Anomalous Quartic Gauge Couplings (aQGCs)

• Anomalous quartic gauge couplings are deviations from the Standard Model in four-boson interactions, emerging from higher-dimensional EFT operators.
• They are analyzed using SMEFT and HEFT frameworks, with dimension-eight operators categorized into S-, M-, and T-type classes.
• Collider experiments and vector-boson scattering probe these effects, addressing rapid energy growth and enforcing EFT consistency via unitarity bounds.

Searching arXiv for recent and foundational papers on anomalous quartic gauge couplings, SMEFT/HEFT bases, positivity, and collider probes. Anomalous quartic gauge couplings (aQGCs) are deviations of four-gauge-boson self-interactions from the Standard Model prediction. In the Standard Model, quartic gauge couplings are fixed by the non-Abelian $SU(2)_L\times U(1)_Y$ gauge structure and the Higgs mechanism; in a linearly realized SMEFT, genuine quartic interactions with no new cubic gauge interactions first arise from dimension-eight operators, whereas in HEFT genuine quartic gauge operators already appear at ${\cal O}(p^4)$ (Durieux et al., 2024, Eboli et al., 2023). Because aQGC contributions grow rapidly with energy and directly affect vector-boson scattering and related multi-boson processes, they are a central probe of electroweak symmetry breaking, heavy new physics, and the consistency constraints of quantum field theory.

1. Gauge-theory origin and physical meaning

In the Standard Model electroweak sector, the gauge group is

$SU(2)_L \times U(1)_Y.$

Because this structure is non-Abelian, the gauge fields self-interact. This produces triple gauge couplings such as $WWZ$ and $WW\gamma$, and quartic gauge couplings such as $WWWW$, $WWZZ$, $WWZ\gamma$, $WW\gamma\gamma$, $ZZZZ$, ${\cal O}(p^4)$0, ${\cal O}(p^4)$1, and ${\cal O}(p^4)$2 after electroweak symmetry breaking (Mukherjee et al., 4 May 2026, Yang et al., 2021). In vector boson scattering, these quartic couplings appear already at leading order and are crucial to preserve perturbative unitarity at high energies; in the Standard Model, cancellations among quartic vertices, trilinear gauge couplings, and Higgs exchange tame the high-energy behavior of the amplitudes (Mukherjee et al., 4 May 2026).

An aQGC is a deviation of these quartic interactions from the Standard Model value. The term “anomalous” is not used in the sense of gauge anomalies; it denotes non-standard interactions induced by heavy new physics and encoded in EFT Wilson coefficients (Yang et al., 2021). Experimentally and phenomenologically, aQGCs are therefore interpreted as a model-independent description of new electroweak dynamics in processes such as VBS, triboson production, exclusive ${\cal O}(p^4)$3, and neutral multi-photon channels.

2. EFT descriptions and operator bases

In a linearly realized SMEFT, the effective Lagrangian is organized as

${\cal O}(p^4)$4

and the genuine aQGC sector is conventionally parameterized by dimension-eight operators of ${\cal O}(p^4)$5-, ${\cal O}(p^4)$6-, and ${\cal O}(p^4)$7-type (Durieux et al., 2024, Bi et al., 2019). In this language, ${\cal O}(p^4)$8-type operators are built from Higgs covariant derivatives only, ${\cal O}(p^4)$9-type operators mix Higgs derivatives with gauge field strengths, and $SU(2)_L \times U(1)_Y.$0-type operators are quartic in gauge field strengths. The standard notation writes the quartic sector as

$SU(2)_L \times U(1)_Y.$1

or, more explicitly,

$SU(2)_L \times U(1)_Y.$2

Class	Generic structure	Typical role
$SU(2)_L \times U(1)_Y.$3-type	$SU(2)_L \times U(1)_Y.$4	Longitudinal sector
$SU(2)_L \times U(1)_Y.$5-type	$SU(2)_L \times U(1)_Y.$6	Mixed longitudinal/transverse sector
$SU(2)_L \times U(1)_Y.$7-type	$SU(2)_L \times U(1)_Y.$8	Transverse sector, including neutral photonic vertices

The LHC EFT Working Group note gives a definitive dimension-eight basis for operators that generate quartic but no cubic electroweak gauge interactions, and distinguishes CP-even from CP-odd structures (Durieux et al., 2024). In that basis, the CP-even sector contains $SU(2)_L \times U(1)_Y.$9-, $WWZ$0-, and $WWZ$1-type operators, while there are no $WWZ$2-type CP-odd operators; CP-odd aQGCs occur only in the $WWZ$3- and $WWZ$4-type sectors. The note also states that earlier AEG bases omitted two operators that are C-odd, P-odd, but CP-even, namely $WWZ$5 and $WWZ$6, and provides mappings to the experimental FS/FM/FT conventions and to UFO implementations (Durieux et al., 2024).

After electroweak symmetry breaking, these operators map onto physical quartic vertices. For example, $WWZ$7 and $WWZ$8 contribute to $WWZ$9, $WW\gamma$0, $WW\gamma$1, and $WW\gamma$2, while $WW\gamma$3 affect only neutral quartic couplings such as $WW\gamma$4, $WW\gamma$5, $WW\gamma$6, and $WW\gamma$7 (Gurkanli, 2023, Senol et al., 2021). This is why purely neutral final states are especially sensitive to $WW\gamma$8 and $WW\gamma$9.

HEFT provides a different, non-linear organization of the same physics. In the HEFT analysis of genuine QGCs, the leading custodial-preserving operators are $WWWW$0 and $WWWW$1, with custodial-violating companions $WWWW$2, $WWWW$3, and $WWWW$4, and no photons appear at $WWWW$5 because the basic chiral building blocks generate only $WWWW$6 and $WWWW$7 quartic vertices at that order (Eboli et al., 2023). This difference matters when comparing HEFT fits to the more common SMEFT $WWWW$8, $WWWW$9, $WWZZ$0 parameterization.

3. Vertices, high-energy growth, and theoretical consistency

The defining phenomenological property of aQGC operators is their rapid energy growth. In the $WWZZ$1 study, the dimension-eight contribution is described schematically by

$WWZZ$2

with a corresponding pure-new-physics cross section scaling approximately as

$WWZZ$3

until unitarity is approached (Zhang et al., 2024). In the same-sign muon-collider VBS study the same point is expressed as

$WWZZ$4

with the resulting strong improvement when the collider energy is raised from $WWZZ$5 to $WWZZ$6 TeV (Mukherjee et al., 4 May 2026). Operationally, this is why essentially every analysis emphasizes high-$WWZZ$7, high-invariant-mass, or large-separation tails.

This growth also makes EFT consistency a central issue. Partial-wave unitarity is commonly imposed through

$WWZZ$8

either at the subprocess level or through event-level invariant-mass cuts (Zhang et al., 2024, Guo et al., 2019). In $WWZZ$9, the leading unitarity bounds take the form

$WWZ\gamma$0

and, using the 95% event criterion, the 13 TeV study quotes a characteristic $WWZ\gamma$1 TeV with bounds such as $WWZ\gamma$2 TeV$WWZ\gamma$3 and $WWZ\gamma$4 TeV$WWZ\gamma$5 (Guo et al., 2019). Other analyses implement unitarity-safe regions by clipping or cutting on $WWZ\gamma$6, $WWZ\gamma$7, or related subprocess energies (Éboli et al., 4 Jun 2026, Senol et al., 27 Feb 2025).

Beyond unitarity, positivity imposes stringent UV-consistency requirements. The positivity analysis derives 19 linear inequalities, 3 quadratic inequalities, and 1 quartic inequality for the 18-dimensional dimension-eight aQGC parameter space, and finds that they reduce the allowed solid-angle volume to about $WWZ\gamma$8 of the naïve space (Bi et al., 2019). In one-operator benchmarks, some coefficients must be strictly positive, some strictly negative, and some are forbidden when switched on alone. This directly qualifies the widespread one-operator-at-a-time practice: it is a useful experimental projection, but it does not always correspond to a UV-completable direction in EFT parameter space (Bi et al., 2019).

4. Collider channels and the structure of sensitivity

Different processes isolate different quartic vertices. At high-energy muon colliders, $WWZ\gamma$9 is used as a direct probe of $WW\gamma\gamma$0 through VBS-type diagrams $WW\gamma\gamma$1, with triboson topologies more important at lower energies and VBS dominance at higher energies (Zhang et al., 2024). A same-sign muon collider sharpens this logic further: because $WW\gamma\gamma$2 or $WW\gamma\gamma$3 cannot annihilate through a neutral $WW\gamma\gamma$4-channel, VBS becomes the dominant production mechanism, and the analysis can be organized into signal regions such as $WW\gamma\gamma$5, $WW\gamma\gamma$6, $WW\gamma\gamma$7, $WW\gamma\gamma$8, and $WW\gamma\gamma$9, probing charged and neutral QGC subclasses in a single setup (Mukherjee et al., 4 May 2026).

Lepton and lepton-photon colliders are particularly effective for neutral quartic couplings. At CLIC stage 3, $ZZZZ$0 probes $ZZZZ$1 and $ZZZZ$2 vertices using the Weizsäcker–Williams approximation, and the analysis is restricted to $ZZZZ$3 with $ZZZZ$4 because the process is sensitive to neutral-photon-rich vertices (Gurkanli, 2023). The companion channel $ZZZZ$5 at $ZZZZ$6 TeV targets the same neutral quartic structures, with $ZZZZ$7 and $ZZZZ$8 especially sensitive to the $ZZZZ$9 final state (Ari et al., 2021).

Hadron-collider studies distribute sensitivity across a wide process set. ${\cal O}(p^4)$00 constrains ${\cal O}(p^4)$01 and ${\cal O}(p^4)$02 at the HL-LHC, HE-LHC, and FCC-hh (Senol et al., 2021). ${\cal O}(p^4)$03 and ${\cal O}(p^4)$04 isolate mixed and tensor operators in VBS topologies, with polarization effects especially useful for ${\cal O}(p^4)$05 in ${\cal O}(p^4)$06 and for ${\cal O}(p^4)$07, ${\cal O}(p^4)$08 in ${\cal O}(p^4)$09 (Yang et al., 2021, Guo et al., 2020). Electroweak ${\cal O}(p^4)$10 is tailored to ${\cal O}(p^4)$11 and ${\cal O}(p^4)$12, while same-sign ${\cal O}(p^4)$13 VBS at the LHC directly probes ${\cal O}(p^4)$14 using leptonic angular information (Senol et al., 27 Feb 2025, Éboli et al., 4 Jun 2026). Exclusive ${\cal O}(p^4)$15 in pp collisions remains the cleanest channel for the ${\cal O}(p^4)$16 quartic vertex itself (Guo et al., 2019).

5. Observables, polarization, and modern analysis strategies

Across channels, aQGC sensitivity is concentrated in high-energy corners of phase space. Typical discriminants are hard photons, large diboson or diphoton invariant masses, large transverse masses, and VBS-like jet configurations. Examples include ${\cal O}(p^4)$17 and ${\cal O}(p^4)$18 in ${\cal O}(p^4)$19 (Senol et al., 2021), ${\cal O}(p^4)$20 and the polarization-inspired variable

${\cal O}(p^4)$21

in ${\cal O}(p^4)$22 (Yang et al., 2021), and the transverse-mass-like quantity

${\cal O}(p^4)$23

together with ${\cal O}(p^4)$24 in ${\cal O}(p^4)$25-motivated pp analyses (Guo et al., 2019). In same-sign ${\cal O}(p^4)$26 VBS, the main traditional handle is

${\cal O}(p^4)$27

whose high-mass tail is strongly enhanced by dimension-eight operators (Éboli et al., 4 Jun 2026).

Polarization and spin correlation have become a distinct methodological layer. In same-sign ${\cal O}(p^4)$28 scattering, spin-correlation asymmetries provide sensitivity to anomalous ${\cal O}(p^4)$29 interactions comparable to that obtained from the transverse-mass distribution of the ${\cal O}(p^4)$30 system, and combining angular asymmetries with ${\cal O}(p^4)$31 improves one-parameter limits by roughly ${\cal O}(p^4)$32–${\cal O}(p^4)$33 at ${\cal O}(p^4)$34 ab${\cal O}(p^4)$35 (Éboli et al., 4 Jun 2026). In ${\cal O}(p^4)$36, the lepton-projection variable ${\cal O}(p^4)$37 and the two-dimensional ${\cal O}(p^4)$38 structure identify the distinctive transverse-helicity patterns of ${\cal O}(p^4)$39, while in ${\cal O}(p^4)$40 the polarization effect is explicitly noted to highlight the signals of ${\cal O}(p^4)$41 operators (Guo et al., 2020, Yang et al., 2021).

Machine learning has been used in both supervised and unsupervised forms. In electroweak ${\cal O}(p^4)$42, the multivariate analysis is based on Boosted Decision Trees with 850 trees, maximum depth 3, AdaBoost, and 20–22 input variables built from the two leading photons and two leading jets (Senol et al., 27 Feb 2025). Unsupervised anomaly detection has also been deployed: the isolation forest analysis of ${\cal O}(p^4)$43 achieved ${\cal O}(p^4)$44 for ${\cal O}(p^4)$45 and ${\cal O}(p^4)$46 for ${\cal O}(p^4)$47, to be compared with ${\cal O}(p^4)$48 and ${\cal O}(p^4)$49 for the cut-based event-selection strategy (Jiang et al., 2021). Quantum-inspired anomaly detection has been explored in ${\cal O}(p^4)$50 through kernel ${\cal O}(p^4)$51-means with three quantum kernels; in that setup the real-vector quantum kernel gives the best overall performance, outperforming the classical kernel for most of ${\cal O}(p^4)$52, with the classical kernel slightly better only for ${\cal O}(p^4)$53 (Zhang et al., 2024).

6. Representative bounds and projected reach

Representative constraints span a very wide range because they depend on the vertex class, the channel, the collider energy, and whether the result is quoted in terms of ${\cal O}(p^4)$54 or ${\cal O}(p^4)$55. The numbers below are taken directly from the cited studies and illustrate the present scale of the field (Guo et al., 2019, Ari et al., 2021, Gurkanli, 2023, Senol et al., 2021, Senol et al., 27 Feb 2025, Zhang et al., 2024, Mukherjee et al., 4 May 2026).

Channel	Scenario	Representative bound
${\cal O}(p^4)$56 in pp	13 TeV, ${\cal O}(p^4)$57, ${\cal O}(p^4)$58	${\cal O}(p^4)$59 TeV${\cal O}(p^4)$60; ${\cal O}(p^4)$61 TeV${\cal O}(p^4)$62
${\cal O}(p^4)$63	CLIC 3 TeV, ${\cal O}(p^4)$64, unpolarized, ${\cal O}(p^4)$65	${\cal O}(p^4)$66; ${\cal O}(p^4)$67 TeV${\cal O}(p^4)$68
${\cal O}(p^4)$69	CLIC 3 TeV, ${\cal O}(p^4)$70, unpolarized, ${\cal O}(p^4)$71	${\cal O}(p^4)$72; ${\cal O}(p^4)$73 TeV${\cal O}(p^4)$74
${\cal O}(p^4)$75	FCC-hh 100 TeV, ${\cal O}(p^4)$76, ${\cal O}(p^4)$77	${\cal O}(p^4)$78; ${\cal O}(p^4)$79 TeV${\cal O}(p^4)$80
${\cal O}(p^4)$81	FCC-hh 100 TeV, ${\cal O}(p^4)$82, 95% CL, ${\cal O}(p^4)$83	${\cal O}(p^4)$84; ${\cal O}(p^4)$85 TeV${\cal O}(p^4)$86
${\cal O}(p^4)$87	14 TeV, ${\cal O}(p^4)$88, real-vector QKKM, ${\cal O}(p^4)$89	${\cal O}(p^4)$90; ${\cal O}(p^4)$91 TeV${\cal O}(p^4)$92
same-sign ${\cal O}(p^4)$93 VBS	6 TeV, ${\cal O}(p^4)$94, 95% CL combined	${\cal O}(p^4)$95; ${\cal O}(p^4)$96 TeV${\cal O}(p^4)$97

Several broad trends are already clear in the literature. First, neutral photonic operators ${\cal O}(p^4)$98 and ${\cal O}(p^4)$99 are exceptionally well constrained in clean neutral channels: the $SU(2)_L \times U(1)_Y.$00 CLIC study quotes improvements by factors between $SU(2)_L \times U(1)_Y.$01 and $SU(2)_L \times U(1)_Y.$02 over experimental results, with particularly dramatic gains for $SU(2)_L \times U(1)_Y.$03 and $SU(2)_L \times U(1)_Y.$04 (Gurkanli, 2023). Second, future hadron colliders extend the reach substantially: in $SU(2)_L \times U(1)_Y.$05, the FCC-hh projections improve current CMS limits by about one order of magnitude for $SU(2)_L \times U(1)_Y.$06 and two orders of magnitude for $SU(2)_L \times U(1)_Y.$07 (Senol et al., 2021). Third, muon colliders shift the sensitivity frontier most strongly in VBS-dominated settings: the $SU(2)_L \times U(1)_Y.$08 analysis finds projected bounds $SU(2)_L \times U(1)_Y.$09–$SU(2)_L \times U(1)_Y.$10 orders of magnitude stronger than current LHC results, while the same-sign $SU(2)_L \times U(1)_Y.$11TRISTAN projections push many tensor couplings to the $SU(2)_L \times U(1)_Y.$12–$SU(2)_L \times U(1)_Y.$13 TeV$SU(2)_L \times U(1)_Y.$14 level (Zhang et al., 2024, Mukherjee et al., 4 May 2026).

Systematic effects and EFT consistency qualifications remain numerically important. In the CLIC $SU(2)_L \times U(1)_Y.$15 study, the best limits with $SU(2)_L \times U(1)_Y.$16 are approximately improved up to about $SU(2)_L \times U(1)_Y.$17 times better than those obtained with $SU(2)_L \times U(1)_Y.$18, while initial electron beam polarization improves the sensitivity by almost a factor of $SU(2)_L \times U(1)_Y.$19 (Ari et al., 2021). In same-sign $SU(2)_L \times U(1)_Y.$20 scattering at the HL-LHC, imposing invariant-mass cut-offs on the EFT contribution yields explicitly unitarity-safe regions and substantially weakens some coefficients, notably $SU(2)_L \times U(1)_Y.$21 (Éboli et al., 4 Jun 2026). A plausible implication is that future aQGC reporting will increasingly have to separate naïve EFT limits, unitarity-safe limits, and positivity-compatible regions rather than quoting a single interval for each coefficient.