A Possible Way of Connecting The Grassmann Variables And The Number of Generation

We construct a Left-Right symmetric model in which the number of generation is related to Grassmann variables. We introduce two sets of complex Grassmann variables (θ q , θ 2 q), (θ 1 l , θ 2 l ) and associate each variable with leftand right-handed quark and lepton fields, respectively. Expanding quark and lepton fields in powers of the Grassmann variables, we find that there are exactly three generations of quarks and leptons. Integrating out the Grassmann variables, we obtain phenomenologically acceptable fermion mass matrices. 11.10.-z, 11.30.Hv, 12.15-y, 12.15.Ff Typeset using REVTEX 1 How many generations of quarks and leptons are there in the Nature is one of the outstanding problems of particle physics today. Considerations from nuclear synthesis [1] and experimental data on the Z decay width from LEP [2] both indicate that there are only three generations of light neutrinos, but they do not provide information about the number of heavy generations. On the theoretical side the situation is not any better. The Standard Model does not have the answer to the problem. To answer this question one needs to go beyond the standard model. Many theoretical efforts have been made, ranging from the topological properties of compactified spacetimes in string theory to considerations from anomaly cancellation and composite models, to determine the number of generation [3]. But the problem is still far from been solved. In this letter we will study an interesting approach to the generation problem which relates the particle spectrum with Grassmann variables (GVs) [4,5]. It is well known that Taylor expansions in the GVs will have finite terms. This terminating nature of the GVs suggestes a very interesting way to classify particle spectrum when connection between particle fields and the GVs is made. Extensive work has been done with models based on SU(5) grand unification group [5]. In Ref. [5] the particle spectrum, including the gauge transformation properties, are all specified by the GVs. In the follwoing we will study another way of relating the GVs and the number of generation, and to construct a low energy model. The gauge group of our model is the SU(3)C × SU(2)L × SU(2)R ×U(1)B−L Left-Right symmetric group. Under this group the quarks q and leptons l tranform as qL : (3, 2, 1, 1/3) , qR : (3, 1, 2, 1/3) , lL : (1, 2, 1,−1) , lR : (1, 1, 2,−1) . (1) To make connections between the GVs and the number of generation we introduce two sets GVs: θq = (θ 1 q , θ 2 q) and θl = (θ 1 l , θ 2 l ) which transform under a global group, G = SU(2)q×SU(2)l×U(1)f as (2,1,α) and (1,2,α), respectively. We also group qL,R and lL,R into Q = (qL, q c R) and L = (lL, l c R), and let them to transform as (2, 1, 0) and (1, 2, 0), respectively. This way of grouping the quarks and leptons suggestes that the global symmetry is in some 2 way related to the helicity of fermions. The fermion and boson fields are component fields of some super bosonic fields Ei expanded in powers of the GVs. There are two class of expansions of the superfields. One has even powers of the GVs and the other has odd powers. Since the overall superfields Ei are bosonic, it is clear that the component fields with even powers of the GVs in the expansion are boson fileds and the ones with odd powers are fermion fields. This expansion does not constrain the gauge transformation properties of the component fields. The superfields can have non-trivial transformation properties under the gauge group. The Lagrangian density L in the ordinary space-time is obtained by first using the available superfields Ei to form terms L(Ei) which are singlets under both the gauge and global symmetries, and then integrating out the GVs, that is L = ∫ d2θqd 2θld 2θ̄qd 2θ̄lL(Ei) . (2) This procedure will select certain terms in L(Ei) because only the terms with proper θ powers will survive the integral. Let us consider the fermion fields first. The terms with the lowest power in the GVs for fermion fields are E1Q = θ̄qQ1 , E1L = θ̄lL1 . (3) Multiplying (θ̄qθq) (θ̄lθl) b on E1i, we generate all allowed fermion fields with the same quantum numbers under the global symmetry G in this theory. We have E1Q = θ̄qQ1 , E2Q = (θ̄qθq)θ̄qQ2 , E3Q = (θ̄lθl)θ̄qQ3 , E4Q = (θ̄lθl) 2θ̄qQ4 , E5Q = (θ̄qθq)(θ̄lθl)θ̄qQ5 , E6Q = (θ̄qθq)(θ̄lθl) 2θ̄qQ6 , E1L = θ̄lL1 , E2L = (θ̄lθl)θ̄lL2 , E3L = (θ̄qθq)θ̄lL3 , (4) E4L = (θ̄qθq) 2θ̄lL4 , E5L = (θ̄qθq)(θ̄lθl)θ̄lL5 , E6L = (θ̄lθl)(θ̄qθq) 2θ̄lL6 . This set of superfields carries −α of the U(1)f charge. One can generate other fields with different expansions which will have different global symmetry transformation properties. 3 Naively, eq.(4) contains six generations of quarks and leptons. This is, however, not true. Some of the fields are actually the same. To see this let us define the dual fields Ẽi. The dual of a superfield is defined by replacing the powers of the GVs (θ̄q) (θq) (θ̄l) (θl) d to (θ̄q) (θq) (θ̄l) (θl) 2−c [4]. We notice that the correct kinetic term in the Lagrangian Lk for the fermion fields can be generated by Lk = ∫ d2θ̄qd 2θ̄ld 2θqd 2θlĒiγμD Ẽi , (5) if the component field of Ei is the same as the component filed of its dual. We therefore require that the component field is the same as the component field of its dual. It is easy to see that the following fields are dual pairs E1Q ↔ E6Q , E2Q ↔ E4Q , E3Q ↔ E5Q . E1L ↔ E6L , E2L ↔ E4L , E3L ↔ E5L . (6) We, therefore, have only three generations of quarks and leptons. We now turn to possible Higgs scalars Hi which may generate fermion masses through Yukawa terms. The Yukawa terms will have the form EiEjHk. It is clear that in order to couple the Higgs scalars Hi to fermions, Hi should carry two times of the U(1)f charge as Ei but with opposite signs. We have H1 = θqθqh1 , H2 = θ̄lθlθqθqh2 , H3 = (θ̄lθl) 2θqθqh3 , (7) H4 = θlθlh4 , H5 = θ̄qθqθlθlh5 , H6 = (θ̄qθq) 2θlθlh6 . From our previous definition for duals, we find H2 and H5 are self-dual, and H3, H6 are the duals of H1, H4, respectively. hi are singlets under the global symmetry G. The gauge transformation properties are not specified. In order to form gauge singlets with fermions to generate masses, we assign hi to transform as (1, 2, 2, 0) under the gauge group. Notice that since hi are singlets under the global symmetry G, we can also expaind Hi by replacing hi to h̃i = τ2h ∗ i τ2 without change the overall transformation properties of Hi under the gauge and the global symmetries. Therefore h̃i should also been included in eq.(7). With the fields in eq.(7), we find that only the following Yukawa terms will survive the GV integration 4 E1QE1QH̃1 , E3QE3QH1 , E1QẼ2QH1 , E1QE3QH2 , E1LE1LH̃4 , E3LE3LH4 , E1LẼ2LH4 , E1LE3LH5 . (8) Because both hi and h̃i are available to form gauge singlets with fermions, each term in eq.(8) contains two terms. For quarks we have EiQEjQHk ⇒ λij q̄iLhkqjR + λ ′ ij q̄iLh̃kqjR , (9) where λ’s are constants. Similarly for leptons. When hi develop vacuum expectation values, the quarks and leptons obtain their masses. If we now identify q1 = (c, s) , q2 = (u, d) , q3 = (t, b) , l1 = (νμ, μ) , l2 = (νe, e) , l3 = (ντ , τ) , (10) we obtain the following form for the fermion mass matrices