Nambu-Goldstone mode for supersymmetry breaking in QCD and Bose-Fermi cold atom system at BEC phase

Nambu-Goldstone mode for supersymmetry
breaking in QCD and Bose-Fermi cold atom
system at BEC phase
Daisuke Satow (Frankfurt Univ. !)

Collaborators: Jean-Paul Blaizot (Saclay CEA ")

Yoshimasa Hidaka (RIKEN #)

Teiji Kunihiro (Kyoto Univ. #)
QCD: Y. Hidaka, DS, and T. Kunihiro, Nucl. Phys. A 876, 93 (2012).
DS, Phys. Rev. D 87, 096011 (2013).
Cold atom: J-P. Blaizot, Y. Hidaka, DS, Phys.Rev. A 92, 063629 (2015).
J-P. Blaizot, Y. Hidaka, DS, arXiv:1707.05634 [cond-mat.quant-gas].
「君、３フレーバーのNJ
UA(1)アノマリーのQCD真
効果とかの研究をや

Outline
2
• Introduction (Supersymmetry,
Nambu-Goldstone mode for SUSY-breaking)
• Relativistic system (Wess-Zumino model, QCD)
• Cold atom system (No BEC)
• Cold atom system (BEC)
• Summary
←Main topic

3
Symmetry related to interchanging
a boson and a fermion
Supersymmetry (SUSY)
fb fb
=
Same energy

4
Supercharge operator: annihilate one fermion
and create one boson (and its inverse process)
SUSY and supercharge
Mathematically, [Q, H]=0
fb
Q~bf: supercharge
fb

5
SUSY breaking
As long as the system is not at vacuum,
the states always contain double degeneracy.
examples: High Temperature (QCD), Fermi sea+BEC (cold atom)
ground state
Q
Q2=0Q2=0

6
NG “fermion” which is related to SUSY breaking
Generally, degeneracy generates zero energy
excitation (Nambu-Goldstone (NG) mode).
degeneracy
order parameter
NG mode
We expect that NG mode appears in SUSY case.
No NG mode

7
“When the order parameter is ﬁnite, the propagator
in the left-hand side has a pole at p→0.”
Order parameterNG mode
Broken symmetry
Nambu-Goldstone theorem:
(fermion ver.)
Jµ: supercurrent
Q=J0: supercharge
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989)
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i

8
If we set O=Q, order parameter is energy-momentum
tensor (T µν) in the relativistic system.
single fermion intermediate state. Hence, we can write, in momentum space
Γµν
JJ (k) ≡ d4
x eik(x−y)
⟨TJµ
(x)Jν
(y)⟩
= Γµ
JΨ(k)S(k)Γν
ΨJ (k).
By making use of the Ward-Takahashi identity (3.39), we ﬁnd
−ikµΓµν
JJ (k) = −im⟨A⟩Γν
ΨJ (k).
This can now be calculated by using our earlier results for Γµ
JΨ(k).
First, in the limit of high temperature, we make use of (3.37) and obtai
the left hand side of the Ward-Takahashi identity as
−ikµΓµν
JJ (k) = m2
⟨A⟩2 π2
g2
γ0
−1
3 γi
=
π2
4
T4
γ0
−1
3 γi
.
For the right hand side, we must evaluate the thermal expectation valu
of the energy-momentum tensor. In a thermal equilibrium state, it reads
⟨Tµν
⟩ = diag(ρ, p, p, p),
where in the high-temperature limit the pressure is given by p = 1
3ρ, an
the energy density can be calculated as
In the presence of medium, SUSY is always broken.
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989)
Broken symmetry
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i

9
SUSY breaking
SUSY is fermionic symmetry,
so the NG mode is fermion (Goldstino).
Rare fermionic zero mode
fb
Q
fb
NG mode appears in <QQ>.
NG mode
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i

Outline
10
• Relativistic system (Wess-Zumino model,
QCD)
• Summary

Goldstino in Wess-Zumino model
11
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989).
In Wess-Zumino model, the goldstino exists.
3.4. The phonino
a b
ψ
Q~(A,B)×Ψ
A, B
(Weyl fermion+Scalar and Pseudo-scalar bosons)
<QQ>~
Pole appears in the Green function.

Goldstino in Wess-Zumino model
12
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989).
Through the fermion-boson coupling,
it appears also in fermion propagator.
3.4. The phonino
a b
Figure 3.6: Typical one- and two-loop co
energy
ψ
gg
dispersion relation Reω＝p/3
Residue g2/π2

13
(Quasi) SUSY in QCD
V. V. Lebedev and A. V. Smilga, Annals Phys. 202, 229 (1990)
There is SUSY approximately if we neglect the
interaction.
q g
=
Both of the quark and the gluon are regarded as
massless at high T

Quasi-goldstino in QCD
14
In weak coupling regime, we established the
existence of the (quasi) goldstino in QCD.
Dispersion relation Reω＝p/3
Damping rate Imω=ζq+ζg=O(g2T)
Residue
as follows: We expand the vertex function in terms of ˜p/g T as in Eq. (2.26). The zeroth
order solution is
A =
1
1 + Cf λ
1, B(k) =
1
k0
2Cf λ
1 + Cf λ
γ0
, C(k) = 0. (2.49)
Here A, B(k), and C(k) are defined by Eq. (2.27), and λ is defined by Eq. (2.7). The
self-consistent equation which determines the vertex function at the first order δΓµ(p, k)
is written as
δΓµ
(p, k) = Cf
d4k′
(2π)4
˜X(k′
)
kνγµ + γνk′µ
k · k′
/k′
PT
νρ(k′
) −2
˜p · k′
δm2
Aγρ
+ δΓρ
(p, k′
) .
(2.50)
Then δΠ(p) defined by Eq. (2.33) becomes
δΠ(p) =
16π2A2λ3Cf
g2
(γ0
(p0
+ iζ) + vp · γ). (2.51)
Owing to Eq. (2.43), the retarded quark self-energy is found to be
ΣR
(p) = Cf
d4k
(2π)4
˜X(k)
γµ/kPT
µν(k)Γν(p, k)
1 + 2˜p · k/δm2
= −
1
Z
(γ0
(p0
+ iζ) + vp · γ),
(2.52)
where the expression of the residue Z will be given shortly. We note that this expression
is the same as that in QED except for the numerical factor. Thus, the expression for the
pole position of the ultrasoft mode in QCD is the same as in QED, while the residue of
that mode is not: The residue is
Z =
g2
16π2λ2Cf
(1 + Cf λ)2
=
g2N
8π2(N2 − 1)
5
6
N +
1
2N
+
2
3
Nf
2
. (2.53)
We can also show that the analytic solution of the self-consistent equation satisfies
the WT identity in the leading order as in QED, by checking that the counterparts of
Eqs. (2.40) and (2.41) are satisfied.

Outline
15
• Summary

Experimental detection
16
How can we detect goldstino in experiment?
So far, detecting quark spectrum in heavy ion collision
has not been done…
Picture: UrQMD group, Frankfurt.

17
SUSY in Cold Atom System
Picture: Ferlaino group, Innsbruck.
cf: T. Ozawa, Nature Physics 11, 801 (2015),
Wess-Zumino model: Y. Yu, and K. Yang, PRL 105, 150605 (2010),
Dense QCD: K. Maeda, G. Baym and T. Hatsuda, PRL 103, 085301 (2009),
Relativistic QED: Kapit and Mueller, PRA 83, 033625 (2011).
Cold atom system is
easier to realize, and its experiment is cleaner.
(Test site of many-body physics)

18
SUSY in Cold Atom System Y. Yu and K. Yang, PRL 100, 090404.
Prepare Bose-Fermi mixture.
fb
1. Same mass: m=mb=mf
2. Same interaction: U=Ubb=Ubf
o achieve an exact SUSY mixture, the system parameters
t be ﬁne-tuned, which requires elaborate experimental
ps and then loses the generality. In this article, we explore
to observe the SUSY response by means of a spectroscopy
surement, even if the mixture deviates slightly from the
Y and the bosons do not condense to form a whole ordered
e. This can resolve the ﬁne-tuning restraints in measuring
SUSY response. On the other hand, the explicit breaking
e SUSY may create new excitations, the bosonic particle–
mionic hole individual continuous excitations, other than
collective goldstino-like mode. Although our theory is
elativistic, the creation of these new excitations due to
Y explicit breaking should be quite general. This may be
lpful point in the study of SUSY in relativistic theory.
We consider a mixture of Bose molecules b and Fermi atoms
ith on-site interaction in a d-dimensional optical lattice
= 2, 3) [see Fig.1(a)]. With properly tuned interactions and
ping amplitudes, this b-f mixture may become SUSY [3].
are interested in a special kind of molecule b, a bound state
, and another species of Fermi atom F with binding energy
and we restrict our analysis to the normal phase of the b-f
mode [3]. In this sense, we regard these excitations a
SUSY responses. The PA spectrum is directly related t
the molecular formation rate varying as the detuning
faithfully describes these two types of excitations. The po
of peak in the PA spectrum determines the frequency o
collective zero-momentum mode. This molecular form
rate is measured by the number variation of the F atom
time. Experimentally, the number counting of atoms is
simpler than detecting the single atom spectrum.
Model setup. The system illustrated in Fig. 1(a) is desc
by a Hamiltonian H = H0 + Hex, where H0 = Hbf +
with Hbf = Hb + Hf + V . By means of the Feshbach
onance [9], the scattering lengths between F and the
mixture can be adjusted to negligibly small. In the
binding approximation, one has
Hα = −
⟨ij⟩
tαa
α†
i aα
j − µα
i
a
α†
i aα
i ,
V =
Ubb
2 i
nb
i nb
i − 1 + Ubf
i
nb
i n
f
i ,
-2947/2010/81(1)/011604(4) 011604-1 ©2010 The American Physical S
=Ubb Ubf

19
SUSY in Cold Atom System Y. Yu and K. Yang, PRL 100, 090404.
Possible candidates: 6Li-7Li, 173Yb-174Yb
(mb/mf =1.17, U is easy to tune) (Ubb/Ubf =1.32, m is almost same)
m=mb=mf
Ubb =Ubf
(Q=∫dx b†
xfx, Q†=∫dx bx f †
x)
[Q, H]=0
S. Endo, private communication.

20
If we set O=Q†, NG mode appears in <QQ†>.
Order parameter is density (<{Q, Q†}>=ρ) in this case.
Goldstino in cold atom systems
Q =bf †
Q† =b†f
SUSY is always broken when ρ is ﬁnite.
Broken symmetry
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i

21
Explicit breaking of SUSY Y. Yu and K. Yang, PRL 100, 090404.
Finite density causes explicit SUSY breaking.
[Q, H-µf Nf -µb Nb]=-ΔµQ
µf - µbGrand Canonical Hamiltonian
Q
Q†
Δµ
Gapped Goldstino
(ω=-Δµ)

22
E
nf
E
nb
Q†
E
nb
εk
(T=0, weak coupling)
Only this state (Fermi sea+BEC)
is realized as ground state.
Explicit breaking of SUSY
δE=Δµ=µf -µb>0
µf=εf, µb=0
E
nf
εk
X
k
Q
δE=-Δµ<0

Goldstino spectrum
23
Calculate the spectrum of the Goldstino.
2
he generic
(2.1)
(2.2)
(2.3)
x)nf (x) ,
(2.4)
x) are the
s hamilto-
wavelength
[12]. Note
where the angular brakets denote an average over the
ground state of the system. Its Fourier transform is writ-
ten as
GR
(p) = i
Z
dt
Z
d3
x ei!t ip·x
✓(t)h{q(t, x), q†
(0)}i.
(2.8)
Here we have introduced a 4-vector notation, to be used
throughout: xµ
⌘ (t, x) and pµ
⌘ (!, p). The frequency
! is assumed to contain a small positive imaginary part
✏ (! ! ! + i✏) in order to take into account the retarded
condition. Such a small imaginary part will not be indi-
cated explicitly in order to simplify the formulae. In fact,
we shall also most of the time drop the superscript R, and
indicate it only when necessary to avoid confusion.
Let us recall some general features of this Green’s func-
tion by looking at its spectral representation in terms of
the excited states n and m that can be reached from
conservation law and the canonical (anti-) commuta-
n relations [12, 13]. It does not depend on the details
the Hamiltonian. This is the Goldstino’s counter part
gapped Nambu-Goldstone modes [20–22]. In the fol-
wing, we shall often refer to G(!, p) as the Goldstino
pagator.
We shall also be interested in the associated Goldstino
ctral function
(!, p) = 2Im G(!, p). (2.12)
is spectral function obeys simple sum rules [13]. The
t sum rule determines the zeroth moment of the spec-
l function. It is valid regardless of the details of the
miltonian, and reads
Z
d¯!
(p) = ⇢. (2.13)
in ord
Th
depen
sume
two p
the b
U. T
the fe
shall
ble or
have
Ferm
for th
to th
U⇢ h
can b
with

24
Goldstino spectrum (free case)
Free case (U=0)
qp~fkb†
k+p
f
b
q†
p~f†
kbk+p
⌘ k2
/(2m), ¯! = !+ µ0 with µ0 = k2
F /(2m),
= ⌦⇢b, while np ⌘ ✓(kF p) denotes the fermion
on number.
q†q
q†
q
The one-loop diagrams contributing to G0
. The full
line represents a fermion (boson) propagator. The
n used for these diagrams, and those below that con-
ws is as follows. The time ﬂows from left to right.
pointing to the right indicates a “particle”, while an
nting to the left indicates a “hole”. The boson hole
Cut
(ω, p)

2525
Landau damping
Continuum.
Width is Δω=pkF/m
-1~10~kF
E
nf
E
nb
E
nb
εk+p
E
nf
εk
qp~Σk fkb†
k+p
εk=k2/2m, |k|<kF
(-ω, -p) (-ω+k2/2m, -p+k)
should be on-shell: ω=-p2/2m-p k cosθ/m

26
Pole, No width.
εk=0, k=0
(ω, p)
(ω, p)
should be on-shell: ω=p2/2m
Other value of ω is not allowed!
Dispersion Relation ω＝p2/2m
Strength ρb
q†
p~Σk f†
k-pbk
E
nf
E
nb
E
nb
E
nf
ε-p
k=0
nf(1+nb)+(1-nf)nb
PoleContinuum

27
Continuum+Pole.
is
4.
n-
ds
he
ve
1
d-
it
m
ee
ce
is
by
st
he
n-
4)
he
stino. The process in the left represents q†
, which replace a
boson with a fermion, while the one in the right represents
q, which replaces a fermion with a boson. The blue square
represents the Fermi sea.
-2
-1
0
1
2
0 0.2 0.4 0.6 0.8 1 1.2
ω
-
/εF
p/kF
FIG. 3: The continuum (red shaded area) and the pole (blue
! =
p2
2m
! =
p2
2m
⇢f
⇢b

28
These pole and continuum satisfies the NG theorem.
tly related to the mag-
F . The range of this
d its shape is in Fig. 4.
nit of momentum (en-
Eq. (4.1) corresponds
h turns a boson in the
0) into a fermion above
m |p| kF (see Figs. 1
pole contribution hid-
t line of Eq. (4.1): it
sea with a momentum
lls the condensate (see
plified by the presence
N0 accompanying this
bution is cancelled by
first term in the first
s the first term of the
lds the following con-
p2
2m
◆
. (4.4)
FIG. 2: Particle-hole excitations contributing to the Gold-
stino. The process in the left represents q†
, which replace a
boson with a fermion, while the one in the right represents
q, which replaces a fermion with a boson. The blue square
represents the Fermi sea.
-2
-1
0
1
2
0 0.2 0.4 0.6 0.8 1 1.2
ω
-
/εF
p/kF
! =
p2
2m
! =
p2
2m
⇢f
⇢b

Goldstino spectrum (interacting case)
29
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
Switch on the interaction.
For simplicity, we start with two-dimension case, in which there are no BEC.
Mean ﬁeld approximation
fermion: Uρb boson: Uρf +2Uρb
Q, Q†
Uρ
Different MF correction
Gap in goldstino spectrum?

30
G0
(p) =
Z
d2
k
(2⇡)2
nF (✏f
k) + nB(✏b
k+p)
! + [2k · p + p2]/2m + U⇢
implies
ution of
pectrum
entually
an su↵er
densities
eraction
calcula-
ch occur
we note
plies
(3.3)
where ✏0
k ⌘ k2
/(2m), ¯! = !+ µ0 with µ0 = k
and N0 = ⌦⇢b, while np ⌘ ✓(kF p) denotes the
occupation number.
q
q† q†q
q†
q
FIG. 1: The one-loop diagrams contributing to G0
.
(dashed) line represents a fermion (boson) propaga
convention used for these diagrams, and those below
tain arrows is as follows. The time ﬂows from left
An arrow pointing to the right indicates a “particle”
arrow pointing to the left indicates a “hole”. The b
propagator is disconnected, and represented by the
Actually, it is the case in Green function
It contradicts with the exact result (Gapless NG mode).
We should have missed something…
At p=0
~U -1
⇢
! + U⇢

31
All ring diagrams contributes at the same order.
We need to sum up inﬁnite ring diagrams.
Random Phase Approximation
U-1 ×U ×U-1=U-1U-1
7
+ + +...=
+= + +...q†
+= + +...q
: The ring diagrams that are summed in the RPA
tion of GRPA
, Eq. (4.12). Note that the propagators
mean ﬁeld propagators. The interaction joining two
ive bubbles is the one in the second line of Eq. (4.11),
H4.

32
Result 1. Goldstino Pole
At p=0
GRPA(p) =
1
[G0(p)] 1 + U
G0
=
⇢
! + U⇢
Gap disappears!
GRPA(!, 0) =
⇢
!

33
Dispersion Relation ω＝-Δµ +αp2/2m
Strength
(p=0: maximum value allowed by sum rule.
The sum rule is saturated by the pole)
Z = ⇢ p2 1
4⇡
✓
✏F
U⇢
◆2
Expression at finite p
10
h
Z = ⇢
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
↵ ⌘
⇢b ⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
= ↵s +
4
5
⇢f
⇢
"F
U⇢
. (4.30)
se formulae reduce to Eqs. (4.17) and (4.18) [(4.19)]
n ⇢b = 0 [⇢f = 0], as they should. Also that the ex-
sion for ↵ is the same as that obtained in the absence
EC [13].
he location of the Goldstino pole obtained numer-
y is plotted in Fig. 11, and compared to the ap-
ximate expression ¯! = ↵p2
/(2m). The interaction
ngth is set to a small value, U⇢f /✏F = 0.1, or kF a =
/4 ' 0.24 in terms of a, for which the weak-coupling
ysis is reliable. One sees on Fig. 11 that the approxi-
e expression is accurate as long as |p| . 0.16kF . This
deed the expected range of validity of the expansion,
ely U⇢ kF |p|/m, as can be seen from the denomi-
or of the expression of GMF
cont, Eq. (4.9). This condition
s to |p| ⌧ U⇢m/kF ' 0.15kF for the current values
he parameters. Note that because the continuum is
ted down by the MF correction U⇢, as compared to
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
FIG. 11: The range of the continuum (red shed area), the
numerical result for the pole position of the Goldstino prop-
agator (black solid line), and the pole position obtained from
the small |p| expansion, Eq. (4.30) (blue long-dashed line).
For illustration of the “level repulsion”, the pole positions of
GMF
pole (green dashed line) and GRPA
(magenta dotted line) are
also plotted. Note that at p = 0 the tip of the continuum
corresponds to the fictitious pole at ¯! = U⇢, carrying no
spectral weight. The densities are the same as in Sec. IV A,
i.e., ⇢b = 2⇢f , and the interaction strength is U⇢f /✏F = 0.1.
↵ =
⇢b ⇢f
⇢
+
✏F
U⇢
⇢f
⇢
ysis is reliable. One sees on Fig. 11 that the approxi-
expression is accurate as long as |p| . 0.16kF . This
deed the expected range of validity of the expansion,
ely U⇢ kF |p|/m, as can be seen from the denomi-
r of the expression of GMF
to |p| ⌧ U⇢m/kF ' 0.15kF for the current values
e parameters. Note that because the continuum is
ed down by the MF correction U⇢, as compared to
ree case (4.2), the Goldstino pole remains out of the
nuum as long as |p| is smaller than ⇠ 0.21kF .
so plotted in Fig. 11 are the dispersion relations cor-
nding to the poles of GMF
pole (Eq. (4.21)) and GRPA
(4.12)). This illustrates the e↵ect of the level re-
on already discussed in the case |p| = 0, yielding
ually the distribution of spectral weight between
ontinuum and the Goldstino pole. Of course, the
rsymmetry plays a crucial role here in putting the
stino pole at ¯! = 0 for p = 0.
e spectral function is analyzed in more details in
12. The contributions to the zeroth moment of
the pole and the continuum are displayed in the up-
anel of this figure. At small momenta, |p| . 0.11kF ,
are well accounted for by the expansion (4.29). In
ame plot, we see that the continuum contribution
ppressed for small momentum, with all the spec-
weight being carried there by the Goldstino. The
panel of Fig. 12 reveals large cancellation between
the small |p| expansion, Eq. (4.30) (blue lo
For illustration of the “level repulsion”, the
GMF
(magenta
also plotted. Note that at p = 0 the tip o
corresponds to the fictitious pole at ¯! =
spectral weight. The densities are the same
i.e., ⇢b = 2⇢f , and the interaction strength is
contribution vanishes since the pole is a
continuum (see Fig. 11). The sum of th
continuum contributions to the zeroth
⇢, as it should because of the sum rule
plies that the spectral weight of the conti
rapidly around the momentum at which
sorbed, which is demonstrated in Fig. 12
iors, namely the suppression (enhanceme
tinuum at small |p| (above |p| ' 0.21kF
also from the spectral function plotted
V. PHENOMENOLOGICAL IMP
The strong coupling between the fermio
stino may o↵er a possibility to infer the p
f F F
ms of a, for which the weak-coupling
One sees on Fig. 11 that the approxi-
curate as long as |p| . 0.16kF . This
d range of validity of the expansion,
m, as can be seen from the denomi-
n of GMF
kF ' 0.15kF for the current values
Note that because the continuum is
MF correction U⇢, as compared to
e Goldstino pole remains out of the
|p| is smaller than ⇠ 0.21kF .
. 11 are the dispersion relations cor-
oles of GMF
pole (Eq. (4.21)) and GRPA
lustrates the e↵ect of the level re-
ussed in the case |p| = 0, yielding
bution of spectral weight between
he Goldstino pole. Of course, the
s a crucial role here in putting the
= 0 for p = 0.
tion is analyzed in more details in
butions to the zeroth moment of
e continuum are displayed in the up-
e. At small momenta, |p| . 0.11kF ,
ted for by the expansion (4.29). In
e that the continuum contribution
all momentum, with all the spec-
rried there by the Goldstino. The
2 reveals large cancellation between
uum contributions to the first mo-
function. This can be understood
momentum, the pole contribution is
m) by using Eq. (4.29). On the other
(2.14) requires the sum of the pole
GMF
contribution vanishes since the pole is absorbed by the
continuum (see Fig. 11). The sum of the pole and the
continuum contributions to the zeroth moment equals
⇢, as it should because of the sum rule (2.13). It im-
plies that the spectral weight of the continuum increases
rapidly around the momentum at which the pole is ab-
sorbed, which is demonstrated in Fig. 12. These behav-
iors, namely the suppression (enhancement) of the con-
tinuum at small |p| (above |p| ' 0.21kF ), can be seen
also from the spectral function plotted in Fig. 13.
V. PHENOMENOLOGICAL IMPLICATION
The strong coupling between the fermion and the Gold-
stino may o↵er a possibility to infer the properties of the
Goldstino from the study of the fermion propagator. This
is what we explore in this section.

34
Result 2. Continuum is shifted.
Cut
shift: Uρ
10
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
+
4
5
⇢f
⇢
"F
U⇢
. (4.30)
to Eqs. (4.17) and (4.18) [(4.19)]
as they should. Also that the ex-
me as that obtained in the absence
Goldstino pole obtained numer-
ig. 11, and compared to the ap-
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
(εk=k2/2m+Uρb, k)
(-ω, -p) (-ω+k2/2m+Uρb, -p+k)
should be on-shell: ω=-p2/2m-p k cosθ/m+Uρ

35
• Continuum+Pole (as U=0 case), but the continuum is
shifted so that the pole is out of the continuum at small p.
• At p=0, all the spectral weights are given to the pole.
Summary
10
⇢
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
⇢b ⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
↵s +
4
5
⇢f
⇢
"F
U⇢
. (4.30)
ce to Eqs. (4.17) and (4.18) [(4.19)]
0], as they should. Also that the ex-
same as that obtained in the absence
he Goldstino pole obtained numer-
Fig. 11, and compared to the ap-
n ¯! = ↵p2
mall value, U⇢f /✏F = 0.1, or kF a =
ms of a, for which the weak-coupling
One sees on Fig. 11 that the approxi-
ccurate as long as |p| . 0.16kF . This
ed range of validity of the expansion,
/m, as can be seen from the denomi-
MF
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
GMF

Outline
36
• Summary

37
What happens in BEC phase?
In free case, no difference. Let us consider interacting case.
-Uρb Uρf +2Uρb
Mean ﬁeld approximation
Fermion particle-Boson hole
excitation (Continuum)
E
nf
E
nb
E
nb
εk+p
E
nf
εk
qp~Σk fkb†
k+p
At p=0
G0
(p) =
⇢f
! + U⇢

38
Fermion hole-Boson particle excitation (Pole)
At p=0
q†
p~Σk f†
k-pbk
E
nf
E
nb
E
nb
E
nf
ε-p
k=0
GMF
pole(p) =
⇢b
! + U⇢f
Uρb -Uρ

39
RPA
At p=0
The gap remains!
Inconsistent with the NG theorem,
so we should have missed something again…
GRPA(p) =
1
[G0(p)] 1 + U
G0
(p) =
⇢f
! + U⇢
GRPA(p) =
⇢f
! + U⇢b

40
Three-point coupling due to BEC
Mixing between
Fermion particle-Boson hole excitation (Continuum)
and Fermion hole-Boson particle excitation (Pole)
+ +
FIG. 8: The diagrams containing the mixing b
RPA MF
b !
p
⇢b
+ +
+ +
FIG. 8: The diagrams containing the mixing betw
the RPA diagrams (GRPA
) and GMF
pole contributin
blob represents the RPA diagrams.

41
Taking into account the mixing
+ + +...
+ + +...
+ + +...
FIG. 8: The diagrams containing the mixing between between
the RPA diagrams (GRPA
) and GMF
pole contributing to ˜G. The
blob represents the RPA diagrams.
case, it is corrected by the diagrams in Fig. 8. The sec-
ond diagram in this figure is of order U2
⇥ U 2
⇥ U 1
,
where the factor U2
comes from the two vertices, the fac-
tor U 2
from the two bubbles, and U 1
from the mean
field propagator near ¯! = |p| = 0, i.e.,
FIG
RPA
FIG
+ + +...case, it is corrected by the diagrams in Fig. 8. The sec-
ond diagram in this figure is of order U2
⇥ U 2
⇥ U 1
,
where the factor U2
comes from the two vertices, the fac-
tor U 2
from the two bubbles, and U 1
from the mean
field propagator near ¯! = |p| = 0, i.e.,
GMF
pole(p) =
⇢b
¯! ✏0
p + U⇢f
⇡
⇢b
U⇢f
. (4.21)
This diagram has the same order of magnitude ⇠ U 1
as the RPA diagrams, and the same holds for the entire
family of diagrams displayed in Fig. 8. Their sum yields
˜G(p) =
1
[GRPA
(p)] 1 U2GMF
pole(p)
, (4.22)
where GRPA
(p) is given by Eq. (4.12). At zero momen-
tum, it reduces to
˜G(!, 0) =
"
⇢2
f
⇢
1
¯!
+
⇢f ⇢b
⇢
1
¯! + U⇢
#
. (4.23)
Here we have one pole with no gap, and another one
with a finite gap (¯! = U⇢), whose existence is due to
the presence of a BEC. One may interpret this result
+
FIG. 10: The d
tween the RPA
G3. There is an
tex (the black
identical contrib
The remaini
is G3(p). It is
are connected
finite momentu
The resulting e
which reduces
G3(!, 0)
At p=0
Gap disappears!
GRPA(p) =
⇢f
! + U⇢b
GMF
pole(p) =
⇢b
! + U⇢f
˜G(p) '
1
⇢
⇢2
f
!

42
Level Repulsion
10
= ⇢
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
↵ ⌘
⇢b ⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
= ↵s +
4
5
⇢f
⇢
"F
U⇢
. (4.30)
reduce to Eqs. (4.17) and (4.18) [(4.19)]
= 0], as they should. Also that the ex-
the same as that obtained in the absence
of the Goldstino pole obtained numer-
in Fig. 11, and compared to the ap-
ssion ¯! = ↵p2
a small value, U⇢f /✏F = 0.1, or kF a =
terms of a, for which the weak-coupling
e. One sees on Fig. 11 that the approxi-
is accurate as long as |p| . 0.16kF . This
ected range of validity of the expansion,
|p|/m, as can be seen from the denomi-
ession of GMF
⇢m/kF ' 0.15kF for the current values
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
GMF
RPA
Pole
Result at ﬁnite p
At small p, 2-peak structure due to level repulsion
(Goldstino+continuum)

43
Mixing from the other point of view
9
...
...
+...
ween
The
sec-
U 1
,
fac-
mean
+ + +...
+ + +...
+ + +...
FIG. 9: The diagrams containing the mixing between the
RPA diagrams (GRPA
) and GMF
pole contributing to GS.
+ + +...
+ + +...
+ + +...
FIG. 10: The diagrams containing the mixing between be-
tween the RPA diagrams (GRPA
) and GMF
pole contributing to
Fermion spectrum is also signiﬁcantly aﬀected
by the mixing with the supercharge!
Novel feature in BEC phase.
Very similar to QCD!
3.4. The phonino
a
ψ
gg

44
12
fermion self-energy at the two-loop order, which
ke into account.
ve the comparable spectral weights, as can be
ig. 17. This is quite di↵erent compared with
oldstino propagator that we discussed in the
ction, in which the continuum is suppressed
|. Here the continuum ends at p = 0 in a
carries a fraction ⇢f /⇢ of the spectral weight.
r” pole carries a fraction ⇢b/⇢, as can be de-
Eq. (5.3). The total spectral weight is equal
agreement with the well-known sum rule,
Z
d¯!
2⇡
S(p) = 1. (5.5)
momentum exceeds ⇠ 0.21kF , the pole is ab-
he continuum, and the whole spectral weight
ed by the continuum. The width of the peak
nuum is decreasing function of |p| for |p| &
s is to be expected since, when |p| becomes
0
0.2
0.4
-0.4
-0.2
0
0.2
20
40
p/kF
ω
-
/εF
FIG. 16: The fermion spectral function S as a function of
|p| and ¯!. The unit of S is 1/✏F . Densities and coupling are
the same as in Fig. 11. At p = 0, the continuum ends in a
pole with spectral weight ⇢f /⇢, while the spectral weight of
the other pole is ⇢b/⇢.
0.8
1
Similar result to the goldstino spectrum
Small p: Goldstino pole + Continuum
Large p: Free particle pole
10
.29)
.30)
19)]
ex-
ence
mer-
ap-
tion
a =
pling
roxi-
This
sion,
omi-
tion
lues
m is
d to
the
cor-
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
GMF
! =
p2
2m
µb Z = 1
! = ↵
p2
2m
Z =
⇢b
⇢
Z =
⇢f
⇢

Possible Experimental Detection
45
1. Fermion spectrum
At small p, it is quite diﬀerent from the free result.
(2-peak structure: Goldstino+continuum)
12
y at the two-loop order, which
spectral weights, as can be
te di↵erent compared with
or that we discussed in the
e continuum is suppressed
nuum ends at p = 0 in a
⇢f /⇢ of the spectral weight.
raction ⇢b/⇢, as can be de-
tal spectral weight is equal
he well-known sum rule,
p) = 1. (5.5)
0
0.2
0.4
-0.4
-0.2
0
0.2
20
40
p/kF
ω
-
/εF
FIG. 16: The fermion spectral function S as a function of
|p| and ¯!. The unit of S is 1/✏F . Densities and coupling are
the same as in Fig. 11. At p = 0, the continuum ends in a
pole with spectral weight ⇢f /⇢, while the spectral weight of
the other pole is ⇢b/⇢.It can be detected via the spectroscopy?

46
2. Fermion distribution
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
-2
-1
0
1
2
ω/εF
0
0.5
1
0
nf
2 2
Because the fermion spectrum is modiﬁed, the fermion
distribution in momentum space is also changed.
Free case
Only one branch in the spectrum
ω<0 states are occupied (p<kF).

47
pproxi-
F . This
ansion,
enomi-
ndition
values
uum is
ared to
t of the
F .
ons cor-
d GRPA
evel re-
yielding
etween
se, the
ing the
tails in
nt of
the up-
GMF
continuum (see Fig. 11). The sum of the pole and the
continuum contributions to the zeroth moment equals
⇢, as it should because of the sum rule (2.13). It im-
plies that the spectral weight of the continuum increases
rapidly around the momentum at which the pole is ab-
sorbed, which is demonstrated in Fig. 12. These behav-
iors, namely the suppression (enhancement) of the con-
tinuum at small |p| (above |p| ' 0.21kF ), can be seen
also from the spectral function plotted in Fig. 13.
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
Weak coupling case
Goldstino pole+Continuum
Near kF, the pole energy becomes positive.
Almost same as free case,
because the weight of the pole near kF
is almost one.

48
0
0.5
0 0.5 1 1.5 2
p/kF
-3
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
nf
U⇢f /✏F = 2/3
Strong coupling case
Goldstino pole and Continuum is
separated, since the distance (Uρ)
becomes large in strong coupling.
The energy at which the pole energy
becomes positive is diﬀerent from kF.
Fermi sea is distorted.

• We analyzed the spectral properties of the goldstino in the
absence/presence of interaction with RPA.

• We observed the crossover from small p to large p region
(from interaction dominant to free case).

• In BEC phase, the importance of the mixing process between
Fermion particle-Boson hole excitation and the Fermion
hole-Boson particle excitation was emphasized.

• We discussed the possibility for experimental detection of
the goldstino.
Summary
49

Nambu-Goldstone mode for supersymmetry breaking in QCD and Bose-Fermi cold atom system at BEC phase

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