750 GeV Diphotons: Implications for Supersymmetric Unification II
7750 GeV Diphotons:Implications for Supersymmetric Unification II
Lawrence J. Hall,
1, 2
Keisuke Harigaya,
1, 2 and Yasunori Nomura
1, 21
Department of Physics, University of California, Berkeley, California 94720, USA Theoretical Physics Group, Lawrence BerkeleyNational Laboratory, Berkeley, California 94720, USA
Abstract
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 GeV diphotonresonance, interpreted as a singlet multiplet S = ( s + ia ) / √
2. One of these has a full generation ofvector matter and a unified gauge coupling α G ∼
1. The diphoton signal rate is enhanced by loopsof 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 largeapparent width if the mass splitting from s and a arises from loops of vector matter. The widthdepends sensitively on A parameters and phases of the vector squark and slepton masses. Vectorquarks and/or squarks are expected to be in reach of the LHC. If the SH u H d coupling is present, a leads to a narrow diphoton resonance, while a second resonance with decays s → hh, W + W − , ZZ islikely to be discovered at future LHC runs. In some of the theories a non-standard origin or runningof the soft parameters is required, for example involving conformal hidden sector interactions. a r X i v : . [ h e p - ph ] M a y . INTRODUCTION Data from both ATLAS and CMS experiments show evidence for a diphoton resonancenear 750 GeV [1–4]. We have previously explored the consistency of this data with per-turbative 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] viathe superpotential interaction λ i S Φ i ¯Φ i . A sufficient diphoton signal results only if λ i takevalues close to the maximum allowed by perturbativity, and hence we take them to be de-termined by renormalization group flow, yielding a highly predictive theory. The diphotonresonance has been further explored in this minimal supersymmetric theory [6] as well asin other supersymmetric theories involving a singlet with vector matter in complete unifiedmultiplets [7–13].In this paper we further explore the diphoton resonance in minimal supersymmetrictheories. In addition to λ i S Φ i ¯Φ i we allow for the interaction of S = ( s + ia ) / √ 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 thephenomenology, mixing s with the doublet Higgs boson h so that there is a further resonanceto be discovered at the LHC of s decaying to pairs of Higgs bosons or electroweak gaugebosons: 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: “( + ) N ” theoriescontain N = 1 , , + ” theory contains a singlevector 10-plet, and the “ + ” theory contains a full generation of vector quarks andleptons. In fact without threshold corrections the ( + ) and + theories becomenon-perturbative just before the gauge couplings unify. We include these theories and studythe form of the threshold corrections required to allow precision perturbative gauge couplingunification. Indeed we find the + theory to be particularly interesting: supersymmetrictheories with 4 or less generations have gauge couplings α a much less than unity at theunification scale, while those with 6 or more generations become non-perturbative far belowthe unification scale. The case of 5 generations, here interpreted as three chiral generationsand one vector generation, is unique, offering the possibility of α a ∼ i , ¯Φ i ); such contributions were ignored in Ref. [5] but were studied for 4 of our6 theories in Ref. [6]. For each theory, the rates are computed for two cases corresponding towhether the supersymmetric mass terms of the vector matter, µ i , satisfy unified boundaryconditions. The corrections from the scalar loops become substantial and are importantfor large A i terms. This is particularly important for the case of unified mass relationsfor µ i , since in the absence of the scalar contributions the rates are frequently marginal orinadequate to explain the data. For example, the + theory with unified mass rela-tions is only viable with large A i . Although the scalar mass parameters introduce furtherparameters, the unification of µ i reduces the parameter space.In general, contributions from multiplets (Φ i , ¯Φ i ) to the diphoton amplitude add withrandom phases, or random signs if CP is conserved, typically significantly reducing thesignal rate. We introduce theories where the mass terms for the vector matter arise purelyfrom a condensate of S , giving µ i ∼ λ i (cid:104) S (cid:105) , which has the effect of aligning the amplitudesfrom 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 offreedom 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 diphotonresonance. Here we extend the analysis to include A i terms as well as CP violation in theholomorphic scalar mass terms of (Φ i , ¯Φ i ).We order our analysis as follows. In the next section we compute the diphoton rate, withseparate subsections for the cases of λ H = 0 and λ H (cid:54) = 0. In the latter case, in addition tohaving 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 theresonance for λ H = 0. In section IV we switch to λ H (cid:54) = 0 and study the diboson LHC signalthat results from one component of S mixing with the light Higgs boson. The condition onthreshold corrections for perturbative unification in ( + ) and + theories is studiedin section V and theories with µ i ∼ λ i (cid:104) S (cid:105) are introduced in section VI.3 I. 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 and ¯Φ i around the TeV scale, and take the most general superpotential couplings and massterms W ⊃ S (cid:88) i λ i Φ i Φ i + λ H SH u H d + (cid:88) i µ i Φ i Φ i + µ H H u H d + µ S S + κ S . (1)The coupling κ flows to small values at low energies and is unimportant for the analysis of thispaper. In section VI we briefly mention its possible role in stabilizing a vacuum expectationvalue (vev) for S . We consider the complete set of possible theories with perturbativegauge coupling unification: the “( + ) N ” theory containing N = 1 , , D, ¯ L ) + ( D, L ), the “ + ” theory containing ( Q, U, E ) + ( ¯ Q, ¯ U , ¯ E ), and the “ + ”theory that contains a full generation of vector quarks and leptons. In the ( + ) and + theories, the standard model gauge couplings near the unification scale M G are inthe strong coupling regime if all super particles are below 1 TeV. We discuss the running ofthe gauge couplings and the threshold corrections around the TeV scale for these theoriesin section V.The diphoton signal is explained by the production of the scalar component(s) of S via gluon fusion and the subsequent decay into diphotons, which are induced by the loopcorrection of Φ i and ¯Φ i . For λ H (cid:54) = 0, s mixes with doublet Higgs and efficiently decays intoa pair of standard model Higgs or gauge bosons, and does not contribute to the diphotonsignal. Thus, we consider the cases with λ H = 0 and λ H (cid:54) = 0 independently. For λ H (cid:54) = 0,the LHC signal of s → hh, W + W − , ZZ is discussed in section IV. A. Vanishing Higgs coupling: λ H = 0 Let us first discuss the size of λ i and µ i . As we have shown in Ref. [5], as long as λ i arelarge enough at high energies, they flow into quasi-fixed points and their low energy valuesare insensitive to the high energy values. In Table I, we show the prediction for λ i (TeV) ineach theory, which we assume in the following. If µ i unify at the unification scale, their The predicted values in ( + ) and + are different from those in Ref. [5]. In these theories, as wewill see in section V, the gauge coupling unification requires moderate threshold correction around the µ i to be real by phase rotations of Φ i and ¯Φ i , λ i are in general complex. Weassume that λ i have a common phase. This is automatic for ( + ), ( + ) and ( + )theories with unified λ i and µ i at the unification scale. 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 S = 1 √ s + ia ) , (2)and refer to “ s ” and “ a ” as “scalar” and “pseudoscalar”, respectively. They may be degen-erate so that both contribute to the diphoton excess at 750 GeV, which we assume unlessotherwise stated. The mass splitting between the two scalars is discussed in section III. D L Q U E ( + ) λ i µ i /µ L + ) λ i µ i /µ L + ) λ i µ i /µ L + ) λ i µ i /µ L + λ i — — 0.87 0.71 0.26 µ i /µ E + λ i µ i /µ E λ i (TeV) and physical mass ratios µ i /µ L,E at one loop level assuming λ H = 0. The mass ratios assume a common value for µ i at M G (cid:39) × GeV.
TeV scale, which is not taken into account in Ref. [5]. This changes the gauge couplings above the TeVscale as well as the predictions of λ i . If we instead assume large threshold corrections at the unificationscale, the predictions of Ref. [5] hold. The alignment is also guaranteed if µ i are solely given by a vev of S . See section VI. σ S Br γγ at the LHC with √ s = 13 TeV as a function of µ L for ( + ) i and µ E for ( + ) and ( + ), assumingthat 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 thiscase, only the ( + ) , , theories can explain the observed diphoton excess and require lightvector matter. Vector quark masses are predicted in the range 700 − σ and 2 σ regions for a signalrate of σ Br γγ = (4 . . − .
1) fb from combined fits to the experimental data [14].Once we relax the assumption of the unification of µ i at the unification scale, the possi-bilities for explaining the 750 GeV excess are greatly expanded. This occurs in theories inwhich boundary conditions in extra dimensions break the unified symmetry [15]. It can alsooccur in four dimensional theories if these masses pick up unified symmetry breaking effectsat an O (1) level. In the upper left panel of Figure 2, we show the prediction for σ S Br γγ atthe LHC with √ s = 13 TeV as a function of degenerate vector quark masses with vectorlepton masses fixed at µ L,E = 380 GeV. We again assume that the scalar components of Φ i and ¯Φ i are sufficiently heavy not to contribute. Now all theories can explain the observeddiphoton excess. For ( + ) , the masses of the vector quarks are as low as 500 GeV. Thissatisfies the lower bound on the vector quark mass, if it dominantly decays into first orsecond generation quarks [16]. For ( + ) , , 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 and ¯Φ i , which isalso investigated in Ref. [6]. The trilinear couplings between the scalar components of S , Φ i and ¯Φ i are given by −L tri = λ i µ i ( S + S ∗ ) (cid:0) | Φ i | + | ¯Φ i | (cid:1) + (cid:0) λ i ( µ ∗ S S ∗ + A i S ) Φ i ¯Φ i + h . c . (cid:1) , (3)where A i are soft trilinear couplings. We take A i to be real by phase rotations of the scalarcomponents of Φ i and ¯Φ i , and we neglect the trilinear couplings proportional to µ S . Themass terms of Φ i and ¯Φ i are given by V mass = m i | Φ i | + m i | ¯Φ i | + (cid:0) B i µ i Φ i ¯Φ i + h . c . (cid:1) . (4)Assuming m i = m i ≡ m i, , the mass eigenbasis, (Φ i + , Φ i − ), is given by Φ i ¯Φ ∗ i = √ e − iθ i √ − e iθ i √ √ Φ i + Φ i − , (5)6 + ) ( + ) ( + ) ( + ) +
10 15 +
400 450 500 550 600 650 700024681012 μ L or μ E ( GeV ) σ ( pp → S → γγ )( f b ) Unified massscalars decoupled
400 450 500 550 600 650 700024681012 μ L or μ E ( GeV ) σ ( pp → S → γγ )( f b ) Unified mass m DQU,02 =( ) , m LE,02 =( ) A DQU = A LE = B DQU = B LE = θ i =
400 450 500 550 600 650 700024681012 μ L or μ E ( GeV ) σ ( pp → S → γγ )( f b ) Unified mass m DQU, - = m LE, - = A i - , θ i =
400 450 500 550 600 650 700024681012 μ L or μ E ( GeV ) σ ( pp → s → γγ )( f b ) Unified mass m DQU, - = m LE, - = A i - , θ i = FIG. 1. Theories with unified mass relations and λ H = 0: Prediction for σB γγ at √ s = 13 TeVas a function of the lightest vector lepton mass, with scalar partners decoupled (upper left), softmasses indicated in the figure (upper right), and the maximal possible A i − terms (lower left). Inthe lower right panel, the contribution only from the scalar s is depicted with the maximal possible A i − term. with masses m i ± = µ i + m i ± | B i | µ i . (6)Here, θ i is the phase of B i , B i = e iθ i | B i | . The trilinear couplings in the mass eigenbasis aregiven by −L tri = λ i √ (cid:0) A si + s | Φ + | + A si − s | Φ − | + A ai + a | Φ + | + A ai − a | Φ − | (cid:1) , (7) A si ± ≡ ∓ A i cos θ i + 2 µ i , A ai ± ≡ ∓ A i sin θ i , (8)where we neglect couplings proportional to Φ + Φ ∗− , which are irrelevant for the diphotonsignal. 7 + ) ( + ) ( + ) ( + ) +
10 15 +
500 1000 1500 2000024681012 μ D , Q , U ( GeV ) σ ( pp → S → γγ )( f b ) μ L , μ E =
380 GeV scalars decoupled
500 1000 1500 2000024681012 μ D , Q , U ( GeV ) σ ( pp → S → γγ )( f b ) μ L , μ E =
380 GeV m DQU,02 =( ) , m LE,02 =( ) A DQU = A LE = B DQU = B LE = θ i =
500 1000 1500 2000024681012 μ D , Q , U ( GeV ) σ ( pp → S → γγ )( f b ) μ L , μ E =
380 GeV m DQU, - =
700 GeV, m LE, - =
380 GeVmaximal A i - , θ i =
500 1000 1500 2000024681012 μ D , Q , U ( GeV ) σ ( pp → s → γγ )( f b ) μ L , μ E =
380 GeV m DQU, - =
700 GeV, m LE, - =
380 GeVmaximal A i - , θ i = FIG. 2. Theories without unified mass relations and λ H = 0: Prediction for σB γγ at √ s = 13 TeVas a function of the degenerate vector quark mass for vector lepton masses at 400 GeV, with scalarpartners decoupled (upper left), soft masses indicated in the figure (upper right), and the maximalpossible A i − terms (lower left). In the lower right panel, the contribution from only the scalar s isdepicted with the maximal possible A i − term. In the upper right panels of Figures 1 and 2, we show the diphoton signal rate includingthe scalar loop contributions. We take reference values of the soft masses shown in thefigures, with moderate values of A i = (1 ,
2) TeV for vector (leptons, quarks). The boundson the vector quark/lepton masses are relaxed typically by 100 GeV. Larger A i can furtherrelax the bound [6]. In the lower left panels of Figures 1 and 2, we take the maximal A i − allowed by stability of the vacuum, m DQU − = 700 GeV, m LE − = 380 GeV, and decoupledΦ + . For the size and derivation of the maximal A i , see appendix A and Ref. [6]. All theoriescan explain the diphoton excess. Note, however, that large A i typically generate a largemass splitting between s and a by quantum corrections (see section III). Both s and a can8ontribute to the diphoton signal at 750 GeV because the phases θ i allow cancellations inthe mass splitting, although tuning is required for a narrow width of the 750 GeV resonance.Alternatively, in the lower right panels of Figures 1 and 2, we assume that the masses of thescalar and the pseudoscalar are sufficiently split that only the scalar s contributes to the750 GeV excess. Even in this case, due to large A i , all theories except ( + ) can explainthe diphoton excess. B. Non-vanishing Higgs coupling: λ H (cid:54) = 0 Let us now turn on the coupling between S and the Higgs multiplet, λ H . The existenceof λ H slightly changes the renormalization running of couplings. In Table II, we show theprediction for λ i (TeV) in each theory. Here we assume that λ H is also large at a high energyscale. 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 D L Q U E H ( + ) λ i µ i /µ L + ) λ i µ i /µ L + ) λ i µ i /µ L + ) λ i µ i /µ L + λ i — — 0.86 0.69 0.26 0.24 µ i /µ E + λ i µ i /µ E λ i (TeV) and physical mass ratios µ i /µ L,E at one loop level with λ H (cid:54) = 0.The mass ratios assume a common value for µ i at M G . β , and the CP conservation in the couplings between S and the Higgs mul-tiplet, the mass eigenstate is approximately given by the heavy Higgs states ( H , A , H ± )composed of H d , the singlet pseudoscalar a , and the mixture of the standard model likeHiggs h and the singlet scalar s . (With θ i (cid:54) = 0 , π , quantum corrections inevitably inducemixing between s and a ; see section III. The mixing is suppressed for sufficiently large m s .The pseudo-scalar a mixes with the heavy CP-odd Higgs A through the A term couplingbetween S and the Higgs multiplet. This leads to the decay of a into a pair of bottomquarks. The decay mode does not affect the diphoton signal rate for a sufficiently largeheavy Higgs mass, a sufficiently small A term, and/or not very large tan β .)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 explainedby 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 ( + ) i and µ E for ( + ) and ( + ), assumingthat µ i unify at the unification scale. In the upper left panel, the contribution from thescalar components of Φ i and ¯Φ i are ignored, while it is taken into account in other panels.All theories except for ( + ) can explain the diphoton excess without large A i terms.Vector quarks are as heavy as 600 − 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 quarkmasses, with vector lepton masses and the Higgsino mass fixed at 380 GeV. The vectorquark masses can be as large as 2 TeV without large A i terms. III. WIDE DIPHOTON RESONANCE FOR λ H = 0 AND SMALL B S µ S In this section, we discuss a possible way to obtain a “wide width resonance” from thescalar S . As we have pointed out in Ref. [5], the mass difference of a few tens of GeV betweenthe scalar s and the pseudoscalar a can be naturally obtained by a threshold correction atthe TeV scale from Φ i and ¯Φ i . Then s and a are observed as a single wide resonance. Herewe 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 othersoft masses, and hence µ S should be suppressed. This requires that the soft mass squared of10 + ) ( + ) ( + ) ( + ) +
10 15 +
400 450 500 550 600 650 700024681012 μ L or μ E ( GeV ) σ ( pp → a → γγ )( f b ) Unified massscalars decoupled μ H =
380 GeV
400 450 500 550 600 650 700024681012 μ L or μ E ( GeV ) σ ( pp → a → γγ )( f b ) Unified mass μ H =
380 GeV m DQU,02 =( ) , m LE,02 =( ) A DQU = A LE = B DQU = B LE = θ i = π /
400 450 500 550 600 650 700024681012 μ L or μ E ( GeV ) σ ( pp → a → γγ )( f b ) Unified mass μ H =
380 GeV m DQU, - = m LE, - = A i - , θ i ≠ FIG. 3. Theories with unified mass relations and λ H (cid:54) = 0: Prediction for σ a B γγ at √ s = 13 TeVas a function of the lightest vector lepton mass, with scalar partners decoupled (upper left panel)and soft masses indicated in the figure (other panels). S , m 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 theother 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∆ V = 12 (cid:16) s a (cid:17) ∆ ss ∆ sa ∆ sa sa , (9)∆ ss = 132 π (cid:88) i λ i (cid:20) µ i ln m i + m i − µ i + 4 µ i A i cos θ i ln m i + m i − + A i cos2 θ i (cid:18) − m i + + m i − m i + − m i − ln m i + m i − (cid:19)(cid:21) , (10)∆ sa = 132 π (cid:88) i λ i A i sin θ i (cid:20) µ i ln m i + m i − + A i cos θ i (cid:18) − m i + + m i − m i + − m i − ln m i + m i − (cid:19)(cid:21) , (11)11 + ) ( + ) ( + ) ( + ) +
10 15 +
500 1000 1500 2000024681012 μ D , Q , U ( GeV ) σ ( pp → a → γγ )( f b ) μ L , μ E , μ H =
380 GeV scalars decoupled
500 1000 1500 2000024681012 μ D , Q , U ( GeV ) σ ( pp → a → γγ )( f b ) μ L , μ E , μ H =
380 GeV m DQU,02 =( ) , m LE,02 =( ) A DQU = A LE = B DQU = B LE = θ i = π /
500 1000 1500 2000024681012 μ D , Q , U ( GeV ) σ ( pp → a → γγ )( f b ) μ L , μ E , μ H =
380 GeV m DQU, - =
700 GeV, m LE, - =
380 GeVmaximal A i - , θ i ≠ FIG. 4. Theories without unified mass relations and λ H (cid:54) = 0: Prediction for σ a B γγ at √ s = 13 TeVas a function of the degenerate vector quark mass, for vector lepton masses and the Higgsino massat 380 GeV, with scalar partners decoupled (upper left panel) and soft masses indicated in thefigure (other panels). where the correction ∆ aa is absorbed into the soft mass squared of S . Note that in thesupersymmetric limit, where A i = 0 and m i + = m i − = µ i , the mass difference vanishes. InFigure 5, the mass difference is shown for each theory as a function of the size of the A i terms, with the mass parameters shown in the table. The mass difference can be few tensof GeV.If µ S = 0, the ˜ s mass arises at one loop from virtual vector matter and ˜ s may be thelightest 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 investigatewhether a small SH u H d coupling that provides this mixing also gives a small enough mixingbetween s and the doublet Higgs boson so that s still contributes to the diphoton resonance.12 + ) ( + ) ( + ) ( + ) +
10 15 + x m s - m a θ i = µ DQU µ LE . m DQU, (1 TeV) m LE, (0 . B DQU B LE . A DQU × x TeV A LE × x TeV x Δ m θ i = π / - - - x m s - m a θ i = π FIG. 5. The mass difference between the scalar s and the pseudoscalar a (or the two mass eigen-states of S in the case of CP violation) for three different values of the phase of B i µ i . The horizontalaxis, x , represents the size of the A i terms as indicated in the table. If so, the predominantly ˜ s dark matter may have a mass allowing the observed abundancevia 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 (cid:54) = 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 θ hs (cid:39) √ λ H vµ H m s = 0 . × λ H . µ H
400 GeV (cid:16) m s (cid:17) − , (12)where m s is the mass of s and v (cid:39)
246 GeV is the vev of the standard model Higgs. Themeasurement of the Higgs production cross section restricts the mixing, θ hs < . m s , m s >
350 GeV (cid:18) λ H . (cid:19) / (cid:16) µ H
380 GeV (cid:17) / . (13)In the limit m s (cid:29) m h,Z,W , the decay width of s into pairs of the standard model Higgsbosons, W bosons, and Z bosons can be evaluated by the equivalence theorem:Γ( s → hh ) (cid:39) Γ( s → ZZ ) (cid:39)
12 Γ( s → W + W − ) (cid:39) λ H π µ H m s = 0 .
13 GeV × (cid:18) λ H . (cid:19) (cid:16) m s TeV (cid:17) − (cid:16) µ H
400 GeV (cid:17) . (14)Through mixing with the standard model Higgs, s decays into a pair of top quarks with arate Γ( s → t ¯ t ) (cid:39) y t λ H π v µ H m s = 0 .
023 GeV × (cid:18) λ H . (cid:19) (cid:16) m s TeV (cid:17) − (cid:16) µ H
400 GeV (cid:17) . (15)For large m s , the scalar s also decays into a fermionic component of Φ i and ¯Φ i with a decayrateΓ( s → Φ i ¯Φ i ) (cid:39) λ H N i m s π (cid:18) − µ i m S (cid:19) / = 1 . × (cid:18) λ i . (cid:19) N i m s TeV (cid:18) − µ i m S (cid:19) / . (16)For simplicity, we assume that the scalar components of Φ i and ¯Φ 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 ( + ) i and µ E for ( + )and ( + )) are 380 GeV and µ i unify at M G . The signal is depleted for m s >
760 GeVsince the decay mode into a pair of vector leptons is open. In Figure 7 we show a similar plotbut assuming µ L = µ E = µ H = 380 GeV with the masses of the vector quarks determinedso that σ a Br γγ = 4 . σ ( pp → s → W W, ZZ ) canbe estimated by the equivalence theorem. In both cases, the cross section is predicted to be O (100 −
1) fb for m s = (400 − V. SEMI-PERTURBATIVE UNIFICATION AND TEV SCALE THRESHOLDS
In this section, we discuss gauge coupling unification in ( + ) and ( + ) theories. Inthese theories, gauge couplings α i become O (1) around the unification scale, and unify in a14
00 600 800 1000 1200 14000.10.5151050100 m s ( GeV ) σ ( pp → s → hh )( f b ) Unified mass μ L or E = μ H =
380 GeV ( + ) ( + ) ( + ) ( + ) + + FIG. 6. Theories with unified mass relations: Prediction for σ s B hh at √ s = 13 TeV as a functionof m s . Predictions for σ s B W W,ZZ can be estimated by the equivalence theorem.
400 600 800 1000 1200 14000.5550500 m s ( GeV ) σ ( pp → s → hh )( f b ) μ L , μ E , μ H =
380 GeV σ a Br γγ = ( + ) ( + ) ( + ) ( + ) + + FIG. 7. Theories without unified mass relations: Prediction for σ s B hh at √ s = 13 TeV as afunction of m s . Predictions for σ s B W W,ZZ can be estimated by the equivalence theorem. semi-perturbative regime. Nevertheless, as we will show, precision gauge coupling unificationis successfully achieved with moderate threshold corrections around the TeV scale.In Figure 8, we show the running of the standard model gauge couplings for ( + ) and( + ) with the NSVZ beta function [18], evaluating anomalous dimensions at the one-loop level. Here we assume that the masses of all MSSM particles and vector quarks/leptonsare 1 TeV. It can be seen that the SU (3) c gauge coupling enters the non-perturbative regimebefore unification. The perturbative unification of gauge couplings requires large thresholdcorrections at a high energy scale or smaller threshold corrections at the TeV scale.To assess the required threshold corrections at the TeV scale, we solve the renormalizationgroup equation from the unification scale down to the electroweak scale. In Figure 9, we15 μ ( GeV ) π / α i ( μ ) ( + ) U ( ) Y SU ( ) L SU ( ) c m SUSY = μ i = λ i ( × GeV )= λ H = μ ( GeV ) π / α i ( μ ) ( + ) U ( ) Y SU ( ) L SU ( ) c m SUSY = μ i = λ i ( × GeV )= λ H = FIG. 8. Running of the standard model gauge couplings for ( + ) and ( + ), with massesof all MSSM particles and vector quarks/leptons of 1 TeV. show ∆b i , the difference between the predicted and observed gauge couplings at the weakscale ∆b i ≡ πα i ( m Z ) (cid:12)(cid:12)(cid:12)(cid:12) prediction − πα i ( m Z ) (cid:12)(cid:12)(cid:12)(cid:12) observed , (17)as a function of the unification scale M G , with various α G . Here we assume that the massesof all MSSM particles and vector quarks/leptons are 1 TeV and λ i ( M G ) = 2.In each panel of Figure 9, the couplings come close to unifying in the region of M G ∼ (5 × – 10 ) GeV, where ∆b i are all positive and typically 3 – 5. These are not very largeand hence can be countered by TeV scale threshold corrections. As superpartner and/orvector quark and lepton masses are increased above 1 TeV, the predicted gauge couplingsat M Z become larger and hence the lines in Figure 9 are lowered, so that raising thesemasses produces threshold corrections of the required sign. For precision unification thethree curves must intersect at a point where ∆b i = 0. For ( + ), this means that thecurve for SU (2) must be lowered more than the curve for SU (3)—the masses for particleswith SU (2) L charge must be raised further than the masses for colored particles.Such a mass spectrum is difficult to achieve in conventional supersymmetric unificationscenarios, where boundary conditions at the unification scale and renormalization runningtypically lead to colored particles heavier than non-colored particles. Precision unificationin ( + ) calls for a non-conventional scenario, such as unified symmetry breaking byboundary conditions in extra dimensions. For example, the masses of superparticles andvector quarks/leptons in Table III, with wino heavier than gluino, predict gauge couplings16 ( ) Y SU ( ) L SU ( ) c × × × × × - - M G / GeV Δ b i ( + ) α G = m SUSY = μ i = × × × × × - - M G / GeV Δ b i ( + ) α G = m SUSY = μ i = × × × × × - - M G / GeV Δ b i ( + ) α G = m SUSY = μ i = × × × × × - - M G / GeV Δ b i ( + ) α G = m SUSY = μ i = FIG. 9. The difference between the predicted and observed gauge couplings at the weak scale as afunction of the unification scale M G , with various α G . Here, the masses of all MSSM particles andvector quarks/leptons are 1 TeV and λ ( M G ) = 2. at M Z in agreement with the observed values, as shown in Figure 10. VI. VACUUM EXPECTATION VALUE FOR S AND SOFT OPERATORSA. Vacuum expectation value for S In estimating the diphoton signal rate, we assumed that the phases of λ i are aligned witheach other in the basis where µ i have a common phase. Even with CP conservation in thesuperpotential, we have assumed that the signs of λ i µ i are independent of i . This alignmentmaximizes the diphoton rate and, while it is not necessary for large A i and non-unifiedmasses, in other cases it is helpful in obtaining a sufficient diphoton rate. This alignment isnaturally achieved if µ i are forbidden by some symmetry, under which S is charged, and are17 × × × × × - - M G / GeV Δ b i ( + ) α G = U ( ) Y SU ( ) L SU ( ) c FIG. 10. Precision gauge coupling unification in ( + ) with masses of superparticles and vectorquarks/leptons shown in Table III.TABLE III. A sample mass spectrum of MSSM superparticles (upper row) and vectorquarks/leptons (lower row). Here, m QUDLE ≡ √ m QUDLE + m QUDLE − . With this mass spectrum,the prediction for the gauge couplings at the weak scale is improved, as is shown in Figure 10. m ˜ q m ˜ u m ˜ d m ˜ l m ˜ e m H µ H m ˜ g m ˜ w m Q m U m D m L m E µ Q µ U µ D µ L µ E solely given by the vev of S . Thus, instead of Eq. (1) we may start with the much simplersuperpotential W ⊃ S (cid:88) i λ i Φ i Φ i + λ H SH u H d . (18)In this case µ i = λ i (cid:104) S (cid:105) 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 showthat only the ( + ) , , theories explain the diphoton resonance. However, the lower panelsshow that the scalar contribution with large A i allows all theories to explain the diphotonresonance. After electroweak symmetry breaking the soft trilinear scalar interaction propor-tional to A H leads to a linear term in S , which will therefore develop a vev. However, even18n the large A H limit this is too small to give sufficient mass to the vector matter. Someother 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 termin 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 ∝ (cid:104) S (cid:105) , the vev of S is given by | (cid:104) S (cid:105) | = 1 κ (cid:114) m s + m a > . κ , (19)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 smallerthan the one at the unification scale. In ( + ) theory, κ (TeV) = 0 . λ i ( M G ) = 1 and κ ( M G ) = 3. The corresponding lower bound on µ L (TeV) is 940 GeV, which is too large toexplain the diphoton excess. In other theories, the lower bound is severer. Theories with µ i,H,S generated from (cid:104) S (cid:105) can explain the diphoton signal if the superpotential couplings λ i,H,S become strong at scales of (10 − ) TeV, since then κ (TeV) can be sufficientlylarge [13]. However, for perturbative couplings to the unification scale, µ i cannot arise from (cid:104) S (cid:105) of Eq. (19).Another possibility is that (cid:104) S (cid:105) 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 i ¯Φ i (4). (Constructionof a similar mechanism with a non- R symmetry is straightforward.) In any supersymmetrictheory, the superpotential, which has an R charge of 2, must have a non-zero vev to cancel thecosmological constant induced by supersymmetry breaking. We denote the chiral operatorof R charge 2 that condenses and generates the superpotential vev as O . The tadpole termof S is given by K = O S + h . c .. (20)In gravity mediation, this term generates a tadpole term ∼ (TeV) S and hence (cid:104) S (cid:105) = O (1) TeV. On the other hand, the superpotential term W ∼ O Φ i ¯Φ i is forbidden (exceptfor Z R ). It is essential that there is no chiral operator, ¯ O , having R charge − O ; otherwise, the superpotential term W ∼ ¯ O Φ i ¯Φ i generates µ i independent of (cid:104) S (cid:105) . Such a chiral operator is actually absent when R symmetry is broken19y a gaugino condensation. This mechanism leads to Eq. (18) with S having a vev of orderthe supersymmetry breaking scale, providing the messenger scale of order the Planck mass.The R symmetry forbids both S and S interactions, so that at tree-level µ S = 0. (Fora discrete Z R symmetry S is allowed. In this case, the degeneracy of s and a cannotbe naturally explained.) The fermionic component of S , ˜ s , is massless at tree-level and isexpected to be the lightest supersymmetric particle. Since the R symmetry is broken by thesupersymmetry breaking interactions, the ˜ s mass appears at the TeV scale from integratingout Φ i and ¯Φ i at one loop. These radiative contributions to m ˜ s are proportional to µ i and B i µ i , and are of order O (10 − s neutralino dark matter results from annihilation via the Z or Higgs pole. B. The scale of soft operators and fine-tuning
Consider the mass scale of the soft supersymmetry breaking at low energies. For afixed value of the gaugino masses, for example close to the experimental limit, as morevector quarks/leptons are added to the theory, the gaugino mass at the unification scalebecomes larger for a high messenger scale. This raises the overall soft mass scale for thescalar superpartners, leading to fine-tuning to obtain a singlet scalar at 750 GeV and scalarvector quarks/leptons sufficiently light to contribute to the diphoton signal. For ( + ) ,( + ) and ( + ) theories, the required fine-tuning to obtain the 750 GeV massamounts to O (1)%. The fine-tuning is severer, typically by a factor of 10, if the masssquared of S , m S , at the TeV scale is required to be positive (see sections III and VI).This is because the renormalization of soft masses makes m S negative at the TeV scale,unless m S is positive and large at the unification scale. To avoid the tachyonic masses ofthe vector squarks/sfermions, their soft masses must be also large enough at a high energyscale, which raises soft mass scales further. Such fine-tuning can be avoided by introducingnon-standard low scale mediation of supersymmetry breaking or non-standard running ofsoft operators, for example induced by conformal hidden sector interactions [19–21]. Such aconformal sector also has the potential to yield large A terms [21], which are favored by thediphoton signal and the Higgs mass of 125 GeV.In section V we found that, for precision gauge coupling unification in ( + ), non-standard soft operators at the TeV scale were also required.20 II. SUMMARY AND DISCUSSION
Following an initial study in Ref. [5], we confirm that the reported 750 GeV diphotonresonance can be explained by supersymmetric theories that add a gauge singlet S = ( s + ia ) / √ i , ¯Φ i ) to the minimal set of particles: there are 6 possibilitiesfor vector matter that allow perturbative gauge coupling unification, and the case of a fullgeneration of vector matter is particularly interesting as it leads to α G ∼
1. For each ofthese 6 possibilities there are two versions of the theory with different Higgs phenomenology,depending on whether λ H SH u H d is included. For λ H = 0 ( (cid:54) = 0) the theory should be viewedas vector matter added to the MSSM (NMSSM).For λ H (cid:54) = 0, a narrow 750 GeV resonance arises from a → γγ and we predict a secondresonance decaying to dibosons s → hh, ZZ, W + W − , with a rate typically accessible infuture LHC runs as shown in Figures 6 and 7. For λ H = 0, there is no mixing of s with theHiggs boson so there are two diphoton resonances arising from a, s → γγ . If one of theseproduces the observed resonance at 750 GeV, the other may be of much higher mass, andboth would be narrow. Alternatively, if the mass splitting between s and a is small theymay both contribute to the observed diphoton signal, leading to an apparent width of orderthe mass splitting.The diphoton event rate depends on several factors: the quantum numbers and massesof the vector quarks and leptons, the masses of the vector squarks and sleptons (whichdepend on A parameters and phases), whether the vector quark and lepton masses obeyunified relations, and whether the resonance is produced by a , s or both. For unified vectorquark and lepton mass relations and decoupled vector squarks and sleptons the event rateis sufficient only for ( + ) , , theories, whether λ H is zero or not; and even these theoriesrequire vector lepton masses below (400 − A terms [6]. Indeed, maximal values of A consistent with vacuum stabilityallow vector quarks to be decoupled in some theories, with the signal arising from vectorleptons, sleptons and squarks. However, these A terms are very large and, for moderatevalues of A in the 1 − S acquiring a vev. In section VI we introduce a theory with an R symmetry thataccomplishes this in a way that explains why all the superpotential mass parameters have ascale 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 toshow that the corresponding width of the diphoton resonance is sensitive to A terms andCP violating phases.While perturbative supersymmetric unified theories can easily account for the diphotonsignal, we find it likely that some scheme beyond gravity mediation is needed for soft oper-ators and their running. The extra matter makes the gluino mass very large at unificationscales 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 ( + ) and( + ) theories we also find that for the gauge couplings to remain perturbative we needeither 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. ACKNOWLEDGMENTS
This work was supported in part by the Department of Energy, Office of Science, Officeof High Energy Physics, under contract No. DE-AC02-05CH11231, by the National ScienceFoundation under grants PHY-1316783 and PHY-1521446, and by MEXT KAKENHI GrantNumber 15H05895. LJH thanks the Simons Foundation and the Institute for TheoreticalStudies, ETH, Zurich.
Appendix A: Maximal A Terms
In this appendix, we estimate the bound on the size of the A i terms from vacuum stability.We consider cases with θ i (cid:39) , π . (The constraint for other θ i can be obtained by taking22nto account the appropriate factors of cos θ i and sin θ i as well as the dynamics of a ). Thestrongest constraint comes from the tunneling involving Φ i − and s . Hereafter we drop thesubscripts i and − . The scalar potential of Φ and s is given by V ( s, Φ) = λ | Φ | + λ s | Φ | − λ √ As | Φ | + 12 m s s + m | Q | . (A1)We take A > s → m s σ/λ , Φ → m s φ/ √ λ and x µ → ξ µ /m s , the action is given by λ S = (cid:90) d ξ (cid:20) ∂σ∂σ + 12 ∂φ∂φ − V ( σ, φ ) (cid:21) , (A2)where V = 116 φ + 14 φ σ − r A √ φ σ + 12 σ + 12 r φ φ , r A ≡ Am s , r φ ≡ m φ m s . (A3)We consider the tunneling path with the minimum potential barrier, in which σ = r A √ φ φ / . (A4)Along this path, the potential is given by V eff ( φ ) = 116 φ + 12 r φ φ − r A φ φ / , (A5)and the canonically normalized field is given by φ c ≡ (cid:115) r A φ (1 + φ / φ. (A6)We numerically obtain the bounce action [22] solving the equation of motion of φ c . InFigure 11, we show the size of the bounce action, S B , as a function of A for m φ = 380 GeVand 700 GeV. We require that the lifetime of the vacuum is longer than the age of theuniverse, S B > A i − for each theory. Theresult is consistent with the one presented in Ref. [6] within a few tens of percent. [1] M. Kado, “ATLAS results” Talk at ATLAS and CMS physics results from Run 2, CERN,Switzerland, December 15 (2015); ATLAS Collaboration, “Search for new physics decaying totwo photons,” ATLAS-CONF-2015-081.
000 4000 6000 8000 10 0001.01.52.02.53.03.5 A ( GeV ) l og λ S B m ϕ =
700 GeV380 GeV
FIG. 11. The bounce action as a function of A for m φ = 380 GeV and 700 GeV.TABLE IV. The upper bound on A i − in GeV, taking m DQU, − = 700 GeV and m LE, − = 380 GeV. λ H = 0 ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) A D − A L − A Q − A U − A E − λ H (cid:54) = 0 ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) A D − A L − A Q − A U − A E −
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