Dirac s Observables for the SU(3)xSU(2)xU(1) Standard Model. Abstract

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1 Dirac s Observables for the SU(3xSU(xU( Standard Model. Luca Lusanna Sezione INFN di Firenze L.go E.Fermi (Arcetri 505 Firenze, Italy LUSANNA@FI.INFN.IT and Paolo Valtancoli Dipartimento di Fisica Universita di Firenze L.go E.Fermi (Arcetri 505 Firenze, Italy VALTANCOLI@FI.INFN.IT Abstract The complete, missing, Hamiltonian treatment of the standard SU(3xSU(xU( model with Grassmann-valued fermion fields in the Higgs phase is given. We bypass the complications of the Hamiltonian theory in the Higgs phase, resulting from the spontaneous symmetry breaking with the Higgs mechanism, by studying the Hamiltonian formulation of the Higgs phase for the gauge equivalent Lagrangian in the unitary gauge. A canonical basis of Dirac s observables is found and the reduced physical Hamiltonian is evaluated. Its self-energy part is nonlocal for the electromagnetic and strong

2 interactions, but local for the weak ones. Therefore, the Fermi 4-fermion interaction reappears at the nonperturbative level. June 996 This work has been partially supported by the network Constrained Dynamical Systems of the E.U. Programme Human Capital and Mobility. Typeset using REVTEX

3 I. INTRODUCTION In two previous papers [,] (referred to as I and II we made the complete canonical reduction of the Higgs model with fermions and with spontaneous symmetry breaking in the Abelian (I and non-abelian SU( (II cases. In both cases there is an ambiguity in solving the Gauss law first class constraints, which reflects the existence of disjoint sectors of solutions of the Euler-Lagrange equations. While in the Abelian case there are two sectors of solutions, the electromagnetic and the Higgs phases, in the non-abelian SU( case the sectors correspond to six phases, one of which is the Higgs phase and one to the SU(- symmetric phase [the remaining four phases have partially broken SU( symmetry and are not SU( covariant]. The Dirac observables and the physical Hamiltonians and Lagrangians of the Higgs phase have been found in both cases. In the Hamiltonian, the self-energy term turns out to be local, but not polynomial, and contains a local four-fermion interaction. Therefore, the nonrenormalizability of the unitary gauge (our method of canonical reduction is similar to it, but without the introduction of gauge-fixings is confirmed. In this paper we will give a complete Hamiltonian formulation of the Higgs sector of the standard SU(3xSU(xU( model of elementary particles with Grassmann-valued fermion fields together with its canonical reduction. Namely, using the results of Refs. [,] and those of Ref. [3], we will find a complete set of canonical Dirac s observables and the reduced physical Hamiltonian. This will be done in the case of a trivial principal SU(3xSU(xU(- bundle (so that there are no monopole solutions over a fixed x o, R 3 sliceofa3decomposition of Minkowski spacetime, without never going to Euclidean space. Since the reduction is non covariant, the next step will be to covariantize the results by reformulating the theory on spacelike hypersurfaces foliating Minkowski spacetime and, then, by restricting the description to the Wigner hyperplanes orthogonal to the total 4-momentum of the system (assumed timelike. In this way the standard model will be described in the covariant relativistic rest-frame instant form of the dynamics, which was defined in Ref. [4,5] for the system of N charged scalar particles (with Grassmann-valued electric charges plus 3

4 the electromagnetic field [for this system one found the Dirac s observables, the physical Hamiltonian with the Coulomb potential extracted from field theory (with the Coulomb self-energies regularized by the property Q i = 0 of the Grassmann-valued electric charges, the second order equations of motion for the field and the particles and the Lienard-Wiechert potentials]. In this form of the dynamics there is a universal breaking of Lorentz covariance connected with the description of the center of mass of the isolated system, but all the other variables have Wigner covariance. This implies that the relative dynamics with respect to the center of mass on the Wigner hyperplane is naturally Euclidean : under a Lorentz transformation the hyperplane is rotated in Minkowski spacetime (and the canonical center of mass transforms noncovariantly like the Newton-Wigner position operator, i.e. it has only the rotational covariance implied by the little group of massive Poincaré representations, but the relative Wigner-covariant variables inside it only feel induced Wigner SO(3 rotations. The Wigner hyperplane seems to be the natural candidate to solve the Lorentz covariance problem of lattice gauge theory. It is also possible to formulate covariant -time relativistic statistical mechanics on this hyperplane [4]. Moreover, the noncovariance of the center of mass identifies a classical unit of lenght (the Møller radius ρ = W /cp = S /c P to be used as a ultraviolet cutoff in a future attempt to quantize these nonlocal and nonpolynomial reduced field theories. In Ref. [6] the results of Ref. [4] were extended to N scalar particles with Grassmann-valued color charges plus the SU(3 color Yang-Mills field (pseudoclassical relativistic scalar-quark model. The Dirac observables, the physical Hamiltonian with the interquark potential and the second order equations of motion for both the field and the particles have been found. In the N= (meson case, a form of the requirement of having only color singlets, suitable for a field-independent quark model, implies a pseudoclassical asymptotic freedom and a regularization of the quark self-energies. To reformulate the standard model in this way, one needs the completion of the description of Dirac and chiral fields and of spinning particles on spacelike hypersurfaces [7] by adapting the method of Refs. [8] for the canonical description of fermion fields in curved spacetimes to spacelike hypersurfaces in Minkowski spacetime. 4

5 See Refs. [9] for a review of the full research program and of its achievements till now. To apply the results of Ref. [3], we must assume that the SU(3 gauge potentials and gauge transformations belong to a suitable weigthed Sobolev space [0,], so that any form of Gribov ambiguity is absent. Instead, it is not necessary that the SU(xU( gauge potentials and gauge transformations belong to the same special spaces, because the Hamiltonian reduction associated with the Gauss laws is purely algebraic and does not require to solve elliptic equations as in the case of the Gauss laws of SU(3. However, if one wishes to have homogeneous Hamiltonian boundary conditions for the various fields [and also to have the possibility to try to make the reduction also of the other non-higgs phases], one has to work in those special spaces for the whole SU(3xSU(xU(. In Section II a review of the standard SU(3xSU(xU( model is given to fix the notations. In Section III we give the Lagrangian density in the unitary gauge and we introduce the mass eigenstates for the fermions. In Section IV we give the Euler-Lagrange equations, the Hamiltonian and the primary and secondary constraints. Also the energy-momentum tensor and the Hamiltonian boundary conditions for the standard model are given. At the end of the Section we show that, if we try to reformulate the Hamiltonian theory in terms of the vector boson fields rather than in terms of the original gauge fields, the constraints change nature and the theory becomes extremely complicated. Therefore, in Section V we study the Hamiltonian formulation of the Lagrangian in the unitary gauge. Now we get only primary and secondary constraints, with those referring to the vector boson being of second class. In Section VI we find the electromagnetic and color Dirac observables. In Section VII a canonical basis of Dirac observables of the standard model is found and the physical reduced Hamiltonian is given. Its self-energy part is nonlocal for the electromagnetic and strong interactions, but local for the weak ones. Therefore, the Fermi 4-fermions interaction reappears at the nonperturbative level. 5

6 Finally, in Section VIII we evaluate the physical Hamilton equations. In the Conclusions some remarks are made. II. THE LAGRANGIAN OF THE STANDARD MODEL In this Section we shall make a brief review of the standard SU(3xSU(xU( model to fix the notations. The standard model is described by the following Lagrangian density [] [see also Refs. [3 6]] L(x = G µν 4gs A (xg Aµν(x W µν 4gw a (xw aµν(x V µν (xv 4gy µν (x [D µ (W,V φ(x] D (W,V µ φ(x λ(φ (xφ(x φ o (l Li (xiγ µ ( µ W aµ (xt a w V µ (xy w ψ (l Li (x (l Ri(xiγ µ ( µ V µ (xy w ψ Ri(x (l (l Li (x φ(x M (l ij ψ Rj(x (l (l φ o Ri(xM (l ij φ (x φ o ψ (l Lj(x Li (xiγµ ( µ W aµ (xt a w V µ(xy w G Aµ (xt A s ψ Li (x Ri (xiγµ ( µ V µ (xy w G Aµ (xts A ψ Ri (x ψ Ri (xiγµ ( µ V µ (xy w G Aµ (xts A ψ Ri (x Li (x φ(x φ o φ(x Li (x φ o M ij ψ Rj (x M ij ψ Ri (xm ij φ (x φ o (x ψ φ Rj Ri (x M (x ij θ g s 3π G Aµν(x G µν A (x = =... G Aµ (xj sa(xv µ µ J µ Y w (xw aµ (xj wa(x, µ φ o ψ Lj (x ψ Lj (x J µ sa(x = J µ Y w (x = Li (xγ µ it A s ψ Li (x (l Li (xγ µ iy w ψ (l Li(x Ri (xγ µ it A s ψ Ri (x ψ Ri (xγ µ it A s (l Ri(xγ µ iy w ψ (l Ri(x Ri (xγ µ iy w ψ Ri (x ψ Ri (xγµ iy w ψ Ri (x, ψ Ri (x, Li (xγ µ iy w ψ Li (x J wa µ (l (x = Li (xγµ itw a ψ(l Li (x Li (xγµ itw a ψ Li (x. ( 6

7 The fields G Aµ (x =g s GAµ (x [A=,..,8], W aµ (x =g w Waµ (x [a=,,3] and V µ (x = g y Ṽ µ (x are the SU(3, SU( and U( gauge potentials respectively; here g s,g w and g y are the associated strong (color, weak isospin and weak hypercharge coupling constants. A The generators of the Lie algebras su(3 of color [Ĝµ = G Aµ ˆT s ] and su( of weak isospin a [Ŵµ = W aµ ˆT w ] in the adjoint representation [8-dimensional for SU(3 and 3-dimensional for SU(] and of the Lie algebra u( of weak hypercharge are [c ABC are the SU(3 totally antisymmetric structure constants] ˆT A s = ˆT A s, ( ˆT A s BC = c ABC, [ ˆT A s, ˆT B s ]=c ABC ˆT C s, ˆT a w = ˆT a w, ( ˆT a w bc = ɛ abc, [ ˆT a w, ˆT b w ]=ɛ abc ˆT c w, Y w = i y = iy. ( The field strengths and the covariant derivatives associated with G Aµ,W aµ,v µ are G Aµν (x = µ G Aν (x ν G Aµ (xc ABC G Bµ (xg Cν (x, W aµν (x = µ W aν (x ν W aµ (xɛ abc W bµ (xw cν (x, V µν (x = µ V ν (x ν V µ (x, (ˆD (G µ AC = δ AC µ c ABC G Bµ =( µ Ĝµ AC, ( ˆD (W µ ac = δ ac µ ɛ abc W bµ =( µ Ŵµ ac, ˆD (V µ = µ V µ Y w, (3 and the gauge transformations are defined as [Ĝµν = G Aµν ˆT A s, Ŵ µν = W aµν ˆT a w ] Ĝ µ (x ĜU µ (x =U s (xĝµ(xu s (xus (x µ U s (x= =Ĝµ(xUs (x( µ U s (x[ĝµ(x,u s (x], Ŵ µ (x Ŵ U µ (x =U w (xŵµ(xu w (xuw (x µ U w (x= =Ŵµ(xUw (x( µ U w (x[ŵµ(x,u w (x], V µ (xy w V U µ (xy w = V µ (xy w U y (x µ U y (x =[V µ (x µ Λ y (x]y w, 7

8 Ĝ µν (x ĜU µν Ŵ µν (x Ŵ U µν (x =U s (x =U w (xĝµν(xu s (x =Ĝµν(xUs (x[ĝµν(x,u s (x], (xŵµν(xu w (x =Ŵµν(xUw (x[ŵµν(x,u w (x], V µν (x V U µν(x =V µν (x. (4 Here U s,u w,u y =e ΛyYw are the realizations in the adjoint representation of the SU(3, SU( and U( gauge transformations respectively. The last term in Eq.( is the topological θ-term [it is the source of strong CP-violation, whose experimental absence requires θ 0 0 ]; in it G µν A = ɛµναβ G Aαβ is the dual field strength ( is the Hodge star operator and one has G Aµν G µν A = E 4 A B A = µ ɛ µναβ (G Aν G Aαβ 6 c ABCG Aν G Bα G Cβ. The field φ(x is a complex Higgs field in the fundamental doublet representation of the weak isospin SU( φ (x φ(x = φ 0 (x =eθa(xta w 0 φ o H(x φ(x=φ c (x=it w φ (x =iτ φ (x = = e θa(xta w φ 0(x φ (x= φ (x 0 vh(x,, (5 where the lower subscript of the components denotes the electric charge. The field φ c is the charge conjugate of the Higgs field. The su( generators in the doublet representation are [τ a are the Pauli matrices] T a w = iτa, [T a w,t b w]=ɛ abc T c w, T a wt b w T b wt a w = δab. (6 If Ũw is the realization of the SU( gauge transformations in the doublet representation and W µ = W aµ T a w, W µν = W aµν T a w, then in analogy with Eqs.(4 one has W µ W U µ = Ũ w W µ Ũ w Ũ w µ Ũ w, W µν Wµν U = Ũ w W µν Ũ w. The constant φ o = φ o appearing in the Higgs potential V (φ = λ(φ φ φ o is real; the three phases θ a (x parametrize the absolute minima φ φ = φ o of V (φ. At the quantum 8

9 level <φ>=φ o 0 is the gauge not-invariant formulation of symmetry breaking. The covariant derivative of the Higgs field and their gauge transformations are D (W,V µab φ b (x=[δ ab ( µ V µ (xy w (W cµ (xtw c ab]φ b (x, φ(x φ U (x =Ũ w (xu y (xφ(x. (7 The relation between the fields V µ, W 3µ, and the electromagnetic, A µ, and neutral vector boson, Z µ,fieldsis V µ =g y Ṽ µ =g y [ sin θ w Zµ cos θ w à µ ]=A µ tg θ w Z µ, W 3µ = g w W3µ = g w [cos θ w Zµ sin θ w à µ ]=A µ cot θ w Z µ, A µ = eãµ = e[cos θ w Ṽ µ sin θ w W3µ ]= [g gwg y w V µgy W 3µ], Z µ =e Z µ =e[ sin θ w Ṽ µ cos θ w W3µ ]= g wg y (W gw 3µ V µ. (8 g y Here θ w is the Weinberg angle and e is the unit of electric charge; their relation to the original coupling constants g w,g y is tg θ w = g y, e = g wg y, g w gw gy g y = The charged vector boson fields are W ±µ = (W µ iw µ, T ± w = (T w it w, W µ = (W µ W µ, W µ = i (W µ W µ, e, g w = e. (9 cos θ w sin θ w W aµ T a w V µ Y w = W µ T w W µ T w A µ (T 3 w Y w Z µ (cot θ w T 3 w tg θ w Y w = =W µ Tw W µ T w iq em A µ iq Z Z µ = = i ( ya µ (cot θ w ytg θ w Z µ W µ W µ ( ya µ (cot θ w ytg θ w Z µ 9. (0

10 The last line of this equation defines the electric Q em = i(tw 3 Y w= y 0 0 y and neutral Q Z = i(cot θ w Tw 3 tg θ wy w = cot θ w ytg θ w 0 charge 0 cot θ w ytg θ w generators for the doublet SU( representation [in the singlet SU( representation one has Q em = iy w = y = Y and Q Z = itg θ w Y w = tg θ wy and V µ Y w = iq em A µ iq Z Z µ = i y(a µ tg θ w Z µ ] Q em = i(t 3 w Y w= (τ3 y= τ3 Y, Q Z = i(cot θ w T 3 w tg θ wy w = (cot θ wτ 3 tg θ w y, it 3 w = sin θ w(sin θ w Q em cos θ w Q Z, iy w = cos θ w (cos θ w Q em sin θ w Q Z. ( φ For the Higgs field φ = one has the assignements Y = [y=] and Q φ em = φ o φ o φ, Q φ Z = ( sin θ w φ sin θ w. 0 φ o φ o The Grassmann-valued fermion fields ψ.iα(x,ψ (l.iα(x represent leptons and quarks respectively; α is a spinor index, while the index i=,,3, denotes the families. The fields ψ (. Liaα(x, ψ (. Riα(x denoteleftandrightfields[ψ L = ( γ 5ψ,ψ R = ( γ 5ψ, L = ( γ 5, R = ( γ 5, ψ = L ψ R R ψ L, γ µ ψ = L γ µ ψ L R γ µ ψ R ]: the left fields belong to the doublet representation of the weak isospin SU(, while the right ones are SU( singlets. The quark fields ψ LiAaα (x,ψ RiAα (x also belong to the fundamental triplet representation of the color SU(3, whose generators are [λ A are the 3 3 Gell-Mann matrices; d ABC are totally symmetric coefficients] T A s = i λa, Trλ A λ B =δ AB, c ABC T B s T C s = 3 T A s, [T A s,tb s ]=c ABCT C s, TA s TB s TB s TA s = 3 δ AB id ABC T C s, 0

11 (Ts A ab (Ts A cd = A 6 δ abδ cd δ adδ bc (a, b, c, d =,,3. ( The SU(3 Casimirs for a representation R, namely C (R = A[Ts(R] A and C 3 (R = ABC Ts A(RT s B(RT s C(R, have the values C (3 = 4, C 3 3(3 = 0 for the triplet (R=3 and 9 C (8 = 3, C 3 (8 = 0, for the adjoint (R=8 representations respectively. The covariant derivatives and gauge transformations [Ũw, Ũs are their realizations in the SU( doublet and SU(3 triplet representations respectively] of the fermion fields are [G µ = G Aµ T A s,w µ =W aµ T a w ] D (W,V µab ψ (l Lib =[δ ab ( µ V µ Y w (W µ ab ] ψ Lib, (l D (V µ ψ (l Ri =( µ V µ Y w ψ (l Ri, D (W,V,G µabab ψ LiBb =[δ abδ AB ( µ V µ Y w δ AB (W µ ab δ ab (G µ AB ]ψ LiBb, D (V,G µabψ RiB =[δ AB ( µ V µ Y w (G µ AB ]ψ RiB same for ψ RiA, ψ (l Li ψ(lu Li = Ũ w U y ψ (l Li, ψ (l Ri ψ (lu Ri = Uy ψ Ri, (l ψ Li ψ U Li = Ũ w U y Ũs ψ Li, ψ Ri ψ U Ri = U y Ũs ψ Ri, same for ψ Ri. (3 The known leptons [electron, muon, tau and the associated massless left neutrinos (right ones are absent] and quarks [six flavours: up, down, charme, strange, top, bottom, each one with three color components] are described by the following fermion fields (weak interaction or gauge eigenstates ψ L(x (l = ψ (l ν el (x e L (x, ψ (l ψ(l L(x= ν µl (x µ L (x ψ (l, R(x=e R (x R(x=µ R (x R3(x=τ R (x, ψ L(x= u L (x, c L (x ψ L(x=, d L (x s L (x ν τl (x ψ(l L3(x= τ L (x ψ L3(x= t L (x, b L (x

12 ψ R(x =u R (x, ψ R(x=d R (x, ψ R(x=c R (x, ψ R(x=s R (x, ψ R3(x=t R (x, ψ R3(x=b R (x. (4 The charge assignements for quarks are ( ν el ν µl ν τl : Q em =0, Y = ( e L µ L τ L : Q em =, Y = ( e R µ R τ R [y = ], Q Z = sin θ w, [y = ], Q Z = sin θ w sin θ w, : Q em =, Y = [y = ], Q Z = sin θ w sin θ w, ( u L c L t L : Q em = 3, Y = 6 ( d L s L b L : Q em = 3, Y = 6 ( u R c R t R : Q em = 3, Y = 3 ( d R s R b R : Q em = 3, Y = 3 Due to Eq.(0, Eqs.(3, (7 and (3 imply [y = 3 ], Q Z = 4 3 sin θ w sin θ w, [y = 3 ], Q Z = 3 sin θ w sin θ w, [y = 4 3 ], Q Z = 4sin θ w 3sin θ w, [y = 3 ], Q Z = sin θ w 3sin θ w. (5 ( ˆD (W µ ab =[ µ W µˆt w W µˆt w (A µ cot θ w Z µ ˆT 3 w ] ab, ˆD (V µ = D (V µ = µ (A µ tg θ w Z µ Y w, D (W,V µab =[ µ W µ T w W µ T w iq em A µ iq Z Z µ ] ab, D (W,V,G µabab = δ AB D (W,V µab δ ab (G µ AB, D (V,G µab = δ AB D (V µ (G µ AB. (6 The Lagrangian density ( has the following form in terms of the fields A µ = eãµ, Z µ = e Z µ, W ±µ = e W±µ sin θ w [we define the following Abelian field strengths: A µν = µ A ν ν A µ, Z µν = µ Z ν ν Z µ, W ± µν = µ W ± ν ν W ± µ ] ˆL(x = G µν 4gs A (xg Aµν (x 4e Aµν (xa µν (x 4e Zµν (xz µν (x sin θ w W µν e (xw µν (xie[a µν (xcot θ w Z µν (x]w µ (xw ν (x ie[w µν (xw µ (x W µν (xw µ (x][a ν (xcot θ w Z ν (x]

13 sin θ w [W e (xw (x (W (x W (x ] sin θ w [A(xcot θ e w Z(x] W (x W (x sin θ w W e (x [A(xcot θ w Z(x]W (x [A(xcot θ w Z(x] [( µ (W µ (xtw W µ (xt w iq em A µ (x iq Z Z µ (xφ(x] [( µ (W µ (xt w W µ (xt w iq em A µ (x iq Z Z µ (xφ(x] λ(φ (xφ(x φ o (l Li (xiγ µ [ µ (W µ (xt w W µ (xt w iq em A µ (x iq Z Z µ (x]ψ (l Li (x (l Ri(xiγ µ [ µ (A µ (x tg θ w Z µ (xy w ]ψ Ri(x (l (l Li (x φ(x M (l ij ψ (l (l φ Rj(x Ri(xM (l φ (x ij ψ o φ Lj(x (l o Li (xiγµ [ µ (W (xt µ w Wµ (xt w iq em A µ (x iq Z Z µ (xg Aµ (xt A s ]ψ Li (x Ri (xiγ µ [ µ (A µ (x tg θ w Z µ (xy w G Aµ (xts A ]ψ Ri (x ψ Ri (xiγ µ [ µ (A µ (x tg θ w Z µ (xy w G Aµ (xts A ] ψ Ri (x Li (x φ(x φ o Li (x φ(x φ o M ij ψ Rj (x M ij Ri (xm ij ψ Rj (x ψ Ri (x M ij φ (x φ o φ (x φ o ψ Lj (x ψ Lj (x θ g s 3π G Aµν(x G µν A (x = =... G Aµ (xj sa(xa µ µ (x j µ (em (xz µ(x j µ (NC (x [W µ (x j µ (CC (xw µ(x j µ (CC ], J µ sa(x = Li (xγµ its A ψ Li (x Ri (xγµ its A (x ψ ψ Ri Ri (xγµ its A ψ Ri (x, j µ (em (x =Jµ w3(xj µ Y w (x= = (l Li (xγ µ Q em ψ (l Li (x (l Ri(xγ µ iy w ψ (l Ri(x Ri (xγ µ iy w ψ Ri (x ψ Ri (xγ µ iy w ψ Ri (x, Li (xγ µ Q em ψ Li (x 3

14 j µ (NC (x=cot θ w J µ w3(x tg θ w J µ Y w (x = = (l Li (xγ µ Q Z ψ (l Li (x tg θ w (l Ri(xγ µ iy w ψ (l Ri(x tg θ w Ri (xγ µ iy w ψ Ri (x tg θ w ψ Ri (xγ µ iy w ψ Ri (x, j µ (CC (x= [J µ w(x±ij µ w(x] = Li (xγ µ Q Z ψ Li (x = (l Li (xγ µ it w ψ (l Li (x Li (xγ µ it w ψ Li (x, [alternative notation j µ (CH = j µ (CC, j µ (CH = j µ (CC]. (7 As a consequence of the spontaneous symmetry breaking with the Higgs mechanism, one can make a field-dependent SU( gauge transformation U (θ w (x =eθa(xˆt w a,ũ w (θ(x= e θa(xta w, to the (not renormalizable unitary gauge where the original SU(3 SU( U( gauge symmetry is broken to SU(3 U em ( describing the remaining massless color and electromagnetic interactions. The explicit action of this gauge transformation is φ(x φ (x =Ũ(θ w (xφ(x= 0 = φ o H(x vh(x W aµ (xtw a W aµ (xt w a = e θa(xt w a (Wbµ (xtw b µ e θc(xt w c, ψ (l Li (x ψ(l Li ψ Li (x ψ (x =Ũ(θ w (xψ (l Li (x, Li (x =Ũ(θ w 0, (xψ Li (x. (8 III. THE LAGRANGIAN DENSITY IN THE UNITARY GAUGE Since the Lagrangian density ( is invariant under this field-dependent SU( gauge transformation, we can rewrite it in terms of the new fields φ of the not trasformed ones G Aµ, V µ, ψ (l Ri, ψ Ri, ψ Ri (or H, W aµ, ψ(l Li, ψ Li,and. The new Lagrangian density does not depend on the three would-be Goldstone bosons θ a (x, which are absorbed to generate the mass terms for the vector bosons W ±µ, Z µ, but not for the electromagnetic field A µ [Z µ and A µ are obtained from Eq.(8 with W 3µ ]. One gets 4

15 D (W,V µab φ b = i (v HW µ [ µ H i(vh, sin θ w Z µ] λ[φ φ φ o ] = λv H ( H v = m H H ( e sin θ w m Z H, where (v = φ o [D (W,V µ φ ] D (W,V µ φ = = µh µ H 4 (v H [W µw µ = µh µ H 4 g wv [ W µ W µ Z sin θ µz µ ]= w Z cos θ Z µ µ ]( H w v = = µh µ H [m W W µ W µ m Z Z µ Z µ ]( e sin θ w m Z H, (9 m H = v λ =φ o λ, mw = vg w = φ o g w, m Z = m W cos θ w = φ o g w g y, v = φ o = sin θ w m Z e = GF, λ = e m H e m H. =. (0 8sin θ w m W sin θ w m Z where G F is the Fermi constant. Therefore g w,g y,φ o,λare replaced by e, θ w [or G F ], m Z,m H, while m W = m Z cos θ w is a derived quantity (m Z is known with a better accuracy; experimentally one has α =(e /4π = ± [in Heaviside-Lorentz units andwith h=c=, so that e is adimensional], G F = g w 4 m W = φ o = (.6639 ± GeV /(93GeV, sin θ w = 0.3, m Z = (9.884 ± 0.00GeV, m H > 65.GeV (95CL, so that m W =(80.6 ± 0.6GeV [m W = m Z cos θ w only at the tree level; radiative corrections give a six percent contribution], <φ>=φ o = v = 3/4 G F = 46.GeV, and, if ρ = m W /m Zcos θ w =,ρ = ρ=(4.± Let us remark that the range of the electromagnetic force is infinite since the electromagnetic field remains massless; at the quantum level the renormalized electromagnetic coupling constant is α(r =α/( α log( h (m 3π m erc e is the electron mass, so that α(r α if r is much higher of the electron Compton wavelength (r >> h/m e cand α(r if one probes distances r h m e 3π/α ec m. Instead for the strong 5

16 color force, where α s = gs /4π is the coupling constant, the QCD renormalization gives α s (r =α s / αs( N 4π 3 flog hc (N Λ sr f =6andN c = 3 are the number of quark flavours and colors respectively; Λ s Gev is the hadronic color energy scale; forn f =6<33/ the sign in the denominator is opposite to the electromagnetic one, so that for r 0 one has the asymptotic freedom of quarks α s (r 0, while α s (r (breakdown of QCD perturbative expansion for r R s = hc Λ s 0 5 m (the range of strong color interactions signalling the confinement of quarks and gluons. The range of weak interactions is determined by the Compton wavelength of the W vector boson, R w = h.5 m W c 0 8 m.. Forcomparison the distance at which the standard description of the (infinite range gravitational interaction is supposed to break down is the Planck length R p = hgn c 3 where G N is the Newton constant. = cm., After the field-dependent SU( gauge transformation U w (θ (x, the gauge invariant Lagrangian density ( becomes [remember that A µ = eã µ, Z µ = e Z µ, W aµ = e sin θ w W aµ, V µ = e cos θ w Ṽ µ, m W = m Z cos θ w, and that φ = iτ φ = v H ; in the terms 0 W µν 4 a W aµν 4Ṽ µν Ṽ µν Eqs.(8, (0 have been used] L (x = G µν 4gs A (xg Aµν (x 4à µν (xã µν(x 4 Z µν (x Z µν(x W µν (x W µν(x [m W W µ(x W µ (x m Z Z µ(x Z µ e (x]( H(x sin θ w m Z µν ie(ã (xcot θ w Z µν (x W µ(x W ν(x ie[ W µν (x W µ(x W µν (x W µ(x](ã ν(xcot θ w Z ν(x e [ sin θ W (x W (x ( W (x W (x ] w e (õ (xcot θ w Z µ (x(ã µ (xcot θ Z w µ (x W (x W (x e W µ (x(ã µ (xcot θ w Z µ (x W ν (x(ã ν (xcot θ w Z ν (x µh(x µ H(x m HH (x( (l Li (xiγ µ [ µ e sin θ w m Z H(x e sin θ w ( W µ(xt w W µ(xt w 6

17 ieq em à µ (x ieq Z Z µ (x]ψ(l Li (x (l Ri (xiγµ [ µ e(ã µ (x tg θ Z w µ (xy w]ψ (l Ri (x e ( H(x sin θ w m Z (l [ Li (x 0 M (l ij ψ (l Rj(x ( (l Ri(xM (l ij 0 ψ (l Lj (x] Li (xiγ µ [ µ e ( sin θ W µ(xt w W µ(xt w w ieq em à µ (x ieq Z Z µ (xg Aµ(xTs A ]ψ Li (x Ri (xiγ µ [ µ e(ã µ (x tg θ w Z µ (xy w G Aµ (xt A s ]ψ Ri (x ψ Ri (xiγµ [ µ e(ã µ (x tg θ Z w µ (xy w G Aµ (xts A e ( H(x sin θ w m Z [ Li (x Li (x 0 M ij M ij 0 ψ Rj (x ] ψ Ri (x ( Ri (xm ij 0 ψ Lj (x ( ψ Rj (x ψ Ri (x M ij 0 ψ Lj (x] θ g s 3π G Aµν(x G µν A (x = =... G Aµ (xj µ sa(xeã µ(x j µ (em (xe Z µ(x j µ (NC (x e [ sin θ W µ(x j µ (CC (x W µ(x j µ (CC (x], w J µ sa(x = Li (xγ µ it A s ψ Li (x Ri (xγ µ its A ψ Ri (x ψ Ri (xγµ its A ψ Ri (x, j µ (l (em(x = Li (xγµ Q em ψ (l (l Li (x Ri (xγµ iy w ψ (l Ri (x Li (xγµ Q em ψ Li (x Ri (xγµ iy w ψ Ri (x ψ Ri (xγµ iy w ψ Ri (x, j µ (l (NC(x= Li (xγ µ Q Z ψ (l Li (x tg θ (l w Ri(xγ µ iy w ψ Ri(x (l tg θ w Ri (xγ µ iy w ψ Ri (x tg θ w ψ Ri (xγ µ iy w ψ Ri (x, j µ (l (CC (x= Li (xγ µ itw ψ (l Li (x Li (xγ µ Q Z ψ Li (x Li (xγ µ it w ψ Li (x. ( The complete set of fermionic currents is [these equations define the currents J µ wa and 7

18 J µ Y w ] j µ wa (x= g w e J µ wa (x = sin θ w J µ wa (x = = (l j µ w (x = [ j µ (CC (x j µ (CC (x], j µ w (x = Li (xγµ itw a ψ(l Li (x Li (xγµ itw a ψ Li (x, i [ j µ (CC (x j µ (CC (x], j µ w 3(x = J µ sin θ w3(x =sin θ w [sin θ w j µ (em (x cos θ w j µ (NC (x] = w (l Li (xγ µ itwψ 3 (l Li (x Li (xγ µ itwψ 3 Li (x, = j µ Y w (x = J µ Y cos θ w (x =cos θ w [cos θ w j µ (em (x sin θ w j µ (NC (x] = w (l Li (xγ µ iy w ψ (l (l Li (x Ri(xγ µ iy w ψ Ri(x (l Li (xγµ iy w ψ Li (x Ri (xγµ iy w ψ (x ψ Ri Ri (xγµ iy ψ w Ri (x, j µ (em (x =cos θ w j µ Y w (xsin θ w j µ w3(x, j µ (NC (x= sin θ w j µ Y w (xcos θ w j µ w3(x, j µ (CC± (x= [ j µ w(x i j µ w(x]. ( One can also present the following quartic terms in a different way: e (õ (x cot θ w Z µ (x(ã µ (x cot θ w Z µ (x W (x W (x e W µ (x(ã µ (x cot θ w Z µ (x W ν (x(ã ν (xcot θ w Z ν (x = e {à (x W (x W (x W (x à (x W (x à (xcot θ w ( Z (x W (x W (x W (x Z (x W (x Z (x cot θ w (à (x Z (x W (x à (x W (x Z (x W (x Z (x W (x à (x}. It can be shown that in the unitary gauge the complex mass matrices containing the Yukawa couplings (replacing the not-gauge-invariant Dirac mass terms can be diagonalized by means of unitary left and right matrices [S (l L,R ( e (l H(x[ Li (x sin θ w m Z 0 M (l ij ψ (l Rj(x = S(l L,R, S L,R = S L,R, S L,R = S L,R ] ( (l Ri(xM (l ij 0 ψ (l Lj (x] = 8

19 =( ( ē (m R =( ( ( e H(x[ sin θ w m Z ē (m L (x µ (m L (x τ (m L (x m e m µ m τ e (m L (x µ (m L (x ]= e (m R (x µ (m R (x τ (m R (x m e 0 0 (x µ (m R (x τ (m R (x 0 m µ m τ τ (m L (x e H(x[m e ē (m (xe (m (xm µ µ (m (xµ (m (xm τ τ (m (xτ (m (x] sin θ w m Z e H(x[ Li (x sin θ w m Z Li (x 0 M ij ψ Rj (x ψ 0 M ij Ri (x M ij ψ Rj (x ( 0 ( Ri (xm ij 0 ψ Lj (x ψ Lj (x] = m d 0 0 d (m R (x ( e =( H(x[ d (m sin θ w m L (x s (m (m L (x b L (x 0 m s 0 s (m R (x Z 0 0 m b b (m R (x m u 0 0 u (m R ( (x ū (m L (x c (m L (x t (m L (x 0 m c 0 c (m R (x 0 0 m t t (m R (x m d 0 0 d (m L ( (x d (m R (x s (m (m R (x b R (x 0 m s 0 s (m L (x 0 0 m b b (m L (x m u 0 0 u (m L ( (x ū (m R (x c (m R (x t (m R (x 0 m c 0 c (m L (x ]= 0 0 m t t (m L (x e =( H(x[m sin θ w m d(m d (xd (m (xm s s (m (xs (m (xm b b(m (xb (m (x Z m u ū (m (xu (m (xm c c (m (xc (m (xm t t (m (xt (m (x], (3 9

20 where the mass eigenstates of leptons and quarks are defined by e (m L e L e (m R e R µ (m L = S (l L µ L, µ (m R = S (l R µ R, τ (m L d (m L s (m L b (m L u (m L c (m L t (m L = = S L S L τ L d L s L b L u L c L t L,, τ (m R d (m R s (m R b (m R u (m R c (m R t (m R = S R = S L τ R d R s R b R u R c R t R, ν (m el ν (m µl ν (m τl = S (l L ν el ν µl, ν τl. (4 The parameters m e,m µ,m τ,m d,m s,m b,m u,m c,m t [m νe = m νµ = m ντ = 0], are called lepton and current quark masses. For leptons they coincide with the asymptotic free (onshell states, which however do not exist for quarks according to the confinement hypothesis. For quarks, at the quantum level, these parameters are thought to be running with the renormalization scale, m q (µ, usually in the MS renormalization scheme; for the light quarks u, d, s, one chooses µ =Gev, while for c and b one can choose m q = m q (µ = m q due to perturbative QCD (m t is still a preliminary result. The chiral symmetry properties of u, d, s, allow to fix in a scale independent way (QCD does not feel flavour the ratios m s /(m d m u =.6±3.3, (m d m u /(m d m u =0.5 ± For heavy quarks one can define the mass m pole q on-shell mass like for leptons, m pole q m pole t associated with a perturbative quark propagator (a kinematical = m q (µ = m pole q [ 4 α 3π s(m pole q (αs ] (note that m t (µ = m t =7Gev. For the study of light hadrons (bound states of quarks one uses also the constituent quark masses, m const q = m q Λ s /c,sinceλ s gives the order of magnitude of the quark kinetic energy; in this way, even if one sends to zero the current mass of u, d, s, quarks, one still has for the proton and the neutron m p m const u The experimental values of the lepton and current quark masses are m const d m n. m e =( ± MeV, m µ = ( ± MeV, 0

21 m τ = (777.0 ± 0.3MeV, m νe < 7.0eV (95CL, m νµ < 0.7MeV (90CL, m ντ < 4MeV (95CL, m d (GeV =(8.5±.5MeV, m s (GeV = (80 ± 5MeV, m b =(4.5 ± 0.0GeV, m u (Gev =(5.0±.5MeV, m c =(.5 ± 0.05GeV, m t = (75 ± 6GeV. (5 Finally, by using the mass eigenstates, the unitary gauge Lagrangian density ( becomes L (x = G µν 4gs A (xg Aµν (x 4à µν (xã µν(x 4 Z µν (x Z µν(x W µν (x W µν (x [m W W µ (x W µ (x m Z Z µ (x Z µ e (x]( H(x sin θ w m Z µν ie(ã (xcot θ w Z µν (x W µ(x W ν(x ie[ W µν (x W µ(x W µν (x W µ(x](ã ν(xcot θ w Z ν(x e [ sin θ W (x W (x ( W (x W (x ] w e (õ (xcot θ w Z µ (x(ã µ (xcot θ Z w µ (x W (x W (x e W µ (x(ã µ (xcot θ Z w µ (x W ν (x(ã ν (xcot θ Z w ν (x µh(x µ H(x e m H H (x( H(x sin θ w m Z ν e (m (x ( ν e (m (x ν µ (m (x ν τ (m (x iγ µ µ ( γ 5 ν µ (m (x ν τ (m (x ( ē (m (x µ (m (x τ (m (x m e 0 0 e (m (x [iγ µ e µ ( H(x 0 m sin θ w m µ 0 ] µ (m (x Z 0 0 m τ τ (m (x ( ū (m (x c (m (x t (m (x

22 m u 0 0 u (m (x [iγ µ e µ ( H(x 0 m sin θ w m c 0 ] c (m (x Z 0 0 m t t (m (x ( d (m (x s (m (x b (m (x m d 0 0 d (m (x [iγ µ e µ ( H(x 0 m sin θ w m s 0 ] s (m (x Z 0 0 m b b (m (x G Aµ (xj µ sa(xeã µ(x j µ (em (xe Z µ(x j µ (NC (x e ( sin θ W µ (x j µ (CC (x W µ (x j µ (CC (x w θ g s 3π G Aµν(x G µν A (x, (6 with the electromagnetic, neutral, charge changing and strong currents defined by the following equations e (m (x ( j µ (em (x = ē (m (x µ (m (x τ (m (x γ µ µ (m (x τ (m (x d (m (x ( d 3 (m (x s (m (x b (m (x γ µ s (m (x b (m (x u (m (x ( ū 3 (m (x c (m (x t (m (x γ µ c (m (x, t (m (x j µ (NC (x= ( ν e (m (x ν (m µ (x ν τ (m (x γ µ ( γ 5 sin θ w ν e (m (x ν (m µ (x ν τ (m (x

23 e (m (x ( (sin θ w ē (m (x µ (m (x τ (m (x γµ γµ γ 5 µ sin θ (m (x w τ (m (x u (m (x ( ( ū (m (x c (m (x t (m (x 4 3 sin θ w γ µ γµ γ 5 c sin θ (m (x w t (m (x d (m (x ( ( d (m (x s (m (x b (m 3 (x sin θ w γµ γµ γ 5 s sin θ (m (x, w b (m (x ν e (m (x ν µ (m (x ν τ (m (x d (m (x ( ū (m (x c (m (x t (m (x γ µ ( γ 5V CKM s (m (x, b (m (x e (m (x ν e (m (x ν µ (m (x ν τ (m (x γ µ ( γ 5 µ (m (x τ (m (x u (m (x ( d (m (x s (m (x b (m (x γ µ ( γ 5V CKM c (m (x, t (m (x ( j µ (CC (x= ē (m (x µ (m (x τ (m (x γ µ ( γ 5 j µ (CC (x= ( ( J sa(x µ = ū (m (x c (m (x t (m (x γ µ it A s u (m (x c (m (x t (m (x 3

24 ( d (m (x s (m (x b (m (x γ µ it A s d (m (x s (m (x. (7 b (m (x The neutral current j µ (NC (x is also written in the alternative forms ( fis the sum over all fermions: j µ (NC (x= sin θ w f (m f (x(g f v γµ g a (f γµ γ 5 ψ (m f (x = f (m f (x(v f γ µ a f γ µ γ 5 ψ (m f (x, where g v = sin θ w v = i(tw 3 sin θ w Q em, g a = sin θ w a = itw 3 [the fermion assignements are: g v (νe,νµ,ντ = g a (νe,νµ,ντ = ; g(e,µ,τ v = (4sin θ w, g a (e,µ,τ = ; g (u,c,t v = ( 8 3 sin θ w, g (u,c,t a = ; g(d,s,b v = ( 4 3 sin θ w, g (d,s,b a = ]. In the charge-changing currents of the V-A type (V=vector γ µ, A=axial-vector γ µ γ 5, the Cabibbo-Kobayashi-Maskawa matrix V CKM = S L S L appears; it can be shown that it depends on three angles θ = θ C,θ 3,θ 3 giving the mixing of the quarks d, s, b, of the three families [c ij = cos θ ij 0,s ij = sin θ ij 0] and a complex phase e iδ 3 [0 δ 3 π], unique source of the weak CP-violation observed in the K system. With only two families, only the Cabibbo angle θ C remains, which is enough to explain the GIM mechanism (absence of flavour changing neutral currents since dd ss = d C d C s C s C with d C = cos θ C d sin θ C s, s C = sin θ C d cos θ C s and the different strength of hadronic S =0and S= Q= processes (S is the strong strangeness. One has c c 3 s c 3 s 3 e iδ 3 V CKM = s c 3 c s 3 s 3 e iδ 3 c c 3 s s 3 s 3 e iδ 3 s 3 c 3 s s 3 c c 3 s 3 e iδ 3 c s 3 s c 3 s 3 e iδ 3 c 3 c 3 ; (8 the matrix of the moduli has the following form and the moduli have the following experimental range of values V ud V us V ub V CKM = V cd V cs V cb = V td V ts V tb 4

25 λ λ Aλ 3 (ρ iη = λ λ Aλ O(λ 4 = Aλ 3 ( ρ iη Aλ to to to = 0.8 to to to 0.046, ( to to to where λ = V us =0.05 ± 0.008, A = V cb /λ =0.80 ± 0.04, ρ η = V ub /λ V cb = 0.36 ± 0.0; one has s =0.9 to 0.3, s 3 =0.036 to 0.046, s 3 =0.00 to The total number of free parameters of the standard model is 9: the nine masses (or Yukawa couplings m e,m µ,m τ,m d,m s,m b,m u,m c,m t ; the three mixing angles θ = θ C,θ 3,θ 3 ; the phase δ 3 [weak CP-violation]; the electromagnetic coupling α; theweinberg angle θ w ; the vector boson mass m Z [or m W ]; the Higgs mass m H ; the strong coupling α s (m ZortheQCDscaleΛ s ;theθ-angle [strong CP-violation]. The unitary gauge Lagrangian density has the following exact global (st Noether theorem and local (nd Noether theorem symmetries; The global groups U (l i (: ψ (l i (x e iα(l i ψ (l i (x, whose conserved quantities are the lepton numbers N i of the three lepton families. The associated conserved currents are J µ (l Ni (x= Li (xγµ ψ (l (l Li (x Ri (xγµ ψ (l Ri (x [Jµ N (x=ē(m (xγ µ e (m (x ν (m so on], µ J µ Ni(x =0, where = means evaluated on the equations of motion. el γµ ν (m el (x and The U sv ( global group [the matrix V CKM mixes the quark families]: ψ Li (x e iα ψ Li (x, ψ Ri (x e iα ψ Ri (x, ψ iα Ri (x e ψ Ri (x, whose conserved quantity is the baryon number B. The associated conserved current is J B(x µ = 3 i= [ Li (xγµ ψ Li (x Ri (xγµ ψ (x ψ Ri Ri (xγµ ψ Ri (x] = d (m (xγ µ d (m (x s (m (xγ µ s (m (x b (m (xγ µ b (m (xū (m (xγ µ u (m (x c (m (xγ µ c (m (x t (m (xγ µ t (m (x 3 The local strong color group SU(3, G Aµ (x ˆT A U s (xg Aµ (x ˆT A s U s (x U s (x µ U s (x, ψ Li (x U s (xψ Li (x, ψ Ri (x U s (xψ Ri (x, ψ Ri (x Us (x ψ Ri (x, giving the conservation of the non-abelian SU(3 charges Q A (improper con- 5

26 servation law from the nd Noether theorem and Gauss theorem. The associated conserved current is J µ sa (x of Eqs.(7. 4 The local electromagnetic gauge group U em ( giving the conservation of the electric charge (improper conservation law from the nd Noether theorem and Gauss theorem. It is called the custodial symmetry. The associated gauge transformations are A µ (x A µ (x U em (x µu em (x, ψ (l Li (x Uem (xψ(l Li (x, ψ(l Ri (x U U em (xψ Li (x, ψ Ri (x U em (xψ Ri (x, ψ Ri (x U em em (xψ(l Ri (x, ψ Li (x (x ψ Ri (x. As we shall see in Section VI, in the Higgs sector at each instant there is a su(xu( algebra of non conserved charges in the electroweak sector. Moreover, the standard model has approximate global symmetries Strong chiral symmetry a If we put m u = m d = m s =0,θ 3 = θ 3 = δ 3 =0(θ = θ c, and rearrange the q (x u (m (x u (m (x, d (m (x, s (m (x, quark fields in the triplet form q(x = q (x = d (m (x,the q 3 (x s (m (x Lagrangian density (6 is invariant under the (strong interactions global Noether transformations associated with an U sv ( U sa ( SU sv (3 SU sa (3 group, whose infinitesimal formis[α V,α A,α V, Ā, α A, Ā are the constant parameters; λā are the SU(3 Gell-Mann matrices in the fundamental triplet representation] q i (x q i (xiα V q i (x, i =,,3, q i (x q i (xiα V, Ā( λā ijq j (x, q i (x e i[α λā V α V,Ā ] q i (x, q i (x q i (xiα A γ 5 q i (x, q i (x e iγ 5[α A α A, Ā λā ] q i (x, q i (x q i (xiα A, Ā( λā ijγ 5 q j (x, (30 whose associated conserved Noether vector and axial-vector currents and charges are V µ (x =i q(xγ µ q(x, Q V = d 3 xv o ( x, x o, A µ (x =i q(xγ µ γ 5 q(x, Q A = d 3 xa o ( x, x o, 6

27 V µ (x =i q(xγµλā Ā q(x, Q V,Ā = A µ Ā (x =i q(xγµ γ 5 λā q(x, Q A,Ā = d 3 xv ō A ( x, xo, d 3 x A o Ā ( x, xo. (3 Q V is the part of baryon number B = d 3 xj o B ( x, xo containing the u (m, d (m, s (m, quarks, while Q V, Ā are the global approximatively conserved (the scale of the breaking is given by m s Gell-Mann flavour charges of the standard quark model of hadrons. They are: strong isospin T a s = λa, a=,,3; strong hypercharge Y s = 3 λ 8, strangeness S s = Y s B, electric charge Q em = T 3 s Y s;u-spinu s = λ6,u s = λ7,u 3 s = 4 ( 3λ 8 λ 3 = 3 4 Y s T3 s ; V-spin Vs = λ4, Vs = λ5, Vs 3 = ( 3λ 8 λ 3 = 3Y 4 4 s T3 s ; the quark assignements are B T s Ts 3 Q em Y s S s U s Us 3 V s Vs 3 u (m /3 / / /3 / / / d (m /3/ / /3 /3 0 / / 0 0 s (m /3 0 0 /3 4/3 / / / /. Since we have {Q V, Ā,Q V, B} = c Ā B CQ V, C, {Q V, Ā,Q A, B} = c Ā B CQ A, C, {Q A, Ā,Q A, B} = c Ā B CQ V, C, we can define the left and right charges Q R = (Q V Q A, Q V = Q L Q R, Q L = (Q V Q A, Q A = Q L Q R, Q R, Ā = (A V,Ā Q A, Ā, Q V, Ā = Q L, Ā Q R, Ā, Q L, Ā = (Q V,Ā Q A, Ā, Q A, Ā = Q L, Ā Q R, Ā, {Q R,Ā,Q R, B} = c Ā B CQ R, C, {Q R, Ā,Q L, B} =0, {Q L, Ā,Q L, B} = c Ā B CQ L, C. (3 In this form the group U sv ( U sa ( SU sv (3 SU sa (3 is replaced by the global strong chiral group U sr ( U sl ( SU sr (3 SU sl (3. At the quantum level one has: 7

28 A The vector current V µ (x is still conserved. B The axial-vector current A µ (x is no more conserved due to the global chiral anomaly [U sa (-anomaly]. On one side, this phenomenon explains the otherwise forbidden decay π 0 γ, but on the other side it constitutes the U sa (-problem, because one cannot invoke a dynamical spontaneous symmetry breaking mechanism, since the associated Goldstone boson should be the η pseudoscalar boson, which has too big a mass. The way out seems to be topological, i.e. connected with the θ vacuum and its strong CP problem, for which there are various interpretations (existence of the axion,... Letusremarkthatwith3colors,N c = 3, there is no local SU(3 SU( U( chiral anomaly, which would spoil the renormalizability of the standard model. C SU sr (3 SU sl (3 is supposed to be dynamically spontaneously broken to the diagonal SU slr (3 = SU sv (3 approximate flavour Gell-Mann symmetry group (valid also for m u = m d = m s 0 by the formation of a quark condensate < q(xq(x > 0 [instead a gluon condensate gs < 0 F 4π Aµν(xF µν A (x 0 >= gs < 0 π A( B A(x E A(x 0 > should correspond to a magnetic color configuration of the vacuum, responsible for the confinement of the electric flux between quarks and for the string tension k (450 Mev (the coefficient of the linear confining potential]. This condensate of quarks pairs breaks chirality [< 0 q i (xq i (x 0 > (0 Mev for each i] with a nonperturbative dynamical mechanism (for instance Nambu-Jona Lasinio. The quark condensate dynamically generates the constituent mass for the quarks, much larger than the current mass [m const u m const d 300Mev, m const s 450Mev], to be used in the quark model as an effective mass. In the limit of exact SU sv (3, the SU sv (3 octet of pseudoscalar mesons π, K, η, would be massless and would correspond to the eight Goldstone bosons associated with the spontaneous symmetry breaking. b One could also put m u = m d = m s = m c = m b = m t = 0 and study the approximate SU sl (6 SU sr (6 global symmetry, but it is much less interesting due to the big breaking of this symmetry measured by the value of m t. Weak chiral symmetry 8

29 If we put m e = m µ = m τ =0andm u =m d =m s =m c =m b =m t = 0 (i.e. all the leptons and quarks are massless, one has the global Noether symmetry SU wl ( SU wr ( which should be spontaneously broken to the weak isospin SU wlr ( = SU w ( global custodial symmetry. See Ref. [9]. 3 Heavy quark symmetry: for this approximate symmetry see Ref. [0]. IV. EULER-LAGRANGE EQUATIONS AND CONSTRAINTS FROM L(X The Euler-Lagrange equations deriving from the Lagrangian density ( are [V (φ = λ(φ φ φ o is the Higgs potential] L (G µ A = gs( L L ν = G Aµ ν G Aµ (G ˆD νabg νµ B gsj µ sa=0, L (W µ a J µ sa = i Li γ µ Ts A ψ Li i Ri γ µ Ts A ψ Ri i ψ Ri γµ Ts A = gw ( L L (W ν = ˆD νab W νµ b g W aµ ν W wĵµ wa=0, aµ ψ Ri, Ĵ µ (l wa = i Li γ µ Twψ a (l Li i Li γ µ Twψ a Li φ [TwD a (W,V µ D (W,V µ Tw]φ a = = J wa µ φ [TwD a (W,V µ D (W,V µ Tw]φ, a L (V µ = gy ( L L ν = ν V νµ g V µ ν V yĵµ Y w =0, µ Ĵ µ (l Y w = i Li γµ Y w ψ (l (l Li i Ri γµ Y w ψ (l Ri i Li γµ Y w ψ Li i ψ Ri γ µ Y ψ w Ri φ [Y w D (W,V µ D (W,V µ Y w ]φ = = J µ Y w φ [Y w D (W,V µ D (W,V µ Y w ]φ, i Ri γµ Y w ψ Ri L φa = L L µ = [D (W,V µ D (W,V φ a µ φ a (l [ φ LiaM (l ij ψ (l Rj o LiaM ij µ φ] a ψ Rj ψ Ri V (φ φ a M ij (itw 3 ab ψ Ljb ] =0, 9

30 L φ a = L φ µ a (l [ φ o RiM (l ij L µ φ a ψ (l Lia = [D (W,V µ D µ (W,V V (φ φ] a φ a Ri M ij ψ Lja Lib (it 3 w ba M ij ψ Rj ] =0, L (l ψli = L (l Li = L (l ψri = L (l Ri = L ψli = L Li = L ψri = L Ri = L ψri = L ψri = L ψ (l Li L (l Li L ψ (l Ri L (l Ri L ψ Li µ µ µ µ µ Rj M φ ji L Li L µ ψ (l Li L µ (l Li L µ ψ (l Ri L µ (l Ri L µ ψ Li ψ Rj φ o L µ µ Li (l = Li [ i( µ W aµ T a w V µy w γ µ ] =[γ µ i( µ W aµ T a wv µ Y w ]ψ (l Li φ φ o M (l (l = Ri[ i( µ V µ Y w γ µ ] (l Lj φ M (l ji φ o (l Rj M (l φ ji φ o =0, ij ψ (l Rj=0, =0, =[γ µ i( µ V µ Y w ]ψ (l Ri M (l φ ij ψ (l Lj=0, φ o = M ji φ M ij ψ Rj φ φ M ij o φ o L ψ Ri L Ri L ψ Ri L ψ Ri µ µ µ µ L µ ψ Ri L µ Ri L µ ψ Ri L µ ψ Ri Li [ φ φ o =0, i( µ W aµ T a w V µy w G Aµ T A s γµ ] =[γ µ i( µ W aµ T a w V µy w G Aµ T A s ]ψ Li ψ Rj =0, = Ri[ i( µ V µ Y w G Aµ Ts A γ µ ] Lj φ M ji φ o =[γ µ i( µ V µ Y w G Aµ Ts A ]ψ Ri M φ ij ψ Lj =0, φ o = ψ i( µ V µ Y w G Aµ Ts A γµ ] Ri [ =[γ µ i( µ V µ Y w G Aµ Ts A ] ψ Ri L (õ = L L ν = ν Ã νµ ie ν ( W µ W ν W ν W µ õ ν à µ ie( W νµ ν W W Wνµ ν e W W (õ cot θ Zµ w e ( W µ W ν W µ W ν (Ãν cot θ w Zν ie{[q em φ(x] [( µ g w ( W µ (xt w W µ (xt w Lj M ij φ φ o M ji =0, φ ψ Lj =0, φ o =0, 30

31 ieq em à µ (x ieq Zµ Z (xφ(x] [( µ g w ( W (xt µ w W (xt µ w ieq em à µ (x ieq Zµ Z (xφ(x] [Q em φ(x]} e j µ (em =0, j µ (em = (l Li γ µ Q em ψ (l Li (l Riγ µ iy w ψ (l Ri Li γ µ Q em ψ Li Ri γ µ iy w ψ Ri ψ Ri γµ iy ψ w Ri, L ( Zµ = L L Z ν = ν Z νµ iecot θ w ν ( W µ W ν W ν W µ µ ν Zµ ie{[q Z φ(x] [( µ g w ( W (xt µ w W (xt µ w ieq em à µ (x ieq Z Zµ (xφ(x] [( µ g w ( W µ (xt w W µ (xt w ieq em à µ (x ieq Z Zµ (xφ(x] [Q Z φ(x]} iecot θ w ( W W νµ ν W W νµ ν e cot θ w W W (õ cot θ w Z µ e cot θ w ( W µ W ν W µ W ν (à ν cot θ Z w ν e j µ (NC =0, j µ (l (NC = Li γµ Q Z ψ (l Li tg θ (l w Ri γµ iy w ψ (l Ri Li γµ Q Z ψ Li tg θ w Ri γ µ iy w ψ ψ Ri tg θ w Ri γ µ iy ψ w Ri, L ( W ± µ = L L W νµ ν = ν W ±µ ν W±µ g w {[Tw φ(x] [( µ g w ( W (xt µ w W (xt µ w ieq em à µ (x ieq Z Zµ (xφ(x] [( µ g w ( W µ (xt w W µ (xt w ieq em à µ (x ieq Z Zµ (xφ(x] [T w φ(x]} ie ν [ W ν (õ cot θ w Zµ W µ (Ãν cot θ w Zν ] ie[(ãµν cot θ w Zµν W ν W µν (Ãν cot θ w Zν ] 3

32 e [ sin θ W µ W ± W µ W W ] e µ W (à cot θ Z w w e (õ cot θ Zµ w W (à cot θ Z e w j µ (CC =0, sin θ w j µ (CC = (l Li γ µ it w ψ (l Li Li γ µ it w ψ Li. (33 In the last lines we added the Euler-Lagrange equations for õ = e A µ, Zµ = e Z µ, W ±µ = e sin θ w W ±µ, obtained from Eq. (7. The canonical momenta implied by the Lagrangian density ( are π (Go A (x = L(x o G Ao (x =0, π (Gk A (x= L(x o G Ak (x = g s π a (Wo (x= L(x o W ao (x =0, π (Wk a (x= L(x o W ak (x = g π (Vo (x= L(x o V o (x =0, π (Vk (x= G ok A (x =gs w Wa ok (x =g E (Gk A (x, w E a (Wk L(x o V k (x = g y V ok (x =gy E (Vk (x, (x, π φa (x= L(x o φ a (x =[D(W,V o φ(x] a, π φ a(x = L(x o φ =[D(W,V o φ(x] a (x a, π (l ψliaα (x = π (l Liaα (x = π (l ψriα (x = π (l Riα (x = L(x o ψ (l Liaα (x = i (l ( Li (xγ o aα, L(x Liaα(x = i (γ oψ (l Li (x aα, o (l L(x o ψ (l ( (l Ri (xγ o α, Riα(x = i L(x (l o Riα (x = i (γ oψ (l Ri(x α, 3

33 π ψliaaα (x = π LiAaα (x = π ψriaα (x = π RiAα (x = π ψriaα (x = L(x o ψ LiAaα(x = i L(x o LiAaα (x = i (γ oψ L(x o ψ ( Li (xγ o Aaα, Li (x Aaα, RiAα(x = i ( Ri (xγ o Aα, L(x RiAα(x = i (γ oψ Ri (x Aα, L(x RiAα(x = i ( ψ Ri (xγ o Aα, o o ψ π ψriaα (x = L(x o ψ RiAα(x They satisfy the standard Poisson brackets = i (γ o {G Aµ ( x, x o,π (Gν B ( y, xo } = δ AB δµδ ν 3 ( x y, {W aµ ( x, x o,π (Wν b ( y, x o } = δ ab δµ ν δ3 ( x y, {V µ ( x, x o,π (Vν ( y, x o } = δ ν µ δ3 ( x y, ψ Ri (x Aα. (34 {φ a ( x, x o,π φb ( y, x o } = {φ a( x, x o,π φ b( y, x o } = δ ab δ 3 ( x y, {H( x, x o,π H ( y, x o } = δ 3 ( x y, {θ a ( x, x o,π θb ( y, x o } = δ ab δ 3 ( x y, {ψ (l Liaα ( x, xo,π (l ψljbβ ( y, xo (l } = { Liaα ( x, xo,π (l Ljbβ ( y, xo } = = δ ij δ ab δ αβ δ 3 ( x y, {ψ Riα( x, (l x o,π ψrjβ( y, (l x o (l } = { Riα( x, x o,π (l Rjβ ( y, xo } = = δ ij δ αβ δ 3 ( x y, {ψ LiAaα( x, x o,π ψljbbβ ( y, xo } = { LiAaα( x, x o,π LjBbβ ( y, xo } = = δ ij δ AB δ ab δ αβ δ 3 ( x y, {ψ RiAα( x, x o,π ψrjbβ ( y, xo } = { RiAα( x, x o,π RjBβ ( y, xo } = = δ ij δ αβ δ 3 ( x y, 33

34 { ψ RiAα ( x, xo,π ψrjbβ ( y, xo } = { ψ RiAα ( x, xo,π ψrjbβ ( y, xo } = = δ ij δ αβ δ 3 ( x y. (35 All the fermionic momenta generate second class constraints of the type π ψ (x i ((xγ o 0, π (x iγ oψ(x 0, which are eliminated [3] by going to Dirac brackets; then the surviving variables ψ(x, (x satisfy (for the sake of simplicity we still use the notation {.,.} for the Dirac brackets {ψ Liaα( x, (l x o (l, Ljbβ ( y, xo } = iδ ij δ ab (γ o αβ δ 3 ( x y, {ψ Riα( x, (l x o (l, Rjβ ( y, xo } = iδ ij (γ o αβ δ 3 ( x y, {ψ LiAaα ( x, xo, LjBbβ ( y, xo } = iδ ij δ AB δ ab (γ o αβ δ 3 ( x y, {ψ RiAα ( x, xo, RjBβ ( y, xo } = iδ ij δ AB (γ o αβ δ 3 ( x y, { ψ RiAα ( x, xo, ψ RjBβ ( y, xo } = iδ ij δ AB (γ o αβ δ 3 ( x y. (36 The resulting Dirac Hamiltonian density is [after allowed integrations by parts; λ (G Ao (x,λ (W ao (x,λ(v o (x, are Dirac multipliers; B (Gk A (x = ɛkij G ij (W k A(x, B a (x = ɛkij W ij a (x, B (V k (x = ɛkij V ij (x are the magnetic fields for the corresponding interactions; α = γ o γ, β = γ o ] H D (x = [gs π(g A (xgs B (G A (x] A [g (W w π a (xgw B a (W (x] a [g y π (V (x B (V (x] π φ (xπ φ (x[ D (W,V φ(x] [ D (W,V φ(x] λ(φ (xφ(x φ o ψ(l Li (x[ α ( W a (xtw a V (xy w ( W a (xtw a V (xy w α]ψ (l Li (x i ψ(l Ri (x[ α ( V (xy w ( V (xy w α]ψ Ri(x (l (l Li(x φ(x M (l ij ψ (l (l φ Rj(x Ri(xM (l φ ij ψ o φ Lj(x (l o i ψ Li (x[ α ( W a (xtw a V (xy w G A (xts a ( W a (xtw a V (xy w G A (xts a α]ψ Li (x i ψ Ri (x[ α ( V (xy w G A (xts A ( V (xy w G A (xts A α]ψ Ri (x 34

35 i ψ Ri (x[ α ( V (xy w G A (xts A ( V (xy w G A (xts A α] ψ Ri (x Li (x φ(x φ o Li (x φ(x φ o G Ao (x[ ˆ (G D M ij M ij AB π(g B ψ Rj ψ (x Ri Rj (x ψ Ri iψ Ri (xts A ψ Ri (xi ψ Ri (xts A φ (x (xm ij (x M ij φ o φ (x φ o (xiψ Li (xts A ψ Li (x ψ Ri (x] W ao (x[ ˆ (W D ab π (W b (xiψ (l Li (xt w a ψ(l (π φ (xt a w φ(x φ (xt a w π φ (x] ψ Lj (x ψ Lj (x Li (xiψ Li (xt a w ψ Li (x V o (x[ π (V (xiψ (l Li (xy wψ (l Li (xiψ(l Ri (xy wψ (l Ri (x iψ Li (xy w ψ Li (xiψ Ri (π φ (xy w φ(x φ (xy w π φ (x] θ g3 s 8π π(g A (x (G B A (x λ Ao (xπ (Go A (xλ ao (xπ (xy w ψ Ri (xi ψ Ri (W o a (xy w ψ Ri (x (xλ (V o (xπ (Vo (x. (37 We get λ Ao (x = G x o Ao (x,λ a (x = W x o ao (x,λ o (x = V x o o (x. The time constancy of the primary constraints π (Go A (x 0, π yields the Gauss law secondary constraints (W o a (x 0,π (Vo (x 0, Γ (G A (x =g s L(Go A (x= π (G A (x c ABCG B (x π (G C (x iψ Li (xts A ψ Li (xiψ Ri (G = ˆ D AB (x π(g B (xjo sa (x 0, (xts A ψ Ri (xi ψ Ri (xts A ψ Ri (x = Γ a (W (x =g w L (Wo a (x= π a (W (x ɛ abcwb (x π c (W (xiψ (l Li (xtwψ a (l Li (x iψ Li (xtwψ a Li (x (π φ (xtwφ(x a φ (xtwπ a φ (x = = ˆ (W D ab (x π (W b (x (π φ (xtwφ(x a φ (xtwπ a φ (x Jwa(x o 0, Γ (V (x =gy L (Vo (x= π (V (xiψ (l Li (xy w ψ (l Li (xiψ (l Ri (xy w ψ Ri(x (l iψ Li (xy w ψ Li (xiψ Ri (π φ (xy w φ(x φ (xy w π φ (x = (xy w ψ Ri (xi ψ Ri 35 (xy w ψ Ri (x

36 = π (V (x (π φ (xy w φ(x φ (xy w π φ (x J o Y w (x 0. (38 The secondary constraints are constants of the motion and the 66=4 primary and secondary constraints are all first class with the only nonvanishing Poisson brackets {Γ (G A ( x, x o, Γ (G B ( y, x o } = c ABC Γ (G C ( x, x o δ 3 ( x y, {Γ (W a ( x, x o, Γ (W b ( y, x o } = ɛ abc Γ (W c ( x, x o δ 3 ( x y. (39 Let us make a digression on the choice of the boundary conditions on the various fields. The conserved energy-momentum and angular momentum tensor densities and Poincaré generators are [σ µν = i [γµ,γ ν ], σ i = ɛijk σ jk ; D (. µ are defined as in Eq.(3 with a change of sign in the fields] Θ µν (x =Θ νµ (x=g s (G µα A (xg Aα ν (x 4 ηµν G αβ A (xg Aαβ (x g w µα (W a (xw aα ν (x 4 ηµν Wa αβ (xw aαβ(x g y (V µα (xv α ν (x 4 ηµν V αβ (xv αβ (x [D (W,V µ φ(x] [D (W,V ν φ(x] [D (W,V ν φ(x] [D (W,V µ φ(x] η µν [[D (W,V α φ(x] [D (W,V αφ(x] V (φ] i (l Li (x[γ µ D (W,V ν D (W,V ν γ µ ]ψ (l Li (x i (l Ri (x[γµ D (V ν D (V ν γ µ ]ψ (l Ri (x η µν (l [ i Li (x φ(x φ o Li (x[γµ D (W,V,Gν M (l ij ψ (l Rj(x (l Ri(xM (l ij φ (x φ o D (W,V,G ν γ µ ]ψ Li (x i Ri (x[γµ D (V,Gν D (V,G ν γ µ ]ψ Ri (x i ψ Ri (x[γ µ D (V,Gν D (V,G ν γ µ ] ψ Ri (x η µν [ Li (x φ(x M ij ψ Rj (x ij φ o η µν [ Li (x φ(x φ o M ij Ri (xm ψ Rj (x ψ Ri (x M ij φ (x φ o φ (x φ o ψ (l Lj(x] ψ Lj (x] ψ Lj (x], M µαβ (x =x α Θ µβ (x x β Θ µα (x 4 (l Li (x(γµ σ αβ σ αβ γ µ ψ (l Li (x 36

37 (l 4 Ri(x(γ µ σ αβ σ αβ γ µ ψ Ri(x (l 4 Li (x(γ µ σ αβ σ αβ γ µ ψ Li (x 4 Ri (x(γ µ σ αβ σ αβ γ µ ψ Ri (x ψ Ri (x(γ µ σ αβ σ αβ γ µ ψ Ri (x, 4 ν Θ νµ (x =0, µ M µαβ (x =0, P µ = J µν = d 3 xθ oµ ( x, x o, d 3 xm oµν ( x, x o, P o = d 3 x { [g s π (G A A [g (W w π a ( x, x o gw a ( x, x o gs B (G A ( x, x o ] B (W a ( x, x o ] [g y π(v ( x, x o gy B (V ( x, x o ] π φ ( x, x o π φ ( x, x o [ D (W,V φ( x, x o ] [ D (W,V φ( x, x o ] V (φ( x, x o i ψ(l Li ( x, xo [D (W,V o D (W,V o ]ψ (l Li ( x, xo i ψ(l Ri ( x, x o [D (V o D (V o ]ψ Ri( x, (l x o (l Li (x φ(x φ o M (l ij ψ (l Rj(x i ψ Li ( x, x o [D (W,V,Go (l Ri(xM (l ij φ (x φ o D (W,V,G o ]ψ Li ( x, xo i ψ Ri ( x, x o [D (V,Go D (V,G o ]ψ Ri ( x, x o ψ i Ri ( x, x o [D (V,Go D (V,G o ] ψ Ri ( x, x o Li (x φ(x φ o M ij ψ Rj (x Li (x φ(x M ij φ o P i = d 3 x {( π (G A ( x, x o B (G A ( π (V ( x, x o B (V ( x, x o i Ri (xm ij ψ Rj (x ψ Ri (x M ij φ (x φ o φ (x φ o ψ (l Lj(x ψ Lj (x] ψ Lj (x}, ( x, x o i ( π (W a ( x, x o B (W a ( x, x o i π φ ( x, x o D (W,V i φ( x, x o [D (W,V i φ( x, x o ] π φ ( x, x o 37