The trilinear couplings of squarks and sleptons to the Higgs bosons can give
rise to a spectrum of bound states with exotic quantum numbers, for example,
those of a leptoquark.Comment: 8 pages, 2 eps figures, latex, epsf; published version (minor changes
  in wording and referencing

Alexander Kusenko

Claudson

Igor I Tkachev

Kusenko

Vadim Kuzmin

Physics Letters B

English

arXiv

Alexander  Kusenko

Crossref

Bound states and resonances in the scalar sector of the MSSM

The trilinear couplings of squarks and sleptons to the Higgs bosons can give rise to a spectrum of bound states with exotic quantum numbers, for example, those of a leptoquark.The trilinear couplings of squarks and sleptons to the Higgs bosons can give rise to a spectrum of bound states with exotic quantum numbers, for example, those of a leptoquark

Kusenko, Alexander

Kuzmin, Vadim

Tkachev, Igor I.

CERN Document Server

CERN-TH/98-34PURD-TH-98-03hep-ph/9801405Bound states and resonances in the scalar sector ofthe MSSMAlexander Kusenko,a;1 Vadim Kuzmin,b;2 and Igor I. Tkachevb;c;3aTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandbInstitute for Nuclear Research, 60th October Anniversary Prospekt 7a, Moscow,Russia 117312cDepartment of Physics, Purdue University, West Lafayette, IN 47907, USAAbstractThe trilinear couplings of squarks and sleptons to the Higgs bosons can give riseto a spectrum of bound states with exotic quantum numbers, for example, those of aleptoquark.CERN-TH/98-34PURD-TH-98-03January, 19981email address: Alexander.Kusenko@cern.ch2email address: kuzmin@ms2.inr.ac.ru3email address: tkachev@physics.purdue.eduThe scalar sector of the Minimal Supersymmetric Standard Model (MSSM) com-prises the Higgs elds and the superpartners of quarks and leptons. The gauge inter-actions of these particles are well understood because the charges of the scalars, byvirtue of supersymmetry, are the same as those of their fermionic partners. In addi-tion, the squarks and sleptons can have some scalar interactions. The quartic couplingof the scalar elds are related to the Yukawa couplings and the gauge couplings bysupersymmetry. However, the strength of the trilinear interactions is to a large ex-tent unconstrained, except indirectly, from the requirement of vacuum stability. It ispossible (and, in some models, desirable) that these dimensionful couplings be large incomparison to the masses of some scalar particles. The main focus of our analyses is onsuch trilinear scalar interactions because they are eectively \attractive" and can causethe appearance of bound states in the theory. These bound states can be observed asresonances in high-energy experiments and can, for example, cause an increase in thecross-section qualitatively similar to that reported last year at HERA [1].We denote the SU(2)-doublet chiral superelds of quarks and leptons as QLand LLrespectively, and use the same notation for their scalar components. The correspondingright-handed SU(2)-singlets are uR, dR, and lR. Here and below the Greek letters standfor the SU(2) indices, the color SU(3) indices are suppressed. When relevant, theadditional flavor indices, in Latin characters, will indicate the generations of (s)quarksand (s)leptons.The superpotential of the MSSM includes the following termsW =  [yuQLH2 uR + ydQLH1 dR + ylLLH2 lR − H1 H2 ] + :::; (1)and generates, in particular, the trilinear interactions of the formV3; = 2yH1 QLuR+ h:c: (2)that preserve supersymmetry. In addition, the trilinear termsV3;A = AH2 QLuR+ h:c:; (3)1which couple the same squark bilinear QLuRto the \wrong" Higgs, appear in the scalarpotential as a consequence of supersymmetry breaking. The potential can be writtenas followsV = V2;H + V2 + V3; + V3;A + V4; (4)where V2;H comprises the terms quadratic in the Higgs elds, V2 contains the massterms of the squarks and sleptons, V4 comprises the gauge D-terms and the terms ofthe form y2Q2q2, y2Q2H2, etc.The trilinear coupling can play a crucial role in creating the bound states andresonances of squarks and sleptons through the exchange of the Higgs elds. Forclarity, we will assume a particular form for the scalar potential that will retain allthe relevant features of the general scalar interaction in the MSSM but will greatlysimplify the discussion. First, we make use of a well-known MSSM prediction thatthere is a light neutral Higgs h0. It is reasonable to neglect the propagation of heavierHiggs scalars in the ladder diagrams. Second, we will assume that the squarks and thesleptons have degenerate masses around m0 (this assumption could be motivated bythe constraints on the FCNC, although the latter can be satised without the squarkdegeneracy). In addition, we assume the equality of the trilinear couplings for thesquarks and sleptons in question. One can easily generalize on all these assumptions,none of which is crucial to the main conclusions of our analysis. However, the algebraicentanglement involved in tracking a large number of parameters may unnecessarilycomplicate the discussion. We now want to examine the possibility of a bound state oftwo squarks, two sleptons, or a squark and a slepton, that exchange the lightest Higgsboson. The relevant interaction can be described by the approximate potential writtenin the flavor-diagonal basis of squarks and sleptons:Va = m20(jQj2 + jqj2 + jLj2 + jlj2) +12m2HjHj2−AHQq−AHLl+ :::; (5)The quartic couplings V4 can be neglected as long as the interaction relevant for the2Bethe-Salpeter equation is dominated by the large trilinear terms, the case in whichwe are mainly interested.Theories of the kind described by the potential in equation (5) have historicallybeen the testing ground for solving the Bethe{Salpeter equation [2],"12E + p2+m20# "12E − p2+m20# (p) =4iA2(2)4Zd4k (k)(p− k)2 +m2H; (6)where  (p) is the wave function, E is the bound state energy, and mHis the physicalmass of the (lightest) Higgs. The potential in equation (5) can be re-written in the formthat matches exactly the interaction studied in Ref. [4] if one diagonalizes the squarkand slepton bilinears that enter into the cubic terms by a unitary transformation. Wecan, therefore, use the results of Refs. [2, 3, 4] for the energy spectrum of the boundstates.For xed energy E, equation (6) is a Fredholm equation that has a discrete spectrumof eigenvalues   A2 that depend on E. In other words, for a given value of thecoupling  one looks for such energy E that makes  an eigenvalue. Then the boundstate energy E is characterized by a discrete spectrum. If mH= 0 and  is small, then’th bound state has energy [2, 3]En = 2m0 1−28n2!; (7)where =116A2m20: (8)The bound states exist for any value of  if the Higgs eld H is massless. In theladder approximation, the bound state energy approaches zero at  = n(n+1), whichcan be used as a semi-quantitative reference point for the strength of the attractiveinteraction. The exchange of a scalar eld with mass mHcreates a bound state only if > min  1:68(mH=m0) [3].3It is interesting to juxtapose these bound states and Q-balls [5]. Bosons can forma coherent state that allows a semiclassical description in terms of non-topologicalsolitons if the number of particles is suciently large. The connection between sparticlebound states and Q-balls claries the role of the trilinear terms. The existence of smallQ-balls [6] in the MSSM relies on the requisite trilinear couplings [7] that make itpossible for the energy of the coherent state with a given baryon or lepton number tobe less than the combined mass of the free particles that carry the same charge. Thesame trilinear term in the scalar potential can be viewed as an attractive interactionbetween the constituent squarks that exchange the Higgs elds. The latter descriptionis more appropriate in the few-body limit, where the semiclassical description of Q-ballsbreaks down. Nevertheless, it is tempting to compare the expression for the groundstate energy E1 in equation (7) to the mass of a small Q-ball. One can suspect thata small Q-ball is merely an alternative description of some bound states. In somecases, such point of view is justied. For example, a bound state of two leptons canbe thought of as a Q-ball with charge Q = 2 associated with the lepton number U(1)Lsymmetry. A bound state of a slepton and a squark can be seen as a U(1)B−L soliton,and so on. Of course, only a subset of sparticle bound states can be linked to Q-balls.The masses of small Q-balls (those, for which the thin-wall approximation is not valid)for a potential (5) with mH= m0 = m were calculated in Ref. [6]:EQ= Qm241− Q254S2  A2m2!235 ; (9)where S = 4:85 is a quantity found numerically. The expression (9) was obtainedin a semiclassical approximation that becomes unreliable when Q  1. The energy inequation (7) was calculated [2, 3] in the ladder approximation from the Bethe-Salpeterequation. Although there is no reason to expect a good agreement between the twoapproximations, one of which (9) is pushed beyond its limit of validity, we notice thatfor n = 1 and Q = 2 they give the same dependency on the parameters A and m. Inaddition, formula (9) would give the same quantitative result if constant S assumed4a somewhat higher value.We believe that taking into account the detailed structure of the mass terms inthe MSSM and the inclusion of all degrees of freedom in the Higgs sector is unlikelyto alter one’s conclusions with respect to the existence of the bound states. Somegeneralizations of formula (7) to the case of the scalar elds with dierent masses andunequal trilinear couplings can be found in [4]. We conclude that the squarks andsleptons of the MSSM can form bound states. The binding energy is determined bythe coupling  in equation (8). If the squarks and the sleptons have dierent massesm1 and m2, and couple to the light Higgs with the couplings A1 and A2 respectively,then the relevant coupling is ~ = (1=16)A1A2=m1m2 [4].Phenomenologically, the trilinear coupling A in equation (3) can be as large as afew TeV. The upper limit on the value of A comes from considerations of vacuumstability with respect to tunneling into a possibly lower color and charge breaking(CCB) minimum in the scalar potential [8, 9]. The strongest limit of this kind exists forthe third generation trilinear coupling, At, because the tunneling into a CCB minimumassociated with a small Yukawa coupling is suppressed [10]. The empirical formulainferred from the numerical analyses [9] is(At=yt)2 + 32 < 7:5(m2~tL +m2~tR); (10)where m~tL and m~tR are the squark masses and yt  1 is the top Yukawa coupling4.Clearly, the ratio At=m~tR can be of order 10 if the right-handed stop is very light.While the masses and widths of the bound states may vary, their quantum numbersare determined by the particle content of the MSSM. Many of them can produce reso-nances at present and future experiments. For example, let us consider a \leptoquark"bound state that can show up as a resonance in an electron-proton collider. The cor-responding diagram is shown in Fig. 1. This resonance can be observed through an4If one requires that the color and charge conserving vacuum be the global minimum of the po-tential, the coecient 7.5 in equation (10) is replaced by 3 [8]. We note that At in Refs. [8, 9] diersfrom ours because we absorbed the Yukawa coupling into the denition of A.5                                                                                                                                            H~HL ~Q ~u ,d u ,dZ,W,eqL LQLLeL L~ ~eγ~γ~ ~L~~Z,W,~~Figure 1: The scalar \leptoquark" bound state can create a resonance at an electron-proton collider. The SU(2)-doublets ~Q and ~L are the rst-generation squark andslepton if the gaugino is ~Z or ~γ. They can be of a dierent flavor (for example,stop and ~) for the ~W exchange. Only one of many possible diagrams is shown.increase in the cross-section, much like that reported by HERA experiments [1] at highQ2. The experimental status of these events remains uncertain but will undoubtedly beclaried in the near future. A squark with mass  200 GeV and with R-parity violatingcouplings [11] has been proposed as an explanation of the HERA events. We emphasizethat the leptoquark resonances can exist in the MSSM with conserved R-parity. Theycorrespond to the bound states of squarks and sleptons of the type illustrated in Fig. 1.Perhaps, the resonances in the 200 GeV mass range can account for the HERA eventsif their coupling to the light fermions is suciently large. Of course, the parameters ofsuch resonances are model-dependent.We expect a variety of resonances to be detectable at a lepton collider, for instance,at LEP, NLC, or a muon collider (Fig. 2).The two-particle bound states can have two squarks (including a \positronium"state that comprises a squark and an anti-squark), two sleptons, or a slepton and asquark. Depending on the trilinear terms, the leptoquark resonances can have dierentquantum numbers that correspond to ed, eu, etc. In addition, there can be colorlessthree-particle states bound by the exchange of the Higgs elds (cf. Ref. [4]) as well asgluons. The states of greatest interest are, presumably, those that correspond to thelarge trilinear couplings and have, therefore, a greater binding energy. Some of these6                                                                                                                                                                                variousfinalstatesZq ,le , µe , µ~q ,l~~ ~+ +- -+ +--HHFigure 2: An example of a resonance at the e+e− or +− collider.multi-particle states may also show up as resonances. We leave the details for futurepublication.In summary, we have shown that large trilinear couplings in the scalar sector ofthe MSSM can give rise to a new family of bound states and resonances, in particularthose with the exotic quantum numbers, that may be observed in experiment.We thank S. Lola and M. Shaposhnikov for helpful discussions. V. A. Kuzminand I. I. Tkachev thank Theory Division at CERN for hospitality. I. I. Tkachev wassupported in part by the U.S. Department of Energy under Grant DE-FG02-91ER40681(Task B) and by the National Science Foundation under Grant PHY-9501458.References[1] C. Adlo it et al., H1 Collab, Z. Phys. C 74 (1997) 191; J. Breitweg et al., ZEUSCollaboration, Z. Phys. C 74 (1997) 207.[2] G. C. Wick Phys. Rev. 96 (1954) 1124; Cutkosky, Phys. Rev. 96 (1954) 1135.[3] L. D. Leo and J. W. Darewych, Can. J. Phys. 70 (1992) 412.[4] L. D. Leo and J. W. Darewych, Can. J. Phys., 71 (1993) 365.[5] G. Rosen, J. Math. Phys. 9 (1968) 996; R. Friedberg, T. D. Lee, and A. Sirlin,Phys. Rev. D13 (1976) 2739; S. Coleman, Nucl. Phys. B262 (1985) 263.7[6] A. Kusenko, Phys. Lett. B 404 (1997) 285.[7] A. Kusenko, Phys. Lett. B 405 (1997) 108.[8] For a recent review of the CCB minima in the MSSM, see, e. g., J. A. Casas,hep-ph/9707475, and references therein.[9] A. Kusenko, P. Langacker, and G. Segre, Phys. Rev. D 54 (1996) 5824; A. Kusenkoand P. Langacker, Phys. Lett. B 391 (1997) 29.[10] M. Claudson, L. J. Hall, and I. Hinchlie, Nucl. Phys. B 228, 501 (1983).[11] D. Choudhury and S. Raychaudhuri, Phys. Lett. B 401 (1997) 54; G. Altarelli,J. Ellis, G. F. Giudice, S. Lola, and M. L. Mangano, Nucl. Phys. B 506 (1997)3; H. Dreiner and P. Morawitz, Nucl. Phys. B 503 (1997) 55; J. Kalinowski,R. Ruckl, H. Spiesberger, and P. M. Zerwas, Z. Phys. C 74 (1997) 595; T. Konand T. Kobayashi, Phys. Lett. B 409 (1997) 265.8

Bound states and resonances in the scalar sector of the MSSM

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