The only allowed Higgs superpotential term at stringy tree level in the
string derived Singlet Extensions of the Minimal Supersymmetric Standard Model
(SEMSSM) is h S H_d H_u, which leads to an additional global U(1) symmetry in
the Higgs potential. We propose the string inspired SEMSSM where the global
U(1) symmetry is broken by the additional superpotential terms or supersymmetry
breaking soft terms that can be obtained naturally due to the instanton effects
or anomalous U(1)_A gauge symmetry. In these models, we can solve the \mu
problem and the fine-tuning problem for the lightest CP-even Higgs boson mass
in the MSSM, generate the baryon asymmetry via electroweak baryogenesis, and
predict the new Higgs physics which can be tested at the LHC and ILC.Comment: RevTex4, 5 pages, 2 tables, comments and references added, version to
  appear in PL

Abel

Amaldi

Barger

Blumenhagen

Bouchard

Braun

Carena

Chen

Cremades

Cvetic

Dedes

Derendinger

Dermisek

Dimopoulos

Dreiner

Einhorn

Ellis

Erler

Fayet

Frere

Giunti

Green

Haber

Ibanez

Langacker

Marciano

Nilles

Okada

Panagiotakopoulos

Pietroni

Sakai

Tianjun Li

Witten

Physics Letters B

English

arXiv

Li, Tianjun

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:hep-ph/0612359v3  28 Aug 2007MIFP-06-26String Inspired Singlet Extensions of the Minimal Supersymmetric Standard ModelTianjun Li1, 21George P. and Cynthia W. Mitchell Institute for Fundamental Physics,Texas A&M University, College Station, TX 77843, USA2 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, ChinaThe only allowed Higgs superpotential term at stringy tree level in the string derived SingletExtensions of the Minimal Supersymmetric Standard Model (SEMSSM) is hSHdHu, which leads toan additional global U(1) symmetry in the Higgs potential. We propose the string inspired SEMSSMwhere the global U(1) symmetry is broken by the additional superpotential terms or supersymmetrybreaking soft terms that can be obtained naturally due to the instanton effects or anomalous U(1)Agauge symmetry. In these models, we can solve the µ problem and the fine-tuning problem for thelightest CP-even Higgs boson mass in the MSSM, generate the baryon asymmetry via electroweakbaryogenesis, and predict the new Higgs physics which can be tested at the LHC and ILC.PACS numbers: 11.25.Mj, 12.10.Kt, 12.60.FrIntroduction – The Minimal Supersymmetric Stan-dard Model (MSSM) can solve the gauge hierarchy prob-lem elegantly due to supersymmetry, has neutralinoas cold dark matter candidate, and accommodates thegauge coupling unification [1, 2]. So, it is the most nat-ural extension of the Standard Model (SM). However,there are a few problems within the MSSM. The bilinearsupersymmetric Higgs mass term µHdHu in the superpo-tential, where Hd and Hu are one pair of Higgs doublets,does not violate supersymmetry and gauge symmetry.Then the natural scale for µ is about Planck scale butnot the TeV scale, which leads to the µ problem. More-over, in order to have the lightest CP-even Higgs bosonmass larger than the low bound 114 GeV from the LEPexperiment, there exists a few percent fine-tuning [3].To solve the µ problem, the Next to the Minimal Su-persymmetric Standard Model (NMSSM) was proposedin which a SM singlet S and a Z3 discrete symmetry areintroduced [4]. The µHdHu term is forbidden by the Z3symmetry, and the superpotential in the NMSSM isW = hSHdHu +κ3!S3 , (1)where h and κ are Yukawa couplings. After S obtainsa vacuum expectation value (VEV), the effective µ termµeff = h〈S〉 is generated. Also, the F-term of S will giveadditional contribution to the Higgs quartic coupling,and then can increase the lightest CP-even Higgs bosonmass. In addition, the lightest CP-even Higgs boson inthe NMSSM can have mass around 100 GeV because ofits invisible decay [5], and the above fine-tuning problemfor its mass can be solved. Moreover, the observed baryonasymmetry can be generated via electroweak baryogen-esis because there are extra CP violating phases in thesupersymmetry breaking soft parameters and the trilin-ear soft term AhhSHdHu can give us strong first orderelectroweak phase transition [6]. Similar results hold forthe nearly MSSM [7] and the U(1)′-extended supersym-metric Standard Models [8]. Therefore, the Singlet Ex-tensions of the Minimal Supersymmetric Standard Model(SEMSSM) is very interesting from phenomenologicalpoint of view.On the other hand, string theory may be the onlyknown theory which can correctly describe the quantumgravity. In string models, we may solve the µ prob-lem in the MSSM [9], and the doublet-triplet splittingproblem in the Grand Unified Theories (GUTs) [10, 11].Thus, how to indirectly test string models at the LargeHadron Collider (LHC) and the International Linear Col-lider (ILC) is a pretty interesting question.As we know, in the string model building, the renor-malizable terms in the superpotential, which arise fromthe Chern-Simmons terms in the heterotic string com-pactification [12] or instanton effects (triangles formed bythe intersections of D6-branes) in Type IIA intersectingD6-brane models [13], have the following generic trilinearform at stringy tree levelW = yφφ1φ2φ3 , (2)where yφ is the Yukawa coupling, and φi are differentfields. Thus, the µHdHu term in the MSSM and theκS3/3! term in the NMSSM do not exist at stringy treelevel in the string derived models. And only the firstterm hSHdHu in the superpotential in Eq. (1) is allowedwhere S is a modulus. With only hSHdHu term in thesuperpotential, we have two global U(1) symmetries inthe Higgs potential in which one of them is U(1)Y gaugesymmetry. So, there is one global U(1) symmetry, andthen there exists one massless Goldstone boson with S,H0d and H0u mixing components, which is excluded fromthe known experiments.In this letter, we propose the string inspired SEMSSM.The global U(1) symmetry in the Higgs potential is bro-ken by the additional superpotential terms or supersym-metry breaking soft terms. The extra superpotentialterms can be realized in the string derived models via in-stanton effects [9]. With anomalous U(1)A gauge symme-try [14], we construct four simple and concrete SEMSSM.In these models, we can naturally solve the µ problem and2the fine-tuning problem for the lightest CP-even Higgsboson mass in the MSSM. We also calculate the Higgsboson masses, chargino masses and neutralino masses attree level, and predict the new Higgs physics which canbe tested at the LHC and ILC. A more detail discussionswill be presented elsewhere [15].Model Building – Let us consider the most generalSEMSSM. The generic superpotential isW = hSHdHu + µHdHu +m2S +µ′2!S2 +κ3!S3 , (3)where µ, m2, and µ′ are mass parameters. The corre-sponding F -term scalar potential isVF = |hHdHu +m2 + µ′S +κ2!S2|2+|hS + µ|2|Hu|2 + |hS + µ|2|Hd|2 . (4)And the D-term scalar potential isVD =G28(|Hu|2 − |Hd|2)2, (5)where G2 = g2Y + g22; gY and g2 are respectively thecoupling constants for U(1)Y and SU(2)L. Moreover, weintroduce the supersymmetry breaking soft terms V Isoftand V IIsoft as followsV Isoft = m2Hd|Hd|2 +m2Hu |Hu|2 +m2S |S|2 , (6)V IIsoft = −(AhhSHdHu +BµBHdHu +AXm2XS+12!B′µ′BS2 +13!AκκXS3 +H.C.), (7)where m2Hd , m2Hu, and m2S are supersymmetry breakingsoft masses, Ah, B, µB, AX , m2X , B′, µ′B , and Aκ aresupersymmetry breaking soft mass parameters, and κXis the coupling constant. In addition, if µ 6= 0, m2 6= 0,µ′ 6= 0, or κ 6= 0, we assume µB = µ, m2X = m2, µ′B =µ′, or κX = κ, respectively. However, even if µ = 0,m2 = 0, µ′ = 0, or κ = 0, we can show that µB, m2X , µ′B,or κX might not be zero in general, so the global U(1)symmetry in the Higgs potential can be broken by thesupersymmetry breaking soft terms [15].In the string derived models, the terms µHdHu,µ′S2/2!, and κS3/3! in superpotential in Eq. (3), whichare forbidden at stringy tree level, might be generateddue to the instanton effects. And the effective µ isaboutMstringe−A whereMstring is the string scale around1017 GeV. So, the µ problem in the MSSM is solved ifA ∼ 33 [9]. Similar result holds for µ′. However, theκS3/3! term generated from instanton effects might behighly suppressed.To construct the string inspired SEMSSM, we con-sider the models with an anomalous U(1)A gauge sym-metry [14]. In string model building, there genericallyexists one anomalous U(1) gauge symmetry in the het-erotic string model building [14] or up to four in theType II orientifold model building [16]. The correspond-ing anomalies are cancelled by the (generalized) Green-Schwarz mechanism [17]. We introduce a SM singlet fieldφ with U(1)A charge −1. To cancel the Fayet-Iliopoulosterm of U(1)A, we assume that φ obtains a VEV sothat the U(1)A D-flatness and supersymmetry can bepreserved. Interestingly, 〈φ〉/MPl is about 0.22, whereMPl is the Planck scale [14]. Moreover, to break the su-persymmetry, we introduce a hidden sector superfield Zwhose F component acquires a VEV around 1021 GeV2.We assume that the U(1)A charges for S and Z aren + p/q and m + p′/q′, respectively, where m and nare integers, (p, q) and (p′ and q′) are relatively primepositive integers, or p/q or p′/q′ is zero. To have thehSHdHu term in superpotential, we also assume thatthe total U(1)A charges for Hd and Hu are −n − p/q,but we do not give the explicit charges for Hd and Huwhich are irrelevant to our discussions. Moreover, ifm + p′/q′ is non-zero, the gaugino masses can not begenerated via F-terms ZWαWα/MPl. To have the gaug-ino masses, we can introduce another U(1)A-unchargedhidden-sector superfield Z ′ whose F component acquiresa VEV. In fact, in the string model building, both dilatonand moduli fields can break the supersymmetry due totheir F-component VEVs. In addition, the supersymme-try breaking soft mass terms in V Isoft can be generatedvia D-term operators∫d4xd2θd2θZZM2Pl(|S|2 + |Hd|2 + |Hu|2), (8)where for simplicity we neglect the coefficients of theseoperators in such kind of discussions in this paper. Thefirst term AhhSHdHu in VIIsoft can be generated via thefollowing F-term operator∫d4xd2θZ (or Z ′)MPlhSHdHu +H.C. . (9)Model A – We choose the following U(1)A charges forS and Zm+ n = 47, p/q = 1/5 , p′/q′ = 4/5 . (10)Then the U(1)A allowed renormalizable superpotential isW = hSHdHu . (11)The additional supersymmetry breaking soft term V IIsoftcan be generated via the following operator (the otheroperators are forbidden by U(1)A or negligible)∫d4xd2θMstringZS(φMPl)48+H.C. . (12)So, we haveV IIsoft = −(AhhSHdHu +AXm2XS +H.C.), (13)3where Ah ∼ AX ∼ 102 GeV, and m2X ∼ 104−6 GeV2.Interestingly, the global U(1) symmetry in the Higgs po-tential is indeed broken by the supersymmetry breakingsoft term AXm2XS.Model B – We choose the following U(1)A charges forS and Zn = −22, p/q = 0 , m = 0, p′/q′ = 0 . (14)The additional relevant F-term and D-term operators are∫d4xd2θ (MstringHdHu + ZHdHu)(φMPl)22+∫d4xd2θd2θ(ZS +ZZSMPl)(φMPl)22+H.C. . (15)Thus, the superpotential in Model B isW = hSHdHu + µHdHu +m2S , (16)where µ ∼ 102−3 GeV and m2 ∼ 104−6 GeV2. And thesupersymmetry breaking soft terms in V IIsoft areV IIsoft = − (AhhSHdHu +BµBHdHu+AXm2XS +H.C.), (17)where Ah ∼ B ∼ µB ∼ AX ∼ 102−3 GeV, and m2X ∼104−6 GeV2.Model C – In Model B, we consider the gauge mediatedsupersymmetry breaking scenario where the VEV of Fcomponent of Z can be about 1010 GeV2. And then thetadpole termm2S in the superpotential can be neglected.Thus, the superpotential in Model C isW = hSHdHu + µHdHu . (18)And the supersymmetry breaking soft terms in V IIsoft areV IIsoft = − (AhhSHdHu +BµBHdHu) . (19)Model C can also be considered as the string derivedmodel with hSHdHu superpotential term where the extraµHdHu term arises from instanton effects [9].Model D – We choose the following U(1)A charges forS and Zn = 11, p/q = 1/2 , m = 0, p′/q′ = 0 . (20)The additional relevant F-term operators are∫d4xd2θ(MstringS2 + ZS2)( φMPl)23+H.C. . (21)So, the superpotential isW = hSHdHu +µ′2!S2 , (22)TABLE I: The Higgs VEVs, and the charged, CP-even, andCP-odd Higgs boson masses in GeV at tree level.Model 〈H0d〉 〈H0u〉 〈S〉 H± H01 H02 H03 A01 A02A 119 127 213 205 67 196 210 127 251B 123 123 188 179 45 184 206 142 214C 123 123 161 165 66 148 171 31 214D 120 126 167 176 67 145 181 39 225where µ′ ∼ 102 GeV. And the supersymmetry breakingsoft terms in V IIsoft areV IIsoft = −(AhhSHdHu +12!B′µ′BS2 +H.C.), (23)where Ah ∼ B′ ∼ µ′B ∼ 102−3 GeV.Model D can be considered as the string derivedmodel with hSHdHu superpotential term where the extraµ′S2/2! term arises from instanton effects [9]. However,there exists a Z4 symmetry in Model D, where Hd andHu have charge 1, and S has charge 2. To avoid thedomain wall problem after symmetry breaking, we canturn on tiny instanton effects to break the Z4 symme-try by generating small high-dimensional operators, andthen we can dissolve the domain wall.Phenomenological Consequences – We shall cal-culate the Higgs boson masses, the chargino and neu-tralino masses at tree level in our models where we ne-glect the loop corrections for simplicity. The input pa-rameters with dimensions of mass or mass-squared arechosen in arbitrary units. After finding an acceptableminimum they are rescaled so that√〈H0d〉2 + 〈H0u〉2 ≃174.1 GeV. For Model A, we choose: h = 0.7, m2Hd =−0.1, m2Hu = −0.2, m2S = 0.1, Ah = 1.0, AX = 0.68,m2X = 0.6. And the VEVs for the Higgs fields atthe minimum are 〈H0d〉 = 0.7031, 〈H0u〉 = 0.75, and〈S〉 = 1.2563. For Model B, we choose: h = 0.7,µ = −0.2, m2 = −0.3, m2Hd = −0.1, m2Hu= −0.1,m2S = 0.1, Ah = 0.6, B = −0.1, µB = −0.2, AX = −1.9,m2X = −0.3. And the VEVs for the Higgs fields are〈H0d〉 = 0.8625, 〈H0u〉 = 0.8625, and 〈S〉 = 1.3156. ForModel C, we choose: h = 0.7, µ = −0.1, m2Hd = −0.1,m2Hu = −0.1, m2S = −0.6, Ah = 2.0, B = −0.6,µB = −0.1. And the VEVs for the Higgs fields are〈H0d〉 = 1.5875, 〈H0u〉 = 1.5875, and 〈S〉 = 2.075. ForModel D, we choose: h = 0.7, µ′ = −0.3, m2Hd = −0.1,m2Hu = −0.4, m2S = −0.68, Ah = 2.0, B′ = −0.6,µ′B = −0.3. And the VEVs for the Higgs fields are〈H0d〉 = 1.6375, 〈H0u〉 = 1.7203, and 〈S〉 = 2.275.We present the Higgs VEVs, the charged Higgs bo-son (H±) mass, CP-even Higgs boson (H01 , H02 , and H03 )masses, and CP-odd Higgs boson (A01 and A02) masses4TABLE II: The chargino and neutralino masses in GeV.Model Mi χ˜±1χ˜±2χ˜01 χ˜02 χ˜03 χ˜04 χ˜05A > 0 115 334 68 88 175 217 336A < 0 163 314 68 156 169 217 314B > 0 75 328 56 81 167 184 330B < 0 118 315 81 125 156 184 316C > 0 76 329 58 80 167 185 330C < 0 120 315 80 127 156 185 316D > 0 87 330 60 68 169 200 331D < 0 132 315 61 138 156 200 315in Table I. Interestingly, the couplings of the CP-evenHiggs boson H01 and H02 with Z0 gauge boson almostvanish, and only the heaviest CP-even Higgs boson H03can couple to Z0 [15]. Thus, the fine-tuning problemfor the lightest CP-even Higgs boson mass in the MSSMfrom the LEP constraints can be completely relaxed, andwe have new Higgs physics at the LHC and ILC becauseH03 has mass around 190 GeV. Moreover, to calculate thechargino and neutralino masses, we choose the positiveand negative gaugino masses M1 and M2 for U(1)Y andSU(2)L: (1) M1 = 150 GeV, and M2 = 300 GeV; (2)M1 = −150 GeV, and M2 = −300 GeV. The massesfor charginos (χ˜±1 and χ˜±2 ) and neutralinos (χ˜0i withi = 1, 2, ..., 5) are given in Table II.Conclusions – In the string derived SEMSSM,hSHdHu is the only allowed superpotential term atstringy tree level. Then there exist an additional globalU(1) symmetry in the Higgs potential, and the axionproblem. We propose the string inspired SEMSSM inwhich the global U(1) symmetry is broken by the addi-tional superpotential terms or supersymmetry breakingsoft terms. The extra superpotential terms can be ob-tained via instanton effects in the string derived models.With anomalous U(1)A gauge symmetry, we present foursimple and concrete SEMSSM. In these models, we cannaturally solve the µ problem and the fine-tuning prob-lem for the lightest CP-even Higgs boson mass in theMSSM, and generated the observed baryon asymmetryvia electroweak baryogenesis. In addition, we calculatethe Higgs boson masses, chargino masses and neutralinomasses at tree level, and predict the new Higgs physicswhich can be tested at the LHC and ILC. Confirmationone of these models at the future colliders might give usindirect implication of string theory.Acknowledgments – This research was supported inpart by the Cambridge-Mitchell Collaboration in Theo-retical Cosmology.[1] S. Dimopoulos and H. Georgi, Nucl. Phys. B 193, 150(1981); S. Dimopoulos, S. Raby and F. Wilczek, Phys.Rev. D 24, 1681 (1981); N. Sakai, Z. Phys. C 11, 153(1981); L. E. Ibanez and G. G. Ross, Phys. Lett. B 105,439 (1981); M. B. Einhorn and D. R. T. Jones, Nucl.Phys. B 196, 475 (1982); W. J. Marciano and G. Sen-janovic, Phys. Rev. D 25, 3092 (1982).[2] U. Amaldi, W. de Boer and H. Furstenau, Phys.Lett. B 260, 447 (1991); J. R. Ellis, S. Kelley andD. V. Nanopoulos, Phys. Lett. B 249, 441 (1990); P. Lan-gacker and M. X. Luo, Phys. Rev. D 44, 817 (1991);C. Giunti, C. W. Kim and U. W. Lee, Mod. Phys. Lett.A 6 (1991) 1745.[3] Y. Okada, M. Yamaguchi and T. Yanagida, Prog.Theor. Phys. 85, 1 (1991); J. R. Ellis, G. Ridolfi andF. Zwirner, Phys. Lett. B 257, 83 (1991); H. E. Haberand R. Hempfling, Phys. Rev. Lett. 66, 1815 (1991);M. Carena, J. R. Espinosa, M. Quiros and C. E. M. Wag-ner, Phys. Lett. B 355, 209 (1995).[4] P. Fayet, Nucl. Phys. B 90, 104 (1975); H. P. Nilles,M. Srednicki and D. Wyler, Phys. Lett. B 120, 346(1983); J. M. Frere, D. R. T. Jones and S. Raby,Nucl. Phys. B 222, 11 (1983); J. P. Derendinger andC. A. Savoy, Nucl. Phys. B 237, 307 (1984).[5] R. Dermisek and J. F. Gunion, Phys. Rev. Lett. 95,041801 (2005).[6] M. Pietroni, Nucl. Phys. B 402 (1993) 27.[7] C. Panagiotakopoulos and K. Tamvakis, Phys. Lett. B469, 145 (1999); C. Panagiotakopoulos and A. Pilaftsis,Phys. Rev. D 63, 055003 (2001); A. Dedes, C. Hugonie,S. Moretti and K. Tamvakis, Phys. Rev. D 63, 055009(2001).[8] M. Cvetic, D. A. Demir, J. R. Espinosa, L. L. Everett andP. Langacker, Phys. Rev. D 56, 2861 (1997) [Erratum-ibid. D 58, 119905 (1998)]; J. Erler, P. Langacker andT. Li, Phys. Rev. D 66, 015002 (2002); V. Barger, P. Lan-gacker, H. S. Lee and G. Shaughnessy, Phys. Rev. D 73,115010 (2006).[9] R. Blumenhagen, M. Cvetic and T. Weigand, Nucl. Phys.B 771, 113 (2007); L. E. Ibanez and A. M. Uranga, JHEP0703, 052 (2007).[10] V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, Phys.Lett. B 618, 252 (2005); V. Bouchard and R. Donagi,Phys. Lett. B 633, 783 (2006); V. Bouchard, M. Cveticand R. Donagi, Nucl. Phys. B 745, 62 (2006).[11] C. M. Chen, T. Li and D. V. Nanopoulos, Nucl. Phys. B751, 260 (2006).[12] E. Witten, Phys. Lett. B 155, 151 (1985); T. Li,J. L. Lopez and D. V. Nanopoulos, Phys. Rev. D 56,2602 (1997).[13] D. Cremades, L. E. Ibanez and F. Marchesano, JHEP0307, 038 (2003); M. Cvetic and I. Papadimitriou, Phys.Rev. D 68, 046001 (2003) [Erratum-ibid. D 70, 029903(2004)]; S. A. Abel and A. W. Owen, Nucl. Phys. B 663,197 (2003).[14] H. K. Dreiner, H. Murayama and M. Thormeier, Nucl.Phys. B 729, 278 (2005), and references therein.[15] T. Li, in preparation.[16] L. E. Ibanez, F. Marchesano and R. Rabadan, JHEP0111, 002 (2001).[17] M. B. Green and J. H. Schwarz, Phys. Lett. B 149, 1175(1984).

String inspired singlet extensions of the minimal supersymmetric Standard Model

Crossref

Texas A&M University

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MIFP-06-26
String Inspired Singlet Extensions of the Minimal Supersymmetric Standard Model
Tianjun Li1, 2
1George P. and Cynthia W. Mitchell Institute for Fundamental Physics,
Texas A&M University, College Station, TX 77843, USA
2 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China
The only allowed Higgs superpotential term at stringy tree level in the string derived Singlet
Extensions of the Minimal Supersymmetric Standard Model (SEMSSM) is hSHdHu, which leads to
an additional global U(1) symmetry in the Higgs potential. We propose the string inspired SEMSSM
where the global U(1) symmetry is broken by the additional superpotential terms or supersymmetry
breaking soft terms that can be obtained naturally due to the instanton effects or anomalous U(1)A
gauge symmetry. In these models, we can solve the µ problem and the fine-tuning problem for the
lightest CP-even Higgs boson mass in the MSSM, generate the baryon asymmetry via electroweak
baryogenesis, and predict the new Higgs physics which can be tested at the LHC and ILC.
PACS numbers: 11.25.Mj, 12.10.Kt, 12.60.Fr
Introduction – The Minimal Supersymmetric Stan-
dard Model (MSSM) can solve the gauge hierarchy prob-
lem elegantly due to supersymmetry, has neutralino
as cold dark matter candidate, and accommodates the
gauge coupling unification [1, 2]. So, it is the most nat-
ural extension of the Standard Model (SM). However,
there are a few problems within the MSSM. The bilinear
supersymmetric Higgs mass term µHdHu in the superpo-
tential, where Hd and Hu are one pair of Higgs doublets,
does not violate supersymmetry and gauge symmetry.
Then the natural scale for µ is about Planck scale but
not the TeV scale, which leads to the µ problem. More-
over, in order to have the lightest CP-even Higgs boson
mass larger than the low bound 114 GeV from the LEP
experiment, there exists a few percent fine-tuning [3].
To solve the µ problem, the Next to the Minimal Su-
persymmetric Standard Model (NMSSM) was proposed
in which a SM singlet S and a Z3 discrete symmetry are
introduced [4]. The µHdHu term is forbidden by the Z3
symmetry, and the superpotential in the NMSSM is
W = hSHdHu +
κ
3!
S3 , (1)
where h and κ are Yukawa couplings. After S obtains
a vacuum expectation value (VEV), the effective µ term
µeff = h〈S〉 is generated. Also, the F-term of S will give
additional contribution to the Higgs quartic coupling,
and then can increase the lightest CP-even Higgs boson
mass. In addition, the lightest CP-even Higgs boson in
the NMSSM can have mass around 100 GeV because of
its invisible decay [5], and the above fine-tuning problem
for its mass can be solved. Moreover, the observed baryon
asymmetry can be generated via electroweak baryogen-
esis because there are extra CP violating phases in the
supersymmetry breaking soft parameters and the trilin-
ear soft term AhhSHdHu can give us strong first order
electroweak phase transition [6]. Similar results hold for
the nearly MSSM [7] and the U(1)′-extended supersym-
metric Standard Models [8]. Therefore, the Singlet Ex-
tensions of the Minimal Supersymmetric Standard Model
(SEMSSM) is very interesting from phenomenological
point of view.
On the other hand, string theory may be the only
known theory which can correctly describe the quantum
gravity. In string models, we may solve the µ prob-
lem in the MSSM [9], and the doublet-triplet splitting
problem in the Grand Unified Theories (GUTs) [10, 11].
Thus, how to indirectly test string models at the Large
Hadron Collider (LHC) and the International Linear Col-
lider (ILC) is a pretty interesting question.
As we know, in the string model building, the renor-
malizable terms in the superpotential, which arise from
the Chern-Simmons terms in the heterotic string com-
pactification [12] or instanton effects (triangles formed by
the intersections of D6-branes) in Type IIA intersecting
D6-brane models [13], have the following generic trilinear
form at stringy tree level
W = yφφ1φ2φ3 , (2)
where yφ is the Yukawa coupling, and φi are different
fields. Thus, the µHdHu term in the MSSM and the
κS3/3! term in the NMSSM do not exist at stringy tree
level in the string derived models. And only the first
term hSHdHu in the superpotential in Eq. (1) is allowed
where S is a modulus. With only hSHdHu term in the
superpotential, we have two global U(1) symmetries in
the Higgs potential in which one of them is U(1)Y gauge
symmetry. So, there is one global U(1) symmetry, and
then there exists one massless Goldstone boson with S,
H0d and H
0
u mixing components, which is excluded from
the known experiments.
In this letter, we propose the string inspired SEMSSM.
The global U(1) symmetry in the Higgs potential is bro-
ken by the additional superpotential terms or supersym-
metry breaking soft terms. The extra superpotential
terms can be realized in the string derived models via in-
stanton effects [9]. With anomalous U(1)A gauge symme-
try [14], we construct four simple and concrete SEMSSM.
In these models, we can naturally solve the µ problem and
2the fine-tuning problem for the lightest CP-even Higgs
boson mass in the MSSM. We also calculate the Higgs
boson masses, chargino masses and neutralino masses at
tree level, and predict the new Higgs physics which can
be tested at the LHC and ILC. A more detail discussions
will be presented elsewhere [15].
Model Building – Let us consider the most general
SEMSSM. The generic superpotential is
W = hSHdHu + µHdHu +m
2S +
µ′
2!
S2 +
κ
3!
S3 , (3)
where µ, m2, and µ′ are mass parameters. The corre-
sponding F -term scalar potential is
VF = |hHdHu +m
2 + µ′S +
κ
2!
S2|2
+|hS + µ|2|Hu|
2 + |hS + µ|2|Hd|
2 . (4)
And the D-term scalar potential is
VD =
G2
8
(
|Hu|
2 − |Hd|
2
)2
, (5)
where G2 = g2Y + g
2
2; gY and g2 are respectively the
coupling constants for U(1)Y and SU(2)L. Moreover, we
introduce the supersymmetry breaking soft terms V Isoft
and V IIsoft as follows
V Isoft = m
2
Hd
|Hd|
2 +m2Hu |Hu|
2 +m2S |S|
2 , (6)
V IIsoft = −
(
AhhSHdHu +BµBHdHu +AXm
2
XS
+
1
2!
B′µ′BS
2 +
1
3!
AκκXS
3 +H.C.
)
, (7)
where m2Hd , m
2
Hu
, and m2S are supersymmetry breaking
soft masses, Ah, B, µB, AX , m
2
X , B
′, µ′B , and Aκ are
supersymmetry breaking soft mass parameters, and κX
is the coupling constant. In addition, if µ 6= 0, m2 6= 0,
µ′ 6= 0, or κ 6= 0, we assume µB = µ, m
2
X = m
2, µ′B =
µ′, or κX = κ, respectively. However, even if µ = 0,
m2 = 0, µ′ = 0, or κ = 0, we can show that µB, m
2
X , µ
′
B,
or κX might not be zero in general, so the global U(1)
symmetry in the Higgs potential can be broken by the
supersymmetry breaking soft terms [15].
In the string derived models, the terms µHdHu,
µ′S2/2!, and κS3/3! in superpotential in Eq. (3), which
are forbidden at stringy tree level, might be generated
due to the instanton effects. And the effective µ is
aboutMstringe
−A whereMstring is the string scale around
1017 GeV. So, the µ problem in the MSSM is solved if
A ∼ 33 [9]. Similar result holds for µ′. However, the
κS3/3! term generated from instanton effects might be
highly suppressed.
To construct the string inspired SEMSSM, we con-
sider the models with an anomalous U(1)A gauge sym-
metry [14]. In string model building, there generically
exists one anomalous U(1) gauge symmetry in the het-
erotic string model building [14] or up to four in the
Type II orientifold model building [16]. The correspond-
ing anomalies are cancelled by the (generalized) Green-
Schwarz mechanism [17]. We introduce a SM singlet field
φ with U(1)A charge −1. To cancel the Fayet-Iliopoulos
term of U(1)A, we assume that φ obtains a VEV so
that the U(1)A D-flatness and supersymmetry can be
preserved. Interestingly, 〈φ〉/MPl is about 0.22, where
MPl is the Planck scale [14]. Moreover, to break the su-
persymmetry, we introduce a hidden sector superfield Z
whose F component acquires a VEV around 1021 GeV2.
We assume that the U(1)A charges for S and Z are
n + p/q and m + p′/q′, respectively, where m and n
are integers, (p, q) and (p′ and q′) are relatively prime
positive integers, or p/q or p′/q′ is zero. To have the
hSHdHu term in superpotential, we also assume that
the total U(1)A charges for Hd and Hu are −n − p/q,
but we do not give the explicit charges for Hd and Hu
which are irrelevant to our discussions. Moreover, if
m + p′/q′ is non-zero, the gaugino masses can not be
generated via F-terms ZWαWα/MPl. To have the gaug-
ino masses, we can introduce another U(1)A-uncharged
hidden-sector superfield Z ′ whose F component acquires
a VEV. In fact, in the string model building, both dilaton
and moduli fields can break the supersymmetry due to
their F-component VEVs. In addition, the supersymme-
try breaking soft mass terms in V Isoft can be generated
via D-term operators
∫
d4xd2θd2θ
ZZ
M2Pl
(
|S|2 + |Hd|
2 + |Hu|
2
)
, (8)
where for simplicity we neglect the coefficients of these
operators in such kind of discussions in this paper. The
first term AhhSHdHu in V
II
soft can be generated via the
following F-term operator
∫
d4xd2θ
Z (or Z ′)
MPl
hSHdHu +H.C. . (9)
Model A – We choose the following U(1)A charges for
S and Z
m+ n = 47, p/q = 1/5 , p′/q′ = 4/5 . (10)
Then the U(1)A allowed renormalizable superpotential is
W = hSHdHu . (11)
The additional supersymmetry breaking soft term V IIsoft
can be generated via the following operator (the other
operators are forbidden by U(1)A or negligible)
∫
d4xd2θMstringZS
(
φ
MPl
)48
+H.C. . (12)
So, we have
V IIsoft = −
(
AhhSHdHu +AXm
2
XS +H.C.
)
, (13)
3where Ah ∼ AX ∼ 10
2 GeV, and m2X ∼ 10
4−6 GeV2.
Interestingly, the global U(1) symmetry in the Higgs po-
tential is indeed broken by the supersymmetry breaking
soft term AXm
2
XS.
Model B – We choose the following U(1)A charges for
S and Z
n = −22, p/q = 0 , m = 0, p′/q′ = 0 . (14)
The additional relevant F-term and D-term operators are
∫
d4xd2θ (MstringHdHu + ZHdHu)
(
φ
MPl
)22
+
∫
d4xd2θd2θ
(
ZS +
ZZS
MPl
)(
φ
MPl
)22
+H.C. . (15)
Thus, the superpotential in Model B is
W = hSHdHu + µHdHu +m
2S , (16)
where µ ∼ 102−3 GeV and m2 ∼ 104−6 GeV2. And the
supersymmetry breaking soft terms in V IIsoft are
V IIsoft = − (AhhSHdHu +BµBHdHu
+AXm
2
XS +H.C.
)
, (17)
where Ah ∼ B ∼ µB ∼ AX ∼ 10
2−3 GeV, and m2X ∼
104−6 GeV2.
Model C – In Model B, we consider the gauge mediated
supersymmetry breaking scenario where the VEV of F
component of Z can be about 1010 GeV2. And then the
tadpole termm2S in the superpotential can be neglected.
Thus, the superpotential in Model C is
W = hSHdHu + µHdHu . (18)
And the supersymmetry breaking soft terms in V IIsoft are
V IIsoft = − (AhhSHdHu +BµBHdHu) . (19)
Model C can also be considered as the string derived
model with hSHdHu superpotential term where the extra
µHdHu term arises from instanton effects [9].
Model D – We choose the following U(1)A charges for
S and Z
n = 11, p/q = 1/2 , m = 0, p′/q′ = 0 . (20)
The additional relevant F-term operators are
∫
d4xd2θ
(
MstringS
2 + ZS2
)( φ
MPl
)23
+H.C. . (21)
So, the superpotential is
W = hSHdHu +
µ′
2!
S2 , (22)
TABLE I: The Higgs VEVs, and the charged, CP-even, and
CP-odd Higgs boson masses in GeV at tree level.
Model 〈H0d〉 〈H
0
u〉 〈S〉 H
± H01 H
0
2 H
0
3 A
0
1 A
0
2
A 119 127 213 205 67 196 210 127 251
B 123 123 188 179 45 184 206 142 214
C 123 123 161 165 66 148 171 31 214
D 120 126 167 176 67 145 181 39 225
where µ′ ∼ 102 GeV. And the supersymmetry breaking
soft terms in V IIsoft are
V IIsoft = −
(
AhhSHdHu +
1
2!
B′µ′BS
2 +H.C.
)
, (23)
where Ah ∼ B
′ ∼ µ′B ∼ 10
2−3 GeV.
Model D can be considered as the string derived
model with hSHdHu superpotential term where the extra
µ′S2/2! term arises from instanton effects [9]. However,
there exists a Z4 symmetry in Model D, where Hd and
Hu have charge 1, and S has charge 2. To avoid the
domain wall problem after symmetry breaking, we can
turn on tiny instanton effects to break the Z4 symme-
try by generating small high-dimensional operators, and
then we can dissolve the domain wall.
Phenomenological Consequences – We shall cal-
culate the Higgs boson masses, the chargino and neu-
tralino masses at tree level in our models where we ne-
glect the loop corrections for simplicity. The input pa-
rameters with dimensions of mass or mass-squared are
chosen in arbitrary units. After finding an acceptable
minimum they are rescaled so that
√
〈H0d〉
2 + 〈H0u〉
2 ≃
174.1 GeV. For Model A, we choose: h = 0.7, m2Hd =
−0.1, m2Hu = −0.2, m
2
S = 0.1, Ah = 1.0, AX = 0.68,
m2X = 0.6. And the VEVs for the Higgs fields at
the minimum are 〈H0d〉 = 0.7031, 〈H
0
u〉 = 0.75, and
〈S〉 = 1.2563. For Model B, we choose: h = 0.7,
µ = −0.2, m2 = −0.3, m2Hd = −0.1, m
2
Hu
= −0.1,
m2S = 0.1, Ah = 0.6, B = −0.1, µB = −0.2, AX = −1.9,
m2X = −0.3. And the VEVs for the Higgs fields are
〈H0d〉 = 0.8625, 〈H
0
u〉 = 0.8625, and 〈S〉 = 1.3156. For
Model C, we choose: h = 0.7, µ = −0.1, m2Hd = −0.1,
m2Hu = −0.1, m
2
S = −0.6, Ah = 2.0, B = −0.6,
µB = −0.1. And the VEVs for the Higgs fields are
〈H0d〉 = 1.5875, 〈H
0
u〉 = 1.5875, and 〈S〉 = 2.075. For
Model D, we choose: h = 0.7, µ′ = −0.3, m2Hd = −0.1,
m2Hu = −0.4, m
2
S = −0.68, Ah = 2.0, B
′ = −0.6,
µ′B = −0.3. And the VEVs for the Higgs fields are
〈H0d〉 = 1.6375, 〈H
0
u〉 = 1.7203, and 〈S〉 = 2.275.
We present the Higgs VEVs, the charged Higgs bo-
son (H±) mass, CP-even Higgs boson (H01 , H
0
2 , and H
0
3 )
masses, and CP-odd Higgs boson (A01 and A
0
2) masses
4TABLE II: The chargino and neutralino masses in GeV.
Model Mi χ˜
±
1
χ˜±
2
χ˜01 χ˜
0
2 χ˜
0
3 χ˜
0
4 χ˜
0
5
A > 0 115 334 68 88 175 217 336
A < 0 163 314 68 156 169 217 314
B > 0 75 328 56 81 167 184 330
B < 0 118 315 81 125 156 184 316
C > 0 76 329 58 80 167 185 330
C < 0 120 315 80 127 156 185 316
D > 0 87 330 60 68 169 200 331
D < 0 132 315 61 138 156 200 315
in Table I. Interestingly, the couplings of the CP-even
Higgs boson H01 and H
0
2 with Z
0 gauge boson almost
vanish, and only the heaviest CP-even Higgs boson H03
can couple to Z0 [15]. Thus, the fine-tuning problem
for the lightest CP-even Higgs boson mass in the MSSM
from the LEP constraints can be completely relaxed, and
we have new Higgs physics at the LHC and ILC because
H03 has mass around 190 GeV. Moreover, to calculate the
chargino and neutralino masses, we choose the positive
and negative gaugino masses M1 and M2 for U(1)Y and
SU(2)L: (1) M1 = 150 GeV, and M2 = 300 GeV; (2)
M1 = −150 GeV, and M2 = −300 GeV. The masses
for charginos (χ˜±1 and χ˜
±
2 ) and neutralinos (χ˜
0
i with
i = 1, 2, ..., 5) are given in Table II.
Conclusions – In the string derived SEMSSM,
hSHdHu is the only allowed superpotential term at
stringy tree level. Then there exist an additional global
U(1) symmetry in the Higgs potential, and the axion
problem. We propose the string inspired SEMSSM in
which the global U(1) symmetry is broken by the addi-
tional superpotential terms or supersymmetry breaking
soft terms. The extra superpotential terms can be ob-
tained via instanton effects in the string derived models.
With anomalous U(1)A gauge symmetry, we present four
simple and concrete SEMSSM. In these models, we can
naturally solve the µ problem and the fine-tuning prob-
lem for the lightest CP-even Higgs boson mass in the
MSSM, and generated the observed baryon asymmetry
via electroweak baryogenesis. In addition, we calculate
the Higgs boson masses, chargino masses and neutralino
masses at tree level, and predict the new Higgs physics
which can be tested at the LHC and ILC. Confirmation
one of these models at the future colliders might give us
indirect implication of string theory.
Acknowledgments – This research was supported in
part by the Cambridge-Mitchell Collaboration in Theo-
retical Cosmology.
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String Inspired Singlet Extensions of the Minimal Supersymmetric
  Standard Model

String Inspired Singlet Extensions of the Minimal Supersymmetric Standard Model

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