New and Useful Gauge Extension of the MSSM

A new nontrivial U(1) gauge extension of the Minimal Supersymmetric Standard Model (MSSM) is proposed which automatically conserves baryon number and lepton number, and solves the μ problem. Naturally small Dirac neutrino masses are also possible in this context. • Talk at TH-2002, Paris, July 2002. If the minimal standard model of quarks and leptons (SM) is extended to include supersymmetry, 3 problems are known to appear. (1) Whereas baryon number B and lepton number L are automatically conserved in the SM as the consequence of the assumed SU(3)C × SU(2)L × U(1)Y gauge symmetry and its representation content, they are no longer so in its most general supersymmetric extension. (2) The necessity of 2 Higgs doublet superfields requires the term μφ̂1φ̂2, but there is no understanding of why μ should be of the order of the supersymmetry breaking scale MSUSY instead of some much larger unification scale. (3) Neutrino masses are not required by the gauge symmetry, which is of course also a shortcoming of the SM. Problem (1) is usually solved by imposing R parity, i.e. R ≡ (−1), which is of course the defining hypothesis of the Minimal Supersymmetric Standard Model (MSSM). Problem (2) is usually ignored or solved by replacing μ by a singlet superfield Ŝ with 〈S̃〉 ∼ MSUSY . Problem (3) is usually sidestepped as in the SM by adding trivial N̂ c singlets. Each of the above solutions is independent of the other two. Is there a single principle which solves all 3 problems? The answer is yes, as I show in this talk, using a new nontrivial U(1) gauge symmetry, as recently proposed [1]. Consider the gauge group SU(3)C × SU(2)L × U(1)Y × U(1)X . The usual quark and lepton (left-handed) chiral superfields transform as follows: (û, d̂) ∼ (3, 2, 1/6;n1), û c ∼ (3∗, 1,−2/3;n2), d̂ c ∼ (3∗, 1, 1/3;n3), (1) (ν̂, ê) ∼ (1, 2,−1/2;n4), ê c ∼ (1, 1, 1;n5), N̂ c ∼ (1, 1, 0;n6). (2) They are supplemented by the two Higgs doublet superfields φ̂1 ∼ (1, 2,−1/2;−n1 − n3), φ̂2 ∼ (1, 2, 1/2;−n1 − n2), (3) with n1 + n3 = n4 + n5, and n1 + n2 = n4 + n6 to allow for the usual Yukawa couplings of the quarks and leptons as in the MSSM. However, the μ term is replaced by the trilinear 2 interaction χ̂φ̂1φ̂2, where χ̂ is a Higgs singlet superfield transforming as χ̂ ∼ (1, 1, 0; 2n1 + n2 + n3). (4) Thus 2n1 + n2 + n3 6= 0 is required so that the effective μ parameter of this model is determined by the U(1)X breaking scale, i.e. 〈χ̂〉. Furthermore, n4 6= −n1 − n3 would imply L conservation, and n2 + 2n3 6= 0 would imply B conservation. In contrast to the SM, the existence of N̂ c is required here as long as n6 6= 0. The big question is now whether there is an anomaly-free U(1)X gauge symmetry with the above properties. The answer is yes if I add two copies of the singlet quark superfields Û ∼ (3, 1, 2/3;n7), Û c ∼ (3∗, 1,−2/3;n8), (5) and one copy of D̂ ∼ (3, 1,−1/3;n7), D̂ c ∼ (3∗, 1, 1/3;n8), (6) with n7 + n8 = −2n1 − n2 − n3, so that their masses are also determined by the U(1)X breaking scale. Consider now the various conditions for the absence of the axial-vector anomaly [2]. The [SU(3)]U(1)X anomaly is absent automatically because 2n1 + n2 + n3 + n7 + n8 = 0. The [SU(2)]U(1)X and [U(1)Y ] U(1)X anomalies are both absent if n2 + n3 = 7n1 + 3n4. The U(1)Y [U(1)X ] 2 anomaly is the sum of 8 quadratic terms in ni, but it factorizes remarkably into 6(3n1 + n4)(2n1 − 4n2 − 3n7) = 0. (7) The condition 3n1+n4 = 0 contradicts 2n1+n2+n3 6= 0, so the condition 2n1−4n2−3n7 = 0 has to be chosen. The [U(1)X ] 3 anomaly is the sum of 11 cubic terms in ni, but it factorizes even more remarkably into − 36(3n1 + n4)(9n1 + n4 − 2n6)(6n1 − n4 − n6) = 0. (8) 3 Thus there are 2 solutions, as summarized in Tables 1 and 2. Table 1: Solutions (A) and (B) where ni = an1 + bn4. (A) (B) a b a b n2 7/2 3/2 5 0 n3 7/2 3/2 2 3 n5 9/2 1/2 3 2 n6 9/2 1/2 6 –1 n7 –4 –2 –6 0 n8 –5 –1 –3 –3 −n1 − n3 –9/2 –3/2 –3 –3 −n1 − n2 –9/2 –3/2 –6 0 2n1 + n2 + n3 9 3 9 3 Note that solutions (A) and (B) are identical if n4 = n1. This turns out to be also the condition [3] for U(1)X to be orthogonal to U(1)Y . The condition 3n1 + n4 6= 0 also forbids Q̂Q̂Q̂L̂ and ûûd̂ê which are allowed by R parity. Thus proton decay is more suppressed here than in the MSSM. There are two more anomalies to consider. The global SU(2) chiral gauge anomaly [4] is absent because the number of SU(2)L doublets is even. The mixed gravitational-gauge anomaly [5] is proportional to the sum of U(1)X charges, i.e. 3(6n1 + 3n2 + 3n3 + 2n4 + n5 + n6) + 3(3n7 + 3n8) +2(−n1 − n3) + 2(−n1 − n2) + (2n1 + n2 + n3) = 6(3n1 + n4), (9) which is not zero. This anomaly may be tolerated if gravity is neglected. On the other hand, it may be rendered zero by adding U(1)X supermultiplets as follows: one with charge 3(3n1 +n4), three with charge −2(3n1 +n4), and three with charge −(3n1 +n4). Hence they