Rg Seminar

8/10/2019 Rg Seminar

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Introduction to theExact Renormalization Group

Informal Seminar

Bertram Klein, GSI

literature: J. Berges, N. Tetradis, and C. Wetterich [hep-ph/0005122].

lectures H. Gies, UB Heidelberg.

D. F. Litim, J. M. Pawlowski [hep-th/0202188].

[Wetterich (1993), Wegner/Houghton (1973), Polchinski (1984)].

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Outline

motivation

exact RG

scale-dependent effective action

one-loop flow equations for effective action

hierarchy of flow equations for n-point functions truncations

connection to perturbative loop expansion

O(N)-model in a derivative expansion: flow equations

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Motivation

need to cover physics across different scales

microscopic theory macroscopic (effective) theory bridge the gap between microscopic theory and effective macroscopic description (in

terms of effective/thermodynamic potentials, . . . )

loose the irrelevantdetails of the microscopic theory

How do we decide what is relevant and what is not?

important role of fluctuations: long-range in the vicinity of a critical point

How do we treat long-range flucutations?

universality: certain behavior in the vicinity of a critical point independentfrom the

details of the theory (e.g. critical exponents)

often additional complications: need to go from one set of degrees of freedom (at the

microscopic level) to a different set (at the macroscopic level)

here: we want to use anaverage effective action

in the macroscopic description

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Exact RG Flows

What do we mean by exact renormalization group flows?

derived from first principles

connects (any given) initial action(classical action) with full quantum effective action

exact flow reproduces standard perturbation theory

flow in theory space: trajectory is scheme-dependent, but end point is not

truncations project true flow onto truncated action

[]

S[]

[fig. nach H. Gies]

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Goal: A scale-dependent effective action

Our goal is an averaged effective action k[] which is ...

. . . a generalization of the effective action which includes onlyfluctuations with q2

k2

. . . a coarse-grained effective action, averaged over volumes 1kd

(i.e. quantum

flucutations on smaller scales are integrated out!)

...for large k (small length scales) very similar to the microscopic actionS[] (since

long-range correlations do notyet play a role)

...for small k (large length scales) includes long-range effects (long-range correlations,

critical behavior, . . . )

. . . and which can be derived from the generating functional.

How does this look in practice?

we look at derivation of such an effective action starting from the generating functional for

n-point correlation functions

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Derivation of the scale-dependent effective action [1]

scalar theory, fields a, a= 1, . . . , N , d Euclidean dimensions

start from the generating functional of the n-point correlation functions (path integral rep.)

Z[J] =

D exp

S[] +

x

J

define a scale-dependent generating functional by inserting a cutoff term

Zk[J] = D exp S[] + x

J Sk[] define scale-dependent generating functional Wk[J] for the connectedGreens functions by

Zk[J] = exp [Wk[J]]

cutoff term for a scalar theory:

Sk[] = 1

2

q

(q)Rk(q)(q)

[cutoff term quadratic in the fields ensures that a one-loop equation can be exact (Litim)]

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Intermezzo: Properties of the cutoff function

required properties for Rk(q):

1. Rk(q) 0 for k 0 at fixedq(so that Wk0[J] =W[J] and thus k0[] = [])2. Rk(q) (divergent) for k (k for ) (so that k[] = [] =S[])

3. Rk(q)> 0 for q2 0 (e.g. Rk(q) k2 for q2 0) (must be an IR regulator, after all!)

examples for popular cutoff functions:

1. without finite UV cutoff

Rk(q) = q2 1

exp

q2

k2 1

2. with a finite UV cutoff

Rk(q) = q2 1

expq2

k2

exp

q2

2

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Derivation of the scale-dependent effective action [2]

exchange dependence on the source J for dependence on expectation value

(x) =

Wk[J]

J(x) (x) =ak[J(x)]

employ a modified Legendre transformationand define the scale dependent effective action as

k[] = Wk[J] + xJ(x)(x) Sk[] ()

(): cutoff term depends on expectation value : crucial for connection to the bare (classical)

action S[] at the UV scale, and to quench only fluctuations around the expectation value!

Variation condition on the action/equation of motion for (x)

k[](x)

= y

Wk[J]J(y)

J(y)(x)

+y

J(y)(x)

(y) =0

+J(x) Sk[]

(x)

= J(x)

(x)

Sk[] =J(x) (Rk)(x)

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Derivation of Flow equation [1]

Flow equation: It describes the change of the scale-dependent effective action at scale k with a

change of the RG scale, and thus howthe effective actions on different scales are connected.

to derive the flow equation we need

modified Legendre transform

scale-dependent generating functional of the connected Greens functions

take the derivative with regard to the scale of the modified Legendre transformation

(introduce t= log (k/) t=kk):

tk[] = tWk[J]

x

Wk[J]

J(x)

=(x)tJ+

x

(x)(tJ)

=0

tSk[] = tWk[J] tSk[]

derivative of the cutoff term (remember that is the independent variable in k[])

tSk[] = t1

2 q (q)Rk(q)(q) =

1

2 q (q)(tRk(q))(q)

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we need the scale derivative ofWk[J]

first express the derivative in terms of exp(Wk[J])

tWk[J] = exp(Wk[J]) exp(Wk[J]) =1

tWk[J] = exp(Wk[J]) (tWk[J]) exp(Wk[J])

= exp(Wk[J]) (texp(Wk[J]))

now go back to the path integral representation: scale dependence appears only in cutoff term

tWk[J] = exp(Wk[J])t

D exp

S[] +

x

J Sk[]

= exp(Wk[J]) D(tSk[]) expS[] + x J Sk[]= exp(Wk[J])

D

1

2

q

(q)(tRk(q))(q)

exp

S[] +

x

J Sk[]

= 1

2 q(tRk(q)) exp(Wk[J]) D (q)(q) expS[] + x J Sk[]

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Express this in terms of the connected Greens functions:

exp(Wk[J])

2

J(q)J(q) exp(Wk[J]) =

2

Wk[J]J(q)J(q)+ Wk[J]J(q) W

k[J]J(q)

= (q)(q)k,connected+ (q)(q)

= Gk(q, q) + (q)(q)

we find for the flow ofWk[J]

tWk[J] = 1

2

q

(tRk(q)) (Gk(q, q) + (q)(q))

= 1

2 q(tRk(q))Gk(q, q) 1

2 q (q)(tRk(q))(q)

= 1

2

q

(tRk(q))Gk(q, q) tSk[]

insert this into the flow equation for k . . .

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. . . result for tWk[J] into flow equation:

tk[] = tWk[J] tSk[]

= 1

2

q

(tRk(q))Gk(q, q) + tSk[] tSk[]

The result for the flow equation for the effective action is

tk[] = 1

2

q

(tRk(q)) Gk(q, q)

This should now be expressed as a functional differential equation for the effective action.

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Inversion of scale-dependent propagator [1]

What remains to do in order to obtain a (functional) differential equation for the

scale-dependent effective action is to express G(p, q) in terms of this effective action

G(p, q) = 2Wk[J]

J(p)J(q), (q) =

Wk[J]

J(q)

use variation condition on effective action (from modified Legendre transformation)

k[](q)

= J(q) (q)Rk(q)

second variation with respect to (q)

2k[]

(q)(q) =

J(q)

(q) Rk(q)(q q

)

J(q)

(q) =

2k[]

(q)(q)+ Rk(q)(q q

).

Now start from an identity to show that this is the inverse ofG(q, q)

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Inversion of scale-dependent propagator [2]

start from the identity

(q)

(q) = (q

q)

=

(q)

Wk[J]

J(q) =

q

2Wk[J]

J(q)J(q)

J(q)

(q)

use expression for J/ established above

= (q q) =

q

2Wk[J]

J(q)J(q)

2k[]

(q)(q)+ Rk(q) (q

q)

the scale dependentinverse propagator is given by

G(q, q) = 2k[]

(q)(q)+ Rk(q)(q q

)1

(result is as expected, but it is necessary to establish the particular form of the scale

dependence of the propagator)

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Result for the Flow equation

tk[] =

1

2q (

tRk(q)) 2k[](q)(q)+ Rk(q)1

Graphical representation (insertion stands for the derivative of the cutoff function tRk)

k

1

2

The line represents the fullpropagator (which includes the complete field dependence).

In a more abstract representation (where in general the trace also involves any internal indices)

tk[] = 1

2Tr

(tRk)

(2)k [] + Rk

=t

1

2Tr log(

(2)k [] + Rk)

Note that this is notequal to the totalderivative of a one-loop effective action(since the terms

t(2)

k

[] are missing), although it is a one-loop flow equation!

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Flow Equation: Exact?

In principle, there are many different ways to introduce some sort of cutoff function into the

path integral, and to obtain in this way a flow equation of type

tk[] =Fk[(2)k ]

with some functional Fk[(p, q)].

We needed to show two propertiesin order to estalish that the flow is exact:

1. Is the action at scale k related to the full effective quantum action for k 0? Are they in

fact connected as k0[] = []?

2. Is the action at scale k related to the classical/initial action for k ? Are they in fact

connected as k

[] =S[]?

[as an analogy, one can think of a proof by induction: one needs to prove the induction step

from n to n + 1 (here: flow equation), but also the induction premise, the validity of the

statement for n= 1 (here: connection to classical action)]

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Exactness [1]

We begin by answering question 1 (which is the easier one):

How is the scale-dependent action related to the quantum effective action?

From the properties of the cutoff function we have

limk0

Rk(q) = 0 limk0

Sk[] = 0

and therefore for the scale-dependent generating functional

limk0

Zk[J] =Z[J]

thus, by the properties we require from the cutoff, it is trivially true that

limk0 k[] = Wk0[J] + x J Sk0[]= W[J] +

x

J= []

is the complete quantum effective action!

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Exactness [2]

We now answer the second question:

How is the effective scale dependent action related to the trival (classical) action?

This turns on the properties required of cutoff function and modified Legendre transform.

start from the modified Legendre transform (this is where the necessity of the modification

really comes in) and exponentiate:

exp(k[]) = expx

J + Sk[] exp(Wk[J])= exp

x

J + Sk[]

D exp

S[] +

x

J Sk[]

= D expS[] + x J( ) + Sk[] Sk[] exp(Wk[J]) has been replaced by the path integral representation ofZk[J].

now use as a background field: = + ,

D

D

(no assumptions necessary regarding its relation to the minimum of the classical action S[])

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Exactness [3]

multiply out the square in the cutoff term (no approximation)

Sk[ +

] = Sk[] + (Rk) + 12 (Rk) introduce this into expression

exp(k[]) =

D exp

S[ + ] +

J Sk[ +

] + Sk[]

=

D exp

S[ + ] +

J

(Rk) ()

Sk[] Sk[] + Sk[]

use equation for to simplify term linear in (remains unconstrained)

k[]

= J (Rk)

find finally

exp(k[]) = D expS[ + ] + k[] Sk[]19


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Exactness [4]

now, in the limit k , the cutoff function diverges by requirement

cutoff term diverges as

lim0

exp

1

2

()2

exp(Sk[

]) []

exponential becomes a -functional (w/ appropriate normalization)! In the path integral

limk

exp(k[]) = limk

D exp(S[ + ] + k[] Sk[])=

D exp(S[ + ] +

k[]

) []

= exp(S[])

[] = S[].

This proves that the scale-dependent effective action coincides with the classical action at the

UV scale and that the RG flow actualy connects the action at any scale k to the classical

action, and thus concludes the proof of the exactness of the ERG flow.

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Properties of the flow equation

What are the properties of this flow equation?

By definition, it describes the change of the effective action k with a change of the RG scale k.

Now, at this point, what have we obtained?

An exact (no approximations so far!) renormalization group flow equation for the effective

action . . .

. . . which is a nonlinear functional differential equation(since it involves the functional

derivatives (2)k [] of k[]!) . . .

. . . and which is of course in its most general form completely unsolvable!

So how do we solve this?

There are two questions that need to be asked:

1. How do we obtain correlation functions for a larger number of fields from this? [How does

it sprout legs?] (hierarchy question important for truncations)

2. This does look like a one-loop equation: are higher loop orders indeed contained in this?

Can we recover ordinary perturbation theory?

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Question 1: Higher n-point functions

How do we obtain flow equations for the higher n-point functions?

Simply take the appropriate number ofderivativesof the

flow equation for the effective action(-dependence in (2)

k []):

tk[] = 1

2Tr

(tRk)[(2)k [] + Rk]

1

take derivatives

tk[] = 12

Tr(tRk)(1)[(2)k + Rk]1 (2)k [(2)k + Rk]1=

1

2Tr

(tRk)[(2)k + Rk]

1 (3)k [

(2)k + Rk]

1

Graphical representation:

t(1)k =

12 tRk

(3)k

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Higher n-point functions [2]

one more derivative to get the flow equation for the two-point function:

2

tk[] = 2

1

2 Tr(tRk)[(2)k + Rk]1 (3)k [(2)k + Rk]1 (3)k [(2)k + Rk]1

1

2Tr

(tRk)[(2)k + Rk]

1 (4)k [

(2)k + Rk]

1

Graphical representation [first graph implies a factor 2]:

t(2)k =

12

tRk(3)k

(3)k

12

(4)k

tRk

What does this imply?

To find flow equation for (2)k , we need

(3)k and

(4)k !

In general, for flow of (n)k , need

(n+1)k ,

(n+2)k :

hierarchy of flow equations! How can we do meaningful calculations?

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Truncations

Problem: In order to calculate the flow of (n)k , we need

(n+1)k and

(n+2)k .

Solution: We need to truncate the effective action and restrict it to correlators ofnmax fields.

But: then this is no longer a closed systemof equations!

in principle, need to write down most general ansatzfor the effective action

this ansatz will contain all invariantsthat are compatible with the symmetriesof the theory

then one truncates by reducing higher n-point functions to contact terms, or to a simplified

momentum dependence

one neglects even higher correlations outright

This is not an expansion in some small parameter (although of course the assumption is thathigher order operators will be irrelevant and suppressed due to the existence of a large scale)

For practical applications, this is obviously the most problematic part, and it requires a lot of

physical insight to make the correct physical choices.

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Question 2: Comparison to perturbation theory

Claim: although the ERG flow equation is a one-loop RG flow equation,

tk[] =1

2(tRk)pq[

(2)

k

[] + Rk]1

qp

, shorthand: ApqBqp = q A(p, q)B(q, p)it contains effects to arbitrary highloop order: expect reproduction of higher loop order

perturbation theory! [in the arguments here, we follow Litim/Pawlowski]

Do a loop expansionof the effective action and compare to perturbation theory:

= S+n=1

n

In terms of the flow equation, contributions of different loop orders can be identified

tk =

n=1

tn,k

notation: m-point correlation function to n-loop order at scale k

(m)n,k

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Comparison to PT: scheme of calculation

Our goal: two-loop result for the effective action (first non-trivial loop order)

As a roadmap for the expansion in loop order, here is an outline of the calculation:

1. start from effective action atn-loop level and calculate the two-point function

2. insert the n-loop two-point function into the (one-loop) flow equation

3. isolate the (n + 1)-loop correction to effective action

4. integrate flow equation to obtainn + 1-loop correction to effective action

n,k2

(2)n,k =

(2)n1,k+

(2)n,k

into flow eq.

(tRk)

(2)

n,k+ Rk

= (tRk) 1

(2)

n1,k+ (2)

n,k+ Rkisolate

tn+1,k = (tRk) 1

(2)n1,k+ Rk

(2)n,k

1

(2)n1,k+ Rk

integrate w.r.t. k: n+1,k = n,k+ n+1,k

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Comparison to PT: effective action at one loop

Effective action at one loop is calculated using the tree-level two-point function

k,1 = S+ k,1(2)k,0 = S

(2)

Flow equation for the one-loop correction to the effective action:

t1,k =

1

2 (tRk)pq S(2) + Rk1

qp

Integrate this to get the one-loop correction:

1,k = k

dk1

kt

1

2[log(S(2) + Rk)]pp

One-loop result corresponds with ordinary result (R similar to Pauli-Villars regulator):

1,k = S+1

2

log(S(2) + Rk)

pp

1

2

log(S(2) + R)

pp

=S+1

2

log(S(2) + Rk)

pp

k

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Comparison to PT: two-point function at one loop

find the one-loop contribution to the two-point function

it is obtained by taking the variation of the one-loop effective action

(2)1,k

qq

= 2

(q)(q)1,k =

1

2

2

(q)(q)

log(S(2) + Rk)

pp

k

= 1

2

GppS

(4)ppqq GpqS

(3)qqqGqpS

(3)ppq

k

where one uses

(q)Gpp =

(q)

S(2) + Rk

1

pp= (1)GpqS

(3)qqqGqp

Graphical representation:

[ ]1

2

double line: UV regularization from the cutoff

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Comparison to PT: notation for regularization

the double line represent the regularization through the presence of the UV cutoff

=

=

k

k

k

R has a similar effect as a Pauli-Villars regulator

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Comparison to PT: flow of two-loop correction to action

Now we need to find the two-loop correction to the flow equation for the effective action.

do this by first inserting the correction to the propagator into the flow equation . . .

. . . and then isolating the two-loop part

t2,k = 1

2(tRk)qp

(2)1,k+ Rk

1

pq=

1

2(tRk)qp

S(2) +

(2)1,k+ Rk

1

pq

= 1

2

(tRk)qp S(2) + Rk1

pq+

+1

2(tRk)qp(1)

S(2) + Rk

1

pq

(2)1,k

qq

S(2) + Rk

1

qq+ . . .

= t1,k+ t2,k+ . . .

Therefore we find for the correction

t2,k = 1

2(tRk)qpGpq

(2)1,k

qq

Gqq

This needs to be integrated over all scales k.

In order to do the k-integration, we need particular properties ofGpq.

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Comparison to PT: k-derivative of two-point function

The tree-level propagator (with cutoff at scale k) is from here on abbreviated as

Gpq = S(2) + Rk

1

pq

Look at derivatives of the propagator w.r.t. the RG scale k:

tGpq = t

S(2) + Rk

1

pq= (1)

S(2) + Rk

1

pq(tRk)qq

S(2) + Rk

1

qq

= (1)Gpq(tRk)qqGqq

Graphically: t =

This seems trivial, but is a very important result (and why we recover perturbation theory)

makes it possible to re-write terms in the flow equations as total derivativeswith the correct

combinatorial factorsthat come from inserting (tRk) in all possible propagators!

with appropriate renaming of indices:

GppS(4)ppqq(tG)qq =

1

2t

GppS

(4)ppqqGqq

GppS

(3)ppqGpqS

(3)qpq(tG)qq =

1

3t

GppS(3)ppqGpqS

(3)qpqGqq

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Comparison to PT: integrand as total derivative

using the result for the scale derivative ofGpq, we can write for the two-loop flow correction

t2,k =

1

2 (tRk)qpGpq (2)1,kqq Gqq

= 1

2

(2)1,k

qq

(tG)qq

now insert the expression for the one-loop propagator correction

use the results for tGpq to write this as a total derivative (note combinatorial factors!)

1

2

1

2

GppS

(4)ppqq GppS

(3)ppqGpqS

(3)qpq

k

(tG)qq

= 1

2

1

2

t 1

2

GppS(4)ppqqGqq

1

3

GppS(3)ppqGpqS

(3)qpqGqq

now perform the scale integration (and look at the graphical representation, to make this a

bit more transparent) . . .

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Comparison to PT: effective action at two loops

integrate with regard to the renormalization scale k

2,k

= k

dk1

k

1

2

1

2 GppS(4)ppqq GppS(3)ppqGpqS(3)qpqk

(

tG)

qq

=

k

dk1

k1

2

1

2t

1

2GppS

(4)ppqqGqq

1

3GppS

(3)ppqGpqS

(3)qpqGqq

1

4[ ] result of the integration (up to regularization terms)

2,k = 1

8

GppS(4)ppqqGqq

1

12

GppS(3)ppqGpqS

(3)qpqGqq

k

]1 121[8

ren.

this is indeed the correct perturbative two-loop result! [Figures are taken from Litim/Pawlowski (2002)]

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O(N)-model: simple example

Action for O(N)-symmetric scalar theory:

S[] = x12 (a)(a) + 12 m22 +14 (2)2given at some scale

= (a), a= 1, . . . , N , d Euclidean dimensions

given in terms of couplings at scale allows for spontaneous symmetry breaking and light modes

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O(N)-model: flow equation for effective potential

The flow equation for the effective potential is the lowest order term in the derivative expansion

t Uk() =

1

2 q t Rk(q) 1M1 + N 1M0 M0(, q

2) = Zk(, q2)q2 + Uk() + Rk(q)

M1(, q2) = Zk(, q

2)q2 + Yk(, q2)q2 + Uk() + 2U

k () + Rk(q)

where =

1

2

a

a

most interesting: regions with light degrees of freedom spontaneous symmetry breaking.

Note that in order to keep the rescaling invariance, we need the wave function renormalization

in the cutoff function (scale argument as the other momenta)

Rk(q) = Zkq2

exp(q2/k2) exp(q2/2)

Observe: as they stand, this set of equations is not closed!

Missing: flow equation for the wave function renormalizations!

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O(N)-model: anomalous dimension

the anomalous dimension is given in terms of the wave function renormalization Zk:

=

d

dtlog Zk(0(k), q

2

= 0)

the wave function renormalization can be calculated from the two-point function

(2)k (;p, q) = [Zk(, q

2)q2 + M2](p + q) (neglect explicit q-dependence ofZk()):

Zk() = limq20

q2

2

(q)(q) k[2

]

= limq20

q2(2)k (

2; q, q)

we need a flow equation for the two-point function as well . . .

. . . which in turn depends on the three- and four-point functions! (Hierarchyof flow equations!)

[why anomalous dimension?

Its the anomalous dimension of the propagator, as in (2)k q

2( q2

k2 + c)/2

presence of relevant scale k allows for scaling different from canonical dimension!]

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O(N)-model: results

need to close the equations

possible approach: obtain higher n-point functions from RG-improved flow equations (they

represent total derivative terms which can be integrated w.r.t. k)

reproduce perturbative function for four-point coupling to two loops

[Papenbrock/Wetterich hep-th/9403164]

result with uniform (no momentum, field dependence) wave function renormalization:

=N+ 8

162 2

17.26N+ 75.95

(162)2 3

need momentum and field dependence of wave function renormalization Zk(, q2)

result there in d= 4 coincides with the perturbative result:

= N+ 8

162 2

9N+ 42

(162)23

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O(N)-model: more details on -functions

to find critical behavior, re-write flow equations for couplings in scale-invariant form

starting from

t

Uk() = 1

2

q

t

Rk(q)N 1

M0+ 1

M1

(neglect tZk from Rk/Zk justified for small anomalous dimension)

1

2

q

t

Rk(q)N 1

M0

1

2

ddq(2)d

tRk

Zk

(N 1)q2 + Rk

Zk+ k2

Uk()

Zkk2

l

d

0(w) =

1

4

1

vdkd q tRkZk 1q2 + RkZk + k2w , 1

vd = 2

d+1

d/2

(d/2), l

d

n(w) =

w l

d

n1(w)

w is a dimensionless variable

functions ldn(w) are threshold functions, since they cut off modes with masses m2 k2

theory becomes theory of effective light modes!

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O(N)-model: -functions [2]

in terms of the threshold functions flow equation

tUk() = 2vdkd (N 1)ld0 U

k()

Zkk2 + ld0 Uk() + 2U

k (

Zkk2 Now take lowest possible approximation (for symmetry breaking): quartic potential

Uk() = k

2 ( 0(k))

2

flow equation for minimum 0(k) from minimum condition:

d

dtUk(0(k)) = tU

k(0(k)) + U

k (0(k))t0(k) 0

t0(k) = 2vdkd2Z1k 3 ld1

20(k)k

Zkk2 + (N 1)ld1(0)

flow equation for coupling k from second derivative 2

2tUk() (note U

k 0 )

tk = 2vdkd4Z2k

2k

9 ld2 20(k)k

Zkk2 + (N 1)ld2(0)

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O(N)-model: -functions [3]

introduce couplings rescaled according to canonical dimensions:

0=Zkk2d0(k) = Z

2k k

d4k

-functions: flow equations of the dimensionless couplings (note where = tlog Zkappears)

t0 = = (2 d )0+ 2vd

3ld1(2) + (N 1)l

d1(0)

t = = (d 4 + 2) + 2vd2 9ld2(2) + (N 1)ld2(0) need in principle anomalous dimension to solve this! As expected, related to long-range

correlations, so it has to be obtained from two-point correlator.

can already analyze fixed point structure (as a function of the dimension d) in this

approximation!d= 4: reproduce one-loop =

N+8162

2, find trivial fixed point ( 0 fork 0)

d= 3: scaling solution, critical point, phase transition.

d= 2: N= 1 phase transition/critical point, N 3 no fixed point/phase transition, N = 2

special: Kosterlitz-Thouless-transition!

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O(N)-model: more details on 2-point function

We want to derive the anomalous dimension: need to get it from the two-point function

We need to derive the flow equation of the two-point function.

Use that in our ansatz the inverse propagator is

G1k (, q2) = Zk(, q

2)q2 + Uk() =M0(, q2) Rk(q)

ansatz for the propagator (fields a are constant expectation values):

(2)k =12

q

(Uk() + Zk(, q

2)q2)a(q)a(q) +12

ab(2Uk () + Yk(, q2)q2)a(q)b(q)

ansatz for the higher couplings [simplified] that takes momentum dependence of couplings into

account in the form in which it appears in the two point function(momentum conserved):

(3)k = 1

2

q1

q2

a(1)k (; q1, q2)

a(q1)b(q2)

b(q1 q2) + . . .

(4)k = 1

8

q1

q2

(2)k (; q1, q2, q3)

a(q1)a(q2)

b(q3)b(q1 q2 q3) + . . .

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O(N)-model: 2-point function [2]

need to express couplings in terms of two-point function couplings

require a continuity condition between the couplings in (3)k, (4)k and in (2)k (roughly:

they have to coincide if one (3-point) or two (4-point) momenta vanish)

(1)(; q1, q2) = U

k () + q2 (q1+ q2)Z

k(, q2 (q1+ q2)) +1

2q21Yk(, q

21) + . . .

(2)(; q1, q2, q3) = U

k () q2 q1Z

k(, q2 q1) q4 q3Z

k(, q4 q3)

+ 12 (q1+ q2)2Yk(, (q1+ q2)2) + . . .

additional approximation: neglect the extra terms (to close equations)!

Now need to insert this into the general flow equation for the two-point function!

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O(N)-model: 2-point function [3]

Flow equation for the two-point function in the couplings introduced above

(-dependence ofMi(p2) suppressed):

tGk(, q2) =

12

p

tRk(p)

{4M21 (p2)M10 ((p + q)

2)((1)k (p, q))

2 + 4M20 (p2)M11 ((p + q)

2)((1)k (p q, q))

2

M20 (p2)[(N 1)

(2)k (q, q, p) + 2

(2)k (q,p, q)] M

21 (p

2)(2)k (q, q, p)}

use continuity conditions from above to replace (1,2)

k !

get flow equation for wave function renormalization

tZk(, q2) =

1

q2

(tGk(, q2) tU

k()) = k(, q2)Zk(, q

2)

actual anomalous dimension from this

= d

dtlog Zk(0(k), k

2) =k(0, k2) 2

k2

Zk(0, k2)

q2Zk(0, q

2)

q2=k2

Zk(0, k

2)

Zk(0, k2)t0

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Summary

What I hope you will take away from todays talk

to cover physics across different scales, it is important to have a systematic scheme of

integrating out quantum fluctuations

the so-called ERG is an exact RG scheme in the following sense: its RG flow equation

connects a classical action at some UV scale to the full quantum effective action

however, a solution relies on some truncation of the effective action result of acalculation is not exact!

the one-loop RG equation reproduces ordinary perturbation theory to arbitrary order (we

have shown this up to second order/the first nontrivial order)

example: works for scalar O(N)-model

different truncations possible: to get (important) anomalous dimension, flow equation for

two-point function is necessary

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Rg Seminar

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