Observability of the Higgs boson decay to a photon and a dark photon

Many collider searches have attempted to detect the Higgs boson decaying to a photon and an invisible massless dark photon. For the branching ratio to this channel to be realistically observable at the LHC, there must exist new mediators that interact with both the standard model and the dark photon. In this paper, we study experimental and theoretical constraints on an extensive set of mediator models. We show that these constraints limit the Higgs branching ratio to a photon and a dark photon to be far smaller than the current sensitivity of collider searches.


I. INTRODUCTION
The dark photon is a hypothetical Abelian gauge boson that can mix with the photon [1] and has been extensively studied both experimentally and theoretically. It could, for example, act as a portal to a dark sector or introduce dark matter self-interactions that can potentially solve the small-scale structure problems [2] and the XENON1T anomaly [3,4]. Depending on its decoupling temperature during the evolution of the Universe, the dark photon can affect the big bang nucleosynthesis by altering the effective number of thermally excited neutrino degrees of freedom [5,6]. The dark photon may also modify the stellar energy transport mechanism and thus the cooling of, for example, neutron stars [7].
A potential discovery channel of the dark photon that has received considerable attention is the decay of the Higgs boson to a photon and a massless dark photon. The interest in this channel stems in large part from early phenomenological studies [8][9][10][11][12], which indicated that a corresponding branching ratio as large as 5% was at the time compatible with experimental constraints [8]. This has motivated several collider searches for the Higgs boson decaying to a photon and a dark photon [13][14][15][16]. Presently, the most precise collider bound on this branching ratio comes from Ref. [15] by ATLAS, which uses 139 fb −1 of integrated luminosity at a center-of-mass energy of 13 TeV. This study has obtained an upper limit of 1.8% at 95% confidence level (CL).
For the Higgs boson to be able to decay to a photon and a dark photon, there must exist some mechanism that allows interactions between the dark photon and standard model (SM) particles. In principle, SM particles themselves could interact at tree level with the dark photon. For example, this could be because of kinematic mixing between the weak hypercharge and the gauge boson of a new U (1) ′ gauge group. The problem with this scenario, however, is that the branching ratio of the Higgs boson to photons is small and that interactions between SM particles and a new light gauge boson are constrained to be very small (see, e.g., [17][18][19][20][21][22]) or can sometimes outright be rotated away [23]. The branching ratio of the Higgs boson to a photon and a dark photon would therefore be far smaller than what could realistically be observed at the LHC. A potentially discoverable branching ratio of the Higgs boson to a photon and a dark photon then requires the introduction of new particles that can mediate interactions between the SM sector and the dark photon. Therefore, an observation of this channel would not only verify the existence of the dark photon, but also provide indirect evidence of more new particles.
In this paper, we study constraints on mediators that enable the Higgs boson to decay to a photon and a massless invisible dark photon. Crucially, we demonstrate that these constraints restrict this branching ratio to values considerably lower than current collider limits. To do this, we consider constraints from the Higgs signal strengths, electroweak precision tests, the electric dipole moment (EDM) of the electron, and unitarity. This paper is an extended and more detailed version of Ref. [24].
Despite previous claims, it is not practically possible to obtain a bound on the branching ratio of the Higgs boson to a photon and a dark photon that is completely modelindependent. Certain crucial constraints like the Higgs signal strengths are too elaborate and can be affected by too many factors for a bound to be strictly universal. To counteract this, we will consider an extensive and representative set of mediator models. They constitute the complete set of possible models that respect some very minimal assumptions, which we will present below. These models are, of course, ultimately only benchmarks, but they will clearly illustrate the reasons why, given the above-mentioned constraints, building models that lead to a large branching ratio of the Higgs boson to a photon and a dark photon would be extremely challenging.
We find the following results. In all models considered, the branching ratio of the Higgs boson to a photon and a dark photon is constrained to be below ∼ 0.4%. Furthermore, for many models, this number is only technically allowed because of the absence of collider searches for certain hard-to-miss signatures and would be considerably lower given the existence of such searches. For certain mediators, the constraints are even considerably stronger than 0.4%. 1 This paper is organized as follows. In Sec. II, we introduce our assumptions and the models they allow. Sections III and IV present the constraints and results for the fermion and scalar mediators, respectively. Concluding remarks are presented in Sec. V, including a discussion on the effects of relaxing our assumptions.

II. ASSUMPTIONS AND MODELS
We begin by presenting our assumptions on how the dark photon and the SM can communicate. Of course, there would be ways to violate them and we discuss the implications of this in the conclusion. The models they admit are presented here in a schematic way. The dark photon is referred to as A ′ , the photon as A, and the Higgs boson as h.
Assume a new U (1) ′ gauge group whose gauge boson is A ′ and that all SM particles are neutral under this group. Assume a set of mediators charged under both SM gauge groups and the U (1) ′ . Then, the conditions that we require our mediator models to satisfy are as follows: 1. The Lagrangian is renormalizable and preserves all the gauge symmetries.
2. The Higgs decay to AA ′ can occur at one loop.
3. The mediators are neutral under QCD.
4. The mediators are either complex scalars or vectorlike fermions.
5. There are no more than two new fields.
6. There are no mediators that have a nonzero expectation value or mix with SM fields.
Several comments are in order: • Assumption 1 is a standard requirement of beyond the standard model (BSM) physics. In addition, a non-renormalizable Lagrangian would allow for tree-level decay of the Higgs boson to AA ′ , which would render the analysis trivial but leave unclear whether a reasonable UV completion is possible.
• Assumption 2 is required such that BR(h → AA ′ ) be sufficiently large to observe. If this decay were to take place at an even higher loop level, the branching ratio would generally simply be too small. In principle, a large BR(h → AA ′ ) could be obtained without this assumption being satisfied in the presence of a new non-perturbative sector, but the complicated nature of this would make the analysis less definite and we do not consider it further. This decay is forbidden by gauge invariance to take place at tree level for a renormalizable Lagrangian.
• Assumption 3 is made for two reasons. First, a mediator charged under QCD would be forced to have a mass at the TeV scale, which would make it difficult to obtain a large BR(h → AA ′ ). Second, even if a large BR(h → AA ′ ) could somehow be obtained, it would unavoidably imply a large modification of the Higgs interactions with the gluons. This would be in tension with experimental measurements, especially considering the gluon-fusion cross section is known at O(10%) precision.
• Assumption 4 is made since there are not many well-motivated BSM models containing mediators with spin higher than 1/2 that lead to a sizable BR(h → AA ′ ).
• Assumptions 4-6 are not, in principle, mandatory, but are introduced to keep the number of possible models at a manageable level.
The most important consequence of these assumptions is that they require the Lagrangian to contain a term coupling the Higgs doublet to mediators at tree level. There are only five generic forms such a term can take while respecting our assumptions. Each class of mediator models then corresponds to a different form of the Higgs coupling to mediators. The classes consist of a single fermion class and four scalar classes. The possibilities for these interaction terms are in a schematic form: Fermion: where H is the Higgs doublet, and the different ψ's are vector-like fermions and ϕ scalars. All indices are suppressed and the details will be explained in Secs. III and IV. For each class, the fields can take different quantum numbers and this is why we say that they are classes of models. These models could, of course, be combined, but we will always consider one model at a time for the sake of definiteness and manageability. In the rest of this paper, we will elaborate on each model and study how large of a BR(h → AA ′ ) they can potentially lead to.

III. FERMION MEDIATORS
We present in this section the only class of fermion mediator models that respect the assumptions listed in Sec. II. The different constraints and the allowed BR(h → AA ′ ) are also discussed.

A. Field content, Lagrangian, and mass eigenstates
We begin by introducing the relevant fields and Lagrangian. Consider a vector-like fermion ψ 1 that transforms under a representation of SU (2) L of dimension p = n ± 1, has a weak hypercharge of Y p = Y n + 1/2 and a charge Q ′ under U (1) ′ . Consider another vectorlike fermion ψ 2 that transforms under a representation of SU (2) L of dimension n, has a weak hypercharge of Y n , and a charge Q ′ under U (1) ′ . The Lagrangian that determines the masses of the fermions is (2) where a, b, and c are SU (2) L indices and are summed from 1 (corresponding to the highest weight state) to the size of the corresponding multiplet. We will write explicitly SU (2) L indices and sums when they are nontrivial. The SU (2) L tensord pn abc is uniquely fixed by gauge invariance and given by the Clebsch-Gordan coefficient where (4) We will assume throughout this paper that, as is most commonly the case, the phase convention of the Clebsch-Gordan coefficients is chosen such that they are always real. We will further assume that the Clebsch-Gordan coefficients that correspond to non-physical spin combinations are zero. These two assumptions will lighten the notation. The parameters A L and A R can be complex, and it is easy to verify that there is a complex phase that cannot generally be reabsorbed via field redefinition. Such a physical complex phase will lead to CP-violating interactions. Once the Higgs field obtains a vacuum expectation value (VEV), the Lagrangian L m will contain the mass terms where v ≈ 246 GeV is the VEV of the Higgs field. Introduce the convenient notation and otherwise.
The mass Lagrangian can be written more succinctly as where the mass matrix is The mass matrix can then be diagonalized by introducing the fieldsψ a via where R L and R R are unitary matrices that diagonalize M † M and M M † , respectively. Of course, these matrices only mix particles of identical electric charges and all entries that correspond to mixing of particles of different charges are zero. The fieldsψ a are the mass eigenstates of mass m a , and there are p + n of them. Finally, the interactions of the Higgs boson with the mass eigenstatesψ a are controlled by the terms where Ω is generally neither Hermitian nor diagonal and given by

B. Gauge interactions
We now discuss all relevant gauge interactions of the mass eigenstatesψ a . The interactions of the A/A ′ with ψ a are controlled by where e ′ is the gauge coupling constant of U (1) ′ andQ is the diagonal charge matrix as QED is a vector-like interaction, wherê with (T p 3 ) ab = (p + 1 − 2a)δ ab /2 and similarly for T n 3 . In practice,Q is identical toQ except for a potential reordering of the diagonal elements.
The interactions between the Z boson andψ a is controlled by the terms where B L and B R are, in general, non-diagonal but Hermitian and given by The interactions between the W boson andψ a is controlled by the terms wherê with (T p + ) ab = a(p − a)δ a,b−1 and similarly for T n + .

C. Relevant Higgs decays
The interactions of the mediatorsψ a lead to contributions to the amplitude of the Higgs decay to AA ′ , but also to those of the experimentally constrained decays to AA and A ′ A ′ . The relevant diagrams are shown in Fig. 1a. Irrespective of the mediators, gauge invariance forces the amplitudes to take the forms where p 1 and p 2 are the momenta of the two gauge bosons. The S coefficients are CP conserving, while theS coefficients are CP violating. For the fermion mediators, the coefficients are given at one loop by where S h→AA with C 0 (s 1 , s 12 , s 2 ; m 0 , m 1 , m 2 ) being the scalar threepoint Passarino-Veltman function [26]. 2 The decay widths are then given by where a symmetry factor of 1/2 has been included for the AA and A ′ A ′ final states. The decay of the Higgs boson to a Z boson and either A or A ′ is shown in Fig. 1b and has an amplitude similar in form to Eq. (21), albeit with much more complicated coefficients S h→ZA , S h→ZA , S h→ZA ′ , andS h→ZA ′ due to the possibility of two different mediators running in the loop. There are several crucial results of this section that need to be discussed. First, the presence of the Levi-Civita tensor in the amplitudes is due to the γ 5 Dirac matrix in the Higgs to two mediators vertex. There will be no such term for the scalar cases.
Second, the amplitudes of Eq. (20) are all highly correlated. Because of this, a large BR(h → AA ′ ) will generally lead to either a large branching ratio of the Higgs boson to invisible particles, a large modification of the coupling of the Higgs to photons, or both. The Higgs signal strengths will therefore impose strong constraints on BR(h → AA ′ ). Third, the decay width into two photons will in general contain terms that come from the interference between the SM amplitudes and the mediator amplitudes. When present, these cross terms typically dominate the modifications of the decay width. The presence of cross terms could, in principle, be avoided in two ways. The first way would be for the mediators to only provide a purely imaginary contribution to S h→AA . Since the SM contribution to S h→AA is almost purely real, there would be essentially no interference. However, this turns out to be impossible for the fermion mediators. As seen in Eq. (21), all constants appearing in the S h→AA are real. In addition, the kinematic function S a must be purely real, as bounds from the Large Electron-Positron collider (LEP) prevent charged mediators from being sufficiently light to be able to "cut" the diagram [28,29]. The second way to avoid interference terms would be for the mediators to only contribute toS h→AA . This can indeed be done for fermion mediators and the signal strength constraints can therefore mostly be evaded. However, a large BR(h → AA ′ ) will in this case lead to a large EDM for the electron. The limits on the EDM will force the complex phase to be small and will close this loophole.
Fourth, it is possible at this point to perform a naive estimate of the upper limit on BR(h → AA ′ ) allowed by the Higgs signal strengths. A sufficiently large BR(h → AA ′ ) will imply large Yukawa couplings. This will generally split the masses of the differentψ a and lead to a single mediator dominating the amplitude. Assume as justified above that Im(Ω ii ) = Im(S a ) = 0. Call ∆BR(h → AA) the deviation of BR(h → AA) from its SM value and assume that it is small. Then, the following approximate relation holds: We will see that this approximation holds well, though other constraints will often force BR(h → AA ′ ) to be even smaller. Of course, Eq. (24) assumes that the amplitude is dominated by a single mediator, which may not be a good approximation in more complicated models. Deviations will be observed when multiple mediators contribute comparably to BR(h → AA ′ ), but the limits will remain of the same order of magnitude barring large fine-tuning or elaborate model building.

D. Higgs signal strengths
The constraints associated with the Higgs signal strengths are taken into account by using the κ formalism [30]. Assume a production mechanism i with cross section σ i or a decay process i with width Γ i . The parameter κ i is defined such that where σ SM i and Γ SM i are the corresponding SM quantities. The only two Higgs couplings that are affected at leading order are those associated with AA and AZ. The corresponding κ's are The invisible (A ′ A ′ ) and semi-invisible (A ′ A and A ′ Z) decays of the Higgs boson are taken into account by properly rescaling the signal strengths. The resulting global reduction of the Higgs signal strengths renders constraints from searches for the Higgs decay to invisible particles mostly superfluous and we do not impose them [31]. Constraints are then applied by using the Higgs signal strength measurements of Ref. [32] by CMS, which uses 137 fb −1 of integrated luminosity at 13 TeV centerof-mass energy, and Ref. [33] by ATLAS, which uses 139 fb −1 of integrated luminosity at also 13 TeV. These studies conveniently provide all the information necessary (measurements, uncertainties, and correlations) to produce our own χ 2 fit. The two searches are assumed to be uncorrelated. As we will be interested in two-dimensional scans, a point of parameter space will be considered excluded at 95% CL if its χ 2 satisfies χ 2 − χ 2 min > 5.99, where χ 2 min is the best fit of the model. 3

E. Electron EDM
As explained in Sec. III C, most constraints from the Higgs signal strengths can be evaded by having almost purely imaginary Ω aa couplings. This is, however, constrained by the fact that they would contribute to the EDM of the electron through Barr-Zee diagrams, which can easily be computed by adapting results from the literature. First, the Barr-Zee diagram involving the photon and the Higgs boson of Fig. 2a can be computed by adapting the results of Ref. [34]. This diagram and its variations lead to a contribution of where j(r, s) = 1 r − s r ln r r − 1 − s ln s s − 1 .
Second, the contribution from the diagrams involving a Z boson and a Higgs boson like that of Fig. 2b can also be computed using Ref. [34] and lead to where 3 Technically speaking, it is possible for the mediator contributions to S h→AA to be about −2S h→AA

SM
. This would mostly avoid the bounds from the Higgs signal strengths and could in principle lead to a larger BR(h → AA ′ ). It however requires very exotic circumstances and a very large amount of fine-tuning. As such, we will not consider such cases. with and Third, the contribution from the diagrams involving two W bosons like in Fig. 2c can be computed by adapting the results of Ref. [35]. The only subtlety is that the fermions can potentially flow in both directions in the fermion loop. This was not the case in Ref. [35]. Thankfully, both diagrams can easily be related and the sum of the two diagrams is where r a = m 2 a /m 2 with The total EDM is then the sum of Eqs. (27), (29), and (34). In practice, the contribution from Eq. (27) generally overwhelmingly dominates when BR(h → AA ′ ) is close to its maximally allowed value. In this case, the diagram of Fig. 2b is suppressed by the fact that g V ee is small due to an accidental partial cancellation and the diagram of Fig. 2c is suppressed because the Yukawa couplings A L and A R are simply much larger than g. The upper limit on the electron EDM that we use is |d e | < 4.1×10 −30 e cm at 90% CL [36]. 4 Finally, when BR(h → AA ′ ) is close to the upper limit that we find, the electron EDM generally constrains A L and A R to have a phase difference very close to 0 (or π). This makes it essentially redundant to consider any other CP-violating observable.

G. Unitarity
As a final constraint, the parameters A R and A L are bounded by unitarity. Consider a given scattering between mediators via Higgs exchange and its amplitude M. The latter can be expanded in partial waves as where P ℓ (cos θ) are the Legendre polynomials.
In the high energy limit, we can work directly with ψ 1 and ψ 2 . It is then simply a question of computing the a 0 factor of every possible scattering ψ Then the matrix of a 0 for the scattering ψ where each row corresponds to the same incoming ψ pair, each column to the same outgoing ψ a 1 ψ b 2 pair, and F cd ab is a block given by and corresponds to a 0 for different combinations of helicity in the basis of (↑↑, ↓↓). Call a eig 0 the set of eigenvalues of a mat 0 . Unitarity then imposes max Re a eig This can be verified to reduce to the surprisingly simple requirement that  [28,29]. Taken and expanded from Ref. [24].

H. Results
Having explained how the different constraints are imposed, we now present the limits on BR(h → AA ′ ) for the fermion case. The parameter space is sampled using a Markov chain with the Metropolis-Hastings algorithm. As the results are only dependent on Q ′ and e ′ via the product Q ′ e ′ , the limits on BR(h → AA ′ ) are independent of the choice of Q ′ and we require |Q ′ e ′ | < √ 4π. To maximize the number of points near the limits and thus reduce the necessary number of simulations, a prior proportional to BR(h → AA ′ ) 2 is assumed. We have verified that the results are independent of the sampling algorithm (and prior), assuming it covers all of the relevant parameter space and sufficient statistics.
The limits on BR(h → AA ′ ) are shown in Fig. 3 for different p, n, and Y n as a function of the mass of the lightest electrically charged mediator of mass m min c . As can be seen, there is an upper bound of about 0.4% that is never crossed. This limit comes from the Higgs signal strengths and is consistent with the estimate of Sec. III C. For many mediators, the limit on BR(h → AA ′ ) suddenly starts to drop from its maximal value above a certain threshold in m min c . This is simply where the constraints from the oblique parameters or unitarity become more powerful than those of the Higgs signal strengths. This is because increasing the masses of the mediators requires larger couplings to maintain a large BR(h → AA ′ ), which conflicts with these constraints. Some mediators are also sufficiently constrained by other bounds that the lower mass plateau is never reached. Obtaining a large BR(h → AA ′ ) requires |Q ′ e ′ | to be considerably larger than the |Q aa e| of a mediatorψ a . This is difficult for mediators with large electric charges, as |Q ′ e ′ | has an upper limit, and is why models with mediators of large electric charges are very constrained. Without the constraints from the electron EDM, the limits on BR(h → AA ′ ) would be far less stringent as interference terms with the SM contributions could be avoided in the decay of the Higgs to two photons. As explained in Sec. III C, the electron EDM forces the complex phase to be close to 0 (or π) which imposes the presence of interference terms.
An important point to mention is that the limits on BR(h → AA ′ ) depend on the mass of the mediators. In theory, there should be a lower limit on the mass of the charged mediators coming from collider searches. If this mass were higher than the threshold in m min c , the bound on BR(h → AA ′ ) would be tightened. In principle, the mediators could decay to exotic channels that have not been probed yet. It is therefore technically impossible to determine a model-independent bound on their masses. However, it would be very difficult for a charged particle of less than a few hundred GeV not to have been observed at the LHC by now. As such, obtaining a large BR(h → AA ′ ) requires a charged light particle that somehow would have avoided detection.

IV. SCALAR MEDIATORS
We now proceed to analyze the four scalar cases of Sec. II. With a few exceptions, the treatment is similar to the fermion case up to minor technical details.

A. Field content, Lagrangian, and mass eigenstates
We begin by introducing in more detail the four scalar models. To each model will correspond a series of masses m a , a rotation matrix R, and a matrix of Higgs couplings Ω. The results of the subsequent sections will be expressed in terms of these quantities.

Scalar case I
Consider a complex scalar ϕ 1 that transforms under a representation of SU (2) L of dimension p = n±1, and has a weak hypercharge Y p = Y n + 1/2 and a charge under U (1) ′ of Q ′ . Consider another complex scalar ϕ 2 that transforms under a representation of SU (2) L of dimension n and has a weak hypercharge of Y n and a charge under U (1) ′ of Q ′ . The Lagrangian that controls the masses of the scalars is (46) The SU (2) L tensord pn abc is given bŷ where (48) The parameter µ can be made real by a field redefinition. Once the Higgs field obtains a VEV, the Lagrangian L 1 m will contain the mass terms (49) Let us introduce the notation and otherwise.
(51) The mass Lagrangian can be written as where the mass matrix is The mass matrix can be diagonalized by introducinĝ The fieldsφ a are the mass eigenstates, and there are p+n of them. Their interactions with the Higgs boson are then described by where Ω is a Hermitian matrix given by

Scalar case II
Consider a complex scalar ϕ that transforms under a representation of SU (2) L of dimension n, and has a weak hypercharge Y n and a charge under U (1) ′ of Q ′ . The Lagrangian that controls the masses of the scalars is The SU (2) L tensord nr abcd is given bŷ where M is summed over {−J, −J + 1, −J + 2, ..., +J} and (59) There are generally two possible ways to contract the SU (2) L indices, and each possible contraction is taken into account by its own coefficient λ r . The only exception to this is when ϕ is a singlet, in which case only the r = n + 1 term leads to a non-zerod nr abcd tensor. The parameters λ r are always real. Once the Higgs obtains a VEV, the Lagrangian L 2 m will contain the mass terms With the notation the mass Lagrangian can be written as where the mass matrix is The mass matrix can be diagonalized by introducing The fieldsφ a are the mass eigenstates, and there are n of them. Their interactions with the Higgs boson are then described by where Ω is a real diagonal matrix given by We note that the mixing matrix could simply be taken as the identity in this case, as there are never multiple states with the same electric charge. We keep R to maintain a uniform notation and order the particles by mass.

Scalar case III
Consider a complex scalar ϕ 1 that transforms under a representation of SU (2) L of dimension p ∈ {n − 2, n, n + 2}, and has a weak hypercharge Y p = Y n and a charge under U (1) ′ of Q ′ . Consider another complex scalar ϕ 2 that transforms under a representation of SU (2) L of dimension n and has a weak hypercharge Y n and a charge under U (1) ′ of Q ′ . The Lagrangian that controls the masses of the scalars is where R = {n − 1, n + 1} ∩ {p − 1, p + 1}. The SU (2) L tensord pnr abcd is given bŷ where M is summed over {−J, −J + 1, −J + 2, ..., +J} and (69) If p = n, there are two possible contractions of the SU (2) L indices, the only exception being if ϕ 1 and ϕ 2 are both singlets in which case only the r = p + 1 term leads to a non-zerod pnr abcd . When p and q differ by 2, only one term is allowed. One λ r can be made real by field redefinition. Therefore, there will be a complex phase that cannot generally be reabsorbed when p = n, but not otherwise. Once the Higgs field obtains a VEV, the Lagrangian L 3 m will contain the mass terms The mass Lagrangian can be written as where the mass matrix is (74) The mass matrix can be diagonalized by introducinĝ The fieldsφ a are the mass eigenstates, and there are p+n of them. Their interactions with the Higgs boson are then described by where Ω is a Hermitian matrix given by Scalar case IV Consider a complex scalar ϕ 1 that transforms under a representation of SU (2) L of dimension p ∈ {n − 2, n, n + 2}, and has a weak hypercharge Y p = Y n +1 and a charge under U (1) ′ of Q ′ . Consider another complex scalar ϕ 2 that transforms under a representation of SU (2) L of dimension n and has a weak hypercharge Y n and a charge under U (1) ′ of Q ′ . Assume that n and p are not both 1. The Lagrangian that controls the masses of the scalars is (78) The SU (2) L tensord pn abcd is given bŷ where M 1 is summed over {−1, 0, 1} and There is only a single possible contraction, as the two Higgs doublets can only be combined in a single nontrivial way. This does not occur for cases II and III because they contain both the Higgs doublet and its conjugate. Because there is only one coefficient, λ can always be made real by a field redefinition. Once the Higgs field obtains a VEV, the Lagrangian L 4 m will contain the mass terms where the mass matrix is The mass matrix can be diagonalized by introducinĝ The fieldsφ a are the mass eigenstates, and there are p+n of them. Their interactions with the Higgs boson are then described by where Ω is a Hermitian matrix given by

B. Gauge interactions
The interactions of A/A ′ withφ a are controlled by whereQ is a diagonal charge matrix given bỹ and Cases I, III, IV: with (T p 3 ) ab = (p + 1 − 2a)δ ab /2 and similarly for T n 3 . The interactions between the Z boson andφ a are controlled by the terms where B is Hermitian but, in general, non-diagonal and given by Cases I, III, IV: Case II: The interaction amongφ a , a A/A ′ boson, and a Z boson is controlled by the terms where we note that [Q, B] = 0. The interactions between the W boson andφ a is controlled by the terms with (T p + ) ab = a(p − a)δ a,b−1 and similarly for T n + .

C. Relevant Higgs decays
The mediatorsφ a lead to an amplitude for the Higgs decaying to AA, AA ′ and A ′ A ′ . The relevant diagrams are shown in Figs. 4a and 4b. Labelling the momenta of the two gauge bosons as p 1 and p 2 , the amplitudes once again take the form with the coefficients now given at one loop by The decay of the Higgs boson to a Z boson and either A or A ′ is shown in Figs. 4c and 4d and has a similar form to Eq. (97), albeit with far more complicated coefficients.
We reiterate that none of these models contribute tõ S h→AA . In addition, the contributions to S h→AA are all forced to be real. It therefore means that these models will unavoidably lead to interference terms with the SM contributions. As such, they will not be able to circumvent the constraints of the Higgs signal strengths like the fermion mediators potentially could have. As such, we will not study the electron EDM for the scalar models.

D. Higgs signal strengths
The constraints associated with the Higgs signal strengths are applied in the same way as for the fermion case.
(101) The constraints are applied as for the fermion mediators.

F. Unitarity
The unitarity constraints coming from scattering two Higgs bosons to two ϕ's can be obtained for the fermion mediators, albeit they are much easier to account for because of the absence of polarization for scalars. Including in a 0 a factor of 1/ √ 2 for identical incoming particles, the constraints are simply given by Note that, for case IV, since there is a single coefficient, the constraint can be simplified to |λ| < 8π 6 p .
No unitarity bound can be generally applied on µ for case I. Furthermore, a bound can be set on Q ′ e ′ by adapting the results of Ref. [46]. This gives where q = n + p for cases I, III and IV and q = n for case II.

G. Results
Having introduced the relevant constraints, we now discuss the limits on BR(h → AA ′ ) for the scalar models. The sampling is performed as for the fermion mediators. Note that it is sometimes possible for a mediator to obtain a negative mass square. This would result in the breaking of U (1) ′ and potentially the electromagnetic group. Such points are discarded because they do not correspond to the desired massless dark photon scenario and are excluded if they break electromagnetism.
The plots in Fig. 5 show the upper bounds on BR(h → AA ′ ) for models I-IV for different combinations of their gauge quantum numbers. Several comments are in order.
• As can be seen, BR(h → AA ′ ) again often exhibits a plateau in the low mass regime. In the cases considered, it is still at best 0.4%. However, the plateau is sometimes lower because of multiple particles having similar masses. Since all mediators have identical U (1) ′ charges but not all of them are always electrically charged, this tends to lead to a larger Higgs branching ratio to invisible particles for a given BR(h → AA ′ ).
• Some models are subject to far stronger constraints because of the oblique parameters or unitarity.
• Case I is similar to the fermion mediators and the limits are qualitatively similar.
• Case II can potentially avoid contributions to the oblique parameters by an appropriate choice of couplings. In practice, this is by only including δ ab δ cd H a † H b ϕ c † ϕ d . This scenario, however, leads to particles of similar masses that all contribute to the invisible decay of the Higgs boson. It is easy to verify that degenerate masses would have which is ≲ 0.4%. However, Eq. (105) results in a limit of 0 when Y = 0. In this case, obtaining a large BR(h → AA ′ ) requires breaking the mass degeneracy which reintroduces the limits from the oblique parameters. For a general Y , it is not trivial whether degenerate or non-degenerate masses lead to a larger BR(h → AA ′ ).
• Case III is mostly similar to case II. The only difference is when p ̸ = n. In this case, the bounds from the oblique parameters cannot be evaded and BR(h → AA ′ ) is strongly constrained.
• Case IV is especially constrained because it leads to a negative contribution to the T parameter. This is why we only include two examples.

V. CONCLUSION
Many collider searches have been performed in the hope of observing the Higgs boson decaying to a photon and a dark photon. For this decay to have a branching ratio realistically observable at the LHC, there must exist new mediators that communicate between the SM particles and the dark photon. In this paper, we have studied the constraints from the Higgs signal strengths, oblique parameters, EDM of the electron, and unitarity on a large set of mediator models. The models are only asked to satisfy a very minimal set of requirements. We have found that for these models, BR(h → AA ′ ) is generally constrained to be below 0.4%, which is far lower than the current collider limit of 1.8%. Furthermore, obtain-ing this 0.4% requires relatively light charged mediators that would have somehow evaded existing searches. For some models, the bounds are even more stringent.
In addition to these constraints, a large BR(h → AA ′ ) imposes several requirements on models that might not be subjectively very pleasing. First, it requires some couplings to be very large. In hindsight, this is unsurprising. The top loop contributes relatively little to the h → AA decay width compared to W loops and BR(h → AA) is still only of O(0.1%). This is despite the fact that the top has a Yukawa coupling with the Higgs of ∼ 1 and a mass of only ∼ 173 GeV. Therefore, obtaining a large BR(h → AA ′ ) requires large couplings between the Higgs and the mediators and also a large dark electric charge e ′ . In the case of Yukawa couplings, they can be of an order of a few or larger. Second, this also leads to the presence of a Landau pole for U (1) ′ at low energies, sometimes as low as the TeV scale.
Because of both these model requirements and the experimental constraints, we believe it would be very challenging to observe the Higgs boson decaying to a photon and a massless dark photon at the LHC.
Nonetheless, there could, in principle, be a few ways to obtain an observable BR(h → AA ′ ) by breaking some of our assumptions. Namely, it could be possible to have multiple mediators with different electric charges, U (1) ′ charges or couplings with the Higgs boson. In this case, it might be possible to have destructive interference in channels that are particularly constrained, like h → AA and h → A ′ A ′ , but constructive one in h → AA ′ . This could, in principle, alleviate the signal strength constraints. However, reaching an unexcluded BR(h → AA ′ ) as high as 1.8% would surely require a considerable amount of fine-tuning. Whether a channel that requires considerable fine-tuning to simply be potentially observable is worth dedicated experimental searches is certainly debatable.