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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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Derivation of Flow equation [2]
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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Derivation of Flow equation [3]
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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Derivation of Flow equation [4]
. . . 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
= 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
S(2) + Rk
1
qq+ . . .
= t1,k+ t2,k+ . . .
Therefore we find for the correction
t2,k = 1
2(tRk)qpGpq
(2)1,k
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
= (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
(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)
=
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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