UCRHEP-T483 Gauged B − x$_{i}$L origin of R parity and its implications Hye-Sung Lee Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA Ernest Ma Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA (Dated: January, 2010) Gauged B − L is a popular candidate for the origin of the conservation of R parity, i.e. R = ( − ) 3 B + L +2 $^{j}$, in supersymmetry, but it fails to forbid the effective dimension-five terms arising from the superfield combinations QQQL , u $^{c}$u$^{c}$d$^{c}$e$^{c}$, and u $^{c}$d$^{c}$d$^{c}$N$^{c}$, which allow the proton to decay. Changing it to B − x$_{i}$L , where x$_{e}$ + x$_{µ}$ + x$_{τ}$ = 3 (with x$_{i}$ = 1) for the three families, would forbid these terms while still serving as a gauge origin of R parity. We show how this is achieved in two minimal models with realistic neutrino mass matrices, and discuss their phenomenological implications. imal models with realistic neutrino mass matrices. We will examine their phenomenological constraints and the prognosis of their verification at the Large Hadron Col- lider (LHC). In the Minimal Supersymmetric Standard Model (MSSM) of particle interactions, the imposition of R par- ity, i.e. R = ( − ) 3 B + L +2 $^{j}$, where B , L , and j stand for baryon number, lepton number, and spin, respectively, serves two purposes. One is to establish a candidate for the dark-matter of the Universe, because the lightest su- persymmetric particle (LSP) is absolutely stable, being odd under R . The other is to forbid the otherwise allowed renormalizable superfield terms LH$_{u}$ , LLe $^{c}$, LQd $^{c}$, and u $^{c}$d$^{c}$d$^{c}$, so that the proton does not decay as a result of these interactions. However, the higher-dimensional quadrilinear terms QQQL and u $^{c}$u$^{c}$d$^{c}$ec are still allowed, giving rise thus to effective dimension-five terms in the Lagrangian which would also induce fast proton decay [1]. This is a serious problem for grand unification, even at a scale as high as 10 16 GeV. With the addition of three neutral singlet superfields N $^{c}$, desirable for neutrino masses, gauged B − L becomes possible. In that case, R parity is better understood the- oretically as a discrete remnant of B − L breaking. How- ever, the offending quadrilinear terms remain, together with the new term u $^{c}$d$^{c}$d$^{c}$N $^{c}$. To get rid of these terms, we propose a simple solution. Instead of gauged B − L , we adopt a flavor-dependent variation, i.e. B − x$_{i}$L , where the usual leptons still have L = 1, but x$_{e,µ,τ}$ for the three families are not equal to 1. In fact, the idea that there is just one N c and that x$_{e,µ,τ}$ = (3 , 0 , 0) or (0 , 3 , 0) or (0 , 0 , 3) has already been explored in the past [2–6]. If x$_{i}$ = 1, then all the unwanted quadrilinear terms are forbidden by gauged B − x$_{i}$L so that proton stability is assured as shown for (0 , 0 , 3) case in Ref. [3]. Here we assume that there are three N $^{c}$, with each x$_{i}$ nonzero. Unlike the previous (0 , 0 , 3) type of models, this opens up a new possibility that the spontaneous breaking of B − x$_{i}$L by suitably chosen Higgs singlet superfields S 1 , 2 will result in an exact discrete residual symmetry which is just the usual R parity. In the following, we will discuss the conditions on x$_{i}$ and construct two min- Both are satisfied if The usual B − L is recovered for x$_{e}$ = x$_{µ}$ = x$_{τ}$ = 1, and B − 3 L$_{τ}$ [2] is obtained for x$_{e}$ = x$_{µ}$ = 0 and x$_{τ}$ = 3. The anomaly-free conditions for the addition of U (1)$_{X}$ to the standard SU (3)$_{C}$ × SU (2)$_{L}$ × U (1)$_{Y}$ gauge group are easily written down. The [ SU (3)$_{C}$] $^{2}$U (1)$_{X}$ and [ U (1)$_{X}$] 3 anomalies are automatically zero because of the vectorial nature of SU (3)$_{C}$ and U (1)$_{X}$. The [gravity] $^{2}$U (1)$_{X}$ anomaly also vanishes for the same rea- son. The [ U (1)$_{X}$] $^{2}$U (1)$_{Y}$ anomaly is zero because the sum of Y charges is zero separately for each family of quarks and for each family of leptons. The remaining two conditions, i.e. [ SU (2)$_{L}$] $^{2}$U (1)$_{X}$ and [ U (1)$_{Y}$ ] $^{2}$U (1)$_{X}$, are given respectively by Our minimal model consists of the usual particle con- tent of the MSSM plus three singlet superfields N c i with L = − 1 and two singlet superfields S 1 , 2 with L = ± 2. Under the gauged U (1)$_{X}$ symmetry of B − x$_{i}$L , quarks have charges 1/3 and leptons have charges − x$_{i}$ . The two usual Higgs doublet superfields have charges 0 and the Higgs singlets S 1 , 2 have charges ∓ 2 x$_{S}$ . The vari- ous x$_{i}$ and x$_{S}$ will be determined by the requirements of anomaly cancellation and a realistic neutrino mass ma- trix. Model Introduction arXiv:1001.0768v3 [hep-ph] 6 May 2010 (3)(3)(1 / 3) − x$_{e}$ − x$_{µ}$ − x$_{τ}$ = 0 , (1) (3)(3)[2(1 / 6) 2 − (2 / 3) 2 − ( − 1 / 3) $^{2}$](1 / 3) + [2( − 1 / 2) 2 − ( − 1) $^{2}$]( − x$_{e}$ − x$_{µ}$ − x$_{τ}$ ) = 0 . (2) x$_{e}$ + x$_{µ}$ + x$_{τ}$ = 3 . (3)