Higgs-Stoponium Mixing Near the Stop-Antistop Threshold

Supersymmetric extensions of the standard model contain additional heavy neutral Higgs bosons that are coupled to heavy scalar top quarks (stops). This system exhibits interesting field theoretic phenomena when the Higgs mass is close to the stop-antistop production threshold. Existing work in the literature has examined the digluon-to-diphoton cross section near threshold and has focused on enhancements in the cross section that might arise either from the perturbative contributions to the Higgs-to-digluon and Higgs-to-diphoton form factors or from mixing of the Higgs boson with stoponium states. Near threshold, enhancements in the relevant amplitudes that go as inverse powers of the stop-antistop relative velocity require resummations of perturbation theory and/or nonperturbative treatments. We present a complete formulation of threshold effects at leading order in the stop-antistop relative velocity in terms of nonrelativistic effective field theory. We give detailed numerical calculations for the case in which the stop-antistop Green's function is modeled with a Coulomb-Schr\"odinger Green's function. We find several general effects that do not appear in a purely perturbative treatment. Higgs-stop-antistop mixing effects displace physical masses from the threshold region, thereby rendering the perturbative threshold enhancements inoperative. In the case of large Higgs-stop-antistop couplings, the displacement of a physical state above threshold substantially increases its width, owing to its decay width to a stop-antistop pair, and greatly reduces its contribution to the cross section.


I. INTRODUCTION
In extensions of the standard model (SM), new heavy particles typically appear. For example, supersymmetric extensions of the SM include heavy stop quarks (stops), which are the supersymmetric partners of a top quark [1][2][3][4][5]. A stop quark and a stop antiquark (antistop) can bind to form a spectrum of stoponium states. The decays of these states into two photons potentially provide a clean signal for their detection. However, stoponium states typically have rather small gluon-fusion cross sections times branching ratios into two photons [6][7][8][9][10][11][12].
Extensions of the SM can also contain heavy Higgs bosons [13]. The presence of new Higgs doublets is motivated by weak-scale extensions of the SM that aim to address the disparity between the electroweak and Planck scales and to provide explanations of the origins of flavor and of the matter-antimatter asymmetry. In these theories, the SM description is recovered in the so-called decoupling regime, in which the masses of the heavy Higgs bosons become large. In such a regime, the heavy neutral Higgs bosons may decay into pairs of third-generation fermions, including top quarks. Owing to the presence of these tree-level decays, the branching ratio of loop-induced decay processes is suppressed, making it difficult to observe the decay of the heavy Higgs boson to two photons.
The interplay of a heavy-stop-antistop system with a heavy Higgs boson whose mass is near the stop-antistop production threshold results in interesting and intricate new phenomena. Loop-induced processes may be enhanced in the presence of heavy quarks or squarks that are strongly coupled to the heavy Higgs boson. In supersymmetric extensions of the SM, the dimensionful coupling of stops to the heavy Higgs bosons is governed by the Higgsino mass parameter µ. Loop-induced processes may be significantly modified if the heavy quarks or squarks have masses that are comparable to that of the heavy Higgs boson. The modification to loop-induced processes may be especially important if there is a production threshold for heavy particle-antiparticle pairs that is close to the Higgs mass [14].
Near threshold, QCD perturbation-expansion contributions of relative order n may receive 1/v n enhancements, where v is half of the relative velocity between the stop and the antistop in the stop-antistop center-of-momentum (CM) frame. Such enhanced contributions are typically proportional to α n s (mtv)/v n , where α s is the strong-interaction running coupling and mt is the stop-quark mass. The presence of these enhanced contributions requires a resummation of the perturbative contributions, which, among other things, takes into account the formation of stop-antistop bound states.
The issue of gluon-fusion Higgs production and decay to diphotons at the stop-antistop production threshold has been addressed recently in Ref. [15]. The authors of Ref. [15] recognized the possibility that Higgs-stoponium mixing and the formation of stop-antistop bound states could have a significant effect on the gg → H → γγ rate. They pointed out, as is stressed in Ref. [16], that stoponium effects lead to Higgs-digluon (Hgg) and Higgs-diphoton (Hγγ) form factors that are enhanced relative to the form factors that are obtained in fixed-order perturbative calculations. We note, though, that the description in Refs. [15,16] does not explicitly account for all of the Higgs-stoponium mixing effects [6].
In this article, we provide a detailed analysis of the interplay between a heavy Higgs boson and a heavy stop-antistop pair for Higgs masses that are close to the stop-antistop production threshold. 1 We take into account threshold enhancements to the stop-antistop amplitudes by means of the analogues for scalar quarks of the effective field theories nonrelativistic QED (NRQED) [17] and nonrelativistic QCD (NRQCD) [18,19]. In this framework, we are also able to give a description of Higgs-stop-antistop mixing near threshold that incorporates fully the effects that are contained in the stop-antistop Green's function. 2 In order to make contact with recent numerical work on the Higgs-stop-antistop system we choose, in our numerical work, a stop mass of 375 GeV, and we assume that the Higgs mass is near the stop-antistop production threshold, which is 750 GeV. Much of the recent numerical work in the literature was motivated by initial results from the ATLAS and CMS experiments that showed excesses in rates for the process pp → γγ at diphoton masses around 750 GeV [20,21]. (For an extensive list of theoretical work that is related to this 1 In the cases of SM extensions that contain more than one stop quark, we will assume that the heaviest stop mass eigenstate is significantly heavier than the lightest one. This implies that the Higgs mass is far below the threshold for production of a heaviest stop and a heaviest antistop. Therefore, the heaviest-stop contribution becomes subdominant with respect to the lightest-stop contribution. We will consider the heaviest-stop contribution to be a perturbation to the rates that are computed in this work. It should be taken into account in precision studies. For large values of the trilinear Higgs-stop-antistop coupling, the coupling of the heavy stop to the Higgs boson is of the same order as and opposite in sign to the coupling of the lightest stop to the Higgs boson. Hence, the contribution of the heavy stop to the diphoton rate may be non-negligible in the regime of strong Higgs-stop-antistop coupling. 2 In Ref. [14], the effects of mixing between the Higgs boson and discrete stoponium resonances were considered. signal, see Ref. [22].) However, we emphasize that our work is not tied to any particular phenomenological model and that it is aimed at understanding the general features of a Higgs-stop-antistop system near threshold.
We find that effects of Higgs-stop-antistop mixing go beyond the modification of the Hgg and Hγγ form factors and significantly change the diphoton production rate near the stop-antistop threshold with respect to the rate that would be obtained from the simple addition of the Higgs and stoponium contributions. For Higgs masses near threshold, we find that several mechanisms that arise from Higgs-stop-antistop mixing suppress the diphoton rate relative to the rates that are obtained in perturbative calculations: (1)  In some of our numerical work, we employ rather large values of the Higgs-stop-antistop coupling. In some specific models of the Higgs-stop-antistop system, such large values of the coupling could lead to the presence of color-symmetry-breaking minima in the potential for the vacua [23][24][25]. Since, in our work, we are not focused on any specific model realization of the Higgs-stop-antistop system, we will not study these possible constraints. However, they would have to be taken into account in the construction of detailed models.
The remainder of this article is organized as follows. In Sec. II, we describe the effectivefield-theory approach that we employ. We also give formulas for the gg → γγ amplitude that account fully for the Higgs-stop-antistop mixing. Section III contains a discussion of a simplified model in which the stop-antistop states are replaced by a single Breit-Wigner resonance. That model exhibits a number of the features of the full theory. In Sec. IV, we present and discuss our numerical results for the gg → γγ amplitude and cross section as a function of the Higgs mass, relative to the stop-antistop threshold. Section V contains our conclusions.

A. NRQED/NRQCD analogues for scalar quarks
We wish to compute the amplitude for gg → γγ in the presence of a Higgs boson with mass m H that couples to a heavy stop quark and a heavy antistop quark, each of which have mass mt. We are concerned with the situation in which m H is near 2mt, the threshold for stop-antistop production. Because the amplitude is computed near threshold, there can be important effects from the binding or near binding of the stop-antistop pair that are not captured in fixed-order perturbation theory. It is convenient to take these effects into account by making use of the analogues for scalar quarks of the effective field theories NRQED [17] and NRQCD [18,19]. 3 We will carry out the effective-field-theory computation at the leading nontrivial order in the heavy-stop velocity v in the stop-antistop CM frame, where v is given by Here p is the 3-momentum of the stop in the stop-antistop (or γγ) CM frame, is the nonrelativistic CM energy of the stop-antistop system, √ŝ is the partonic CM energy, and m γγ is the γγ mass.
The effective field theory is an expansion in powers of v. Hence, our calculation should be valid as long as v is much less than 1. We expect corrections to our calculation of the gg → γγ amplitude to be of relative order v 2 .
The heavy-stop part of the effective Lagrangian density that we will use, which is valid 3 In Ref. [26], an effective field theory for stoponium systems was developed, and a resummation of threshold logarithms in the stoponium production cross section was carried out by making use of soft-collinear effective theory [27]. However, the formalism in Ref. [26] does not address the possibility of stoponium-Higgs mixing.
at the leading order in v, is where mt is the stop pole mass, ψ is the field that annihilates a stop, χ is the field that creates an antistop, G a µ is the gluon field with adjoint color index a, G a µν is the gluon field strength with adjoint color index a, A µ is the electromagnetic field, and F µν is the electromagnetic field strength. The covariant derivative contains both the electromagnetic field and the gluon field: where e is the electromagnetic coupling, et is the stop-quark charge, g is the stronginteraction coupling, and t a is an SU(3) matrix in the adjoint representation that is normalized to Tr In the calculations in this paper, we ignore the couplings of stops and antistops to the electromagnetic field, except in the annihilation of a stop-antistop pair into two photons.
The C i 's are short-distance coefficients, which will be determined by matching the effective theory with full QED and full QCD at the stop-antistop threshold. The short-distance In this section, we compute the short-distance coefficients in Eq. (4) by matching the effective theory to the full theory. Because our focus is on the formulation of the calculation and on the qualitative features of the threshold physics, we work at the lowest nontrivial order in the electromagnetic and strong couplings. Therefore, one should take care in comparing our numerical results with those in the literature, which often are performed at next-toleading order, and, therefore, include two-loop effects in the couplings of the Higgs boson to digluons and diphotons [28][29][30][31]. 4 We compute the short-distance coefficients that appear at the Born level by evaluating the corresponding amplitude in the full theory at the stop-antistop threshold. We compute the short-distance coefficients that appear at one-loop level by evaluating the one-loop amplitude in the full theory at threshold and subtracting the corresponding one-loop amplitude in the effective theory.
The short-distance coefficients C ggtt , C γγtt , and Ctt Htt are easily obtained by carrying out Born-level calculations in full QCD at the stop antistop threshold. They are given by where α = e 2 /(4π), et is the stop electromagnetic charge, α s = g 2 /(4π), and N c = 3 is the number of SU(3) colors. We have chosen the effective-field-theory factorization scale to be mt, which accounts for the argument of α s . We have also taken the color-singlet projection of the stop-antistop pairs, making use of the projector where i and j are the squark and antiquark color indices, respectively. 4 An expansion of in powers of α s is valid for the short-distance coefficients, since they contain no 1/v enhancements. However, in the computation of the effective-field-theory amplitudes in Sec. II C, the expansion in powers of α s can fail because there are contributions to the stop-antistop Green's function in order α n s that are enhanced by factors 1/v n . We compute these contributions to all orders in α s .
The short-distance coefficient C Htt is simply the Born-level Higgs-stop-antistop coupling, g Htt , rescaled by a factor of √ N c : where we have normalized the Higgs coupling to stops in terms of the stop mass, with κ being an adjustable parameter.
The short-distance coefficients C ggH and C Hγγ are generated by quark loops and stop loops. We take into account the b-quark, t-quark, and stop loops, which give the most important contributions. In full QCD, we have the amplitude [32][33][34] iM where and Here, (k 1 , ǫ 1 ) and (k 2 , ǫ 2 ) are the (momentum, polarization) of the initial gluons, a and b are the gluon color indices, m b and e b are the bottom-quark mass and electric charge, m t and e t are the top-quark mass and electric charge, Similarly, in the case of H → γγ, we have the amplitude [34][35][36][37] iM where and (k 3 , ǫ 3 ) and (k 4 , ǫ 4 ) are the (momentum, polarization) of the final photons.
The corresponding quantities in the effective theory are produced by the stop loop that is generated by the C Htt , C ggtt , and C γγtt terms in the effective action. In the modifiedminimal-subtraction (MS) scheme, we obtain The effective-theory amplitudes in Eq. (17) vanish at the stop-antistop threshold (E = 0). Therefore, the short-distance coefficients C ggH and C γγH are obtained simply by evaluating A gg→H and A H→γγ at stop-antistop threshold: Finally, as we have mentioned, the short-distance coefficient ImTtt →gg→tt is obtained by computing the contribution from a two-gluon intermediate state to the imaginary part of the stop-antistop forward T -matrix, evaluated at the stop-antistop threshold. By making use of unitarity, one can obtain this quantity simply from a cut diagram. The result is C. Computation of the gg → γγ amplitude In the nonrelativistic effective theory, the stop-antistop interactions can be taken into account by considering the stop-antistop Green's function Note that the fields in the Green's function are evaluated at zero spatial separation and that color-singlet projections of the initial stop and antistop and the final stop and antistop have been taken by making use of the projectors P ij P kl [Eq. (7)]. The Green's function contains all of the effects of the 1/v n enhancements that we have mentioned.
The Green's function Gtt(ŝ) can be evaluated in a systematic expansion in powers of v by considering a reformulation of the nonrelativistic effective theory in terms of an effective theory that is the scalar-squark analogue of potential NRQCD [38]. In this effective theory, We can take into account the four-fermion terms in the effective action that are proportional to Ctt Htt and ImTtt →gg→tt by replacing Gtt(ŝ) with In the case of a stoponium state, this replacement accounts for the decay width into two gluons, which, for small values of the stop width, is the dominant stoponium decay width. can also decay with a significant rate into a pair of 125 GeV Higgs bosons. This occurs, for instance, in the minimal supersymmetric standard model, for large values of the stop mixing parameters [39]. In our work, we have assumed that this coupling takes moderate values and, consequently, that the decay width of the stoponium state into pairs of SM-like Higgs bosons is much smaller than its decay width into gluon pairs.
The coupling of the Higgs-boson to a stop-antistop pair leads to several contributions to the gg → γγ amplitude. We write these contributions to the amplitude as There is a contribution in which the initial gg pair transitions to the Higgs boson and the Higgs boson transitions to a γγ pair: Here, is the Higgs propagator (that is, the Higgs Green's function in the absence of Higgs-stopantistop interactions), m H is the Higgs pole mass, and Γ H is the Higgs width.
There is also a contribution in which the initial gg state transitions to a Higgs boson, which transitions to a stop-antistop pair, which transitions to a γγ pair: There is a contribution in which the initial gg pair transitions to a stop-antistop pair, which transitions to a Higgs boson, which transitions to a γγ pair: There is a contribution in which the initial gg pair transitions to a stop-antistop pair, which transitions to a Higgs boson, which transitions to a stop-antistop pair, which transitions to a γγ pair: Finally, there is a contribution that does not involve the Higgs boson: We note that the amplitudes A 4 and A 5 can be combined to give a simpler expression: The form of A ′ 4 (gg → γγ) in Eq. (29b) is the same as the form of A 1 (gg → γγ) in Eq. (23b), but with the rôles of the Higgs boson and the stop-antistop pair interchanged.
We also note that the total amplitude can be obtained from the matrix expression where the matrix whose inverse is taken in Eq. (31a) is (−i) times the effective Hamiltonian for the Higgs-stop-antistop system. The expression in Eq. (31a) is a generalization of Eq. (7) in Ref. [15].

D. Coulomb-Schrödinger Green's function
The stop-antistop Green's function Gtt(ŝ) can be computed at the leading nontrivial order in v, by allowing the stop and antistop to interact only through the potential of the leading order in v, which is called the static potential. One could take the static potential from lattice data, which are well described by a Coulomb-plus-linear potential (Cornell potential [40]) with a roll-off to a flat potential above the stop-antistop threshold. The Green's function corresponding to such a potential could, in principle, be evaluated numerically. In this paper, we choose instead to deal with a completely analytic Green's function, which, we believe, illustrates the qualitative features of the Higgs-stop-antistop system. In particular, we make use of a modified Coulomb-Schrödinger Green's function, which we describe in detail below.
We obtain the relationship between Gtt(ŝ) and the Schrödinger Green's function as follows. Potential interactions are independent of the relative momentum p 0 of the stop quark and the antistop quark. Therefore, we integrate the effective-field-theory stop and antistop propagators over p 0 to obtain dp 0 2π where Γt is the stop-quark width and G S (E, p) is the Schrödinger propagator (Schrödinger Green's function in the absence of interactions). Hence, we conclude that, in the case of a Coulomb potential, where G C−S (0, 0, E + iΓt) is the Coulomb-Schrödinger Green's function evaluated at zero spatial separation between the initial stop and antistop and zero spatial separation between the final stop and antistop.
In the MS scheme, G C−S (0, 0, E + iΓt) is given by [41,42] µ is the MS scale, γ E is Euler's constant, and ψ(1 − λ) is the digamma function. We take µ = mt. 6 The first and second terms in Eq. (34a) correspond to zero and one Coulomb interaction, respectively. These are the only contributions that are ultraviolet divergent and that, therefore, depend on the renormalization scheme. The sum in Eq. (34a) contains the contributions involving two or more Coulomb interactions. When Γt = 0, the nth term in the sum contains the pole |ψ n (0)| 2 /(E n − E), where E n is the energy of the nth bound state and ψ n (0) is the wave function at the origin of the nth bound state. The nth term in the sum also contains nonpole contributions. It is best not to separate the pole and nonpole contributions, as either of them alone produces a spurious logarithmic singularity at E = 0 (threshold).
In this work, we wish to capture the essential features of the QCD Green's function, which we expect to contain only a few bound-state poles below threshold. Therefore, we modify G C−S (0, 0, E) by retaining only a few terms in the sum in Eq. (34a). We expect that, for large mt, the lowest-lying bound states will be given approximately by the bound states of the Coulomb potential, and so the modified Coulomb Green's function should give a qualitative description of the system. 7 In computing the Coulomb Green's function, we set α s to a constant by requiring that This equation should be approximately valid for the low-lying bound states. It yields α s ≈ 0.13.

E. Higgs-boson form factors
The perturbative form factor for the coupling of the Higgs boson to diphotons through a stop loop is given by 7 Lattice measurements of the static quark-antiquark potential [43], which is identical to the static squarkantisquark potential, suggest that the static potential is predominantly Coulombic at short distances, as would be expected from asymptotic freedom. A recent lattice calculation [44] indicates that the stoponium ground-state wave functions at the origin, for 100 GeV ≤ mt ≤ 750 GeV, may have values that are substantially larger than those that are obtained from potentials that match QCD perturbation theory at short distances [45]. There is not, as yet, an independent confirmation of this surprising result.
The amplitude A 1 + A 2 is proportional to this form factor, and, hence, this form factor is built into our formalism. There is a similar form factor for the Higgs coupling to two gluons through a stop loop: The amplitude A 1 + A 3 is proportional to this form factor, and, so, this form factor is also built into our formalism. The amplitude A 4 is proportional to the cross term between the Higgs-to-diphoton form factor and the Higgs-to-digluon form factor: The factorsGtt in the second terms of the form factors in Eqs. (36) and (37) and in the form-factor contribution in Eq. (38) give enhancements of the total amplitude whenŝ is near a threshold peak inGtt. Such enhancements are already present in the perturbative calculations that make use of the first and second terms of the Coulomb Green's function [Eq. (34a)], and they become stronger when one takes into account the additional effects that are associated with the stop-antistop bound states [16]. However, as we will illustrate in detail in Sec. III, when the Higgs mass is close to stop-antistop threshold, the effect of Higgs-stop-antistop mixing is to displace the physical mass peaks away from threshold.
Consequently, when the full effects of Higgs-stop-antistop mixing are taken into account, this enhancement effect is not operative.

III. CASE OF A SINGLE BREIT-WIGNER RESONANCE IN THE STOP-ANTISTOP GREEN'S FUNCTION
We now discuss the situation in which we modelGtt with the simplified form of a Breit-Wigner resonance. 8 This is the only modification to the effective theory that we make. In particular, we use the formulas for the short-distance coefficients that are given above.

A. Structure of the Breit-Wigner amplitude
We consider the situation in whichGtt is given bỹ where mtt is the bound-state mass, Γtt is the bound-state width, and ψ(0) is the bound-state wave function at the origin. The bound-state width is given by Here, Γtt gg , is the width of the bound state to two gluons : For the Coulomb ground state, |ψ(0)| 2 = 8α 3 s (mtv)m 3 t /(27π). Now, we can write Eq. (31a) as whereĈ ggtt = NttC ggtt ,Ĉ γγtt = NttC γγtt , andĈ Htt = NttC Htt . We can diagonalize the matrix in Eq. (42) by making use of a similarity transformation: which implies that and The masses and widths are given by (Recall that, in our definition,Ĉ Htt is purely imaginary.) Note that these values of Approximate expressions for the masses and widths are In the approximate forms in Eq. (46), we have neglected terms of higher order in (m H Γ H − mttΓtt) 2 /∆ 2 , which is less than 2% for the values of m H Γ H , mttΓtt, andĈ 2 Htt that we use in our cross-section calculations in Sec. III B.
The matrix S(θ) is given by where In the approximation in Eq. (48a), we have again neglected terms of higher order in ( Note that m 2 + and m 2 − are always centered at the average of m 2 H and m 2 tt and separated from each other by a nonzero amount, namely, ∆. When 2|Ĉ Htt | is small in comparison with |m 2 H − m 2 tt |, θ approaches 0 or π/2, and the masses and widths approach their original values. On the other hand, when |m 2 H − m 2 tt | is negligible in comparison with 2|Ĉ Htt |, mixing is maximal, and θ is very close to π/4. 9 In this case of maximal mixing, ∆ approaches its minimum value, 2|Ĉ Htt |, and In the case of maximal mixing, the widths are given by That is, the widths Γ + and Γ − become approximately the average of Γ H and Γtt. Consequently, the mass separations are also much greater than Γ ± . That is, the physical resonances are well separated for any value of m H relative to mtt.
In Sec. II E, we discussed the threshold enhancements that are present in the form factors for the Higgs couplings to gg and γγ. We now see that those enhancements are rendered inoperative because, when m H approaches the stop-antistop threshold, the physical masses are displaced from threshold by a sizable amount. In the next section, we will provide a detailed numerical analysis of this effect.

B. Qualitative features of the Breit-Wigner cross section
In this section, we present numerical results for the short-distance coefficients and also for cross sections at the LHC at a CM energy √ s of 13 TeV. We note that the gg contribution to the total cross section is given by the expression where f g is the gluon distribution and √ŝ min and √ŝ max are the lower and upper limits, respectively, of the range in √ŝ that includes all relevant contributions to the cross section.
In the computations of σ tot in the remainder of this paper, we take √ŝ min = 600 GeV and √ŝ max = 900 GeV.
As we have already mentioned, the Breit-Wigner resonances are always well separated in comparison to their widths. Therefore, we can approximate the cross section in Eq. (51) as a sum of the individual contributions of the two resonances. If we also neglect the dependences of the gluon distributions onŝ over the width of the resonance, then we obtain the narrow-resonance approximation where F (m j ) is the gluon flux factor: As can be seen from Higgs couplings to top-quark, bottom-quark, and τ -lepton pairs that occur in the decoupling limit of large Higgs mass [13]. We will make use of an intermediate value of tan β, setting tan β = m t (2mt)/m b (2mt). At this value of tan β, the Higgs decay width into third generation fermions is minimized, and our estimate is Γ H ≃ 1.2 GeV. 11 We have converted the result in Ref. [50], which is expressed in terms of quark pole mass, to an expression in terms of the quark modified-minimal-subtraction (MS) mass by adding 2 + (3/2) log[4/(1 − β 2 )] to ∆ H in Eq. (2.26) of Ref. [50]. However, we remind the reader that the nonrelativistic approximation that we use in our calculations is valid only when |m H − mtt| ≪ mtt. The structure that appears in the quantities |C gg± C γγ± | 2 is a consequence of the fact that the Higgs coefficients contain real and imaginary parts that are comparable in magnitude. The mixing then produces a complicated pattern of interference. We emphasize that the peak that appears in the upper line is not produced by a resonance or by a threshold enhancement from the Higgs to gg or γγ form factors. Rather, it is entirely a consequence of interference effects in the short-distance coefficients.
Now, let us consider the behavior of the cross section as a function of m H . In the right panel of Fig. 2, we show the contributions of the larger-mass eigenstate and the smaller-mass eigenstate to the cross section times the branching ratio into γγ in the narrow-resonance approximation, the sum of those contributions, and the exact cross section times the branching ratio into γγ. We also show the exact cross section in the absence of mixing so that one can judge the importance of the mixing effects.
We can understand the qualitative features of the cross section times the branching ratio into γγ from the formula for the narrow-resonance approximation to the cross section in Eq. (52a). As can be seen from In Fig. 3, we display the values of |C γγ± C gg± | 2 as functions of m H . Again, we also show |C γγtt C ggtt | 2 and |C γγH C ggH | 2 , so that one can judge the importance of the mixing effects.
As can be seen by comparing the left panel of Fig. 3 with the left panel of Fig. 2, the effect of mixing on the short-distance coefficients in the case of large stop width is essentially the same as in the case of small stop width. However, the stop-antistop width is now Γtt ≈ 0.2 GeV, which is not far from the Higgs width Γ H ≈ 1.2 GeV. It follows that the width of the narrowest resonance goes from about 0.2 GeV for minimal mixing to about 0.7 GeV for maximal mixing, which is a much smaller range than in the case of a small stop width. Consequently, as can be seen from the right panel of Fig. 3, the effect of mixing on the resonance widths has a much less dramatic effect on the shape of the cross section times the branching ratio than in the small-stop-width case. The shape of the cross section times the branching ratio in the thick, dashed, red line now exhibits a peak that corresponds to the peak in the sum of the magnitudes of the products of short-distance coefficients. Comparing with the situation for small stop width, we see that the cross section times the branching ratio away from threshold is significantly smaller, owing to that fact that the stoponium width is now much larger. For the same reason, the cross section times the branching ratio away from threshold more closely approaches the cross section in the absence of mixing. Again, we note that the result from the narrow-resonance approximation for the cross section times the branching ratio into γγ agrees well with the exact result. As can be seen from the left panel of Fig. 5, the effect of mixing on the short-distance coefficients is essentially the same as in the case of κ = 8 and Γt = 0.1 MeV (Fig. 4). Again, owing to the dominance of the imaginary parts of C Hgg and C Hγγ , the effects of mixing small effect on the cross section times branching ratio at minimal mixing, where the Higgs contribution is dominant, and changes the resonance widths only slightly at maximal mixing.
Consequently, as in the case of κ = 8 and Γt = 0.1 MeV, the cross section times branching ratio changes very little from minimal mixing to maximal mixing. This can be seen in the right panel of Fig. 4. In this case, the contributions from the two eigenstates sum to produce a total of the cross section times the branching ratio that deviates very little from the total of the cross section times the branching ratio in the the absence of mixing. Again, the result from the narrow-resonance approximation for the cross section is essentially featureless, and it agrees well with the exact result. Third, the Coulomb-Schrödinger Green's function contains multiple bound-state poles.
As we will see, the additional poles beyond the ground-state pole do not have a dramatic effect on the cross section times the branching ratio to γγ. Specifically, the cross-section contribution is proportional to |Z ± | 2 /Γ ± .

IV. CASE OF THE COULOMB-SCHRÖDINGER GREEN'S FUNCTION
In this section, we present numerical results for the gg → γγ amplitudes and the asso- In computing cross sections, we take into account only the gluon-gluon initiated process, which has been the focus of our discussion. As we have mentioned, the true stop-antistop Green's function likely contains only a few bound states below threshold. Therefore, we give results that are obtained by taking into account only the first term or the first three We remind the reader that, as we have explained in Sec. III, the cross-section contribution of a resonance whose amplitude can be approximated by a Breit-Wigner form is proportional, in the narrow-width approximation, to the square of the maximum height of the absolute value of the amplitude times the full width at half maximum of the absolute value of the amplitude. in the Coulomb-Schrödinger case. Note that, at maximal mixing, both physical peaks are much broader than the unmixed stoponium peak.
In Fig. 7, we show the total diphoton production cross section σ tot as a function of m H .
For comparison, we also show σ bare H (which corresponds to A bare H ) and σ barẽ tt (which corresponds to A barẽ tt ) as functions of m H . Figure 7 shows that there are only small quantitative changes in σ tot as one includes additional stoponium poles in the stop-antistop Green's function.
A comparison of Fig. 7 with the right panel of Fig. 2 shows that the total cross section σ tot in the Coulomb-Schrödinger case has the same qualitative features as in the Breit-Wigner case. The Higgs cross section σ bare H is much less than the stop-antistop cross section σ barẽ tt . The cross section is dominated by the width of the narrowest peak. At minimal mixing, this narrowest peak corresponds to the stoponium peak. At maximal mixing, the width of the narrowest physical peak is much greater than at minimal mixing, and the height shrinks roughly as the inverse of the width, resulting in a suppression of the cross section.
As in the Breit-Wigner case (Sec. III B 1), this suppression is so great that it overwhelms the peaking effect that results from mixing of the short-distance coefficients. We see that, even at m H = 720 GeV and m H = 780 GeV, mixing broadens the narrowest peak and reduces its height sufficiently that the total cross section is well below σ barẽ tt .
B. κ = 1, Γt = 0.1 GeV In this case, the Higgs boson still couples weakly to the stop, but the stop width is much closer to the Higgs width than in the previous example. where mixing is maximal, we see that the physical masses are displaced slightly relative to the Higgs and stoponium masses. Comparison with Fig. 6 shows that there is a marked increase in the width of the stoponium peak in A barẽ tt , owing to the increase in the stop width. Because of mixing effects, the width of the narrowest peak in A tot is much larger than the width of the stoponium peak in A barẽ tt . The width of the narrowest peak in A tot is larger in Fig. 8  In Fig. 9, we display σ tot , σ bare H , and σ barẽ tt , as functions of m H . As can be seen, there are only small quantitative differences that are associated with the inclusion of additional stoponium poles in the stop-antistop Green's function.
A comparison of Fig. 9 with the right panel of Fig. 3 shows that the total cross section σ tot in the Coulomb-Schrödinger case again has the same qualitative features as in the We see that, in contrast with the small-stop-width case, the cross section now displays a peak at maximal mixing. As we explained in Sec. III B 2, this peak arises from the effects of mixing on the short-distance coefficients and is unrelated to threshold-enhancement effects.
A peak persists in σ tot because, owing to the large stop width, the width of the narrowest peak in A tot does not change sufficiently between minimal and maximal mixing to reverse the peaking effect from the short-distance coefficients. Large values of the Higgs-stop-antistop coupling have a very large impact on the diphoton production rate, as has been emphasized in the context of perturbative calculations in Ref. [15]. As we will see, for κ = 8, the Higgs-stop-antistop mixing effects become dramatic, and it is essential to include those effects, which go beyond the effects that are contained in fixed-order perturbation theory, in order to compute the diphoton rate reliably.
In Fig. 10 the Breit-Wigner amplitude. At maximal mixing, the larger-mass peak that is present in the Breit-Wigner amplitude has almost disappeared in the Coulomb-Schrödinger amplitude, owing to the large decay width of the larger-mass physical state into a stop-antistop pair.
There is again a structure near threshold that appears only in the Coulomb-Schrödinger amplitude that arises from the logarithmic term in Eq. (34), but in this large-stop-width case, the structure has nearly disappeared. We also see that both the height and the width of the lower-mass peak are significantly smaller in the Coulomb-Schrödinger amplitude than in the Breit-Wigner amplitude. These changes in the lower-mass peak result in a greatly reduced contribution of the lower-mass peak to the cross section. In this larger-stop-width case, the stoponium peak in |A bare H | is much broader than in the smaller-stop-width case and is so small as to be nearly invisible.
In Fig. 13, we show σ tot , σ bare H , and σ barẽ tt , as functions of m H . A comparison with the right panel of Fig. 5 shows that the shape of σ tot in the Coulomb-Schrödinger case is again similar to the shape of σ tot in the Breit-Wigner case. The cross section is featureless for the reasons that we mentioned in the discussion of the cross section for the narrower stop width. For this larger stop-width, the cross section is much smaller in the Coulomb-Schrödinger case than in the Breit-Wigner case. As in the case of the smaller stop width, the higher-mass peak has disappeared in the Coulomb-Schrödinger amplitude, owing to the large width of the higher-mass peak into a stop-antistop pair. Furthermore, the reduction in both the height and the width of the lower-mass peak has resulted in an additional reduction of σ tot , relative to its values in the smaller-stop-width case.
Again, we see that the larger value of the Higgs-stop-antistop coupling greatly enhances

V. CONCLUSIONS
The system of a heavy Higgs boson that is coupled to a stop-antistop pair exhibits some interesting field-theoretic phenomena near the stop-antistop production threshold. This system has attracted interest in the context of the production of a heavy Higgs boson near the stop-antistop threshold and its subsequent decay to two photons [15], owing to the perturbative enhancement of the Higgs couplings to photons and gluons near threshold. However, as is well known, the appearance of Coulomb infrared singularities near threshold invalidates the use of fixed-order perturbation theory. These Coulomb singularities first appear at two-loop order in the Higgs-to-diphoton and Higgs-to-digluon form factors. They occur when v (one-half the stop-antistop relative velocity in the stop-antistop CM frame) goes to zero, and they are a manifestation of the general phenomenon of 1/v enhancements of two-particle amplitudes near threshold. These enhancements require an all-orders treatment, and they lead, among other things, to the formation of stoponium bound states.
A discussion of nonperturbative threshold effects is given in Ref. [15] and focuses on the enhancements of the Higgs-to-diphoton and Higgs-to-digluon form factors that are induced by the stop-antistop bound states [16]. The discussion in Ref. [15] suggests that nonpertur- 13 At κ = 5, the cross-section results are qualitatively similar to those at κ = 8, except that σ bare H is reduced relative to σ tot , and, so, there is a mild enhancement of σ tot relative to σ bare H , except in a small region of m H between 750 and 770 GeV. bative threshold effects produce an enhancement of the digluon-to-diphoton cross section, relative to the predictions for the cross section that are based on perturbative treatments of the Higgs-to-diphoton and Higgs-to-digluon form factors near threshold. However, a correct treatment of the threshold effects also requires a complete analysis of Higgs-stop-antistop mixing effects.
In this paper, we have formulated the calculation of the threshold enhancements to the digluon-to-diphoton cross section in terms of the scalar-quark analogues of the effective field theories NRQED and NRQCD. Our treatment is valid up to corrections of relative order v 2 . The effective theory gives a complete accounting of Higgs-stop-antistop mixing in the threshold region. We have studied these enhancement and mixing effects numerically by making use of a model Green's function for the stop-antistop system, namely, the Coulomb-Schrödinger Green's function. The Coulomb-Schrödinger Green's function does not correctly account for the QCD confining potential, which should be Coulombic only for the lowest-lying stoponium bound states. Therefore, we have considered the case in which the expression for the Coulomb-Schrödinger Green's function is truncated so that it contains only a few bound states. This approach retains only bound states for which the Coulombic approximation is expected to be valid, and it is in keeping with the actual stoponium spectrum, which is expected have only a few bound states. At a qualitative level, we have checked that the results that we have obtained are independent of the number of bound states that we have retained. Moreover, the quantitative differences that are associated with the inclusion of heavier bounds states are small, giving us confidence that our conclusions are not dependent on the specifics of the model Green's function that we have chosen.
We have also investigated a simplified model in which the stop-antistop Green's function is represented by a simple Breit-Wigner resonance. This simplified model exhibits some, but not all, of the qualitative features of the more complicated Coulomb-Schrödinger model.
We have found that the Higgs-stop-antistop mixing produces three general effects that are very significant. First, the mixing leads to mass eigenstates whose widths are larger than the widths of the stoponium states. For a single stoponium state and for large values of the Higgs-stop-antistop coupling, the widths of the mass eigenstates at threshold approach the average of the Higgs and stoponium widths. These increases in the widths, and the concomitant reductions in the peak heights, reduce the contributions to the cross section relative to the contribution that would be obtained from a narrow stoponium state. Second, the physical masses are shifted from the input Higgs and stoponium masses, and, when the Higgs mass is near threshold, the physical masses are displaced away from the threshold region. This effect is particularly important for large values of the Higgs-stop-antistop coupling and can render the perturbative threshold enhancements inoperative. Third, when the Higgs-stop-antistop coupling is large, the displacement of the mass of the higher-mass physical state to a point above threshold can give that state a very large width into a stop-antistop pair, resulting in a drastic reduction of its contribution to the cross section.
In addition, to these general effects, there are some effects that depend on the details of the couplings of the Higgs boson and the stop-antistop pair to photons and gluons and on the details of the Coulomb-Schrödinger Green's function. For example, the couplings can mix in such a way as to produce a peak near threshold that has nothing to do with the threshold enhancements that are associated with the perturbative Higgs-digluon and Higgsdiphoton form factors. The Coulomb-Schrödinger Green's function can also lead to changes in the heights and widths of the physical peaks in the amplitudes, relative to their heights and widths in the simple Breit-Wigner model. These effects are driven largely by the term of lowest order in α s , in the Coulomb-Schrödinger Green's function. That term is universal in that it is independent of the nature of the squark-antiquark static potential. However, the details of the effects that arise from it seem to depend on nonuniversal features of the Coulomb-Schrödinger Green's function.
In general, for large values of the heavy-Higgs coupling to the stop-antistop pair, the mixing effects result in suppressions of the digluon-to-diphoton cross section at threshold relative to the cross section that is predicted in one-loop perturbation theory. The precise suppression factor depends not only on the Higgs-stop-antistop coupling but also on the stop width. We remind the reader that, because our focus is on the formulation of the calculation and on the qualitative features of the threshold physics, we have computed the Higgs couplings to digluons and diphotons at the one-loop level, and, so, one should take care in comparing our numerical results with those in the literature, which often include two-loop effects.
Although we have concentrated on the case of the Higgs-stop-antistop interaction, the theoretical framework that we have developed is applicable to the coupling of Higgs bosons to other scalar particles in the region near the particle-antiparticle threshold. It can also be generalized easily to the case of a Higgs boson coupled to heavy fermions and to calculations of rates to different final states. For example, one could study the case of a τ + τ − final state by replacing the γγ short-distance coefficients in Eq. (31a) with the corresponding τ + τ − short-distance coefficients. 14 We reserve the study of these additional cases for a separate publication.
Appendix A: Diagonal form of the amplitude in the general case In the general case, which includes the example of the Coulomb-Schrödinger Green's function, we can write Eq. (31a) as whereĜtt(ŝ) =Gtt(ŝ)/N 2 tt , with N 2 tt given in Eq. (40). 15 The eigenvalues of the matrix in Eq. (A1) whose inverse is taken are given by and the tangent of the rotation angle of the similarity transformation that diagonalizes that matrix is given by We see that both the eigenvalues and the rotation angle now depend onŝ.
The physical-state poles are located at the valuesŝ =ŝ ± for which α ± (ŝ) vanishes. [Note that there may be more than one value ofŝ ± for which α ± (ŝ) vanishes.] Near a pole, the eigenvalues of the inverse matrix that appears in Eq. (A1) are where m 2 ± = Re(ŝ ± ), (A4b) and the tangent of the rotation angle is given by