Multiplicative Conservation of Baryon Number and Baryogenesis

In the canonical seesaw mechanism of neutrino mass, lepton number is only multiplicatively conserved, which enables the important phenomenon of leptogenesis to occur, as an attractive explanation of the present baryon asymmetry of the Universe. A parallel possibility, hitherto unrecognized, also holds for baryon number and baryogenesis. This new idea is shown to be naturally realized in the context of a known supersymmetric string-inspired extension of the Standard Model, based on E(6) particle content, and having an extra U(1)_N gauge symmetry. Within this framework, two-loop radiative neutrino masses are also possible, together with a new form of very long-lived matter.

The Universe has an imbalance of matter over antimatter, called the baryon asymmetry [1]. An elegant way of understanding this is the phenomenon of leptogenesis [2,3], by which a lepton asymmetry is established from the decays of heavy Majorana singlet fermions N i and gets converted [4] into a baryon asymmetry through sphalerons [5] during the electroweak phase transition in the early Universe. In this scenario, lepton number is only multiplicatively conserved and neutrinos acquire small Majorana masses through the famous canonical seesaw mechanism [6].
In this paper, a parallel and equally elegant possibility, i.e. multiplicatively conserved baryon number and baryogenesis, is proposed and shown to be naturally realized in the framework of a known supersymmetric string-inspired extension [7] of the Standard Model (SM), as detailed below.
In leptogenesis, the only interactions of N i are with the lepton doublets (ν i , l i ) and the Higgs doublet (φ + , φ 0 ). As the Universe expands and cools, the out-of-equilibrium decays of N 1 (i.e. the lightest N i ) into l − φ + and l + φ − establish a lepton (L) asymmetry. This engenders a baryon (B) asymmetry through sphaleron interactions which change B + L but not B − L.
Consider now the following extension of the SM. Leth,h c be heavy singlet scalar quarks of charge Q = ∓1/3 and baryon number B = ∓2/3; and let N c 1,2 be heavy singlet neutral fermions of B = 1. As a result, the new interactions where Q = (u, d), are allowed. Suppose also that N c 1,2 are Majorana so that baryon number is only multiplicatively conserved. Then the decays generate a baryon asymmetry under the same conditions as in leptogenesis. Again N c 2 is needed to obtain the required CP violation from the interference of the tree and one-loop diagrams as shown in Fig. 1. As an illustration, let m N c ∼ 10 6 GeV and d c N c 2h couplings ∼ 10 −2 , then a decay asymmetry of order 10 −6 may be established, enabling a baryon asymmetry η B ∼ 10 −10 to be obtained. The out-of-equilibrium condition requires however that the d c N c 1h couplings be less than about 10 −5 . Sphaleron interactions will modify this pure B asymmetry into a B − L asymmetry in exact analogy to what happens to a pure L asymmetry in the case of leptogenesis.
This new idea of multiplicatively conserved baryon number and baryogenesis turns out to be naturally realized in the context of a supersymmetric string-inspired extension of the SM proposed some time ago [7], with many interesting features in its own right [8,9]. It is based on E 6 with matter content given by three 27 representations and with gauge interactions of the SM plus those of U(1) N , which is a linear combination of U(1) ψ and U(1) χ in the decomposition: In terms of the maximal subgroup SU given by where T 3L,3R and Y L,R are the usual quantum numbers of the SU(2) × U(1) decompositions of SU(3) L,R . The particle content of a 27 multiplet of E 6 is shown in Table 1.
There are eleven possible generic trilinear terms invariant under Five are necessary for fermion masses, namely for m u , m d , m e , m E , m h respectively. The other six are some of which must be absent to prevent rapid proton decay. Hence all such models require an additional discrete symmetry, the simplest of which is of course a single Z 2 , resulting in eight generic possibilities, as shown already many years ago [10]. I consider here instead an exactly conserved (−) L × (−) 3B symmetry as shown in Table 2.
As a result, all terms of Eq. (7) are allowed as well as the following from Eq. (8):   Since (−) 3B is conserved, the proton is stable, but the deuteron is not, and neutronantineutron oscillations are allowed. The latter two processes are based on the same mechanism responsible for baryogenesis, i.e. Eqs. (2) and (3), as shown in Fig. 2. As an illustration, let m N c ∼ 10 6 GeV, mh ∼ 10 3 GeV, d c N ch couplings ∼ 10 −2 , and udh couplings ∼ 10 −5 , then Since N c is odd under (−) 3B but even under (−) L , there is no LN cĒ coupling and N c does not play the role of a singlet right-handed neutrino as is normally assumed. This means that neutrino masses remain zero, contrary to present data on neutrino oscillations. To remedy this shortcoming, a very interesting variation of the above scenario is described below.
There are three sets of E,Ē, S superfields, but only one is required to break SU(2) L × U(1) Y × U(1) N and to endow all fermions (except the neutrinos) with masses. Let them thus be divided as in Table 3. In that case, the generic SEĒ couplings are restricted to S 1 E 1,2,3Ē1,2,3 , S 2,3 E 1Ē2,3 , S 2,3 E 2,3Ē1 , and the important new couplings are allowed. The decays of N c 1 into l −Ẽ+ and l +Ẽ− now also generate a lepton asymmetry. Thus remarkably, both B and L asymmetries are established in the decays of N c 1 , and at energies below its mass, the theory conserves both B and L. Sphaleron interactions then convert both asymmetries into a B − L asymmetry, the baryon component of which is observed today.
from supersymmetry breaking, seesaw neutrino masses are now generated in two loops as shown in Fig. 3. This has the same structure discussed in Ref. [9]. However, no extra Z 2 symmetry beyond (−) L and (−) 3B is assumed here. For N c of order 10 6 GeV, realistic neutrino masses of order 0.1 eV may be obtained. This is well below the bound of 10 9 GeV on the reheating temperature of the Universe for avoiding the overproduction of gravitinos [11].
Since (−) 3B+L is still the same, i.e. even, for all E,Ē, S superfields, R parity is unchanged in this scenario. However, the lightest particle contained in E 2,3 ,Ē 2,3 , and S 2,3 must decay through N c , so its lifetime is very long, say of order 10 6 seconds. An example of such a decay is shown in Fig. 4. Depending on their masses, there may even be two such long-lived particles, one with R parity even and the other odd. The mass of N c 1 (the lightest N c i ) may be of order 10 6 GeV.
(3) The proton is stable, but the deuteron is not, and neutron-antineutron oscillations are allowed.
(4) This scenario is naturally realized in a known supersymmetric string-inspired extension of the SM, i.e. SU(3) C × SU(2) L × U(1) Y × U(1) N with particle content given by three 27 representations of E 6 .
(5) If kinematically allowed, the U(1) N gauge boson Z N will be discovered with ease at the LHC because it has both quark and lepton couplings (see Table 1).
(6) Two-loop radiative seesaw neutrino masses are also possible in an interesting variation of the model, where both B and L asymmetries are established by the decays of N c 1 , to be converted into a B − L asymmetry by sphaleron interactions.
(7) The lightest particle odd under R = (−) 3B+L+2j is a candidate for the dark matter of the Universe as in the MSSM. However, there is now at least one particle which is very long-lived, and will also appear as missing energy at the LHC.
This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837.