750 GeV Diphotons: Implications for Supersymmetric Unification II

Perturbative supersymmetric gauge coupling unification is possible in six theories where complete SU(5) TeV-scale multiplets of vector matter account for the size of the reported $750~{\rm GeV}$ diphoton resonance, interpreted as a singlet multiplet $S = (s+ia)/\sqrt{2}$. One of these has a full generation of vector matter and a unified gauge coupling $\alpha_G \sim 1$. The diphoton signal rate is enhanced by loops of vector squarks and sleptons, especially when the trilinear $A$ couplings are large. If the $SH_u H_d$ coupling is absent, both $s$ and $a$ can contribute to the resonance, which may then have a large apparent width if the mass splitting from $s$ and $a$ arises from loops of vector matter. The width depends sensitively on $A$ parameters and phases of the vector squark and slepton masses. Vector quarks and/or squarks are expected to be in reach of the LHC. If the $SH_uH_d$ coupling is present, $a$ leads to a narrow diphoton resonance, while a second resonance with decays $s \rightarrow hh, W^+ W^-,ZZ$ is likely to be discovered at future LHC runs. In some of the theories a non-standard origin or running of the soft parameters is required, for example involving conformal hidden sector interactions.


I. INTRODUCTION
Data from both ATLAS and CMS experiments show evidence for a diphoton resonance near 750 GeV [1][2][3][4]. We have previously explored the consistency of this data with perturbative gauge coupling unification in supersymmetric theories by adding a singlet field S and vector matter (Φ i ,Φ i ) to the minimal supersymmetric standard model (MSSM) [5] via the superpotential interaction λ i SΦ iΦi . A sufficient diphoton signal results only if λ i take values close to the maximum allowed by perturbativity, and hence we take them to be determined by renormalization group flow, yielding a highly predictive theory. The diphoton resonance has been further explored in this minimal supersymmetric theory [6] as well as in other supersymmetric theories involving a singlet with vector matter in complete unified multiplets [7][8][9][10][11][12][13].
In this paper we further explore the diphoton resonance in minimal supersymmetric theories. In addition to λ i SΦ iΦi we allow for the interaction of S = (s + ia)/ √ 2 with Higgs doublets via λ H SH u H d , giving the next-to-minimal supersymmetric standard model (NMSSM) with vector matter. This additional interaction makes significant changes to the phenomenology, mixing s with the doublet Higgs boson h so that there is a further resonance to be discovered at the LHC of s decaying to pairs of Higgs bosons or electroweak gauge bosons: s → hh, W + W − , ZZ. In this case the diphoton resonance is produced by a alone, and is narrow.
As in Ref. [5] we consider the complete set of 6 possibilities for vector matter that fills SU (5) multiplets and allows perturbative gauge coupling unification: "(5 + 5) N 5 " theories contain N 5 = 1, 2, 3 or 4 copies of vector 5-plets, the "10 + 10" theory contains a single vector 10-plet, and the "15 + 15" theory contains a full generation of vector quarks and leptons. In fact without threshold corrections the (5 + 5) 4 and 15 + 15 theories become non-perturbative just before the gauge couplings unify. We include these theories and study the form of the threshold corrections required to allow precision perturbative gauge coupling unification. Indeed we find the 15+15 theory to be particularly interesting: supersymmetric theories with 4 or less generations have gauge couplings α a much less than unity at the unification scale, while those with 6 or more generations become non-perturbative far below the unification scale. The case of 5 generations, here interpreted as three chiral generations and one vector generation, is unique, offering the possibility of α a ∼ 1 at the unification scale.
We compute the contribution to the diphoton signal from loops containing scalar superpartners of (Φ i ,Φ i ); such contributions were ignored in Ref. [5] but were studied for 4 of our 6 theories in Ref. [6]. For each theory, the rates are computed for two cases corresponding to whether the supersymmetric mass terms of the vector matter, µ i , satisfy unified boundary conditions. The corrections from the scalar loops become substantial and are important for large A i terms. This is particularly important for the case of unified mass relations for µ i , since in the absence of the scalar contributions the rates are frequently marginal or inadequate to explain the data. For example, the 15 + 15 theory with unified mass relations is only viable with large A i . Although the scalar mass parameters introduce further parameters, the unification of µ i reduces the parameter space.
In general, contributions from multiplets (Φ i ,Φ i ) to the diphoton amplitude add with random phases, or random signs if CP is conserved, typically significantly reducing the signal rate. We introduce theories where the mass terms for the vector matter arise purely from a condensate of S, giving µ i ∼ λ i S , which has the effect of aligning the amplitudes from each multiplet and maximizing the signal rate. In addition, the resulting values for µ i correspond to the case of unified masses. Thus while the theories become more predictive, large A i are needed in some theories for a sufficient signal rate.
We explore the possibility that the mass splitting between the two scalar degrees of freedom in S arise from loops containing (Φ i ,Φ i ). It was argued in Ref. [5] that when λ H = 0 such splittings could lead to an apparent width of 10s of GeV for the diphoton resonance. Here we extend the analysis to include A i terms as well as CP violation in the holomorphic scalar mass terms of (Φ i ,Φ i ).
We order our analysis as follows. In the next section we compute the diphoton rate, with separate subsections for the cases of λ H = 0 and λ H = 0. In the latter case, in addition to having Higgsino loop contributions, the diphoton rate arises from only one scalar mode of S, as the other mixes with the light Higgs boson. In section III we discuss the width of the resonance for λ H = 0. In section IV we switch to λ H = 0 and study the diboson LHC signal that results from one component of S mixing with the light Higgs boson. The condition on threshold corrections for perturbative unification in (5 + 5) 4 and 15 + 15 theories is studied in section V and theories with µ i ∼ λ i S are introduced in section VI.

II. 750 GEV DIPHOTON RESONANCE
In this section, we discuss an explanation of the diphoton excess observed at the LHC [1-4]. We introduce a singlet chiral multiplet S and pairs of SU (5) charged chiral multiplets Φ i andΦ i around the TeV scale, and take the most general superpotential couplings and mass The coupling κ flows to small values at low energies and is unimportant for the analysis of this paper. In section VI we briefly mention its possible role in stabilizing a vacuum expectation value (vev) for S. We consider the complete set of possible theories with perturbative gauge coupling unification: the "(5 + 5) N 5 " theory containing N 5 = 1, 2, 3 or 4 copies of (D,L) + (D, L), the "10 + 10" theory containing (Q, U, E) + (Q,Ū ,Ē), and the "15 + 15" theory that contains a full generation of vector quarks and leptons. In the ( Let us first discuss the size of λ i and µ i . As we have shown in Ref. [5], as long as λ i are large enough at high energies, they flow into quasi-fixed points and their low energy values are insensitive to the high energy values. In Table I, we show the prediction for λ i (TeV) in each theory, which we assume in the following. 1 If µ i unify at the unification scale, their 1 The predicted values in (5 + 5) 4 and 15 + 15 are different from those in Ref. [5]. In these theories, as we will see in section V, the gauge coupling unification requires moderate threshold correction around the relative values are fixed by the renormalization group running, which are also shown in Table I.
After taking µ i to be real by phase rotations of Φ i andΦ i , λ i are in general complex. We assume that λ i have a common phase. This is automatic for (5 + 5), (10 + 10) and (15 + 15) theories with unified λ i and µ i at the unification scale. 2 By a phase rotation of S, we take λ i to be real and positive. In this basis, we decompose the scalar components of S as and refer to "s" and "a" as "scalar" and "pseudoscalar", respectively. They may be degenerate so that both contribute to the diphoton excess at 750 GeV, which we assume unless otherwise stated. The mass splitting between the two scalars is discussed in section III. TeV scale, which is not taken into account in Ref. [5]. This changes the gauge couplings above the TeV scale as well as the predictions of λ i . If we instead assume large threshold corrections at the unification scale, the predictions of Ref. [5] hold. 2 The alignment is also guaranteed if µ i are solely given by a vev of S. See section VI.
The upper left panel of Figure 1, shows the prediction for σ S Br γγ at the LHC with √ s = 13 TeV as a function of µ L for (5 + 5) i and µ E for (10 + 10) and (15 + 15), assuming that µ i unify at the unification scale. We also assume that the scalar components of Φ i and Φ i are heavy enough that their loop corrections do not contribute to the signal. For this case, only the (5 + 5) 2,3,4 theories can explain the observed diphoton excess and require light vector matter. Vector quark masses are predicted in the range 700 − 1200 GeV, which can be observed at the LHC. In Figures 1 -4 we show shaded 1σ and 2σ regions for a signal rate of σBr γγ = (4.7 + 1.2 − 1.1) fb from combined fits to the experimental data [14]. second generation quarks [16]. For (5 + 5) 3,4 , vector quark masses can be as large as 2 TeV.
Next, let us take into account the effect of the scalar components of Φ i andΦ i , which is also investigated in Ref. [6]. The trilinear couplings between the scalar components of S, Φ i andΦ i are given by where A i are soft trilinear couplings. We take A i to be real by phase rotations of the scalar components of Φ i andΦ i , and we neglect the trilinear couplings proportional to µ S . The mass terms of Φ i andΦ i are given by Assuming   with masses Here, θ i is the phase of B i , B i = e iθ i |B i |. The trilinear couplings in the mass eigenbasis are given by where we neglect couplings proportional to Φ + Φ * − , which are irrelevant for the diphoton signal.    Let us now turn on the coupling between S and the Higgs multiplet, λ H . The existence of λ H slightly changes the renormalization running of couplings. In Table II, we show the prediction for λ i (TeV) in each theory. Here we assume that λ H is also large at a high energy scale. The low energy couplings are slightly smaller than those in the theory with λ H = 0.
The mixing between the Higgs multiplet and S is as follows. Assuming the decoupling The mass ratios assume a common value for µ i at M G .
limit, large tanβ, and the CP conservation in the couplings between S and the Higgs multiplet, the mass eigenstate is approximately given by the heavy Higgs states (H 0 , A 0 , H ± ) composed of H d , the singlet pseudoscalar a, and the mixture of the standard model like Higgs h and the singlet scalar s. (With θ i = 0, π, quantum corrections inevitably induce mixing between s and a; see section III. The mixing is suppressed for sufficiently large m 2 s . The pseudo-scalar a mixes with the heavy CP-odd Higgs A 0 through the A term coupling between S and the Higgs multiplet. This leads to the decay of a into a pair of bottom quarks. The decay mode does not affect the diphoton signal rate for a sufficiently large heavy Higgs mass, a sufficiently small A term, and/or not very large tanβ.) The scalar s efficiently decays into the standard model Higgs, W boson, and Z boson, and hence does not contribute to the 750 GeV excess. The excess can be still explained by the pseudoscalar a. In Figure 3, we show the prediction for σ a Br γγ at the LHC with √ s = 13 TeV as a function of µ L for (5 + 5) i and µ E for (10 + 10) and (15 + 15), assuming that µ i unify at the unification scale. In the upper left panel, the contribution from the scalar components of Φ i andΦ i are ignored, while it is taken into account in other panels.
All theories except for (15 + 15) can explain the diphoton excess without large A i terms.
Vector quarks are as heavy as 600 − 1200 GeV, which is expected to be within the reach of the LHC. The bound is, however, relaxed by large A i terms, as shown in the lower panel.
In Figure 4, we show the prediction for σ a Br γγ as a function of degenerate vector quark masses, with vector lepton masses and the Higgsino mass fixed at 380 GeV. The vector quark masses can be as large as 2 TeV without large A i terms.
In this section, we discuss a possible way to obtain a "wide width resonance" from the scalar S. As we have pointed out in Ref. [5], the mass difference of a few tens of GeV between the scalar s and the pseudoscalar a can be naturally obtained by a threshold correction at the TeV scale from Φ i andΦ i . Then s and a are observed as a single wide resonance. Here we explore the dependence of the mass splitting on (A i , θ i ).
This explanation requires that the holomorphic supersymmetry breaking soft mass of S, the B S µ S term, is small. In gravity mediation, the size of the B S term is as large as other soft masses, and hence µ S should be suppressed. This requires that the soft mass squared of S, m 2 S , is positive at the low energy scale. Otherwise, the vev of S is large (see section VI) and hence fine-tuning is required to obtain small enough µ i . In gauge mediation, on the other hand, the B S µ S term is given by a three loop effect and hence is suppressed even if µ S is unsuppressed.
The quantum correction to the mass matrix is given by  where the correction ∆ aa is absorbed into the soft mass squared of S. Note that in the supersymmetric limit, where A i = 0 and m i+ = m i− = µ i , the mass difference vanishes. In If µ S = 0, thes mass arises at one loop from virtual vector matter ands may be the lightest supersymmetric particle. For this to be interpreted as "singlet-doublet" dark matter, a mixing with the Higgsino should be introduced. Further work is needed to investigate whether a small SH u H d coupling that provides this mixing also gives a small enough mixing between s and the doublet Higgs boson so that s still contributes to the diphoton resonance. If so, the predominantlys dark matter may have a mass allowing the observed abundance via freezeout annihilation on the Z or Higgs pole.

IV. SIGNAL OF S DECAY TO STANDARD MODEL DIBOSONS
In this section, we discuss the signal from s → hh, W + W − , ZZ at the LHC for λ H = 0.
The scalar s is produced via gluon fusion and decays into pairs of standard model particles.
If it is heavy enough, it also decays into a pair of vector quarks/leptons.
The mixing between the standard model like Higgs h and the singlet scalar s given by where m s is the mass of s and v 246 GeV is the vev of the standard model Higgs. The measurement of the Higgs production cross section restricts the mixing, θ 2 hs < 0.
In the limit m s m h,Z,W , the decay width of s into pairs of the standard model Higgs bosons, W bosons, and Z bosons can be evaluated by the equivalence theorem: Through mixing with the standard model Higgs, s decays into a pair of top quarks with a rate Γ(s → tt) For large m s , the scalar s also decays into a fermionic component of Φ i andΦ i with a decay rate For simplicity, we assume that the scalar components of Φ i andΦ i are heavy enough that s does not decay into them. Inclusion of these decay modes is straightforward.
In Figure 6, we show the prediction for σ s Br hh at the 13 TeV LHC as a function of m s , assuming that µ H and the lightest vector-lepton mass (µ L for (5 + 5) i and µ E for (10 + 10) and (   semi-perturbative regime. Nevertheless, as we will show, precision gauge coupling unification is successfully achieved with moderate threshold corrections around the TeV scale. In Figure 8, To assess the required threshold corrections at the TeV scale, we solve the renormalization group equation from the unification scale down to the electroweak scale. In Figure 9, we   show ∆b i , the difference between the predicted and observed gauge couplings at the weak In each panel of Figure 9, the couplings come close to unifying in the region of M G ∼ (5×10 16 -10 17 ) GeV, where ∆b i are all positive and typically 3 -5. These are not very large and hence can be countered by TeV scale threshold corrections. As superpartner and/or vector quark and lepton masses are increased above 1 TeV, the predicted gauge couplings at M Z become larger and hence the lines in Figure 9 are lowered, so that raising these masses produces threshold corrections of the required sign. For precision unification the three curves must intersect at a point where ∆b i = 0. For (15 + 15), this means that the curve for SU (2) must be lowered more than the curve for SU (3)-the masses for particles with SU (2) L charge must be raised further than the masses for colored particles.  Table III, with wino heavier than gluino, predict gauge couplings  at M Z in agreement with the observed values, as shown in Figure 10.

A. Vacuum expectation value for S
In estimating the diphoton signal rate, we assumed that the phases of λ i are aligned with each other in the basis where µ i have a common phase. Even with CP conservation in the superpotential, we have assumed that the signs of λ i µ i are independent of i. This alignment maximizes the diphoton rate and, while it is not necessary for large A i and non-unified masses, in other cases it is helpful in obtaining a sufficient diphoton rate. This alignment is naturally achieved if µ i are forbidden by some symmetry, under which S is charged, and are  Table III. the prediction for the gauge couplings at the weak scale is improved, as is shown in Figure 10.
1500 1000 1000 1000 500 800 800 800 380 380 solely given by the vev of S. Thus, instead of Eq. (1) we may start with the much simpler superpotential In this case µ i = λ i S and the spectrum of vector matter is given by the "Unified" case, with the diphoton rate given in Figure 1. In the absence of large A i , the upper panels show that only the (5 + 5) 2,3,4 theories explain the diphoton resonance. However, the lower panels show that the scalar contribution with large A i allows all theories to explain the diphoton resonance. After electroweak symmetry breaking the soft trilinear scalar interaction proportional to A H leads to a linear term in S, which will therefore develop a vev. However, even in the large A H limit this is too small to give sufficient mass to the vector matter. Some other origin for a large vev must be found.
One idea for achieving this is to give a negative mass squared to S, with a restoring term in the potential for S arising from the superpotential coupling κ of Eq. (1). Assuming that B S µ S is negligible, as occurs if µ S ∝ S , the vev of S is given by where we have used m s = 0 and m a = 750 GeV to obtain the last inequality. The coupling κ, however, receives large renormalization and its size at the low energy scale is much smaller than the one at the unification scale. In ( Another possibility is that S arises from a positive mass squared and a tadpole term. One may wonder whether the mechanism to yield the tadpole term in general generates µ i terms independent of the vev of S. This is avoided by the so-called SUSY-zero. Consider, for example, an R symmetry with a charge assignments S(−2) and Φ iΦi (4). (Construction of a similar mechanism with a non-R symmetry is straightforward.) In any supersymmetric theory, the superpotential, which has an R charge of 2, must have a non-zero vev to cancel the cosmological constant induced by supersymmetry breaking. We denote the chiral operator of R charge 2 that condenses and generates the superpotential vev as O. The tadpole term of S is given by In gravity mediation, this term generates a tadpole term ∼ (TeV) 3 S and hence S = O(1) TeV. On the other hand, the superpotential term W ∼ OΦ iΦi is forbidden (except for Z 4R ). It is essential that there is no chiral operator,Ō, having R charge −2 and a similar expectation value as O; otherwise, the superpotential term W ∼ŌΦ iΦi generates µ i independent of S . Such a chiral operator is actually absent when R symmetry is broken by a gaugino condensation. This mechanism leads to Eq. (18) with S having a vev of order the supersymmetry breaking scale, providing the messenger scale of order the Planck mass.
The R symmetry forbids both S 2 and S 3 interactions, so that at tree-level µ S = 0. (For a discrete Z 6 R symmetry S 2 is allowed. In this case, the degeneracy of s and a cannot be naturally explained.) The fermionic component of S,s, is massless at tree-level and is expected to be the lightest supersymmetric particle. Since the R symmetry is broken by the supersymmetry breaking interactions, thes mass appears at the TeV scale from integrating out Φ i andΦ i at one loop. These radiative contributions to ms are proportional to µ i and  III and VI). This is because the renormalization of soft masses makes m 2 S negative at the TeV scale, unless m 2 S is positive and large at the unification scale. To avoid the tachyonic masses of the vector squarks/sfermions, their soft masses must be also large enough at a high energy scale, which raises soft mass scales further. Such fine-tuning can be avoided by introducing non-standard low scale mediation of supersymmetry breaking or non-standard running of soft operators, for example induced by conformal hidden sector interactions [19][20][21]. Such a conformal sector also has the potential to yield large A terms [21], which are favored by the diphoton signal and the Higgs mass of 125 GeV.
In section V we found that, for precision gauge coupling unification in (15 + 15), nonstandard soft operators at the TeV scale were also required.

VII. SUMMARY AND DISCUSSION
Following an initial study in Ref. [5], we confirm that the reported 750 GeV diphoton resonance can be explained by supersymmetric theories that add a gauge singlet S = (s + The diphoton event rate also depends on whether the amplitudes from the various vector matter multiplets add coherently. This occurs automatically if the vector matter masses arise from S acquiring a vev. In section VI we introduce a theory with an R symmetry that accomplishes this in a way that explains why all the superpotential mass parameters have a scale governed by supersymmetry breaking.
There is an interesting possibility that for λ H = 0 the mass splitting between s and a arises dominantly from loops of vector matter [5]. In Figure 5 we extend our analysis to show that the corresponding width of the diphoton resonance is sensitive to A terms and CP violating phases.
While perturbative supersymmetric unified theories can easily account for the diphoton signal, we find it likely that some scheme beyond gravity mediation is needed for soft operators and their running. The extra matter makes the gluino mass very large at unification scales which then typically leads to masses for the scalar superpartner that are too large.
This problem is strengthened as more vector multiplets are added, and in the (5 + 5) 4 and (15 + 15) theories we also find that for the gauge couplings to remain perturbative we need either non-standard boundary conditions or running of the soft parameters. Furthermore, in the theory introduced to align the phases of the amplitudes for the diphoton resonance, vacuum stability also suggests non-standard running of soft operators.
into account the appropriate factors of cos θ i and sin θ i as well as the dynamics of a). The strongest constraint comes from the tunneling involving Φ i− and s. Hereafter we drop the subscripts i and −. The scalar potential of Φ and s is given by We take A > 0 without loss of generality. After a change of variables, s → m s σ/λ, Φ → m s φ/ √ 2λ and x µ → ξ µ /m s , the action is given by where V = 1 16 We consider the tunneling path with the minimum potential barrier, in which Along this path, the potential is given by and the canonically normalized field is given by We numerically obtain the bounce action [22] solving the equation of motion of φ c . In Figure 11, we show the size of the bounce action, S B , as a function of A for m φ = 380 GeV and 700 GeV. We require that the lifetime of the vacuum is longer than the age of the universe, S B > 400. In Table IV, we show the upper bound on A i− for each theory. The result is consistent with the one presented in Ref. [6] within a few tens of percent.