Phenomenology of the Utilitarian Supersymmetric Standard Model

We study the 2010 specific version of the 2002 proposed $U(1)_X$ extension of the supersymmetric standard model, which has no $\mu$ term and conserves baryon number and lepton number separately and automatically. We consider in detail the scalar sector as well as the extra $Z_X$ gauge boson, and their interactions with the necessary extra color-triplet particles of this model, which behave as leptoquarks. We show how the diphoton excess at 750 GeV, recently observed at the LHC, may be explained within this context. We identify a new fermion dark-matter candidate and discuss its properties. An important byproduct of this study is the discovery of relaxed supersymmetric constraints on the Higgs boson's mass of 125 GeV.


Since the recent announcements [1, 2] by the ATLAS and CMS Collaborations at the Large
Hadron Collider (LHC) of a diphoton excess around 750 GeV, numerous papers [3] have appeared explaining its presence or discussing its implications. In this paper, we study the phenomenology of a model proposed in 2002 [4], which has exactly all the necessary and sufficient particles and interactions for this purpose. They were of course there for solving other issues in particle physics. However, the observed diphoton excess may well be a first revelation [5] of this model, including its connection to dark matter.
This 2002 model extends the supersymmetric standard model by a new U (1) X gauge symmetry. It replaces the µ term with a singlet scalar superfield which also couples to heavy color-triplet superfields which are electroweak singlets. The latter are not ad hoc inventions, but are necessary for the cancellation of axial-vector anomalies. It was shown in Ref. [4] how this was accomplished by the remarkable exact factorization of the sum of eleven cubic terms, resulting in two generic classes of solutions [6]. Both are able to enforce the conservation of baryon number and lepton number up to dimension-five terms. As such, the scalar singlet and the vectorlike quarks are indispensible ingredients of this 2002 model. They are thus naturally suited for explaining the observed diphoton excess. In 2010 [7], a specific version was discussed, which will be the subject of this paper as well. An important byproduct of this study is the discovery of relaxed supersymmetric constraints on the Higgs boson's mass of 125 GeV. This is independent of whether the diphoton excess is confirmed or not.

Model
Consider the gauge group SU (3) C × SU (2) L × U (1) Y × U (1) X with the particle content of Ref. [4]. For n 1 = 0 and n 4 = 1/3 in Solution (A), the various superfields transform as shown in Table 1. There are three copies of Q, u c , d c , L, e c , N c , S 1 , S 2 ; two copies of U, U c , S 3 ; and one copy of φ 1 , φ 2 , D, D c . The only allowed terms of the superpotential are thus trilinear, Table 1: Particle content of proposed model.
The absence of any bilinear term means that all masses come from soft supersymmetry breaking, thus explaining why the U (1) X and electroweak symmetry breaking scales are not far from that of supersymmetry breaking. As S 1,2,3 acquire nonzero vacuum expectation values (VEVs), the exotic (U, U c ) and (D, D c ) fermions obtain Dirac masses from S 3 , which also generates the µ term. The singlet N c fermion gets a large Majorana mass from S 1 , so that the neutrino ν gets a small seesaw mass in the usual way. The singlet S 1,2,3 fermions themselves get Majorana masses from their scalar counterparts S 1,2,3 through the S 1 S 2 S 3 terms. The only massless fields left are the usual quarks and leptons. They then become massive as φ 0 1,2 acquire VEVs, as in the minimal supersymmetric standard model (MSSM).
Because of U (1) X , the structure of the superpotential conserves both B and (−1) L , with i.e. R ≡ (−) 2j+3B+L , and is conserved. Note also that the quadrilinear terms QQQL and u c u c d c e c (allowed in the MSSM) as well as u c d c d c N c are forbidden by U (1) X . Proton decay is thus strongly suppressed. It may proceed through the quintilinear term QQQLS 1 as the S 1 fields acquire VEVs, but this is a dimension-six term in the effective Lagrangian, which is suppressed by two powers of a very large mass, say the Planck mass, and may safely be allowed.

Gauge Sector
The new Z X gauge boson of this model becomes massive through S 1,2,3 = u 1,2,3 , whereas φ 0 1,2 = v 1,2 contribute to both Z and Z X . The resulting 2 × 2 mass-squared matrix is given Since precision electroweak measurements require Z −Z X mixing to be very small [9], v 1 = v 2 , i.e. tan β = 1, is preferred. With the 2012 discovery [10,11] of the 125 GeV particle, and identified as the one Higgs boson h responsible for electroweak symmetry breaking, tan β = 1 is not compatible with the MSSM, but is perfectly consistent here, as shown already in Ref. [7] and in more detail in the next section.
Consider the decay of Z X to the usual quarks and leptons. Each fermionic partial width is given by where c L,R can be read off under U (1) X from Table 1. Thus This will serve to distinguish it from other Z models [12].
At the LHC, limits on the mass of any Z boson depend on its production by u and d quarks times its branching fraction to e − e + and µ − µ + . In a general analysis of Z couplings to u and d quarks, where f = u, d. The c u , c d coefficients used in an experimental search [13,14] of Z are then given by where l = e, µ. In this model To estimate B(Z → l − l + ), we assume Z X decays to all SM quarks and leptons with effective zero mass, all the scalar leptons with effective mass of 500 GeV, all the scalar quarks with effective mass of 800 GeV, the exotic U, D fermions with effective mass of 400 GeV (needed to explain the diphoton excess), and one pseudo-Dirac fermion from combiningS 1,2 (the dark matter candidate to be discussed) with mass of 200 GeV. We find B(Z → l − l + ) = 0.04, and for g X = 0.53, a lower bound of 2.85 TeV on m Z X is obtained from the LHC data based on the 7 and 8 TeV runs.

5
Consider the scalar potential consisting of φ 1,2 and S 1,2,3 , where only the S 1,2,3 scalars with VEVs are included. The superpotential linking the corresponding superfields is Its contribution to the scalar potential is where φ 1 has been redefined to Φ 1 = (φ + 1 , φ 0 1 ). The gauge contribution is The soft supersymmetry-breaking terms are In addition, there is an important one-loop contribution from the t quark and its supersymmetric scalar partners: where is the well-known correction which allows the Higgs mass to exceed m Z .
Let φ 0 1,2 = v 1,2 and S 1,2,3 = u 1,2,3 , we study the conditions for obtaining a minimum of the scalar potential We then require that this solution does not mix the Re(φ 1,2 ) and Re(S 1,2,3 ) sectors. The additional conditions are Hence The 2 × 2 mass-squared matrix spanning where For λ 2 v 2 << κ, the Higgs boson h Re(φ 0 1 + φ 0 2 ) has a mass given by whereas its heavy counterpart H Re(−φ 0 1 + φ 0 2 ) has a mass given by The conditions for obtaining the minimum of V in the S 1,2,3 directions are ] is given by The 5×5 mass-squared matrix spanning has two zero eigenvalues, corresponding to the would-be Goldstone modes (1, 1, 0, 0, 0) and for the Z and Z X gauge bosons. One exact mass eigenstate is with mass given by Assuming that v 2 << u 2 2,3 , the other two mass eigenstates are A −Im(φ 0 1 ) + Im(φ 0 2 ) and A S [u 3 Im(S 1 ) + √ 2u 3 Im(S 2 ) + √ 2u 2 Im(S 3 )]/ u 2 2 + 3u 2 3 /2 with masses given by respectively. The charged scalar H ± = (−φ ± 1 + φ ± 2 )/ √ 2 has a mass given by In the MSSM without radiative corrections, where tan β = v 2 /v 1 . For v 1 = v 2 as in this model, m h would be zero. There is of course the important radiative correction from Eq. (14), but that alone will not reach 125 GeV. Hence the MSSM requires both large tan β and large radiative correction, but a significant tension remains in accommodating all data. In this model, as Eq. (23) shows, m 2 where v = 123 GeV. This is a very interesting and important result, allowing the Higgs boson mass to be determined by the gauge U (1) X coupling g X in addition to the Yukawa coupling f which replaces the µ parameter, i.e. µ = f u 3 . There is no tension between m h = 125 GeV and the superparticle mass spectrum. Since λ 2 0.25 form t 1 TeV, we have the important constraint For illustration, we have already chosen g X = 0.53. Hence f = 0.5 and for u 3 = 2 TeV, f u 3 = 1 TeV is the value of the µ parameter of the MSSM. Let us choose u 2 = 4 TeV, then m Z X = 2.87 TeV, which is slightly above the present experimental lower bound of 2.85 TeV using g X = 0.53 discussed earlier.
As for the heavy Higgs doublet, the four components (H ± , H, A) are all degenerate in mass, i.e. m 2 (4f 2 − 2g 2 X )u 2 3 + (4/3)g 2 X u 2 2 up to v 2 corrections. Each mass is then about 2.78 TeV. In more detail, as shown in Eq. (37), m 2 H ± is corrected by g 2 2 v 2 = m 2 W plus a term due to f . As shown in Eq. (24), m 2 H is corrected by (g 2 1 + g 2 2 )v 2 = m 2 Z plus a term due to f and λ 2 . These are exactly in accordance with Eqs. (38) and (39).
In the S 1,2,3 sector, the three physical scalars are mixtures of all three Re(S i ) components, whereas the physical pseudoscalar A 12 has no Im(S 3 ) component. Since only S 3 couples to U U c , DD c , and φ 1 φ 2 , a candidate for the 750 GeV diphoton resonance must have an S 3 component. It could be one of the three scalars or the pseudoscalar A S , or the other S 3 without VEV. In the following, we will consider the last option, specifically a pseudoscalar χ with a significant component of this other S 3 . This allows the χU U c , χDD c and χφ 1 φ 2 couplings to be independent of the masses of U , D, and the charged higgsino. The other scalars and pseudoscalars are assumed to be much heavier, and yet to be discovered. comes from these couplings as well as S 3 φ 1 φ 2 . In addition, the direct S 1 S 2 S 3 couplings enable the decay of S 3 to other final states, including those of the dark sector, which contribute to its total width. The fact that the exotic U, U c , D, D c scalars are leptoquarks is also very useful for understanding [15]  by this existing model. In the following, we assume that the pseudoscalar χ is the 750 GeV particle, and show how its production and decay are consistent with the present data.
As mentioned earlier, there are 2 copies of S 3 and 3 copies each of S 1,2 . In addition to the ones with VEVs in their scalar components, there are 5 other superfields. One pairS 1,2 may form a pseudo-Dirac fermion, and be the lightest particle with odd R parity. It will couple to χ, say with strength f S which is independent of all other couplings that we have discussed, then the tree-level decay χ →S 1S2 dominates the total width of χ and is invisible.
For m χ = 750 GeV and m S = 200 GeV, we find Γ = 36 GeV if f S = 1.2. These numbers reinforce our numerical analysis to support the claim that χ is a possible candidate for the 750 GeV diphoton excess. Note also that λ g and λ γ have scalar contributions which we have not considered. Adding them will allow us to reduce the fermion contributions we have assumed and still get the same final reuslts.
If we disregard the decay to dark matter (f S = 0), then the total width of χ is dominated by Γ(χ → gg), which is then less than a GeV. Assuming that the cross section for the diphoton resonance is 6.2 ± 1 fb [16],

Scalar Neutrino and Neutralino Sectors
In the neutrino sector, the 2 × 2 mass matrix spanning (ν, N c ) per family is given by the well-known seesaw structure: where m D comes from v 2 and m N from u 1 . There are two neutral complex scalars with odd R parity per family, i.e.ν = (ν R + iν I )/ √ 2 andÑ c = (Ñ c R + iÑ c I )/ √ 2. The 4 × 4 mass-squared matrix spanning (ν R ,ν I ,Ñ c R ,Ñ c I ) is given by In the MSSM,ν is ruled out as a dark-matter candidate because it interacts elastically with nuclei through the Z boson. Here, the A N term allows a mass splitting between the real and imaginary parts of the scalar fields, and avoids this elastic-scattering constraint by virtue of kinematics. However, we still assume their masses to be heavier than that ofS 1,2 , discussed in the previous section.
In the neutralino sector, in addition to the 4 × 4 mass matrix of the MSSM spanning (B,W 3 ,φ 0 1 ,φ 0 2 ) with the µ parameter replaced by f u 3 , i.e.
The two are connected through the 4 × 4 matrix These neutral fermions are odd under R parity and the lightest could in principle be a darkmatter candidate. To avoid the stringent bounds on dark matter with the MSSM alone, we assume again that all these particles are heavier thanS 1,2 , as the dark matter discussed in the previous section.

Dark Matter
The 5 × 5 mass matrix spanning the 5 singlet fermions (S 1 ,S 2 ,S 1 ,S 2 ,S 3 ), corresponding to superfields with zero VEV for their scalar components, is given by Note that the 4 × 4 submatrix spanning (S 1 ,S 2 ,S 1 ,S 2 ) has been diagonalized to form two Dirac fermions. We can choose m 0 to be small, say 200 GeV, and M 1,2,3 to be large, of order TeV. However, because of the mixing terms m 13 , m 23 , the light Dirac fermion gets split into two Majorana fermions, so it should be called a pseudo-Dirac fermion.
The dark matter with odd R parity is the lighter of the two Majorana fermions, call itS, contained in the pseudo-Dirac fermion formed out ofS 1,2 as discussed in Sec. 6. It couples to the Z X gauge boson, but in the nonrelativistic limit, its elastic scattering cross section with nuclei through Z X vanishes because it is Majorana. It also does not couple directly to the Higgs boson h, so its direct detection at underground search experiments is very much suppressed. However, it does couple to A S which couples also to quarks through the very small mixing of A S with A. This is further suppressed because it contributes only to the spin-dependent cross section. To obtain a spin-independent cross section at tree level, the constraint of Eqs. (17) to (19) have to be relaxed so that h mixes with S 1,2,3 .
Let the coupling of h toSS be , then the effective interaction for elastic scattering ofS with nuclei through h is given by where f q = m q /2v = m q /(246 GeV). The spin-independent direct-detection cross section per nucleon is given by where µ DM = m DM M A /(m DM + M A ) is the reduced mass of the dark matter. Using [18] with [19] f p u = 0.023, f p d = 0.032, f p s = 0.020, f n u = 0.017, f n d = 0.041, f n s = 0.020, we find λ p 3.50 × 10 −8 GeV −2 , and λ n 3.57 × 10 −8 GeV −2 . Using A = 131, Z = 54, and M A = 130.9 atomic mass units for the LUX experiment [20], and m DM = 200 GeV, we find for the upper limit of σ SI < 1.5 × 10 −45 cm 2 , the bound < 6.5 × 10 −4 .
We have already invoked the χS 1S2 coupling to obtain a large invisible width for χ.
Consider now the fermion counterpart of χ, call itS , and the scalar counterparts ofS 1,2 , then the couplingsS S 1 S 2 andS S 2 S 1 are also f S = 1.2. Suppose one linear combination of S 1,2 , call it ζ, is lighter than 200 GeV, then the thermal relic abundance of dark matter is determined by the annihilationSS → ζζ, with a cross section times relative velocity given by Setting this equal to the optimal value [21] of 2.2×10 −26 cm 3 /s, we find f ζ 0.62 for m S = 1 TeV, m S = 200 GeV, and m ζ = 150 GeV. Note that ζ stays in thermal equilibrium through its interaction with h from a term in V D . It is also very difficult to be produced at the LHC, because it is an SM singlet, so its mass of 150 GeV is allowed.

Conclusion
The utilitarian supersymmetric U (1) X gauge extension of the Standard Model of particle interactions proposed 14 years ago [4] allows for two classes of anomaly-free models which have no µ term and conserve baryon number and lepton number automatically. A simple version [7] with leptoquark superfields is especially interesting because of existing LHC flavor anomalies.
The new Z X gauge boson of this model has specified couplings to quarks and leptons which are distinct from other gauge extensions and may be tested at the LHC. On the other hand, a hint may already be discovered with the recent announcements by ATLAS and CMS of a diphoton excess at around 750 GeV. It may well be the revelation of the singlet scalar (or pseudoscalar) S 3 predicted by this model which also predicts that there should be singlet leptoquarks and other particles that S 3 must couple to. Consequently, gluon fusion will produce S 3 which will then decay to two photons together with other particles, including those of the dark sector. This scenario explains the observed diphoton excess, all within the context of the original model, and not an invention after the fact.
Since S 3 couples to leptoquarks, the S 3 → l + i l − j decay must occur at some level. As such, S 3 → e + µ − would be a very distinct signature at the LHC. Its branching fraction depends on unknown Yukawa couplings which need not be very small. Similarly, the S 3 couplings to φ 1 φ 2 as well as leptoquarks imply decays to ZZ and Zγ with rates comparable to γγ.
An important byproduct of this study is the discovery of relaxed supersymmetric constraints on the Higgs boson's mass of 125 GeV. It is now given by Eq. (23), i.e. m 2 h (g 2 X + 2f 2 + λ 2 )v 2 , which allows it to be free of the tension encountered in the MSSM. This prediction is independent of whether the diphoton excess is confirmed or not.
Most importantly, since S 3 replaces the µ parameter, its identification with the 750 GeV excess implies the existence of supersymmetry. If confirmed and supported by subsequent data, it may even be considered in retrospect as the first evidence for the long-sought existence of supersymmetry.