ar X iv : h ep - p h / 02 02 22 3 v 3 3 A pr 2 00 3 Strong and Electroweak Interactions , and Their Unification with Noncommutative Space - Time

Quantum field theories based on noncommutative space-time (NCQFT) have been extensively studied recently. However no NCQFT model, which can uniquely describe the strong and electroweak interactions, has been constructed. This prevents consistent and systematic study of noncommutative space-time. In this work we construct a NCQFT model based on the trinification gauge group SU(3)C × SU(3)L × SU(3)R. A unique feature of this model, that all matter fields (fermions and Higgses) are assigned to (anti)fundamental representations of the factor SU(3) groups, allows us to construct a NCQFT model for strong and electroweak interactions and their unification without ambiguities. This model provides an example which allows consistent and systematic study of noncommutative space-time phenomenology. We also comment on some related issues regarding extensions to E6 and U(3)C × U(3)L × U(3)R models. PACS numbers:22.20-z,11.15-q,12.10, 12.60-i Typeset using REVTEX 1 Noncommutative quantum field theory (NCQFT), based on modification of the spacetime commutation relations, provides an alternative to the ordinary quantum field theory. A simple way to modify the space-time properties is to change the usual space-time coordinate x to noncommutative coordinate X̂ such that [1] [X̂, X̂ ] = iθ , (1) where θ is a real anti-symmetric matrix. We will consider the case where θ is a constant and commutes with X̂. NCQFT based on the above commutation relation can be easily studied using the WeylMoyal correspondence replacing the product of two fields A(X̂) and B(X̂) with noncommutative coordinates by product of the same fields but ordinary coordinate x through the star “∗” product, A(X̂)B(X̂) → A(x) ∗B(x) = Exp[i 1 2 θ∂x,μ∂y,ν ]A(x)B(y)|x=y. (2) Properties related to NCQFT have been studied extensively recently [2–11,13,14]. NCQFT for a pure U(1) group is easy to study. Related phenomenology have been studied recently [2]. But it is more complicated for non-abelian groups. Due to the “∗” product nature, there are fundamental differences between ordinary and noncommutative gauge theories and cause many difficulties to construct a unique and consistent model for strong and electroweak interactions based on SU(3)C ×SU(2)L×U(1)Y gauge group [3–11,13,14]. One of the main problems is that SU(N) group can not be simply gauged with “∗” product as will be explain in the following. Another problem is that, naively, the charges of any U(1) gauge group with “∗” product are quantized to only three possible values, 1, 0, -1 which cannot accommodate all the hypercharges for matter fields in the SM. In this work we construct a NCQFT model based on the trinification gauge group SU(3)C × SU(3)L × SU(3)R. We show that NCQFT model for strong and electroweak interactions and their unification can be consistently constructed. This model therefore provides an example which allows a consistent and systematic study of noncommutative space-time phenomenology. 2 With noncommutative space-time there are modifications to the fields compared with the ordinary ones. We indicate the fields in NCQFT with a hat and the ordinary ones without hat. The definition of gauge transformation α̂ of a gauge field Âμ for a SU(N) is similar to the ordinary one but with usual product replaced by the “∗” product. For example δαφ̂ = iα̂ ∗ φ̂, (3) where φ̂ is a fundamental representation of SU(N). We use the notation Âμ =  a μT , α̂ = αT a with T a being the SU(N) generator normalized as Tr(T T ) = δ/2. Due to the noncommutativity of the space-time, two consecutive local transformations α̂ and β̂ of the type in the above, (δαδβ − δβδα) = (α̂ ∗ β̂ − β̂ ∗ α̂), (4) cannot be reduced to the matrix commutator of the generators of the Lie algebra due to the noncommutativity of the space-time. They have to be in the enveloping algebra α̂ = α + α ab : T T b : +...+ α a1...an : T a1 ...T an : +... (5) where : T a1 ...T an : is totally symmetric under exchange of ai. This poses a difficulty in constructing non-abelian SU(N) gauge theories [3]. Seiberg and Witten have shown [5] that the fields defined in noncommutative coordinate can be mapped on to the ordinary fields, the Seiberg-Witten mapping. In Ref. [6] it was shown that this mapping actually can be applied to the “∗” product with any gauge groups. It is possible to study non-abelian gauge group theories. Using the above enveloping algebra, one can obtain the noncommutative fields in terms of the ordinary fields with corrections in powers of the noncommutative parameter, θ , order by order. To the first order in θ noncommutative fields can be expressed as α̂ = α+ 1 4 θ{∂μα,Aν}+ cθ μν [∂μα,Aν ], Aμ = − 1 4 θ{Aα, ∂βAμ + Fβμ}+ cθ ([Aα, ∂μAβ] + i[AαAβ , Aμ]), φ̂ = aθFμνφ− 1 2 θAμ∂νφ+ i( 1 4 + c)θAμAνφ, (6) 3 where Fμν = ∂μAν − ∂νAμ − igN [Aμ, Aν ]. The term proportional to a can be absorbed into the redefinition of the matter field φ. The parameter c can not be removed by redefinition of the gauge field. It must be a purely imaginary number from the requirement that the gauge field be self conjugate. Using the above noncommutative fields, one can construct a gauge theory for SU(N) group. The action S of a SU(N) NCQFT, to the leading order in θ, is given by [6,7], S = ∫ Ldx, L = − 1 2 Tr(FμνF ) + 1 4 gNθ Tr(FμνFρσF ρσ − 4FμρFνσF ) + φ̄(iγDμ −m)φ− 1 4 θφ̄Fαβ(iγ Dμ −m)φ− 1 2 θφ̄γFμαiDβφ, (7) where Dμ = ∂μ−igNT Aμ. We note that the parameter c does not appear in the Lagrangian. The Lagrangian is uniquely determined to order θ. We will therefore work with a simple choice c = 0 from now on. In the above, if φ is a chiral field, m = 0. To obtain a theory which can describe the strong and electroweak interactions such as the Standard Model (SM), one also needs to solve the U(1) charge quantization problem, namely only three possible values, 1, 0, -1 for U(1) charges, as mentioned earlier. It has been shown that this difficulty can also be overcome with the use of the Seiberg-Witten mapping [5] . To solve the U(1) charge quantization problem, one associates each charge gq of the nth matter field a gauge field  μ [8]. In the commutative limit, θ μν → 0,  μ becomes the single gauge field Aμ of the ordinary commuting space-time U(1) gauge theory. But at nonzero orders in θ ,  μ receives corrections [8]. In doing so, the kinetic energy of the gauge boson will, however, be affected. Depending on how the kinetic energy is defined (weight over different field strength of  μ ), the resulting kinetic energy will be different even though the proper normalization to obtain the correct kinetic energy in the commutative limit is imposed [8]. In the SM there are six different matter field multiplets for each generation, i.e. uR, dR, (u, d)L, eR, (ν, e)L and (H , H), a priori one can choose a different gi for each of them. 4 After identifying three combinations with the usual g3, g2 and g1 couplings for the SM gauge groups, there is still a freedom to choose different gauge boson self interaction couplings at non-zero orders in θ . This leads to ambiguities in self interactions of gauge bosons when non-zero order terms in θ are included [8]. This problem needs to be resolved. A way to solve this problem is to have a theory without the use of U(1) factor group. There are many groups without U(1) factor group which contain the SM gauge group and may be used to describe the strong and electroweak interactions. However not all of them can be easily extended to a full NCQFT using the formulation described above. For example one can easily obtain unique gauge boson self interactions in SU(5) theory [9]. But the matter fields require more than one representations 5 and 10 which causes additional complications [10] and will reintroduce the uniqueness problem for kinetic energy. One way of obtaining a consistent NCQFT is to have a theory in which all matter and Higgs fields are in the same representation such that once Seiberg-Witten mapping is used to solve the problem of gauging a SU(N) group, there is no problem with unique determination of kinetic energy. To this end we propose to use the trinification gauge group [15], SU(3)C×SU(3)L×SU(3)R with a Z3 symmetry. This theory leads to unification of strong and electroweak interactions. An important feature of this theory is that the matter and Higgs fields are assigned to (anti-)fundamental representations of the factor SU(3) groups and therefore the formalism described earlier can be readily used. In the trinification model, the gauge fields are in the adjoint representation, 24 = A + A + A = (8, 1, 1) + (1, 8, 1) + (1, 1, 8), (8) which contains 24 gauge bosons. A contains the color gluon bosons, a linear combination of component fields of A and A forms the U(1)Y gauge boson, and A L contains the SU(2)L gauge bosons. The rest are integrally charged heavy gauge bosons which do not mediate proton decays [15]. Each generation of fermions is assigned to a 27, 5 ψ = ψ + ψ + ψ = (1, 3, 3̄) + (3̄, 1, 3) + (3, 3̄, 1), ψ = 