Gap Equation and Schwinger Dyson Equation

Pok Man Lo

University of Pittsburgh

High Energy Journal Club, 9-11-2008

personal information

institution: University of Pittsburgh

advisor: Eric Swanson
outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
dynamically generated mass

The mass of the constituents of the proton is expected to be

1
3 Mproton
dynamically generated mass

The mass of the constituents of the proton is expected to be

1
3 Mproton

It is believed that the (relatively large) constituent quark mass is

due to the fact that chiral symmetry is spontaneously broken
dynamically generated mass

The mass of the constituents of the proton is expected to be

1
3 Mproton

It is believed that the (relatively large) constituent quark mass is

due to the fact that chiral symmetry is spontaneously broken

current quark mass is small

the mass of the quark is dynamically generated
dynamically generated mass

The mass of the constituents of the proton is expected to be

1
3 Mproton

It is believed that the (relatively large) constituent quark mass is

due to the fact that chiral symmetry is spontaneously broken

current quark mass is small

the mass of the quark is dynamically generated

Mass is generated through interaction, and can generally be M (k)

gap equation

Gap equation dictates how the interaction give rise to the

dynamical mass

...that is, how the interaction dress the particle in question

gap equation

Gap equation dictates how the interaction give rise to the

dynamical mass

...that is, how the interaction dress the particle in question

study of spontaneous symmetry breaking mandates the use of

non-perturbative techniques
gap equation

Gap equation dictates how the interaction give rise to the

dynamical mass

...that is, how the interaction dress the particle in question

study of spontaneous symmetry breaking mandates the use of

non-perturbative techniques

non-perturbative physics
non-perturbative physics

exact solution: dream of a theorist...?

Newtonian physics: 3-body problem was insoluble

non-perturbative physics

exact solution: dream of a theorist...?

Newtonian physics: 3-body problem was insoluble

QED: 2-body and 1-body problem was insoluble

non-perturbative physics

exact solution: dream of a theorist...?

Newtonian physics: 3-body problem was insoluble

QED: 2-body and 1-body problem was insoluble

now: 0-body, namely, the vacuum is insoluble

non-perturbative physics

exact solution: dream of a theorist...?

Newtonian physics: 3-body problem was insoluble

QED: 2-body and 1-body problem was insoluble

now: 0-body, namely, the vacuum is insoluble

No body is too many!!

we are tranined to make sensible approximation

illustration of the failure of perturbation
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 2 4 6 8 10

...an innocent looking function

illustration of the failure of perturbation
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 2 4 6 8 10

...an innocent looking function

using Taylor expansion, we write

f (x) = A0 + A1 x + A2 x2 + ...
illustration of the failure of perturbation
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 2 4 6 8 10

...an innocent looking function

using Taylor expansion, we write

f (x) = A0 + A1 x + A2 x2 + ...

but what you will find is ...

illustration of the failure of perturbation
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 2 4 6 8 10

...an innocent looking function

using Taylor expansion, we write

f (x) = A0 + A1 x + A2 x2 + ...

but what you will find is ...

A0 , A1 , A2 , ... are all strightly 0!
illustration of the failure of perturbation

in view of the differential equation...

illustration of the failure of perturbation

in view of the differential equation...

f (x) + x3 f 0 = 0
illustration of the failure of perturbation

in view of the differential equation...

f (x) + x3 f 0 = 0

1
f (x) = exp− x2
illustration of the failure of perturbation

in view of the differential equation...

f (x) + x3 f 0 = 0

1
f (x) = exp− x2

any other method will work, other than perturbation!

different philosophy

Perturbative Vs Non-perturbative:
different philosophy

Perturbative Vs Non-perturbative:

perturbative method: sum all diagrams up to certain order in α

different philosophy

Perturbative Vs Non-perturbative:

perturbative method: sum all diagrams up to certain order in α

non-perturbative method: sum a certain class of diagram to all

order,
different philosophy

Perturbative Vs Non-perturbative:

perturbative method: sum all diagrams up to certain order in α

non-perturbative method: sum a certain class of diagram to all

order, ...or more generally, a truncaton
different philosophy

Perturbative Vs Non-perturbative:

perturbative method: sum all diagrams up to certain order in α

non-perturbative method: sum a certain class of diagram to all

order, ...or more generally, a truncaton

in some sense, non-perturbative method includes the whole

perturbation method as the latter is just a classification of diagram
by the coupling constant α
different philosophy

Perturbative Vs Non-perturbative:

perturbative method: sum all diagrams up to certain order in α

non-perturbative method: sum a certain class of diagram to all

order, ...or more generally, a truncaton

in some sense, non-perturbative method includes the whole

perturbation method as the latter is just a classification of diagram
by the coupling constant α

perturbation is not the only way to make approximation!

different philosophy

Perturbative Vs Non-perturbative:

perturbative method: sum all diagrams up to certain order in α

non-perturbative method: sum a certain class of diagram to all

order, ...or more generally, a truncaton

in some sense, non-perturbative method includes the whole

perturbation method as the latter is just a classification of diagram
by the coupling constant α

perturbation is not the only way to make approximation!

QFT contains more than just the S-matrix and perturbation!

side track: bound state problem

another important area of non-perturbative physics is the

formation of bound state
side track: bound state problem

another important area of non-perturbative physics is the

formation of bound state

easy to motivate the necessity to include all orders in coupling

constant
side track: bound state problem

another important area of non-perturbative physics is the

formation of bound state

easy to motivate the necessity to include all orders in coupling

constant

classical
q mechanics example: mass attached to a spring:
k
A sin( m t)
outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
method of partial sum

summing a particular class of diagrams to all order

method of partial sum

summing a particular class of diagrams to all order

to be explicit, let’s consider the Hamiltonian:

Z Z
H= 3
d xψ~x† γ 0 [−i~γ · ∇ + m]ψ~x − G d3 xd3 yV~x~y ψ~x† T a ψ~x ψ~y† T a ψ~y

this corresonds to the case where the quark exchange an

instantaneous gluon locally

we investigate the diagrams for constructing a propagator

diagrammatics of propagator

in general, to construct the propagator, we need to sum...

diagrammatics of propagator

in general, we cannot sum them all!

diagrammatics of propagator

in general, we cannot sum them all!

but let’s focus on the following class of diagram:

diagrammatics of propagator

in general, we cannot sum them all!

but let’s focus on the following class of diagram:

diagrammatics of propagator

the above procedure should be compared with the summing of

geometric series:

1
= 1 + x + x2 + x3 + ...
1−x
diagrammatics of propagator

inspired by the series, we know we can do better:

diagrammatics of propagator

inspired by the series, we know we can do better:

diagrammatics of propagator

also, to take care of

diagrammatics of propagator
diagrammatics of propagator
diagrammatics of propagator

the above corresponds to the Hartree-Fock approximation of the

Schwinger Dyson Equation
the inclusion of ring diagram

the above is pretty standard, it would be interesting to see how one

can go beyond Hartree-Fock approximation
the inclusion of ring diagram

the above is pretty standard, it would be interesting to see how one

can go beyond Hartree-Fock approximation

for example, we can include the ring diagram...

but where do this come from?

the inclusion of ring diagram

the above is pretty standard, it would be interesting to see how one

can go beyond Hartree-Fock approximation

for example, we can include the ring diagram...

but where do this come from?

the inclusion of ring diagram

in fact the full Schwinger Dyson Equation for the above case reads
like:
the inclusion of ring diagram

we see the ring structure comes out if we expand...

the inclusion of ring diagram

we see the ring structure comes out if we expand...

the inclusion of ring diagram

the Hartree Fock approximation is an approximation in the sense

that it effectively truncate the Schwinger Dyson Equation,
namely, by neglecting the connected four point function
the inclusion of ring diagram

the Hartree Fock approximation is an approximation in the sense

that it effectively truncate the Schwinger Dyson Equation,
namely, by neglecting the connected four point function

in general, the four point function itself depends on some higher

function, hence there will be an infinite tower of Schwinger Dyson
Equation

in perturbation, we approximate by expanding up to a certain order

in α

in Schwinger Dyson approach, we approximate by truncation

note that the sum for each truncaton is up to all order in α, hence
the result is non-perturbative
side track: the case for qed

the Schwinger Dyson Equations for QED are:

the ring diagram in QED

in this case, the ring obviously comes form the expansion of the
dressed photon propagator in the fermion Schwinger Dyson
equation

it is interesting to compare the case we have above: the four point

function do not appear explicitly in the fermion propagator, in
QED, there is no need for that, the dressed photon contain such a
term

of course, at some point, one will need to truncate these towers of

Schwinger Dyson Equation
some comments

1.) the diagrammatic approach is intuitive in the sense that you

can describe the various contribution using physically meaning
Feynman diagrams

2.) but it’s hard to decide which terms to include in the expansion

one needs a more systematic method to generate these equations:

Schwinger Dyson Equation directly from the generating functional

outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
Schwinger Dyson equation

represents an exact relations among n-point functions

Schwinger Dyson equation

represents an exact relations among n-point functions

effectively derives an expression for full propagator

Schwinger Dyson equation

represents an exact relations among n-point functions

effectively derives an expression for full propagator

by expanding Vint in terms of functional derivatives, we can obtain

the complete perturbative expansion of the theoretical quantity
Schwinger Dyson equation

represents an exact relations among n-point functions

effectively derives an expression for full propagator

by expanding Vint in terms of functional derivatives, we can obtain

the complete perturbative expansion of the theoretical quantity

basic idea: Z
d
dx f (x) → 0
dx
Schwinger Dyson equation

R
DψDψDGei(S+ ηψ+ψη+...)
R
Z= R
DψDψDGei(S)
Schwinger Dyson equation

R
DψDψDGei(S+ ηψ+ψη+...)
R
Z= R
DψDψDGei(S)
using the fact
Z
δ i(S+R ηψ+ψη+...)
DψDψDG e =0
δψ
we get

δS δ δ δ
{ [−i , i , −i ] + η}Z = 0
δψ δη δη δjµ
Schwinger Dyson equation

R
DψDψDGei(S+ ηψ+ψη+...)
R
Z= R
DψDψDGei(S)
using the fact
Z
δ i(S+R ηψ+ψη+...)
DψDψDG e =0
δψ
we get

δS δ δ δ
{ [−i , i , −i ] + η}Z = 0
δψ δη δη δjµ
for connected diagram, it is useful to define W by Z = eiW
Schwinger Dyson equation

as an illustration, we derive the Gap Equation for our old

Hamiltonian:

Z Z
H= 3
d xψ~x† γ 0 [−i~γ · ∇ + m]ψ~x + G d3 xd3 yV~x~y ψ~x† T a ψ~x ψ~y† T a ψ~y
Schwinger Dyson Equation

Schwinger Dyson Equation

δ2W
= (iγ · ∂x − m)−1 δ +
δηδη
2
W 2W
V S0 γ 0 T δδηδη T r[γ 0 T δδηδη
R
G ]
2 W 0 δ W 2
G V S0 γ 0 T δδηδη
R
γ T δηδη
δ4 W
G V S0 γ 0 γ 0 T T δηδηδηδη
R

corresponding to the diagrams ...

Schwinger Dyson Equation
gap equation

for Hartree-Fock approximation, we neglect the 4 point function

and get:
gap equation

for Hartree-Fock approximation, we neglect the 4 point function

and get:

Schwinger Dyson Equation

δ2W δ2W 0 a δ2W

Z
(iγ · ∂x − m) + 2G d4 zV γ 0 T a i γ T = δxy
δηx δηy δηx δηz δηz δηy
gap equation for Hartree Fock approximation

gap equation

d3 k 0 M (k~0 ) |k~0 |M (~k)

Z
T r[T T ]
M (~k) = m − G V ~0 ~ [ − k̂ 0 · k̂ ]
Nc (2π)3 k −k Ek~0 E ~0 |~k|
k

it dictates how the mass is dynamically generated

outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
numerical results for gap equation

contact potential: V (~x) = δ (3) (~x)

linear potential: V (~x) = br

contact potential

Dynamical Mass Generation

4
|G| > G_critical
|G| < G_critical
3.5

2.5
M

1.5

0.5

0
0 0.5 1 1.5 2 2.5 3 3.5 4
current mass m
linear potential

gap equation for linear case

0.35

0.3

0.25

0.2
M(k)

0.15

0.1

0.05

0
1e-05 0.0001 0.001 0.01 0.1 1 10
log k
outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
motivation

FTFT is needed to study the physics of QGP, deconfinement and

chiral restoration

when calculating observables in QFT, we only calculate the

vacuum expectation value

at finite temperature, excited states start to contribute, the

interesting quantity should be the thermal average of an
observable

we expect nE~k to enter QFT

partition function dictates the equilibrium Finite Temperature QFT

outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
basics of FTFT

the partition function:

Z
−βH
Z = T r[e ]= Dφ < φ|e−βH |φ >
basics of FTFT

the partition function:

Z
−βH
Z = T r[e ]= Dφ < φ|e−βH |φ >
Z R
−βH
Z = T r[e ]= DψDψe− ψψ
< −ψ|e−βH |ψ >
basics of FTFT

the partition function:

Z
−βH
Z = T r[e ]= Dφ < φ|e−βH |φ >
Z R
−βH
Z = T r[e ]= DψDψe− ψψ
< −ψ|e−βH |ψ >

observables are given by

T r[e−βH O]
hhOii =
T r[e−βH ]
basics of FTFT

Z
−βH
Z = T r[e ]= Dφ < φ|e−βH |φ >
Z R
−βH
Z = T r[e ]= DψDψe− ψψ
< −ψ|e−βH |ψ >

the form of partition function motivates the use of imaginary time

τ = it

a corresponding path integral representation of the partition

function can be obtained, with periodic or antiperiodic
boundary condition

the origin of this is of course due to the statistics of the particle

basics of FTFT

the statistics of the particles, together with the boundary condition

motivates the use of Matsubara Green’s function: G[ωn~k]
basics of FTFT

the statistics of the particles, together with the boundary condition

motivates the use of Matsubara Green’s function: G[ωn~k]

2nπ (2n+1)π
with ωn = β or β
basics of FTFT

the statistics of the particles, together with the boundary condition

motivates the use of Matsubara Green’s function: G[ωn~k]

2nπ (2n+1)π
with ωn = β or β

Z
1
dk 0 → Σω
β n
similarly, one can derive the Schwinger Dyson Equation

techniques in performing the Matsubara frequency sum

similarly, one can derive the Schwinger Dyson Equation

techniques in performing the Matsubara frequency sum

→ which in turns give you the thermodynamics...

similarly, one can derive the Schwinger Dyson Equation

techniques in performing the Matsubara frequency sum

→ which in turns give you the thermodynamics...

in the non-interacting limit, it reduces to statistical mechanics...

outline

1 Gap equation and the study of non-perturbative physics

2 Tools to study the Gap Equation

Diagrammatics
Hartree-Fock approximation
Beyond Hartree-Fock
side track: the case for QED
Schwinger Dyson Equation

3 Numerical Results

4 hidden track: finite temperature gap equation

Survey of basic formalism of Finite Temperature of Field
Theory
Finite Temperature Gap Equation
gap equation in finite temperature

Generalized Gap Equation at Finite Temperature

d3 k 0 M (k~0 ) |k~0 |M (~k)

Z
T r[T T ]
M (~k) = m − G 3
Vk~0 −~k [ − k̂ 0 · k̂ ]
Nc (2π) Ek~0 E ~0 |~k|
k

(1 − 2nE ~0 )
k

1
with nE~k = βE~
e k +1
numerical result for finite temperature gap equation

again, results for contact case and linear potential

contact case

finite temperature contact mass

1
G_eff = 3.6

0.9

0.8

0.7
dynamical mass

0.6

0.5

0.4

0.3

0.2

0.1

0
0 2 4 6 8 10
beta
linear potential case

M(k) for various Temperature

0.35
beta = 500
beta = 100
beta = 30
0.3 beta = 7
beta = 6.85
beta = 6.35

0.25

0.2

0.15

0.1

0.05

0
1e-05 0.0001 0.001 0.01 0.1 1 10
linear potential case

M(0) Vs Temperature
350

300

250

200
M(0) (MeV)

150

100

50

0
0 20 40 60 80 100 120 140 160
Temperature (MeV)
thank you