Light sterile neutrinos, dark matter, and new resonances in a U(1) extension of the MSSM
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Light sterile neutrinos, dark matter, and new resonancesin a U (1) extension of the MSSM A. Hebbar, ∗ G. Lazarides, † and Q. Shafi ‡ Bartol Research Institute, Department of Physics and Astronomy,University of Delaware, Newark, DE 19716, USA School of Electrical and Computer Engineering, Faculty of Engineering,Aristotle University of Thessaloniki, Thessaloniki 54124, Greece (Dated: September 11, 2018)We present ψ ′ MSSM, a model based on a U (1) ψ ′ extension of the minimal supersymmetric stan-dard model. The gauge symmetry U (1) ψ ′ , also known as U (1) N , is a linear combination of the U (1) χ and U (1) ψ subgroups of E . The model predicts the existence of three sterile neutrinos withmasses . . U (1) ψ ′ breaking scale is of order 10 TeV. Their contribution to the effectivenumber of neutrinos at nucleosynthesis is ∆ N ν ≃ .
29. The model can provide a variety of possiblecold dark matter candidates including the lightest sterile sneutrino. If the U (1) ψ ′ breaking scale isincreased to 10 TeV, the sterile neutrinos, which are stable on account of a Z symmetry, becomeviable warm dark matter candidates. The observed value of the standard model Higgs boson masscan be obtained with relatively light stop quarks thanks to the D-term contribution from U (1) ψ ′ .The model predicts diquark and diphoton resonances which may be found at an updated LHC. Thewell-known µ problem is resolved and the observed baryon asymmetry of the universe can be gener-ated via leptogenesis. The breaking of U (1) ψ ′ produces superconducting strings that may be presentin our galaxy. A U (1) R symmetry plays a key role in keeping the proton stable and providing thelight sterile neutrinos. I. INTRODUCTION E grand unified theory (GUT) [1] contains two espe-cially interesting maximal subgroups for model building,namely SU (3) and SO (10) × U (1) ψ . Supersymmetric(SUSY) models based on SU (3) , sometimes referred toas trinification models, have been extensively discussedin the literature. For instance, in SUSY SU (3) , mecha-nisms have been proposed to resolve [2] the minimalsupersymmetric standard model (MSSM) µ problem ormake [3] the proton essentially stable.The subgroup SO (10) × U (1) ψ of E can be decom-posed further, via SU (5), to the MSSM gauge symmetrygroup accompanied by U (1) χ × U (1) ψ [4, 5]. One in-triguing combination of these two U (1)’s, denoted hereas U (1) ψ ′ (also known as U (1) N [4] in the literature),is assumed [6] here to be broken at a scale at least anorder of magnitude greater than the TeV scale of softSUSY breaking. We refer to this extension of the MSSMaccompanied by U (1) ψ ′ as ψ ′ MSSM. The well-knownright handed neutrino contained in the matter 16-plet of SO (10) transforms as a singlet under U (1) ψ ′ . This en-ables the three right handed neutrinos to acquire largemasses, so that the standard seesaw scenarios can applyand high scale leptogenesis [7] can be realized [8]. Notethat the subscript ψ ′ reiterates the essential role playedby U (1) ψ ′ in resolving the MSSM µ problem.Our ψ ′ MSSM model employs in an essential way a ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: shafi@bartol.udel.edu U (1) R symmetry such that dimension five and higherdimensional operators potentially causing proton decayare eliminated. The MSSM µ problem is also resolvedand the usual lightest SUSY particle of MSSM remains[9] a compelling dark matter candidate. More intrigu-ingly perhaps, the model predicts that the three SO (10)singlet sterile neutrino matter fields that it contains canonly acquire tiny masses, on the order of 0 . U (1) ψ ′ is broken around 10 TeV. We estimate that forthis case the effective number of neutrinos during nu-cleosynthesis is changed by ≃ .
29. The lightest sterilesneutrino as well as two more particles, which are stableon account of discrete symmetries, can, under certain cir-cumstances, be additional cold dark matter candidates.If the breaking scale of U (1) ψ ′ is increased to 10 TeVor so, the sterile neutrinos, which happen to be stable onaccount of a Z symmetry, become plausible candidatesfor keV scale warm dark matter.The contribution of the D-term for U (1) ψ ′ to the massof the lightest CP-even neutral Higgs boson of the MSSMcan be appreciable leading, in the so-called decouplinglimit, to the observed value of 125 GeV with relativelylight stop quarks.In addition to the Z ′ gauge boson associated with thebreaking of the U (1) ψ ′ gauge symmetry, the model pre-dicts the existence of diphoton [10] and diquark [11] reso-nances with masses in the TeV range. A high luminosityor high energy (33 TeV) LHC upgrade may be able to findthem. Note that the U (1) ψ ′ breaking produces supercon-ducting strings [12] which presumably survived inflationand should be present in our galaxy. If the breaking scaleis not too high, a 100 TeV collider may be able to makethese strings, which definitely would be exciting.The layout of our paper is as follows. In Sec. II, we in-troduce the model with its field content, symmetries, andcouplings. In Sec. III, we analyze the details of the spon-taneous symmetry breaking of U (1) ψ ′ , while in Sec. IVwe discuss the spontaneous breaking of the electroweaksymmetry. Sec. V is devoted to the diphoton excess andSec. VI to the presentation of a numerical example. InSec. VII, we study the sterile neutrinos. The possiblecomposition of dark matter in the universe is presented inSec. VIII and our conclusions are summarized in Sec. IX. II. THE MODEL
We consider a SUSY model based on the gauge group G SM × U (1) ψ ′ , where G SM = SU (3) c × SU (2) L × U (1) Y is the standard model (SM) gauge group. The GUT-normalized generator Q ψ ′ of the extra local U (1) ψ ′ sym-metry is given by Q ψ ′ = 14 ( Q χ + √ Q ψ ) , (1)where Q χ is the GUT-normalized generator of the U (1) χ subgroup of SO (10) which commutes with its SU (5) sub-group and Q ψ is the GUT-normalized generator of the U (1) ψ subgroup of E which commutes with its SO (10)subgroup. The U (1) ψ ′ symmetry is to be spontaneouslybroken at some scale M and we prefer to implement thisbreaking by a SUSY generalization of the well-knownBrout-Englert-Higgs mechanism.The important part of the superpotential is W = y u H u qu c + y d H d qd c + y ν H u lν c + y e H d le c + 12 M ν c ν c ν c + λ iµ N H iu H id + κS ( N ¯ N − M )+ λ iD N D i D ci + λ iq D i qq + λ iq c D ci u c d c + λ L SL ¯ L + λ αH d ν c ¯ LH αd + λ iN N i N i ¯ N m P , (2)where m P is the reduced Planck mass and y u , y d , y ν , y e are the Yukawa coupling constants with the familyindices suppressed. Here q , u c , d c , l , ν c , e c are theusual quark and lepton superfields of MSSM includingthe right handed neutrinos ν c and H iu , H jd ( i, j = 1 , , SU (2) L doublets with hypercharge Y = 1 / − / N , ¯ N constitute a conju-gate pair of SM singlets, while S is a gauge singlet. Thecoupling λ ijµ N H iu H jd is diagonalized by appropriate rota-tions of H iu and H jd and a discrete Z symmetry underwhich H αu and H αd ( α = 2 ,
3) are odd is imposed. Conse-quently, only H u , H d couple to quarks and leptons andare the standard electroweak Higgs superfields.The superfields D i and D ci ( i = 1 , ,
3) are color tripletsand antitriplets with Y = − / / λ ijD N D i D cj is diagonalized by appropriaterotations of D i and D cj . The superfields N i ( i = 1 , , λ ijN N i N j ¯ N / m P isagain diagonalized by rotating N i and N j . We impose TABLE I: Superfield Content of the Model.Superfields Representations Extra Symmetriesunder G SM Z Z ′ R √ Q ψ ′ Matter Superfields q ( , , /
6) + + 1 / u c ( ¯3 , , − /
3) + + 1 / d c ( ¯3 , , /
3) + + 1 / l ( , , − /
2) + + 0 2 ν c ( , ,
0) + + 1 0 e c ( , ,
1) + + 1 1 H αu ( , , / − + 1 − H αd ( , , − / − + 1 − D i ( , , − /
3) + + 1 − D ci ( ¯3 , , /
3) + + 1 − N i ( , ,
0) + − H u ( , , /
2) + + 1 − H d ( , , − /
2) + + 1 − S ( , ,
0) + + 2 0 N ( , ,
0) + + 0 5¯ N ( , ,
0) + + 0 − SU (2) L Doublet Superfields L ( , , − / − + 0 − L ( , , / − + 0 3 an extra Z ′ symmetry under which the N i ’s are odd.In order to achieve unification of the MSSM gauge cou-pling constants, we introduced an extra conjugate pairof SU (2) L doublets L and ¯ L with Y = − / / Z and to-gether with H αd and H αu ( α = 2 ,
3) form three complete SU (5) multiplets with the color (anti)triplets D i and D ci .Note that the superfields q , u c , d c , l , ν c , e c , H iu , H id , D i , D ci , and N i form three complete fundamental represen-tations of E , while N , ¯ N and L , ¯ L are conjugate pairsfrom incomplete E multiplets.In Table I, we summarize all the superfields of themodel together with their transformation properties un-der the SM gauge group G SM and their charges underthe discrete symmetries Z , Z ′ , the global R symme-try U (1) R , and the local U (1) ψ ′ with GUT-normalizedcharge Q ψ ′ . Note that the discrete symmetries Z , Z ′ donot carry SU (3) c or SU (2) L anomalies.The symmetries of the model allow not only the super-potential terms in Eq. (2), but also the following higherorder terms (divided by appropriate powers of m P ): ν c H αu LN, e c H αd L ¯ N , H u H u ll, H αu H βu ll, H u H αd l ¯ L,H αu H d l ¯ L, H d H d ¯ L ¯ L, H αd H βd ¯ L ¯ L, qu c qd c ¯ N , qu c e c l ¯ N ,qd c ν c l ¯ N, e c ν c LLN, H αu qd c lL, H u H αu lLN,H u H u LLN N, H αu H βu LLN N, H αu qu c l ¯ L ¯ N,H αd qd c l ¯ L ¯ N , ν c H d l ¯ L ¯ L ¯ N , e c H u lLLN, qd c Lqd c L,D ci u c u c ¯ L ¯ L ¯ N, D ci d c d c LLN, e c qd c lLL, H u qd c LLN,H d qu c ¯ L ¯ L ¯ N , H d H αd l ¯ L ¯ L ¯ L ¯ N , H αu e c LLLN N,ν c qu c l ¯ L ¯ L ¯ N ¯ N , qu c qu c ¯ L ¯ L ¯ N ¯ N , e c e c LLLLN N,H αd qu c l ¯ L ¯ L ¯ L ¯ N ¯ N . (3)Note that all the couplings in Eqs. (2) and (3) can bemultiplied by the combinations N ¯ N/m , L ¯ L/m , and¯ Ll ¯ N ¯ Ll ¯ N/m arbitrarily many times and this exhaustsall the possible superpotential couplings compatible withthe symmetries of the model.Assigning baryon number B = − / / D i and D ci , respectively, we see that thebaryon number U (1) B symmetry is automatically presentto all orders in the superpotential and, thus, fast pro-ton decay and other baryon number violating effects areavoided [13].The fundamental representation of E contains two SMsinglets with the quantum numbers of ν c and N i . Letus assume that at high energies the gauge symmetry is G SM × U (1) χ × U (1) ψ . A conjugate pair of Higgs super-fields of the type ν c , ¯ ν c from an incomplete E multipletcan break U (1) χ × U (1) ψ to U (1) ψ ′ at a scale of order theGUT scale. So, at lower energies, only the gauge sym-metry G SM × U (1) ψ ′ of our model survives. The spon-taneous breaking of U (1) ψ ′ at a scale M ∼
10 TeV isthen achieved by a conjugate pair of Higgs superfieldsof the type N , ¯ N from an incomplete E multiplet viathe superpotential terms κS ( N ¯ N − M ). This breakingwill generate a network of local superconducting strings.Their string tension, which is determined by the scale M ,is relatively small and certainly satisfies the most strin-gent relevant upper bound from pulsar timing arrays [14].Note, in passing, that the kinetic mixing of U (1) ψ ′ and U (1) Y is negligible – see fourth paper in Ref. [4].The ‘bare’ MSSM µ term is replaced by a term λ µ N H u H d , so that the µ term is generated after N ac-quires a non-zero vacuum expectation value (VEV) h N i of order 10 TeV. The same VEV gives masses to the tworemaining pairs of SU (2) L doublets H αu , H αd ( α = 2 , λ αµ N H αu H αd as well as to thediquarks D i , D ci ( i = 1 , ,
3) via the terms λ iD N D i D ci .The gauge singlet S acquires a VEV h S i of order TeVfrom soft SUSY breaking [15]. (In the SUSY limit theVEV of S is zero.) This VEV generates masses for theextra doublets L , ¯ L via the term λ L SL ¯ L . Finally, thesterile neutrino fields, which are the fermionic parts of N i , acquire masses of order 10 − eV or so via the terms λ iN N i N i ¯ N / m P .The spontaneous breaking of U (1) ψ ′ implemented withthe fields S , N , ¯ N delivers, in the exact SUSY limit,four spin zero particles all with the same mass given by √ κM . This mass, even for M ≫ κ . We shouldpoint out though that, depending on the SUSY breakingmechanism, these states may end up with significantlydifferent masses. The diquarks D i , D ci may be found [11]at the LHC. III. U (1) ψ ′ BREAKING
We will assume here that the breaking scale of U (1) ψ ′ is much bigger than the electroweak scale. In this case, the spontaneous breaking of U (1) ψ ′ is not affected bythe electroweak Higgs doublets in any essential way andcan be discussed by considering only the superpotentialterms δW = κS ( N ¯ N − M ) (4)in the right-hand side (RHS) of Eq. (2). They give thefollowing scalar potential V = κ | N ¯ N − M | + κ | S | ( | N | + | ¯ N | )+ (cid:0) AκSN ¯ N − ( A − m / ) κM S + H . c . (cid:1) + m ( | N | + | ¯ N | + | S | ) + D − terms . (5)Here the mass parameter M and the dimensionless cou-pling constant κ are made real and positive by fieldrephasing and the scalar components of the superfieldsare denoted by the same symbol. The parameter m / isthe gravitino mass, A ∼ m / is the coefficient of the tri-linear soft terms taken real and positive, and m ∼ m / is the common soft mass of N , ¯ N , and S . We assumed,for definiteness, minimal supergravity. In this case, thecoefficients of the trilinear and linear soft terms are re-lated as shown in Eq. (5). Vanishing of the D-termsimplies that | N | = | ¯ N | , which yields ¯ N ∗ = e iϑ N , whileminimization of the potential requires that ϑ = 0. So, N and ¯ N can be rotated to the positive real axis by a U (1) ψ ′ transformation.We find [15] that the scalar potential in Eq. (5) is mini-mized at h S i = − m / κ X n ≥ c n (cid:16) m / M (cid:17) n (6)and h N i = (cid:10) ¯ N (cid:11) ≡ N √ M X n ≥ d n (cid:16) m / M (cid:17) n , (7)where c n , d n are numerical coefficients of order unity.Assuming that M ≫ m / and keeping in h S i and N terms up to order m / , these formulas can be approxi-mated as follows: h S i ≃ − m / κ , N ≃ M + Am / − m / − m κ . (8)We should point out that the trilinear and linear softterms in the second line of Eq. (5) play an importantrole in our scheme. Substituting N and ¯ N by their VEVs,these terms yield a linear term in S which, together withthe mass term of S , generates [15] a VEV for S of orderTeV. It is then obvious that, substituting this VEV of S in the superpotential term λ L SL ¯ L , the superfields L , ¯ L acquire a mass m L = λ L | h S i | = λ L m / /κ . Moreover,the MSSM µ term is obtained by substituting h N i inthe superpotential term λ µ N H u H d with µ = λ µ N / √ H αu , H αd ( α = 2 ,
3) and D i , D ci acquire masses oforder TeV from the couplings λ αµ N H αu H αd and λ iD N D i D ci respectively. Note that, with D i , D ci , L , ¯ L , and H αu , H αd masses ∼ TeV, the gauge couplings stay in the perturba-tive domain for up to four such pairs of color (anti)tripletsand SU (2) L doublets.The mass spectrum of the scalar S − N − ¯ N systemcan be constructed by substituting N = h N i + δ ˜ N and¯ N = (cid:10) ¯ N (cid:11) + δ ˜¯ N . In the unbroken SUSY limit, we find twocomplex scalar fields S and θ = ( δ ˜ N + δ ˜¯ N ) / √ m S = m θ = √ κM . Soft SUSY breaking can, ofcourse, mix these fields and generate a mass splitting. Forexample, the trilinear soft term AκSN ¯ N yields a mass-squared splitting ±√ κM A with the mass eigenstatesnow being ( S + θ ∗ ) / √ S − θ ∗ ) / √
2. This splittingis small for A ≪ √ κM . IV. ELECTROWEAK SYMMETRY BREAKING
The standard scalar potential for the radiative elec-troweak symmetry breaking in MSSM is modified in thepresent model. A modification originates from the D-term for U (1) ψ ′ : V D = g ψ ′ (cid:2) − | H u | − | H d | + 5 (cid:0) | N | − | ¯ N | (cid:1)(cid:3) , (9)where g ψ ′ is the GUT-normalized gauge coupling con-stant for the U (1) ψ ′ symmetry and H u , H d are the neu-tral components of the scalar parts of the Higgs SU (2) L doublet superfields H u , H d respectively. In order to findthe leading contribution of this D-term to the electroweakpotential, we must integrate out to one loop the heavydegrees of freedom N and ¯ N . To this end, we expressthese complex scalar fields in terms of the canonicallynormalized real scalar fields δN , δ ¯ N , ϕ , ¯ ϕ as follows: N = 1 √ N + δN ) e iϕN , ¯ N = 1 √ N + δ ¯ N ) e i ¯ ϕN . (10)Then the combination | N | − | ¯ N | , which appears in theD-term in Eq. (9), becomes | N | − | ¯ N | = √ N η + ηξ, (11)where η = δN − δ ¯ N √ , ξ = δN + δ ¯ N √ V D = g ψ ′ h E + 10 √ N Eη + 50 N η + · · · i , (13)where E ≡ − | H u | − | H d | . Here we kept only up toquadratic terms in η , ξ , but ignored the mixed quadratic term proportional to ηξ since its coefficient is muchsmaller than the coefficient of the η term assuming that N is much bigger than the electroweak scale.We see, from Eq. (13), that integrating out the heavystates reduces to the calculation of a path integral overthe real scalar field η . To do this, we first need to findthe η dependence of the potential V in Eq. (5). So wesubstitute in this equation N and ¯ N from Eq. (10). Keep-ing only η -dependent terms up to the second order andsubstituting S by its VEV in Eq. (8), we obtain δV ≃ (cid:18) − κ N + m / + m + κ M + Am / (cid:19) η , (14)which, substituting N from Eq. (8), gives δV ≃ m N η with m N ≡ m / + m . (15)Adding δV to the D-term potential in Eq. (13), we obtainthe potential V η = g ψ ′ E + √ g ψ ′ N Eη + m N + 5 g ψ ′ N ! η + · · · , (16)which can be given the form V η = g ψ ′ E g ψ ′ N m N ! − + m N + 5 g ψ ′ N ! × η + g ψ ′ N E √ (cid:18) m N + g ψ ′ N (cid:19) + · · · . (17)The path integral Z ( dη ) e − iV η V , (18)where V is the spacetime volume, can be readily calcu-lated and, besides an irrelevant overall constant factor,we are left with the term δV D ≃ g ψ ′ (cid:2) | H u | + 3 | H d | (cid:3) (cid:18) m Z ′ m N (cid:19) − (19)to be added to the usual electroweak symmetry breakingpotential. Here m Z ′ = √ g ψ ′ N / Z ′ gauge boson associated with U (1) ψ ′ .Another modification of the MSSM electroweak poten-tial comes from the integration of the heavy complex field S with mass √ κM in the exact SUSY limit. The crossF-term F N between the superpotential terms κSN ¯ N and λ µ N H u H d in Eq. (2) together with the mass-squaredterm of S give2 κ M | S | + (cid:16) κS ∗ ¯ N ∗ ˜ λ µ H u H d + H . c . (cid:17) = (cid:12)(cid:12)(cid:12)(cid:12) √ κM S + 1 √ λ µ H u H d (cid:12)(cid:12)(cid:12)(cid:12) −
12 ˜ λ µ | H u H d | , (20)where ˜ λ µ ≡ λ µ . Integrating out S , we then obtain theextra term −
12 ˜ λ µ | H u | | H d | (21)in the electroweak potential. One can show that the in-tegration of all the other heavy fields gives smaller con-tributions, which we ignore.Now the potential for the electroweak symmetry break-ing as can be derived from the superpotential terms κS ( N ¯ N − M ) − ˜ λ µ N H u H d (22)after substituting the VEVs of S , N , and ¯ N from Eq. (8)and adding the D-term in Eq. (19) and the term inEq. (21) is V EW ≃ m H u | H u | + m H d | H d | − B ( H u H d + H . c . )+ λ µ | H u | | H d | + 18 ( g + g ′ )( | H u | − | H d | ) + c ( Q u | H u | + Q d | H d | ) , (23)where m H u = ˜ m H u + µ , m H d = ˜ m H d + µ with ˜ m H u ,˜ m H d being the soft masses of H u , H d and B = ˜ B − m / with ˜ B being the coefficient of the soft trilinear termcorresponding to the second term in Eq. (22). Here λ µ ≡ ˜ λ µ / √ g is the SU (2) L and g ′ the non-GUT-normalized U (1) Y gauge coupling constant, Q u = 2, Q d = 3, and c = g ψ ′ (cid:18) m Z ′ m N (cid:19) − . (24)Note that the potential in Eq. (23) contains the so-called next-to-minimal supersymmetric standard model(NMSSM) term λ µ | H u | | H d | . (25)Minimization of the potential in Eq. (23) yields thefollowing relations: m H u = m A cos β + 12 m Z cos 2 β − λ µ v cos β − cQ u v ( Q u sin β + Q d cos β ) ,m H d = m A sin β − m Z cos 2 β − λ µ v sin β − cQ d v ( Q u sin β + Q d cos β ) . (26)Here v = v u + v d with v u = h H u i and v d = h H d i , tan β = v u /v d , and the expressions m Z = 12 ( g + g ′ ) v , m A = 2 Bµ sin 2 β (27)for the Z gauge boson mass m Z and the CP-odd Higgsboson mass m A are used. Note that the latter is notaffected by the extra terms in the potential V EW sincethey involve only the absolute values of H u , H d . The mass-squared matrix in the CP-even Higgs sector M = M M M M (28)can be constructed by substituting H u = v u + h u / √ H d = v d + h d / √ h u , h d . We find M = m H u + 12 m Z (3 sin β − cos β ) + λ µ v cos β +2 cQ u v (cid:0) Q u sin β + Q d cos β (cid:1) , M = (cid:0) − m A − m Z + 2 λ µ v + 4 cQ u Q d v (cid:1) sin β cos β, M = m H d + 12 m Z (3 cos β − β ) + λ µ v sin β +2 cQ d v (3 Q d cos β + Q u sin β ) . (29)Using the minimization conditions in Eq. (26), M and M can be cast in the form M = m A cos β + ( m Z + 4 cQ u v ) sin β, M = m A sin β + ( m Z + 4 cQ d v ) cos β. (30)The eigenvalues m h and m H of the mass-squared matrixin Eq. (28), which are, respectively, the ‘tree-level’ massessquared of the lightest and heavier neutral CP-even Higgsbosons, can now be constructed: m h,H = 12 Σ ∓ r
14 Σ − ∆ (31)withΣ = m A + m Z + 4 c v ( Q u sin β + Q d cos β ) , ∆ = m A m Z cos β + 4 cv m A ( Q u sin β + Q d cos β ) + λ µ v m A sin β + cv m Z ( Q u + Q d ) sin β + m Z λ µ v sin β − λ µ v sin β − cQ u Q d λ µ v sin β. (32)Let us note that, here, by ‘tree-level’ masses we meanthe masses without the inclusion of the radiative correc-tions in MSSM. It is easy to see that m h , in the so-calleddecoupling limit where m A ≫ m Z , is given by m h = m Z cos β + 4 cv ( Q u sin β + Q d cos β ) + λ µ v sin β. (33) V. DIPHOTON RESONANCES
The real scalar θ and real pseudoscalar θ compo-nents of θ = ( δ ˜ N + δ ˜¯ N ) / √ θ + iθ ) / √ m θ = √ κM in the exact SUSY limit can be produced atthe LHC by gluon fusion via a fermionic D i , D ci loop asindicated in Fig. 1. They can decay into gluons, photons, Z or W ± gauge bosons via the same loop diagram as well FIG. 1: Production of the complex scalar field θ at the LHCby gluon ( g ) fusion and its subsequent decay into photons( γ ). Solid (dashed) lines represent the fermionic (bosonic)component of the indicated superfields. The arrows depict thechirality of the superfields and the crosses are mass insertionswhich must be inserted in each of the lines in the loops. as a similar fermionic H iu , H id loop. The most promisingdecay channel to search for these resonances is into twophotons with the relevant diagrams also shown in Fig. 1.Applying the results of Ref. [16], the cross section ofthe diphoton excess is σ ( pp → θ m → γγ ) ≃ C gg m θ s Γ θ m Γ( θ m → gg )Γ( θ m → γγ ) , (34)where m = 1 , C gg ≃ √ s ≃
13 TeV, Γ θ m is thetotal decay width of θ m , and the decay widths of θ m totwo gluons ( g ) or two photons ( γ ) are given byΓ( θ m → gg ) = m θ α s π h N i X i =1 A m ( x i ) ! , (35)Γ( θ m → γγ ) = m θ α Y cos θ W π h N i " X i =1 A m ( x i )+32 X i =1 A m ( y i ) (cid:18) α tan θ W α Y (cid:19) . (36)Here A ( x ) = 2 x [1 + (1 − x ) arcsin (1 / √ x )], A ( x ) =2 x arcsin (1 / √ x ), x i = 4 m D i /m θ > m D i = λ iD h N i being the mass of D i and D ci , y i = 4 m H i /m θ > m H i = λ iµ h N i being the mass of H iu and H id , and α s , α Y , and α are the strong, hypercharge, and SU (2) L fine-structure constants, respectively.The cross section in Eq. (34) simplifies under the as-sumption that the spin zero fields θ m decay predomi-nantly into gluons, namely Γ θ m ≃ Γ( θ m → gg ). In this FIG. 2: Decay of the complex scalar field θ into a fermionic D i , D ci (a) or H iu , H id (b) pair or a bosonic L , ¯ L pair (c). Thenotation is the same as in Fig. 1.FIG. 3: Decay of the complex scalar field S into a bosonic D i , D ci (a) or H iu , H id (b) pair or a fermionic L , ¯ L pair (c).The notation is the same as in Fig. 1. case, one obtains [17] σ ( pp → θ m → γγ ) ≃ . × Γ( θ m → γγ ) m θ fb . (37)For x i and y i just above unity, which guarantees that thedecay of θ m to D i , D ci and H iu , H id pairs is kinematicallyblocked, A ( x i ) and A ( y i ) are maximized with values A ≃ A ≃ π /
2. So we consider this case. It isalso more beneficial to consider the decay of the pseu-doscalar θ since A ( x ) > A ( x ) for all x >
1. UsingEq. (36), we then find that Eq. (37) gives σ ( pp → θ → γγ ) ≃ . (cid:18) m θ h N i (cid:19) fb ≃ κ fb . (38)In the exact SUSY limit, the complex scalar field θ could decay into a fermionic D i , D ci or H iu , H id pair viathe superpotential terms λ iD N D i D ci or λ iµ N H iu H id if thisis kinematically allowed – see Figs. 2(a) and 2(b). Itcould also decay into a bosonic L , ¯ L pair via the F-term F S between the superpotential couplings κSN ¯ N and λ L SL ¯ L if this is kinematically allowed – see Fig. 2(c).The decay widths in the three cases areΓ θD i = ( λ iD ) π m θ , Γ θH i = ( λ iµ ) π m θ , Γ θL = ( λ L ) π m θ , (39)respectively, where we assumed that the mass of the re-levant D i , D ci , or H iu , H id , or L , ¯ L is much smaller than m θ /
2. Depending on the kinematics the total decaywidth of the resonance could easily lie in the 100 GeVrange. The diphoton, dijet, and diboson decay modes inthis case would be subdominant.Our estimate in Eq. (37) holds provided that the decaywidths of θ into a D i , D ci , or H iu , H id , or L , ¯ L pair are sub-dominant or these decays are kinematically blocked. Thelatter is achieved for m θ ≃ √ κM < m D i ≃ λ iD M ,2 m H i ≃ λ iµ M , and 2 m L ≃ λ L | h S i | ≃ λ L m / /κ ,which implies that κ . √ λ iD , √ λ iµ , λ L m / m θ . (40)Note that the estimate of the maximal cross section of thediphoton excess in Eq. (38) corresponds to saturating thefirst two of the inequalities in Eq. (40). For simplicity andfor not disturbing the MSSM gauge coupling unification,we choose to saturate the third inequality too.The complex scalar field S can decay into a bosonic D i , D ci or H iu , H id pair via the F-terms F N between the super-potential couplings κSN ¯ N and λ iD N D i D ci or λ iµ N H iu H id if this is kinematically allowed – see Figs. 3(a) and 3(b).It could also decay into a fermionic L , ¯ L pair via thesuperpotential coupling λ L SL ¯ L if this is kinematicallyallowed – see Fig. 3(c). The decay widths Γ SD i , Γ SH i , andΓ SL in the three cases are, respectively, equal to the decaywidths Γ θD i , Γ θH i , and Γ θL in Eq. (39). It is obvious that, ifthe inequalities in Eq. (40) are satisfied so as our estimateof the cross section of the diphoton excess in Eq. (38) tohold, these decay channels of S are also blocked. In thiscase, S will decay to lighter particles.Note that, in the exact SUSY limit, the complex scalarfield S cannot be produced at the LHC by gluon fu-sion and, thus, cannot lead to diphoton excess. Thiswould require bosonic D i , D ci loops with mass-squaredinsertions originating from soft trilinear SUSY breakingterms – for such loops see Ref. [18]. As we already men-tioned, the soft SUSY breaking terms generate mixingbetween the scalar fields S and θ . Consequently, we canhave four diphoton resonance states rather than just twofrom the scalar θ alone. Soft SUSY breaking also givesrise to more diagrams contributing to the diphoton ex-cess. However, our estimate of the cross section of thediphoton excess for exact SUSY is the dominant one pro-vided that the scale of U (1) ψ ′ breaking is much biggerthan the soft SUSY breaking scale. Finally, let us notethat demanding that the mass of the Z ′ gauge boson m Z ′ ≃ √ g ψ ′ M/ √ > . g ψ ′ M & . . (41) VI. NUMERICAL ANALYSIS
We can show that the gauge coupling constant g ψ ′ as-sociated with the U (1) ψ ′ gauge symmetry unifies withthe MSSM gauge coupling constants provided that itsvalue at low energies is equal to about 0.45. This valuedepends very little on the exact value of the diquark, theextra SU (2) L doublet, the resonance, and the Z ′ gaugesupermultiplet masses. So the bound in Eq. (41) im-plies that M & .
34 TeV. As an example, we will set
MSSM + D - term + NMSSM termMSSM + NMSSM termMSSM M SUSY ( GeV ) m h ( G e V ) FIG. 4: Higgs boson mass m h in the decoupling limit and formaximal stop quark mixing versus M SUSY for M = 10 TeV,˜ λ µ = 0 .
3, tan β = 20, and m / = 4 TeV. The dotted(red) curve corresponds to MSSM, the dashed (blue) curveto MSSM plus the NMSSM correction, and the continuous(brown) curve to MSSM plus the D-term and NMSSM cor-rections. The experimental value of m h is also depicted bythe bold horizontal line. M = 10 TeV. In addition, we can show that the couplingconstants κ and ˜ λ µ remain perturbative up to the GUTscale provided that they are not much bigger than about0.7. The requirement that the diphoton resonance mass m θ = √ κM is bigger than about 4 . κ & . . & λ iD , λ iµ & . λ iD ≃ λ iµ ≃ .
3, which meansin particular that ˜ λ µ ≃ .
3. This choice implies that κ ≃ . m D i ≃ m H i ≃ µ ≃ m θ ≃ m Z ′ ≃ . m L ≃ κ . .
7, the resonance mass remains below 9 . m h in the decoupling limit versus M SUSY , whichis the geometric mean of the stop quark mass eigenva-lues. We generally assume maximal stop quark mixing,which maximizes m h , and include the two-loop radiativecorrections to m h in MSSM using the package SUSYHD[21]. The NMSSM and D-term contributions to m h arealso included from Eq. (33). In this figure, tan β = 20and m / = 4 TeV. Notice that the NMSSM correctionis very small since ˜ λ µ is relatively small. The D-termcorrection, however, is sizable and allows us to obtainthe observed value of m h with much smaller stop quarkmasses than the ones required in MSSM or NMSSM. In-deed, the inclusion of the D-term from U (1) ψ ′ reduces M SUSY from about 1900 GeV to about 1200 GeV. Note,in passing, that λ L , in this case, is about 0.32.In Fig. 5, we plot m h in the decoupling limit and formaximal stop quark mixing versus tan β for M = 10 TeV,˜ λ µ = 0 . M SUSY = 1200 GeV, and m / = 4 TeV.We see that the experimental value of m h is achieved at MSSM + D - term + NMSSM termMSSM + D - termMSSM
10 20 30 40 50100105110115120125130135 tan β m h ( G e V ) FIG. 5: Higgs boson mass m h in the decoupling limit and formaximal stop quark mixing versus tan β for M = 10 TeV,˜ λ µ = 0 . M SUSY = 1200 GeV, and m / = 4 TeV. Thenotation is the same as in Fig. 4. MSSM + D - term + NMSSM termMSSM + NMSSM termMSSM m / ( GeV ) m h ( G e V ) FIG. 6: Higgs boson mass m h in the decoupling limit andfor maximal stop quark mixing versus m / for M = 10 TeV,˜ λ µ = 0 .
3, tan β = 20, and M SUSY = 1200 GeV. The notationis the same as in Fig. 4. tan β = 20 as it should consistently with Fig. 4. How-ever, as one can see from Fig. 5, the observed m h canbe practically obtained in a wide range of tan β ’s. Notethat, without the inclusion of the D-term contributionfrom U (1) ψ ′ , the Higgs boson mass remains well belowits observed value for all the values of tan β . This againshows the crucial role of the D-term for obtaining the ob-served value of m h with relatively low stop quark masses.Finally, we notice that, for larger tan β ’s, m h decreasesas tan β increases in all three cases depicted in this figure.This is due to the relatively large value of µ .In Fig. 6, we depict m h under the same assumptionsversus m / for M = 10 TeV, ˜ λ µ = 0 .
3, tan β = 20, and M SUSY = 1200 GeV. The observed Higgs boson mass isobtained at m / = 4 TeV consistently with Figs. 4 and 5.We see again that, without the D-term, m h remains wellbelow its observed value for all m / ’s. We also observethat, without the D-term, m h is independent from thevalue of m / as it should. In the present numerical example, the cross section ofthe diphoton excess in Eq. (38) turns out to be equal to1.94 fb. Needless to say that higher cross sections can beobtained for higher values of κ . The diphoton resonancemass, as already discussed, is equal to 6 TeV and thediquark masses about 3 TeV. In conclusion, we see thatour model can predict diphoton and diquark resonanceswhich hopefully can be observed in future experiments. VII. STERILE NEUTRINOS
After the spontaneous breaking of the U (1) ψ ′ symme-try, the fermionic components of the three superfields N i ,which are SM singlets, acquire masses m N i ≃ λ iN M /m P via the last superpotential coupling in Eq. (2). Thesemasses can be . . M ∼
10 TeV and thesefermionic fields, which are stable on account of the Z ′ symmetry in Table I, can act as sterile neutrinos.In the early universe, the sterile neutrinos are kept inequilibrium via reactions of the sort N i ¯ N i ↔ a pair of SMparticles or N i + a SM particle ↔ N i + a SM particle.These reactions proceed via a s- or t-channel exchangeof a Z ′ gauge boson. The thermal average h σv i , where σ is the corresponding cross section and v the relativevelocity of the annihilating particles, is estimated to beof order T /M with T being the cosmic temperature.The interaction rate per sterile neutrino is then given byΓ N i = n h σv i ∼ T M , (42)where n ∼ T is the number density of massless particlesin thermal equilibrium. The decoupling temperature T D of sterile neutrinos is estimated from the conditionΓ N i ∼ H ∼ T m P , (43)where H is the Hubble parameter. This condition impliesthat T D ∼ M (cid:18) Mm P (cid:19) . (44)Here we followed the same strategy as the one used forestimating the SM neutrino decoupling temperature viaprocesses involving weak gauge boson exchange. In thecase of ordinary neutrinos, however, the scale M shouldbe identified with the electroweak scale, which is of order100 GeV, and the decoupling temperature turns out tobe of order 1 MeV. From Eq. (44), we see that T D scaleslike M / . So, in our case and for M ≃
10 TeV, T D isexpected to be of order 460 MeV, which is well above thecritical temperature for the QCD transition.The effective number of massless degrees of freedom inequilibrium right after the decoupling of sterile neutrinosis 61.75. At T ∼ T ν is raised rela-tive to the temperature of the sterile neutrinos T N by afactor (61 . / . / . Consequently, the contributionof the three sterile neutrinos to the effective number ofneutrinos at big bang nucleosynthesis is∆ N ν = 3 × (cid:18) . . (cid:19) ≃ . . (45)This result is perfectly compatible with the Planck satel-lite bound [22] on the effective number of massless neu-trinos N ν = 3 . ± . . (46)Note that although the derivation of our estimate inEq. (45) is somewhat rough, we believe that the resultis quite accurate. This is due to the fact that the ef-fective number of massless degrees of freedom in equili-brium right after the decoupling of sterile neutrinos doesnot change if T D varies between the critical temperatureof the QCD transition, which is about 200 MeV, and themass of the charm quark m c ≃ VIII. DARK MATTER
The scalar component of the superfield N i , which isexpected to have mass of order m / , can decay into afermionic N i and a particle-sparticle pair via a Z ′ gauginoexchange provided that this is kinematically allowed. Anecessary (but not sufficient) condition for this decay tobe possible is that there exist sparticles which are lighterthan the scalar N i . Note that, as a consequence of theunbroken discrete symmetry Z ′ , the decay products ofthe scalar N i should necessarily contain an odd numberof N j superfields.If the decay of the lightest scalar N i (denoted as ˆ N ) iskinematically blocked, this particle can contribute to thecold dark matter in the universe. In the early universe,the scalar ˆ N is kept in equilibrium since, for example,a pair of these scalars can annihilate into a pair of SMparticles via a Z ′ gauge boson exchange. The thermalaverage h σv i in this case and for s-wave annihilation isexpected to be h σv i ∼ m N M , (47)where m ˆ N is the mass of the scalar ˆ N .Following the standard analysis of Ref. [23], we can es-timate the freeze-out temperature T f of the sterile sneu-trino ˆ N as well as its relic abundance Ω ˆ N h in the uni-verse. To this end, we take M ≃ .
34 TeV, which satu-rates the lower bound on m Z ′ [19] mentioned in Sec. V. The requirement that Ω ˆ N h equals the cold dark matterabundance Ω CDM h ≃ .
12 from the Planck satellite data[24] then implies that m ˆ N ≃ .
25 TeV. The freeze-outtemperature T f in this case is about 51 GeV and the cor-responding number of massless degrees of freedom 86 . M require even higher values of m ˆ N .So we see that the SUSY spectrum is pushed up con-siderably if the decay of the lightest sterile sneutrino iskinematically blocked and this particle contributes to thecold dark matter of the universe.The model possesses an accidental lepton parity sym-metry Z lp2 under which the superfields l , e c , ν c , L , ¯ L areodd. Combining this symmetry with the baryon parity Z bp2 subgroup of U (1) B under which q , u c , d c are odd,we obtain a matter parity symmetry Z mp2 under which q , u c , d c , l , e c , ν c , L , ¯ L are odd. A discrete R-paritycan then be generated if we combine this symmetry withfermion parity. The bosonic q , u c , d c , l , e c , ν c , L , ¯ L and the fermionic H iu , H id , D i , D ci , N i , S , N , ¯ N areodd under this R-parity. Note that the decay productsof these particles with the exception, of course, of thefermionic N i cannot contain a single N i because of the Z ′ symmetry. Also, they cannot contain a single L , ¯ L , H αu , H αd except, of course, for the decay products of thebosonic L , ¯ L and fermionic H αu , H αd themselves as a con-sequence of the Z symmetry. The S , N , ¯ N fermions candecay into a Higgs boson-Higgsino pair, while the D i , D ci fermions can decay into a quark-squark pair. So all theparticles with negative R-parity, except the bosonic L , ¯ L and the fermionic H αu , H αd , N i end up yielding the usualstable lightest sparticle of MSSM which can, in principle,participate in the cold dark matter of the universe.The possible fate of the N i superfields has been al-ready discussed. The Z symmetry and R-parity implythat the lightest state in the bosonic L , ¯ L and fermionic H αu , H αd , or in the fermionic L , ¯ L and bosonic H αu , H αd ,which is hopefully neutral, is stable. We thus have twomore candidates for cold dark matter. Their relic abun-dances in the universe depend on details. However, iftheir masses are large, these abundances can be negligi-ble. Finally, let us mention that, if the breaking scale h N i of U (1) ψ ′ is increased to about 10 TeV, the sterile neu-trinos become plausible candidates for keV scale warmdark matter (for a recent review see Ref. [25]). In con-clusion, we see that the model possesses many possiblecandidates for the composition of dark matter.
IX. SUMMARY
We have explored the implications of appending a U (1)gauge symmetry to the MSSM gauge group SU (3) c × SU (2) L × U (1) Y . This U (1) symmetry, referred to hereas U (1) ψ ′ , arises from a linear combination of U (1) χ and U (1) ψ contained in E . The three matter 27-plets in E give rise to three SO (10) singlet fermions N i , called ste-rile neutrinos, which are prevented from acquiring massesvia renormalizable couplings by a combination of sym-0metries, especially a U (1) R symmetry. Thus, for a re-latively low ( ∼
10 TeV or so) breaking scale of U (1) ψ ′ ,these fermionic N i ’s, the lightest of which happens tobe stable, only acquire tiny masses . . U (1) ψ ′ at suitably higher energies, of order 10 TeV orso, would yield keV scale masses for the fermionic N i ’sand thus transform them into plausible warm dark mat-ter candidates. The D-term for U (1) ψ ′ can contributeappreciably to the mass of the lightest neutral CP-evenMSSM Higgs boson. Consequently, the observed value ofthis mass can be obtained in the decoupling limit withrelatively light stop quarks. The spontaneous breaking of U (1) ψ ′ yields superconducting cosmic strings whichpresumably were not inflated away. The model also pre-dicts the existence of diquark and diphoton resonanceswhich may be found at the LHC or its future upgrades.The MSSM µ problem is naturally resolved. The righthanded neutrinos can acquire large masses, which allowsthe standard seesaw mechanism and the leptogenesis sce-nario to be realized. Baryon number is conserved to allorders in perturbation theory rendering a stable proton. Acknowledgments
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