau}>15~{}\text{GeV}$, we find using EDFF gives a cross-
section of 0.594 pb that is 18% smaller than our ChFF baseline of 0.720 pb. We
also compare these kinematic differences in Fig. 7, which provides a
quantitative indication of the theoretical modeling in state-of-the-art
generators. We find these discrepancies are within 10% for
$m_{\tau\tau}\lesssim 200$ GeV and $p_{\text{T}}^{\tau}\lesssim 100$ GeV, but
grow to higher values reaching over 30%. Further estimation of theoretical
uncertainties could be made by comparing with alternative generators such as
SuperChic Harland-Lang  _et al._ (2020). Dissociative processes additionally
have uncertainties arising from the inelastic photon PDF, where we could use
alternative PDF sets such as MMHT2015qed_nlo Harland-Lang  _et al._ (2019c).
Precise evaluation of these systematic uncertainties is beyond the scope of
this phenomenological study. The main text instead discusses in detail the
strategies to experimentally constrain these in situ with data to what we
expect to reach percent-level accuracy.

###  .3 Generator-level validation studies

This section provides further kinematic validation of the simulated Monte
Carlo samples, specifically the generator-level (also called “truth-level” in
the literature) distributions, inspecting the Les Houches Events (LHE) files
using pylhe Heinrich (2018).

To highlight the principal advantage of proton-proton over lead-ion
collisions, Fig. 5 illustrates kinematic distributions of the $NN\to
N(\gamma\gamma\to\tau\tau)N$ process for three beam types $N$ alongside their
corresponding center-of-momentum energies (per nucleon for ions): proton-
proton pp (14 TeV), proton-lead pPb (8.8 TeV), lead-lead PbPb (5.52 TeV). We
normalize the distributions to unity to compare shapes for the ditau invariant
mass $m_{\tau\tau}$, tau-lepton transverse momentum $p_{\text{T}}^{\tau}$ and
rapidity $\eta_{\tau}$. These variables follow the photon flux distributions,
where we consider the charged form factor (ChFF) from gamma-UPC in MadGraph.
The pp collisions reach far higher values in $m_{\tau\tau}$ and
$p_{\text{T}}^{\tau}$ compared those with lead ions. Relative to
$m_{\tau\tau}=30$ GeV, the $m_{\tau\tau}$ spectrum drops by four orders of
magnitude at 200 GeV, 400 GeV and 1000 GeV for PbPb, pPb and pp, respectively.
Similarly, the $p_{\text{T}}^{\tau}$ spectrum for pp is significantly harder
compared to PbPb, where the $p_{\text{T}}^{\tau}$ can probe above 200 GeV for
pp that is inaccessible to PbPb. Moreover, the tau-leptons in elastic
production are back-to-back in the transverse plane with little transverse
boost of the ditau system, so the kinematic boost of the individual tau-
leptons arises from the center-of-mass system of the initial-state photons.
The $\eta_{\tau}$ distribution highlights how the larger beam energy of pp
gives a higher longitudinal boost than PbPb, while the pPb verifies the
expected asymmetric distribution, which illustrate suitability for the LHCb
experimental acceptance $2<|\eta|<5$ and upgraded trackers for ATLAS and CMS
covering $|\eta|<4$.

We study the impact of the different LHC pp center-of-mass energies
$\sqrt{s}\in\\{13,13.6,14\\}$ TeV on distributions of the
$\text{pp}\to\text{p}(\gamma\gamma\to\tau\tau)\text{p}$ process in Fig. 6.
This illustrates the differences in $\sqrt{s}$ are generally small within the
scope of this study. Therefore for simplicity, we generate samples at only 14
TeV and rescale luminosities to estimate LHC and HL-LHC sensitivity in the
main text. This is only relevant in the short term where we anticipate the
ATLAS and CMS collaborations would combine the 13 TeV from Run 2 with the 13.6
TeV dataset from Run 3, while HL-LHC is expected to reach 14 TeV. Crucially,
the slight suppression in rate for 13 TeV compared to 14 TeV at the highest
$m_{\tau\tau}$ bins are around $\mathcal{O}(10\%)$, which remain subdominant
compared to the suppression when using lead ion fluxes (Fig. 5). The
suppression at the highest $|\eta_{\tau}|$ values is expected for the reduced
longitudinal boost of smaller beam energies, but are outside the close to
negligible for the $|\eta|<4$ acceptance of the upgraded ATLAS and CMS
tracker.

To estimate theoretical uncertainties of the photon flux, Fig. 7 shows the
impact on tau-lepton kinematics when considering an alternative photon flux
denoted the Electric Dipole form factor (EDFF) in the gamma-UPC package. In
Ref. Shao and d’Enterria (2022), the differences from the ChFF nominal we
adopt in this paper are discussed and compared with dielectron and dimuon
meausrements. In brief, they found the ChFF flux provides better differential
modelling but slightly overestimates the data by around 10% compared with the
EDFF flux. We find differences of around 10 to 20% at high $m_{\tau\tau}$ and
20 to 30% for $p_{\text{T}}^{\tau}$ tails. Fortunately, experimental analyses
can constraint these differences using data-driven control sample techniques
as discussed in the main text. We expect future measurements of dilepton
standard candles and theoretical work will improve these theoretical
uncertainties.

Figure 8 shows how shifts in magnetic dipole moments $\delta a_{\tau}$ impact
tau-lepton kinematics. These shape differences are modest in the
$m_{\tau\tau}$ and especially striking in the $p_{\text{T}}^{\tau}$ spectra,
rising at high values as expected from effective field theory. Changes in
these spectra also modify the kinematics of the tau-lepton decay products.
Analyses can therefore fit observables such as the electron and muon
$p_{\text{T}}^{\ell}$ spectra in $\tau\to\ell\nu\nu$ decays to substantially
enhance sensitivity to $\delta a_{\tau}$ beyond measuring inclusive cross-
sections alone, as introduced in Ref. Beresford and Liu (2020).

Figure 9 compares how SM, BSM-only and their combined (SM + BSM) diagrams in
the matrix elements impact unit-normalized differential distributions of
$m(\tau\tau)$ and $p_{\text{T}}^{\tau}$. To generate only the linear
(quadratic) pieces with one (two) BSM dipole vertices entering the matrix
element, we set NP^2==1 (NP^2==2) in MadGraph . Displayed are shifts in the
magnetic dipole of $\delta a_{\tau}=0.01$.

Figure 10 shows the enlarged and zoomed axis versions of Fig. 2 for the
$\gamma\gamma\to\tau\tau$ cross-section varying with magnetic dipole shifts
$\delta a_{\tau}$ assuming $\delta d_{\tau}=0$. This highlights the dramatic
impact on the cross-section variations
$\sigma_{\gamma\gamma\to\tau\tau}/\sigma^{\text{SM}}_{\gamma\gamma\to\tau\tau}$
as we consider tighter requirements on $p_{\text{T}}^{\tau}>3,30,100$ GeV,
reserving the 100 GeV requirement only for pp. By contrast, for a given
minimum $p_{\text{T}}^{\tau}$, the different beam types with corresponding
center-of-mass energies (pp, pPb, PbPb) have a subdominant effect. For
$p_{\text{T}}^{\tau}>3,30$ GeV, the cross-section changes by less than 1% for
$-0.003<\delta a_{\tau}<0.002$, whereas imposing $p_{\text{T}}^{\tau}>100$ GeV
provides a percent-level cross-section change even for shifts below per-mille
$-0.0008<\delta a_{\tau}<0.0007$. Table 1 display the absolute total LHC
cross-sections of elastic $\gamma\gamma\to\tau\tau$ production when varying
the magnetic $\delta a_{\tau}$. This considers various beam configurations and
generator requirements on the tau-lepton transverse momentum.

Figure 5: Unit normalized generator-level MadGraph distributions of
$\gamma\gamma\to\tau\tau$ comparing beam types: $pp$ (thin blue), proton-lead
pPb (medium green), lead-lead PbPb (thick orange). A generator-level
requirement of $p_{\text{T}}(\tau)>15$ GeV is imposed using the charged form-
factor photon flux from gamma-UPC. The lower panel shows the ratio relative to
the nominal $pp$.

Figure 6: Unit normalized generator-level MadGraph distributions of
$\gamma\gamma\to\tau\tau$ comparing $pp$ centre-of-mass energy: 14 TeV (thick
blue), 13.6 TeV (medium green), 13 TeV (thin orange). A generator-level
requirement of $p_{\text{T}}(\tau)>15$ GeV is imposed using the charged form-
factor photon flux from gamma-UPC. The lower panel shows the ratio relative to
the nominal 14 TeV.

Figure 7: Unit normalized generator-level MadGraph distributions of
$\gamma\gamma\to\tau\tau$ comparing $pp$ at 14 TeV photon flux choices of
gamma-UPC in MadGraph: charged form factor “ChFF” (thick blue), electric
dipole form factor “EDFF” (medium green). A generator-level requirement of
$p_{\text{T}}(\tau)>15$ GeV is imposed. The lower panel shows the ratio
relative to the nominal ChFF.

Figure 8: Unit normalized generator-level MadGraph distributions of
$\gamma\gamma\to\tau\tau$ comparing magnetic dipole variations $\delta
a_{\tau}$ (upper) for $pp$ at 14 TeV photon flux choices of gamma-UPC in
MadGraph. A generator-level requirement of $p_{\text{T}}(\tau)>15$ GeV is
imposed. The lower panel of each subfigure shows the ratio relative to the SM
nominal.

Figure 9: Unit normalized generator-level MadGraph distributions of
$\text{pp}\to\text{p}(\gamma\gamma\to\tau\tau)\text{p}$ at 14 TeV comparing
the SM-only (very thick blue), SM plus BSM (NP^2<=2, thick pink), only 1 BSM
vertex (NP^2==1, medium orange), only 2 BSM vertices (NP^2==2, thin green) for
$\delta a_{\tau}=0.01$. A generator-level requirement of
$p_{\text{T}}^{\tau}>15$ GeV is imposed. The lower panel shows the ratio
relative to the SM nominal.

Figure 10: Wider (left) and zoomed (right) axis version of Fig. 2 showing cross-sections relative to the SM computed using MadGraph+SMEFTsim for elastic photon-fusion production of tau-lepton pairs $\sigma_{\gamma\gamma\to\tau\tau}/\sigma_{\gamma\gamma\to\tau\tau}^{\text{SM}}$ vs magnetic moment variations $\delta a_{\tau}$. Various beam types and center-of-mass energies are displayed: proton-proton pp 14 TeV (solid), proton-lead pPb 8.8 TeV (dashed), lead-lead PbPb 5.52 TeV (dotted). Different generator requirements on the minimum tau-lepton transverse momentum $p_{\text{T}}^{\tau}$ are shown: 100 GeV (blue squares, only pp), 30 GeV (purple triangles), 3 GeV (orange circles). Note the linear-log scale division at $\delta a_{\tau}=\pm 0.001$. $\sigma^{\mathrm{LHC}}_{\gamma\gamma\to\tau\tau}$ [pb]: | pp 14 TeV | pPb 8.8 TeV | PbPb 5.52 TeV | pp 14 TeV | pPb 8.8 TeV | PbPb 5.52 TeV | pp 14 TeV
---|---|---|---|---|---|---|---
$\delta a_{\tau}$ | $p_{\mathrm{T}}^{\tau}>3~{}\mathrm{GeV}$ | $p_{\mathrm{T}}^{\tau}>3~{}\mathrm{GeV}$ | $p_{\mathrm{T}}^{\tau}>3~{}\mathrm{GeV}$ | $p_{\mathrm{T}}^{\tau}>30~{}\mathrm{GeV}$ | $p_{\mathrm{T}}^{\tau}>30~{}\mathrm{GeV}$ | $p_{\mathrm{T}}^{\tau}>30~{}\mathrm{GeV}$ | $p_{\mathrm{T}}^{\tau}>100~{}\mathrm{GeV}$
0.1 | 61.27 | $1.499\text{\times}{10}^{05}$ | $2.35\text{\times}{10}^{08}$ | 2.181 | 2047 | $3.454\text{\times}{10}^{05}$ | 0.4644
0.05 | 47.23 | $1.181\text{\times}{10}^{05}$ | $1.896\text{\times}{10}^{08}$ | 0.628 | 610.5 | $1.076\text{\times}{10}^{05}$ | 0.1179
0.02 | 41.74 | $1.054\text{\times}{10}^{05}$ | $1.707\text{\times}{10}^{08}$ | 0.1901 | 202.4 | $3.978\text{\times}{10}^{04}$ | 0.02109
0.01 | 40.41 | $1.022\text{\times}{10}^{05}$ | $1.657\text{\times}{10}^{08}$ | 0.1256 | 142.6 | $2.963\text{\times}{10}^{04}$ | 0.007192
0.005 | 39.76 | $1.007\text{\times}{10}^{05}$ | $1.634\text{\times}{10}^{08}$ | 0.109 | 126.3 | $2.69\text{\times}{10}^{04}$ | 0.003687
0.002 | 39.5 | $9.991\text{\times}{10}^{04}$ | $1.622\text{\times}{10}^{08}$ | 0.1038 | 121.2 | $2.602\text{\times}{10}^{04}$ | 0.0027
0.001 | 39.4 | $9.964\text{\times}{10}^{04}$ | $1.618\text{\times}{10}^{08}$ | 0.1029 | 120.2 | $2.586\text{\times}{10}^{04}$ | 0.002556
0.0 | 39.22 | $9.933\text{\times}{10}^{04}$ | $1.611\text{\times}{10}^{08}$ | 0.1024 | 120 | $2.581\text{\times}{10}^{04}$ | 0.002505
$-$0.001 | 39.21 | $9.918\text{\times}{10}^{04}$ | $1.609\text{\times}{10}^{08}$ | 0.1023 | 119.5 | $2.57\text{\times}{10}^{04}$ | 0.002541
$-$0.002 | 39.12 | $9.894\text{\times}{10}^{04}$ | $1.605\text{\times}{10}^{08}$ | 0.1026 | 119.8 | $2.57\text{\times}{10}^{04}$ | 0.002671
$-$0.005 | 38.83 | $9.821\text{\times}{10}^{04}$ | $1.594\text{\times}{10}^{08}$ | 0.1061 | 122.8 | $2.613\text{\times}{10}^{04}$ | 0.003615
$-$0.01 | 38.49 | $9.723\text{\times}{10}^{04}$ | $1.574\text{\times}{10}^{08}$ | 0.1198 | 135.6 | $2.806\text{\times}{10}^{04}$ | 0.007048
$-$0.02 | 37.82 | $9.547\text{\times}{10}^{04}$ | $1.544\text{\times}{10}^{08}$ | 0.1784 | 188.4 | $3.655\text{\times}{10}^{04}$ | 0.02079
$-$0.05 | 37.52 | $9.356\text{\times}{10}^{04}$ | $1.493\text{\times}{10}^{08}$ | 0.5982 | 575.6 | $9.965\text{\times}{10}^{04}$ | 0.1172
$-$0.1 | 41.91 | $1.007\text{\times}{10}^{05}$ | $1.542\text{\times}{10}^{08}$ | 2.119 | 1977 | $3.294\text{\times}{10}^{05}$ | 0.4627
Table 1:  Generator-level LHC cross-sections
$\sigma_{\gamma\gamma\to\tau\tau}^{\text{LHC}}$ in picobarns (pb) for elastic
photon-fusion production of tau-lepton pairs using MadGraph 3.5.0 and the
charged form factor photon fluxes from gamma-UPC. This is interfaced with the
SMEFTsim3 package to evaluate for variations in the anomalous magnetic moment
$\delta a_{\tau}$. The columns show different beam species with their
corresponding center-of-mass energies, and different generator-level
requirements on the tau-lepton transverse moment $p_{\text{T}}^{\tau}$.