Construction of the models of the
dynamical Electroweak symmetry
breaking using the ideas that come
from the effective field 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
abstract
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
Accidents
Excess 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 phase
diquarks:
Color superconductivity, CFL phase
Conclusions
1. 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
Lorentz Group as the source of dynamical
electroweak symmetry breaking
M. Zubkov
ITEP Moscow 2010-2013
1.M.A.Zubkov, arXiv:1301.6971
2. G.E. Volovik, M.A. Zubkov, arXiv:1302.2360, To appear
in JETP lett.
3. M.A.Zubkov,
Mod. Phys. Lett. A, Vol. 25,
No. 34 (2010) pp. 2885-2898
Abstract
1. Dynamics of torsion degrees of freedom in Poincare
gravity can be considered as the gauge theory of
Lorentz group.
2. Suppose this theory is coupled in a nonminimal way
to spinors. The effective four - fermion interactions
appear that may lead to the condensation of fermions.
3. The given construction may provide the dynamical
Electroweak symmetry breaking.
30
Plan
1. From Superconductivity to Technicolor
2. Fermions in Riemann – Cartan space.
3. 4 – fermion interactions.
4. Condensation of fermions.
5. Fermion masses.
31
Superconductivity
Ginzburg _ Landau Model




1
2

3
2
2
3
L

[(


2
iA
)
H
]
(


2
iA
)
Hd
x

(|
H
|
v
)
d
x


2
4
BCS Model

 

g
L

(

i

C

)(

i

C

)
d
x
;C

i


M

0 3
L

i
(


iA
)
m

d
x

L


I

2
T
I
2
5
T
5
Hi5C

T
3
0
2
Weinberg – Salam Model (Higgs sector)

1
2 2


3
23
L

[(


2
iA
)
H
]
(


2
iA
)
Hd
x

(|
H
|

v
)
d
x
 


2
4
h


1


A

SU
(
2
)

U
(
1
);
H



h
2


N J L model
  
g
L

(


)(


)
d
x
M


3

3




L

i
(


iA
)
d
x

i
(


iA
)
d
x

L
L  
L
R 
R
I


2
I
2
a
a3
L
R
R
L
H  R
a
a
L


Technicolor








3

3
L

i
(


iA

iB
)
d
x

i
(


iA

iB
)
d
x
L  
L
R  
R


1
2
3





Tr

B

B
,
B
d
x
[


]


2

2
g
TC
Effective N J L model







g
L

(


)(


)
d
x

...


3

3
L

i
(


iA
)
d
x

i
(


iA
)
d
x

L
L  
L
R  
R
I


I
2
a b b a3
TC
L
R R
L
2
TC
Nonlinear Sigma - Model
a b
3 ab




~


L R
TC

F


a
L

Tr
([
D
U
]
D
U
);
U

exp(
i

F
)

a/
T
4
2
T
250
GeV

N
F

0
.
3
N
N

D
T
D
TC
TC
SM fermions and Technifermions from the same
multiplet of ETC gauge group G

G

...

SU
(
3
)

SU
(
2
)

U
(
1
)
ETC
TC




L
 (



)(



)

(



)(



)






 (



)(



)

...
a
abcd
s
2
ETC
I
b
s
a
b
abcd
t
t
2
ETC

c
s
c
t
d
s
a
b
abcd
t
s
2
ETC
c
s
d
t
d
t



L
 (


)(


)
 (


)(


)
I
a b
abcd
ss
2
ETC

c d
ss
a b
abcd
t t
2
ETC

c d
ss

 (


)(


)

...
a b
abcd
t t
2
ETC

SM mass
c d
t t
 
M
cd
3
aacd
TC
2
ETC


~
1000
GeV
ETC

~
100
GeV
TC
G
(3,1)
TCSO
Idea
1 
1 
 
 
 
 
   2  g 2 


 1 
 1 
2 
2 
Poincare gravity = Gauge theory of Lorentz group +
group of translations
Variables:
a
a b
E
;

E
g
Translational connection = Tetrad field
 abE


Lorentz group connection
 

C

.


 
 so(3,1)
 






1


g

g


g


g
)








2(

b

SS
[E
,
C
 ]
Massless fermions












 
i
4




S

E
{

i
D

[
D
]

i
}
d
x
f

2
   aEa
(S.Alexandrov, Class.Quant.Grav.25:145012,2008)
1ab ab
D




C
)


  (


[
a
b
]
4
 

C

.


 
 






T

D
E

E

E
C
E
C
E











a
a a
a b
a b

E


E


E

E

C
E

0
  
 .b
 .b


~a
a
a b
D
E


E

E
0
[

]
[

]
.
b
[


]
cabcadEbEc[Ed]
d
TabcadEbEcT.



T.



.
.


1


g
(

g


g


g
)









2
a
.
a
[
]
a
[
]
a b
.
b
[
]
a b
.
b
[
]
a b
.
b
[
]
1 
..
C.  (T.
T.. T
)
2
1
ab  (cabcccabccba)Ec
2
1
Cab  (TabcTcabTcba)Ec
2













 
Massless fermions
i
5
5
4




S

E
{

i
D

[
D
]

i
}
d
x
f
L

2



i

;



i

i
S

ijkl
T
jkl
j
T

T
i
.ij
1
1 l
q

T
 (T
T

) 
S
ijk
ijk
j
ik
k
ij
ijkl
3
6








   
i
~ ~

5
5
4




S

E
{

i
D

[
D
]

i
}
d
x
f



2
1
5
d
d
b
b
4





E
{
S

4
T

S

4
T
}
d
x
d
b

8
PARTICULAR CASE






χ= 0



 
i
~ ~

5
5
4




S

E
{

i
D

[
D
]

i
}
d
x
f



2
1
5
d
b
4

E
{
S

4
T
}
d
x
d
b

8
(I.L.Shapiro, Phys.Rep. 357, 113 (2002)
Action for the torsion


1
(
1
)
2
i 4 2
i 4
2
i 4
S
[
C
]


M
S
S
d
x

M
T
T
d
x

2
M
T
S
d
x
T
SS
i
TT
i
TS
i



4
M TS = 0
Flat metric, CP invariance
SO(3,1) gauge field action
ab
ab
S
[
C
]

S
[
C
]

S
[
C
]
T
T
 T 
(
1
)
(
2
)
2
*
M
(
1
)
2 

ab
4 T 

ab
4
S
[
C
]


M
EE
E
G
d
x

EE
E
G
d
x
T
T
a
b
..
 
a
b
..



(S.Holst, Phys.Rev.D. 53, 5966 (1996)
G [D
,D
]
a
.b

D

[E
]

C


a
.b

a
.b

γ →∞
1 ab cd
G 
G
..
cd
..

2
* ab
..









EG
d
x


EA
A
d
x


EG
G
d
x


EG
G
d
x







EG
G
d
x

abcd
4
abcd
4
ab
4
ab
4
S

EG
G
d
x

EG
G
d
x

EG
G
d
x

EG
G
d
x
T
1
.
abcd
2
.
cdab
3
.
ab
4
.
ba




(
2
)
*
*
abcd
4
abcd
4
abcd
4
6 .
abcd
1 .
abcd
2 .
cdab
5
2
4
.
3
* 4
ab
. ab
(E.Elizalde,S.D.Odintsov, Phys.Atom.Nucl.56:409-411,1993)
GabcdEcEdGab
abc
Gac G...
b
GGaa
1
A

(
G

G

G

G

G

G
)
abcd
abcd
acdb
adbc
bcad
bdca
cdab
6
2
*
M
(
1
)
2 

ab
4 T 

ab
4
S
[
C
]


M
EE
E
G
d
x

EE
E
G
d
x
T
T
a
b
..
 
a
b
..




2

1
1
2
2
4~
S

M
E

R

T

S

TS
d
x

S
[
q
]





24
3
3

(
1
)
2
T
T
(S.Mercuri,
Phys. Rev. D 73 (2006) 084016)








   
i
~ ~

5
5
4




S

E
{

i
D

[
D
]

i
}
d
x
f



2
1
5
d
d
b
b
4





E
{
S

4
T

S

4
T
}
d
x
d
b

8
V 
A 5








 














ρ= τ = 0 ;γ →∞
i
~~ 5

5

4




S

E
{

i
D

[
D
]

i
}
d
x
f



2
2



 
4
3
2
2

2
22
2
22
 22
E
{
V



A



2
AV


}
d
x













32
(
1

)
M
T


 

(S.Alexandrov, Class.Quant.Grav.25:145012,2008)
(S.Mercuri,
Phys. Rev. D 73 (2006) 084016)
V 
A 5
(S.Alexandrov, Class.Quant.Grav.25:145012,2008)
ρ= τ = 0 ;γ →∞
~
E
; D

a
a
J.Bijnens, C.Bruno, E. de Rafael, Nucl.Phys. B390 (1993) 501-541
Attractive force between fermions => condensate
SM fields + technifermions














2
4
a
b
a
b
b
a
b
a
4
S

{
G
(
)(
)

G
[(
)(
)

(
L

R
)]
d
x
4
t
S
tL
tR
tR
tL
V
tL
tL
tL
tL
2

N

T
NT  24
2
4

m
 2G

t
S
t
t
N

T 
J.Bijnens, C.Bruno, E. de Rafael, Nucl.Phys. B390 (1993) 501-541
Corrections to the effective action are taken into account
Lattice discretization
Gauge field action
Conclusions
1. 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.
49
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