Hypercharge Gauge-Field Form Factors

• Hypercharge gauge-field form factors are effective coupling coefficients from higher-dimensional operators built from Bₘᵤν, mediating neutral-state interactions.
• They inherently generate correlated photon and Z boson couplings after electroweak symmetry breaking due to the Bₘᵤ mixing with W₃.
• Collider studies exploit these form factors to probe dark-sector models, revealing energy-dependent sensitivities and optimization via beam polarization.

Hypercharge gauge-field form factors are coefficients or matrix-element functions associated with couplings to the Standard Model hypercharge gauge field $B_\mu$ or its field strength $B_{\mu\nu}$, rather than directly to the physical photon. In the effective-field-theory treatments emphasized in recent dark-sector collider studies, this choice embeds neutral-state interactions in the electroweak gauge theory and implies correlated couplings to both the photon and the $Z$ boson after electroweak symmetry breaking (Zhang et al., 18 Jul 2025). In broader electroweak applications, related form factors arise as hypercharge-current form factors in Sudakov and threshold calculations, and as loop-induced Higgs couplings to electroweak gauge fields whose photon component is obtained only after mixing (Assi et al., 2020, Phan et al., 2021).

1. Operator definition at the hypercharge level

A central realization of hypercharge gauge-field form factors is the EFT of an electrically neutral Dirac fermion $\chi$ whose interactions with the Standard Model arise only through higher-dimensional operators built from $B_{\mu\nu}$ or $\partial^\nu B_{\mu\nu}$. In the notation of the dark-state collider analyses, the effective Lagrangian is

$$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$

with $$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$ and $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$ (Zhang et al., 18 Jul 2025). The same operator basis appears in the earlier electron-collider study, which takes $\chi$ to be a complete SM singlet and treats these coefficients as hypercharge form factors generated in some unspecified UV completion (Zhang et al., 2022).

Form factor	Hypercharge operator	Mass dimension
Magnetic dipole moment $B_{\mu\nu}$0	$B_{\mu\nu}$1	5
Electric dipole moment $B_{\mu\nu}$2	$B_{\mu\nu}$3	5
Anapole moment $B_{\mu\nu}$4	$B_{\mu\nu}$5	6
Charge radius $B_{\mu\nu}$6	$B_{\mu\nu}$7	6

In these works, $B_{\mu\nu}$8 and $B_{\mu\nu}$9 are quoted in units of the Bohr magneton $Z$0, while $Z$1 and $Z$2 have mass dimension $Z$3 (Zhang et al., 18 Jul 2025). The implicit EFT interpretation is that dipole couplings scale as $Z$4 and anapole or charge-radius couplings as $Z$5, with no specific UV completion imposed (Zhang et al., 18 Jul 2025).

For self-conjugate Majorana states, the operator content is more constrained. The hypercharge-anapole analysis states that the hypercharge anapole moment is the only allowed $Z$6 gauge-invariant coupling between a self-conjugate Majorana dark matter field and the Standard Model hypercharge gauge boson, while ordinary charge and magnetic or electric dipole moments are forbidden (Choi et al., 2024). For spin-$Z$7 Majorana dark matter, the operator is

$Z$8

and the corresponding vertex is proportional to $Z$9 (Choi et al., 2024).

2. Electroweak embedding and correlated photon–$\chi$0 couplings

The defining feature of hypercharge gauge-field form factors is that they are written in terms of the gauge eigenstate $\chi$1, not the physical photon. After electroweak symmetry breaking,

$\chi$2

so that

$\chi$3

Substituting this into the hypercharge-level EFT yields physical photon and $\chi$4-boson couplings obeying

$\chi$5

for $\chi$6 (Zhang et al., 18 Jul 2025). The earlier collider study presents the same relations in the notation $\chi$7, $\chi$8, and analogously for EDM, anapole, and charge-radius coefficients (Zhang et al., 2022).

This embedding is the principal reason for working at the hypercharge level. The photon is not a fundamental gauge eigenstate of the Standard Model, and the cited analyses explicitly motivate hypercharge operators by electroweak gauge invariance and by the automatic generation of $\chi$9-boson operators alongside photon operators (Zhang et al., 18 Jul 2025, Zhang et al., 2022). In the collider formulation of (Zhang et al., 18 Jul 2025), one cannot switch on a pure photon form factor without simultaneously inducing a $B_{\mu\nu}$0 form factor; the relative normalization is fixed by Standard Model mixing.

At low energies, where the $B_{\mu\nu}$1 decouples, the EFT reduces to the familiar electromagnetic form-factor Lagrangian written solely in terms of $B_{\mu\nu}$2, and the $B_{\mu\nu}$3-superscript is often dropped for brevity (Zhang et al., 18 Jul 2025). The 2022 electron-collider study states that when $B_{\mu\nu}$4, the production rate tends to be the same as the one obtained by considering only dark-sector–photon interactions (Zhang et al., 2022). By contrast, near the $B_{\mu\nu}$5 pole or at higher collider energies, the hypercharge-based description is essential because $B_{\mu\nu}$6-exchange and $B_{\mu\nu}$7-$B_{\mu\nu}$8 interference become non-negligible (Zhang et al., 18 Jul 2025, Zhang et al., 2022).

An analogous logic appears in other settings. The hypercharge-anapole study emphasizes that after electroweak symmetry breaking any $B_{\mu\nu}$9 vertex induces both $\partial^\nu B_{\mu\nu}$0 and $\partial^\nu B_{\mu\nu}$1 vertices, with $\partial^\nu B_{\mu\nu}$2 and $\partial^\nu B_{\mu\nu}$3 coefficients fixed by mixing (Choi et al., 2024). In electroweak form-factor calculations, the hypercharge current $\partial^\nu B_{\mu\nu}$4 contributes to physical photon and $\partial^\nu B_{\mu\nu}$5 amplitudes through the same decomposition of $\partial^\nu B_{\mu\nu}$6 and $\partial^\nu B_{\mu\nu}$7 (Assi et al., 2020).

3. Parametrization, momentum dependence, and form-factor scaling

The dark-state EFT analyses use “form factors” in the sense of effective couplings multiplying local higher-dimensional operators, rather than explicit momentum-space functions $\partial^\nu B_{\mu\nu}$8. The 2025 collider study states that it works directly at the level of the effective operators in configuration space and treats the coefficients as constants up to collider scales in the heavy-mediator or contact limit (Zhang et al., 18 Jul 2025). The 2022 electron-collider paper makes the same point: momentum dependence enters through the kinematics of the process and the structure of the operators, not through an explicit nontrivial $\partial^\nu B_{\mu\nu}$9-dependent form factor (Zhang et al., 2022).

This yields characteristic momentum behavior at the vertex level. For a neutral gauge boson $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$0 with momentum $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$1, dipole interactions are schematically proportional to $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$2 or $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$3, while anapole and charge-radius vertices scale as $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$4 and $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$5 because of $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$6 (Zhang et al., 18 Jul 2025). This is why dimension-6 operators acquire an extra power of the hard scale relative to dimension-5 operators in collider observables (Zhang et al., 18 Jul 2025, Zhang et al., 2022).

In the monophoton analyses, the reduced pair-production cross section $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$7 is written in terms of an operator-dependent factor $$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$8. In the 2025 study,

$$\mathcal{L}_{\chi} = \frac12\mu_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\chi B_{\mu\nu} + \frac{i}{2}d_{\chi}^{B}\overline{\chi}\sigma^{\mu\nu}\gamma^{5}\chi B_{\mu\nu} - a_{\chi}^{B}\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}B_{\mu\nu} + b_{\chi}^{B}\overline{\chi}\gamma^{\mu}\chi\partial^{\nu}B_{\mu\nu},$$9

$$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$0

$$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$1

$$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$2

so dimension-6 operators carry an extra power of $$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$3 (Zhang et al., 18 Jul 2025). The earlier collider study presents the same qualitative separation: its $$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$4 factors distinguish dimension-5 magnetic and electric dipoles from dimension-6 anapole and charge-radius operators, and it explicitly notes that high-energy colliders are far more powerful for dimension-6 operators because of the stronger energy growth (Zhang et al., 2022).

A different but related notion of hypercharge form factor appears in the Sudakov and threshold EFT analysis of electroweak currents. There the hypercharge form factor for a fermion $$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$5 is defined by

$$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$6

and the high-energy behavior is governed by universal Abelian Sudakov logarithms proportional to $$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$7 (Assi et al., 2020). The same paper states that the vector-fermion and vector-scalar form factors in its spontaneously broken $$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$8-Higgs model can be mapped to the Standard Model hypercharge sector by the substitutions $$B_{\mu\nu}\equiv \partial_\mu B_\nu-\partial_\nu B_\mu$$9 and $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$0, with non-Abelian $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$1 pieces removed in the pure $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$2 limit (Assi et al., 2020).

4. Collider probes of hypercharge form factors

The most developed phenomenology of hypercharge gauge-field form factors is based on monophoton searches at electron–positron colliders. The signal process is

$$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$3

with $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$4 produced through $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$5-channel $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$6 exchange induced by the hypercharge operators and the observed photon radiated from the initial electron or positron line (Zhang et al., 18 Jul 2025). The ISR-factorized differential cross section used in the 2025 analysis is

$$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$7

with $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$8, $$\sigma_{\mu\nu}\equiv \frac{i}{2}[\gamma_\mu,\gamma_\nu]$$9, $\chi$0, and the improved Altarelli–Parisi radiator function

$\chi$1

(Zhang et al., 18 Jul 2025). The 2022 study uses the same ISR structure for BESIII, STCF, Belle II, LEP, and CEPC (Zhang et al., 2022).

A distinctive collider consequence of the hypercharge formulation is the role of beam polarization. Because the $\chi$2 couples chirally to electrons, the polarized cross sections depend strongly on the left- and right-handed electron couplings $\chi$3 and $\chi$4, while the dominant irreducible background $\chi$5 is strongly suppressed by right-handed electrons and left-handed positrons (Zhang et al., 18 Jul 2025). The 2025 analysis reports that at $\chi$6 TeV ILC, fully right-handed electrons and fully left-handed positrons, $\chi$7, enhance the signal cross section by a factor $\chi$8 and suppress the neutrino background by a factor $\chi$9 compared to unpolarized beams; for realistic ILC polarizations $B_{\mu\nu}$00, the configuration $B_{\mu\nu}$01 is optimal, while at CLIC the preferred option is $B_{\mu\nu}$02 (Zhang et al., 18 Jul 2025).

The same study defines a $B_{\mu\nu}$03 statistic

$B_{\mu\nu}$04

with 95% C.L. limits determined by $B_{\mu\nu}$05 (Zhang et al., 18 Jul 2025). Its benchmark results include, at ILC with $B_{\mu\nu}$06 TeV, $B_{\mu\nu}$07, and $B_{\mu\nu}$08, the limits $B_{\mu\nu}$09 and $B_{\mu\nu}$10 without systematics, or $B_{\mu\nu}$11 and $B_{\mu\nu}$12 for $B_{\mu\nu}$13. Combining all four ILC polarization modes at 1 TeV with a total of $B_{\mu\nu}$14 improves the sensitivity to about $B_{\mu\nu}$15 for dimension-5 operators and about $B_{\mu\nu}$16 for dimension-6 operators. At CLIC with $B_{\mu\nu}$17 TeV and a total of $B_{\mu\nu}$18, the projected sensitivities are $B_{\mu\nu}$19 for EDM and $B_{\mu\nu}$20 for AM. Over most of the accessible mass range, the study finds that ILC and CLIC can probe electromagnetic form factors roughly one to two orders of magnitude below current limits (Zhang et al., 18 Jul 2025).

The earlier electron-collider survey extends the same hypercharge-form-factor framework to lower-energy machines. It finds that BESIII, STCF, and Belle II, operating at several GeV, have leading sensitivity on the corresponding electromagnetic form factors for the mass-dimension 5 operators with dark states lighter than several GeV, but cannot provide competitive upper limits for the mass-dimension 6 operators. Future CEPC, operated on and beyond the $B_{\mu\nu}$21-boson mass with competitive luminosity, can probe unexplored parameter space for mass-dimension 5 operators in the mass region $B_{\mu\nu}$22 GeV and for mass-dimension 6 operators in the mass region $B_{\mu\nu}$23 GeV (Zhang et al., 2022).

A recurrent collider signature of the hypercharge construction is the $B_{\mu\nu}$24-resonant structure of the monophoton spectrum. The 2025 study identifies a resonance at

$B_{\mu\nu}$25

arising because the same hypercharge operator induces both $B_{\mu\nu}$26 and $B_{\mu\nu}$27 channels in $B_{\mu\nu}$28 and $B_{\mu\nu}$29 (Zhang et al., 18 Jul 2025). The 2022 study reaches the same conclusion from a complementary direction: invisible $B_{\mu\nu}$30-decay constraints exist precisely because hypercharge form factors imply $B_{\mu\nu}$31, a channel absent in a photon-only EFT at leading order (Zhang et al., 2022).

5. Generalizations: Majorana, higher spin, and electroweak current form factors

Hypercharge gauge-field form factors extend beyond the Dirac-fermion dark-state EFT. The higher-spin anapole analysis constructs general $B_{\mu\nu}$32 gauge-invariant three-point vertices for two identical massive Majorana particles of spin $B_{\mu\nu}$33, $B_{\mu\nu}$34, $B_{\mu\nu}$35, and $B_{\mu\nu}$36 coupled to the hypercharge gauge boson (Choi et al., 2024). For half-integer spin the minimal leading structure is axial and anapole-like; for integer spin there are two independent derivative structures, one with a Levi-Civita tensor and one without (Choi et al., 2024). After electroweak symmetry breaking, all of these hypercharge vertices induce correlated $B_{\mu\nu}$37 and $B_{\mu\nu}$38 interactions with the same $B_{\mu\nu}$39 and $B_{\mu\nu}$40 relations as in the spin-$B_{\mu\nu}$41 Dirac case (Choi et al., 2024).

That study also provides a combined phenomenological analysis using relic abundance, direct detection, collider searches, and a naive perturbativity bound. Its abstract reports that the scenario with higher-spin dark matter is more stringently constrained than a lower-spin scenario, primarily because of the reduced annihilation cross section and/or the enhanced rate of LHC mono-jet events; it further states that the spin-2 anapole dark matter scenario is almost entirely excluded, while the high-luminosity LHC exhibits high sensitivities in probing spin-1 and spin-$B_{\mu\nu}$42 scenarios except for a tiny parameter range of dark matter mass around 1 TeV (Choi et al., 2024).

Another generalization concerns the ordinary electroweak currents of fermions and scalars. The two-loop EFT analysis of Sudakov and threshold form factors computes vector, scalar, and tensor form factors in a spontaneously broken $B_{\mu\nu}$43-Higgs model and states that its results are mappable to the Standard Model hypercharge sector by simple substitutions $B_{\mu\nu}$44 and Casimirs $B_{\mu\nu}$45 (Assi et al., 2020). In that setting, the basic hypercharge form factor is not a higher-dimensional dark-state operator but the on-shell matrix element of $B_{\mu\nu}$46 between external fermion or scalar states. The paper emphasizes that the EFT structure—hard matching at $B_{\mu\nu}$47, SCET running, and low-scale matching—organizes universal double and single Sudakov logarithms and includes scalar/Higgs contributions at two loops (Assi et al., 2020).

A related but distinct electroweak realization appears in the calculation of $B_{\mu\nu}$48 form factors. That paper computes one-loop off-shell Higgs–photon form factors in $B_{\mu\nu}$49 gauge, but it explicitly frames the result from the viewpoint that the photon is a linear combination of the hypercharge field $B_{\mu\nu}$50 and the neutral weak field $B_{\mu\nu}$51 (Phan et al., 2021). Its discussion states that the $B_{\mu\nu}$52-boson loop encodes the Higgs coupling to the electroweak gauge fields $B_{\mu\nu}$53 and $B_{\mu\nu}$54, and after diagonalizing to $B_{\mu\nu}$55 and $B_{\mu\nu}$56 this yields effective form factors for $B_{\mu\nu}$57, $B_{\mu\nu}$58, and $B_{\mu\nu}$59, with the latter two not computed in that work but structurally analogous (Phan et al., 2021). This is not a hypercharge form factor in the same operator sense as the dark-sector EFTs, but it is an adjacent usage in which electroweak mixing is again indispensable.

6. Effective and geometric realizations beyond local dark-sector EFTs

In broader model-building, hypercharge gauge-field form factors can also refer to effective modifications of hypercharge-like currents and propagators generated by new gauge structure. In the deconstructed-hypercharge model with gauge group

$B_{\mu\nu}$60

the product $B_{\mu\nu}$61 is broken to the diagonal subgroup identified with Standard Model hypercharge, and the orthogonal combination is a massive $B_{\mu\nu}$62 (Davighi et al., 2023). The effective hypercharge coupling obeys

$B_{\mu\nu}$63

while the $B_{\mu\nu}$64 couplings are family non-universal,

$B_{\mu\nu}$65

in family space (Davighi et al., 2023).

That paper explicitly interprets the low-energy theory obtained by integrating out $B_{\mu\nu}$66 as an effective-form-factor description. At energies $B_{\mu\nu}$67, exchange of the heavy boson generates current-current operators

$B_{\mu\nu}$68

and the authors describe this as an EFT “form factor” for the hypercharge-like current, matched onto Warsaw-basis SMEFT operators such as $B_{\mu\nu}$69, $B_{\mu\nu}$70, and the bosonic coefficients

$B_{\mu\nu}$71

(Davighi et al., 2023). In that usage, four-fermion operators encode current-current form factors, Higgs–fermion operators encode vertex form factors, and $B_{\mu\nu}$72 or $B_{\mu\nu}$73 encode oblique form factors of the hypercharge/$B_{\mu\nu}$74 propagator (Davighi et al., 2023).

An even more geometric usage appears in F-theory. There, hypercharge is embedded in the Cartan of $B_{\mu\nu}$75 GUT as

$B_{\mu\nu}$76

and a hypercharge flux must break $B_{\mu\nu}$77 without generating a Stückelberg mass for the hypercharge gauge boson (Braun et al., 2014). The construction achieves this by choosing fluxes that are nontrivial on the GUT divisor but trivial when pushed forward to the bulk. In the explicit compact $B_{\mu\nu}$78 model, the hypercharge $B_{\mu\nu}$79-flux is

$B_{\mu\nu}$80

where each $B_{\mu\nu}$81 is a combination of Cartan $B_{\mu\nu}$82-fibrations over curves $B_{\mu\nu}$83 on the GUT divisor (Braun et al., 2014). The paper states that this four-cycle is trivial in the ambient space but nontrivial in the resolved fourfold, ensuring massless hypercharge and nontrivial GUT breaking (Braun et al., 2014). In this geometric context, what is being controlled is the masslessness, charge assignment, and chiral coupling structure of the hypercharge gauge field rather than a local momentum-space vertex function.

Taken together, these works show that “hypercharge gauge-field form factors” is not a single narrowly defined object but a family of related constructions tied together by a common principle: the hypercharge gauge field $B_{\mu\nu}$84 is the electroweakly consistent starting point, and any physically observable photon or $B_{\mu\nu}$85 interaction must descend from that structure after mixing. In dark-sector EFTs this principle fixes the relation between photon and $B_{\mu\nu}$86 couplings and sharply shapes collider phenomenology (Zhang et al., 18 Jul 2025, Zhang et al., 2022); in higher-spin Majorana theories it constrains the allowed operator basis to anapole-type vertices (Choi et al., 2024); in electroweak perturbation theory it governs the organization of hypercharge-current form factors and their Sudakov evolution (Assi et al., 2020); and in extended gauge or geometric models it reappears as correlated effective operators or flux data controlling the low-energy behavior of hypercharge itself (Davighi et al., 2023, Braun et al., 2014).