Construction of the models of the Construction of the models of the dynamical Electroweak symmetry dynamical Electroweak symmetry

breaking using the ideas that breaking using the ideas that come from the effective field come from the effective field

model of Helium - 3 superfluid.model of Helium - 3 superfluid.

1. Nambu sum rule

2. Lorentz group as a source of DESB.

The Nambu sum rule and the relation between the masses of composite Higgs bosons

G.E.Volovik (Aalto U& ITP, Moscow)M.A.Zubkov (ITEP, Moscow)ArXiv:1302.2360, to appear in JETP lett.arXiv:1209.0204

abstractabstract It may appear that the recently found resonance at 125 GeV is not the only Higgs boson. The masses of

the Higgses may be related to the mass of t — quark as

(The sum is within a group of the scalar excitations.) This rule is the analogue of the sum rule in He3-B

(for the excitations of the same total momentum J). It was originally proposed by Nambu for He3-B. We call it the Nambu sum rule.

Plan 1. Introduction: experimental situation; accidents;

the essence of the Nambu sum rule.

2. Review of bosonic excitations in He3-B,A and thin films He3-a,b.

3. Consideration of the toy model for the top — quark condensation

4. Review of bosonic excitations in dense QCD

5. Conclusions

Nambu Sum rule

QCD

He3-B

AccidentsExcess of events at the Tevatron and CMS in 2011 at

325 GeV

"A 325 GeV scalar resonance seen at CDF?", Krzysztof A. Meissner,

Hermann Nicolai, arXiv:1208.5653

"Probing Minimal Supersymmetry at the LHC with the Higgs Boson Masses", L. Maiani, A. D. Polosa, V. Riquer, arXiv:1202.5998

Excess of events at ATLAS in 2011 at 245 GeV

He-3

Bosons

He3-B

derivation

He3-b thin film

He3-a thin film

Triply degenerated

He3-A

Goldstones:hidden symmetry: there are three terms that depend on with definite these terms are transformed to each other by

New massless modes:

He3-A

Bosonic modes (triply degenerated):

and six massless models

Relativistic models of top quark condensation

Particular case

bosons

C is diagonal

T-anti t and q — anti q channels

Channels with different quarks

Nambu Sum Rule

If one of the quarks dominates

Veltman identity vs Nambu Sum Rule

The number of colors is important

Hadronic phasediquarks:

Color superconductivity, CFL phase

Conclusions1. The notion of the Nambu Sum rule is introduced. It works in:

- Helium superfluid 3-B, 3-A, in thin films 3-b,3-a

- QCD at finite chemical potential (normal phase, small T, small mu),

- color superconductor (CFL phase)

- the considered models of top — quark condensation

2. Lessons for the TeV — scale physics:

- The 125 GeV Higgs cannot be the Nambu partner of the neutral

Goldstone boson

- If there are two twice degenerated Higgs bosons, then the Namby partner of the 125 GeV Higgs should have mass 210 GeV

-If there are only two states, then the second Higgs should have mass 325 GeV

-If there are two Higgs bosons of equal masses, these masses are around 245 GeV

M. ZubkovM. Zubkov

ITEP Moscow 2010-2013ITEP Moscow 2010-2013

1.1.M.A.Zubkov, arXiv:1301.69712. G.E. Volovik, M.A. Zubkov, arXiv:1302.2360, To appear in

JETP lett.3. 3. M.A.Zubkov, M.A.Zubkov, Mod. Phys. Lett. A, Vol. 25, Mod. Phys. Lett. A, Vol. 25,

No. 34 (2010) pp. 2885-2898No. 34 (2010) pp. 2885-2898

Lorentz Group as the source of dynamical Lorentz Group as the source of dynamical electroweak symmetry breaking electroweak symmetry breaking

3030

AbstractAbstract

1. Dynamics of torsion degrees of freedom in Poincare 1. Dynamics of torsion degrees of freedom in Poincare gravity can be considered as the gauge theory of gravity can be considered as the gauge theory of Lorentz group. Lorentz group.

2. Suppose this theory is coupled in a nonminimal way 2. Suppose this theory is coupled in a nonminimal way to spinors. The effective four - fermion interactions to spinors. The effective four - fermion interactions appear that may lead to the condensation of fermions. appear that may lead to the condensation of fermions.

3. The given construction may provide the dynamical 3. The given construction may provide the dynamical Electroweak symmetry breaking. Electroweak symmetry breaking.

3131

PlanPlan1. From Superconductivity to Technicolor2. Fermions in Riemann – Cartan space.3. 4 – fermion interactions. 4. Condensation of fermions.5. Fermion masses.

203

552

2

30

;))((

)(

iCxdCiCiMgL

LxdmiAiL

TTI

I

xdvHxHdiAHiAL 32223 )|(|4

)2(])2[(21

Ginzburg _ Landau ModelGinzburg _ Landau ModelSuperconductivitySuperconductivity

BCS ModelBCS Model

CiH T5

xdMgL

LxdiAixdiAiL

aLRRL

aI

IRRLL

32

2

33

))((

)()(

2

1

32223

);1()2(

)|(|4

)2(])2[(21

hh

HUSUA

xdvHxHdiAHiAL

Weinberg – Salam Model (Higgs sector)Weinberg – Salam Model (Higgs sector)

N J L modelN J L model

RaL

aH

xdBBBTrg

xdiBiAixdiBiAiL

TC

RRLL

32][2

33

,21

)()(

TCTCDTD NNFNGeV 3.0250

abTC

bR

aL 3~

TechnicolorTechnicolor

Effective N J L modelEffective N J L model

...))((

)()(

32

2

33

xdgL

LxdiAixdiAiL

aLR

bbRL

a

TC

TCI

IRRLL

)/exp();]([4

2

Ta

aT FiUUDUDTrFL

Nonlinear Sigma - ModelNonlinear Sigma - Model

GeVTC 100~

)1()2()3(... USUSUGG TCETC

...))((

))(())((

2

22

dtt

cbtt

a

ETC

abcd

dts

cbst

a

ETC

abcddss

cbss

a

ETC

abcdIL

SM fermions and Technifermions from the same SM fermions and Technifermions from the same multiplet of ETC gauge groupmultiplet of ETC gauge group

SM massSM mass

...))((

))(())((

2

22

dtt

cbtt

a

ETC

abcd

dss

cbtt

a

ETC

abcddss

cbss

a

ETC

abcdIL

2

3

ETC

TCaacdcdM

GeVETC 1000~

)1,3(so

)1,3(SOGTC

2

1

2

1

2

1

2

1

g

Idea Idea

Poincare gravity = Gauge theory of Lorentz Poincare gravity = Gauge theory of Lorentz group + group of translationsgroup + group of translations

],[ CESS b

gEEE baab

a ;

.C

Variables: Variables:

Translational connection = Tetrad fieldTranslational connection = Tetrad field

Lorentz group connection Lorentz group connection

)(21

gggg

bab

bab

bab

aaa

bab

aa

bab

bab

aaa

ECECEEEDT

EEED

ECEEEE

][.][.][.][][.

][.][][

..

0~0

aaE

][)(41

baabab CD

Massless fermions Massless fermions

(S.Alexandrov, (S.Alexandrov, Class.Quant.Grav.25:145012,2008)Class.Quant.Grav.25:145012,2008)

...

.

][

T

TEET

EEEcd

cbadabc

dcbadabc

)(21

gggg

xdiDDiEiS f4}][{

2

ccbacababcab

ccbacababcab

ETTTC

Eccc

TTTC

)(21

)(21

)(21 ...

...

.C

ii ;

Massless fermions Massless fermions

xdiDDiEiS Lf455 }][{

2

lijklijkikjijkijk

jiji

jklijkli

STTTq

TT

TS

61)(

31

.

xdTSTSE

xdiDDiEiS

bbb

ddd

f

45

455

}44{81

}]~[~{2

Action for the torsion Action for the torsion (I.L.Shapiro, Phys.Rep. 357, 113 (2002)(I.L.Shapiro, Phys.Rep. 357, 113 (2002)

xdSTMxdTTMxdSSMCS ii

TSii

TTii

SST424242)1( 2

41][

PARTICULAR CASEPARTICULAR CASE

xdTSE

xdiDDiEiS

bbd

d

f

45

455

}4{81

}]~[~{2

= 0

M TS=0

Flat metric, CP invariance Flat metric, CP invariance

ab

ab CED .. ][ ],[. DDGa

b

SO(3,1) gauge field action SO(3,1) gauge field action

(S.Holst, Phys.Rev.D. 53, 5966 (1996)(S.Holst, Phys.Rev.D. 53, 5966 (1996)

aa

abcb

ac

abdcabcd

GG

GG

GEEG

...

xdGEEEMxdGEEEMCS abba

TabbaTT

4..

*2

4..

2)1( ][

)(61

cdabbdcabcadadbcacdbabcdabcd GGGGGGA

cdabcd

ab GG ......*

21

xdGEG

xdGEGxdGEGxdAEAxdEG

xdGEGxdGEGxdGEGxdGEGS

abab

cdababcd

abcdabcd

abcdabcd

baab

abab

cdababcd

abcdabcd

T

4*.3

4*.2

4*.1

4.6

42.5

4.4

4.3

4.2

4.1

)2(

][][][ )2()1( abT

abTT CSCSCS

(E.Elizalde,S.D.Odintsov, (E.Elizalde,S.D.Odintsov, Phys.Atom.Nucl.56:409-411,1993Phys.Atom.Nucl.56:409-411,1993))

→∞

xdTSTSE

xdiDDiEiS

bbb

ddd

f

45

455

}44{81

}]~[~{2

((S.Mercuri, Phys. Rev. D 73 (2006) 084016S.Mercuri, Phys. Rev. D 73 (2006) 084016))

5

A

V

xdGEEEMxdGEEEMCS abba

TabbaTT

4..

*2

4..

2)1( ][

][~31

241

32 4222)1( qSxdTSSTREMS TT

((S.Alexandrov, Class.Quant.Grav.25:145012,2008S.Alexandrov, Class.Quant.Grav.25:145012,2008))

xdAVAVEM

xdiDDiEiS

T

f

422222222

2

455

}222{)1(32

3

}]~[~{2

= = 0 ; → ∞

((S.Mercuri, Phys. Rev. D 73 (2006) 084016S.Mercuri, Phys. Rev. D 73 (2006) 084016))

5

A

V

((S.Alexandrov, Class.Quant.Grav.25:145012,2008S.Alexandrov, Class.Quant.Grav.25:145012,2008)) = = 0 ; → ∞

Attractive force between fermions => Attractive force between fermions => condensatecondensate

DE aa ~;

J.Bijnens, C.Bruno, E. de Rafael, Nucl.Phys. B390 (1993) 501-541J.Bijnens, C.Bruno, E. de Rafael, Nucl.Phys. B390 (1993) 501-541

xdRLGGN

S atL

btL

btL

atLV

atL

btR

btR

atLS

Tt

42

2

4 )]}())([())(({4

SM fields + technifermionsSM fields + technifermions

24TN

J.Bijnens, C.Bruno, E. de Rafael, Nucl.Phys. B390 (1993) 501-541J.Bijnens, C.Bruno, E. de Rafael, Nucl.Phys. B390 (1993) 501-541

ttST

t GN

m

2

24

Corrections to the effective action are taken into account

Lattice discretization

Gauge field action

4949

ConclusionsConclusions1. The dynamical SO(3,1) gauge theory with the scale >1000

TeV and the torsion mass ~ this scale is coupled in a nonminimal way to fermions.

2. Due to the effecitve 4 – fermion interactions fermions are condensed and provide the DEWSB.

3. In zero order approximation all fermions acquire equal masses.

4. When the corrections due to the other gauge fields are taken into account, the hierarchy of fermion masses appears.

5. The given gauge theory is unusual since it should provide chiral symmetry breaking but cannot be confining.

Summary of two talks:1. Nambu sum rules in the NJL models: from Helium-3 to top

quark condensation. G.E. Volovik, M.A. Zubkov, ArXiv:1302.2360, to appear in JETP lett.; arXiv:1209.0204.

2. Numerical investigation of the effective field model of graphene monolayer using lattice field theory technique.M.V.Ulybyshev, M.A. Zubkov, Solid State Comm. http://dx.doi.org/10.1016/j.ssc.2012.12.030

3. Calculation of Euler — Heisenberg effective lagrangian for the effective field model of multilayer graphene.

M.I. Katsnelson, G.E. Volovik, M.A. Zubkov, Annals Phys. 331 (2013) 160

4. Momentum space topology in 4D relativistic quantum field theory.M.A. Zubkov, G.E. Volovik, Nucl.Phys. B860 (2012) 295-309, M.A. Zubkov, Phys.Rev. D86 (2012) 034505

5. Numerical investigation of Weinberg — Salam model in lattice regularization. M.A. Zubkov, Phys.Rev. D85 (2012) 073001.

6. Gauge theory of Lorentz group as a source of the dynamical Electroweak symmetry breaking. M.A. Zubkov. arXiv:1301.6971