CONTENTS 1 Hubbard-Stratonovich transformations to self-energies with coset decomposition to anomalous pair condensates for the standard model of electroweak interactions (Classical field theory with self-energy matrices of the irreducible propagator parts) Bernhard Mieck1 0 (article finished at 25 November 2009) 1 0 Abstract 2 The standard model of the strong and electroweak interactions is transformed from the ordinary path n a integral with the Lagrangians of quarks and leptons and with the Abelian and non-Abelian gauge fields to J correspondingself-energies. WeapplythepreciseformulationintermsofmasslessMajoranaFermifieldswith 6 ’Nambu’doublingwhichnaturallyleadstotheappropriateHST’s(Hubbard-Stratonovich-transformations)of the self-energies and to the subsequent coset decomposition for the SSB (spontaneous symmetry breaking). ] h The total coset decomposition of the Fermi fields is given by the dimension N0 = 90 for the symmetry p breaking SO(N0,N0)/U(N0)⊗U(N0) where the densities of fermions, related to the invariant subgroup n- U(N0), are contained in a background functional for the remaining SO(N0,N0)/U(N0) coset field degrees e of freedom of total Fermi fields which are composed of quark and lepton pairs. We find that the Higgs g field partially enters into a sum with the self-energies for the gauge field strength tensors so that it may be s. very difficult to observe pure, solely Higgs fields without a considerable contribution from the Abelian and c non-Abelian gauge field strength self-energies. The self-energy matrices of the standard model, representing i s the irreducible terms of propagation, allow rigorous derivations for the so-called ”effective”, classical field y theoriesasthe Skyrme-likemodels fromgradientexpansionsofthe determinantremainingfromthe bilinear, h anomalous doubled, fermionic field integrations. p [ Keywords : ’Nambu’ doubling of fields, Hubbard-Stratonovich transformations to self-energies, coset de- 1 composition for pair condensates, standard model of strong and electroweak forces. v 4 8 PACS : 12.10.Dm , 12.39.Fe , 12.39.Dc , 11.15.Tk 8 0 . 1Contents 0 0 1 Introduction 2 1 : 1.1 The SO(90,90)/U(90) ⊗ U(90) self-energy matrix for the total standard model . . . . . . . . . 2 v i X 2 Anomalous doubling of Fermi fields within the standard model 3 r 2.1 Symmetry breaking source actions and anomalous pair condensates . . . . . . . . . . . . . . . . 3 a 2.2 The Lagrangian of the standard model in terms of currents and auxiliary Higgs fields . . . . . . 8 3 Self-energies of gauge field strength tensors and quartic Fermi fields 12 3.1 Gaussian transformations with self-energy matrices of the gauge fields . . . . . . . . . . . . . . 12 4 HST and coset decomposition of anomalous doubled Fermi fields 15 4.1 Transformation of current-current terms or quartic, bilinear Fermi fields to self-energies . . . . 15 5 Summary and conclusion 18 5.1 The combination of the Higgs field with the gauge self-energies . . . . . . . . . . . . . . . . . . 18 1e-mail: ”[email protected]”;freelanceactivity2009;currentlocation: ZumKohlwaldfeld16,D-65817Eppstein,Germany. 2 1 INTRODUCTION 1 Introduction 1.1 The SO(90,90)/U(90) ⊗ U(90) self-energy matrix for the total standard model Although the standard model with the SU (3) × SU (2) × U (1) gauge interactions contains numerous un- c L Y determined parameters, it gives profound insight into the spontaneous symmetry breaking (SSB) of gauge symmetries with the Higgs mechanism [1, 2, 3]. The latter phenomenon causes the various masses of quarks and leptons which initially enter into the Lagrangian only as massless Majorana Fermi fields. The presented treatment starts out from the path integral for thestandard model with axial gauge constraints and transforms the anti-commuting Fermi fields of leptons and quarks and also the gauge fields with their gauge field strength tensorstocorrespondingself-energiesaftersuitableHSTs(Hubbard-Stratonovich-transformations). Apartfrom the Lagrangians of the standard model, we introduce symmetry breaking condensate ’seeds’ for fermionic co- herent wavefunctions and for the even-numbered pairs of ’Nambu’ or ’anomalous’ doubled Fermi fields [4, 5]. The self-energies of the gauge field strength tensors only contribute as background fields in common with the density parts of the total self-energy of the Fermi fields whereas the parts of the total self-energy, replacing the anomalous doubled Fermi field constituents, are the remaining field degrees of freedom in a coset decomposi- tion. The various, prevailing transformations in this paper follow in analogy to the coset decomposition for the strong interaction case [6]. However, one has to extend the coset decomposition SO(N ,N )/U(N )⊗U(N ) 0 0 0 0 from the dimension N = 24 for the QCD case with up-down (isospin) quark fields to N = 90 under inclusion 0 0 of the electroweak force. The dimension N = 90 is attained from counting the independent components of 0 quark and lepton Fermi fields with consideration of missing right-handed neutrinos. We finally achieve the effective generating function (1.1) of coset matrices Tˆ (x ) = exp{−Yˆ(x )} for anomalous doubled Fermi (Ψ) p p fields Ψ(x ) = {ψ(x ); ψ∗(x )} of leptons and quarks with the ’Nambu’ metric tensor Sˆ for the process of p p p doubling fields 2 Z Tˆ (x );Jˆ;J ;Jˆ = d[Tˆ−1(x ) dTˆ (x )] ∆ Tˆ−1;AB (x ),TˆAB (x );JˆAB (x ) (1.1) (Ψ) p ψ ψψ (Ψ) p (Ψ) p (Ψ);Mψ;Nψ p (Ψ);Mψ;Nψ p ψψ;Mψ;Nψ p Z (cid:16) (cid:17) (cid:2)×× DMˆET−M1hψ′;M;BˆN′ψAM′′′ABψ′′;(Nxψp(,x(cid:3)ypq′,′y)q−)iˆ11BM/2′ψ′′A;eN′x′ψ′′pδ(cid:26)pq−′ δ2(ı4)Z(Cxpd4−xpyqd′e′4)yq JhψTHˆ;,MASˆψi−(x1p)NASˆ′′′A′A;NB′′′(yTˆq′(B′Ψ,′)yB;Mq′′)ψ;TMˆ(Aψ′Ψ′()Bx;Npψ′);ZNCψ(dy4qy)q′′Jψ×B;Nψ(yq) ; ψ ψ (cid:27) h i (cid:16) (cid:17) MˆAB (x ,y )= ˆ1AB δ δ(4)(x −y )+ (1.2) Mψ;Nψ p q Mψ;Nψ pq p q + hHˆ i−1 δHˆ Tˆ−1,Tˆ + hHˆ i−1 Tˆ−1 Sˆ JˆTˆ AB (x ,y ) ; Sˆ Sˆ (Ψ) (Ψ) Sˆ (Ψ) (Ψ) p q Mψ;Nψ h i hHˆ i = Sˆ hHˆi; (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (1.3) Sˆ −−−−−−−→ δHˆSˆ Tˆ(−Ψ1),Tˆ(Ψ) = Tˆ(−Ψ1) Sˆ hHˆi Tˆ(Ψ) − Sˆ hHˆi = exp [Yˆ , ...]− Sˆ hHˆi − Sˆ hHˆi . (1.4) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) (cid:8) (cid:9) The path integral (1.1) on the non-equilibrium time contour with the two branches ’p,q = ±’ consists of the effective kinetic part hHˆ i with effective gauge terms from a saddle point computation of a background Sˆ averaging functional(denotedbyh...i)withtheself-energy ”densities”of Fermifieldsastheinvariant subgroup in the coset decomposition for the SSB. In correspondence to appendix D of [6], one can perform a gradient 2Sincethe precise derivation of (1.1) involvesseveral typesof gauge-field-’dressed’ coset matrices, we mark thefinal, remaining coset matrix Tˆ(Ψ)(xp)=exp{−Yˆ(xp)} by the subscript ’(Ψ)’ for the total, anomalous doubled Fermi fields. The part ∆e(...) (first linein (1.1)) denotesa condensate’seed’ functional with a condensate’seed’ matrix JˆAB (x ) for ’Nambu’pairsof fermions. ψψ;Mψ;Nψ p 3 expansion with the gradient term (1.4) up to fourth order for an effective Lagrangian (Derrick’s theorem [7]); hence, one has obtained a reliable, convenient, non-perturbative, classical method for the dynamics within the total standard model from the spacetime evolution of the coset matrices Tˆ (x ) (compare e. g. the ’Skyrme (Ψ) p model’ [8, 9]). It is also possible to calculate the energy momentum tensor of (1.1-1.4) or of a gradient- expanded version from the infinitesimal spacetime variations (with inclusion of the variations of the coset integration measure d[Tˆ−1(x )dTˆ (x )], SO(90,90)/U(90)) in order to couple to gravity; the corresponding (Ψ) p (Ψ) p energy-momentum tensor of (1.1-1.4) has then to act as a source tensor in the classical Einstein-field equations [10, 11]. In this manner one can encompass all known interactions and their non-perturbative dynamics into the spacetime evolution of the coset matrices Tˆ (x ) where the locally Euclidean spacetime coordinates dxµ (Ψ) p p −1/2 µ in the generating function (1.1) are related by the inverse square root gˆ (x ) dx = dz of the coordinate µν p p p,ν metric tensor gˆ (x ) to the curved spacetime dz which are determined from the Einstein field equations. µν p p,µ According to chap. 5 (”Derivation of a nontrivial topology and the chiral anomaly”) in [6], one can also conclude for Hopf invariants Π (S2)= Z from the quaternion eigenvalues and matrix elements of the generator 3 Yˆ(x )withinthecosetmatrixTˆ (x );however,thereoccursananomalycancellationwithinthetotalstandard p (Ψ) p model so that the total sum of Hopf invariants of the combined lepton and quark sectors should add to zero. Thisvanishingsumof Hopfinvariants Π (S2)= Zshouldthereforeconstrainscatteringanddecays ofcombined 3 quark and lepton fields, as e. g. in neutron decay n → p+e− +ν. The nontrivial topological configuration of fields can be rather involved for the case of the Hopf invariants; as one considers the pre-image of a point from the S2 sphere within the internal space of fields to the compactified three dimensional coordinate space, one can attain closed, one dimensional loops or toroidal configurations [12]. In the case of the original Skyrme model, one assigns nontrivial configurations or ’solitons’ of homotopy Π (S3) = Z to Baryons whereas our 3 precise derivation [6] for the solely strong interaction leads to nonzero Hopf invariants Π (S2) = Z from the 3 non-vanishing axial anomaly of QCD; in consequence, the transformed path integral in [6] with coset matrices is more closely related to the Skyrme-Faddeev model [13]. It is therefore also of interest for the total standard modeltowhatextent nontrivial topological configurations (as therestrictive, vanishingsumof Hopf invariants) can determine prevailing field combinations as baryons or mesons with the leptonic sector of electrons and neutrions. 2 Anomalous doubling of Fermi fields within the standard model 2.1 Symmetry breaking source actions and anomalous pair condensates Theappliedpathintegral forthestandardmodelisgiven bycontour timeintegrals (2.1)forforwardη = +1 p=+ and backward η = −1 propagation which is considered by the contour time metric (2.2), the contour time p=− coordinates xµ (2.3) and contour time derivatives ∂ˆ within the actions p=± p=±,µ +∞ −∞ +∞ +∞ d4x ... = d3~x dx0 ...+ dx0 ... = d3~x dx0 ...− dx0 ... p + − + − ZC ZL3 (cid:18)Z−∞ Z+∞ (cid:19) ZL3 (cid:18)Z−∞ Z−∞ (cid:19) +∞ = d3~x dx0 η ... ; (2.1) p p ZL3 (cid:18)p=±Z−∞ (cid:19) X η = +1 ; −1 ; η = +1 ; −1 ; ”p”, ”q” = ± ; (2.2) p q p=+ p=− q=+ q=− (cid:8) (cid:9) (cid:8) (cid:9) |{z} |{z} |{z} |{z} 4 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MODEL xµ = x0, ~x ; xµ = x0 , ~x ; xµ = x0 , ~x ; p p + + − − (2.3) ∂ˆp,µ = (cid:0) ∂∂x0p ,(cid:1)∂∂~x ; ∂ˆ+,µ = (cid:0) ∂x∂0+ ,(cid:1)∂∂~x ; ∂ˆ−,µ = (cid:0) ∂x∂0− ,(cid:1)∂∂~x . (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) The appropriate description of the standard model starts out from massless Majorana fields of fermions which obtain their corresponding masses from the spontaneous symmetry breaking with the Higgs phenomenon. We combine the strongly interacting quark fields q (x ) and lepton fields l (x ) of the three basic families m p m p m = 1,2,3 into one single fermionic spinor field ψ (x ) with the distinguishing labels ψ = ”l”, ”q” where the m p e tilde ’ ’ over l (x ) indicates the missing field degree of freedom for the right-handed neutrinos ν (x ) ≡ 0. m p m,R p The derivation of the nonlinear sigma model SO(90,90)/U(90) ⊗ U(90) within this paper folelows according to the foermulatieon of Ref. [14] (C.P. Burgess and G.D. Moore, ”The Standard Model : A Primer”); however, we emphasize with our derivation one additional, important point which concerns the anomalous doubling(or also ’Nambu’ doubling [4, 5]) within the standard model given in terms of left-handed and right-handed Majorana fields. Therefore, we double the total, two spin-component field ψ (x ) (2.4) of leptons l (x ) and quarks m p m p q (x ) by their complex conjugates l∗ (x ), q∗ (x ) or in total by ψ∗ (x ) which is taken into account by the m p m p m p m p first uppercase letters A,B,C = 1,2 (2.8) of the Latin alphabet. Hence, one has anomaloeus doubled Fermi fields LA(x ) (2.6), QA(x ) (2.7) or ien total ΨA(x ) (2.5) with the corresponding capital letters ′L′, ′Q′ or ′Ψ′ m p m p m p T e ψ (x ) = l (x ) ; q (x ) ;ψ = ”l”, ”q” ; e (2.4) m p m p m p (cid:16) e (cid:17) e ψ=”l” ψ=e”q” T ΨA(=1,2)(x ) = l (x ), l∗ (x ) ; q (x ), q∗ (x ) ; (2.5) m p m p m p m p m p (cid:16)zA=1 }| A=2{ zA=1 }| A=2{(cid:17) e e T LA(=1,2)(x ) = l| {(xz }), l|∗{(xz }) |; {z } | {z } (2.6) m p m p m p (cid:16) A=1 A=2 (cid:17) e e e T QA(=1,2)(x ) = |q {(zx}),|q∗{(zx}) ; (2.7) m p m p m p (cid:16) A=1 A=2 (cid:17) A,B,C, ... = 1, 2 . (2.8) | {z } | {z } According to the formulation of Ref. [14], we point out the additional fact of the standard model that it is entirely specified in terms of ’Nambu’ or anomalous doubled Fermi fields ΨA(x ), LA(x ), QA(x ) [4, 5]. In m p m p m p consequence this ’Nambu’ doubling naturally leads to a coset decomposition for anomalous doubled pairs of Fermi fields with a removal of the self-energy densities as background fields. e Wedefineinrelation(2.9)thetotalgaugegroupSU (3)⊗SU (2)⊗U (1)(correspondingtoRef. [14])with c L Y the first Greek letters α,β,γ = 1,...,8 for the eight gluon fields Gα(x ), with the first lowercase Latin letters µ p a,b,c = 1,2,3 for the SU (2) gauge boson fields Wa(x ) and with the Greek letters κ,λ,µ,ν,ρ of spacetime L µ p indicesfortheweak hyperchargegaugebosonfieldB (x ). Furthermore, theindicesr,s = 1,2,3 andf,g = 1,2 µ p denote the matrix elements (λˆα) of (gluon) Gell-Mann matrices and the Pauli-iso-spin matrices (τˆa) for the rs fg electroweak doublets. However, we depart from the formulation of Ref. [14] concerning the 4×4 Dirac gamma matrices (γˆµ) , (I,J,K = 1,...,4) and separate these into ’Nambu’ doubled parts A,B = 1,2 consisting of IJ 2 × 2 Pauli spin matrices (~σˆ) (i ,j =↑,↓) for the massless Majorana Fermi fields. Consequently, the iAjB A B 2×2 Pauli spin matrices within the 4×4 Dirac gamma matrices transform the upper ’↑’ and lower ’↓’ spin components of the Majorana fields. The particular list of definitions is described in (2.10) to (2.16) where we especially hint to the split of the 4×4 Dirac gamma matrices (γˆµ) , (I,J,K = 1,...,4) into the 2×2 Pauli IJ 2.1 Symmetry breaking source actions and anomalous pair condensates 5 spin matrices for the anomalous doubling of Fermi fields within the off-diagonal blocks ’A 6= B’ of (γˆµ)AB iAjB SU (3) × SU (2) × U (1) c L Y ↓ ↓ ↓ 8Gα(x ) 3 Wa(x ) B (x ) µ p µ p µ p α, β, γ = 1,...,8 a, b, c= 1,2,3 κ,λ,µ,ν,ρ = 0,1,2,3 ↓ ↓ ↓ (2.9) (λˆα) (τˆa) (γˆµ) = (γˆµ)AB rs fg IJ iAjB r, s = 1,2,3 f, g = 1,2 I, J, K = 1,2,3,4 ; A, B, C = 1,2 i , j , k =↑,↓ . A B C λˆα : (gluon) Gell-Mann matrices ; (2.10) τˆa : Pauli (iso)-spin matrices of the weak interaction ; (2.11) ηˆ := diag −1, +1, +1, +1 ; ε0123 = +1; (mostly ’+’ convention, cf. [14]); (2.12) µν (cid:0) 0 −ˆı (cid:1) 0 ˆ1 0 −ı~σˆ γˆµ IJ : γˆ0 = (cid:18) −ˆı 0 (cid:19)IJ ; βˆ= ıγˆ0 = (cid:18) ˆ1 0 (cid:19)IJ ; ~γˆ = ı~σˆ 0 !IJ;(I,J=1,...,4);(2.13) (cid:0) (cid:1) ~σˆ := σˆ , σˆ , σˆ = Pauli spin-matrices ; 1 2 3 γˆµ AB : γ(cid:0)ˆ0 = (cid:1)0 −(ˆı)iAjB ABβˆ= ı γˆ0 AB ; ~γˆ = 0 −ı(~σˆ)iAjB AB (2.14) iAjB (cid:18) −(ˆı)iAjB 0 (cid:19); iAjB ı(~σˆ)iAjB 0 !; (cid:0) (cid:1) (cid:0) (cid:1) ˆ (~σ)iAjB := (σˆ1)iAjB , (σˆ2)iAjB, (σˆ3)iAjB = Pauli spin-matrices ; (iA,jB=↑,↓) ; ψ(xp) = (cid:0)ψ†(xp)βˆ; (cid:1) (2.15) ˆ1 0 γˆ := −ıγˆ0γˆ1γˆ2γˆ3 = ; (2.16) 5 0 −ˆ1 (cid:18) (cid:19) Pˆ = ˆ1+γˆ5 = ˆ1 0 ; Pˆ = ˆ1−γˆ5 = 0 0 L 2 0 0 R 2 0 ˆ1 (cid:0) (cid:1) (cid:18) (cid:19) (cid:0) (cid:1) (cid:18) (cid:19). InthefollowingthedetailedlabelingwiththevariousindicesisdefinedfortheleptonandquarksectorsofFermi fields; this also allows to conclude for the dimension N = 90 of the coset decomposition SO(N ,N )/U(N ) ⊗ 0 0 0 0 U(N ) for the total self-energy SO(N ,N ) with coset matrices Tˆ (x ) SO(N ,N )/U(N ) for anomalous 0 0 0 (Ψ) p 0 0 0 pairs of fields and the unitary subgroup U(N ) for self-energy densities as the vacuum or background states. 0 The left-handed ′H = L′, two-component spin ′i =↑,↓′ lepton fields (2.17-2.20) consist of the three families A m,n = 1,2,3 with the left-handed neutrino l = ”ν ” and left-handed electron l = ”e ”; the latter distinction L l L L is also redundantly contained within the Pauli-iso-spin indices f,g = 1,2 for the neutrino f,g = 1 and electron f,g = 2 where both notations will convenienetly be applied in parallel. We thereefore attain the dimension (m = 1,2,3)×(f = 1,2)×(H = L)×(i =↑,↓) = 3·2·1·2 = 12withintheleft-handedleptonsector. Theright-handed A sector of leptons (2.19,2.20) lacks the right-handed neutrino field degree of freedom l = ”ν ” ≡ 0 f=1,H=R H=R with the remaining right-handed electron part (2.20). As we count the contributing dimension of the right- handed lepton sector, we are left with a total of (m = 1,2,3)×(f = 2)×(H = R)×(ie =↑,↓) = 3·1·1·2 = 6 A components l (x ) = ν (x ) ; (or l = ”ν ”) ; (2.17) m,f=1,H=L,iA=↑,↓ p m,H=L,iA=↑,↓ p L L l (x ) = e (x ) ; (or l = ”e ”) ; (2.18) em,f=2,H=L,iA=↑,↓ p m,H=L,iA=↑,↓ p eL L e e 6 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MODEL l (x ) = ν (x )≡ 0 ; (or l = ”ν ”≡ 0) ; (2.19) m,f=1,H=R,iA=↑,↓ p m,H=R,iA=↑,↓ p f=1,H=R H=R l (x ) = e (x ) ; (or l = ”e ”) . (2.20) em,f=2,H=R,iA=↑,↓ p m,H=R,iA=↑,↓ p R e R The electroweak doublet structure of quarks q = ”u”, q = ”d” (2.21,2.22) has equal numbers of right- and e e left-handed parts within the three families m = 1,2,3 where the notation of the 2×2 iso-spin matrices with f,g = 1,2 equivalently refers to the u(p)- and d(own)-quark components. In comparison to the lepton sector, one has also to include the indices r,s = 1,2,3 of the Gell-Mann matrices for the gluons so that one acquires a total of (m = 1,2,3)×(f = 1,2)×(r = 1,2,3)×(H = L,R)×(i =↑,↓) = 3·2·3·2·2 = 72 components A within the quark sector q (x ) = u (x ); (or q = ”u”) ; (2.21) m,f=1,r=1,2,3,H=L,R,iA=↑,↓ p m,r=1,2,3,H=L,R,iA=↑,↓ p q (x ) = d (x ) ; (or q = ”d”) . (2.22) m,f=2,r=1,2,3,H=L,R,iA=↑,↓ p m,r=1,2,3,H=L,R,iA=↑,↓ p As we add the 18 components of the lepton sector to the 72 components of the quark sector to N = 90 and as 0 weconsider the’Nambu’metric tensor SˆAB (2.23)of theanomalous doubling, onefinally achieves thetotal self- energy matrix SO(N ,N ) for the standard model to be decomposed into the coset part SO(N ,N )/U(N ) of 0 0 0 0 0 anomalous pairs and the unitary sub-group part U(N ) of self-energy densities. Note that the ’Nambu’ metric 0 tensor SˆAB (2.23) has off-diagonal entries compared to previous investigations [15, 16, 6]. This is caused by the changeoftheanomalousdoubleddensityoffieldsfromψ∗(x )·ψ(x )= 1 (Ψ†,A(x )SˆA=BΨB(x ))withSˆA=B = p p 2 p p diag{ˆ1; −ˆ1} of Refs. [15, 16, 6] to the considered case with ψ∗(x )·ψ(x ) = 1(ΨT,A(x )SˆA6=BΨB(x )) which p p 2 p p contains the off-diagonal metric SˆA6=B (2.23) and transposition ΨT,A(x ) instead of the hermitian conjugation p Ψ†,A(x ) as in previous investigations [15, 16, 6] p ˆ0 −ˆ1 AB SˆAB = . (2.23) ˆ1 ˆ0 (cid:18) (cid:19) Apart from the Lagrangian for the dynamics of Fermi and gauge boson fields, we introduce a symmetry break- ing source action A [Jˆ,J ,Jˆ ] (2.24) of the fermionic part with three source fields JˆAB (y ,x ), JA (x ), S ψ ψψ Mψ;Nψ q p ψ;Mψ p JˆAB (x ) where the bilinear parts have even, complex, commuting sources JˆAB (y ,x ), JˆAB (x ) ψψ;Mψ;Nψ p Mψ;Nψ q p ψψ;Mψ;Nψ p for anomalous doubled Fermi fields and where the linear symmetry breaking part is caused by anti-commuting, doubled source fields JA (x ) for coherent macroscopic wavefunctions 3. The source matrix JˆAB (y ,x ) is ψ;Mψ p Mψ;Nψ q p usedforgeneratingbilinearobservablesofFermifieldsbydifferentiation whereasthematrixfieldJˆAB (x ) ψψ;Mψ;Nψ p withanti-symmetric sub-matricesˆj (x ),ˆj† (x )acts as acondensate seedforanomalous paired ψψ;Mψ;Nψ p ψψ;Mψ;Nψ p fermionic fields after setting the matrix source field to equivalent values JˆψAψB;Mψ;Nψ(x+) = JˆψAψB;Mψ;Nψ(x−) on the two branches p = ± of the time contour. Similar considerations hold for the anti-commuting source field JA (x ) which can be chosen to generate an odd number of Fermi fields for observables and for condensate ψ;Mψ p seeds of macroscopic wavefunctions from equivalent values JψA;Mψ(x+) = JψA;Mψ(x−) on the two time contour branches. In relations (2.24-2.33), we can therefore list the source action A [Jˆ,J ,Jˆ ] (2.24) with the par- S ψ ψψ ticular property of missing right-handed neutrinos which is separately specified for the source fields JA (x ) ψ;Mψ p and JˆAB (x ) in eqs. (2.26-2.33) and is also denoted by the tilde ’ ’ over the lepton sector l (x ) ψψ;Mψ;Nψ p Mel p 1 A [Jˆ,J ,Jˆ ] = d4x JT,A (x ) SˆAB ΨB (x )+ΨT,A(x ) SˆAeB JB (x ) + e (2.24) S ψ ψψ 2 p ψ;Mψ p Mψ p Mψ p ψ;Mψ p ZC (cid:18) (cid:19) 3Wesummarizethevariousindicesforlepton(2.17-2.20)andquarkfields(2.21,2.22)bycollectiveindices’M ’,’N ’forbrevity, ψ ψ cf. e.g. relations (2.34-2.36). 2.1 Symmetry breaking source actions and anomalous pair condensates 7 1 ˆj† (x ) 0 AB + d4x ΨT,A(x ) ψψ;Mψ;Nψ p ΨB (x )+ 2 p Mψ p 0 ˆj (x ) Nψ p ZC (cid:18) ψψ;Mψ;Nψ p (cid:19) JˆψAψB;Mψ;Nψ(xp) 1 | {z } + d4x d4y ΨT,A(y ) JˆAB (y ,x ) ΨB (x ) ; 2 p q Mψ q Mψ;Nψ q p Nψ p ZC T ΨA (x ) = l (x ), l∗ (x ) ; q (x ), q∗ (x ) ; (2.25) Mψ p Mel p Mel p Mq p Mq p (cid:18) (cid:19) T e e JψA;Mψ(xp) = (cid:18)jel;Mel(xp), jel∗;Mel(xp) ; jq;Mq(xp), jq∗;Mq(xp)(cid:19) ; (2.26) 0 ≡ l (x )= ν (x ); (2.27) m,f=1,H=R,iA p m,H=R,iA p 0 ≡ ejel;m,f=1,H=R,iA = jν;m,H=R,iA(xp) ; (2.28) jelel;m,f,H1,iA;n,g,H2,iB(xp) = − jelTel;m,f,H1,iA;n,g,H2,iB(xp); (2.29) j (x ) = − jT (x ) ; (2.30) qq;m,f,r,H1,iA;n,g,s,H2,iB p qq;m,f,r,H1,iA;n,g,s,H2,iB p jelq;m,f,H1,iA;n,g,s,H2,iB(xp) = − jelTq;m,f,H1,iA;n,g,s,H2,iB(xp); (2.31) jelel;m,f=1,H1=R,iA;n,g,H2,iB(xp) = jνel;m,H1=R,iA;n,g,H2,iB(xp) ≡ 0 ; (2.32) jelq;m,f=1,H1=R,iA;n,g,s,H2,iB(xp) = jνq;m,H1=R,iA;n,g,s,H2,iB(xp)≡ 0 . (2.33) Since one has to assign many indices for the lepton and quark sectors with its various subspaces, we have defined collective indices Mψ, Nψ or Mel, Nel and Mq, Nq (2.34-2.36) with the extension that a bar over an additionally listed index, as e.g. in M (r,i ) (2.36), denotes the omittance of these over-barred indices in the q A prevailing total collection of these M := Mψ=el := {l,m,f,H,iA} without right-handed neutrinos (2.34) ψ M := {q,m,f,r,H,i } (cid:26) ψ=q A e ′ ′ N := Nψ=el′ := {l ,n,g,H ,jB} without right-handed neutrinos (2.35) ψ ′ ′ (cid:26) Nψ=q′ := {q ,n,g,s,H ,jB} e M (r,i ) := {q,m,f, r ,H, i }= {q,m,f,H,} ; etc. further examples . (2.36) q A A canceled canceled SimilartothesourcetermA [Jˆ,J ,Jˆ |]{(z2}.24)of|F{ezr}mifields,wealsoincludeasourceactionA [ˆj(Gˆ),ˆj(Wˆ),ˆj(Bˆ)] S ψ ψψ sg α a (2.37)forthegaugefieldstrengthtensorswithanti-symmetric, even-andreal-valuedsourcematricesˆj(Bˆ)µν(x ), p ˆj(Wˆ)µν(x ),ˆj(Gˆ)µν(x ) (2.39); however, we omit an anomalous kind of doubling as in the case of the Fermi fields a p α p whereasweallowfortherathergeneralextensionofthespacetimemetrictensorηˆ ηˆ (2.38)bycorresponding µλ νρ θ-terms for possible, nontrivial θ vacua (see the description for possible, relevant changes by these θ-terms in Refs. [17]) A [ˆj(Gˆ),ˆj(Wˆ),ˆj(Bˆ)] = d4x ˆj(Gˆ)µν(x ) ηˆ(g3,θ3) Gˆλρ(x )+ (2.37) sg α a p α p µν,λρ α p ZC (cid:18) + ˆj(Wˆ)µν(x ) ηˆ(g2,θ2) Wˆ λρ(x )+ˆj(Bˆ)µν(x ) ηˆ(g1,θ1) Bˆλρ(x ) ; a p µν,λρ a p p µν,λρ p (cid:19) 8 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MODEL g2 θ g2θ ηˆ(g1,θ1) = ηˆ ηˆ + 1 1 εˆ ; ηˆ(g2,θ2) = ηˆ ηˆ + 2 2 εˆ ; (2.38) µν,λρ µλ νρ 16π2 µνλρ µν,λρ µλ νρ 16π2 µνλρ g2 θ ηˆ(g3,θ3) = ηˆ ηˆ + 3 3 εˆ ; µν,λρ µλ νρ 16π2 µνλρ ˆj(Bˆ)µν(x ) = −ˆj(Bˆ)νµ(x ) ; ˆj(Wˆ)µν(x ) = −ˆj(Wˆ)νµ(x ) ; (2.39) p p a p a p ˆj(Gˆ)µν(x ) = −ˆj(Gˆ)νµ(x ). α p α p Finally, we can state the general structure of the generating function Z[Jˆ,J ,Jˆ ;ˆj(Gˆ),ˆj(Wˆ),ˆj(Bˆ)] (2.43) for the ψ ψψ standard model with the source actions A [Jˆ,J ,Jˆ ] (2.24), A [ˆj(Gˆ),ˆj(Wˆ),ˆj(Bˆ)] (2.37), with the integration S ψ ψψ sg α a µ over the anti-commuting fields of leptons and quarks (2.6,2.7) and the integration over the gauge fields G (x ), α p Wµ(x ), Bµ(x ) whose chosen axial gauge conditions (2.40-2.42) are specified by the auxiliary, real fields a p p s(G)(x ), s(W)(x ), s(B)(x ) within the standard Fourier integral representation of corresponding delta functions α p a p p containing constant vectors n , n and n (G)µ (W)µ (B)µ δ n Bµ(x ) = d[s(B)(x )] exp ı d4x s(B)(x ) n Bµ(x ) ; (2.40) (B)µ p p p p (B)µ p Z n ZC o 3 (cid:0) (cid:1) δ n Wµ(x ) = d[s(W)(x )] exp ı d4x s(W)(x )nµ Wa(x ) ; (2.41) (W)µ a p a p p a p (W) µ p aY=1 (cid:0) (cid:1) Z n ZC o 8 δ n Gµ(x ) = d[s(G)(x )] exp ı d4x s(G)(x )nµ Gα(x ) ; (2.42) (G)µ α p α p p α p (G) µ p αY=1 (cid:0) (cid:1) Z n ZC o Z[Jˆ,J ,Jˆ ;ˆj(Gˆ),ˆj(Wˆ),ˆj(Bˆ)] = d[l (x ),l∗ (x )]d[q (x ),q∗ (x )] × (2.43) ψ ψψ Mel p Mel p Mq p Mq p Z × d[φ†(ex ),φ(xe)] exp ı d4x L (x )+L (x ) −ı A [Jˆ,J ,Jˆ ] p p p ff p H p S ψ ψψ Z (cid:26) ZC (cid:16) (cid:17) (cid:27) × d[Gµ(x ),s(G)(x )] d[Wµ(x ),s(W)(x )] d[Bµ(x ),s(B)(x )] α p α p a p a p p p Z Z Z × exp ı d4x L (x )+L (x ) −ıA [ˆj(Gˆ),ˆj(Wˆ),ˆj(Bˆ)] . p gj p gg p sg α a (cid:26) ZC (cid:16) (cid:17) (cid:27) 2.2 The Lagrangian of the standard model in terms of currents and auxiliary Higgs fields It remains to describe the detailed form of the kinetic, fermionic part L (x ) and Higgs part L (x ) within ff p H p the total Lagrangian L (x ) where the gauge boson - current coupling L (x ) determines the interaction of tot p gj p fermions and where one also has the additional, non-Abelian self-interaction in L (x ) among the gauge fields gg p L (x ) = L (x )+L (x )+L (x )+L (x ). (2.44) tot p ff p H p gj p gg p Although we follow the detailed description and specification of Lagrangian parts according to [14], we again outline the various parts of the standard model Lagrangian L (x ), in order to emphasize the natural ap- tot p pearance of an anomalous doubling of the Fermi fields, due to the proper formulation with massless Majorana Fermi fields and symmetry breaking terms [4, 5]. Therefore, one is guided by the formulation, corresponding to Ref. [14], to the coset decomposition into ’Nambu’ doubled pairs within the self-energy and remaining sub- group parts of the self-energy densities as background or vacuum state. We define from [14] left-, right-handed, vectorial and axial gamma matrices (γˆµ )AB (2.45,2.46), (γˆµ )AB (2.47,2.48) which separate into a H=L,R iAjB 5,H=L,R iAjB 2.2 The Lagrangian of the standard model in terms of currents and auxiliary Higgs fields 9 symmetric (anti-symmetric) block structureof Pauli spin matrices in the off-diagonal blocks A6= B of vectorial (γˆµ )AB (axial(γˆµ )AB )gammamatrices,respectively. Thevectorial gammamatrices(γˆµ )AB H=L,R iAjB 5,H=L,R iAjB H=L,R iAjB enter into the kinetic part L (x ) (2.49) of fermions with an additional ’−ıSˆAB ε ’ term (2.50) for further ff p p analytic properties and convergence of non-equilibrium Green functions T AB 0 −ˆı, ı~σ γˆµ AB = iAjB ; (2.45) H=L iAjB −ˆı, ı~σ iAjB (cid:0) 0 (cid:1) ! (cid:0) (cid:1) (cid:0) (cid:1) T AB 0 −ˆı, −ı~σ γˆµ AB = iAjB ; (2.46) H=R iAjB −ˆı, −ı~σ iAjB (cid:0) 0 (cid:1) ! (cid:0) (cid:1) (cid:0) 0 (cid:1) − −ˆı, ı~σ T AB γˆµ AB = iAjB ; (2.47) 5,H=L iAjB −ˆı, ı~σ iAjB (cid:0) 0 (cid:1) ! (cid:0) (cid:1) (cid:0) 0(cid:1) −ˆı, −ı~σ T AB γˆµ AB = iAjB ; (2.48) 5,H=R iAjB − −ˆı, −ı~σ iAjB (cid:0) 0 (cid:1) ! (cid:0) (cid:1) T,A B L (x ) = −1 (cid:0) lMel(xp) (cid:1) γˆµ AB ∂ˆ −ıSˆAB ε δ δ lNel(xp) +(2.49) ff p 2 lM∗ e(xp) ! H iAjB p,µ p iAjB Mel(iA);Nel(jB) lN∗e(xp) ! e l (cid:16)(cid:0) (cid:1) (cid:17) e l T,A B 1 q (x ) q (x ) − Meq p γˆµ AB ∂ˆ −ıSˆAB ε δ δ eNq p 2 (cid:18) qM∗q(xp) (cid:19) H iAjB p,µ p iAjB Mq(iA);Nq(jB)(cid:18) qM∗q(xp) (cid:19) (cid:16) (cid:17) 1 (cid:0) (cid:1) = − ΨT,A(x ) γˆµ AB ∂ˆ −ıSˆAB ε δ δ ΨB (x ) ; 2 Mψ p H iAjB p,µ p iAjB Mψ(iA);Nψ(jB) Nψ p εp = ηp ε+ ; ε+ >(cid:16)(cid:0)0 ; (cid:1)ε+ → 0+ . (cid:17) (2.50) The Higgs part L (x ) (2.51), whose non-zero vacuum value of φa(x )= (φ (x ), φ (x ))T causes mass terms H p p 1 p 2 p of fermions and couplings among quarks according to the Kobayashi-Maskawa matrix [1], is also given in correspondence to [14]; however, we reformulate this part by introducing an anti-symmetric matrix FˆAB ψψ;Mψ;Nψ (2.52) for the ’Nambu’ doubled Fermi fields ΨT,A(x ) FˆAB (x )ΨB (x ), similar to the kinetic partwhich Mψ p ψψ;Mψ;Nψ p Nψ p is also in total anti-symmetric, due to the anti-symmetric derivative operator ∂ˆ and symmetric, vectorial p,µ gamma matrices (2.45,2.46) L (x ) = − ∂ˆ φ(x ) † ∂ˆµφ(x ) +µ2 φ†(x )φ(x ) −λ φ†(x )φ(x ) 2−λ µ2 2+ (2.51) H p p,µ p p p p p p p 2λ (cid:16) (cid:17) (cid:16) (cid:17) − φ(cid:0)(x ) ν† ((cid:1)x(cid:0)) fˆ e (cid:1) (x )+(cid:0) u† (x )hˆ (cid:1) d (x )−u† (x ) gˆ d (x ) + 1 p m,L p mn n,R p m,L p mn n,R p m,R p mn n,L p − φ∗(x )(cid:16)e† (x ) fˆ ν (x )+d† (x )hˆ u (x )−d† (x )gˆ u (x )(cid:17)+ 1 p m,R p mn n,L p m,R p mn n,L p m,L p mn n,R p − φ (x )(cid:16)e† (x ) fˆ e (x )+d† (x )hˆ d (x )+u† (x )gˆ u (x )(cid:17)+ 2 p m,L p mn n,R p m,L p mn n,R p m,R p mn n,L p − φ∗(x )(cid:16)e† (x ) fˆ e (x )+d† (x ) hˆ d (x )+u† (x ) gˆ u (x )(cid:17) 2 p m,R p mn n,L p m,R p mn n,L p m,L p mn n,R p ; (cid:16) (cid:17) 2 µ2 2 L (x ) = − ∂ˆ φ(x ) †· ∂ˆµφ(x ) +µ2 φ†(x ) · φ(x ) −λ φ†(x ) · φ(x ) −λ + (2.52) H p p,µ p p p p p p p 2λ (cid:0) (cid:1)T,A(cid:0) (cid:1) (cid:0) (cid:1) (cid:16) B (cid:17) (cid:16) (cid:17) l (x ) l (x ) − Mel p FˆAB φ (x ),φ (x );fˆ Nel p + lM∗ e(xp) ! Mel;Nel 1 p 2 p mn lN∗e(xp) ! e l (cid:16) (cid:17) e l e e 10 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MODEL T,A B q (x ) q (x ) − Mq p HˆAB φ (x ),φ (x );hˆ +GˆAB φ (x ),φ (x );gˆ Nq p q∗ (x ) Mq;Nq 1 p 2 p mn Mq;Nq 1 p 2 p mn q∗ (x ) (cid:18) Mq p (cid:19) (cid:18) (cid:19)(cid:18) Nq p (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) = − ∂ˆ φ(x ) †· ∂ˆµφ(x ) +µ2 φ†(x ) · φ(x ) −λ φ†(x ) · φ(x ) 2−λ µ2 2+ p,µ p p p p p p p 2λ − 1Ψ(cid:0)T,A(x ) Fˆ(cid:1)AB (cid:0) φ ((cid:1)x ),φ (cid:0)(x ) ΨB (x ).(cid:1) (cid:16) (cid:17) (cid:16) (cid:17) 2 Mψ p ψψ;Mψ;Nψ 1 p 2 p Nψ p (cid:16) (cid:17) In analogy, the Lagrangian L (x ) (2.53) with anomalous doubled Fermi fields within the various currents gj p (φ)µ (φ)µ (apartfromthepureHiggscurrentsj (x )andj (x ))followstheprinciple”(transposed’Nambu’doubled B p W;a p FermifieldΨT,A(x ))×(anti-symmetricmatrixoroperator)AB ×(anomalousdoubledFermifieldΨB (x ))” Mψ p Mψ;Nψ Nψ p as in L (x ) (2.49) or L (x ) (2.52) ff p H p g g L (x ) = − 1 B (x ) j(ax.)µ(x )+j(φ)µ(x ) − 2 Wa(x ) j(vec.)µ(x )+j(ax.)µ(x )+j(φ)µ(x ) + gj p 2 µ p B p B p 2 µ p W;a p W;a p W;a p g (cid:16) (cid:17) (cid:16) (cid:17) − 3 Gα(x ) j(vec.)µ(x )+j(ax.)µ(x ) . (2.53) 2 µ p G;α p G;α p (cid:16) (cid:17) (−) (+) We therefore include anti-symmetric and symmetric Pauli-isospin-matrices τˆ , τˆ (2.54) and the eight a a Gell-Mann (gluon) matrices λˆ(−), λˆ(−) (2.55) α α τˆ −τˆT τˆ +τˆT τˆ(−) = a a ; τˆ(+) = a a ; (2.54) a fg 2 fg a fg 2 fg (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) λˆ −λˆT (cid:0) (cid:1) λˆ +λˆT λˆ(−) = α α ; λˆ(+) = α α , (2.55) α rs 2 rs α rs 2 rs (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) µ µ which combine with the vectorial and axial gamma matrices (γˆ ) (2.45,2.46), (γˆ ) (2.47,2.48) to completely H 5,H anti-symmetric matrix couplings among the bilinear anomalous doubled Fermi fields. Relations (2.56-2.65) encompass all the various currents of the standard model (according to Ref. [14]) which, however, are entirely reformulated interms of bilinear, anomalous doubledFermi fieldscoupledto anti-symmetric matrices. Further- (Y) (Y) more, one has to incorporate the weak hyper-charges e , e (2.58) and left- and right-handed charge values el;H q;H (G) e (2.65) of strongly interacting quarks, in order to attain the proper couplings for the standard model. Note q;H (vec.)µ (ax.)µ that one has only left-handed current components within j (x ) (2.60), j (x ) (2.61) according to the W;a p W;a p SU (2) gauge group of weak interactions L (vec.)µ j (x ) ≡ 0 ; (2.56) B p T,A B (ax.)µ 1 lMe(xp) µ AB (Y) lNe(xp) j (x ) = − l ı γˆ e δ l + (2.57) B p 2 lM∗ e(xp) ! 5,H iAjB ˜l,H Mel(iA);Nel(jB) lN∗e(xp) ! e l (cid:0) (cid:1) e l T,A 1 q (x ) q (x ) − Meq p ı γˆµ AB e(Y) δ Neq p 2 (cid:18) qM∗q(xp) (cid:19) 5,H iAjB q,H Mq(iA);Nq(jB)(cid:18) qN∗q(xp) (cid:19) 1 (cid:0) (cid:1) = − ΨT,A(x ) ı γˆµ AB e(Y) δ ΨB (x ); 2 Mψ p 5,H iAjB ψ,H Mψ(iA);Nψ(jB) Nψ p e(Y) = −1 ; e(Y) = +2 ;(cid:0) e((cid:1)Y) = +1/3 ; e(Y) = −4/3 ; e(Y) = +2/3 ; (2.58) ˜l,H=L ˜l,H=R q,H=L q=u,H=R q=d,H=R (cid:0) j(φ)µ(x ) = ı φ†(x ) · ∂ˆ(cid:1)µφ(x(cid:0) ) − ∂ˆµφ(x ) † · φ(x ) ; (cid:1) (2.59) B p p p p p p p h i (cid:0) (cid:1) (cid:0) (cid:1)