Mixed Higgs-Radion States at the LHC -- a Detailed Study
Amit Chakraborty Ushoshi Maitra Sreerup Raychaudhuri Tousik Samui






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     Mixed Higgs-Radion States at the LHC – a Detailed Study1.05
    
    Amit Chakraborty[E-mail:  amit@post.kek.jp]
     [2mm]
    
    Theory  Center,  Institute  of  Particle  and  Nuclear  Studies,  
    
    KEK,  1-1  Oho,  Tsukuba,  Ibaraki 305-0801, Japan
    
    Ushoshi Maitra[E-mail:  ushoshi@theory.tifr.res.in], 
    Sreerup Raychaudhuri[E-mail:  sreerup@theory.tifr.res.in] and 
    Tousik Samui[E-mail:  tousik@theory.tifr.res.in]
     [2mm]
    
    Department of Theoretical Physics, Tata Institute of Fundamental Research,  
     
    1, Homi Bhabha Road, Mumbai 400 005, India.1.15December 30, 2023 Abstract
    
    1.05 Light radions constitute one of the few surviving possibilities for
    observable new particle states at the sub-TeV level which arise in 
    models with extra spacetime dimensions. It is already known that the 
    125 GeV state discovered at CERN is unlikely to be a pure radion state, since its decays resemble those of the Standard Model Higgs boson too closely. However, due to experimental errors in the measured decay 
    widths, the possibility still remains that it could be a mixture of 
    the radion with one (or more) Higgs states. We use the existing LHC 
    data at 8 and 13 TeV to make a thorough investigation of this possibility. Not surprisingly, it turns out that this model is already constrained quite
    effectively by direct LHC searches for an additional scalar heavier than 125 GeV. 
    We then make
    a detailed study of the so-called `conformal point', where this
    heavy state practically decouples from (most of) the Standard Model 
    fields. Some projections for the future are also included.PACS Nos:  04.60.Bc, 12.60.Fr, 14.80.Cp, 13.85.Rm1.25
    
    
     1. Introduction
    
    The 2012 discovery<cit.>, at the LHC, of a weakly-interacting light scalar state 
    — which appears from all current indications to be an elementary Higgs particle — revives the old question of how the mass of such a scalar can remain stable against large electroweak corrections in a theory with a momentum cutoff at some very high scale. This, as is well-known, goes by 
    the name of the gauge hierarchy problem, or, alternatively, as the fine-tuning problem. It has also been known for several decades that any solution to this problem must invoke new physics beyond the Standard Model (SM) of strong and electroweak interactions. 
    
    One of the most elegant solutions of the hierarchy problem is that devised 
    in 1999 by L. Randall and R. Sundrum (RS)<cit.>. They considered a world with one 
    extra space dimension, having the topology of a circle folded about a 
    diameter (𝕊^1/ℤ_2), at either end of which lies 
    a pair of four-dimensional manifolds – called `branes' – containing 
    matter. One of these is the
    so-called infra-red (IR) brane, where all the SM fields lie, and the other 
    is the so-called ultra-violet (UV) brane, where we have field elements 
    comprising a theory of strong[Here `strong' means comparable to 
    electroweak strength.] gravity. 
    One can then tune the cosmological constant on the 
    two branes, as well as that in the 𝕊^1/ℤ_2 bulk, to 
    obtain a solution of the five-dimensional Einstein equations in the form 
    of a `warped' metric
    
        ds^2 = e^-2 K R_c ϕη_μν dx^μ dx^ν - R_c^2 dϕ^2
    
    where the 𝕊^1/ℤ_2 `throat' is characterised by the 
    compactification radius R_c, an angular coordinate ϕ and a curvature parameter
    K. It can then be shown that the mass of the Higgs scalar is 
    generated on the UV brane at a value close to the bulk Planck mass M_5
    (itself a little smaller than the four-dimensional Planck mass M_P = 
    (ħ c/G_N)^1/2), and projected on the IR brane through the 
    expanding `throat', thereby acquiring the much smaller value 
    
        M_H ∼ e^-π K R_c M_5
       
    If we can now tune K R_c ≃ 11.6, we recover the correct ballpark
    for the mass of the discovered scalar. This constitutes a neat solution to the hierarchy problem in terms of spacetime geometry, without having recourse to any parameters which are unnaturally large or small. In fact, the Planck scale
    is the only fundamental mass scale in this theory.   
    
    It is fair to ask, however, whether the parameter K R_c is protected 
    against small dynamical fluctuations, for
    
        δ M_H/M_H≈ 11.6π δ R_c/R_c
    
    i.e. small fluctuations in the inter-brane distance would lead to magnified
    fluctuations in the Higgs boson mass. As the latter is now known to an accuracy 
    of about 2%, it follows that the inter-brane distance must be stable to an 
    accuracy of about 5× 10^-4— for which the minimal RS model  
    has  no provision.
    
    A brilliant solution to this was devised by Goldberger and Wise 
    (1999)<cit.>. If one allows for fluctuations in the size of the extra dimension, 
    we can rewrite the metric in Eq. (<ref>) as
    
        ds^2 = e^- 2T(x) ϕη_μν dx^μ dx^ν 
        - [ T(x)/ K]^2 dϕ^2
    
    where the dynamic T(x) replacing KR_c is known as a modulus field. In the minimal RS model, this
    is a free field and hence, as mentioned above, there is no constraint at all 
    on K R_c = ⟨ T(x) ⟩. Goldberger and Wise then augmented 
    the model by the introduction of a bulk scalar B(x,y), with a mass M_B 
    and quartic self-interactions on the IR and UV branes, with vacuum expectation values V_IR and V_UV respectively – all 
    these mass-dimension quantities being in the ballpark of the Planck mass. They were then able 
    to show that the scalar modulus field T(x) develops a potential 
    with a minimum at
    
        ⟨ T(x) ⟩ =  K R_c ≃4/π( K/M_B)^2 lnV_UV/V_IR
    
    which can be easily tuned to the required value 11.6 by varying the unknowns
    M_B, V_IR and V_UV without having recourse to unnaturally large 
    or small numbers. This is consistent with the general philosophy of the RS model.
    
    The modulus field T(x), which is like a dilaton in the fifth dimension, can 
    be parametrised as a  radion
        φ(x) = Λ_φ  e^-π{T(x) -  KR_c}
    
    which has a vacuum
    expectation value
    
        Λ_φ = √(24M_5^3/ K) e^-π KR_c
    
    and a mass
    
        M^2_φ = 2 K^2/M_5^3(V_UV - V_IR)^2 
        e^-2π KR_c
    
    Because of the warp factor e^-π KR_c , both the radion mass M_φ 
    and the radion vacuum
    expectation value Λ_φ lie at or around the electroweak scale. Hence, 
    it is easier, for phenomenological purposes, to treat them as the free parameters 
    in the theory, rather than the set { K, M_5, V_UV, V_IR}. 
    It is also worth noting that if we let V_UV = V_IR, in which 
    case Eq. (<ref>) tells us that the radion is massless, we would also 
    have R_c = 0 from Eq. (<ref>), i.e. the two branes would coalesce
    and M_H immediately shoot up to M_5— which takes us back to the Standard Model 
    and the hierarchy problem. We conclude, therefore, that V_UV > V_IR and hence 
    the radion must be massive. 
    
    The interactions of the radion with matter on the IR brane will naturally 
    follow those of the dilaton (which it is a variant of) and can be written as
    
        L_ int(φ) = 1/Λ_φ φ( T_μ^μ 
        +  A_T )
     
    where T_μν is the tree-level energy-momentum tensor and A_T 
    is the trace anomaly. For on-shell particles, the tree-level T_μ^μ has 
    the explicit form 
    
        T_μ^μ  = ∑_f m_f f̅ f + M_H^2 H^2 - 2M_W^2 W^+μW^-_μ 
        - M_Z^2 Z^μ Z_μ
    
    where the sum runs over all fermions f. This, apart from the 
    A_T term,  is exactly like the coupling of 
    the Higgs boson, except that the SM vacuum expectation value v is replaced 
    by the radion vacuum expectation value Λ_φ. Not surprisingly, 
    radion phenomenology is very similar to Higgs boson phenomenology. It differs,
    however, in the anomaly term
    
        A_T = ∑_i β(g_i)/2g_i  F^μν i F_μν^i
    
    where β(g_i) is the beta function corresponding to the coupling g_i
    of the gauge field A_i which has the field strength tensor F_μν^i.
    The sum over i runs over all the gauge fields in the SM, including photons,
    gluons and W^± and Z bosons. The A_T term induces 
    substantial couplings of the 
    radion to γγ and gg pairs, which are completely absent in 
    Eq. (<ref>). On the other hand, similar anomaly-induced 
    contributions to radion couplings with W^+W^- and ZZ pairs are 
    usually negligible 
    compared to the corresponding terms in Eq. (<ref>), because
    of the large masses of these particles, and only become significant when their tree-level couplings to one of the scalars vanishes.  
    
    Like the Higgs boson, the tree-level radion couplings in Eqn. (<ref>) would
    be subject, in addition to the trace anomaly contributions, to radiative corrections,
    especially from loops involving the top quark. Moreover, it is worth mentioning that there could be large brane corrections to the above couplings if the mass of the radion 
    is comparable to the Kaluza-Klein scale <cit.>, determined by the mass of the 
    lightest graviton mode in the minimal RS construction. To avoid this, we require a radion 
    which is comparatively light, and this requires a modest level of fine tuning 
    <cit.>. The discussions in this article are, therefore, subject to this assumption.
    
    As remarked above, the phenomenological behaviour of such a light radion is rather
    similar to that of the Higgs boson. This naturally leads one to ask whether
    these two low-lying elementary scalar states can  mix, since they carry the 
    same set of conserved quantum numbers, once the electroweak symmetry has been broken. In fact, this is possible, as was first pointed out in
    Ref. <cit.> and has been discussed by many others<cit.>. 
    Before proceeding further, it may be noted that there are several  phenomenological models with fermions and gauge bosons accessing the bulk<cit.>, which have better control over the flavour problem. In these models, the top quark remains close to the TeV brane along with the Higgs field while the other fermions are close to the UV brane. This suppresses the 
    higher-dimensional operators contributing to flavour-changing neutral currents, since the effective interaction of fermions with the Higgs field is governed by the overlap of their profiles and hence this scenario 
    naturally generates the 
    pattern of fermion masses and mixings. These models predict heavy Kaluza-Klein particles on the TeV brane having masses in the range of a TeV. However, the radion and Higgs fields, being still close to the TeV brane, 
    mix more-or-less without bulk effects <cit.>. 
    Hence, the mixing can be understood fairly 
    accurately using a minimal model 
    where all the relevant particles are confined to the TeV brane[The only caveat to this is the fact that heavy Kaluza-Klein excitations of the
    top quark may contribute to Higgs production at a hadron collider through loop diagrams. However, if these excitations are at the level of a TeV, the corresponding loop
    contributions are not more than a few percent and may be safely neglected
    — as we have done in this work.], for this is, after all, no more than
    approximating a sharply-peaked function by a delta function. 
    
    In the following section, therefore, we briefly discuss, following Refs. <cit.> how the radion-Higgs field mixing may be 
    described in terms of a single
    mixing parameter ξ. The next section then describes constraints on the
    mixed Higgs-Radion scenario, as obtained using all experimental inputs
    currently available, especially those from the LHC. For easy
    comparison, we include projections of the discovery reach of the LHC alongside
    the current constraints. Before concluding, we include a short section on
    the so-called `conformal point' near ξ = 1/6, which has unique features. 
    While some of the observations in this paper echo previous ones<cit.>, the data
    used are current, leading to new bounds, and, for ease of reading, we have
    presented our findings in a manner such that this paper can be read as far
    as possible independently of the preceding literature.  
    
    
     2. Radion-Higgs mixing
    
    Mixing of the radion field φ(x) with the Higgs scalar h(x) of the SM
    has been discussed by several authors <cit.>, with the same broad
    features, but we choose to closely follow the formalism of 
    Ref. <cit.>.
    
    The mixing occurs through the kinetic terms
    
        L = 1/2∂^μ h  ∂_μ h 
        - 1/2 M_h^2 h^2                                                                                                                                                                                                                                                                                                        + β/2∂^μφ ∂_μφ - 1/2 M_φ^2 φ^2                                                                                                                                                                                                                                                                                                        
         + 6γξ ∂^μφ ∂_μ h
    
    where γ≡ v/Λ_φ, v being the SM Higgs vacuum
    expectation value. In this
    formalism, the mixing parameter appears twice – once in the mixing term
    6γξ ∂^μφ ∂_μ h, and once in the
    non-canonical normalisation β≡ 1 + 6γ^2ξ of the radion 
    kinetic term. As is usual, the Higgs boson mass is given by 
    M_h^2 = 2λ v^2, 
    where λ is the Higgs quartic coupling and v is the Higgs vacuum
    expectation value. 
    
    We note that the presence of the non-canonical normalisation β means that
    the identification of physical states H and Φ will involve a scaling
    as well as a rotation of states, i.e. a non-unitary transformation. Hence, we 
    write the unphysical states φ, h as linear combinations of the
    physical ones Φ, H, with real coefficients A,B,C and D, thus
    
        φ    =     A Φ + B H 
        
        h     =     C Φ + D H  ,
    
    where the coefficients A,B,C and D are given by
    
        A     =     -1/Zcosθ              
        B   =  1/Zsinθ
        
        C     =    sinθ + 6γξ/Zcosθ        
        D   =   cosθ - 6γξ/Zsinθ
    
    in terms of
    
        Z^2 = β - (6γξ)^2
    
    and a mixing angle θ, defined by
    
        tan 2θ = 12γξ Z M_h^2/M_φ^2 - M_h^2 (Z^2 - 36γ^2ξ^2)
    
    The mixing parameter ξ is immediately constrained by the requirement that
    Z^2 > 0 to get a real mixing angle.
    The mass eigenvalues of the physical eigenstates Φ and H 
    are now given by
    
        M^2_Φ,H = 1/2Z^2( M^2_φ + β M^2_h
        ±√((M^2_φ + β M^2_h)^2 - 4Z^2 M^2_φ M^2_h))
    
    where the sign is chosen to ensure that M_H < M_Φ. We identify the
    lighter state H as the scalar state of mass around 125 GeV which was 
    discovered at the CERN LHC in 2012, while the other state Φ is
    a heavier scalar state predicted in the model.
    From these formulae, it is clear that the free parameters in question are
    M_h, M_φ, Λ_φ and ξ, everything else being 
    computable in terms of them. We also note in passing that since M^2_h = 
    2λ v^2, this makes 
    the Higgs quartic coupling λ an unknown quantity in this model, just 
    as it used to 
    be in the Standard Model before the identification of the 125 GeV scalar
    with the Higgs boson[This is a reflection of the fact that we
    still do not have a direct measurement of λ. All that we have is the 
    estimate λ = (125  GeV)^2/2v^2 ≃ 0.129— which is true only if the 125 GeV state is purely a SM Higgs boson without any admixture of new states.]. 
    
    Instead of the Lagrangian parameters M_h and M_φ, however, we 
    find it 
    more convenient to use the physical masses M_H and M_Φ, which can 
    be traded for the previous two by some simple algebra, leading to
    
        M_φ^2 = Z^2/2[ M_Φ^2 + M_H^2 +
        √((M_Φ^2 + M_H^2)^2 - 4β M_Φ^2 M_H^2/Z^2)]
        
         
        M_h^2 = Z^2/2β[ M_Φ^2 + M_H^2 -
        √((M_Φ^2 + M_H^2)^2 - 4β M_Φ^2 M_H^2/Z^2)]
    
    Since we 
    identify M_H = 125 GeV, we are left with a set of only three independent 
    parameters, viz. M_Φ, Λ_φ and ξ. The rest of our 
    analysis will be presented in terms of these variables.
    
    We now have another theoretical constraint, apart from Z^2 > 0. This
    is the requirement that the parameters M_φ and M_h be real 
    (to keep the Lagrangian Hermitian), which automatically means that
    
        (M_Φ^2 + M_H^2)^2 > 4β M_Φ^2 M_H^2/Z^2
    
    Imposing both these constraints reduces the possible range of ξ,
    for a given M_Φ and Λ_φ, quite significantly 
    (see below).
    
    Since the mixing of the h and the φ to produce the physical H and
    the Φ is non-unitary, we define two mixing indicators as follows.
    We first invert Eq. (<ref>) to write
    
        Φ    =     a φ + b h 
        
        H     =     c φ + d h  ,
    
    where
    
        ( [ a b; c d ]) 
        = ( [ A B; C D ])^-1 .
    
    In terms of this, we now define indicators
    
        f_φ/H = |c|/|c|+|d|        
        f_h/Φ = |b|/|a|+|b|
    
    which, in a sense, indicate the fraction of radion φ in the light state H, and the fraction of Higgs boson h in the heavy state Φ. These, together with the mixing angle θ defined in Eq. (<ref>), are plotted in Fig. <ref>, as a function of the mixing parameter ξ. 
    
    
    
    
    
    
    
    In each of the three panels in Fig. <ref>, we have four boxes placed
    one above the other, corresponding to choices of four
    different values of the radion vacuum
    expectation value, viz. Λ_φ = 1, 5, 10 and 20 TeV respectively (marked in the respective boxes). Within each box, the curves
    are colour-coded, with black, green, red and blue indicating benchmark 
    choices of the
    heavy scalar mass as M_Φ = 250 GeV, 500 GeV, 750 GeV and 1 TeV
    respectively (indicated at the top of the figure). Each curve ends abruptly
    at some maximum and minimum values of the mixing parameter ξ– this is 
    a reflection of the theoretical limitations (see above). As may be seen from the different plots, this restriction
    is extremely stringent when Λ_φ is small, and even when we push 
    Λ_φ  as high as 20, does not permit the value of |ξ| to exceed
    15. If we consider the panel on the left, it is clear that we get significant 
    values of the mixing angle θ only when the heavy Φ state is as light 
    as around 250 GeV. For values of M_Φ of 500 GeV or greater, θ does 
    not exceed 10^0. However, since the mixing is not unitary, the smallness of 
    θ is not necessarily an indicator of small mixing. This becomes clear
    if we look at the central and right panels of Fig. <ref>, which
    tell us the proportion of the radion in the 125 GeV state, and the proportion of
    the Higgs boson in the heavier state respectively. In each case, as |ξ|
    increases, the mixing becomes more, starting from zero when |ξ| = 0 to
    about equal mixtures when |ξ| reaches its maximum theoretically-permitted value. The purpose of this paper is, as explained above, to 
    see how far such large mixings are allowed in the light of current experimental data. 
      
    We next consider the effect of mixing on the couplings of the two scalar
    states to the SM fields. As shown in Ref. <cit.>, the tree-level
    couplings 
    of the heavy Φ state to pairs of SM fields XX̅ (except
    X = H) have the form
    
        g_Φ XX̅ = g_φ XX̅( C + γ A )
        ≡ c_Φ g_φ XX̅
    
    where g_φ XX̅ can be read off from Eqs. (<ref>–<ref>), and c_Φ = C + γ A is a scaling factor. 
    Similarly, the couplings of the light 125 GeV state have the form
    
        g_H XX̅ = g_h XX̅( D + γ B )
        ≡ c_H  g_h XX̅
     
    where g_h XX̅ are the SM couplings and c_H = D + γ B is
    a scaling factor. Very different from these is the coupling of the heavy
    scalar to a pair of light scalars, since all three fields 
    are mixed states, and this
    can be written <cit.> for a Φ(p)-H(k_1)-H(k_2) vertex, as
    
        g_HH    =    1/Λ_φ[ (k_1^2 + k_2^2 )
        { AD^2 + 6ξ B( CD + γ AD + γ BC ) }.  
               + . D{ 12γξ AB + 2BC + (6ξ - 1)AD  } p^2
        - 4M_h^2 D(AD + 2BC) - 3 M_h^2CD^2/γ]
    
    The couplings of the scalars H and Φ with other particles are
    conveniently listed in the Appendix of Ref. <cit.>. 
     
    
    
    
    To get a feeling of how these couplings are affected by the variation
    in the basic parameters ξ, Λ_φ and M_Φ, we plot
    them in Fig. <ref> on a scheme similar to that in Fig. <ref>. The three panels show, from left to right, the 
    scaling factors c_Φ and c_H, and the coupling g_HH respectively.
    As in Fig. <ref> it is immediately clear that for ξ = 0,
    c_Φ is very small (small enough to appear as zero on this scale), 
    as befits a radion with a small coupling to matter, whereas c_H = 1 indicating that the lighter scalar is the SM Higgs boson. Similarly,
    for ξ = 0, 
    the g_HH coupling is very small (small enough to appear as zero on 
    this scale), indicating that the heavy scalar couples only weakly to a 
    pair of light scalars. There are also genuine zeroes in the couplings, which
    are discussed in more detail in Section 4.     
     
    An interesting feature of both Fig. <ref> and Fig. <ref> is the fact that the variation in parameters
    is rather slow for smaller values of ξ, but is very sharp
    for larger values just before the unphysical region. These larger
    values of the scaling factor and Φ HH coupling are likely to
    have phenomenological consequences at observable levels, and hence
    are more likely to be constrained by experimental data. In the
    next section, we shall see that this is indeed the case.   
    
    
    
     3. Experimental Constraints
    
    We are now in a position to apply the experimental constraints to
    this model. Since the two scalars H and Φ are the crucial elements, 
    the main constraints will come from
    
    
      (a) the measured signal strengths μ_XX of the 125 GeV
    scalar in its decay channels to XX̅ pairs – these are known to
    match reasonably closely to the SM predictions, leaving only limited room
    for a mixed state;
    
      (b) the lack of signals for a heavy scalar in the range of a few hundred
    GeV to about a TeV – by implication, any new scalar would be very heavy
    and mix only marginally with the SM Higgs boson. 
    
    
    In principle, the scalars could also contribute as virtual states to 
    any neutral current processes. However, as most of these are suppressed
    by the small masses of the initial states (either e^± or u and d 
    quarks), we do not really get any useful constraints from these processes.
    Constraints from electroweak precision tests are not very strong 
    <cit.>.
    In the rest of this sections, therefore, we concentrate on the two
    issues listed above.
    
    
    
    
    We first take up the signal strengths of the 125 GeV scalar H. This decays
    into several channels
    
        H ⟶ X + X̅
    
    where X = ℓ^-, u, d, s, c, b, W, Z, γ, g with one of X or 
    X̅
    being off-shell in the case of W and Z. At the LHC, the H is produced 
    dominantly through gluon-gluon fusion[In our numerical analysis, we 
    have also included the vector boson fusion mode.]. Hence, we can define signal 
    strengths μ_XX as
    
        μ_XX = 
        σ(pp → gg → H)_ exp ℬ(H → XX̅)_ exp/σ(pp → gg → H)_ SM ℬ(H → XX̅)_ SM
    
    where σ and B stand for cross-section and branching 
    ratio respectively, and the subscripts `SM' and `exp' mean the SM 
    prediction and the experimental 
    value respectively. If we are making a theoretical prediction, then `exp' will
    stand for the expected value in the theoretical model in question — in the 
    present case, the model with radion-Higgs mixing. Of course, in an experiment
    only the entire numerator on the right side of Eq. (<ref>) can be 
    measured and not the individual factors. By this definition, then, all the
    SM signal strengths are normalised to unity, and experimental deviations from 
    it constitute the leeway for new physics. These allowed experimental deviations 
    are given in Table <ref>.
    
    
    
    
    Obviously, for zero mixing, the signal strengths predicted for the H
    scalar will be the same as the SM values, i.e. unity. As ξ increases,
    we  should expect deviations from unity, and indeed that is what happens,
    as illustrated, in Fig. <ref>. The three panels, from
    left to right, correspond to choices of M_Φ = 250, 500 and 750 GeV 
    respectively. The graph for M_Φ = 1 TeV is very similar to that for 
    M_Φ = 750 GeV, and hence we do not show it explicitly. Likewise,
    the actual graphs for μ_γγ are slightly different, but not 
    enough to show up on a plot at this scale. Each curve
    in the panels corresponds to the value of Λ_φ, in TeV, written
    alongside, i.e. 1, 2, 3, 5 and 10 TeV respectively. The steepness of
    the curves decreases with increasing Λ_φ, for which we also
    have larger permitted ranges in ξ, as we have earlier shown in
    Fig. <ref>. Horizontal broken lines in 
    Fig. <ref> represent the useful 95% C.L. constraints from
    the signal strengths in Table. <ref>, and are marked on the 
    right side of the figure.  
    
    The behaviour of the predicted signal strengths with increasing ξ is 
    quite as expected, remaining close to the SM value for small ξ and
    showing large deviations near the edge of the theoretically-allowed range.
    This, as we have seen earlier, is due to the large deviations of the coupling
    of the H from the SM coupling at such values of ξ. It is thus obvious
    that the present constraints from signal strengths will only affect narrow
    strips of the parameter space adjacent to the theoretically-disallowed region,
    and this, in fact, is what we find (see below). It may be noted in passing
    that a region of the parameter space where D + γ B ≃ 0 would be
    very strongly constrained from the signal strengths, but this does not happen
    anywhere inside the region allowed by theoretical considerations.  
    
    When we turn to the heavy Φ state, once again the main production mode 
    is through gluon-gluon fusion, but now there is no analogous SM prediction and hence one looks for the direct signals in
    the various decay channels of the Φ. As in the case of the light scalar, the
    potentially observable ones are Φ→γγ<cit.>,
    WW<cit.>,
    ZZ<cit.> and 
    τ^+τ^-<cit.> to which we can now add Φ→ tt̅ and Φ→ HH<cit.>.
    The bb̅<cit.> signal would be difficult to distinguish from the QCD
    background, unless the mass of the Φ scalar is very well known, as in the 
    case of the H scalar. The behaviour of all these branching ratios, as
    functions of the scalar mass M_Φ is shown in Fig. <ref>,
    where Λ_φ is fixed to 5 TeV and the panels, from left to right,
    correspond to ξ = 0 (no mixing), and ξ = 1, 2 and 3 respectively.
    The relevant decay channel is marked alongside each curve. These curves
    terminate at the left end where they correspond to theoretically-disallowed 
    regions in the parameter space. 
    
    
    
    
    One feature which is immediately obvious from these curves is the fact
    that the scalar Φ decays dominantly through the WW and ZZ channels.
    When the mixing is low, the HH channel is also competitive, but as ξ
    rises, it gets suppressed. In any case, the signals from the WW and ZZ
    channels are leptonic and clean, whereas the signals arising from HH,
    dominantly leading to 4b final states, are hadronic, as are those arising
    from the direct decays of 
    the Φ into quark pairs. These hadronic channels are generally suppressed 
    compared to WW and ZZ, and, in any case, would be plagued by large QCD 
    backgrounds. It may be still possible to investigate the tt̅ and HH
    channels, using jet substructure-based tagging methods for boosted particles, 
    but such experimental searches are still not competitive 
    <cit.>. Thus, in principle, we get 
    constraints from every decay channel of the Φ, but the most
    useful ones will arise from  the ATLAS and CMS search results for a heavy 
    scalar resonance decaying
    to WW and ZZ pairs, which are equally applicable to the Φ scalar
    in the model under consideration. As is well-known, the experimental results are all negative, and hence the 95% C.L. upper limits on the cross-section 
    are given in Table <ref>.
    
    
    
    
    We are now in a position to compare these data with the predictions of our
    theory. As in the case of the H state, the cross section for 
    pp →Φ→ VV, where V = W,Z, can be written
    
        σ(pp →Φ→ VV) = σ(pp → gg →Φ) 
         ℬ(Φ→ VV)
    
    where ℬ(Φ→ VV) is the branching ratio of the Φ to a 
    VV pair. These can be calculated in terms of the free parameters 
    ξ, M_Φ and Λ_φ respectively. Our results are shown
    in Fig. <ref>. 
    
    
    
    The four upper panels of Fig. <ref> represent the cross-section, in pb, for the process
    pp →Φ→ WW and the lower four panels represent the process
    pp →Φ→ ZZ. In each row the panels correspond, from left to 
    right, to M_Φ = 250 GeV, 500 GeV, 750 GeV and 1 TeV, respectively.
    Within each panel, the curves show the variation of the cross-section
    with the mixing parameter ξ, for different values of the radion 
    vacuum
    expectation value, corresponding to different colours, as marked in the legend above
    the panels. The horizontal solid lines correspond to the CMS bounds from
    the 13 TeV data, as shown in Table. <ref>, while the broken lines
    correspond to the ATLAS 13 TeV data. 
    
    All the curves have a distinct 
    minimum at a small value of ξ varying from 0.2 to 2 — this corresponds
    to a minimum in the cross-section σ(pp → gg →Φ) where
    there is maximal cancellation in the amplitude for gg →Φ
    due to the top quark loop and the trace anomaly term. In this region, the heavy scalar can be produced in association with a W^±/Z and it further decays to WW or ZZ pairs,
    leading to a final state with three gauge bosons or their decay products. 
    In view of the low production cross-sections for higher values of Λ_φ, one has to consider hadronic decays of one or more of these gauge
    bosons, and this immediately invites a large QCD background at the LHC. 
    However, the region can be successfully probed at a high energy 
    e^+ e^- collider (such as the proposed ILC) with √(s) = 1 TeV <cit.>.
    
    In addition to the dip described above, there is a very sharp minimum,
    very close to the vertical axis, which corresponds to the so-called 
    `conformal' point, where c_Φ→ 0. We defer the discussion of 
    this point to the next section and focus here on the constraints obtainable
    from the rest of the parameter space. Here, as in the case of signal 
    strengths the constraints rule out larger values of ξ, with the exact
    bound depending on the other two parameters of the theory.  
    
    From Figs. <ref> and <ref> we can
    draw some general conclusions. The first is that the effect of increasing
    the mixing parameter ξ becomes weaker and weaker as the vacuum
    expectation value Λ_φ keeps increasing. This is true both for the signal strengths in 
    Fig. <ref> as well the cross-section in Fig. <ref> and is easy to track down as due to the limiting
    case γ→ 0. A similar argument may be made for the parameter
    M_Φ– at least numerically – though the parameter dependence here is
    much more complicated.
    
    
     We may argue, therefore, that for a fixed ξ,
    the region with small M_Φ and small Λ_φ is more constrained
    — which also corresponds to the commonsense argument that if these 
    parameters are small, radion-mediated processes are large and vice versa.
    These expectations are corroborated by our results shown in 
    Fig. <ref>. Here we show the Λ_φ–M_Φ plane
    for four different values of ξ, viz. ξ = -0.5, 0, 1 and 1.5, 
    as marked
    on each panel. As indicated in the key at the top, the region shaded grey
    corresponds to the theoretically disallowed region, and includes all values of 
    Λ_φ < 1 TeV, except in the panel on the top left, 
    marked ξ = 0, which corresponds to the case of an un-mixed radion
    of mass M_Φ.  
    Here, though values of Λ_φ < 1
    TeV are theoretically permitted, the experimental constraints do not allow them, 
    as is apparent from the figure. In all the panels, the dark grey shaded region
    is ruled out by the signal strengths at Runs 1 and 2 and the hatched regions by the
    ATLAS and CMS searches for a heavy scalar at Run-2 of the LHC. 
    These are the strongest 
    constraints and represent the state of the art as far as current 
    experimental data are concerned[We have, in fact, considered
    constraints from all the channels separately, but the others are
    subsumed in the ones shown in the figure, and hence are not shown in
    order to have uncluttered figures.]. The jagged shape of the curves reflects
    the fact that the LHC has, till now, collected quite a small amount 
    of data for rare processes like the decay of a heavy scalar. 
    However, the LHC has the potential to
    search much further, and this is shown by the red and yellow-shaded regions,
    which represent, respectively, the expectations from the signal strength measurements if μ_XX = 1 ± 0.05 for all X, and the ATLAS and
    CMS discovery limits at 95% C.L. for the heavy Φ if the
    LHC were to run at 14 TeV and collect 3000 fb^-1 of data<cit.>— which
    may not be too far from the reality. For the panel with ξ = 0, there
    are no constraints from the signal strengths, since the H is completely
    SM-like; but the constraints from the heavy scalar searches are quite strong
    because that scalar is a pure radion. A comparative study of the four plots
    indicates that the value ξ≈ 1 would permit the largest part of
    the parameter space to survive consistently negative results from LHC, while
    negative values of ξ are better suited to a discovery of the heavy scalar
    predicted in this theory.     
    
    
    
    
    Coming to constraints on ξ, it is clear from Figs. <ref> 
    and <ref> that ξ = 0, which 
    corresponds to the 125 GeV scalar being the Standard Model Higgs boson —
    not surprisingly — is always allowed by the signal strength data. For given 
    values of M_Φ and Λ_φ, 
    ξ can range on the positive and negative side, but when its magnitude
    grows larger, all new physics effects grow and, at some point, higher
    magnitudes of ξ get disallowed – first by the experimental constraints
    and then by the requirement of theoretical consistency. For low values of 
    Λ_φ and M_Φ, we arrive at this point for fairly low values of
    ξ. As both these parameters increase, however, the allowed range
    grows, creating a funnel-like shape, which grows wider as Λ_φ and 
    M_Φ increase. This is illustrated in Fig. <ref>, where we 
    show the Λ_φ-ξ plane for the same choices of M_Φ as in
    the earlier figures. The shading and hatching conventions of this figure
    are exactly the same as those of Fig. <ref>. It is immediately
    obvious that for low values of Λ_φ close to 1 TeV, the range of
    ξ is severely constrained by theoretical consistency alone. A heavy 
    scalar of mass 250 GeV is also rather severely constrained, except for 
    a narrow cone, which will shrink further when the LHC finishes its run.
    Constraints ease up for a heavier scalar, since that is much more difficult
    to find. It is interesting that even if LHC completes its run without
    finding any evidence for a heavy scalar up to 1 TeV, there will be a range
    of parameter space where this model is still allowed. However, for these
    parameters, the 125 GeV will be so similar to the SM Higgs boson, and the
    interactions of the heavy scalar will be so heavily suppressed that the
    model may no longer be interesting, at least from a phenomenological point 
    of view.
    
    An interesting feature of all the plots in Fig. <ref> is the
    needle-thin sliver of allowed parameter space which appears in every graph
    close to the vertical axis. This corresponds, in every case, to the 
    `conformal point' mentioned above, where all constraints from a heavy
    scalar search weaken considerably. This region – though extremely fine-tuned – 
    is interesting in its own right, and therefore we carry out a detailed study in
    the next section. 
    
    
     4. The Conformal Point
    
    As explained before, for every choice of M_Φ and Λ_φ, there is 
    a fixed value ξ = ξ_0 which satisfies the equation c_Φ = 0, and 
    hence
    
        C(ξ) + γ A(ξ) = 0
     
    and this is known as the `conformal' point[From this stage we drop
    the quotes on `conformal'.]. It corresponds to the case when the tree-level
    couplings g_Φ XX̅ of both the fermions and gauge bosons – generically
    denoted X– with the heavy scalar Φ vanish. This is a curious situation
    and corresponds to the case when the mixing is fine-tuned to be such that the 
    parts of the coupling arising from the SM h and the radion φ cancel 
    each other. Like all fine-tuned situations, if this is the reality, it can
    hardly
    be a random effect, and must represent some deeper structure in the theory, which
    is not addressed in our present formulation. Nevertheless, it is interesting to 
    explore the phenomenological implications of this scenario. In this section,
    therefore, we investigate the conformal point and see how it can be constrained using current and projected data, just as the other points can. 
    It is important to note that though most of the tree-level couplings
    of the Φ to pairs of SM particles vanish at the conformal point (except
    for the coupling to HH pairs), there exist one-loop couplings to pairs
    of gauge bosons through the trace anomaly. This makes the pattern of 
    branching ratios at the conformal point very different from that in
    other regions of parameter space. The most important feature of this is
    the fact that the decays Φ→ gg and Φ→γγ are 
    considerably enhanced with respect to the others – in fact the former
    is the dominant decay mode. This behaviour is nicely exhibited in 
    Fig. <ref>, where we exhibit the behaviour of the relevant 
    branching ratios in the immediate vicinity of the conformal point. 
    
    
     
    In Fig. <ref>, it is immediately apparent that for the particular
    value ξ = ξ_0, the tree-level decay modes of Φ→ XX̅,
    where X is a massive gauge boson or a fermion, drop sharply by many orders of
    magnitude. This is particularly true for the cases X = t, b and H,
    with the minimum for  the last case occurring at a slightly displaced
    point from the others (best seen in the zoomed panel on the right). 
    On the other hand, the branching ratios for the purely
    one-loop decays, viz. Φ→ gg and Φ→γγ exhibit
    a growth at the same point, attributable to their partial decay widths being
    finite, whereas the others drop almost to zero. However, the decays to WW and
    ZZ states do not disappear altogether because they too have anomaly
    contributions. Naturally the decay Φ→ gg dominates the others because
    of the appearance of the strong coupling as well as the colour factor. The
    decay Φ→γγ also shows a gentle increase, but is intrinsically
    much more rare than the digluon mode. At the conformal point, therefore, constraints on the model will have to be sought in a different 
    fashion. One obvious way is to consider Higgs boson signal
    strengths, for if the couplings of the Φ vanish that does not mean that 
    the couplings of the H will also vanish. 
    Accordingly, there will be contributions to the signal
    strengths and these can be used to constrain the model. In fact, even the
    heavy scalar searches, i.e. pp → S → VV, where V = W,Z can be used
    to a limited extent, since the branching ratios Φ→ VV, though small
    at ξ = ξ_0, are not absolutely negligible. However – and this is a
    distinct feature of the conformal point – the strongest bounds come from 
    diphoton searches, which is not entirely surprising, given that this mode is 
    considerably enhanced at the conformal point.  
    
    In trying to understand how the conformal point is constrained by the data,
    we need to recognise that the conformal point ξ_0 is not unique, but a 
    function of M_Φ and Λ_φ, with the dependence on the
    former being much stronger than that on the latter. Its variation with M_Φ is shown
    in the upper panel of  Fig. <ref>,  where the thickness
    of the line corresponds to variation of Λ_φ
    from 1 TeV to 20 TeV.  
    This plot shows that the variation flattens out as M_Φ
    grows above 500 GeV, and has a very weak dependence on Λ_φ. Nevertheless, we have scanned a sizeable portion
    of the M_Φ–Λ_φ plane
    and calculated the values of ξ_0 at every point by solving 
    Eq. (<ref>).
    
    
    
    
    
    
    improvement can be obtained if these 
    measurements yield results much closer to the SM prediction.
    The shaded yellow region represents the predictions from ZZ decay modes of
    a heavy scalar for the LHC running at 14 TeV with 3000 fb^-1<cit.> of data  
    (which is all that is currently available), and it does worse than the Run-2 data.
    It may be expected that diphoton searches would provide better discovery limits — when the Run-2 projections become available. 
    
    All in all, we can conclude that the 
    conformal point is somewhat less constrained than the rest of the parameter
    space. It was this narrow window which had been used <cit.> to 
    explain the purported discovery of a heavy 750 GeV scalar during 2015-2016 
    <cit.>,
    though that proto-signal did not survive the test of time<cit.>. 
    
    
     5. Summary and Outlook
    
    The minimal Randall-Sundrum model continues to be one of the most elegant ways of
    solving the hierarchy problem, and it works best if there is a Goldberger-Wise
    stabilisation, which works best if there is a light radion state. Though
    there are strong constraints on such a light radion per se, there remains room 
    for a light radion mixed with the SM Higgs boson to survive. In this article, 
    we have explored this possibility, using an existing formalism, in the light of 
    current data from the LHC Runs 1 and 2. Our findings are summarised below.
    
    The possibility of a radion-Higgs mixing arises essentially because we have no
    independent measurement of the Higgs boson self coupling λ, so
    that the SM formula M_h^2 = 2λ v^2 is open to other interpretations.
    One of these is the mixed radion-Higgs scenario, where the lighter eigenstate 
    is identified with the 125 GeV scalar discovered at the LHC. 
    In this model, there are three free parameters,
    viz. the mixing parameter ξ, the mass M_Φ of the heavy scalar Φ,
    and the radion vacuum expectation value Λ_φ. However, 
    self-consistency of the theory imposes fairly stringent constraints on the 
    choices of the mixing parameter ξ. These, as we show,
    are further constrained by (a) the signal strengths measured for the decays
    of the 125 GeV scalar at the LHC, and (b) the search for a heavy scalar
    decaying into a pair of electroweak vector bosons, be they W's, Z's or 
    photons. These lead to further bounds on the parameter space, essentially
    pushing Λ_φ above a TeV (and hence reducing all radion-mediated
    effects) and M_Φ to values closer to a TeV, though here some avenues
    for a lighter M_Φ remain. 
    
    In addition to the current data, we have tried to predict discovery limits
    at the LHC in two ways. One way is to use the signal strengths, and assume
    that they will eventually converge within 5% of the SM prediction. This
    leads to modestly enhanced bounds on the radion-Higgs mixing scenario. The
    other way is to use the projected discovery limits from the ATLAS and CMS 
    Collaborations for a heavy scalar in Run-2, where we identify that heavy
    scalar with our heavier eigenstate Φ. This, in fact, is very effective
    for most choices of the mixing parameter ξ and is sensitive to rather high 
    values of M_Φ and Λ_φ. The only exception is at the so-called
    conformal point, which is a peculiar feature of this model, involving a value of 
    the mixing parameter where the heavy scalar essentially decouples from SM fields.
    Even this is constrained, however, by the signal strengths and by the diphoton
    decay mode, which, being generated by the trace anomaly, survives the vanishing
    of tree-level couplings. However, the smallest values of M_Φ and 
    Λ_φ are, indeed, allowed if this scenario were to be true.  
    
    It is interesting to ask how our results would be modified if we replace
    the simplistic model used above with a more phenomenologically-relevant
    model where the fields can access the bulk. As explained in the Introduction,
    the radion and Higgs fields, being still close to the TeV brane, mix in the same manner <cit.>. The decay of the radion to the 
    light quarks is severely suppressed because of the small overlap <cit.> of their wavefunctions in the bulk. Decays of the radion to massive gauge bosons are governed by an additional coupling that can be safely neglected for Λ_φ 1 TeV.  
    Radions decaying to massless gauge boson pairs (especially to diphotons) 
    is significantly enhanced, however, due 
    to the tree-level coupling in the case of bulk scenario. However, this doesn't really effect our region of interest <cit.>. We
    feel, therefore, that the results of this work are robust against
    more realistic variations of the minimal model and may be safely adopted 
    in such cases.   
    
    To conclude, then, we have shown that a mixed radion-Higgs scenario is quite
    consistent with the current experimental data at the LHC, and there is every
    possibility that the heavy scalar predicted in this model could be discovered
    as the LHC continues to run at its present energy of 13 TeV. Discovery of this
    would certainly be one of the most exciting things to happen in the near future,
    and, if, the branching ratios turn out to be consistent with this model, could
    provide a powerful insight into the nature of spacetime itself. Such a happy
    consummation is to be devoutly hoped for, but, for the present, we must reconcile
    ourself to a fairly long wait as the Run-2 of the LHC continues. 
    
     Acknowledgements: The authors acknowledge useful discussions with Debjyoti 
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