The assumption that the Weinberg rotation between the gauge fields associated
with the third component of the “weak isospin” (T3) and the hypercharge (Y ) proceeds in a natural way from a global homomorphism of the SU(2)
U(1) gauge group in some locally isomorphic
group (which proves to be U(2)), imposes strong restrictions so as to fix the single value sin2  W = 1/2. This result can be thought of only as being an asymptotic limit corresponding to an earlier stage of the Universe. It also lends support to the idea that e2/g2 and 1−M2W /M2Z are in principle unrelated quantities.V.A. is grateful to M. Asorey, J. Julve and A. Tiemblo, and all of us to J.M. Cerveró, J. Navarro-Salas, M. Navarro and A. Romero for valuable discussions

Aldaya Valverde, Víctor

Calixto Molina, Manuel

Navarro, Miguel

International Journal of Theoretical Physics

English

Repositorio Digital de la Universidad Politécnica de Cartagena

GROUP-THEORETICAL DETERMINATION OFTHE MIXING ANGLE IN THE ELECTROWEAKGAUGE GROUP∗V. Aldaya1,2 M. Calixto1 J. Guerrero1,3Abstract. The assumption that the Weinberg rotation between the gauge fields associatedwith the third component of the “weak isospin” (T3) and the hypercharge (Y ) proceeds in anatural way from a global homomorphism of the SU(2)⊗U(1) gauge group in some locally iso-morphic group (which proves to be U(2)), imposes strong restrictions so as to fix the single valuesin2 θW = 1/2. This result can be thought of only as being an asymptotic limit correspondingto an earlier stage of the Universe. It also lends support to the idea that e2/g2 and 1−M2W /M2Zare in principle unrelated quantities.There are two basic ingredients in the constitution of a model to describe the unifiedelectroweak interactions, the Weinberg-Salam-Glashow Standard Model, which deserve1Instituto de F́ısica Teórica y Computacional Carlos I, Facultad de Ciencias, Universidad de Granada,Campus de Fuentenueva, Granada 18002, Spain2IFIC, Centro Mixto Universidad de Valencia-CSIC, Burjasot 46100-Valencia,Spain.3Departamento de F́ısica Teórica y del Cosmos, Facultad de Ciencias, Universidad de Granada, Cam-pus de Fuentenueva, Granada 18002, Spain∗Work partially supported by the DGICYT.0further study and which lessen the (mathematical) beauty of the theory as a whole. Oneis the way in which the W-Z-bosons acquire mass, the Higgs mechanism, and the otheris the rotation between the gauge fields associated with the third component of weakisospin (A3µ) and the hypercharge (A4µ), intended to define the proper electromagneticfield, without any (apparent) connection to the “weak” Gell’man-Nishijima relationQ = T3 +12Y , (1)meant to define a proper electric charge in the Lie algebra. We shall focus on the latterquestion.In this paper we wish to explore the restrictions that appear on the mixing angle θWas a consequence of the natural consistence requirement that the rotation in the gaugefieldsZ0µ = cos θW A3µ − sin θW A4µ (2)Aµ = sin θW A3µ + cos θW A4µcomes from an exponentiable (linear) transformation on the Lie algebra of SU(2)⊗U(1).Since the gauge group is not simply connected, it is not true that any automorphism ofthe Lie algebra can be realized as the derivative of a global group homomorphism or, inother words, a differentiable mapping between two locally isomorphic groups providinga given automorphism of the (commom) Lie algebra, can in general destroy the globalgroup law.To analyse the set of global homomorphism from SU(2)⊗U(1) to a locally isomorphicgroup we can proceed in two different ways: either we study the set of discrete normalsubgroups of SU(2) ⊗ U(1), which are the possible kernels of those homomorphisms, orwe write the explicit group law of SU(2) ⊗ U(1), perform an arbitrary homomorphism1and analyze the conditions under which the group law is not destroyed. We shall followthe second approach although some comments on the first one will be added at the end.Let us parametrize the group SU(2) in a co-ordinate system adapted to the Hopffibration SU(2) → S2, the sphere S2 being parametrized by stereographic projection.The SU(2) ⊗ U(1) group law in the local chart at the identity, which nevertheless keepsthe global character of the toral subgroup, is:η′′ =z′′1|z′′1 |=η′η − η′η∗C ′C∗√(1 − η∗2C ′C∗)(1 − η2CC∗′)C ′′ =z′′2z′′1=Cη2 + C ′η2 − C ′C∗ (3)C∗′′ =z′′2∗z′′1∗ =C∗η−2 + C∗′η−2 − C∗′Cζ ′′ = ζ ′ζwhere η ∈ U(1) ⊂ SU(2), ζ ∈ U(1), C ∈ C and z1, z2 characterize a SU(2) matrixz1 z2−z2∗ z1∗. The commutation relations between the (right) generators T+ ≡ XC∗ , T− ≡XC , T3 ≡ Xη and Y ≡ Xζ are:[T3, T±] = ±2T±[T+, T−] = T3 (4)[Y, all] = 0We shall consider transformations induced by an homomorphism of the torus:η̃ = ηpζp′2ζ̃ = ηqζq′(5)C̃ = C , C̃∗ = C∗,where the parameters p, p′, q, q′ have to be integers for the univalueness requirement.After we apply this transformation the group law becomes:η̃′′ =η̃′1p η̃1p − η̃′ 1p η̃−d+2qp′dp ζ̃2p′d C̃∗C̃ ′√(1 − η̃− 2q′d ζ̃2p′d C̃ ′C̃∗)(1 − η̃ 2q′d ζ̃−2p′d C̃C̃∗′)pC̃ ′′ =C̃η̃2q′d ζ̃−2p′d + C̃ ′η̃2q′d ζ̃−2p′d − C̃ ′C̃∗(6)C̃∗′′=C̃∗η̃−2q′d ζ̃2p′d + C̃∗′η̃−2q′d ζ̃2p′d − C̃∗′C̃ζ̃ ′′ = ζ̃ ′ζ̃(η̃′′η̃′−1η̃−1)qp ,where d is the determinant of the matrixp p′q q′, and this group law is well-behaved if2p′d= m,2q′d= n,qp= k, m, n, k ∈ Z (7)which, in particular, imply p = ±1,±2. This particular result simply states the well-known fact that the only invariant subgroups of SU(2) itself are I (the identity) and Z2,respectively.The commutation relations between the new generators (with a definition analogousto that given above),[T̃3, T̃±]= ±2q′dT̃±[Ỹ , T̃±]= ±−2p′dT̃± (8)[T̃+, T̃−]= pT̃3 + qỸ ,3can be obtained directly from (7) or by applying the tangent mapping to (5) to the oldones. This transformation gives:T̃3 =q′dT3 −qdYỸ =−p′dT3 +pdY (9)and provides a generalized Gell’Mann-Nishijima relation and its counterpart, which nowappear quantized.Let us now examine the transformation induced by (5) in the (3rd − 4th internalcomponents of the) gauge fields. It is given by:Ã3µÃ4µ =1r̃00 1r̃′p p′q q′r 00 r′A3µA4µ (10)where r, r′ are the original coupling constants associated with isospin and hyperchargerespectively, and r̃, r̃′ are the final ones. In fact, the covariant derivative Dµ = ∂µ −igki TkAiµ, where i, k run over 1,2,3,4 (T4 ≡ Y ) goes to D̃µ = ∂µ−ig̃ki T̃kÃiµ = Dµ. Therefore,Ãlµ = (g̃−1)lj ajk gki Aiµ (11)where ajk is the transformation matrix changing co-ordinates in the Lie algebra, whichcontains the central matrix in (10) as a box, and g = diag(r, r, r, r′) and g̃ = diag(r̃, r̃, r̃, r̃′)are the initial and final (bare) coupling constants matrices.We now impose the requirement that the complete transformation (10), rather thanthe central matrix in it, be the Weinberg rotation (3) (Z0µ ≡ Ã3µ, Aµ ≡ Ã4µ). This resultsinr̃2r̃′2= −pp′qq′,r2r′2= −q′p′qp, tan2 θW = −qp′pq′, r̃ =pcos θWr , (12)4which contain a further restriction: the product of the four integers pp′qq′ < 0, a conditionafterwards necessary to have a (non-trivial) rotation. If the transformation (5) is anautomorphism of the torus (d = ±1), then the only possible rotations between the gaugefields are the trivial ones (tan2 θW = 0,∞), so that the final group has to be the quotientof SU(2)⊗U(1) by a non-trivial normal (discrete) subgroup. Adding (12) to (7) we arriveat the final result:{p = ±1 and (p′ = −kq′, k = ±1)} ⇒{tan2 θW = 1 , d = ±2q′}(13)For these values of p, p′, q, q′ the kernel of the homomorphism (see the transformation (5))is the normal subgroupHd ≡{(C, C∗, η; ζ) = (0, 0, 1; ei2sd2π), (0, 0,−1; ei 2s+1d 2π)/s = 0, 1, ..., |d|2− 1}(14)which is isomorphic, as a group, to Z|d|. All these homomorphisms lead to the same valuefor tan2 θW (= 1) and indeed, all can be written as:p p′q q′ =±1 −kq′±k q′ =1 −kk 1±1 00 q′ (15)where the second factor has determinant ±q′, and represents a transformation fromSU(2) ⊗ U(1) to SU(2) ⊗ (U(1)/Z|q′|), and the first one has determinant 2 and wouldtake SU(2) ⊗U(1) to (SU(2)⊗ U(1))/H2 ≈ U(2) by itself. The second factor affects thequotient between the original coupling constants (not the final one), as can be seen in(12), and the generalized Gell’Man-Nishijima relation (9). Among the possible values forq′ only q′ = ±1 provides us with a proper electric charge; the choice of the signs of p, q, p′is a matter of convention and will define either T̃3 or Ỹ as ± the electric charge Q. Thecorresponding homomorphism has Kernel H2 = {(0, 0, 1; 1), (0, 0,−1;−1)} and U(2) asthe image group (true gauge group) [2, 3, 4].5With the usual choice of multiplets in the Lagrangian of the Standard Model (see e.g.[5]) T3 and Y have the expressionsT3 =1 0 00 −1 00 0 0, Y =1 0 00 1 00 0 2(16)which agrees with the usual expressions if the U(1) subgroups are trivially reparametrizedby α = −2i ln η, β = i ln ζ (T3 → 12T3, Y → −Y ). The particular choice of signs p = p′ =q′ = −q = −1 yields:Q = Ỹ =0 0 00 −1 00 0 −1, T̃3 =−1 0 00 0 00 0 −1(17)The first surprising result is the fact that only one value of tan2 θW is allowed, whichmeans only one coupling constant (the electric charge, essentially, i.e. e ≡ r̃′ =√2r ≡g/√2), even though the gauge group (U(2)) is not a simple group. According to gen-eral settings [6], however, the theory must contain a coupling constant for each simpleor abelian term in the Lie algebra decomposition. An immediate conclusion is that theassignment of constants should be done according to factors in the direct product decom-position of the group, rather than the algebra.The second result is the particular structure of the neutral weak current derived fromthe expression of T̃3 above, according to which the gauge field Z0 interacts with the (left-handed) neutrino and the right-handed electron only; i.e. the neutral weak current is pureV-A for the neutrino and pure V+A for the electron.Last, but not least, is the striking value of sin2 θW =12(tan2 θW = 1), far from theexperimental value ≈ 0.23 [7]. In the light of this result, only the hope remains thatour theoretical value of θW really corresponds to that state of the Universe in which the6electroweak interaction was not yet spontaneously broken, and that the process of spon-taneous symmetry breaking, not fully understood (at least from a pure group-theoreticalpoint of view) could relax the strong conditions (7). For instance, breaking down theSU(2) group law (7) and preserving the Uem(1), leads to tan2 θW = − qq′ , allowing anyrational value.In any case, the discrepancy between our theoretical value for sin2θW as given by aratio of coupling constants, e2g2= r̃′2(2r)2, and the experimental one obtained through theexpression 1 − M2WM2Z, lends support to the idea that in principle both quantities are notrelated [8].Acknowledgements V.A. is grateful to M. Asorey, J. Julve and A. Tiemblo, andall of us to J.M. Cerveró, J. Navarro-Salas, M. Navarro and A. Romero for valuablediscussions.7References[1] S. Weinberg, Phys. Rev. 19, 1264 (1967)[2] C.J. Isham, J. Phys. A 14, 2943 (1981)[3] L. O’Raifeartaigh, Group structure of gauge theories, Cambridge University Press(1986)[4] J. LaChapelle, J. Math. Phys. 35, 2186 (1994)[5] L.H. Ryder, Quantum Field Theory, Cambridge University Press (1985)[6] C. Itzykson and J.B. Zuber, Quantum Field Theory, MacGraw-Hill International Edi-tion (1985)[7] M. Aguilar-Benitez et al., Phys. Rev. D45, Part 2 (Review of Particle Properties)(1992)[8] G. Passarino and M. Veltman, Phys. Let. B237, 537 (1990)8

Group-Theoretical Determination of the Mixing Angle in the Electroweak Gauge Group

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Group-Theoretical Determination of the Mixing Angle in the Electroweak Gauge Group

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