Upper bound on all products of R - parity violating couplings λ ′ and λ ′ ′ from proton decay

We prove that any product of R-parity violating couplings λ (L-violating) and λ (B-violating) can be strongly restricted by proton decay data. For any pair λ and λ the decay exists at least at one loop level. For squark masses below 1 TeV we find the conservative bounds |λ′ · λ′′| < 10 in absence of squark flavor mixing, and |λ′ · λ′′| < 10 when this mixing is taken into account. We study the dependence of the bounds on the flavor basis in which R-parity breaking couplings are determined. 1. The proton decay gives very strong bounds on the products of the R-parity violating couplings λ (L-violating) and λ (B-violating) involving light generations [1]. The decay takes place at tree level, and for squark masses about 1 TeV one gets [2]: |λ · λ| ∼< 10. (1) What are the bounds on the couplings involving heavy generations? Whether there are unrestricted couplings? These questions are important not only for phenomenology [3, 4, 5, 6, 7, 8, 9] but also for understanding the origin of the R-parity, as well as for the unification of interactions. Notice, for instance, that in stringy unified models the R-parity violation could be the only mechanism of the proton decay. In the context of the Grand Unified Theories it was shown that the quark-lepton symmetry usually leads to separate strong bounds on λ and λ [10]. The situation can be different in absence of Grand Unification. It was claimed that certain products of λ and λ constants may be large without inducing proton decay at observable level [11]. In this Letter we prove that operators relevant to the proton decay are always present in the one loop effective lagrangian, and they imply strong bounds on any product of the couplings λ · λ. 2. Let us introduce the L-violating coupling constant λijk and the B violating coupling constant λ mnp, where i, j, k,m, n, p = 1, 2, 3 are the generation indices, as the constants of the R-parity violating interactions: λijk D cα i (νj S d klD α l − Ej S kl U l ) + λ mnp ǫαβγ D cα m D cβ n U cγ p (2) which are consistent with the Standard Model symmetry. Here, Ei, νi, D c i , Di, U c i , Ui are the superfields with charged leptons, neutrinos, downand up-type-quarks, and α, β, γ are color indices. Notice that in (2) the sum is over the flavor index l only. The couplings are written in terms of superfields whose fermionic components coincide with the mass eigenstates (in all orders of perturbation theory). The unitary matrices S and S connect the fermionic states of the original basis in which the couplings are defined with the mass states. The product SS = V (3) gives the Cabibbo-Kobayashi-Maskawa matrix V .