Abstract
Motivated by the bulk mixing between a massive radion and a bulk scalar Higgs in warped extra dimensions, we construct an effective four dimensional action that—via the correspondence—describes the most general mixing between the only light states in the theory, the dilaton and the Higgs. Due to conformal invariance, once the Higgs scalar is localized in the bulk of the extradimension the coupling between the dilaton and the Higgs kinetic term vanishes, implying a suppressed coupling between the dilaton and massive gauge bosons. We comment on the implications of the mixing and couplings to Standard Model particles. Identifying the recently discovered GeV resonance with the lightest Higgslike mixed state , we study the phenomenology and constraints for the heaviest radionlike state . In particular we find that in the small mixing scenario with a radionlike state in the mass range GeV, the diphoton channel can provide the best chance of discovery at the LHC if the collaborations extend their searches into this energy range.
Radion/DilatonHiggs Mixing Phenomenology
in Light of the LHC
Peter Cox, Anibal D. Medina, Tirtha Sankar Ray, Andrew Spray
ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics,
University of Melbourne, Victoria 3010, Australia
1 Introduction
Among the most popular models that extend the Standard Model (SM) of particle physics and solve the gauge hierarchy problem are warped extra dimensions [1] and composite scenarios [2, 3] where the Higgs is identified as a pseudoNambuGoldstone boson (PNGB) of a broken shift symmetry [4]. In the case of warped extra dimensions, the nonfactorizable geometry that leads to a slice of space is responsible for effectively reducing quadratic contributions to the Higgs mass once the Higgs fivedimensional (5D) scalar is localized on (brane Higgs) or near (bulk Higgs) the infrared (IR) brane. Thus, even though there is no implicit symmetry that leads to a light Higgs, once we assume that such a light mass is generated, it remains natural. On the other hand, explicit calculable realizations of composite PNGB Higgs were first found in warped scenarios in what are known as GaugeHiggs Unification (GHU) models. Here the SM gauge group is enlarged to a gauge group G in the bulk of the extra dimension and broken via boundary conditions to the subgroups H (IRbrane) and (UVbrane) on the branes. In this way, the fifth component of the gauge field that belongs to the coset group has the right quantum numbers to be the Higgs. Though it is protected by the gauge symmetry at treelevel, it acquires a potential at loop level that successfully leads to electroweak symmetry breaking (EWSB) and provides a light Higgs mass protected from the UVphysics [4, 5]. Due to the correspondence and through the language of holography, it was realized that these kinds of models are particular realizations of composite Higgs scenarios where the Higgs is a PNGB arising from the spontaneous breaking of a global shift symmetry G, and where SM particles have a degree of compositeness determined by their coupling to operators that reside in the conformal sector.
In both realizations, the conformal sector is spontaneously broken and a corresponding Goldstone mode is expected in the theory. In the 5D picture, this mode is known as the radion and is associated with the spin0 fluctuations of the metric. In order to stabilize the extra dimension the radion is coupled to an additional scalar field [6]; the gravityscalar system can provide a stabilizing potential and a mass for the radion, which is expected to be light. From the 4D point of view this can be accomplished if the corresponding Goldstone mode, the dilaton, couples to a nearly marginal operator of dimension [7, 8]. Since, besides the other wellknown particles of the SM, the LHC has recently discovered what seems to be a light scalar state, it seems reasonable to take the approach that the only new light states accessible at the moment at the LHC are the Higgs and the radion/dilaton. Given that these two light scalar states posses the same quantum numbers, mixing between them is expected, which can have important consequences in the phenomenology of this effective two scalar system.
In this work we begin by studying the less known case of a bulk scalar Higgs in a warped extra dimension that mixes via a term with the radion. We show how in this way one can arrive at an effective Lagrangian that describes the different mixing possibilities encountered in these kinds of models. We also show that moving the Higgs from the brane into the bulk of the extra dimension can already have important consequences on how the radion couples to SM particles, leading to a different radion phenomenology compared to the brane Higgs case [9, 10, 11, 12, 15]. In particular we show that due to the geometry/conformal symmetry of the radion in the bulk, its couplings to 4D scalar kinetic terms vanish, and therefore the radion coupling to massive gauge bosons is suppressed. The 5D construction is used as a tool to obtain the dependence of the radion and Higgs mixing and couplings on the masses , and energy scales GeV, radion decay constant which is taken to be of the order of the conformal breaking scale , TeV; we allow for freedom in the specific numerical values of dimensionless parameters.
Once the mixing is taken into account, we perform a numerical scan over the relevant parameters satisfying the most recent constraints from the LHC on Higgs physics and exotic resonant searches. By matching the lightest mixed state’s mass and signal strengths to those measured at the LHC for the 125 GeV resonance, we are able to predict the branching fractions and crosssections for the most relevant decays of the heaviest mixed state. Interestingly, we find that in some regions the production of two light mixed states via the decay of the heaviest mixed state can contribute as much as 30 to the total production cross section. Furthermore we find that in the case of negligible mixing, a light scalar state with mass in the range GeV with a sizeable crosssection into diphotons is still allowed by LHC constraints, providing a very interesting motivation to look in the diphoton channel at larger invariant masses than is currently done at the LHC by ATLAS and CMS.
The paper is organized as follows. In sections 2 and 3 we introduce the radion and a simplified model of a bulk Higgs in warped 5D space and compute the different mixing terms that arise in the presence of the bulk term. In sections 4 and 5 we write the effective Lagrangian describing the mixing between the radion/dilaton and the Higgs and provide the relevant couplings and branching fractions. Sections 6 and 7 contain the LHC constraints used and the phenomenological study of the heaviest mixed state. Our conclusions are given in section 8.
2 The Radion
We are interested in a 5D background that preserves 4D Lorentz symmetry, which can always be written in the form,
(1) 
where is the extra dimensional coordinate and a convex function of . In the RandallSundrum solution, where is the curvature scale, and the space reduces to a slice of with boundaries at (UVbrane) and (IRbrane). By an appropriate gauge choice, one can decouple the spin0 (radion) from the spin2 (graviton) fluctuations of the metric Eq. (1). The spin0 fluctuations are given by,
(2) 
In the absence of a stabilizing mechanism, the radion is massless and it is simple to check that it consists of a single state with a profile in the extradimension given by
(3) 
where we have used that , with the 4D reduced Planck mass.
In order to stabilize the extradimension, it is customary to introduce an additional scalar in the bulk of the extra dimension with corresponding bulk and brane potentials such that the gravity and scalar sectors mix. The backreaction of the scalar on the geometry provides a mass for the physical state associated with the radion. This will produce deviations from the pure solution for the geometry; however, if the backreaction is not large the deviations tend to be small and the approximate form for the radion profile holds [16]. We will comment in section 3.3 on the consequences of the backreaction on the radionHiggs mixing, which are important once the Higgs is moved to the bulk of the extra dimension.
3 RadionHiggs mixing
Light radion/dilaton phenomenology and mixing with an IRbrane localized Higgs has been studied extensively in the literature [10, 11, 12, 13, 14]. It has been found that current LHC measurements, in particular of the Higgs mass and signal strengths, already put significant constraints on the parameter space of these models [17]. In this paper we study the consequences of moving the Higgs into the bulk of the extra dimension and mixing it with the gravity sector via a bulk term . Such a bulk mixing term was also considered in [18] but in the context of higher curvature GaussBonnet terms. We motivate an effective 4D low energy action that describes all the possible mixing terms that one may encounter between the two light states in the model, the radion and the Higgs. We also derive the parametric size of these mixing terms. In this context let us briefly survey the possible localization of the Higgs, what this implies for the radionHiggs mixing in the theory, and the possibility of a Higgs as a PNGB of a shift symmetry.
3.1 The brane Higgs scenario
In this case one can simply write the Higgs part of the Lagrangian as follows:
(4) 
where is the induced metric on the boundary. After the Higgs gets a vev , one can perform a Taylor expansion of the potential,
(5) 
The mass mixing term that can arise from the term in the above equation vanishes exactly due to the minimization condition. No mixing of any type arises from the kinetic term, as . This is the reason for the absence of any mass mixing in brane Higgs models. Only kinetic mixing via the usual term is expected.
3.2 The bulk Higgs scenario
Let us now consider a scenario where the Higgs and the SM fields can access the 5D bulk. We use this model as a tool to motivate our effective action in section 4 and therefore we briefly describe the process of EWSB and Higgs mass generation. Technical details of the calculation that are similar to those of [19] are deferred to appendix A. In this case the full Higgsradion action may be written as
(6) 
where is the 5D bulk potential ( a dimensionless localization parameter), are the 4D brane potentials, is the induced metric, and denotes the jump in the extrinsic curvature across the brane. Note that in adding the direct coupling between the Higgs and the scalar curvature in the bulk, we must also modify the GibbonsHawking term to ensure the correct cancellation of boundary terms. EWSB is induced on the IR brane by taking
(7) 
where and are dimensionless quantities. On the UV brane, we simply add a mass term
(8) 
To simplify our analysis, we assume that the Higgs back reaction on the metric is negligible. This requires that the Higgs vev satisfy
(9) 
The explicit mixing terms of Eq. (3.2) contribute to the effective bulk and brane masses for the Higgs. It is straightforward to solve for the Higgs vev . Expressing it in terms of the physical observable , we find
(10) 
where . The explicit relation between and the 5D parameters is
(11) 
where and we have neglected terms suppressed by additional powers of . Inserting our expression for into Eq.(9), we find that the backreaction is negligible for values of , , and , provided that both and .
The Higgs fluctuation , with mass , has a more complex form. In the limit that the Higgs mass is small compared to the RS scale , we find that the profile is approximately proportional to the vev:
(12) 
Using the IR b.c. one also can determine the mass . The resultant equation is complicated in the general case. However, in our limiting case , one can obtain an approximate analytical expression for the lightest mode given by
(13) 
To investigate the mass mixing induced by the bulk Lagrangian Eq. (3.2), we expand the scalar curvature using the metric and the replacement :
(14) 
Now using Eq. (3) and we find that the expressions reduce to
(15) 
The nonderivative terms linear in the radion vanish. There could be a residual mass mixing that arises from the product of the constant terms in Eq. (15) with the linear fluctuation in the volume element. However, as discussed below Eq. (9), these constant terms are effective bulk and brane masses for the Higgs, and are more naturally associated with the mixing from the potentials. Indeed, one can explicitly redefine the Lagrangian mass parameters to absorb these constant terms:
(16) 
This will naturally lead to modifications to the definition of , and the relation between and the Lagrangian parameters. Finally we compute the mass mixing that might arise from the potential terms in the bulk and on the brane and the kinetic term in the bulk,
[Bulk potential]  
[IR brane potential]  
[Higgs kinetic term]  (17) 
We have neglected the UV potential as the Higgs is localized near the IR brane.^{1}^{1}1One can show that the contribution from the UV potential cancels with additional exponentially suppressed terms that have been omitted in Eq. (3.2). We find that these contributions cancel exactly and leave us with no mass mixing between the Higgs and the radion.
The derivative terms in Eq. (15) lead to a kinetic mixing, as in the brane Higgs case. A quantitative difference from the brane scenario is that the size of the induced mixing is dependent. Specifically,
(18) 
The mixing term also gives contributions to the radion kinetic term. One contribution arises when the linear derivative term combines with the linear term in . Another contribution of the same order comes from the terms in quadratic in the radion. The net result is
(19) 
3.3 Bulk Higgs with back reaction
In the above discussion we did not consider the back reaction of the Higgs and the radion on the metric. This will modify the bulk profiles of the Higgs, the Higgs vev and the radion. The Higgs back reaction can be assumed to be small as already argued; even if we include its effect, it can at most induce a mass mixing proportional to the Higgs mass . A mass mixing of this order can also arise if we include the differences between the Higgs and Higgs vev bulk profiles ; that is, if we expand the Bessel functions in Eq. (73) to include subleading terms in .
A larger contribution to the mass mixing may arise due to the back reaction of the radion. Let us assume that the radius stabilizing mechanism results in a small perturbation in the bulk profile of the fields. We can write the following ansatz for the perturbed radion bulk profile and the metric:
(20) 
where ; is the radion vev on the TeV (Planck) brane introduced to stabilize the bulk; and The equation of motion for the radion field can be solved using the above ansatz as an expansion in [11]. Expanding up to we obtain
(21) 
One can now solve for the normalization factor for the radion at this order of the expansion,
(22) 
Solving for the Higgs vev in the same approximation yields,
(23) 
where is given by,
(24)  
Finally we will assume that,
(25) 
Thus including the radion back reaction, the Higgsradion mixing action at leading order can now be written as,
(26) 
where is the radion mass given by[11],
(27) 
As expected we find that the mass mixing terms arising from the radion back reaction are proportional to the radion mass.
3.4 Composite Higgs models
As mentioned in the introduction, the 5D analogue of the PNGB composite Higgs is the GHU scenario where the Higgs is identified as the fifth component of a 5D gauge boson belonging to the coset group . The higherdimensional gauge symmetry translates to a 4D shift symmetry of the Higgs. In a slice of , the sector of the gauge boson kinetic term of the bulk Lagrangian is
(28)  
Notice that the higherdimensional gauge symmetry prevents a treelevel mass for both in the bulk and on the brane. Also, the antisymmetric nature of the field strength tensor prevents a term like . This immediately implies that a composite Higgs cannot have mass mixing with the radion even when the back reaction is considered. The only possible mixing can be introduced on the brane after the shift symmetry is explicitly broken by the Yukawa and SM gauge interactions to develop a potential. The relevant brane term reads,
(29) 
One can estimate the size of by noticing that PNGB potentials are generated at loop level primarily through the top Yukawa which is also responsible for the Higgs developing a potential. Naive dimensional analysis suggests that
(30) 
where is the compositeness scale given by and is a generic function of . Thus, we expect the kinetic mixing induced by this term to be very small.
4 Effective action
Up to this stage we have worked with a 5D warped scenario, considering two particular examples of EWSB, in both of which the Higgs resides in the bulk of the extra dimension. In this way we have been able to determine the possible induced mixing terms between the radion and the Higgs. Though we have determined these mixing terms for a particular scenario we expect their dependence on physical quantities to be general. In fact, from the 4D point of view through the correspondence, we are describing a scenario of a conformal sector that is spontaneously broken leading to a light pseudoNambuGoldstone boson known as the dilaton. This light state can mix with the other light state in the theory, the Higgs, via the conformally covariant generalization of the gauge covariant derivative[20, 21]:
(31) 
where is the Higgs conformal weight, is the gauge covariant derivative and we have included an additional term as suggested in Ref. [21] in order to account for the breaking of the special conformal symmetries, which in the 5D picture corresponds to the case . This interaction leads to kinetic mixing as found in the previous section. This mixing is always present due to the remnant shift symmetry of the model, and it is the only possible type of mixing allowed when the CFT is broken spontaneously, as we also saw in our simplified 5D calculation. An explicit breaking of the conformal symmetry is signalled by the presence of a nonvanishing dilaton mass and consequently the possibility of a mass mixing term between the dilaton and the Higgs field. This is represented in the 5D picture by the deformation from space due to back reaction effects responsible for the stabilization of the extra dimension and thus, for the generation of the radion mass. As we saw, this explicit breaking of the conformal symmetry leads to mass mixing between the radion and the Higgs in the 5D picture as described in Eq. (26).
It then becomes clear that from a pure 4D perspective, we can represent the most general effective phenomenological Lagrangian describing the light degrees of freedom of a spontaneously broken conformal sector by,
(32)  
where , and are numerical coefficients, and we use the terms radion/dilaton interchangeably. From this point onwards we focus on this phenomenological Lagrangian to describe the possible mixing scenarios that may arise:

The no mass mixing scenario, . From the 5D point of view, this case corresponds to a pure slice where the back reaction on the geometry from the radion potential that stabilizes the extra dimension can be neglected. Strictly speaking, it is not compatible to have a massive radion and no mass mixing unless a tuning of the parameters is involved such that . From the 4D point of view, this corresponds to no explicit conformal breaking parameter in the dilaton self interactions and thus to a CFT that is not badly broken.

The generic scenario where corresponds from the 5D point of view to considering the leading back reaction contributions of the radion potential and from the 4D point of view to explicit conformal breaking terms in the dilaton potential.

The gaugeHiggs unification/pseudoNambuGoldstone composite Higgs 5D/4D scenarios correspond to and when explicit sources of conformal breaking are neglected.
Despite the fact that a brane or a bulk Higgs may enter in the same mixing category, the phenomenology can very different due to the way in which in the conformal breaking is felt as we will see in the next section. For the study of the radionHiggs mixing and its effect on both the Higgs and radion phenomenology at colliders we shall consider and as free parameters. We will see in the next section that the GHU/PNGB composite Higgs scenario reduces phenomenologically to the case of a brane Higgs with , and has therefore been covered by previous radion studies [17]. Thus we focus our scans on covering all possible values for and for a bulk scalar Higgs, which provides phenomenologically distinct signatures with respect to the brane Higgs case.
One can diagonalize the kinetic term in Eq. (32) by going to a new basis and , where
(33) 
This transformation decouples the kinetic mixing but introduces additional mass mixing terms. The mass matrix in the basis then takes the form
(34) 
The mass eigenbasis is obtained by the orthogonal transformation
(35) 
where and
(36) 
There are correspondingly two eigenstates; a lighter one and a heavier one , with masses:
(37) 
The gauge basis is related to the mass basis according to:
(38) 
5 Higgs and Radion couplings, mixing and branching ratios
5.1 Higgs and Radion couplings
Though we motivated our effective theory by studying a particular 5D scenario, we ultimately focused on an effective 4D picture wherein we consider two types of Higgs sector: i) the Higgs is identified with a light scalar doublet charged under the gauge group ; or ii) the Higgs field is identified with a composite PNGB of an enlarged broken global group that contains as a subgroup. In both cases there is an associated conformal sector that is spontaneously broken at an energy scale and that in our effective theory translates into the existence of a possible light state; the dilaton. One may be worried about possible contributions to the Higgs couplings arising from mixing or loopeffects involving resonances of the conformal sector. However, notice that in case i) the only symmetry additional to those already found in the SM is the spontaneously broken conformal symmetry. We expect any possible additional composite resonances besides the dilaton to have masses of the order , with the strong coupling from the conformal sector, making their effects on the Higgs couplings strongly suppressed. In case ii) due to the enlarged global group in which SM particles are embedded and due to the shift symmetry protection of the Higgs, there is a relationship between the Higgs mass and light top fermionic resonances of the form . Therefore in order to reproduce a light Higgs mass, one usually finds the existence of light fermionic resonances that couple strongly to the Higgs, with masses . This can have significant effects, in particular for Higgs couplings to gluons or photons. It has been shown nonetheless that due to the pseudoNambuGoldstone nature of the Higgs, the resonant fermionic loop contributions cancel against the top quark modified Yukawa coupling, and lead to modifications in the coupling to gluons that are suppressed by the ratio [22]. Therefore, in the two Higgs scenarios considered, we do not expect sizeable deviations of the Higgs couplings from their SM values, and thus for simplicity we restrict the couplings to SM values.^{2}^{2}2In the case of generic warped extra dimensional scenarios the mass scale of the lowest lying KK fermions A naive estimate of the of shift in the Higgs coupling to gluons due to the KK towers of the SM fermions is as follows, where is the quadratic Casimir of the KK states and . This translates into a lower limit on the mass of the lightest KK state that may be as large as 3.2 TeV for a shift in the decay width, which is the resolution of current experimental data. A detailed calculation of this is rather model dependent [23] and beyond the mandate of this paper.
Allowing for the possibility of a bulk Higgs implies that some of the known radion couplings to SM fields are modified, in particular those involving radion couplings to the Higgs field itself as well as to massive gauge bosons. We use the 5D language as an easy tool to calculate the couplings and assume a given warp factor that solves the hierarchy problem, though our results are general with the replacement . As was shown in Ref. [24], the bulk radion couples at linear order to SM fields through the bulk stress energy tensor as
(39) 
where the conformal coordinate is related to the extradimensional coordinate as , and is the bulk stress energy tensor which can be written as
(40) 
Focusing on the coupling to SM gauge bosons (massive or massless), using Eq. (39), one can easily show that due to the bulk kinetic terms for the gauge fields, there will always be a nonvanishing coupling of the form
(41) 
where we used that in this case and therefore . As was argued in [24], the fact that this tree level coupling is nonvanishing implies that loop effects merely renormalize this treelevel operator. Therefore, loop effects are prominent on the branes where no treelevel coupling is allowed, being stronger on the IR brane where the radion is usually closely localized. This provides the main mechanism of radion production through gluon fusion as is usual in radion scenarios. We refer the reader to Ref. [24] for the appropriate expressions for the radiondigluon and radiondiphoton couplings, including fermion and gauge boson loops as well as QCD and QED trace anomalies respectively.
In addition, via electroweak symmetry breaking (EWSB), there is in principle a possibly large additional coupling of the radion to a pair of massive gauge bosons, which is dominant in the case of a branelocalized Higgs. As is wellknown, the gauge bosons acquire their mass through the kinetic term of the Higgs field, which in the case of a bulk Higgs scalar leads to mass terms for the gauge bosons of the form
(42) 
It follows that the contribution to the stress energy tensor is , which implies that , where the last index is contracted using the Minkowski metric. Now , and therefore , which exactly compensates the contribution from . Thus the linear radion coupling to the electroweak gauge boson mass terms vanishes in the case of a bulk Higgs. This result can also be checked by simply expanding the metric in its spin0 fluctuations in the matter action
(43) 
where the index on the r.h.s is contracted using the Minkowski metric. Therefore we also see in this way that the coupling vanishes. Notice that this result is general for the kinetic term of any scalar. We can understand this result from the 4D point of view as follows: as we just noticed, the vanishing of this particular coupling is geometrical from the 5D point of view. As a matter of fact we can take both the UV and IR branes to infinity, and the results would still hold in pure space. In that particular case, it is clear that the conformal symmetry is exact. If we look at the 4D picture this implies that the 4Danalogue of the radion, the dilaton field, can only couple derivatively to conformally invariant operators, in particular to , where is a 4Dscalar field. Therefore from Lorentz invariance we see that no linear coupling can be written that derivatively couples the radion to . This has important consequences for the radion phenomenology when the Higgs is a scalar in the bulk, since then its coupling to pairs of massive SM gauge bosons only comes from Eq. (39) and is highly suppressed.
In the case of gaugeHiggs unification scenarios, the Higgs field is identified with the fifth component of a gauge field belonging to the coset of an enlarged gauge group that is broken down to the subgroup via boundary conditions. In that case the equivalent of the scalar kinetic term is given by
(44) 
where the index . Due to the extra index in the kinetic term, there is a nonvanishing radion coupling proportional to the EWSB induced masses
(45) 
where the index on the r.h.s is contracted using the Minkowski metric. Thus, in these kinds of scenarios the radion coupling to massive SM gauge bosons is similar to that encountered for a localized Higgs scalar on the IR brane.
Another potential difference with respect to the brane Higgs scenario may arise in the Yukawa induced SM fermionradion interactions with the Higgs field which tend to dominate for heavy fermions with respect to other radionfermion interactions that are momentum suppressed. For that reason we focus on the term
(46) 
where is the 5D Yukawa coupling. We again expand the spin0 fluctuations of the metric and use that the lefthanded and righthanded fermion wellnormalized zero mode profiles are given by
(47) 
where and satisfy
(48) 
The upper and lower signs correspond to and respectively, while are the fermion bulk mass parameters defined by . Using Eq. (10) for the Higgs vev, we can obtain an expression for the SM fermion masses by integrating the zeromode profiles for the fermions and the Higgs vev along the extra dimension. In that case we see that we can express the fermion mass as
(49)  
The interaction Eq. (46), once expanded in the spin0 fluctuation of the metric, takes the form
(50) 
where in the last line we assume that the fermion and Higgs profiles are IR localized and satisfy . So contrary to the gaugeboson case, we notice that the coupling of the radion to, in particular, the top quark can be nonnegligible, similar to the case with a localized Higgs field.
Finally we look at the coupling of the radion to two Higgs. For this coupling there is a kinetic mixing contribution coming from as well as contributions from the Higgs kinetic term, bulk Higgs mass and important boundary contributions from the IRbrane potential . The Higgs kinetic and bulk mass contributions cancel against some of the IRbrane contributions and after replacing in terms of and using Eq. (13), one can write the radiondiHiggs coupling in the form
(51) 
We have also used the radion equation of motion . Given Eq. (51), we do not expect large differences arising in comparison with the brane localized Higgs counterpart.
To summarize, after studying the radion couplings to SM particles, we expect the largest modifications in the phenomenology of the bulk scalar Higgs scenario to arise due to the vanishing of the radionmassive diboson coupling proportional to the gauge boson mass. We list for completeness in Table 1 the most relevant couplings of the unmixed Higgs and radion states, where , and are the usual integrals over fermion and gauge boson states running in the loop and and are the function coefficients.
5.2 Mixing and branching ratios
Most of the interactions between the radion and SM particles, except those with massive gauge bosons and to the Higgs itself, have the same structure as those of the SM Higgs to fermions and gauge bosons. So one can easily obtain most of the decay rates of the mixed states by inspecting the wellknown expressions for the Higgs decay rates (see for example [25]) and using the replacements: and , where from Eq. (38),
(52) 
with and the Higgs and radion couplings to SM particles respectively.
The interactions that have a structure different than those of the Higgs to SM particles are those of the mixed states to massive gauge bosons and among the mixed states themselves. In this case, the decay rate of the mixed states into massive gauge bosons can be written as
(53)  