We construct a new minimal extension of the Minimal Supersymmetric Standard
Model (MSSM) by promoting the $\mu$-parameter to a singlet superfield. The
resulting renormalizable superpotential is enforced by a $\mathcal{Z}_5$
$R$-symmetry which is imposed on the non-renormalizable operators as well. The
proposed model provides a natural solution to the $\mu$-problem and is free
from phenomenological and cosmological problems.Comment: 5 page

Abel

C Panagiotakopoulos

Ciafaloni

K Tamvakis

Panagiotakopoulos

Physics Letters B

English

arXiv

Crossref

New minimal extension of MSSM

We construct a new minimal extension of the Minimal Supersymmetric Standard Model (MSSM) by promoting the $\mu$-parameter to a singlet superfield. The resulting renormalizable superpotential is enforced by a $\mathcal{Z}_5$ $R$-symmetry which is imposed on the non-renormalizable operators as well. The proposed model provides a natural solution to the $\mu$-problem and is free from phenomenological and cosmological problems.We construct a new minimal extension of the Minimal Supersymmetric Standard Model (MSSM) by promoting the $\mu$-parameter to a singlet superfield. The resulting renormalizable superpotential is enforced by a $\mathcal{Z}_5$ $R$-symmetry which is imposed on the non-renormalizable operators as well. The proposed model provides a natural solution to the $\mu$-problem and is free from phenomenological and cosmological problems.We construct a new minimal extension of the Minimal Supersymmetric Standard Model (MSSM) by promoting the $\mu$-parameter to a singlet superfield. The resulting renormalizable superpotential is enforced by a $\mathcal{Z}_5$ $R$-symmetry which is imposed on the non-renormalizable operators as well. The proposed model provides a natural solution to the $\mu$-problem and is free from phenomenological and cosmological problems.We construct a new minimal extension of the Minimal Supersymmetric Standard Model (MSSM) by promoting the $\mu$-parameter to a singlet superfield. The resulting renormalizable superpotential is enforced by a $\mathcal{Z}_5$ $R$-symmetry which is imposed on the non-renormalizable operators as well. The proposed model provides a natural solution to the $\mu$-problem and is free from phenomenological and cosmological problems.We construct a new minimal extension of the minimal supersymmetric standard model (MSSM) by promoting the μ -parameter to a singlet superfield. The resulting renormalizable superpotential is enforced by a Z 5 R -symmetry which is imposed on the non-renormalizable operators as well. The proposed model provides a natural solution to the μ -problem and is free from phenomenological and cosmological problems

Panagiotakopoulos, C.

Tamvakis, K.

CERN Document Server

New Minimal Extension of MSSMC. Panagiotakopoulosa and K. TamvakisbaTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandandPhysics Division, School of TechnologyAristotle University of Thessaloniki, 54006 Thessaloniki, GreecebPhysics Department, University of Ioannina45110 Ioannina, GreeceAbstractWe construct a new minimal extension of the Minimal Supersymmetric Standard Model(MSSM) by promoting the µ-parameter to a singlet supereld. The resulting renormalizablesuperpotential is enforced by a Z5 R-symmetry which is imposed on the non-renormalizableoperators as well. The proposed model provides a natural solution to the µ-problem and isfree from phenomenological and cosmological problems.August 1999The Minimal Supersymmetric extension of the Standard Model (MSSM) [1], dened bypromoting each standard eld to a supereld, doubling the higgs elds and imposing R-parityconservation, seems to be preferred by the low energy data which support unication of thegauge couplings in the supersymmetric case. The most viable scenario for the breaking ofsupersymmetry at some low scale ms, no larger than  1 TeV, is the one based on sponta-neously broken supergravity. The breaking of supergravity takes place in some hidden sectorand is communicated to the visible sector through gravitational interactions. The resultingtheory with broken supersymmetry contains, independently of the details of the underlyinghigh energy theory, a number of soft supersymmetry (susy) breaking terms proportional topowers of the scale ms. Probably the most attractive feature of the MSSM is that it realizes aversion of \dimensional transmutation" where radiative corrections generate the electroweakscale MW from the susy-breaking scale ms. Unfortunately, a realistic implementation ofradiative symmetry breaking [2] in MSSM requires the presence of a coupling µH1H2 in-volving the higgs elds H1 and H2, the so called µ term, with values of the theoreticallyarbitrary parameter µ close to ms  MW . This nullies all merits of radiative symmetrybreaking since it amounts to introducing the electroweak scale by hand. Of course, thereexist scenarios to account for the origin of the µ term, alas, all in extended settings [3].A straightforward solution to the µ-problem would be to enlarge the eld content ofMSSM by adding a massless gauge singlet eld S that couples to the higgs elds as λSH1H2and acquires a vacuum expectation value (vev) of the order of ms  MW . Such a modelwith a purely cubic renormalizable superpotential containing a self-coupling of S as wellbecame known as the \Next to Minimal" SSM or NMSSM [4]. At the renormalizable levelthe model possesses a Z3 symmetry under which all superelds are multiplied by e2pii/3whose spontaneous breaking leads to the formation of cosmologically catastrophic domainwalls unless the discrete symmetry is not respected by higher order (non-renormalizable)operators. The existence of higher order operators 1 violating the Z3 symmetry, however,was shown [5] to be intimately related to the generation of quadratically divergent tadpolesfor the singlet [6]. Their generic contribution to the eective potential, cut-o at the Planckscale MP , isδV  ξMPm2sS + h.c., (1)where ξ is a factor depending on the loop order in which the tadpole is generated. Suchterms tend to destabilize the gauge hierarchy through a vev for the light singlet S muchlarger than the electroweak scale.Recently we have found [7] a simple resolution to the above problems of NMSSM byimposing a Z2 R-symmetry on the non-renormalizable operators under which all supereldsas well as the superpotential flip sign. Thus, the potentially harmful to the gauge hierarchyoperators [8] are forbidden but a harmless tadpoleδV  ξm3sS + h.c. (2)breaking the Z3 symmetry and making the walls disappear can still be generated.1These non-renormalizable terms appear either as D-terms in the Ka¨hler potential or as F -terms in thesuperpotential. The natural setting for these interactions is N = 1 Supergravity spontaneously broken by aset of hidden sector elds.1Our purpose in the present note is to get rid of the cubic superpotential self-coupling ofthe singlet S thereby constructing the simplest extension of the MSSM. To accomplish ourgoal we should, of course, nd substitutes for the twofold role played by the S3 coupling, asthis trilinear coupling contributes to the mechanism generating the vev of S through the softsusy-breaking terms and explicitly breaks the unwanted Peccei-Quinn symmetry present inits absence.The renormalizable superpotential of the proposed modelWren = λSH1H2 + Y (u)QU cH1 + Y (d)QDcH2 + Y (e)LEcH2 (3)possesses the global symmetriesU(1)B : Q(13), U c(−13), Dc(−13), L(0), Ec(0), H1(0), H2(0), S(0);U(1)L : Q(0), Uc(0), Dc(0), L(1), Ec(−1), H1(0), H2(0), S(0);U(1)PQ : Q(−1), U c(0), Dc(0), L(−1), Ec(0), H1(1), H2(1), S(−2);U(1)R : Q(1), Uc(1), Dc(1), L(1), Ec(1), H1(0), H2(0), S(2),where the charge of the supereld under the corresponding symmetry is given in parenthesis.U(1)B and U(1)L are the usual baryon and lepton number symmetries, U(1)PQ is an anoma-lous Peccei-Quinn symmetry whereas U(1)R is a non-anomalous R-symmetry under whichthe renormalizable superpotential Wren has charge 2. The soft trilinear susy-breaking termsbreak the continuous R-symmetry U(1)R down to its maximal non-R Z2 subgroup whichis the usual matter-parity. The U(1)PQ, which remains unbroken by the soft susy-breakingterms, could be broken by a linear eective potential term of the type given by Eq. (2) withξ  1 arising from non-divergent tadpoles. It is, however, quite dicult to achieve such anunsuppressed value of ξ. Thus, we rather have to resort to divergent tadpole contributionscut-o at MP which occur at very high order such that ξMP  ms. Finally, U(1)B and U(1)Lremain unbroken by both the susy-breaking terms and the tadpole but might be violatedby some non-renormalizable operators hopefully of suciently high order. Consequently, itis sucient to nd a symmetry which ensures the renormalizable superpotential of Eq. (3)and allows the generation of an adequately suppressed tadpole. All unwanted symmetrieswill then be broken and a vev < S > ms will readily be generated by combining the softsusy-breaking mass-squared term  m2sSS with the above linear in S contribution to theeective potential.A continuous symmetry enforcing the form of Wren in Eq. (3) is the U(1) R-symmetryobtained as the linear combination R0 = 3R + PQ of U(1)R and U(1)PQ :U(1)R′ : Q(2), Uc(3), Dc(3), L(2), Ec(3), H1(1), H2(1), S(4)under which the superpotential W has charge 6. U(1)R′ is broken by the trilinear softsusy-breaking terms down to its maximal non-R subgroup Z6 which is the product of aZ2 and a Z3 subgroup. The Z2 is essentially the usual matter parity (up to a SU(2)Lelement reversing the sign of all doublets) which leaves the tadpole invariant. Under the Z32(which is a subgroup of U(1)PQ) instead, S transforms non-trivially and the tadpole doesnot remain invariant. Thus, we should avoid imposing the whole U(1)R′ symmetry or one ofits subgroups which contains the aforementioned Z3 if we want a tadpole to be generated.A subgroup of U(1)R′ which is completely broken by the trilinear soft susy-breakingterms and is suciently large to enforce Wren of Eq. (3) but suciently small to allow thegeneration of a sizeable tadpole is the Z5 subgroup Zr5 of U(1)R′ generated byZr5 : (H1, H2)! α(H1, H2), (Q, L)! α2(Q, L), (U c, Dc, Ec)! α3(U c, Dc, Ec),S ! α4S, W ! αWwith α = e2pii/5. To examine the generation of the tadpole we bear in mind that the poten-tially harmful non-renormalizable terms are either even superpotential terms or odd Ka¨hlerpotential ones. Moreover, a tadpole diagram is divergent if an odd number of such \dan-gerous" non-renormalizable terms is combined with any number of renormalizable ones.Respecting the above rules [8] and the Zr5 R-symmetry we were able to show, not withoutsome eort, that divergent tadpoles rst appear at six loops. One example of such a di-vergent six-loop tadpole diagram is obtained by combining the non-renormalizable Ka¨hlerpotential terms λ1S2H1H2/M2P + h.c. and λ2S(H1H2)3/M5P + h.c. with the renormalizablesuperpotential term λSH1H2 (four times). The so generated linear eective potential termδV  (16pi2)−6λ1λ2λ4MP m2sS + h.c.is of the desired order of magnitude.Notice that the Z5 R-symmetry Zr5 , although it does not contain the usual matter parity,still manages to adequately stabilize the proton since, in addition to all d = 4 baryon andlepton number violating operators, it also forbids the dangerous QQQL and U cU cDcEc d = 5ones.It is very interesting that non-zero light neutrino masses are readily incorporated in themodel by simply introducing gauge singlet states νc transforming like Ec under all globalsymmetries. The allowed large majorana mass terms for these states break U(1)L down toits Z2 subgroup and generate small ordinary neutrino masses through the standard see-sawmechanism.In conclusion, we have shown that the µ term of MSSM can be generated by promotingthe parameter µ to a singlet supereld and imposing a Z5 R-symmetry. The resulting modelis a truly minimal extension of MSSM.AcknowledgementsWe acknowledge support by the TMR network \Beyond the Standard Model". We also wishto thank A. Pilaftsis for his valuable comments on the manuscript. C.P. wishes to thank S.Abel for many useful discussions.3References[1] H.-P. Nilles, Phys. Rep. 110 (1984) 1;H. E. Haber and G. L. Kane, Phys. Rep. 117 (1985) 75;A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145 (1987) 1.[2] L. E. Iba~nez and G. G. Ross, Phys. Lett. 110 (1982) 215; K. Inoue, A. Kakuto, H.Komatsu and S. Takeshita, Progr. Theor. Phys. 68 (1982) 927, 71 (1984) 96 ; J. Ellis,D. V. Nanopoulos and K. Tamvakis, Phys. Lett. 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New Minimal Extension of MSSM

New Minimal Extension of MSSM

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