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 /
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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
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 2 /
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Introduction
INTRODUCTION
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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
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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.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 5 /
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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.
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Dynamical Mass Generation : NJL model
DYNAMICAL MASSGENERATION : NJL model
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Dynamical Mass Generation : NJL model
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 :
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Dynamical Mass Generation : NJL model
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.
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Dynamical Mass Generation : NJL model
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−.
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Dynamical Mass Generation : NJL model
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.
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Dynamical Mass Generation : NJL model
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).
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Supersymmetric NJL models
SUPERSYMMETRIC NJLMODELS
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Supersymmetric NJL models
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−
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Supersymmetric NJL models
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.
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Supersymmetric NJL models
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.
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Supersymmetric NJL models
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.
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Supersymmetric NJL models
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)
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Supersymmetric NJL models
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 .
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Supersymmetric NJL models
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 .,
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Supersymmetric NJL models
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
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Supersymmetric NJL models
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. )
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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.
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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)].
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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.
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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.
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GAP EQUATION ANALYSIS
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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.
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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.
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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 .
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Gap equation : Dimension-6, SNJL
DIMENSION-6 SNJL
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Gap equation : Dimension-6, SNJL
Dimension-6, SNJL
Interaction :
g2
∫
d4θΦ†+Φ
†−Φ+Φ− (1− m2
C θ2θ2)
Diagram :
Φ+ Φ−
Φ†+Φ
†−
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Gap equation : Dimension-6, SNJL
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:
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Gap equation : Dimension-6, SNJL
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|
)]
.
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Gap equation : Dimension-6, SNJL
-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.
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Gap equation : Dimension-6, SNJL
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
]
.
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Gap equation : Dimension-6, SNJL
-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.
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Gap equation : Dimension-6, SNJL
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
)
.
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Gap equation : Dimension-6, SNJL
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
) .
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Gap equation : Dimension-6, SNJL
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
)
.
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Gap equation : Dimension-6, SNJL
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.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 41 /
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Gap equation : Dimension-6, SNJL
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 = .....
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 42 /
59
Gap Equation : Dimension-5, HSNJL
DIMENSION-5 HSNJL
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59
Gap Equation : Dimension-5, HSNJL
Gap Equation : Dimension-5, HSNJL
Interaction :
− G
2
∫
d4θΦ+Φ−Φ+Φ− (1 + Bθ2) δ2(θ) + h.c . .
Diagram :
Φ+ Φ−
Φ+Φ−
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 44 /
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Gap Equation : Dimension-5, HSNJL
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 .
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 45 /
59
Gap Equation : Dimension-5, HSNJL
-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
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 46 /
59
Gap Equation : Dimension-5, HSNJL
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
) .
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 47 /
59
Symmetry Structures
SYMMETRY STRUCTURES
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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−.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 49 /
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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− ,
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 50 /
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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) .
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 51 /
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!
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 52 /
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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.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 53 /
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.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 54 /
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.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 55 /
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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.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 56 /
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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).
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Summary and Prospect
SUMMARY & PROSPECT
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Summary and Prospect
Summary and Prospect
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.
Dong-Won JUNG (KIAS) NJL, SNJL and HSNJLJun. 8, 2013 @NRF Workshop, Yonsei U. 59 /
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