Scalar decays to $\gamma\gamma$, $Z\gamma$, and $W\gamma$ in the Georgi-Machacek model

We compute the decay widths for the neutral and singly-charged Higgs bosons in the Georgi-Machacek model into the final states $\gamma\gamma$, $Z\gamma$, and $W\gamma$. These decays are most phenomenologically interesting for the fermiophobic custodial fiveplet states $H_5^0$ and $H_5^{\pm}$ when their masses are below threshold for decays into $WW$, $ZZ$, or $WZ$. We study the allowed branching ratios into these final states using scans over the allowed parameter space, and show how the model can be constrained by LEP searches for a fermiophobic Higgs boson decaying to two photons. The calculation involves evaluating one-loop diagrams in which the loop contains particles with two different masses, some of which do not appear in the existing literature. We give results for these diagrams in a form convenient for numerical implementation using the LoopTools package.


I. INTRODUCTION
Since the discovery of a Standard Model (SM)-like Higgs boson at the CERN Large Hadron Collider (LHC) [1], there has been considerable interest in models with extended Higgs sectors to be used as benchmarks for LHC searches for physics beyond the SM. One such model is the Georgi-Machacek (GM) model [2,3], which adds isospin-triplet scalar fields to the SM in a way that preserves custodial SU (2) symmetry. This model is interesting because the isospin triplets can make a non-negligible contribution to electroweak symmetry breaking. Its phenomenology has been studied extensively , and its parameter space has been constrained using the perturbativity and vacuum stability of the scalar potential [8,15,23], the electroweak oblique parameters [16,17,21,27], Z-pole and B-physics observables [7,15,21,27], and direct collider searches [25,[35][36][37][38][39]. The GM model has also been incorporated into Little Higgs [40,41], supersymmetric [42][43][44], and neutrino seesaw [26] models. Extensions with an additional isospin doublet [45] and a singlet scalar dark matter candidate [46] have also been considered, as have generalizations of the model to include higher-isospin scalars [13,37,[47][48][49].
The most distinct phenomenological feature of the GM model is the presence of a custodial fiveplet of scalars, (H ++ 5 , H + 5 , H 0 5 , H − 5 , H −− 5 ). These scalars are fermiophobic and couple at tree level to W or Z boson pairs with a strength proportional to the isospin-triplet scalar fields' vacuum expectation value (vev). Direct searches at the LHC for these custodial-fiveplet scalars have so far focused on scalar masses above 200 GeV [35,38,39] (see also Refs. [50,51]), where they decay predominantly into pairs of on-shell vector bosons. For lower masses, the tree-level decays are forced off-shell and the loop-induced decays of H ± 5 → W ± γ and H 0 5 → γγ, Zγ can become important. These final states offer sensitive new experimental probes. The diphoton decay mode can also be used to take advantage of existing limits on the production of scalars decaying to photon pairs from the CERN Large Electron-Positron (LEP) collider [52] and the LHC [53].
Our goal in this paper is to compute the loop-induced decay widths of the scalars in the GM model and study their behavior over the model's parameter space, focusing on scalar masses below 200 GeV. This is made nontrivial by the fact that some diagrams appear in the decays H 0 5 → Zγ and H ± 5 → W ± γ that have not previously been computed in the literature. Some of these new diagrams also appear in the custodial-triplet scalar decay H ± 3 → W ± γ; we discuss this process for completeness although it is of less phenomenological interest because decays of H ± 3 to fermion pairs tend to dominate its branching ratios.
The challenge is diagrams in which the loop contains particles with two different masses. Such "heterogeneous" loop diagrams are forbidden by gauge invariance in the familiar decays of the SM Higgs boson to two photons or two gluons; they are absent in the SM Higgs decay to Zγ due to custodial symmetry. Heterogeneous diagrams appear in two Higgs doublet models in the decay H ± → W ± γ; these have been computed in Refs. [54][55][56]. 1 In the two Higgs doublet model, the contributing diagrams involve top and bottom quarks, H ± and a neutral scalar h 0 or H 0 , and W ± and a neutral scalar h 0 or H 0 . Explicit results for these loop diagrams have been given in Ref. [56] as integrals over Feynman parameters. For ease of numerical implementation, we recalculate them here in terms of the one-loop Passarino-Veltman integrals [59] in the notation used by the LoopTools package [60]. Our results agree with those of Ref. [56].
The GM model admits additional heterogeneous diagrams not present in two Higgs doublet models. These include diagrams that involve W ± and Z, Z and H ± 5 , W ± and H ±± 5 , and W ± and H ± 5 . These contribute to the decays H + 5 → W + γ, H + 3 → W + γ, and H 0 5 → Zγ. By contributing to H 0 5 → Zγ, the new diagrams can affect the branching ratio of H 0 5 → γγ (though we find that the effect is numerically small). We compute these new loop diagrams and give explicit results as integrals over Feynman parameters as well as in terms of the one-loop Passarino-Veltman integrals in the notation used by the LoopTools package.
With the new loop diagrams in hand, we implement the full one-loop decays H 0 5 → Zγ, H ± 5 → W ± γ, and H ± 3 → W ± γ into a private code based on GMCALC 1.2.0 [61] (all other decays to γγ and Zγ are already implemented in the public version of the code) and perform parameter scans to study the allowed range of branching ratios after imposing the theoretical and experimental constraints on the model. We show that a large fraction of the parameter space with H 5 masses below about 110 GeV is excluded by LEP searches for fermiophobic Higgs production in e + e − → ZH 0 5 with H 0 5 → γγ [52]. Our results for the H 0 5 → γγ branching ratio can also be combined with scalar pair-production cross sections to impose limits from LHC diphoton searches as in Ref. [53]; we leave this to future work. These one-loop decays will be included in GMCALC 1.3.0 and higher. This paper is organized as follows. In Sec. II we present the results of the one-loop diagram calculations for the decays of the GM scalars to V γ. In Sec. III we assemble the familiar loop contributions with these new diagrams to compute the decay amplitudes for neutral scalars into γγ and Zγ and for singly-charged scalars into W γ in the GM model. In Sec. IV we present numerical scans over the viable GM parameter space and apply the LEP limit on fermiophobic Higgs decays into two photons to constrain the model. We summarize our conclusions in Sec. V. For completeness, in Appendix A we review the Lagrangian and physical spectrum of the GM model, in Appendix B we collect the Feynman rules for the GM model scalars that we use in this paper, and in Appendix C we summarize the LoopTools conventions for the one-loop Passarino-Veltman integrals used in our results. Finally in Appendix D we give some details of the calculations in the 't Hooft-Feynman gauge of processes involving Goldstone bosons or ghosts.
is forced by electromagnetic gauge invariance to take the form [56] where q and k are the momenta and ε µ (q) and ε ν (k) are the polarization vectors of the photon and the gauge boson V , respectively. The resulting decay partial width is where V = γ, Z, or W + . Here η V is a symmetry factor that accounts for identical particles in the final state, with η γ = 2 and η Z = η W = 1.
In calculating the scalar formfactor S, we follow the approach used by Ref. [56] for the calculation of the one-loop amplitudes contributing to H + → W + γ in the Yukawa-aligned two Higgs doublet model (2HDM) [62]. Ref. [56] employed the clever strategy of computing only the coefficient of k µ q ν in order to determine the form factor S. Neglecting all terms proportional to g µν significantly reduces the complexity of the calculations, as it reduces the number of Feynman diagrams that must be considered to those illustrated in Fig. 1 and removes the need for renormalization. The pseudoscalar formfactorS receives contributions only from fermions in the loop as shown in the first diagram of Fig. 1.
To fix the signs of the charges of the particles appearing in the triangle diagrams of Fig. 1, we adopt the convention that H i is the incoming parent scalar and V is an outgoing final-state vector boson. The particle in the loop with subscript 1 propagates from the H i vertex to the V vertex, while the particle in the loop with subscript 2 propagates from the V vertex, through the photon vertex, back to the H i vertex.
The first diagram in Fig. 1 has been computed in Ref. [56]. The second diagram has been computed in Ref. [56] for the special case that H i and s 2 have the same mass. The fourth diagram has been computed in Ref. [56] for the special case that X 2 and V have the same mass. To our knowledge, the remaining diagrams have not appeared in the literature.
In what follows we give our results for each diagram in the context of the GM model. The results given in terms of integrals over Feynman parameters were computed in Unitarity gauge, while the results given in terms of LoopTools functions were computed in 't Hooft-Feynman gauge including all relevant additional diagrams involving Goldstone bosons or ghosts. We used dimensional regularization to handle divergences, which cancel in the final results. The LoopTools conventions for the three-point integrals are summarized in Appendix C. In each case we checked numerically that the two approaches agree to within the (percent-level) precision of our numerical integration over the Feynman parameters.
The decays of the scalars H 0 5 → Zγ, H 0 3 → Zγ, and H + 5 → W + γ have also been checked numerically using MadGraph5 aMC@NLO [63] with the GM model renormalized by NLOCT [64]. It should be noted that these tools compute the full amplitude, including the coefficient of the g µν term in Eq. (1), and so the electroweak renormalization of the model including tadpole and mixing counterterms is needed to obtain finite results. Again, the numerical results agree at the percent level. The decay H + 3 → W + γ has not been checked using MadGraph5 aMC@NLO because further development in the handling of contributions with on-shell cuts is still needed.

A. Fermion loop diagram
The first diagram in Fig. 1 contributes to H + 3 → W + γ with f 1 = t, f 2 = b, and with f 1 =b, f 2 =t. The calculation is exactly as in Ref. [56] with a translation from the Yukawa-aligned 2HDM coupling notation to the appropriate FIG. 1. One-loop Feynman diagrams contributing to the process Hi → V γ in the GM model. Here V = γ, Z or W ± , f1,2 denote fermions, s1,2 denote scalars, and X1,2 denote W or Z bosons.
cot θ H dependence in the GM model (see Appendix A). The appropriate couplings are as given for the Type I 2HDM in Ref. [56] with cot β → tan θ H , i.e., where k 2 = M 2 W and q 2 = 0 are the final-state particles' invariant masses. We can obtain a check of these formulas by artificially setting m b = m t ≡ m f . In this limit the H +t b coupling becomes purely pseudoscalar and the CP-even formfactor A H + 3 W γ f f vanishes, while the CP-odd formfactor reduces tõ where in this case τ f = 4m 2 f /m 2 3 , λ f = 4m 2 f /k 2 , and I 2 is the function that appears in the usual calculation of the fermion loop contribution to a CP-odd scalar decaying to Zγ [6] [see Eq. (39)].

B. Scalar loop diagram
The second diagram in Fig. 1 contributes with two different scalar masses in the loop to H + 3 → W + γ. (For H 0 5 → Zγ and H + 5 → W + γ the scalar loop diagrams involve three scalars with the same masses.) We define triple-scalar and vector-scalar-scalar couplings, with all particles incoming, in terms of the Feynman rules such that the triple-scalar vertex Feynman rule is −iC His * 1 s2 and the vector-scalar-scalar vertex Feynman rule is ieC V * s1s * 2 (p 1 − p 2 ) µ , where p 1 and p 2 are the incoming momenta of s 1 and s * 2 , respectively, and V * is the incoming particle corresponding to outgoing vector boson V . The photon coupling to two scalars is fixed by the Feynman rule ie 2 Q s (p s − p s * ) µ , where p s is the incoming momentum of the incoming scalar s, p s * is the incoming momentum of outgoing scalar s (or incoming s * ), and Q s is the electric charge in units of e of the scalar s. Explicit formulas for these couplings in the GM model are given in Appendix B.
The formfactor is given as an integral over Feynman parameters by where This agrees with the corresponding result of Ref. [56] for the special case m s2 = m Hi .
In terms of the LoopTools functions, the scalar loop contribution is given by where k 2 = M 2 V and q 2 = 0 are the final-state particles' invariant masses.
In the limit m s1 = m s2 ≡ m s , this expression reduces to where in this case τ s = 4m 2 s /m 2 Hi , λ s = 4m 2 s /k 2 , and I 1 is the function that appears in the usual calculation of the scalar loop contribution to a CP-even scalar decaying to Zγ [6] [see Eq. (38)].

C. Vector-scalar-scalar loop diagram
The third diagram in Fig. 1 contributes to H + 3 → W + γ, H + 5 → W + γ, and H 0 5 → Zγ. For this diagram we need the scalar-vector-vector coupling, which is defined by the Feynman rule ie 2 C s * XV * g µν , again with all particles incoming. Explicit expressions are given in Appendix B.
The formfactor is given as an integral over Feynman parameters by where In terms of the LoopTools functions, the vector-scalar-scalar loop contribution is given by where k 2 = M 2 V and q 2 = 0 are the final-state particles' invariant masses.

D. Scalar-vector-vector loop diagram
The fourth diagram in Fig. 1 contributes to H + 3 → W + γ, H + 5 → W + γ, and H 0 5 → Zγ. The formfactor is given as an integral over Feynman parameters by where This agrees with the corresponding result of Ref. [56] for the special case In terms of the LoopTools functions, the scalar-vector-vector loop contribution is given by where k 2 = M 2 V and q 2 = 0 are the final-state particles' invariant masses.

E. Vector loop diagram
The fifth and sixth diagrams in Fig. 1 contribute with two different gauge boson masses in the loop to H + 5 → W + γ.

The masses and couplings are given by m Hi
The formfactor is given as an integral over Feynman parameters by where The numerator can be equivalently expressed in the following, equally horrible, form, In terms of the LoopTools functions, the vector loop contribution to H + 5 → W + γ is given by where k 2 = M 2 W and q 2 = 0 are the final-state particles' invariant masses. In this case we have computed the specific diagrams appearing in H + 5 → W + γ rather than a more generic case because we had to use the relations among the gauge, Goldstone, and ghost couplings to simplify the result of the 't Hooft-Feynman gauge calculation. We note also that these simplifications make use of the final-state W boson mass, so that this result is good for on-shell decays only.

III. SCALAR DECAYS TO V γ IN THE GM MODEL
With the loop functions in hand, we now assemble all the contributions to compute the amplitudes for decays of the neutral scalars in the GM model into γγ and Zγ and the singly-charged scalars into W γ.

A. Decays to two photons
In the GM model, the neutral scalars h, H, H 0 3 , and H 0 5 can decay to two photons through the usual loop-induced processes. Electromagnetic gauge invariance ensures that only a single particle runs around the loop in each diagram, so that the decay amplitudes S andS can be expressed in terms of the familiar functions given, e.g., in Ref. [6]. The contributing particles in the loop are summarized in Table I. Note that h, H, and H 0 5 are CP-even and hence decay via the formfactor S, while H 0 3 is CP-odd and decays via the formfactorS. The CP-odd formfactor receives contributions only from fermions in the loop. The state H 0 5 is fermiophobic and hence the fermion loop does not contribute to its decay. Hi where N cf = 3 for quarks and 1 for leptons, Q j is the electric charge of particle j in units of e, and the sums run over all fermions and scalars that can propagate in the loop for the parent scalar H i . In practice the charged scalars are s = {H + 3 , H + 5 , H ++ 5 } and we keep only the top quark contribution to the fermion loop, f = t. The coupling coefficients β Hi j are defined as, for a propagating W boson, fermion f , and scalar s, respectively. The couplings C ijk are given in Appendix B. In the case of the W boson and fermion loops, these factors β Hi W,f are equal to the usual ratios κ Hi W,f of the scalar coupling to W W or ff normalized to the corresponding SM Higgs coupling as described in Ref. [65]. Note that β The loop factors are given in terms of the usual functions for particles of spin 0, 1/2 and 1 [6], where τ j = 4m 2 j /m 2 Hi and with η ± = 1 ± √ 1 − τ . For the CP-odd scalar H 0 3 , the decay is described by Eq. (2) with S = 0 and [6] where is defined as in Eq. (30). Again we will include only the top quark in the loop,

B. Decays to Zγ
The neutral scalars h, H, H 0 3 , and H 0 5 can also decay to Zγ through a loop. For this decay, loops involving particles with two different masses can appear, because the Z boson (unlike the photon) can couple to two different-mass particles. These new diagrams arise only in the decay of the custodial-fiveplet scalar H 0 5 → Zγ, because custodial symmetry is enough to forbid them in the decays of the custodial-singlet scalars h and H, and the CP-odd scalar H 0 3 → Zγ decay receives contributions only from loops of fermions, whose couplings to the Z boson are flavor-diagonal. As before, the CP-even scalars h, H, and H 0 5 decay via the formfactor S, while the CP-odd scalar H 0 3 decays via the formfactorS. The state H 0 5 is fermiophobic and hence the fermion loop does not contribute to its decay. The contributing particles in the loop are summarized in Table II.
For the custodial-singlet CP-even scalars h and H, the decay is described by Eq. (2) withS = 0 and A HiZγ where T 3L f is the third component of isospin of the left-handed fermion f (T 3L f = 1/2 for the top quark), τ j = 4m 2 j /m 2 Hi , λ j = 4m 2 j /M 2 Z , and the loop functions are The function f (τ ) was given in Eq. (32), and with η ± defined as for f (τ ). The couplings C Zss * are given in Appendix B. 4 Here we have rewritten A in a form that clearly separates the kinematic dependence of the loop diagrams on M W and M Z (encoded in λ W and τ W ) from their dependence on the triple-and quartic-gauge couplings, following Appendix B of Ref. [23].
where the CP-odd fermion loop function is [6] A HiZγ with I 2 given by Eq. (39) and β The singly-charged scalars H + 3 and H + 5 of the GM model can decay to W + γ through a loop. These states are not CP eigenstates and hence their decays generically receive contributions from both S andS in Eq. (2). Indeed, both S andS contribute to H + 3 → W + γ. However, becauseS is generated only by fermions in the loop, the decay of the fermiophobic scalar H + 5 → W + γ receives contributions only from S. For H + 3 → W + γ, the decay is described by Eq. (2) with The particles that contribute to the sums are summarized in Table III Diagram Formfactor Particles s1s2s2

D. Competing decay modes
In order to compute the branching ratios for the loop-induced decays, we need the partial widths for all competing decay modes of the neutral and singly-charged scalars. For this we use the decay partial width calculations for the scalars of the GM model as implemented in GMCALC 1.2.0 [61], which includes the following processes: 1. Tree-level decays to V 1 V 2 , with V = W or Z, including full doubly-offshell effects; 2. Tree-level decays to one scalar and one vector boson, using the two-body expression when kinematically allowed and taking the vector boson off-shell otherwise; 3. Tree-level decays to two scalars (two-body only);

Decays to gluon pairs, including partial QCD corrections at the level of Ref. [66];
5. Decays to fermion pairs (two-body only), including partial QCD corrections at the level of Ref. [66].
In our numerical analysis we will be most interested in H 5 masses below the V V threshold, where the loop-induced decays can obtain large branching ratios. For such H 5 masses our inclusion of the doubly-offshell effects in the competing decays H 0 5 → W W, ZZ and H + 5 → W + Z is essential. Also interesting are H + 3 decays to W + γ below the threshold for H + 3 → tb. We have not included off-shell top quark effects in this competing decay; we leave this improvement to future work.
For very light charged (neutral) scalars below the W (Z) boson mass, off-shell loop-induced decays to W * γ (Z * γ) can become relevant. These could in principle be implemented by taking the W (Z) boson off-shell with a Breit-Wigner distribution as in Ref. [67]. However, this approach neglects non-resonant box diagram contributions to the full H i → ff γ process (so-called Dalitz decays [68][69][70]). Furthermore, our result for the vector loop contribution to H + 5 → W + γ in terms of the LoopTools functions is valid for an on-shell final-state W boson only. For these reasons, in our numerical implementation we compute the scalar decays to W γ (Zγ) as strictly two-body decays. This will result in a counterintuitive resurgence of the off-shell H + 5 → W + Z branching ratio for m 5 < M W , when the two-body H + 5 → W + γ decay is forbidden in our calculation. Our numerical results for H + 5 decay branching ratios are therefore not to be trusted for m 5 < M W . For these masses, one can rely instead on searches for H ++ 5 → W + W + , for which there is no competing decay, and for H 0 5 → γγ, which will dominate over any off-shell H 0 5 → Z * γ contribution. In our numerical analysis in the next section we consider only H + 5 masses above M W and H 0 5 masses above M Z .

IV. NUMERICAL RESULTS
In this section we study the branching ratios of the scalar decays to γγ, Zγ, and W γ in the GM model. We set m h = 125 GeV and scan over the full allowed parameter space of the model using a modified version of GMCALC 1.2.0 [61] into which we have implemented the new one-loop decays. GMCALC 1.2.0 lets us impose the theoretical constraints from perturbative unitarity of the quartic couplings in the Higgs potential and the stability of the correct electroweak vacuum [23], as well as indirect constraints from b → sγ [27]. We also impose an upper bound on sin θ H as a function of m 5 determined in Ref. [25] from an ATLAS measurement of the like-sign W W cross section in vector boson fusion at the 8 TeV LHC [71], which would be increased by H ++ 5 production. 5 Finally we require that m 3 , m 5 ≥ 76 GeV; the lower bound on m 5 was found in Ref. [37] based on an ATLAS search for anomalous like-sign dimuon production at 8 TeV [72,73], and the lower bound on m 3 comes from the LEP search for charged Higgs pair production in the Type I two Higgs doublet model [74], where we require m 3 ≤ m 5 to prevent off-shell decays H + 3 → H + 5 Z, H 0 5 W + . In our scans we require 76 GeV ≤ m 5 ≤ 200 GeV and scan all other parameters over their allowed ranges. The theoretical constraints force m 3 600 GeV in these scans.
Our new result for Γ(H 0 5 → Zγ) allows us to make an accurate calculation of the branching ratio BR(H 0 5 → γγ) for m 5 < 2M W , where the Zγ decay can contribute non-negligibly to the H 0 5 total width. We now use this to apply a new constraint on the GM model from a LEP search for e + e − → ZH 0 5 with H 0 5 → γγ [52]. We take the numerical exclusion limit from HiggsBounds 4.2.0 [75]. The exclusion is shown in the left panel of Fig. 2  The effect of the LEP H 0 5 → γγ constraint on the GM model parameter space can be better understood by studying the m 5 -sin θ H plane, as shown in the right panel of Fig. 2. To illustrate the effects of the other experimental constraints on the model, we show the points excluded by b → sγ in cyan and the points excluded by the ATLAS like-sign W W cross section in green. Again the red points are excluded by the LEP H 0 5 → γγ constraint and the black points are allowed. We see that LEP excludes most of the parameter space for m 5 110 GeV, except for points at low sin θ H for which the e + e − → ZH 0 5 cross section is suppressed and a smattering of points at higher sin θ H for which BR(H 0 5 → γγ) is suppressed due to cancellations among the loop amplitudes. In Fig. 3 we show the branching ratios of H 0 5 → γγ (left panel) and H 0 5 → Zγ (right panel) as a function of m 5 . The black points are allowed. We see that BR(H 0 5 → γγ) can reach several tens of percent for m 5 130 GeV, and be above 1% for a large fraction of the parameter space with m 5 2M W . Similarly, BR(H 0 5 → Zγ) can reach several percent for m 5 ∼ 110-150 GeV, but never surpasses 10%. The rapid decline of BR(H 0 5 → Zγ) for m 5 110 GeV is due to the kinematic suppression from the on-shell Z boson.
In Fig. 4 we study the effect of the new vector-scalar-scalar and scalar-vector-vector contributions to H 0 5 → Zγ. In this figure we plot a "partial" calculation of Γ(H 0 5 → Zγ), obtained by computing only the usual W and scalar loop diagrams for which standard expressions are available [6], normalized to the full calculation of Eq. (41). Over most of the parameter space, neglecting the new vector-scalar-scalar and scalar-vector-vector loop contributions would lead to a result for Γ(H 0 5 → Zγ) about a factor of two smaller than that of the full calculation, except at parameter points where an accidental cancellation among loop amplitudes occurs in either the "partial" or the full result. 5 Other LHC searches for vector boson fusion production of H ++ 5 [35] or H + 5 [38,39] consider only masses above 200 GeV. In Fig. 5 we plot the branching ratios for H + 5 → W + γ (left) and H + 3 → W + γ (right) as a function of m 5 and m 3 , respectively. The black points are allowed. We see that BR(H + 5 → W + γ) can reach a few tens of percent for m 5 130 GeV, and be above 1% for a large fraction of the parameter space with m 5 2M W . BR(H + 3 → W + γ) is typically smaller due to competition with decays to fermion pairs, though it can reach tens of percent for select parameter points. This happens because H + 3 → W + γ receives contributions from scalar loop diagrams (see Table III), which can remain unsuppressed at small sin θ H when the tree-level decays of H + 3 into fermion pairs are suppressed. Finally we comment on the decays of H 0 3 . At low masses, the decays of this state are dominated by ff and gg, as well as decays to Zh, ZH, ZH 0 5 , and/or W ± H ∓ 5 when not too kinematically suppressed. Because H 0 3 is CP-odd, its loop-induced decays to γγ and Zγ receive contributions only from fermions in the loop. Therefore the partial widths for these loop-induced decays, as well as the loop-induced decay to gg and the tree-level decay to ff , all scale with the same H 0 3f f coupling modification factor tan 2 θ H . None of these decays of H 0 3 involve the new one-loop diagrams computed in this paper, and they are already implemented in GMCALC 1.2.0.

V. CONCLUSIONS
In this paper we evaluated the one-loop contributions to H i → V γ from "heterogeneous" loop diagrams involving particles with two different masses propagating in the loop. These are necessary for a full leading-order calculation of the decay widths of H + 3 → W + γ, H + 5 → W + γ, and H 0 5 → Zγ in the GM model. The novel results presented here are for (1) the scalar loop diagram with m Hi = m s1 = m s2 , which contributes to H + 3 → W + γ; (2) the vector-scalarscalar loop diagram, which contributes to H 0 5 → Zγ and H + 5 → W + γ; (3) the scalar-vector-vector loop diagram with M V = M X2 , which contributes to H 0 5 → Zγ; and (4) the vector loop diagram which contributes to H + 5 → W + γ. We gave the results for these diagrams in terms of the LoopTools functions for ease of numerical implementation. We also recalculated the heterogeneous loop diagrams previously computed in Ref. [56] in order to give expressions for them in terms of the LoopTools functions.
Using these results we performed numerical scans over the theoretically-and experimentally-allowed parameter space of the GM model in order to study the behavior of the H i → V γ branching ratios. We showed that a LEP search for e + e − → ZH 0 5 with H 0 5 → γγ strongly constrains the GM model parameter space for m 5 110 GeV. Our results for the loop-induced H i → V γ decays will be implemented into GMCALC 1.3.0 and higher, which will allow the experimental searches for H 0 5 and H + 5 at the LHC to be reliably extended below the V V threshold. The scalar sector of the GM model [2,3] consists of the usual complex doublet (φ + , φ 0 ) T with hypercharge 6 Y = 1, a real triplet (ξ + , ξ 0 , −ξ + * ) T with Y = 0, and a complex triplet (χ ++ , χ + , χ 0 ) T with Y = 2. The doublet is responsible for the fermion masses as in the SM. Custodial symmetry is preserved at tree level by imposing a global SU(2) L ×SU(2) R symmetry on the scalar potential. In order to make this symmetry explicit, we write the doublet in the form of a bidoublet Φ and combine the triplets to form a bitriplet X: The vevs are defined by Φ = v φ √ 2 I 2×2 and X = v χ I 3×3 , where I is the unit matrix and the W and Z boson masses constrain The most general gauge-invariant scalar potential involving these fields that conserves custodial SU(2) is given, in the conventions of Ref. [23], by 7 Here the SU(2) generators for the doublet representation are τ a = σ a /2 with σ a being the Pauli matrices, the generators for the triplet representation are and the matrix U , which rotates X into the Cartesian basis, is given by [8] The physical fields can be organized by their transformation properties under the custodial SU(2) symmetry into a fiveplet, a triplet, and two singlets. The fiveplet and triplet states are given by where the vevs are parameterized by and we have decomposed the neutral fields into real and imaginary parts according to (A8) 6 We use Q = T 3 + Y /2. 7 A translation table to other parameterizations in the literature has been given in the appendix of Ref. [23].
The masses within each custodial multiplet are degenerate at tree level and can be written (after eliminating µ 2 2 and µ 2 3 in favor of the vevs) as 8 Note that the custodial-fiveplet states H 5 consist entirely of the triplet fields, and hence do not couple to fermions at tree level. In contrast, the H 3 states contain a doublet admixture and hence do couple to fermions. The two custodial-singlet mass eigenstates are given by where c α = cos α, s α = sin α, and The mixing angle and masses are given by where we choose m h < m H , and The couplings of the scalars in the GM model that we use in this paper are collected in Appendix B.
Here f stands for any charged fermion, u stands for any up-type quark, and d stands for any down-type quark or charged lepton.
The Feynman rules for the vertices involving a charged scalar and two fermions are given as follows, with all particles incoming: Here V ud is the appropriate element of the CKM matrix and the projection operators are defined as P R,L = (1 ± γ 5 )/2.
We define the coupling coefficients C S and C P used in the work above according to i C S The custodial fiveplet states do not couple to fermions.

Triple scalar couplings
The Feynman rules for the triple-scalar couplings involving incoming scalars s 1 s 2 s 3 are given by −iC s1s2s3 , with all particles incoming. The ordering of the indices does not matter for these couplings. The couplings used in our calculations given as follows:

Scalar-vector-vector couplings
The Feynman rules for the vertices involving a scalar and two gauge bosons are defined as ie 2 C sV1V2 g µν . The couplings used in our calculations are given by

Vector-scalar-scalar couplings
The Feynman rules for the vertices involving two scalars and a single Z boson are defined as ieC Zs1s2 (p 1 − p 2 ) µ , where p 1 (p 2 ) is the incoming momentum of incoming scalar s 1 (s 2 ). The couplings are given by The Feynman rules for the vertices involving two scalars and a single W + boson are defined as ieC W + s1s2 (p 1 − p 2 ) µ , where again p 1 (p 2 ) is the incoming momentum of incoming scalar s 1 (s 2 ). The couplings are given by The couplings for the conjugate processes involving an incoming W − are obtained using

Couplings involving Goldstone bosons
Our calculation of the vector-scalar-scalar, scalar-vector-vector, and vector loop diagrams in the 't Hooft-Feynman gauge require the calculation of diagrams involving Goldstone bosons. We collect the relevant couplings here.
The couplings of Goldstone bosons to other scalars are given by −iC s1s2s3 , where the coefficients used in this paper are The couplings of a pair of Goldstone bosons to the Z or W are given by ieC V s1s2 (p 1 − p 2 ) µ , where p 1 (p 2 ) is the incoming momentum of incoming scalar s 1 (s 2 ) and the coefficients are The couplings of a Goldstone boson and a physical scalar to a single vector boson are given by ieC V s1s2 (p 1 − p 2 ) µ , where p 1 (p 2 ) is the incoming momentum of incoming scalar s 1 (s 2 ). The coefficients used here are given by The couplings of a Goldstone boson to two vector bosons are given by ie 2 C sV1V2 g µν , with

Couplings involving ghosts
Our calculation of the vector loop diagrams in the 't Hooft-Feynman gauge requires the calculation of diagrams involving ghosts. This enters only in the decay H + 5 → W + γ. The relevant term in the ghost Lagrangian involving an incoming H + 5 , incoming ghost c − and outgoing ghost c Z is where ξ = 1 is the gauge-fixing parameter for the 't Hooft-Feynman gauge. The resulting Feynman rule for the Higgs-ghost-ghost vertex is ie 2 C Hic − c Z with In our conventions, the Feynman rules for a pair of ghosts coupling to a vector boson arē where −k is the incoming momentum of the incoming antighost; i.e., k is the outgoing momentum of the outgoing ghost.
The vector and tensor three-point integrals are decomposed into scalar coefficients according to where C 21 = C 12 due to the symmetry of C µν under permutation of Lorentz indices. For compactness, when a sum of three-point integrals with a common set of arguments appears, we have specified the arguments only once at the end of the sum.
To combine these into a single expression, we again examine the Goldstone boson couplings for the actual combinations of parent and internal particles in the decays of interest. For H + 5 → W + γ and H 0 5 → Zγ, s 1 is always an H 5 state and X 2 is always a W boson (of either charge). Because the coupling of two H 5 states to a Goldstone boson is zero by custodial symmetry, S sGG does not contribute to these decays. The remaining pieces are easy to combine using the relations between the Goldstone couplings and the corresponding gauge boson couplings, yielding the first line of Eq. (24) [note that the second line does not contribute for an initial-state H 5 because s 1 is also an H 5 state and hence (m 2 Hi − m 2 s1 ) = 0]. For H + 3 → W + γ, the situation is more complicated. s 1 can be h, H, H 0 5 , or H ++ 5 , and in all cases S sGG is nonzero. For either of the H 5 states in the loop, the combination is again fairly straightforward and yields the expression in Eq. (24), with S sGG giving rise to the terms in the second line. For h or H in the loop, the combination of S sXX and S sGX is again straightforward, yielding the first line in Eq. (24); the simplification of S sGG is non-obvious because of the complicated form of the H + 3 hG − and H + 3 HG − couplings, but it can be verified numerically that it also reduces to the terms in the second line of Eq. (24) for each diagram individually.

Vector loop diagram
In 't Hooft-Feynman gauge, the last diagram in Fig. 1 and its Goldstone boson substitutions do not contribute to the k µ q ν term, so we only need to worry about the fifth diagram and its Goldstone and ghost substitutions. There are nine diagrams: one as shown by the fifth diagram in Fig. 1, three in which a single gauge boson in the loop is replaced by the corresponding Goldstone boson, three in which two of the gauge bosons in the loop are replaced by their corresponding Goldstone bosons, one in which all three gauge bosons in the loop are replaced by Goldstone bosons, and a diagram with ghosts in the loop.
We write the contribution to the amplitude from these nine diagrams as A HiV γ X1X2X2 = S XXX + S GXX + S XGX + S XXG + S GGX + S XGG + S GXG + S GGG + S ghost , where again the subscripts denote the particles in the loop proceeding clockwise from the H i vertex. The diagrams corresponding to S XGX and S GXG do not contribute to the k µ q ν term in the amplitude, so they are zero.
For the remaining diagrams, we specialize to the actual process of interest, H + 5 → W + γ, with X 1 = Z and X 2 = W − . We can then use the explicit expressions for the triple-gauge and ghost vertices. We obtain, The last expression for S ghost includes the contributions of the two ghost diagrams: one with c Z , c − , c − proceeding clockwise around the loop from the H + 5 vertex, and one with c + , c + , c Z proceeding counterclockwise around the loop from the H + 5 vertex (these are distinct because the antiparticle of the ghost c − isc − , not c + ). These two diagrams give identical contributions.
Inserting explicit expressions for all the couplings and combining all the terms is then relatively straightforward, and yields the expression for A H + 5 W γ ZW W given in Eq. (28).