Dynamical Symmetry Breaking: NJL, SNJL and HSJNJLkimcs.yonsei.ac.kr/sub_pages/conference/nrf2013/files/... · 2015. 11. 17. · Dynamical Symmetry Breaking: NJL, SNJL and HSJNJL Dong-Won

Dynamical Symmetry Breaking :

NJL, SNJL and HSJNJL

Dong-Won JUNG

KIAS,

Based on the works collaborating with

Y.M. Dai, Gaber Faisel, J.S. Lee and Otto C. W. Kong

Phys.Rev. D81 (2010) 031701JHEP 1201 (2012) 164

Phys.Rev. D87 (2013) 085033

Jun. 8, 2013 @NRF Workshop, Yonsei U.

Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 1 /

59

Outline

1 Introduction

2 Dynamical Mass Generation : NJL model

3 Supersymmetric NJL models

4 Gap equation : Dimension-6, SNJL

5 Gap Equation : Dimension-5, HSNJL

6 Symmetry Structures

7 Summary and Prospect


59

Introduction

INTRODUCTION


59

Introduction

Introduction-1

Dynamical Mass generation and Symmetry Breaking : One ofthe main concern of the modern particle physics.

Nambu and Jona-Lasinio (NJL) : a strong attractive four-fermiinteraction → Dirac mass.

After the Standard Model (SM), the mechanism for (EW)gauge symmetry breaking has been considered seriously.

ex. Nambu, Miranskiy, Tanabashi, Yamawaki, Marciano,Bardeen, Hill, Lindner, Luty...etc.

Most of top condensate (and the related) models predicted theunrealistic large top quark mass


59

Introduction

Introduction-2

Supersymmetry (SUSY) version of NJL (SNJL) : Buchmullerand Love, ’82.

→ No dynamical mass generation because of exact SUSY.

With soft SUSY breaking mass - Mass generation :Buchmullear and Ellwanger, ’84.

EWSB models with D6 four-fermion interactions (w/ Higgsboson as auxiliary field) were suggested : Clark, Love andBardeen ’90, Carena, Clark, Wagner, Bardeen and Sasaki ’92.

Alternative model with D5 holomorphic terms was suggestedrecently (HSNJL) : Jung, Kong and Lee ’2010.


59

Introduction

Introduction-3

Instead of auxiliary fields method, direct evaluation of the GapEquation is worth doing.

We perform the analyses of TOY models with D6 (SNJL) andD5 (HSNJL) interactions.

Performing SUPERGRAPH calculation, more general featuresare revealed for both SNJL and HSNJL models.


59

Dynamical Mass Generation : NJL model

DYNAMICAL MASSGENERATION : NJL model


59


Simple Example of Nambu–Jona-Lasinio model

The Lagrangian :

L = Lkinetic + Gψ+ψ−ψ+ψ−

Graphical form of the Gap Equation with D6 four-fermioninteractions :


59


Evaluating the diagram,

m = 2Gmi

(2π)4

∫

d4l(

l2 −m2)−1

.

→ G−1 =Λ2

8π2

[

1− m2

Λ2ln

[

Λ2

m2+ 1

]]

,

which has non-zero solution for m when

G ≥ Gc =8π2

Λ2.


59


Auxiliary fields method

NJL Lagrangian

L = i∂mψ+σmψ+ + i∂mψ−σ

mψ− + g2ψ+ψ−ψ+ψ−.

With the auxiliary field φ,

L → L− (φ− gψ+ψ−)(φ− gψ+ψ−)

= i∂mψ+σmψ+ + i∂mψ−σ

mψ− + φ∗φ+ gφψ+ψ− + gφψ+ψ−,

where φ = gψ+ψ−.


59


From the 1-loop effective potential for φ and v = 16π2 VΛ4 , the

symmetry breaking condition is

∂v

∂φ∗= φ

2g2

Λ2

[

1

α− 1 + η ln

(

1

η+ 1

)]

= 0,

where η = g2

Λ2φ∗φ, α = g2Λ2

8π2 .

For the above equation to get the non-zero solution for η,

g2Λ2

8π2> 1.


59


Expanding around the vacuum expectation value φ0, andcalculating the 2-point function, we can get the effective action

with φ(x) =√

12 (σ(x) + iπ(x)),

Γ = Z

∫

d4x

[

1

2σ(x)

(

�− 4m2)

σ(x) +1

2π(x)�π(x)

]

,

where Z = g2

16π2

(

ln Λ2

m2 +O (1))

.

More realistic models were proposed with top(bottom) quarkcondensates by Bardeen et. al.,(1990), Luty(1990), andB.Chung et.al., etc. (2005).


59

Supersymmetric NJL models

SUPERSYMMETRIC NJLMODELS


59


SNJL Model with Dimension-6 operator

The basic model Lagrangian is

L =

∫

d4θ[

Φ+Φ+ + Φ−Φ−

]

+

∫

d4θ[

g2 Φ+Φ−Φ+Φ−

]

.

In componet field, one can check that it contains D6 operator,

Lψ = i ψ+σµ∂µψ+ + i ψ−σ

µ∂µψ− + g2ψ+ψ−ψ+ψ−


59


Two chiral superfields are necessary to rewrite the previousLagrangian,

L =

∫

d4θ[

(Φ+Φ+ + Φ−Φ−)(1 −m2θ2θ2) + Φ1Φ1

]

+

∫

d2θ [µΦ2(Φ1 + gΦ+Φ−) ] + h.c .

The equation of motion for Φ2 gives Φ1 as a composite,Φ1 = −g Φ+Φ−, which yields Φ1Φ1 = g2Φ+Φ−Φ+Φ−

Note that soft SUSY breaking terms are important, since thepotential is zero if the supersymmetry is exact, which means nonon-trivial vacuum.


59


M. Carena et. al. used the Lagrangian,

ΓΛ = ΓYM

+

∫

dV[

Qe2VTQ + T ce−2VT T c + Bce−2VBBc]

(

1−∆2θθ2)

+

∫

dV H1e2VH1H1

(

1−M2Hθθ

2)

−∫

dS(

m0H1H2

(

1 + B0θ2)

− gTH2QTc(

1 + A0θ2))

+ h.c .

The kinetic term for H2 is generated from the 1-loop,

ZH2

∫

dV H2e2VH2H2

(

1 + A0θ2 + A0θ

2 +(

2∆2 + A20

)

θ2θ2)

,

where ZH2=

g2TNc

16π2 ln Λ2

µ2.


59


Rescaling H2 → H2

(

1− A0θ2)

/√

ZH2,

ΓΛ = ΓYM

+

∫

dV[

Qe2VTQT ce−2VT T c + Bce−2VB

]

(

1−∆2θθ2)

+

∫

dV H1e2VH1H1

(

1−M2Hθθ

2)

−∫

dS(

mH1H2

(

1 + B0θ2)

− hTH2QTc(

1 + A0θ2))

+ h.c .

+

∫

dV H2e2VH2H2

(

1 + 2∆2θ2θ2)

,

where m = m0/√

ZH2, hT = gT /

√

ZH2.


59


Renormalization group analysis has been done with the suitableboudary conditions at Λ.

m, hT →∞ while keepinghT

mfixed.

No Yukawa term is incorporated for bottom quark.

Very small tanβ is allowed, which is allowed only in quite smallparameter region. (LEP II)


59


HSNJL Model with Dimension-5 operators

As an alternative for the D6 SNJL, let’s consider theLagrangian with the holomorphic interaction:

L =

∫

d4θ[

Φ+Φ+ + Φ−Φ−

]

−∫

d2θ

[

G

2Φ+Φ−Φ+Φ−

]

+ h.c .


59


With the auxiliary field Φ0,

L =

∫

d4θ[

(Φ+Φ+ + Φ−Φ−)(1−m2θ2θ2)]

+

∫

d2θ[1

2(√µΦ0 +

√GΦ+Φ−)(

√µΦ0 +

√GΦ+Φ−)

−G

2Φ+Φ−Φ+Φ−

]

+ h.c .

=

∫

d4θ[

(Φ+Φ+ + Φ−Φ−)(1−m2θ2θ2)]

+

∫

d2θ[µ

2Φ20 +

√

µGΦ0Φ+Φ−

]

+ h.c .,


59


Realistic Model

Just for the 3rd generation,

W = G εαβQαa3 Uc a

3 Qβb3 Dc b

3 (1 + Aθ2)

Two Higgs superfields are introduced, then

W = −µ(Hd − λtQ3Uc3 )(Hu − λbQ3D

c3 )(1 + Aθ2)

= (−µHdHu + ytQ3HuUc3 + ybHdQ3D

c3 )(1 + Aθ2),

where µλt = yt , µλb = yb, µλtλb = G


59


Equation of motion for Hu gives Hd = λtQ3Uc3 while that for Hd

yields Hu = λbQ3Dc3

It is the MSSM itself with the boundary conditions for Yukawacouplings and µ, at the compositeness scale Λ.

Renormalization Group analysis is possible. (cf. Carena et. al. )


59

RG analysis

Compositeness condition translate to the boundary conditions

yt , yb, µ→∞,while keepingytyb

µ= G .

Constraints :

mt = 171.3 ± 1.6GeV

mb = 4.20+0.17−0.07 GeV

Denote the scales by Λt and Λb (i.e. where y2/4π = 1),respectively.


59

35

40

45

50

55

60

65

70

75

104 106 108 1010 1012 1014 1016

tanβ

Λ [GeV]

Ms = 10 TeV

Ms = 1 TeV

Ms = 200 GeV

Figure: We have included a SUSY threshold correction ǫb of value −0.01 in therunning of yb with [(1 + ǫb tanβ) =

√2mb/(yb v cosβ)].


59

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

103 104 105 106 107 108 109 1010 1011

y t,

y b

µ [GeV]

Ms = 1 TeV

tanβ = 57.8Λb = 104 GeV

tanβ = 42.8Λb = 1010 GeV

yb

yt

Figure: Illustration of yb and yt runnings for a couple of cases.


59

90

95

100

105

110

115

120

125

130

103 104

mh

[GeV

]

Ms [GeV]200

mA ≥ 100 GeV

mA = 140 GeV

mA = 130 GeV

mA = 120 GeV

mA = 110 GeV

mA = 100 GeV

Figure: Prediction for the lightest Higgs mass.


59

GAP EQUATION ANALYSIS


59

Supersymmetric Gap Equation

Lagrangian with Φ±(y , θ) = A±(y) +√2θψ±(y) + θ2F±(y) :

L =

∫

d4θ

[

(

Φ†+Φ+ +Φ†

−Φ−

)

(1−∆) +(

MΦ+Φ− δ2(θ) + h.c .

)

]

+ LI ,

where soft SUSY breaking mass ∆ = m2θ2θ2 and a superfieldDirac mass parameter M = m − θ2η .Effective action is

Γ =

∫

d4p

2π4

∫

d2θΦ+(−p, θ) Γ(2)+−(p, θ2)Φ−(p, θ) + h.c .+ · · · ,

where

Φ±(p, θ) =

∫

d4x e−ip.xΦ±(x , θ) .

The Γ(2)+−(p, θ

2) function again contains in general a scalar partand a part with θ2, in exact analog to the parameter M.


59

Self-consistent Hartree approximation :

LI = −[

MΦ+Φ−δ2(θ) + h.c .

]

+ LTI ,

the mass terms added and subtracted as above, and there areTRUE interaction terms.

One looks for nontrivial solution for M from the equation

Γ(2)+−(p, θ

2)∣

∣

∣

on-shell

= 0

∼ Σ+−(p, θ2)∣

∣

on-shell= 0

∼ −M = Σ(loop)+− (p, θ2)

∣

∣

∣

on-shell

,

where Σ(loop)+− from loop diagrams involving the true interactions.


59

The propagators are

〈T (Φ±(1)Φ†±(2))〉

=−i

p2 + |m|2 δ412 −

i

[(p2 + |m|2 + m2)2 − |η|2](

ηmθ21 + ηmθ12)

δ412

+i [m2(p2 + |m|2 + m2)− |η|2]

(p2 + |m|2)[(p2 + |m|2 + m2)2 − |η|2]

[

|m|2θ21 θ12+

D21 θ

21 θ1

2D

21

16

]

δ412 ,

〈T (Φ+(1)Φ−(2))〉

=i m

p2(p2 + |m|2)D2

1

4δ412

− i

[(p2 + |m|2 + m2)2 − |η|2]

[

ηD21 θ1

2

4− η|m|2 D2

1 θ12

4p2

]

δ412

+i m [m2(p2 + |m|2 + m2)− |η|2]

(p2 + |m|2)[(p2 + |m|2 + m2)2 − |η|2]

[

D21 θ

21 θ1

2

4+θ12θ21 D

21

4

]

δ412 .


59

Gap equation : Dimension-6, SNJL

DIMENSION-6 SNJL


59


Dimension-6, SNJL

Interaction :

g2

∫

d4θΦ†+Φ

†−Φ+Φ− (1− m2

C θ2θ2)

Diagram :

Φ+ Φ−

Φ†+Φ

†−


59


After evaluation, a pair of gap equations is derived.

m = 2mg2I1(|m|2, m2, |η|,Λ2) ,

η = −η g2m2CI2(|m|2, m2, |η|,Λ2) .

Note that the equations for m and η are somewhat decoupled,which we will see is not the case with the D5 interaction.

The analytic forms and generic shapes of the integral functionsare as follows:


59


I1(|m|2, m2, |η|,Λ2) =

∫

d4k

(2π)4[m2(k2 + |m|2 + m2)− |η|2]

(k2 + |m|2)[(k2 + |m|2 + m2)2 − |η|2]

=1

16π2

[

1

2(|m|2 + m2) ln

(|m|2 + m2 + Λ2)2 − |η|2(|m|2 + m2)2 − |η|2 − |m|2 ln (|m|2 + Λ2)

|m|2

+ |η|(

tanh−1 |m|2 + m2 + Λ2

|η| − tanh−1 |m|2 + m2

|η|

)]

,

I2(|m|2, m2, |η|,Λ2) =

∫

d4k

(2π)41

(k2 + |m|2 + m2)2 − |η|2

=1

16π2

[

1

2ln

(|m|2 + m2 + Λ2)2 − |η|2(|m|2 + m2)2 − |η|2

+|m|2 + m2

|η|

(

tanh−1 |m|2 + m2 + Λ2

|η| − tanh−1 |m|2 + m2

|η|

)]

.


59


-50 000 0 50 000

-300

-200

-100

0

100

200

300

Η

m

-50 000

0

50 000

Η

-200

-100

0

100

200

m

-200

0

200

Figure: Contour plot (left) and 3D plot (right) of I1 in the real (η,m) plane.

The blank regions are where |η| > m2 + m2 : tachyonic mass fora scalar.

maximum value at the origin (η = 0,m = 0).

Along the border lines that satisfy |η| = m2 + m2, the value of

function approaches − Λ2

32π2 , definitely negative.


59


The maximum value at the origin is

I1(

0, m2, 0,Λ2)

=m2

16π2log

[

1 +Λ2

m2

]

.

On the η-axis (m = 0), at the both ends, i.e.(

|η| = m2, 0)

, wehave

I1(

0, m2, |η| = m2,Λ2)

=m2

16π2log

[

1 +Λ2

2m2

]

.


59


-50 000 0 50 000

-300

-200

-100

0

100

200

300

Η

m

-50 000

0

50 000

Η

-200

-100

0

100

200

m0.00

0.01

0.02

0.03

Figure: Contour plot (left) and 3D plot(right) of I2 in the real (η,m) plane.

A saddle shaped : concave along the η-axis and convex alongthe m-axis.

Maximum value at(

|η| = m2, 0)

on the centers of the tachyonicexclusion borders (|η| = m2 + m2)

local minimum at the origin.

positive definite.


59


The maximum is given by

I2(

0, m2, |η| = m2,Λ2)

=1

16π2log

[

1 +Λ2

2m2

]

.

The origin is a local minimum, with a value given by

I2(

0, m2, 0,Λ2)

=1

16π2

(

log

[

1 +Λ2

m2

]

− Λ2

Λ2 + m2

)

.


59


The gap equations can be written in simplifed form as

m(

1− 2g2I1)

= 0, η(

1 + g2m2C I2

)

= 0.

When m2C= 0 :

→ Normal SNJL case. ( It is reduced to NJL when m →∞.)

2mg

Λ2

[

8π2

g2Λ2− |m|

2 + m2

Λ2ln

(

1 +Λ2

|m|2 + m2

)

+|m|2Λ2

ln

(

1 +Λ2

|m|2)]

= 0 .

Then the condition for non-zero Dirac mass m is

g2 >8π2

m2 ln(

1 + Λ2

m2

) .


59


From the equation I2 =1

−g2m2C

, at least

m2C < 0

must be satisfied for non-zero η.

Another extreme solution : m = 0, η 6= 0

→ According to the shape of I2, then condition is

1

16π2

[

ln

(

1 +Λ2

m2

)

− Λ2

Λ2 + m2

]

≤ 1

−g2m2C

<1

16π2ln

(

1 +Λ2

2m2

)

.


59


Most generally, m 6= 0, η 6= 0 are possible.

Two distinct configurations according to 1−g2m2

C

(blue).

-40 000 -20 000 0 20 000 40 000

-300

-200

-100

0

100

200

300

Η

m

@ ��

1

2 g2DMIN

@ ��

1

2 g2DMAX

- ��

1

g2 mC

~2

-40 000 -20 000 0 20 000 40 000

-300

-200

-100

0

100

200

300

Η

m

@ ��

1

2 g2DMIN

@ ��

1

2 g2DMAX

� - ��

1

g2 mC

~2

12g2 |max(yellow) and 1

2g2 |min(red) can be derived, in principle.


59


It’s hard to get analyitc form.

For example,

1

2g2|min =

16π2

g2m2C

Λ2 −

Λ2 − 2

e

16π2

g2m2C − 1

m2

ln

2e

16π2

g2m2C −1

Λ2−2

e

16π2

g2m2C −1

m2

Λ2−2

e

16π2

g2m2C −1

m2

32π2

e

16π2

g2m2C − 1

,

1

2g2|max = .....


59

Gap Equation : Dimension-5, HSNJL

DIMENSION-5 HSNJL


59



Interaction :

− G

2

∫

d4θΦ+Φ−Φ+Φ− (1 + Bθ2) δ2(θ) + h.c . .

Diagram :

Φ+ Φ−

Φ+Φ−


59


A pair of gap equations is derived

m =ηG

2I2(|m|2, m2, |η|,Λ2) ,

η = mG I1(|m|2, m2, |η|,Λ2)− ηGB

2I2(|m|2, m2, |η|,Λ2) .

Unlike the D-6 case, those are coupled with each other, i.e.,

m = 0←→ η = 0.

For simplicity, let’s first consier B = 0.

m =ηG

2I2,

η = mG I1.

If there is solution, then that must satisfy

η2

2I2 = m2I1 .


59


-50 000 0 50 000

-1000

-500

0

500

1000

Η

m��

Η2

2I2=m2 I1, H3L

��

m

G= ��

Η

2I2, H1L

��

Η

G=m I1, H2L


59


The condition for non-zero m(η) is

G 2 > G 20 =

512π2

m2 ln(

1 + Λ2

m2

) [

ln(

1 + Λ2

m2

)

− Λ2

m2+Λ2

] .

General B 6= 0 case, it is also hard to write down the condtionexplicitly. Instead, we can derive in the case of small B .

G >√

G 20 + b2 + b ∼ G0 + b ,

where

b = B8π2

m2 ln(

1 + Λ2

m2

) .


59

Symmetry Structures

SYMMETRY STRUCTURES


59

Symmetry Structures

Effective Field Theory model of HSNJL

Condensation in HSNJL,

−G

2

∫

d2θ 〈Φ+Φ−〉Φ+Φ− (1 + Bθ2) −→∫

d2θ (m − η θ2)Φ+Φ− ,

which gives

m (Φ+Φ−)F = −G 〈Φ+Φ−〉A (Φ+Φ−)F ,

η (Φ+Φ−)A = G

(

〈Φ+Φ−〉F + B 〈Φ+Φ−〉A)

(Φ+Φ−)A .

Note that (Φ+Φ−)F = A+F− + F+A− − ψ+ψ−.


59

Symmetry Structures

This can be recasted with auxiliary field,

LeffHSNJL =

∫

d4θ(

Φ†+Φ+ +Φ†

−Φ−

)

(1− m2θ2θ2)

+

{∫

d2θ

[

1

2(√µ0Φ0 +

√GΦ+Φ−)(

√µ0Φ0 +

√GΦ+Φ−)

−G

2Φ+Φ−Φ+Φ−

]

(1 + Bθ2) + h.c .

}

=

∫

d4θ(

Φ†+Φ+ +Φ†

−Φ−

)

(1− m2θ2θ2)

+

{∫

d2θ[µ02Φ20 +

√

µ0GΦ0Φ+Φ−

]

(1 + Bθ2) + h.c .

}

,

where Φ0 is the auxiliary Higgs superfield that comes out as thecomposite

Φ0 = −√

G/µ0Φ+Φ− ,


59

Symmetry Structures

As the Φ0 develops a vacuum expection value (VEV), we havethe mass

M = m − ηθ2 =√

µ0G 〈Φ0〉 (1 + Bθ2)

or

m =√

µ0G 〈Φ0〉A ,

η = −√

µ0G ( 〈Φ0〉F + B 〈Φ0〉A) .


59

Symmetry Structures

Effective Field Theory model of SNJL

Interaction term is given

g2

∫

d4θ⟨

Φ†+Φ

†−

⟩

Φ+Φ− (1− m2C θ

2θ2) .

The component content is given by

g2⟨

Φ†+Φ

†−

⟩

F(Φ+Φ−)F and − g2m2

C

⟨

Φ†+Φ

†−

⟩

A(Φ+Φ−)A .

which means

m = g2⟨

Φ†+Φ

†−

⟩

F,

η = g2m2C

⟨

Φ†+Φ

†−

⟩

A.

→ We need different auxiliary field!


59

Symmetry Structures

Again, the Lagrangian casts

LeffSNJL =

∫

d4θ[(

Φ†+Φ+ +Φ†

−Φ−

)

(1− m2θ2θ2) + Φ†CΦC (1− m2

Cθ2θ2)

]

+

{∫

d2θ µ ΦH (ΦC + gΦ+Φ−) (1 + Aθ2) + h.c .

}

.

whereΦC = −gΦ+Φ− .

A is arbitrary. (cf. B in HSNJL)

ΦC has more complex structure, i.e.,

µFH + AµAH = −∂m∂mA∗C − m2

C A∗C ,

µψH = −iσm∂mψ†C ,

µAH = −F ∗C,

from EOM.


59

Symmetry Structures

the Lagrangian in the presence of nontrivial VEV for ΦH gives

M = m − η θ2 = µg 〈ΦH〉 (1 + Aθ2) .

The result is

m = µg 〈ΦH〉A −→ −g⟨

Φ†C

⟩

F,

η = −µg ( 〈ΦH〉F + A 〈ΦH〉A) −→ −g m2C

⟨

Φ†C

⟩

A.

Note that nonzero η and hence nonzero⟨

Φ†C

⟩

Ais possible even

with A = 0.


59

Symmetry Structures

Symmetry Breaking (for SNJL)

The Symmetry Breaking for (S)NJL : the chiral symmetry

U(1)V × U(1)A → U(1)V .

More generally, U(1)V × U(1)A symmetry is

U(1)+ × U(1)−, or Φ+,Φ− number symmetries.

→ No reason to expect an interaction term to respect. (HSNJLbreaks it explicitly)

So in the SNJL model,

U(1)+ × U(1)− −→ U(1)+ − U(1)−,

dynamically.


59

Symmetry Structures

Symmetry Breaking for HSNJL?

Φ20 term in the simplest HSNJL : Φ0 ,real representation.

Symmetries : Z4 for Φ+ and Φ− with e iπ/2

→ The D5 term (Φ+Φ−)2respects the symmetry while the

Φ+Φ− Dirac mass term is not allowed.

→ Z4 symmetry is dynamically broken by the Φ+Φ− vacuumcondensate just with Z2.


59

Symmetry Structures

Another example :

Φ+, SU(2)-triplet while having an equal and opposite U(1)charge with a singlet Φ−.

→ The Φ+Φ− term is then invariant under the U(1) butremains an SU(2)-triplet.

→ Its condensate breaks the SU(2) symmetry. In that case, thecondensate will be along one of the three SU(2) components.

In the same way, EWSB model was built before by Jung, Kongand Lee, PRD81, 031701 (2010).


59

Summary and Prospect

SUMMARY & PROSPECT


59



Symmetry Breaking structures for the supersymmetric modelswith D6 and D5 are studied.

Most general forms of the Gap Equations are derived withSupergraph technique.

Novel features for SNJL and HSJNL are presented, includingthe distinction of the two.

It can be helpful for the studies for SpontaneousSupersymmetry Breaking?

More complete phenomenological study for the realistc HSNJLwill be worth doing.


59

Documents

Dynamical Symmetry Breaking: NJL, SNJL and HSJNJLkimcs.yonsei.ac.kr/sub_pages/conference/nrf2013/files/... · 2015. 11. 17. · Dynamical Symmetry Breaking: NJL, SNJL and HSJNJL Dong-Won