acqft09_jzinnjustin

8/10/2019 ACQFT09_JZinnJustin

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RENORMALIZATION GROUP: AN INTRODUCTION

J. ZINN-JUSTIN*CEA,IRFU et Institut de Physique Theorique, Centre de Saclay,

F-91191 Gif-sur-Yvette cedex, FRANCE.

Renormalization grouphas played a crucial role in 20th century physics intwo apparently unrelated domains: the theory offundamental interactions

at the microscopic scale and the theory ofcontinuous macroscopic phase

transitions. In the former framework, it emerged as a consequence of the

necessity ofrenormalization to cancel infinities that appear in a straight-forward interpretation of quantum field theory and the possibility to define

the parameters of therenormalized theoryat different momentum scales.

Email : [email protected]


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In the statistical physics of phase transitions, a more general renormal-

ization group, based on recursive averaging over short distance degrees of

freedom, was latter introduced to explain theuniversality propertiesof con-

tinuous phase transitions.

Thefield renormalization groupnow is understood as theasymptotic form

of the general renormalization group in the neighbourhood of the Gaussian

fixed point.

Correspondingly, in the framework of statistical field theories relevant forsimple phase transitions, we review here first the perturbative renormal-

ization group then a more general formulation called functional or exact

renormalization group.


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As general references, cf. for example,

J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Claren-

don Press 1989 (Oxford 4th ed. 2002),

J. Zinn-Justin, Transitions de phase et groupe de renormalisation. EDP

Sciences/CNRS Editions, Les Ulis 2005

English versionPhase transitions and renormalization group, Oxford Univ.

Press (Oxford 2007).

In preparation, the introductionWilson-Fisher fixed pointon the sitewww.scholarpedia.org


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Statistical field theory

In the theory ofcontinuous phase transitions, one is interested in thelarge

distance behaviour or macroscopic properties of physical observables near

the transition temperatureT =Tc. At the critical temperature, the correla-tion length, which defines the scale on which correlations aboveTcdecay ex-

ponentially, diverges and the correlation functions decay only algebraically.

This gives rise to non-trivial large distance properties that are, to a large

extent, independent of the short distance structure, a property called uni-

versality.

Intuitive arguments indicate that even if the initial statistical model is

defined in terms of random variables associated to the sites of a space lattice,

and taking only a finite set of values (like, e.g., the classical spins of the

Ising model), the large distance behaviour can be inferred from astatistical

field theory in continuum space.


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Therefore, we consider a classical statistical system defined in terms of a

random real field(x)in continuum space,x Rd, and a functional measureon fields of the form eH()/Z, whereH() is called the Hamiltonian instatistical physics and

Z is the partition function(a normalization) given

by the field integral (i.e., a sum over field configurations)

Z= [d(x)] eH().

The condition ofshort range interactionsin the statistical system translates

into the property of localityof the field theory: H() can be chosen as aspace-integral over a linear combination of monomials in the field and its

derivatives.


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We assume also space translation and rotation invariance and, to discuss

a specific example, Z2 reflection symmetry:H()=H(). A typical formthen is

H() = ddx 12(x)2 + 12r2(x) + g4! 4(x) +

.

Finally, the coefficients ofH()are regular functions of the temperatureT

near the critical temperatureTc. In the low temperature phaseT < Tc, theZ2 symmetry is spontaneously broken.


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Correlation functions

Physical observables involve field correlation functions (generalized mo-

ments),

(x1)(x2) . . . (xn) 1Z

[d(x)](x1)(x2) . . . (xn) eH().

They can be derived by functional differentiation from the generating func-

tional (generalized partition function) in an external field H(x),

Z(H) =

[d(x)] exp

H() +

ddx H(x)(x)

,

as

(x1)(x2) . . . (xn) = 1Z(0)

H(x1)

H(x2). . .

H(xn)Z(H)

H=0

.


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Connected correlation functions

The more relevant physical observables are theconnected correlation func-

tions W(n)(x1, x2, . . . , xn)(generalized cumulants), which can be obtained

by function differentiation from the free energyW

(H) = lnZ

(H) in the

external field H:

W(n)(x1, x2, . . . , xn) =

H(x1)

H(x2). . .

H(xn)W(H)

H=0

.

Due to translation invariance,W(n)(x1, x2, . . . , xn) =W

(n)(x1+a, x2+a, . . . , xn+a)

for any vectora.

Connectedcorrelation functions have the so-calledcluster property: if one

separates the points x1, . . . , xn in two non-empty sets, connected functions

go to zero when the distance between the two sets goes to infinity. It is the

large distance behaviour of connected correlation functions in the critical

domain near Tc thatmay exhibit universal properties.


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The renormalization group: General formulation

To construct an RG flow, the basic idea is to integrate in the field integral

recursively over short distance degrees of freedom. This leads to the defi-

nition of an effective HamiltonianH function of a scale parameter > 0(such that H1 = H) and of a transformation Tin the space of Hamiltonianssuch that

d

dH =

T [

H] , (1)

an equation calledRG equation(RGE). The appearance of the derivative

d/d = d/d ln reflects the multiplicative character of dilatations. The

RGE thus defines a dynamical process in the timeln . The denomination

renormalization group(RG) refers to the property thatln belongs to theadditive group of real numbers.


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RG equation: General structure, fixed points

We will derive RGE that define astationary Markov process, that is, T[H]depends onH but not on the trajectory that has led fromH=1 toH,and depends on only through

H.

Universality is then related to the existence of fixed points, solution of

T(H) = 0 .We assume also that the mapping

T isdifferentiable, so that near a fixed

point the RG flow can be linearized,

T(H + H) LH,

and is governed by the eigenvalues and eigenvectors of the linear operatorL. Formally, the solution of the linearized equations can be written as

H = H +L (H=1 H) .


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The Gaussian measures

In the spirit of the central limit theorem of probabilities, one could hope that

the universal properties of phase transitions can be described by Gaussian

or weakly perturbed Gaussian measures. The simplest form satisfying allconditions in d space dimensions (we assume d 2), corresponds to thequadratic Hamiltonian (0 0constant)

H(0)() = ddx 12x(x)2 + 1202(x) . (2)One sees immediately that the Gaussian model can only describe the high

temperature phase T

Tc

.

In a Gaussian model, all correlation functions can be expressed in terms

of thetwo-point functionwith the help ofWicks theorem.


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Regularization

However, with this Hamiltonian the Gaussian model has a problem: too

singular fieldscontribute to the field integral in such a way that correlation

functions at coinciding points are not defined. For example,

2(x)

=W(2)(0, 0) =

1

(2)d

ddp

p2 +0,

which diverges in any space dimension d 2. In particular, expectationvalues of perturbations to the Gaussian theory of the form m(x)(powers

of the field at the same point) are not defined.

Thus, it is necessary to modify the Gaussian measure to restrict the field

integration to more regular fields, continuousto definepowers of the field,

satisfyingdifferentiability conditionsto defineexpectation values of the field

and its derivatives taken at the same point, a procedure calledregulariza-

tion.


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This can be achieved by adding to

H(0)() =

ddx

12

x(x)2

+ 1202(x)

,

enough terms with more derivatives (here, periodic boundary conditions

are assumed)

H(0)() HG() = H(0)() +12

2kmaxk=2

k

ddx (x)2kx (x), (3)

where the coefficientk are only constrained by the positivity of the Hamil-

tonian. For example, simplecontinuityrequires2kmax> d.


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The Gaussian two-point function at large distance

The two-point function then reads

W(2)(x, 0) = 1

(2)d d

dp eipx

0+p2 +

k=2 kp2k.

At large distance|x| , for 0= 0, correlations decrease exponentiallyas

W(2)(x, 0)

1

|x|(d1)/2

e|x|/ ,

where is thecorrelation lengthand

1/0 for 0 0 .At thecritical point(T =Tc), the correlation length diverges, which implies

0 = 0and, for d >2, one finds the algebraiccritical behaviour

W(2)(x, 0) |x|

1

|x|d2 .

For d= 2, the Gaussian model is not defined at Tc.


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The Gaussian fixed point

An RG can be constructed that reproduces the properties of the Gaussian

models. The Hamiltonian flow can be implemented by the simple scaling

(x) (2d)/2(x/). (4)

After the change of variables x =x/, one verifies that the Hamiltonian

HG() =1

2

ddx

x(x)2, (5)corresponding to thecritical Gaussian model, is invariant. The RG has

HG

as afixed point. The Hamiltonian flow (4) corresponds in fact to the linear

approximation of the general RGneartheGaussian fixed point.


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The linearized RG flow

The transformation (4) generates the linearized RG flow at the Gaussian

fixed point. Eigenvectors of the linear flow (4) are monomials of the form

On,k() =

ddx On,k(, x),

where On,k(, x)is a product of powers of the field and its derivatives at

pointxwith2npowers of the field (reflection Z2 symmetry) and2k powers

of.Their RG behaviour under the transformation (4) is then given by a

simple dimensional analysis. One defines the dimension ofx as-1and the

(Gaussian) dimension of the field is [] = (d

2)/2. The dimension[

On,k]

ofOn,k is then[On,k] = d+n(d 2) + 2k . (6)

It can be verified thatOn,k scales like [On,k], and the corresponding

eigenvalue ofL

thus is n,k = [On,k].


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DiscussionWhen +,

(i) for n,k > 0 the amplitude ofOn,k() increases; it is a direction ofinstability and in theRG terminology

On,k()is arelevantperturbation;

(ii) for n,k < 0, the amplitude ofOn,k()decreases; it is a direction ofstability andOn,k()is anirrelevantperturbation;

(iii) in the special casen,k = 0, one speaks of amarginalperturbation and

thelinear approximation is no longer sufficient to discuss stability. Loga-rithmic behaviour in is then expected (we omit here unphysical redundant

perturbations).

Since1,0 = 2, ddx 2(x)corresponds always to a direction of instability:

indeed it induces a deviation from the critical temperature and thus a finitecorrelation length.

Ford >4, no other perturbation is relevant and the Gaussian fixed point

isstableon the critical surface (=

).


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Since2,0 = 4 d, atd= 4one term becomes marginal: ddx 4(x), whichbelow dimension four becomes relevant. In dimension d = 4 , > 0small (a notion we define later), it is theonly relevant perturbationand one

expects to be able to describe critical properties with a Gaussian theory to

which this unique term is added.

To summarize, for systems with a Z2 or, more generally, with an O(N)

symmetry, one concludes that

(i) the Gaussian fixed point is stable above space dimension four;(ii) from a next order analysis, one shows that it is marginally stable in

dimension four;

(iii) it is unstable below dimension four.


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RescalingIf what follows, we assume that initially the statistical system is very close

to the Gaussian fixed point. The RG flow is then first governed by the local

linear flow. Therefore, we implement first the corresponding RG transfor-

mation. We introduce a parameter 1and substitute(x) (2d)/2(x/).

After the change of variables x =x/, a monomial

On,k()is multiplied

by [On,k], where [On,k] is the dimension in the sense of the linearizedRG. In the quantum field theory language, this could be called a Gaussian

renormalization. The Gaussian RG dimensions can then be expressed in

terms of: space coordinatesxhave dimension1, derivatives dimension

and the field dimension(d2)/2. The Hamiltonian is dimensionless.

In the context of quantum field theory, since the regularization has the

effect, in Fourier representation, to suppress the contributions of momenta

|p| in Feynman graphs,is called thecut-off.


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Statistical scalar field theory: Perturbation theory

The Gaussian model

After rescaling, the Hamiltonian of the Gaussian model takes the form

HG() =12

ddx

x(x)

2

+022(x) +

k=2k

22k(x)2kx (x)

,

(7)where 0 is the amplitude of the only relevant term for d > 4. Except

for the two-point function at coinciding points, one can take the limit. However, for 0

= 0, to obtain a non-trivial universal large distance

behaviour, it is also necessary to compensate the RG flow by choosing 0

infinitesimal, taking the limit at r = 02 fixed (r is a Gaussianrenormalizedparameter in quantum field theory language). This defines the

critical domain.


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The perturbed Gaussian or quasi-Gaussian modelTo allow forspontaneous symmetry breaking and, thus, to be able to de-

scribe physics belowTc, terms have necessarily to be added to the Gaussian

Hamiltonian to generate a double-well potential for constant fields. The

minimal addition, and the leading term from the RG viewpoint, is of 4

type. This leads to

H() =

HG() +

g

4!

4d ddx 4(x), g 0 .The 4 term generates a shift of the critical temperature. To recover a

critical theory (T =Tc), it is necessary to choose a 2 term with a specific

g-dependent coefficient 1

2

(0)c(g), amass renormalizationin quantum field

theory terminology.

As we have explained, for d > 4 the 4 term then is an irrelevant con-

tribution that does not invalidate the universal predictions of the Gaussian

model.


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Corrections to the Gaussian theory can be obtained by expanding in powersof the coefficient g.

Setting u=g4d, the partition function, for example, is then given by

Z= k=0

(u)k(4!)kk!

ddx 4(x)

kG

.

The Gaussian expectations values

G can then be evaluated in terms of

the Gaussian two-point function with the help of Wicks theorem (Feynman

graph expansion).

By contrast, for anyd


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Renormalization group in dimensiond= 4 Ford


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Dimensional continuation and regularization

To discuss dimensional continuation, it is convenient to introduce the Fourier

representation of correlation functions. Taking into account translation in-

variance, one defines

(2)d(d)

ni=1

pi

W(n)(p1, . . . , pn)

=

ddx1. . . ddxnW

(n)(x1, . . . , xn)expi n

j=1

xjpj , (8)

where, in analogy with quantum mechanics, the Fourier variables pi are

called momenta (and have dimension ). We also introduce the Fourierrepresentation of the two-point function (or propagator)(x), correspond-

ing to the Hamiltonian of the Gaussian model,

(x)

(x)(0)

G=

1

(2)d ddp eipx(p).


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Dimensional continuation

A general representation useful fordimensional continuationof the Gaussian

two-point function is the Laplace representation

(p) =

0

ds (s2)esp2

, (9)

where the function (s) 1when s .Moreover, to reduce the field integration to continuous fields and, thus,

to render the perturbative expansion finite, one needs at least (s) =O(sq)

withq >(d 2)/2for s 0.If, in addition, one wants the expectation values of all local polynomials

to be defined, one must impose to (s)toconverge to zero faster than any

power.


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A contribution to perturbation theory (represented graphically by a Feyn-man diagram) takes, in Fourier representation, the form of a product of

propagators integrated over a subset of momenta. With the Laplace rep-

resentation, all momentum integrations become Gaussian and can be per-

formed, resulting in explicit analytic meromorphic functions of the dimen-

sion parameterd. For example, the contribution of orderg to the two-point

function is proportional to

d = 1(2)d

dp (p) =

1(2)d

dp

0

ds (s2)esp2

= 1

(4)d/2

0

ds sd/2(s2),

which, in the latter form, is holomorphic for2


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Dimensional regularizationFor the theory of critical phenomena, dimensional continuation is sufficient

since it allows exploring the neighbourhood of dimension four, determining

fixed points and calculating universal quantities as = (4

d)-expansions.

However, for practical calculations, but then restricted to the leading

large distance behaviour, an additional step is extremely useful. It can be

verified that if one decreasesRe denough, so that by naive power counting

all momentum integrals are convergent, one can, after explicit dimensional

continuation, take the infinite limit. The resulting perturbative contri-

butions becomemeromorphic functions with poles at dimensions at which

large momentum, and low momentum in the critical theory, divergences

appear. This method of regularizing large momentum divergences is calleddimensional regularization and is extensively used in quantum field the-

ory. In the theory of critical phenomena, it has also been used to calculate

universal quantities like critical exponents, as = (4 d)-expansions.


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Perturbative renormalization group

The renormalization theorem

The perturbative renormalization group, as it has been developed in theframework of theperturbative expansion of quantum field theory, relies on

the so-called renormalization theory. For the 4 field theory it has been

first formulated in space dimension d= 4. For critical phenomena, a small

extension is required that involves an additional expansion in powers of= 4 d, after dimensional continuation.To formulate the renormalization theorem, one introduces a momentum

, called the renormalization scale, and a parameter gr characterizing the

effective4 coefficient at scale, called therenormalized coupling constant.


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One can then find two dimensionless functions Z(/,g) and Zg(/,g)that satisfy (g and/ are the only two dimensionless combinations)

4dg=4dZg(/,g)gr=4dgr + O(g

2), Z(/,g) = 1 + O(g), (10)

calculable order by order in a double series expansion in powers ofg and

, such that all connected correlations functions

W(n)r (pi; gr, , ) =Zn/2(g, /)W(n)(pi; g, ), (11)

calledrenormalized, have, order by order ingr, finite limits W(n)r (pi; gr, )

when

at pi, , gr fixed.

The renormalization constantZ1/2(/,g)is the ratio between the renor-

malization in presence of the 4 interaction and the Gaussian field renor-

malization(d2)/2.


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RemarksThere is some arbitrariness in the choice of the renormalization constants

Z and Zg since they can be multiplied by arbitrary functions of gr. The

constants can be completely determined by imposing threerenormalization

conditionsto therenormalized correlation functions, which are theninde-

pendent of the specific choice of the regularization. This a first important

result: since initial and renormalized correlation functions have the same

large distance behaviour, this behaviour is to a large extentuniversal since

it can, therefore, only depend at most on one parameter, the 4 coefficient

g.


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Critical RG equations

From the relation between initial and renormalized functions (equation (11))

and the existence of a limit , a new equation follows, obtained by

differentiation of the equation with respect toat, gr fixed:

gr, fixed

Zn/2(g, /)W(n)(pi; g, ) 0 . (12)

In agreement with the perturbative philosophy, one then neglects all contri-butions that, order by order, decay as powers of. One defines asymptotic

functions W(n)as. (pi; g, )and Zas.(g, /) as sums of the perturbative con-

tributions to the functions W(n)(pi; g, )and Z(g, /), respectively, that

do not go to zero when . Using the chain rule, one derives fromequation (12)

+(g, /)

g

+n

2

(g, /) W(n)as. (pi; g, ) = 0 .


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The functions and are defined by

(g, /) =

gr,

g , (g, /) =

gr,

ln Zas.(g, /).

Since the functions W(n)as. do not depend on, the functionsandcannot

depend on /, and one finally obtains the RG equations (Zinn-Justin

1973):

+(g)

g+ n

2(g)

W(n)as. (pi; g, ) = 0 . (13)

From equation (10), one immediately infers that (g) = g+O(g2).


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RG equations in the critical domain above Tc

Correlation functions may also exhibit universal properties near Tc when

the correlation length is large in the microscopic scale, here, 1. To

describe universal properties in the critical domain above Tc, one adds the2 relevant term to the Hamiltonian:

Ht() = H() + t2

ddx 2(x),

where t, the coefficient of 2

, characterizes the deviation from the criti-cal temperature: t T Tc. The renormalization theorem leads to theappearance of a new renormalization factor Z2(/,g)associated with the

parametert. By arguments of the same nature as in the critical theory, one

derives a more general RGE of the form (Zinn-Justin 1973)

+(g)

g+

n

2(g) 2(g)t

t

W(n)as. (pi; t,g, ) = 0 , (14)

where a new RG function 2(g), related to Z2(/,g), appears.


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These equations can be further generalized to deal with an external field(a magnetic field for magnetic systems) and the corresponding induced field

expectation value (magnetization for magnetic systems).

Renormalized RG equationsFord= 4 , if one is only interested in the leading scaling behaviour (andthe first correction), it is technically simpler to use dimensional regular-

izationand therenormalized theoryin the so-calledminimal (or modified

minimal) subtraction scheme. The relation (11) between initial and renor-

malized correlation functions is asymptotically symmetric. One thus derives

also (for the critical theory)

+(gr)

gr +n

2(gr)

W(n)r (pi, gr, ) = 0

with the definitions

(gr) =

g

gr, (gr) =

g

ln Z(gr, ) .


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In this scheme, the renormalization constants (10) are obtained by goingto low dimensions where the infinitelimit, at gr fixed, can be taken. For

example,

lim

Z(/,g)

|gr fixed

=Z(gr, ).

Then, order by order in powers of gr, they have a Laurent expansion in

powers of. In theminimal subtraction scheme, the freedom in the choice

of the renormalization constants is used to reduce the Laurent expansion to

the singular terms. For example, Z(gr, )takes the form

Z(gr, ) = 1 +

n=1

n(gr)

n with n(gr) =O(g

n+1r ).

A remarkable consequence is that the RG functions (gr), and 2(gr)

when a2 term is added,become independent ofand (gr)has the simple

dependence(gr) = gr+2(gr),where2(gr) =O(g2r )is also independentof.


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Solution of the RG equations: The epsilon-expansion

RG equations can be solved by the method of characteristics. In the simplest

example of the critical theory, one introduces a scale parameter and two

functions ofg()and ()defined by

d

dg() = g(), g(1) =g , d

dln () = g(), (1) = 1 .

(15)

The function g()is the effective coefficient of the 4 term at the scale .One verifies that equation (13) is then equivalent to

d

dn/2()W(n)

as.pi; g(), /= 0 ,

which implies ( )

W(n)as.

pi; g, =n/2()W(n)

as.p

i; g(), .


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From its definition, one infers that W(n)as. has dimension(d (d+ 2)n/2).Therefore,

W(n)as. pi/; g, =(d+2)n/2d W(n)as. pi; g, =(d+2)n/2dn/2()W(n)as.

pi; g(),

.

These equations show that here the general Hamiltonian flow reduces here

to the flow ofg() and, thus, the large distance behaviour is governed by

the zeros of the function (g). When , since (g) =g+O(g2),ifg > 0 is initially very small, it moves away from the unstable Gaussian

fixed point, in agreement with the general RG analysis.


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If one assumes the existence of another zero g with then (g)>0, theng()will converge toward this fixed point. Sinceg()tends toward the fixed

point valueg,and if(g) is finite, one finds the universal behaviour

W(n)as.pi/; g,

(d+2)n/2d W(n)as.pi; g,

.

For the connected correlation functions in space, this result translates into

W(n)as.

xi; g,

n(d2+)/2W(n)as.

xi; g,

,

for all xi distinct.

The exponentd = (d

2 + )/2is thedimension of the field, from the

point of view of large distance properties.


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Explicit calculationsFrom the perturbative calculation of the two- and four-point functions at

one-loop order, one derives

(g) = g+ 3162

g2 +O(g3, g2).

In the sense of an -expansion, (g) has a zero g with a positive slope

(WilsonFisher 1972)

g =162

3 +O(2), =(g) =+O(2),

which governs the large momentum behaviour of correlation functions.

In addition, the exponent governs the leading correction to the critical

behaviour.


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GeneralizationThe results obtained for models with a Z2 reflection symmetry can easily

be generalized to N-vector models with O(N)orthogonal symmetry, which

belong todifferent universality classes. Their universal properties can then

be derived from anO(N)symmetric field theory with anN-component field(x) and a g(2)2 quartic term. Further generalizations involve theories

with N-component fields but smaller symmetry groups, such that several

independent quartic 4 terms are allowed. The structure of fixed points

may then be more complicate.

Finally, correlation functions of the O(N)model can be evaluated in the

largeN limitexplicitly and the predictions of the -expansion can then be

verified.


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Epsilon-expansion: A few results

From the simple existence of the fixed point and of the corresponding -

expansion,universal properties of a large class of critical phenomena can be

proved to all orders in : this includesrelations between critical exponents,scaling behaviour of correlation functions or the equation of state.

Moreover,universal quantitiescan then be calculated as -expansions.

The scaling equation of state

An example of the general results that can be obtained is provided by the

equation of stateof magnetic systems, that is, the relation between applied

magnetic field H, magnetization M and temperature T. In the relevant

limit|H| 1,|T Tc| 1, RG has proved Widoms conjectured scalingform

H=Mf

(T Tc)/M1/

,

wheref(z)is a universal function (up to normalizations).


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Moreover, the exponents satisfy the relations

=d+ 2 d 2 + , =

12(d 2 +),

where , the correlation length exponent, given by = 1/

2(g

) + 2

,characterizes the divergence of the correlation length at Tc:

|T Tc|.

Other relations can be derived, involving the magnetic susceptibility expo-nent characterizing the divergence of the two-point correlation function

at zero momentum at Tc, or the exponent characterizing the behaviour

of the specific heat:

=(2 ), = 2 d .Note the relations involving the dimension dexplicitly are not valid for the

Gaussian fixed point.


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Critical exponents as -expansions

As an illustration, we give here two successive terms of the -expansion of

the exponents , and for theO(N)models, although the RG functions

of the(2

)2

field theory are known to five-loop order and, thus,the criticalexponents are known up to 5. In terms of the variable v=Ndg where Ndis the loop factor

Nd= 2/(4)d/2(d/2),

the RG functions(v)and2(v)at two-loop order,(v)at three-loop order

are

(v) = v+(N+ 8)6

v2 (3N+ 14)12

v3 +O(v4),

(v) =(N+ 2)

72 v2

1 (N+ 8)

24 v

+O(v4),

2(v) =

(N+ 2)

6 v 1

5

12v +O(v3).


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The fixed point value solution of(v) = 0is then

v() = 6

(N+ 8)

1 +

3(3N+ 14)

(N+ 8)2

+O(3).

The values of the critical exponents

=(v), = 2

2 +2(v), =(v),

follow

= 2(N+ 2)

2(N+ 8)2 1 +(N2 + 56N+ 272)

4(N+ 8)2 +O(

4),

= 1 + (N+ 2)

2(N+ 8)+

(N+ 2)

4(N+ 8)3

N2 + 22N+ 52

2 +O(3),

=

3(3N+ 14)

(N+ 8)2 2 +O(3).


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Though this may not be obvious on these few terms, the -expansion isdivergent for any > 0, as large order estimates based on instanton cal-

culus have shown. Extracting precise numbers from the known terms of

the series requires a summation method. For example, adding simply the

known successive terms for = 1andN= 1yields

= 1.000 . . . , 1.1666 . . . , 1.2438 . . . , 1.1948 . . . , 1.3384 . . . , 0.8918 . . . ,

while the best field theory estimate is = 1.2396 0.0013.


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Summation of the -expansion and numerical values of exponents

We display below (Table 1) the results for some critical exponents of the

O(N) model obtained from Borel summation of the -expansion (Guida

and Zinn-Justin 1998). Due to scaling relations like = (2 ), +2 = d, only two among the first four are independent, but the seriesare summed independently to check consistency. N = 0 corresponds to

statistical properties of polymers (mathematically theself-avoiding random

walk),N= 1, theIsing universality class, to liquid-vapour, binary mixturesor anisotropic magnet phase transitions. N = 2 describes the superfluid

Helium transition, while N= 3correspond toisotropic ferromagnets.

As a comparison, we also display (Table 2) the best available field theory

results obtained from Borel summation ofd= 3perturbative series(Guidaand Zinn-Justin 1998).


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Table 1

Critical exponents of theO(N) model, d= 3, obtained from the-expansion.

N 0 1 2 3

1.1571 0.0030 1.2355 0.0050 1.3110 0.0070 1.3820 0.0090

0.5878 0.0011 0.6290 0.0025 0.6680 0.0035 0.7045 0.0055

0.0315 0.0035 0.0360 0.0050 0.0380 0.0050 0.0375 0.0045

0.3032 0.0014 0.3265 0.0015 0.3465 0.0035 0.3655 0.0035

0.828 0.023 0.814 0.018 0.802 0.018 0.794 0.018


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Table 2

Critical exponents of theO(N) model, d= 3, obtained from the(2)23 field theory.

N 0 1 2 3

gNi 1.413 0.006 1.411 0.004 1.403 0.003 1.390 0.004

1.1596 0.0020 1.2396 0.0013 1.3169 0.0020 1.3895 0.0050

0.5882 0.0011 0.6304 0.0013 0.6703 0.0015 0.7073 0.0035

0.0284 0.0025 0.0335 0.0025 0.0354 0.0025 0.0355 0.0025

0.3024 0.0008 0.3258 0.0014 0.3470 0.0016 0.3662 0.0025

0.812 0.016 0.799 0.011 0.789 0.011 0.782 0.0013


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Functional (or exact) renormalization group

We now briefly describe a general approach to the RG close to ideas ini-

tially developed by Wegner and Wilson, and based on a partial integration

over the large-momentum modes of fields. This RG takes the form offunc-tional renormalization group(FRG) equations that express the equivalence

between a change of a scale parameter related to microscopic physics and a

change of the parameters of the Hamiltonian. Some forms of these equations

are exact and one then also speaks of theexact renormalization group.These FRG equations have been used to recover the first terms of the

-expansion and later by Polchinski to give a new proof of the renormaliz-

ability of field theories, avoiding any argument based on Feynman diagrams

and combinatorics.From the practical viewpoint, several variants of these FRG equations

have led to new approximation schemes no longer based on the standard

perturbative expansion.


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Technically, these FRG equations follow from identities that express theinvariance of the partition function under a correlated change of the propa-

gator and the other parameters of the Hamiltonian. We discuss these equa-

tions, in continuum space, in the framework of local statistical field theory.

It is easy to verify that, except in the Gaussian case, these equations areclosed only if an infinite number of local interactions are included.

It is then possible to infer various RGE satisfied by correlation functions.

Depending on the chosen form, these RGE are either exact or only exact

at large distance or small momenta, up to corrections decaying faster than

any power of the dilatation parameter.

Here, we discuss only the Hamiltonian flow, the RGE for correlation func-

tions requiring some additional considerations.


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Partial field integration and effective Hamiltonian

Using identities that involve only Gaussian integrations, one first proves

equality between two partition functions corresponding to two different

Hamiltonians. This relation can then be interpreted as resulting from a par-

tial integration over some components of the fields. One infers asufficient

condition for correlated modifications of the propagator and interactions, in

a statistical field theory, to leave the partition function invariant.

In what follows, we assume that the field theory is translation invariant.Moreover, all Gaussian two-point functions, or propagators, (x y) aresuch that in the Fourier representation

(x) = 1

(2)d

dd

p eipx

(p),

(p) decreases faster than any power for|p| so that the Gaussianexpectation value of any local polynomials in the field exists.


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Partial integrationOne first establishes a relation between partition functions corresponding

to two different local Hamiltonians in d space dimensions.

The first Hamiltonian depends on a field and we write it in the form

H1() =12

dx dy (x)K1(x y)(y) + V1(), (16)

where K1 is a positive operator and the functionalV1() is expandablein powers of the field , local and translation invariant. To the explicit

quadratic part is associated the propagator1, dz K1(x z)1(z y) =(x y).

The second Hamiltonian depends on two fields 1, 2 in the form

H(1, 2) =12

dx dy [1(x)K2(x y)1(y) +2(x)K(x y)2(y)]

+V1(1+2).


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Again, we define dz K2(x z)2(z y) =(x y),

dz K(x z)D(z y) =(x y).

The kernels K1, K2 and

Kare positive, which is a necessary condition for

the field integrals to exist, at least in a perturbative sense. Moreover, the

properties of the propagators 1, 2 andD (thus also positive) ensuresthe existence of a formal expansion of the field integrals in powers of the

interactionV

1.

Then, if 1 = 2 + D K1 = K2(K2 + K)1K, the ratio of thepartition functions

Z1 = [d] eH1() andZ2 = [d1d2] e

H(1,2) (17)

does not depend onV1:

Z2 = det(D2)

det1 1/2

Z1. (18)


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Another form of the identity. In what follows, we use the compact nota-

tion

dx dy (x)K(x y)(y) (K).We define

eV2() = (det D)1/2

[d]exp

12(K) V1(+)

, (19)

as well as H2() = 12(K2) +V2(). Then, the equivalence takes the moreinteresting form

[d] eH2() = det2det1

1/2

[d] eH1() . (20)The left hand side can be interpreted as resulting from a partial integration

over the fieldsince the propagatorDis positive and, thus, in the sense ofoperators,2


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Differential form

We now assume that the propagator is a function of a real parameter s:

(s). Moreover,(s)is a smooth function with a negative derivative.We define

D(s) = d(s)ds

0 . (21)

Similarly,

K1 =K(s) = 1(s), K2=K(s

), K(s, s) = [D(s, s)]1 >0 . (22)

Since the kernels K(s),K(s, s) are positive, all Gaussian integrals aredefined.


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Finally, we set

V1() = V(, s), V2() = V(, s), H1() = H(, s), H2() = H(, s).

The equivalence then takes the form [d] eH(,s

) =

det(s)

det(s)

1/2 [d] eH(,s),

whereV(, s)is given by

eV(,s) = det D(s, s

)1/2

[d]exp 12K(s, s

) V(+, s) .(23)


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Differential formSetting s = s+, > 0, one expands in powers of 0. Identifyingthe terms of order , after some algebraic manipulations one obtains the

functional equation

d

dsV(, s) = 1

2

dx dy D(s; x y)

2V

(x)(y) V

(x)

V(y)

. (24)

The equation expresses asufficient conditionfor the partition function

Z(s) = (det (s))1/2

[d] eH(,s) with

H(, s) = 12 (K(s)) +

V(, s), (25)

to be independent of the parameter s.

This property relates a modification of the propagator to a modification

of the interaction, quite in the spirit of the RG.


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Remark.(i) A sufficient condition forZ(s) to be independent of s, is that the

equation is satisfied as an expectation value with the measure eH(,s).

One can thus add to the equation contributions with vanishing expectation

value to derive other sufficient conditions (useful for correlation functions).

(ii) Let us set

(, s) =eV(,s) .

Then, the functional equation reduces to

d

ds(, s) = 1

2

dx dy D(s; x y)

2(, s)

(x)(y).

This afunctional heat equationsince the kernelD is positive. One maywonder why we do not consider this equation, which is linear and thus

much simpler. The reason is that,in contrast to(, s),V(, s)is a localfunctional, a property that plays an essential role.


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Hamiltonian evolution

From the evolution equation, for V(, s)one then infers the evolution of theHamiltonian

H(, s) = 1

2

(K(s)) +V

(, s).

Introducing the operator L(s), with kernel L(s; x y), defined by

L(s)

D(s)1(s) =

d ln (s)

ds

, (26)

one finds

d

dsH(, s) =

1

2 dx dy D(s; x y)

2H(x)(y)

H(x)

H(y)

dx dy (x)L(s; x y) H(y)

+1

2tr L(s). (27)


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Formal solution

Since the evolution equation is a first-order differential equation in s, the

form of the functionalV(, s) for an initial value s0 of the parameter sdetermines the solution for all s

s0. From the very proof of equation

(24), its solution can be easily inferred:

eV(,s) = det D(s0, s)1/2

[d]exp 12K(s0, s) V(+, s0) .

(28)The equation implies thatV(, s) is the sum of the connected con-

tributions of the diagrams constructed with the propagatorK1(s0, s) =(s)

(s0)and interactions

V(+, s0).

Only if initially V(, s0)is a quadratic form (a Gaussian model) it remainsso. However, if one adds, for example, a quartic term, then an infinite

number of terms of higher degrees will automatically be generated.


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Field renormalization

In order to be able to find RG fixed-point solutions, it is necessary to intro-

duce a field renormalization. To prove the corresponding identities, we set

(x) =

Z(s)(x), H(, s) = H(, s),whereZ(s)is an arbitrary differentiable function. We then define

(s) = d ln Z(s)

ds . (29)

One then infers from the Hamiltonian flow equation the modified equation

(omitting some constant term)

d

dsH(, s) = 1

2

dx dy D(x y) 2

H(x)(y)

H(x)

H(y)

dx dy (x)L(s; x y) H(y)

+1

2(s)

dx (x)

H(x)

. (30)


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High-momentum mode integration and RGEThese equations can be applied to a situation where the partial integra-

tion over the field corresponds, in the Fourier representation, to a partial

integration over its high-momentum modes, which in position space also

corresponds to an integration overshort-distance degrees of freedom.

In what follows, we specialize to a critical (massless) propagator. A

possible deviation from the critical theory is included inV().

Cut-off parameter and propagator. In the preceding formalism, we nowidentify s ln , whereis a large-momentum cut-off, which also repre-sents the inverse of the microscopic scale. A variation ofsthen corresponds

to a dilatation of the parameter.


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We then choose a regularized propagator of the form

(x) =

ddk

(2)deikx (k) with (k) =

C(k2/2)

k2 .

The function C(t) is regular for t 0, positive, decreasing, goes to 1 fort 0and goes to zero faster than any power for t . It suppresses thefield Fourier components corresponding to momenta much higher thanin

the field integral.

The Fourier transform of the derivative D(x),

D(k) = (k)

=

2

2C(k2/2), (31)

has anessential property: it has no pole atk= 0and thusit is not critical.


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The function

D(x) = (x)

=

ddk

(2)d eikx D(k) =

d2D=1(x), (32)

thus decays for |x| faster than any power ifC(t)is smooth, exponen-tially ifC(t) is analytical. The propagatorD(0, ) =D0 D,0 >,whose inverse now appears in the field integral solving the flow equation,

shares this property.


65/83

RGEWith these assumptions and definitions, the flow equation forV(, )be-comes

d

dV(, ) =1

2

dd

x dd

y D(x y) 2

V(x)(y)

V(x)

V(y) . (33)This equation being exact, one uses also the terminologyexact renormal-

ization group.

Remarks.

Since D(x)decreases faster than any power, ifV()is initially local, itremains local, a property that becomes more apparent when one expands

the equation in powers of.The equation differs from the abstract RGE by the property that the

scale parameter appears explicitly through the function D. We shall

later eliminate this explicit dependence.


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Field Fourier componentsIn terms of the Fourier components (k)of the field,

(x) = ddk eikx (k) (k) =

ddx

(2)

deikx (x),

the equation becomes

d

dV(, ) = 1

2 ddk

(2)d

D(k) 2V(k)(k)

V(k)

V(k) . (34)In the equation, locality translates into regularity of the Fourier compo-

nents. If the coefficients of the expansion ofV(, )in powers of, afterfactorization of the -function, are regular functions for an initial value of = 0, they remain for


67/83

Finally, the flow equation for the complete Hamiltonian expressed interms of Fourier components,

H(, ) =12

(2)d

ddk (k)1 (k)(k) + V(, ),

takes the form (omitting the term independent of)

d

dH(, ) =1

2

ddk

(2)dD(k)

2H

(k)(

k)

H(k)

H(

k)

+

ddk(2)d

L(k) H(k)

(k) (35)

with (equation (26))

L(k) = D(k)/(k). (36)

In equation (35), we have implicitly subtracted both from H(, )and fromthe equation their values at = 0.


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RGE: Standard formAfter a field renormalization, required to be able to reach non-Gaussian

fixed points, the RGE take the form (30) with s= ln :

H(, )

= 1

2

ddx ddy D(x y)

2H

(x)(y) H

(x)

H(y)

+

ddx ddy (x)L(x y) H(y)

+ 2

ddx (x) H

(x). (37)

The function isa prioriarbitrary but with one restriction, it must depend

ononly throughH(, ). It must be adjusted to ensure the existence offixed points.


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The latter equation does not have a stationary Markovian form since Dand L depend explicitly on:

L(x) = 1

(2)d ddk eikx L(k) =

dL1(x), D(x) = d2D1(x).

To eliminate this dependence, we perform a Gaussian renormalization of

the form with

(x) = (2d)/2(x/).

In what follows, we omit the primes. Moreover, we introduce the dilatation

parameter= 0/that relates the initial scale0 to the running scale

and thus

d

d= d

d.


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Then, the RGE take a form consistent with the general RG flow equation:

d

dH(, ) = T [H(, )] ,

with

T [H] = 12

ddx ddy D(x y)

2H

(x)(y) H

(x)

H(y)

ddx

H(x)1

2 (d 2 +) + x

x

(x)

ddx ddy L(x y) H(x)

(y). (38)

This is the form more suitable for looking for fixed points. At a fixed

point, the right hand side vanishesfor a suitably chosen renormalization of

the field , which determines the value of the exponent .


71/83

Expansion in powers of the field: RGE in component formIf one expandsH(, )(and then equation (38)) in powers of(x),

H(, ) =

n=0

1

n!

i

ddxi (xi)

H(n)(x1, . . . , xn; ),

one derives equations for the components. For n = 2(D1 D),

d

dH(n)(xi; ) =1

2 n(d+ 2 ) +j,

xj

xj

H(n)(xi; ) 1

2 ddx ddy D(x y)

H(n+2)(x1, x2, . . . , xn, x , y; )

I

H(l+1)(xi1 , . . . , xil , x; )H(nl+1)(xil+1 , . . . , xin , y; )

,

where the setI

{i1, i2, . . . , il

}describes all distinct subsets of

{1, 2, . . . , n

}.


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In the Fourier representation, the equations take the form

d

dH(n)(pi; ) =

d 1

2n(d 2 +)

j,

pj

pj

H(n)(pi; )

12

ddk(2)d

D(k)H(n+2)(p1, p2, . . . , pn, k, k; )

+1

2 ID(p0)H(l+1)(pi1 , . . . , pil , p0; )H(nl+1)(pil+1 , . . . , pin , p0),(39)

where the momentum p0 is determined by total momentum conservation.


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For n= 2, one finds an additional term in the equation:

d

dH(2)(x1; ) =

d+ 2 +

x1

x1

H(2)(x1; )

1

2

ddx ddy D(x y) H(4)(x1, 0, , x , y; ) 2H2)(x x1; )H(2)(y; ) 2

ddy L(x1 y)H(2)(y; ).

One verifies explicitly that, except in the Gaussian example, all functionsH(n) are coupled. In the Fourier representation,

d

dH(2)(p; ) = 2

p

pH(2)(p; ) 2L(p)H(2)(p; )

12

ddk

(2)dD(k)H(4)(p, p, k, k; ) + D(p)

H(2)(p; )

2. (40)


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Perturbative solutionThe flow equations is any of the different forms, can be solved perturbatively.

One first specifies the form of the Hamiltonian at the initial scale = 1, for

instance,

H() = HG() + g4!

ddx 4(x), u 0 .

In the finite form (28) of the RGE, one can then expand the field integral

in powers of the constant g. Alternatively, in the differential form, one first

expands in powers of the field . This leads to the infinite set of coupled

integro-differential equations (39,40) that one can integrate perturbatively

with the Ansatz that the terms in H(; )quadratic and quartic inare of

orderg and the general term of degree2n of order gn1

.It is also possible to further expand the equations in powers of= 4 d

and to look for fixed points. Theresults of the perturbative RG are then

recovered.


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Fixed points and local flowOnce a fixed pointH has been determined, one can expand equation

(38) in the vicinity of the fixed point:

H() = H+E().One then obtains the linearized RGE

d

dE() = LE(),

where the linear operator L, after an integration by parts, takes the form

L=

ddx (x)

1

2(d+ 2 ) +

x

x

(x)

+

ddx ddy D(x y)1

22

(x)(y)+ H

(x)

(y)

ddx ddy L(x y)(x)

(y). (41)


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We denote bythe eigenvalues andE E(= 1)the eigenvectors ofL:LE =E, (42)

and thus

E() =E(1).

Equation (42) can be written more explicitly in terms of the components

E(n) (pi)ofE as

E(n)

(pi)

=

d 1

2n(d 2 +)

j

D(pj )1(pj )

j,

pj

pj

E(n) (pi)

1

2 ddk

(2)d D(k)E(n+2) (p1, p2, . . . , pn, k, k)

+

I

D(p0)E(l+1) (pi1 , . . . , pil , p0)

H(nl+1) (pil+1 , . . . , pin , p0). (43)


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Gaussian fixed pointAt the Gaussian fixed point, the Hamiltonian is quadratic and = 0. The

local flow at the fixed point is then governed by the operator

L=

ddx (x)

12

(d+ 2) +

x x

(x)

1

2 ddx ddy D(x y)

2

(x)(y)

.

The eigenvectors are obtained by choosingH(n) vanishing for all n largerthan some value N. The coefficient of the term of degree N in then

satisfies the homogeneous equation

E(N) (pi) =

d 1

2N(d 2)

j,pj

pj

E(N) (pi).


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This is an eigenvalue equation identical to the one obtained in the pertur-bative RG. The solutions are homogeneous polynomials in the momenta.

Ifr is the degree in the variables pi, the eigenvalue is given by

=d 1

2N(d 2) r .The other coefficientsH(n), n < N, are then entirely determined by the

equations

1

2(N n)(d 2) +r

j,

pj

pj

E(n) (pi)

=12

d

d

k(2)d D(k)E(n+2) (p1, p2, . . . , pn, k, k). (44)


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One might be surprised by the occurrence of these additional terms. Infact, one can verify that if one sets

E() = exp

1

2

ddx ddy (x y)

2

(x)(y)

(),

the functional()satisfies the simpler eigenvalue equation

() =

ddx (x)

1

2(d+ 2) +

x

x

(x)(),

whose solutions are the simple monomials On,k(, x)found in the pertur-bative analysis of the stability of the Gaussian fixed point. For example, for

() = ddx m(x),

after some algebra, one verifies

=d m(d 2)/2 .


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The linear operator that transforms()into E(),

exp

1

2

ddx ddy (x y)

2

(x)(y)

,

replaces all monomials inthat contribute toby theirnormal products.

We recall that thenormal productof a E(N)()of a monomial of degree

N inis a polynomial with the same term of orderNand is such that, for

all n < N, the Gaussian correlation functions n

i=1(xi)E

(N)()

with the measure eH vanish. Let us point out that the definition of

normal productsdepends explicitly on the choice of the Gaussian measure.

Beyond the Gaussian model: perturbative solution


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Beyond the Gaussian model: perturbative solution

In dimension4(and then ind= 4 ), a non-trivial theory can be definedand parametrized, for example, in terms ofg(), the value ofH(4)(pi, )atp1= =p4 = 0:

g() H(4)

(pi = 0, ). (45)

One then introduces the function(g)defined by

dg

d = g(). (46)This allows substituting in the left hand side of the flow equation

d

d =(g)

d

dg .

One then solves perturbatively the flow equations, with appropriate bound-

ary conditions.

All other interactions are then determined perturbatively as functions of


82/83

ot e te act o s a e t e dete ed pe tu bat ve y as u ct o s o

g, under the assumption that they are at least of order g2. They become

implicit functions ofthrough g(). One suppresses in this way all correc-

tions due to irrelevant operators, keeping only the contributions due to the

marginal operator. The Hamiltonian flow, like in the perturbative renormal-ization group is reduced the flow ofg(), but the fixed point Hamiltonian

is much more complicate, since all subleading corrections to the leading

behaviour are suppressed.

The function (g)is determined by the condition

p2H(2)(p; g)

p=0

= 1 p2

V(2)(p; g)p=0

= 0 , (47)

whichsuppresses the redundant operatorthat corresponds to a change ofnormalization of the field.

The two conditions (45), (47) replace the renormalization conditions of

the usual renormalization theory.

Final remarks


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In practice, the FRG has been used as a starting point for various non-

perturbative approximations, reducing the functional equations to partial

differential equations. Quite interesting results have been obtained. The

main problem is that none of these approximation scheme has a systematiccharacter, or when a systematic method is claimed, the next approximation

is out of reach.

A further practical remark is that since the effective interactionV()appears as the generating functional of some kind of connected Feynmandiagrams, an additional simplification has been obtained by introducing its

Legendre transform, which contains only one-line irreducible diagrams.

acqft09_jzinnjustin

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