NambuJona-Lasinio Model: Harvests and Youngsters Kenji Fukushima - - PowerPoint PPT Presentation

SLIDE 1

Nambu—Jona-Lasinio Model:
 Harvests and Youngsters

Kenji Fukushima




Department of Physics, the University of Tokyo ~ Nambu and Science Frontier ~

1

SLIDE 2

• Nov. 17, 2015 @ Osaka

NJL Model in the Past

2

Most useful example to consider fundamental questions

Dynamical origin of the mass Spontaneous symmetry breaking (Talk by Watanabe/Noumi)

Most successful model to capture essence of QCD physics

Ground states of QCD matter (QCD Critical Point / Color-Super / Inhomogeneity) Application to neutron star (Talk by Masuda) Nucleon structure (Faddeev eq. / Non-topological soliton) We have learnt much about good physics

Harvests from NJL

SLIDE 3

• Nov. 17, 2015 @ Osaka

NJL Model in the Future

3

Improvements to approach QCD dynamics

Interplay with the gauge sector Attempts to “derive” NJL from QCD

Some exotic/unsolved applications

Deconfinement crossover at high density
 Fate of symmetry in curved spacetime Diakonov’s model of spinor quantum gravity We still have so much homework (physics not yet matured)

Youngsters from NJL

SLIDE 4

• Nov. 17, 2015 @ Osaka

NJL: Introduction

4

P H YSI CAL R EVI EW

VOLUME

&22, NUMBER AI RII,

Dynamical Model of Elementary

Particles Based on an Analogy

with Superconductivity.

P

• Y. NAMBU

AND G. JONA-LASINIoj'

The Enrico terms Institute for Nuclear

StuCkes and the Department

• f Physics,

The University

• f Chicago, Chicago,

Illinois (Received October 27, 1960)

It is suggested

that the nucleon mass arises largely as a self-energy

• f some primary

fermion

field through

the same mechanism as the appearance

• f energy gap in the theory of superconductivity.

The idea can be put into a mathematical

formulation utilizing

a generalized

Hartree-Fock

approximation which regards real nucleons as quasi-particle

• excitations. We consider a simplified

model of nonlinear four-fermion

interaction

which allows a p5-gauge group. An interesting consequence

• f the symmetry

is that there arise automatically pseudoscalar zero-mass bound states of nucleon-antinucleon pair which may be regarded as an idealized pion.

In addition,

massive bound states of nucleon number zero and two are predicted in a simple approximation.

The theory

contains two parameters

which can be explicitly related

to observed

nucleon mass and the pion-nucleon coupling

constant.

Some paradoxical aspects of the theory in connection with the p5 trans- formation are discussed in detail.

• I. INTRODUCTION

" 'N this paper we are going to develop

a dynamical

• theory of elementary

particles in which nucleons and

mesons are derived in a unified way from a fundamental spinor field.

In basic physical

ideas, it has thus the

characteristic features

• f a compound-particle

model,

but

unlike

most

• f the

existing theories, dynamical

treatment

• f the interaction

makes up an essential part

• f the theory. Strange particles are not yet considered.

The scheme

is motivated

by the observation

• f an

interesting analogy between the properties

• f Dirac

particles and the quasi-particle excitations

that appear

in the theory of superconductivity, which was originated with great success by Bardeen, Cooper, and Schrieffer, ' and subsequently given an elegant mathematical forlnu-

lation by Bogoliubov. ' The characteristic feature of the

BCS theory is that it produces

an energy gap between the ground

state and the excited states of a supercon-

ductor, a fact w'hich has been confirmed

experimentally.

The gap is caused

due to the fact that the attractive phonon-mediated

interaction between electrons produces correlated pairs of electrons with opposite momenta and

spin near the Fermi surface, and it takes a finite amount

• f energy to break this correlation.

Elementary excitations in a superconductor can be

conveniently described by means of a coherent mixture

• f

electrons and holes, which

• beys

the

following

* Supported

by the U. S. Atomic Energy Commission.

f' Fulbright

Fellow, on leave of absence from Instituto di Fisica dell Universita,

Roma, Italy and Istituto

Nazionale di Fisica Nucleare, Sezione di Roma, Italy.

'A

preliminary version

• f the

work was presented

at the

Midwestern Conference

• n Theoretical

Physics,

April, 1960 (un- published).

See also Y. Nambu,

• Phys. Rev. Letters 4, 380 (1960);

and Proceedings

• f the Tenth

Annual

Rochester

Conference

• n

High-Energy Nuclear Physics, 1960 (to be published).

' J. Bardeen, L. N. Cooper, and J.R. Schrieffer, Phys. Rev. 106,

162 (1957).

3 N. N. Bogoliubov, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 58,

73 (1958) Ltranslation: Soviet Phys. -JETP 34, 41, 51 (1958)g;

• N. N. Sogoliubov,
• V. V. Tolmachev,

and D. V. Shirkov,

A %em

Methodin

the Theory of Supercondlctivity

(Academy of Sciences of U, S.S.R., Moscow, 1958).

equations'

4:

E4~= e lto~+40

(1.1 E0 ~*=

eA ~*—

+44~,

near the Fermi surface. 11„+ is the component

• f the

excitation

corresponding

to an electron state

• f mo-

mentum

P and spin +(up), andri ~*corresponding

to a hole state of momentum

p and spin +, which means an absence of an electron

• f momentum

—

p and spin

—

(down).

eo is the kinetic

energy measured from the

Fermi surface; g is a constant. There

will also be an

equation complex conjugate

to Eq. (1), describing

another type of excitation. Equation

(1) gives the eigenvalues

E„=a (e,'+y')-*'.

(1.2)

The two states of this quasi-particle

are separated

in energy by 2

~ E„~.In the ground

state of the system all

the quasi-particles

should

be in the lower (negative)

energy

states

• f Eq. (2), and it would

take a finite

energy

2)E„~ )~2~&~ to excite a particle

to the upper

• state. The situation

bears a remarkable resemblance

to

the case of a Dirac particle. The four-component

Dirac

equation can be split into two sets to read

EP,=o"Pter+ res,

Egs —

—

—

• "Pigs+ nell r,

E„=W (p'+nt') l,

where

tPt and Ps are the two eigenstates

• f the chirality
• perator ys—

—

y jy2y3y4.

According

to Dirac's

• riginal

interpretation, the

ground state (vacuum) of the world has all the electrons in the negative energy

states,

and to create excited

states (with zero particle number)

we have to supply an energy &

~2m. In the BCS-Bogoliubov theory, the gap parameter

@,

which is absent

for free electrons, is determined es- sentially as a self-consistent

(Hartree-Fock)

representa- tion of the electron-electron interaction eGect.

4 J. G. Valatin, Nuovo cimento 7, 843 (1958).

345

DYNAM ICAL

MODEL

OF ELEMENTARY

PAkTI CI ES

347

we can explore

the whole idea mathematically,

using

essentially

the formulation

developed in reference 8. It is gratifying

that the various field-theoretical

techniques

can be fully utilized. Section 3 will be devoted to intro- duction

• f the Hartree-Fock

equation for nucleon self- energy, which will make the starting point of the theory.

Then we go on to discuss in Sec. 4 the collective modes.

In addition

to the expected pseudoscalar

"pion" states,

we find

• ther

massive mesons

• f scalar

and

vector variety, as well as a scalar "deuteron. " The coupling constants of these mesons can be easily determined. The

relation

• f the pion to the y5 gauge group

will be dis-

cussed in Secs. 5 and 6.

The theory

promises many

practical

consequences.

For this purpose,

however,

it is necessary

to make our

model more realistic by incorporating

the isospin, and

allowing

for a violation

• f ys invariance.

But in doing

so, there arise at the same time new problems concerning

the mass splitting and instability. This refined

model

will be elaborated in Part II of this work, where we shall

also find predictions about strong and weak interactions. Thus the general structure

• f the weak interaction

cur- rents modified by strong interactions can be treated to some degree, enabling

• ne to derive the decay processes
• f various

particles

under simple assumptions.

The

calculation

• f the pion decay rate gives perhaps
• ne of

the most

interesting supports

• f the theory.

Results about strong interactions

themselves

are equally inter-

• esting. We shall find specific predictions

about heavier

mesons, which

are in line with the recent theoretical expectations.

• II. THE PRIMARY INTERACTION

We brieQy discuss the possible nature of the primary interaction between fermions. Lacking any radically

new concepts, the interaction

could be either mediated

by some fundamental

Bose field or due to an inherent

nonlinearity in the fermion field. According

to our

postulate, these interactions must

allow chirality

con- servation in addition

to the conservation

• f nucleon

number.

The chirality X here is defined

as the eigen- value of y5, or in terms of quantized

fields,

Furthermore, the dynamics

• f our theory

would re- quire that the interaction

be attractive between particle and antiparticle

in order to make bound-state

formation

• possible. Under the transformation

(2.3), various tensors

transform as follows:

Vector:

Axial vector:

Scalar:

Pseudoscalar:

Tensor:

sPyuP ~iPyuf,

s4'Vu'Y&4' ~ s4"YuTsf'

Pit —

+~ cos2cr+ifygk

sin2n,

igygk —

+ sPygk cos2n

$P

—

sin2n, Pa u,P -+ Po u,P cos2n+i Pysrru, f sin2cr.

(2.5)

It is obvious that a vector or pseudovector

Bose field

coupled to the fermion field satisfies the invariance. The

vector case would

also satisfy

the dynamical require- ment since, as in the electromagnetic interaction, the forces would be attractive between

• pposite

nucleon

• charges. The pseudovector

field, on the other hand, does

not meet the requirement as can be seen by studying the self-consistent

mass equation discussed later.

The vector field looks particularly attractive

since it can be associated with the nucleon number gauge group.

This idea has been

explored by Lee and Yang, ' and recently by Sakurai. " But since we are dealing with strong interactions, such a field would have to have a finite observed mass in a realistic theory. Whether this is compatible with the invariance requirement is not yet

clear. (Besides,

if the bare mass

• f both

spinor

and vector field were zero, the theory would not contain any parameter

with the dimensions

• f mass. )

The

nonlinear fermion

interaction

seems

to offer

another possibility. Heisenberg and

his

co-workers"

have been developing

a comprehensive

theory

• f ele-

mentary particles along this line. It is not easy, however,

to gain a clear physical insight into their results obtained

by

means

• f highly

complicated mathematical ma- chinery. We would like to choose the nonlinear interaction in this paper. Although this looks similar to Heisenberg' s theory, the dynamical treatment

will be quite different

and more amenable

to qualitative

understanding.

The following

Lagrangian density

will be assumed

(A=c= 1):

X=

fy4ysgd'x

(2.1)

I-= A,~A+gsl (A8—

' (6s4)'7.

—

(2.6) The nucleon

number is, on the other hand

E= ~$74fd'x.

(2.2)

The coupling

parameter

go is positive,

and has dimen-

sions Lmass7 '. The»

invariance property

• f the

interaction

is evident

from Eq. (2.5). According

to the

Fierz theorem, it is also equivalent

to

—

stol:(WvA)' —

(4vuv4)'7

(2 7)

f~ expLin»7$,

g ~ it exp Linys7,

f~ exp/in7$,

P~ $ expl —

su7, where n is an arbitrary

constant phase.

(2.3) (2.4)

These are, respectively, generators

• f the p&- and ordi-

nary-gauge groups

This particular choice of ys-invariant

form was taken without

a compelling

reason,

but has the advantage

• T. D, Lee and C. N. Yang, Phys. Rev. 98, 1501 (1955).

'0 J.J. Samurai, Ann. Phys. 11, 1 (1960).

"W.Heisenberg,

• Z. Naturforsch.

14, 441 (1959).Earlier papers

are quoted there.

Y.

NAlVI8 U AN 0 G. JONA —LAS I N I 0 that it can be naturally

extended to incorporate isotopic spin." Unlike Heisenberg's case, we do not have any theory about the handling

• f the highly

divergent singularities inherent in nonlinear

interactions. So we will introduce, as an additional and independent

assumption, an ad hoc relativistic cutoG or form factor in actual calculations.

Thus the theory may also be regarded as an approxi- mate treatment

• f the intermediate-boson

model with a large eGective mass. As will be seen in subsequent

sections, the nonlinear model makes mathematics particularly easy, at least in the lowest approximation,

enabling

• ne to derive many

interesting quantitative results. IIL THE SELF-CONSISTENT EQUATION FOR NUCLEON

MASS

We will assume that all quantities

we calculate

here

are somewhow convergent,

without asking the reason behind it. This will be done actually by introducing

a

suitable phenomenological cutoff. Without specifying the interaction, let Z be the unrenormalized proper self-energy

part of the fermion,

expressed in terms of observed mass m, coupling con-

stant

g, and cutoff A. A real Dirac particle will satisfy

the equation

energy Lagrangian J„and split J thus

L= (Ls+L,)+(L, L,)—

=Le'+L .

For L, we assume

quite general form (quadratic

• r

bilinear in the fields) such that Ls' leads to linear field

• equations. This will enable one to defi~e a vacuum and a

complete set of "quasi-particle"

states, each particle

being an eigenmode

• f Lo'. Now

we treat J

as per- turbation, and determine J, from the requirement

that

J

shall not yield

additional self-energy

• effects. This

procedure then leads to Eq. (3.2). The self-consistent nature

• f such a procedure

is evident since the self- energy is calculated by perturbation theory with fields which are already subject to the self-energy eGect.

In order to apply the method

to our problem, let us

assume that L,= — mite, and introduce

the propagator

Ss'

&(x) for the corresponding

Dirac particle with mass

• m. In the lowest
• rder,

and using the two alternative forms Eqs. (2.6) and (2.7), we get for Eq. (3.2)

Z= 2gpI TrSs"& i(0)—

ys TrSs't '(0)ys

—

• lv. »7P '"'(0)+lv.v»v. v S '"'(0)3

(34)

in coordinate

space. This is quadratically

divergent,

but with a cutoff can

be made finite. In momentum space we have

zy p+mp+Z(p m gA)=0

for iy p+m=0

• Namel. y

(3 1)

8gpi p m d4p F(p,A), (2zr)4 &

p'+m' ie— (3.5)

m —

mp —

—

Z(p, m, g,A) I,,~ =p.

where F(p,A) is a cutoif factor. In this case the self- energy operator is a constant.

Substituting

2 from Eq.

The g will also be related to the bare coupling

gs by an

(3 5) E (3 2)

(

0)'

equation

• f the type

g/gp —

—

I'(m, g,A).

(3.3)

Equations

(3.1) and (3.2) may be solved by successive

approximation starting from

mo and

• go. It is possible,

however, that there are also solutions which cannot thus

be obtained. In fact, there can be a solution m/0

even in the case where ma=0, and moreover

the symmetry

seems to forbid a Gnite m.

This kind of situation can be most easily examined by

means

• f the

generalized

Hartree-Pock

procedure'"

which was developed

before in connection with the theory of superconductivity.

The basic idea is not new

in field theory, and in fact in its simplest form

the method is identical with the renormalization procedure

• f Dyson,

considered

• nly

in a somewhat different

context.

Suppose a Lagrangian is composed

• f the free and

interaction part: L=Ls+L;. Instead of diagonalizing Ls and treating L; as perturbation,

we introduce

the self-

gpmz

t

d4p

m= —

~

F(p,A).

2zr4 J

ps+ms —

ze

This has two solutions: either m=0, or

(3.6)

gsi

t-

d'p

1= —

F(p,A).

2x-4 ~

P'+ms

ie—

(3 7) (m'

q

• '*m'

(A.'

q

—:

A.

=I —

+I

I—

I.

I —

+I I+— (3»

gsAz

(As

)

A' (ms

]

m

If we use Eq. (3.5) with

an invariant

cutoff at ps=As after the change

• f path: ps —

+ zp, , we get

The first trivial

• ne corresponds

to the ordinary

per- turbative result.

The second,

nontrivial solution

will

determine

m in terms of go and A.

If we evaluate Eq. (3.7) with a straight

noninvariant cutoff at

I pI =A, we get

"This will be done in Part II.

's N. N. Bogoliubov, Uspekhi Fis. Nauk 67, 549 (1959) /trans-

lation: Soviet Phys. -Uspekhi 67, 256 (1959)].

m'

(A'

=1—

in( —

+1 I.

g A'

A.'

&m'

(3.9)

Regularization is a part of
 the model definition
 (→ non-local extension)

SLIDE 5

• Nov. 17, 2015 @ Osaka

NJL: Introduction

5

The most insightful observation Relation between massless and massive Dirac states

DYNAM ICAL

MODEL

OF ELEMENTARY

PARTICLES

Since the right-hand

side of Eq. (3.8) or (3.9) is positive

and ~

&1 for real A/m, the nontrivial

solution exists only if

0 &2~'/g~'&1. (3.10)

(y„c&„+m)P& &(x)=0,

&P'"&(x) =&P& &(x)

for ms=0.

(3.11a) (3.11b) (3.11c)

According

to the standard

procedure,

we decompose

the f's into Fourier components: Equation

(3.9) is plotted

in Fig. 1 as a function

• f

ms/iI&.'. As

gal&&.s increases

• ver the critical value 2s', m

starts rising from 0. The nonanalytic nature of the solu-

tion is evident as ns cannot be expanded

in powers of go.

ln the following

we will

assume

that Eq. (3.10) is

satisfied, so that the nontrivial solution

• exists. As we

shall see later, physically

this means that the nucleon- antinucleon interaction must be attractive

(gI&) 0) and

strong enough to cause a bound pair of zero total mass.

In the SCS theory, the nontrivial

solution corresponds

to a superconductive state,

whereas

the

trivial

• ne

corresponds

to a normal

state,

which is not the true ground

state of the superconductor.

We may expect a

similar situation

to hold in the present case.

In this connection, it must be kept in mind that our

solutions are only approximate

• nes. We are operating

under

the assumption

that the corrections to them are

not catastrophic,

and can be appropriately

calculated

when necessary. If this does not turn out to be so for some solution, such a solution must be discarded. Later we shall indeed find this possibility

for the trivial

solu-

tion, but for the moment

we

will

ignore such

con- siderations.

Let us define then the vacuum

corresponding

to the

two solutions.

Let

&P"& and

&P&

& be quantized

fields

satisfying the equations

2r'

goAs

• Fio. 1. Plot oi the self-consistent

mass equation

(3.9).

'"'(p, )=[l(1+8. )7' "'(p, ) +L-:(1—

&.)7'b""(—

p, s)

b'"'(»s) =[-'(1+&.

)7'b"'(P s)

• [-:(1-~,

)7: l'&'(-p, ),

&.=

I p I/(p'+m'):.

(3.15)

The vacuum

Q&'& or Q'

' with respect to the field

&P"& or P~m& is now defined as a&'& (p,s)Q&"= b&'& (pp) Qi"=0,

a(

&(p,s)Q& &=b& &(p,s)Qi &=0.

(3.16) (3.16')

Both

&P&'&, &P&'& and

&P' ', &P'

' applied to Q"& always create

particles of mass zero, whereas the same applied to 0(

)

create particles of mass m. From Eqs. (3.15) and (3.16) we obtain The operator sets (a''& b&"&) and (a'

& b'

&) are related by

a canonical

transformation because of Eq. (3.11c):

~i'"'(p s) =2 [I-'"'*(p s)&.'"(p,s')&"'(p s')

n, S'

+si-'"'*(p s)t'-""(—

p, s')b'"'(—

p, s') 7 (3.14)

b'-'(p, s)= 2 [o.'-'*(p,s)~.'"(p,s')b"'(p, s')

a, S'

+e-'"'*(p s)&-"'*(—

p s')ii""(— p s') 7.

Using Eq. (1.3), this is evaluated

to give

4-"&(*)=-

p, s

Po =(P'+~')~

[~-'"'(p,s)~"'(p,s)e'"'

Q'-& =II{I!

(1+~.

)7-:

P, S

+e *' (p,s)b ' (p,s)e-'

'7

1

4-""(~)=—,

p, s

po =(@2+F2)&

[I-'"&*(p s)~"&"(p,s)

&&e '"'+t& &'&(p,s)b~'~(p, s)e'" '7,

i=0 or m,

—

[l(1— P.)7'~""(p,s)b""(—

p, s))Q"'

(3 17)

(3 12)

Thus

Q~~& is, in terms of zero-mass particles, a superposi-

tion of pair states. Each pair has zero momentum,

spin and nucleon number,

and carries &2 units of chirality, since chirality equals

minus

the helicity s for massless particles.

Let us calculate

the scalar product

(Q&'&,Q' ') from

• Eq. (3.15):

where

si i'~(p, s),

t& "&(p,s) are

the normalized spinor eigenfunctions for particles and antiparticles,

with mo- mentum

p and helicity s=&1, and

(Q&

& Q~"&)=II [-,

'(1+P„)7-:

P, S

=exp{+ —,

' in[-', (1+P„)7).

(3.18)

P, S

{~"'(p,

s),~""(p',s') )

={

b&'&(y,s),b&"t(y', s') )=b» b„, etc.

(3.13)

For

large

p, p„1—

m'/2P', so that the exponent

DYNAM ICAL

MODEL

OF ELEMENTARY

PARTICLES

Since the right-hand

side of Eq. (3.8) or (3.9) is positive

and ~

&1 for real A/m, the nontrivial

solution exists only if

0 &2~'/g~'&1. (3.10)

(y„c&„+m)P& &(x)=0,

&P'"&(x) =&P& &(x)

for ms=0.

(3.11a) (3.11b) (3.11c)

According

to the standard

procedure,

we decompose

the f's into Fourier components: Equation

(3.9) is plotted

in Fig. 1 as a function

• f

ms/iI&.'. As

gal&&.s increases

• ver the critical value 2s', m

starts rising from 0. The nonanalytic nature of the solu-

tion is evident as ns cannot be expanded

in powers of go.

ln the following

we will

assume

that Eq. (3.10) is

satisfied, so that the nontrivial solution

• exists. As we

shall see later, physically

this means that the nucleon- antinucleon interaction must be attractive

(gI&) 0) and

strong enough to cause a bound pair of zero total mass.

In the SCS theory, the nontrivial

solution corresponds

to a superconductive state,

whereas

the

trivial

• ne

corresponds

to a normal

state,

which is not the true ground

state of the superconductor.

We may expect a

similar situation

to hold in the present case.

In this connection, it must be kept in mind that our

solutions are only approximate

• nes. We are operating

under

the assumption

that the corrections to them are

not catastrophic,

and can be appropriately

calculated

when necessary. If this does not turn out to be so for some solution, such a solution must be discarded. Later we shall indeed find this possibility

for the trivial

solu-

tion, but for the moment

we

will

ignore such

con- siderations.

Let us define then the vacuum

corresponding

to the

two solutions.

Let

&P"& and

&P&

& be quantized

fields

satisfying the equations

2r'

goAs

• Fio. 1. Plot oi the self-consistent

mass equation

(3.9).

'"'(p, )=[l(1+8. )7' "'(p, ) +L-:(1—

&.)7'b""(—

p, s)

b'"'(»s) =[-'(1+&.

)7'b"'(P s)

• [-:(1-~,

)7: l'&'(-p, ),

&.=

I p I/(p'+m'):.

(3.15)

The vacuum

Q&'& or Q'

' with respect to the field

&P"& or P~m& is now defined as a&'& (p,s)Q&"= b&'& (pp) Qi"=0,

a(

&(p,s)Q& &=b& &(p,s)Qi &=0.

(3.16) (3.16')

Both

&P&'&, &P&'& and

&P' ', &P'

' applied to Q"& always create

particles of mass zero, whereas the same applied to 0(

)

create particles of mass m. From Eqs. (3.15) and (3.16) we obtain The operator sets (a''& b&"&) and (a'

& b'

&) are related by

a canonical

transformation because of Eq. (3.11c):

~i'"'(p s) =2 [I-'"'*(p s)&.'"(p,s')&"'(p s')

n, S'

+si-'"'*(p s)t'-""(—

p, s')b'"'(—

p, s') 7 (3.14)

b'-'(p, s)= 2 [o.'-'*(p,s)~.'"(p,s')b"'(p, s')

a, S'

+e-'"'*(p s)&-"'*(—

p s')ii""(— p s') 7.

Using Eq. (1.3), this is evaluated

to give

4-"&(*)=-

p, s

Po =(P'+~')~

[~-'"'(p,s)~"'(p,s)e'"'

Q'-& =II{I!

(1+~.

)7-:

P, S

+e *' (p,s)b ' (p,s)e-'

'7

1

4-""(~)=—,

p, s

po =(@2+F2)&

[I-'"&*(p s)~"&"(p,s)

&&e '"'+t& &'&(p,s)b~'~(p, s)e'" '7,

i=0 or m,

—

[l(1— P.)7'~""(p,s)b""(—

p, s))Q"'

(3 17)

(3 12)

Thus Q~~& is, in terms of zero-mass particles, a superposi- tion of pair states. Each pair has zero momentum,

spin and nucleon number,

and carries &2 units of chirality, since chirality equals

minus

the helicity s for massless particles.

Let us calculate

the scalar product

(Q&'&,Q' ') from

• Eq. (3.15):

where

si i'~(p, s),

t& "&(p,s) are

the normalized spinor eigenfunctions for particles and antiparticles,

with mo- mentum

p and helicity s=&1, and

(Q&

& Q~"&)=II [-,

'(1+P„)7-:

P, S

=exp{+ —,

' in[-', (1+P„)7).

(3.18)

P, S

{~"'(p,

s),~""(p',s') )

={

b&'&(y,s),b&"t(y', s') )=b» b„, etc.

(3.13)

For

large

p, p„1—

m'/2P', so that the exponent

Projection with
 ua(m) and va(m)

Massless Dirac eq.
 Massive Dirac eq.
 Initial condition

SLIDE 6

• Nov. 17, 2015 @ Osaka

NJL: Introduction

6

The most insightful observation Relation between massless and massive Dirac states

DYNAM ICAL

MODEL

OF ELEMENTARY

PARTICLES

Since the right-hand

side of Eq. (3.8) or (3.9) is positive

and ~

&1 for real A/m, the nontrivial

solution exists only if

0 &2~'/g~'&1.

(3.10)

(y„c&„+m)P& &(x)=0,

&P'"&(x) =&P& &(x)

for ms=0.

(3.11a) (3.11b) (3.11c)

According

to the standard

procedure,

we decompose

the f's into Fourier components:

Equation

(3.9) is plotted

in Fig. 1 as a function

• f

ms/iI&.'. As

gal&&.s increases

• ver the critical value 2s', m

starts rising from 0. The nonanalytic nature of the solu-

tion is evident as ns cannot be expanded

in powers of go.

ln the following

we will

assume

that Eq. (3.10) is

satisfied, so that the nontrivial solution

• exists. As we

shall see later, physically

this means that the nucleon- antinucleon interaction must be attractive

(gI&) 0) and

strong enough to cause a bound pair of zero total mass.

In the SCS theory, the nontrivial

solution corresponds

to a superconductive state,

whereas

the

trivial

• ne

corresponds

to a normal

state,

which is not the true ground

state of the superconductor.

We may expect a

similar situation

to hold in the present case.

In this connection, it must be kept in mind that our

solutions are only approximate

• nes. We are operating

under

the assumption

that the corrections to them are

not catastrophic,

and can be appropriately

calculated

when necessary. If this does not turn out to be so for some solution, such a solution must be discarded. Later we shall indeed find this possibility

for the trivial

solu-

tion, but for the moment

we

will

ignore such

con- siderations.

Let us define then the vacuum

corresponding

to the

two solutions.

Let

&P"& and

&P&

& be quantized

fields

satisfying the equations

2r'

goAs

• Fio. 1. Plot oi the self-consistent

mass equation

(3.9).

'"'(p, )=[l(1+8. )7' "'(p, ) +L-:(1—

&.)7'b""(—

p, s)

b'"'(»s) =[-'(1+&.

)7'b"'(P s)

• [-:(1-~,

)7: l'&'(-p, ),

&.=

I p I/(p'+m'):.

(3.15)

The vacuum

Q&'& or Q'

' with respect to the field

&P"& or P~m& is now defined as a&'& (p,s)Q&"= b&'& (pp) Qi"=0,

a(

&(p,s)Q& &=b& &(p,s)Qi &=0.

(3.16)

(3.16')

Both

&P&'&, &P&'& and

&P' ', &P'

' applied to Q"& always create

particles of mass zero, whereas the same applied to 0(

)

create particles of mass m. From Eqs. (3.15) and (3.16) we obtain The operator sets (a''& b&"&) and (a'

& b'

&) are related by

a canonical

transformation because of Eq. (3.11c):

~i'"'(p s) =2 [I-'"'*(p s)&.'"(p,s')&"'(p s')

n, S'

+si-'"'*(p s)t'-""(—

p, s')b'"'(—

p, s') 7 (3.14)

b'-'(p, s)= 2 [o.'-'*(p,s)~.'"(p,s')b"'(p, s')

a, S'

+e-'"'*(p s)&-"'*(—

p s')ii""(— p s') 7.

Using Eq. (1.3), this is evaluated

to give

4-"&(*)=-

p, s

Po =(P'+~')~

[~-'"'(p,s)~"'(p,s)e'"'

Q'-& =II{I!

(1+~.

)7-:

P, S

+e *' (p,s)b ' (p,s)e-'

'7

1

4-""(~)=—,

p, s

po =(@2+F2)&

[I-'"&*(p s)~"&"(p,s)

&&e '"'+t& &'&(p,s)b~'~(p, s)e'" '7,

i=0 or m,

—

[l(1— P.)7'~""(p,s)b""(—

p, s))Q"'

(3 17)

(3 12)

Thus

Q~~& is, in terms of zero-mass particles, a superposi-

tion of pair states. Each pair has zero momentum,

spin and nucleon number,

and carries &2 units of chirality, since chirality equals

minus

the helicity s for massless particles.

Let us calculate

the scalar product

(Q&'&,Q' ') from

• Eq. (3.15):

where

si i'~(p, s),

t& "&(p,s) are

the normalized spinor eigenfunctions for particles and antiparticles,

with mo- mentum

p and helicity s=&1, and

(Q&

& Q~"&)=II [-,

'(1+P„)7-:

P, S

=exp{+ —,

' in[-', (1+P„)7).

(3.18)

P, S

{~"'(p,

s),~""(p',s') )

={

b&'&(y,s),b&"t(y', s') )=b» b„, etc.

(3.13)

For

large

p, p„1—

m'/2P', so that the exponent

DYNAM ICAL

MODEL

OF ELEMENTARY

PARTICLES

Since the right-hand

side of Eq. (3.8) or (3.9) is positive

and ~

&1 for real A/m, the nontrivial

solution exists only if

0 &2~'/g~'&1.

(3.10)

(y„c&„+m)P& &(x)=0,

&P'"&(x) =&P& &(x)

for ms=0.

(3.11a) (3.11b) (3.11c)

According

to the standard

procedure,

we decompose

the f's into Fourier components:

Equation

(3.9) is plotted

in Fig. 1 as a function

• f

ms/iI&.'. As

gal&&.s increases

• ver the critical value 2s', m

starts rising from 0. The nonanalytic

nature of the solu- tion is evident as ns cannot be expanded

in powers of go.

ln the following

we will

assume

that Eq. (3.10) is

satisfied, so that the nontrivial solution

• exists. As we

shall see later, physically

this means that the nucleon- antinucleon interaction must be attractive

(gI&) 0) and

strong enough to cause a bound pair of zero total mass.

In the SCS theory, the nontrivial

solution corresponds

to a superconductive state,

whereas

the

trivial

• ne

corresponds

to a normal

state,

which is not the true ground

state of the superconductor.

We may expect a

similar situation

to hold in the present case.

In this connection, it must be kept in mind that our

solutions

are only approximate

• nes. We are operating

under

the assumption

that the corrections to them are

not catastrophic,

and can be appropriately

calculated

when necessary. If this does not turn out to be so for some solution,

such a solution must be discarded. Later

we shall indeed find this possibility

for the trivial

solu-

tion,

but

for the

moment

we

will

ignore such

con- siderations.

Let us define then the vacuum

corresponding

to the

two solutions.

Let

&P"& and

&P&

& be quantized

fields

satisfying the equations

2r'

goAs

• Fio. 1. Plot oi the self-consistent

mass equation

(3.9).

'"'(p, )=[l(1+8. )7' "'(p, ) +L-:(1—

&.)7'b""(—

p, s)

b'"'(»s) =[-'(1+&.

)7'b"'(P s)

• [-:(1-~,

)7: l'&'(-p, ),

&.=

I p I/(p'+m'):.

(3.15)

The vacuum

Q&'& or Q'

' with respect to the field

&P"& or P~m& is now defined as

a&'& (p,s)Q&"= b&'& (pp) Qi"=0,

a(

&(p,s)Q& &=b& &(p,s)Qi &=0.

(3.16)

(3.16')

Both

&P&'&, &P&'& and

&P' ', &P'

' applied to Q"& always create

particles of mass zero, whereas the same applied to 0(

)

create particles of mass m. From Eqs. (3.15) and (3.16) we obtain The operator sets (a''& b&"&) and (a'

& b'

&) are related by

a canonical

transformation because of Eq. (3.11c):

~i'"'(p s) =2 [I-'"'*(p s)&.'"(p,s')&"'(p s')

n, S'

+si-'"'*(p s)t'-""(—

p, s')b'"'(—

p, s') 7 (3.14)

b'-'(p, s)= 2 [o.'-'*(p,s)~.'"(p,s')b"'(p, s')

a, S'

+e-'"'*(p s)&-"'*(—

p s')ii""(— p s') 7.

Using Eq. (1.3), this is evaluated

to give

4-"&(*)=-

p, s

Po =(P'+~')~

[~-'"'(p,s)~"'(p,s)e'"'

Q'-& =II{I!

(1+~.

)7-:

P, S

+e *' (p,s)b ' (p,s)e-'

'7

1

4-""(~)=—,

p, s

po =(@2+F2)&

[I-'"&*(p s)~"&"(p,s)

&&e '"'+t& &'&(p,s)b~'~(p, s)e'" '7,

i=0 or m,

—

[l(1— P.)7'~""(p,s)b""(—

p, s))Q"'

(3 17)

(3 12)

Thus

Q~~& is, in terms of zero-mass particles, a superposi-

tion of pair states. Each pair has zero momentum,

spin and nucleon number,

and carries &2 units of chirality, since chirality equals

minus

the helicity s for massless particles.

Let us calculate

the scalar product

(Q&'&,Q' ') from

• Eq. (3.15):

where

si i'~(p, s),

t& "&(p,s) are

the normalized spinor eigenfunctions

for particles

and antiparticles,

with mo- mentum

p and helicity s=&1, and

(Q&

& Q~"&)=II [-,

'(1+P„)7-:

P, S

=exp{+ —,

' in[-', (1+P„)7).

(3.18)

P, S

{~"'(p,

s),~""(p',s') )

={

b&'&(y,s),b&"t(y', s') )=b» b„, etc.

(3.13)

For

large

p, p„1—

m'/2P', so that the exponent

Massive vacuum
 is a BCS one

Mixture of massless
 “particle”+”anti-particle”

SLIDE 7

• Nov. 17, 2015 @ Osaka

NJL: Introduction

7

The most insightful observation Relation between massless and massive Dirac states

DYNAM ICAL

MODEL

OF ELEMENTARY

PARTICLES

Since the right-hand

side of Eq. (3.8) or (3.9) is positive

and ~

&1 for real A/m, the nontrivial

solution exists only if

0 &2~'/g~'&1.

(3.10)

(y„c&„+m)P& &(x)=0,

&P'"&(x) =&P& &(x)

for ms=0.

(3.11a) (3.11b) (3.11c)

According

to the standard

procedure,

we decompose

the f's into Fourier components:

Equation

(3.9) is plotted

in Fig. 1 as a function

• f

ms/iI&.'. As

gal&&.s increases

• ver the critical value 2s', m

starts rising from 0. The nonanalytic

nature of the solu- tion is evident as ns cannot be expanded

in powers of go.

ln the following

we will

assume

that Eq. (3.10) is

satisfied, so that the nontrivial solution

• exists. As we

shall see later, physically

this means that the nucleon- antinucleon interaction must be attractive

(gI&) 0) and

strong enough to cause a bound pair of zero total mass.

In the SCS theory, the nontrivial

solution corresponds

to a superconductive state,

whereas

the

trivial

• ne

corresponds

to a normal

state,

which is not the true ground

state of the superconductor.

We may expect a

similar situation

to hold in the present case.

In this connection, it must be kept in mind that our

solutions

are only approximate

• nes. We are operating

under

the assumption

that the corrections to them are

not catastrophic,

and can be appropriately

calculated

when necessary. If this does not turn out to be so for some solution,

such a solution must be discarded. Later

we shall indeed find this possibility

for the trivial

solu-

tion,

but

for the

moment

we

will

ignore such

con- siderations.

Let us define then the vacuum

corresponding

to the

two solutions.

Let

&P"& and

&P&

& be quantized

fields

satisfying the equations

2r'

goAs

• Fio. 1. Plot oi the self-consistent

mass equation

(3.9).

'"'(p, )=[l(1+8. )7' "'(p, ) +L-:(1—

&.)7'b""(—

p, s)

b'"'(»s) =[-'(1+&.

)7'b"'(P s)

• [-:(1-~,

)7: l'&'(-p, ),

&.=

I p I/(p'+m'):.

(3.15)

The vacuum

Q&'& or Q'

' with respect to the field

&P"& or P~m& is now defined as

a&'& (p,s)Q&"= b&'& (pp) Qi"=0,

a(

&(p,s)Q& &=b& &(p,s)Qi &=0.

(3.16)

(3.16')

Both

&P&'&, &P&'& and

&P' ', &P'

' applied to Q"& always create

particles of mass zero, whereas the same applied to 0(

)

create particles of mass m. From Eqs. (3.15) and (3.16) we obtain The operator sets (a''& b&"&) and (a'

& b'

&) are related by

a canonical

transformation because of Eq. (3.11c):

~i'"'(p s) =2 [I-'"'*(p s)&.'"(p,s')&"'(p s')

n, S'

+si-'"'*(p s)t'-""(—

p, s')b'"'(—

p, s') 7 (3.14)

b'-'(p, s)= 2 [o.'-'*(p,s)~.'"(p,s')b"'(p, s')

a, S'

+e-'"'*(p s)&-"'*(—

p s')ii""(— p s') 7.

Using Eq. (1.3), this is evaluated

to give

4-"&(*)=-

p, s

Po =(P'+~')~

[~-'"'(p,s)~"'(p,s)e'"'

Q'-& =II{I!

(1+~.

)7-:

P, S

+e *' (p,s)b ' (p,s)e-'

'7

1

4-""(~)=—,

p, s

po =(@2+F2)&

[I-'"&*(p s)~"&"(p,s)

&&e '"'+t& &'&(p,s)b~'~(p, s)e'" '7,

i=0 or m,

—

[l(1— P.)7'~""(p,s)b""(—

p, s))Q"'

(3 17)

(3 12)

Thus

Q~~& is, in terms of zero-mass particles, a superposi-

tion of pair states. Each pair has zero momentum,

spin and nucleon number,

and carries &2 units of chirality, since chirality equals

minus

the helicity s for massless particles.

Let us calculate

the scalar product

(Q&'&,Q' ') from

• Eq. (3.15):

where

si i'~(p, s),

t& "&(p,s) are

the normalized spinor eigenfunctions

for particles

and antiparticles,

with mo- mentum

p and helicity s=&1, and

(Q&

& Q~"&)=II [-,

'(1+P„)7-:

P, S

=exp{+ —,

' in[-', (1+P„)7).

(3.18)

P, S

{~"'(p,

s),~""(p',s') )

={

b&'&(y,s),b&"t(y', s') )=b» b„, etc.

(3.13)

For

large

p, p„1—

m'/2P', so that the exponent

Massive vacuum
 is a BCS one

The Dirac equation knew that a massive solution
 is a BCS state in terms of the massless solution!

SLIDE 8

• Nov. 17, 2015 @ Osaka

NJL: Introduction

8

The most insightful observation Relation between massless and massive Dirac states

p e(p) p e(p)

he↑e↓i h¯ qqi

BCS NJL

SLIDE 9

• Nov. 17, 2015 @ Osaka

DYNAM ICAL

MODEL

OF ELEMENTARY

PARTICLES

Since the right-hand

side of Eq. (3.8) or (3.9) is positive

and ~

&1 for real A/m, the nontrivial

solution exists only if

0 &2~'/g~'&1. (3.10)

(y„c&„+m)P& &(x)=0,

&P'"&(x) =&P& &(x)

for ms=0.

(3.11a) (3.11b) (3.11c)

According

to the standard

procedure,

we decompose

the f's into Fourier components: Equation

(3.9) is plotted

in Fig. 1 as a function

• f

ms/iI&.'. As

gal&&.s increases

• ver the critical value 2s', m

starts rising from 0. The nonanalytic nature of the solu- tion is evident as ns cannot be expanded

in powers of go.

ln the following

we will

assume

that Eq. (3.10) is

satisfied, so that the nontrivial solution

• exists. As we

shall see later, physically

this means that the nucleon- antinucleon interaction must be attractive

(gI&) 0) and

strong enough to cause a bound pair of zero total mass.

In the SCS theory, the nontrivial

solution corresponds

to a superconductive state,

whereas

the trivial

• ne

corresponds

to a normal state,

which is not the true ground

state of the superconductor.

We may expect a

similar situation

to hold in the present case. In this connection, it must be kept in mind that our

solutions are only approximate

• nes. We are operating

under

the assumption

that the corrections to them are not catastrophic,

and can be appropriately calculated

when necessary. If this does not turn out to be so for some solution, such a solution must be discarded. Later we shall indeed find this possibility

for the trivial

solu-

tion, but for the moment

we

will

ignore such con-

siderations.

Let us define then the vacuum

corresponding

to the

two solutions.

Let

&P"& and

&P&

& be quantized

fields

satisfying the equations

2r'

goAs

• Fio. 1. Plot oi the self-consistent

mass equation

(3.9).

'"'(p, )=[l(1+8. )7' "'(p, ) +L-:(1—

&.)7'b""(—

p, s)

b'"'(»s) =[-'(1+&.

)7'b"'(P s)

• [-:(1-~,

)7: l'&'(-p, ),

&.=

I p I/(p'+m'):.

(3.15)

The vacuum

Q&'& or Q'

' with respect to the field

&P"& or P~m& is now defined as a&'& (p,s)Q&"= b&'& (pp) Qi"=0,

a(

&(p,s)Q& &=b& &(p,s)Qi &=0.

(3.16) (3.16')

Both

&P&'&, &P&'& and &P' ', &P'

' applied to Q"& always create

particles of mass zero, whereas the same applied to 0(

)

create particles of mass m. From Eqs. (3.15) and (3.16) we obtain The operator sets (a''& b&"&) and (a'

& b'

&) are related by

a canonical

transformation because of Eq. (3.11c):

~i'"'(p s) =2 [I-'"'*(p s)&.'"(p,s')&"'(p s')

n, S'

+si-'"'*(p s)t'-""(—

p, s')b'"'(— p, s') 7 (3.14)

b'-'(p, s)= 2 [o.'-'*(p,s)~.'"(p,s')b"'(p, s')

a, S'

+e-'"'*(p s)&-"'*(—

p s')ii""(— p s') 7.

Using Eq. (1.3), this is evaluated

to give

4-"&(*)=-

p, s

Po =(P'+~')~

[~-'"'(p,s)~"'(p,s)e'"'

Q'-& =II{I!

(1+~.

)7-:

P, S

+e *' (p,s)b ' (p,s)e-'

'7

1

4-""(~)=—,

p, s po =(@2+F2)&

[I-'"&*(p s)~"&"(p,s)

&&e '"'+t& &'&(p,s)b~'~(p, s)e'" '7,

i=0 or m,

—

[l(1— P.)7'~""(p,s)b""(—

p, s))Q"'

(3 17) (3 12) Thus

Q~~& is, in terms of zero-mass particles, a superposi-

tion of pair states. Each pair has zero momentum,

spin and nucleon number, and carries &2 units of chirality, since chirality equals minus

the helicity s for massless particles.

Let us calculate

the scalar product

(Q&'&,Q' ') from

• Eq. (3.15):

where

si i'~(p, s),

t& "&(p,s) are

the normalized spinor eigenfunctions for particles and antiparticles, with mo- mentum p and helicity s=&1, and

(Q&

& Q~"&)=II [-,

'(1+P„)7-:

P, S

=exp{+ —,

' in[-', (1+P„)7).

(3.18)

P, S

{~"'(p,

s),~""(p',s') )

={

b&'&(y,s),b&"t(y', s') )=b» b„, etc.

(3.13)

For

large

p, p„1—

m'/2P', so that the exponent

NJL: Introduction

9

The most famous result

Y.

NAlVI8 U

AN 0 G. JONA —LAS I N I 0

that it can be naturally

extended to incorporate

isotopic

spin." Unlike Heisenberg's

case, we do not have any theory about the handling

• f the highly divergent

singularities inherent in nonlinear

interactions. So we will introduce,

as an additional and independent

assumption, an ad hoc

relativistic

cutoG or form factor in actual calculations.

Thus the theory

may also be regarded as an approxi-

mate treatment

• f the intermediate-boson

model with a large eGective mass. As will be seen in subsequent

sections, the nonlinear

model makes mathematics

particularly easy, at least in

the lowest approximation,

enabling

• ne to derive many

interesting

quantitative results. IIL THE SELF-CONSISTENT EQUATION FOR NUCLEON

MASS

We will assume that all quantities

we calculate

here

are somewhow

convergent, without asking

the reason

behind it. This will be done actually

by introducing

a

suitable

phenomenological

cutoff. Without

specifying

the interaction,

let Z be the

unrenormalized

proper

self-energy

part of the fermion,

expressed

in terms of observed

mass m, coupling

con-

stant

g, and cutoff A. A real Dirac particle will satisfy

the equation

energy Lagrangian J„and split J thus

L= (Ls+L,)+(L, L,)—

=Le'+L .

For L, we assume

quite

general form

(quadratic

• r

bilinear

in the fields) such that Ls' leads to linear field

• equations. This will enable one to defi~e a vacuum and a

complete

set of "quasi-particle" states,

each particle

being

an eigenmode

• f Lo'. Now

we treat J

as per- turbation,

and determine J, from the requirement

that

J

shall not yield

additional

self-energy

• effects. This

procedure then leads to Eq. (3.2). The self-consistent

nature

• f such a procedure

is evident since the self- energy is calculated

by perturbation theory with fields

which are already subject to the self-energy

eGect.

In order to apply the method

to our problem,

let us

assume that L,= —

mite, and introduce

the propagator

Ss' &(x) for the corresponding

Dirac particle with mass

• m. In the lowest
• rder,

and using

the two alternative

forms Eqs. (2.6) and (2.7), we get for Eq. (3.2)

Z= 2gpI TrSs"& i(0)—

ys TrSs't '(0)ys

—

• lv. »7P '"'(0)+lv.v»v. v S '"'(0)3

(34)

in coordinate

space. This is quadratically

divergent,

but with a cutoff can

be made finite. In momentum space we have

zy p+mp+Z(p

m gA)=0

for iy p+m=0

• Namel. y

(3 1)

8gpi

p

m

d4p F(p,A),

(2zr)4 &

p'+m'

ie— (3.5)

m —

mp —

—

Z(p, m, g,A) I,,~ =p.

where F(p,A) is a cutoif factor. In this case the self- energy operator is a constant.

Substituting

2 from Eq.

The g will also be related to the bare coupling

gs by an

(3 5) E (3 2)

(

0)'

equation

• f the type

g/gp —

—

I'(m, g,A).

(3.3)

Equations

(3.1) and (3.2) may be solved by successive

approximation starting

from

mo and go. It is possible,

however, that there are also solutions which cannot thus

be obtained. In fact, there can be a solution m/0

even in the case where ma=0, and moreover

the symmetry

seems to forbid a Gnite m.

This kind of situation

can be most easily examined by

means

• f the

generalized

Hartree-Pock

procedure'"

which was

developed

before

in connection with

the theory of superconductivity.

The basic idea is not new

in field theory,

and

in fact in its simplest

form

the method

is identical

with the renormalization

procedure

• f Dyson,

considered

• nly

in a somewhat different

context.

Suppose a Lagrangian

is composed

• f the free and

interaction part: L=Ls+L;. Instead of diagonalizing

Ls

and treating L; as perturbation,

we introduce

the self-

gpmz

t

d4p

m= —

~

F(p,A).

2zr4 J

ps+ms —

ze

This has two solutions:

either m=0, or

(3.6)

gsi

t-

d'p

1= —

F(p,A).

2x-4 ~

P'+ms

ie—

(3 7)

(m'

q

• '*m'

(A.'

q

—:

A.

=I —

+I

I—

I.

I —

+I I+— (3»

gsAz

(As

)

A'

(ms

]

m

If we use Eq. (3.5) with

an invariant

cutoff at ps=As

after the change

• f path: ps —

+ zp, , we get

The first trivial

• ne corresponds

to the ordinary

per- turbative result.

The second,

nontrivial solution

will

determine

m in terms of go and A.

If we evaluate Eq. (3.7) with a straight

noninvariant cutoff at

I pI =A, we get

"This will be done in Part II.

's N. N. Bogoliubov,

Uspekhi Fis. Nauk 67, 549 (1959) /trans-

lation: Soviet Phys. -Uspekhi 67, 256 (1959)].

m'

(A'

=1—

in( —

+1 I.

g A'

A.'

&m'

(3.9)

Gap eq.

=

Y.

NAlVI8 U

AN 0 G. JONA —LAS I N I 0

that it can be naturally

extended to incorporate isotopic spin." Unlike Heisenberg's

case, we do not have any theory about the handling

• f the highly divergent

singularities inherent in nonlinear

interactions. So we will introduce,

as an additional and independent

assumption, an ad hoc

relativistic

cutoG or form factor in actual calculations.

Thus the theory

may also be regarded as an approxi-

mate treatment

• f the intermediate-boson

model with a large eGective mass. As will be seen in subsequent

sections, the nonlinear

model makes mathematics

particularly easy, at least in

the lowest approximation,

enabling

• ne to derive many

interesting

quantitative results. IIL THE SELF-CONSISTENT EQUATION FOR NUCLEON

MASS

We will assume that all quantities

we calculate

here

are somewhow

convergent, without asking

the reason

behind it. This will be done actually

by introducing

a

suitable

phenomenological

cutoff. Without

specifying

the interaction, let Z be the

unrenormalized

proper

self-energy

part of the fermion,

expressed

in terms of observed

mass m, coupling con-

stant

g, and cutoff A. A real Dirac particle will satisfy

the equation

energy Lagrangian J„and split J thus

L= (Ls+L,)+(L, L,)—

=Le'+L .

For L, we assume

quite general form

(quadratic

• r

bilinear

in the fields) such that Ls' leads to linear field

• equations. This will enable one to defi~e a vacuum and a

complete

set of "quasi-particle" states, each particle

being

an eigenmode

• f Lo'. Now

we treat J

as per- turbation,

and determine J, from the requirement

that

J

shall not yield

additional

self-energy

• effects. This

procedure then leads to Eq. (3.2). The self-consistent

nature

• f such a procedure

is evident since the self- energy is calculated

by perturbation theory with fields

which are already subject to the self-energy

eGect.

In order to apply the method

to our problem,

let us

assume that L,= —

mite, and introduce

the propagator

Ss'

&(x) for the corresponding

Dirac particle with mass

• m. In the lowest
• rder,

and using

the two alternative

forms Eqs. (2.6) and (2.7), we get for Eq. (3.2)

Z= 2gpI TrSs"& i(0)—

ys TrSs't '(0)ys

—

• lv. »7P '"'(0)+lv.v»v. v S '"'(0)3

(34)

in coordinate

space. This is quadratically

divergent,

but with a cutoff can

be made finite. In momentum space we have

zy p+mp+Z(p

m gA)=0

for iy p+m=0

• Namel. y

(3 1)

8gpi

p

m

d4p F(p,A),

(2zr)4 &

p'+m'

ie— (3.5)

m —

mp —

—

Z(p, m, g,A) I,,~ =p.

where F(p,A) is a cutoif factor. In this case the self- energy operator is a constant.

Substituting

2 from Eq.

The g will also be related to the bare coupling

gs by an

(3 5) E (3 2)

(

0)'

equation

• f the type

g/gp —

—

I'(m, g,A).

(3.3)

Equations

(3.1) and (3.2) may be solved by successive

approximation starting

from

mo and go. It is possible,

however, that there are also solutions which cannot thus

be obtained. In fact, there can be a solution m/0

even in the case where ma=0, and moreover

the symmetry

seems to forbid a Gnite m.

This kind of situation

can be most easily examined by

means

• f the

generalized

Hartree-Pock

procedure'"

which was

developed

before

in connection with

the theory of superconductivity.

The basic idea is not new

in field theory,

and

in fact in its simplest

form

the method is identical

with the renormalization

procedure

• f Dyson,

considered

• nly

in a somewhat different

context.

Suppose a Lagrangian is composed

• f the free and

interaction part: L=Ls+L;. Instead of diagonalizing

Ls

and treating L; as perturbation,

we introduce

the self-

gpmz

t

d4p

m= —

~

F(p,A).

2zr4 J

ps+ms —

ze

This has two solutions:

either m=0, or

(3.6)

gsi

t-

d'p

1= —

F(p,A).

2x-4 ~

P'+ms

ie—

(3 7)

(m'

q

• '*m'

(A.'

q

—:

A.

=I —

+I

I—

I.

I —

+I I+— (3»

gsAz

(As

)

A'

(ms

]

m

If we use Eq. (3.5) with

an invariant

cutoff at ps=As

after the change

• f path: ps —

+ zp, , we get

The first trivial

• ne corresponds

to the ordinary

per- turbative result.

The second,

nontrivial solution

will

determine

m in terms of go and A.

If we evaluate Eq. (3.7) with a straight

noninvariant cutoff at

I pI =A, we get

"This will be done in Part II.

's N. N. Bogoliubov,

Uspekhi Fis. Nauk 67, 549 (1959) /trans-

lation: Soviet Phys. -Uspekhi 67, 256 (1959)].

m'

(A'

=1—

in( —

+1 I.

g A'

A.'

&m'

(3.9)

If the coupling g0 is large enough,
 there is a solution of the gap eq.

Sharp UV cutoff for
 the four momentum

SLIDE 10

• Nov. 17, 2015 @ Osaka

Critical Coupling in NJL

10

RG picture

! " t "*

" ^ ^

^

l

Amplitude in a
 resonating channel

β = dˆ λ dk = 2ˆ λ + #ˆ λ2 + · · ·

• cf. Thouless criterion

Kondo effect

(ˆ λ = kd−2λ)

What if d = 2 ? (GN)
 (realized by strong B)

SLIDE 11

• Nov. 17, 2015 @ Osaka

Critical Coupling in QCD

11

RG picture

! " t "*

" ^ ^

^

l

Amplitude in a
 resonating channel

Meson and gauge fluctuations pushing down Chiral symmetry is (believed to be)
 broken for any gauge coupling if the temperature T = 0 Review: Jens Braun

SLIDE 12

• Nov. 17, 2015 @ Osaka

Critical Coupling in QCD

12

RG picture

! " t "*

" ^ ^

^

l

Amplitude in a
 resonating channel

Meson and gauge fluctuations pushing down Chiral symmetry is (believed to be)
 broken for any gauge coupling if the temperature T = 0 Review: Jens Braun

What happens at finite T ?

SLIDE 13

• Nov. 17, 2015 @ Osaka

Chiral Phase Diagram I

13

Hatsuda-Kunihiro (1985) suggestively similar to Cabibbo-Parisi (1975) NJL Hagedorn Limit

SLIDE 14

• Nov. 17, 2015 @ Osaka

Chiral Phase Diagram II

14

Asakawa-Yazaki (1986) Discovery of QCD Critical Point

SLIDE 15

• Nov. 17, 2015 @ Osaka

Experimentalists’ Hope

15

QCD
 Critical
 Point?

SLIDE 16

• Nov. 17, 2015 @ Osaka

Theorists’ Picture

16

Lifshitz
 point

20 40 60 80 100 240 260 280 300 320 340 360 T (MeV) µ (MeV)

Review: Buballa-Carignano (2014)

Chiral
 Density
 Wave

Likely candidate from NJL

SLIDE 17

• Nov. 17, 2015 @ Osaka

Theorists’ Picture

17

State-Of-The-Art Work NG mode
 → No long-range order
 Long-range corr. remains

• cf. Hidaka-Kamikado-Kanazawa-Noumi

SLIDE 18

• Nov. 17, 2015 @ Osaka

Schematic QCD Phase Diagram

18

Chemical Potential μ

Nuclear Superfluid B

20 40 60 80 100 240 260 280 300 320 340 360 T (MeV) µ (MeV)

Sometimes called “Quarkyonic”,
 sometimes “Crystalline”, sometimes “Solitonic”, sometimes…

Fukushima-Sasaki

SLIDE 19

• Nov. 17, 2015 @ Osaka

Inhomogeneity in Sakai-Sugimoto

19

Ooguri-Park (2010) Chuang-Dai-Kawamoto-Lin-Yeh (2010) Fukushima-Morales (2013)

Still awaits a confirmation
 by NJL with strong B

SLIDE 20

• Nov. 17, 2015 @ Osaka

Augmented with Gauge Fields

20

Gauged NJL (Polyakov-loop augmented version)

At finite T the p.b.c. makes A0 (or A4) special

SNJL[ψ, ∂] → SNJL[ψ, ∂ + igA4] + Sglue[A4]

For random color distribution quarks averaged away
 (Confinement in the disordered phase)

Qualitative understanding for the (almost) coincidence of
 the chiral phase transition and the deconfinement at high T

SLIDE 21

• Nov. 17, 2015 @ Osaka

Establishment of PNJL

21

0.6 0.8 1 1.2 1.4 1.6 1.8 2 TTc 0.2 0.4 0.6 0.8 nqT3

Μ0.2 Tc Μ0.4 Tc Μ0.6 Tc

100 200 300 1 Temperature [MeV] Order Parameters Chiral Condensate Polyakov Loop

KF: PLB591, 277 (2004)

Ratti-Thaler-Weise: PRD73, 014019 (2006)

Unwanted excitations
 killed by gauge dynamics Baryon number fluct. is
 an effective measure for
 quark deconfinement

SLIDE 22

• Nov. 17, 2015 @ Osaka

NJL derivable from QCD???

22

In principle, yes, once gluons are integrated out

SLIDE 23

• Nov. 17, 2015 @ Osaka

NJL derivable from QCD???

23

In principle, yes, once gluons are integrated out

In practice… the situation is not so bad!

Gluons : massive (“physical” gluons should be confined) Ghost : IR enhanced (responsible for confinement)

Transverse Gluon Dressing Func. Ghost Dressing Func.

SLIDE 24

• Nov. 17, 2015 @ Osaka

NJL derivable from QCD???

24

In principle, yes, once gluons are integrated out

In practice… the situation is not so bad!

Gluons : massive (“physical” gluons should be confined) Ghost : IR enhanced (responsible for confinement)

QCD confinement realized in an NJL-friendly way

Lattice / Functional RG / Dyson-Schwinger (Landau gauge) Gribov-Zwanziger formalism (Landau/Coulomb gauge) Kondo et al. Cho-Faddeev-Niemi decomposition (Gauge Inv.)

SLIDE 25

• Nov. 17, 2015 @ Osaka

NJL-Limit of Sakai-Sugimoto

25

Preis-Rebhan-Schmitt (2011)

Confined Chiral broken

x4 u x4 x4

Deconfined Chiral broken Deconfined Chiral symmetric

uc

L L

0.04 0.08 0.12 0.16 10-2 10-1 100 T log10 µq

Confined Chiral Broken Zero Density Deconfined Chiral Broken Zero Density Confined Chiral Broken Finite Density Deconfined Chiral Broken Finite Density Deconfined / Chiral Symmetric Finite Density

Td T

NJL results are understood
 in the small L limit. (Chiral physics without conf.) Agreement breaks down at finite baryon density.
 We have no enough understanding of introducing density.

SLIDE 26

• Nov. 17, 2015 @ Osaka

Exotic Applications

26

Dynamical Symmetry Breaking in Curved Spacetime

– Four-Fermion Interactions –

Tomohiro Inagaki 1, Information Processing Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739 Taizo Muta 2, Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739 Sergei D. Odintsov 3, Departamento de Fisica, Universidad del Valle, A. A. 25360 Cali, Colombia and

• Dept. Math. and Physics, Tomsk Pedagogical University, 634041 Tomsk, Russia

Review: Inagaki-Muta-Odintsov (1997) Applications include: Quark-Gluon Plasma / Early Universe
 Condensed Matter etc…

SLIDE 27

• Nov. 17, 2015 @ Osaka

Curvature-induced Transition

27

−0.06 −0.04 −0.02 0.02 0.04 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 V/m2.5 σ/m0 R = 3Rcr/2 R = Rcr R = Rcr/2 R = 0 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 V/m2 σ/m0 R = 0 R = −2m−2 R = −4m−2

Chiral Phase Transition Chiral Broken More NJL (mean-field)

SLIDE 28

• Nov. 17, 2015 @ Osaka

Chiral Gap Effect

28

Fukushima-Flachi / Fukushima-Flachi-Vitaliano (2015)

Chiral circle is shifted by a scalar curvature as

ρ → ρ + R 12λ2

−0.06 −0.04 −0.02 0.02 0.04 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 V/m2.5 σ/m0 R = 3Rcr/2 R = Rcr R = Rcr/2 R = 0 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 V/m2 σ/m0 R = 0 R = −2m−2 R = −4m−2

reproducing qualitatively QCD

SLIDE 29

• Nov. 17, 2015 @ Osaka

Diakonov’s Gravity Model

29

Diakonov-Tumanov-Vladimirov (2011)

Riemann : metric gµn Cartan : vierbein eµA + spin connection wµAB

Integrated Out Equivalent

Sign problem → large fluctuations (for bosonic fields) Fermions do not care the stability of potential

eA

µ = 1

2ψ†γArµψ 1 2(rµψ)†γAψ

Akama (1978) Wetterich (2005)

QCD-like theory : y ~ quarks wµAB ~ gluon [SU(2)xSU(2)] Chiral condensate

hψ†ψi 6= 0

→ What happens?

SLIDE 30

• Nov. 17, 2015 @ Osaka

Diakonov’s Gravity Model

30

Diakonov-Tumanov-Vladimirov (2011)

QCD-like theory : y ~ quarks wµAB ~ gluon [SU(2)xSU(2)] SO(16) unification 256 fermions ← 16D metric → 4 generations x 64 fermions in the SM

SO(16) ⊃ (SU(2) × SU(2)) × SU(3)C × SU(2)W × U(1)γ

Spontaneously broken by a fermion condensate

This scenario is not really NJL, but a sprit inherited!

SLIDE 31

• Nov. 17, 2015 @ Osaka

NJL: Summary

31

Best model for beginners to have experiences

SLIDE 32

• Nov. 17, 2015 @ Osaka

NJL: Summary

32

Best model for beginners to have experiences Best model for experts to test new ideas

SLIDE 33

• Nov. 17, 2015 @ Osaka

NJL: Summary

33

Best model for beginners to have experiences Best model for experts to test new ideas Best model for pioneers to do academic exercises Effective Model with Fermions + SSB

ubiquitous in all physics fields!