Towards computing the lepton and quark mass spectra and their consequences Jiří Hošek Department of Theoretical Physics Nuclear Physics Institute Rez (Prague) Czech Republic arXiv: (not a phenomenological paper) Jiří Hošek Department of Theoretic a l Physics, Nuclear Physics Institute Rez (Prague) Czech Republic J.H., in “Strong coupling Gauge Theories in LHC Era”, World Scientific 2011; P. Beneš, J.H. and A. Smetana, arXiv: ; T. Brauner and J.H., to be published

Plan of the talk: I define the SU(3) f gauge flavor dynamics (g.f.d.) intended to replace the Higgs sector of the Standard model. I illustrate how g.f.d. spontaneously generates fermion proper self- energies Σ, hence the wide and wild spectrum of light fermion masses out of one very large mass scale. I analyze necessary consequences which follow from dynamical fermion mass generation by the existence theorem of Goldstone: Composite ‘would-be’ NG bosons hidden in massive chiral gauge bosons of anomaly-free gauged chiral symmetries. Composite pseudo NG bosons of global anomalous Abelian symmetries massive due to instantons: h, a, b. Conclusions and outlook.

A. Gauge SU(3) f flavor dynamics chiral quark and lepton fields of 3 families + 9 ν R ; 8 flavor gluons C µ ; 1 coupling constant h (or scale Λ) - Sterility symmetry U(3) s =U(1) s x SU(3) s : - anomaly freedom A. Smetana, JHEP 1304(2013)139 - Asymptotic freedom at small x, strong coupling at large x - In QCD we trust : g.f.d. is not vector-like and thus nonconfining - Electroweak interactions and QCD introduced by gauging corresponding indices and adding the gauge field terms

B. Spontaneous generation of fermion self energies Σ(p 2 ) Fermion mass terms strictly prohibited because of chiral symmetries of the Lagrangian. Fermion particles can, however, be massive if solutions of field equations are not chirally invariant P. Beneš, Phys. Rev. D81, (2010) Full fermion propagator:

B. Σ(p 2 ) is given by finding UV finite solution of Schwinger-Dyson equation NJL, Phys. Rev. 122(1961)345 H. Pagels, Phys. Rev. D21(1980)2336 At low momenta entirely unknown. Dynamical fermion mass generation is therefore a formidable task. Fermion mass m≡∑(p 2 =m 2 ) is given once Σ(p2) is known

B. Σ(p 2 ) found in separable approximation 1. Integrate only up to Λ i.e., set above Λ. The model thus becomes not asymptotically, but strictly free above Λ. Λ should not be confused with cutoff ! Fix external p=(p,0,0,0) and integrate over angles: where 2. Analyze the separable Ansatz (motivated by BCS) g ab are the unknown effective low-momentum coupling constants

B. Properties of Σ(p 2 ) 1 Solution of the SD equation is 2. Provided the numerical 8 x 8 matrix Г fulfils gap equation 3. Bi-unitary transformation 4. Gap equation becomes where

B. Properties of Σ(p 2 ) for neutrinos 1. To account for Majorana neutrinos introduce 2. Neutrino self-energy is 3. Neutrino gap equation

B. Hand-made world of quarks and leptons 1. Up quarks: Assume : U=V=1 (no mixing) Gap equation easily solved: Where

2. Down quarks: without mixing clearly the d-quark masses come out huge mixing helps: B. Hand-made world of quarks and leptons

3. Charged leptons: Mixing analogous to d-quarks implies 4. Neutrinos: Computer-made world seems necessary: space 12 x 12, only one mixing matrix at hand. At this exploratory stage we ASSUME: three active neutrinos acquire small masses, at least one very heavy sterile neutrino M R ~10 9 GeV. B. Hand-made world of quarks and leptons

Numbers come out reasonable: Axion invisible: Quantitative analysis (expectedly not easy) remains to be done.

C. Fate of gauged chiral symmetries Electroweak case is strightforward: gauge interactions can be switched off as weak external perturbations, and the NG excitations become real. Origin of fermion masses is not electroweak. Case of g.f.d. is generically self-consistent (difficult) : Strong-coupling flavor gluon exchanges both generate the fermion masses and their propagators are influenced by them (flavor gluons become massive): SELF-BREAKING. Because we neglect the effect of ‘would-be’ NG bosons due to flavor gluon self-interactions, We can treat the gauge boson mass generation in both cases on the same footing.

C. Fate of gauged chiral symmetries Mass of a chiral gauge boson is given by the mass of heaviest fermion to which it couples, and by the gauge coupling Illustrative computation: P. Beneš, T. Brauner, JH, Phys.Rev.D 75, (2007) The Lagrangian and symmetry Axial-vector current and Ward-Takahashi identity

C. Fate of gauged chiral symmetries Consequently, the NG pole is seen in Г and interpreted in terms of the effective Yukawa coupling P Where the vector tadpole is (+ of course 1 2)

C. Fate of gauged chiral symmetries Effective Yukawa coupling P computed in terms of Σ Gauge boson mass squared is the residue at the massless pole of the gauge field polarization tensor Π (Schwinger)

C. Fate of gauged chiral symmetries P. Beneš, arXiv: (PhD Dissertation); Pagels-Stokar formula We know Σ(p 2 ) explicitly : m 2 W, m 2 Z ~ g 2 m 2 top m 2 C ~ h 2 M 2 R Simple but subtle computations remain to be performed.

D. Fate of Abelian chiral symmetries 1. Abelian charges are not quantized (unless properly immersed in GUTs). Hence, their appropriate linear combinations become physical. 2. At quantum level the Abelian chiral currents are not conserved even if fermions are massless (G. t’ Hooft) Introduce where X=Y, W, G, F abbreviates four gauge forces : U(1) Y, SU(2) L, SU(3) c and SU(3) f

D. Fate of Abelian chiral symmetries Consider 6 Abelian chiral fermion currents of the model and take into account the quantum effect of axial anomalies

D. Fate of Abelian chiral symmetries 3. Obviously, there are two anomaly-free currents 4. For f=0, e=-2 we get the current of weak hypercharge Y. The electric charge Q is The weak hypercharge current is hidden in vectorial electro- magnetic current coupled to massless A µ. No ‘would-be’ NG boson.

D. Fate of Abelian chiral symmetries 5. For definiteness we fix second anomaly-free current by f≠0, e=0: It creates the true massless composite NG boson coupled to all fermions by calculable couplings. To prevent any conflict with data we better gauge the Y’ symmetry The NG becomes ‘would-be’ and the corresponding Z’ heavy: M Z’ ~ g’’ M R

D. Fate of Abelian chiral symmetries 6. One of four anomalous currents is the vectorial baryon current i.e., no (pseudo) NG boson. 7. It is good to remember that B-L-S is conserved in the present model.

E. Three pseudo NG bosons: Higgs, axion, arion. 1. Charges of anomalous spontaneously broken symmetries create massive pseudo NG bosons 2. There is a freedom in fixing the linear combinations of chiral fermion currents

E. Three pseudo NG bosons: Higgs, axion, arion 3. For definiteness we demand: Charges of four currents are orthogonal Higgs does not interact with gluons : Yukawa couplings not suppressed by M R. Axion is invisible: Axion interacts only strongly. Arion interacts only electroweakly. To get currents completely fixed we set

E. Three pseudo NG bosons: Higgs, axion, arion The result is 1. WT identities fix the effective Yukawa couplings of pseudo NG bosons with fermions: basically pseudoscalar, not universal, suppressed or not. 2. Divergences of currents fix the effective interactions of pseudo NG bosons with gauge fields. There is no way how h could interact with WW and ZZ by SM interactions.

E. Three pseudo NG bosons: Higgs, axion, arion Effective interactions of pseudo NG bosons with fermions and non-Abelian gauge fields give rise to their masses (S. Weinberg, F. Wilczek) 1. Axion(QCD instanton, WW): From its invisibility we fix Λ ~ 10 9 GeV 2. Arion (screened electroweak axion, A. Anselm, N. Uraltsev) m b ~ eV

E. Three pseudo NG bosons: Higgs, axion, arion 3. Higgs massive by screened instanton of gauge flavor dynamics: In Anselm-Uraltsev formula replace m W by m C, g W by corresponding h and get a wishful estimate m h ~ 10 2 GeV Straightforward effective field theory of massive pseudo NG bosons and their interactions (with important details unknown to me) remains to be developed

F. Conclusions and outlook 1. Explicit theoretically consistent framework - the chiral gauge flavor dynamics - for computing chirality-changing fermion self-energies Σ(p 2 ). 2. Anomaly-freedom enforces the existence of a fixed sector of sterile neutrinos. 3. Goldstone theorem implies the existence of various composite NG type excitations. In particular, Higgs, axion and arion are very close relatives. 4. The model offers (according to my personal taste) very good candidates for dark matter. The model correlates great many observed/observable phenomena

F. Conclusions and outlook 5. If, however, it turns out that the CERN 126 GeV “Higgs” is the 0 + scalar with renormalizable couplings to WW and ZZ, I will (maybe) start doing something else. Unfair SM unification: Higgs interacts with photons via loop, whereas with its brothers W and Z directly. “Elegance is an attitude”