0 Some aspects of the nonperturbative
renormalization of the ϕ4 model
J. Kaupužs ∗
Institute of Mathematics and Computer Science, University of Latvia
Raiņa bulvāris 29, LV–1459 Riga, Latvia
November 4, 2018
Abstract
A nonperturbative renormalization of the ϕ4 model is considered. First we inte-
grate out only a single pair of conjugated modes with wave vectors ±q. Then we are
looking for the RG equation which would describe the transformation of the Hamil-
tonian under the integration over a shell Λ − dΛ < k < Λ, where dΛ → 0. We show
that the known Wegner–Houghton equation is consistent with the assumption of a
simple superposition of the integration results for ±q. The renormalized action can
be expanded in powers of the ϕ4 coupling constant u in the high temperature phase
at u → 0. We compare the expansion coefficients with those exactly calculated by the
diagrammatic perturbative method, and find some inconsistency. It causes a question
in which sense the Wegner–Houghton equation is really exact.
1 Introduction
The renormalization group (RG) approach, perhaps, is the most extensively used one in
numerous studies of critical phenomena [1, 2]. Particularly, the perturbative RG approach
to the ϕ4 or Ginzburg–Landau model is widely known [3, 4, 5, 6]. However, the pertur-
bative approach suffers from some problems [7]. Therefore it is interesting to look for a
nonperturbative approach. Historically, nonperturbative RG equations have been devel-
oped in parallel to the perturbative ones. These are so called exact RG equations (ERGE).
The method of deriving such RG equations is close in spirit to the famous Wilson’s ap-
proach, where the basic idea is to integrate out the short–wave fluctuations corresponding
to the wave vectors within Λ/s < q < Λ with the upper (or ultraviolet) cutoff parameter
Λ and the renormalization scale s > 1. The oldest nonperturbative equation of this kind,
originally presented by Wegner and Houghton [8], uses the sharp momentum cutoff. Later,
a similar equation with smooth momentum cutoff has been proposed by Polchinski [9]. The
RG equations of this class are reviewed in [10].
According to the known classification [10, 11], there is another class of nonperturbative
RG equations proposed by [12] and reviewed in [11]. Some relevant discussion can be found
in [10], as well. Such equations describe the variation of an average effective action Γk[φ]
depending on the running cutoff scale k. Here φ(x) = 〈ϕ(x)〉 is the averaged order–
parameter field (for simplicity, we refer to the case of scalar field). According to [12], the
averaging is performed over volume ∼ k−d such that the fluctuation degrees of freedom
E–mail: kaupuzs@latnet.lv
http://arxiv.org/abs/0704.0142v2
with momenta q > k are effectively integrated out. In fact, the averaging over volume
∼ k−d is the usual block–spin–averaging procedure of the real–space renormalization.
At the same time, the fluctuations with q . k are suppressed by a smooth infrared
cutoff. As one can judge from [11], the existence of a deterministic relation between the
configuration of external source {J(x)} and that of the averaged order parameter {φ(x)} is
(implicitly) assumed in the nonperturbative derivation of the RG flow equation. Namely,
it is stated (see the text between (2.28) and (2.29) in [11]) that δJ(x)/δφ(y) is the inverse
of δφ(x)/δJ(y), which has certain meaning as a matrix identity. To make this point
clearer, let us consider a toy example ~J = A~φ, where ~J = (J(x1), J(x2), . . . , J(xN)) and
~φ = (φ(x1), φ(x2), . . . , φ(xN)) are N–component vectors and A is a matrix of size N ×N .
In this case ∂J(xi)/∂φ(xj) is the element Aij of matrix A, whereas ∂φ(xi)/∂J(xj) is the
element
of the inverse matrix A−1. In the continuum limit N → ∞, this toy
example corresponds to a linear dependence between {J(x)} and {φ(x)}. The calculation
of derivative always implies the linearisation around some point, so that the matrix identity
used in [11] (as a continuum limit in the above example) has a general meaning. However,
it makes sense only if there exists a deterministic relation between the configurations of
φ(x) and J(x) or, in a mathematical notation, if there exist mappings f : {J(x)} → {φ(x)}
and f−1 : {φ(x)} → {J(x)}. On the other hand, according to the block–averaging, the
values of φ(x) should be understood as the block–averages. These, of course, are not
uniquely determined by the external sources, but are fluctuating quantities. So, we are
quite sceptical about the exactness of such an approach of averaged effective action.
The integration over fluctuation degrees with momenta q > k does not alter the be-
havior of the infrared modes, directly related to the critical exponents. From this point of
view, the approach based on the equations of Wegner–Houghton and Polchinski type seems
to be more natural. These are widely believed to be the exact RG equations, although, in
view of our currently presented results, it turns out to be questionable in which sense they
are really exact. In any case, the nonperturbative RG equations cannot be solved exactly,
therefore a suitable truncation is used. The convergence of several truncation schemes and
of the derivative expansion has been widely studied in [13, 14, 15, 16, 17]. Here [17] refers
to the specific approach of [12]. A review about all the methods of approximate solution
can be found in [18].
Another problem is to test and verify the nonperturbative RG equations, comparing
the results with the known exact and rigorous solutions, as well as with the results of
the perturbation theory. In [15], the derivative expansion of the RG β–function has been
considered, showing the agreement up to the second order between the perturbative results
and those obtained from the Legendre flow equation, which also belongs to the same class
of RG equations as the Wegner–Houghton and Polchinski equations. It has been stated
in [13] that the critical exponent ν, extracted from the Wegner–Houghton equation in the
local potential approximation, agrees with the ε–expansion up to the O(ε) order, as well
as with the 1/n (1/N in the notations of [13]) expansion in the leading order. However,
looking carefully on the results of [13], one should make clear that “the leading order of
the 1/n expansion” in this case is no more than the zeroth order, whereas the expansion
coefficient at 1/n is inconsistent with that proposed by the perturbative RG calculation
at any fixed dimension d except only d = 4. The inconsistency could be understood from
the point of view that the Wegner–Houghton equation has been solved approximately.
Therefore it would be interesting to verify whether the problem is eliminated beyond the
local potential approximation. One should also take into account that the perturbative
RG theory is not rigorous and, therefore, we think that a possible inconsistency still would
not prove that something is really wrong with the nonperturbative RG equation. In any
case, it is a remarkable fact that correct RG eigenvalue spectrum and critical exponents
are obtained in the local potential approximation at n → ∞ from the Wegner–Houghton
equation [13], as well as from similar RG equations [19], in agreement with the known
exact and rigorous results for the spherical model. It shows that some solutions, being
not exact, nevertheless can lead to exact critical exponents. From this point of view, it
seems also possible that some kind of approximations, made in the derivation of an RG
equation, are not harmful for the critical exponents.
We propose a simple test of the Wegner–Houghton equation: to verify the expansion of
the renormalized action S of the ϕ4 model in powers of the coupling constant u at u → 0
in the high–temperature phase. Such a test is rigorous, in the sense that the natural
domain of validity of the perturbation theory is considered. We think that it would be
quite natural to start with such a relatively simple and straightforward test before passing
to more complicated ones, considered in [13, 15, 19]. We have made this simplest test
in our paper and have found that the Wegner–Houghton equation fails to give all correct
expansion coefficients. We have also proposed another derivation of the Wegner–Houghton
equation (Secs. 2, 3). It is helpful to clarify the origin of the mentioned inconsistency. It
is also less obscure from the point of view that the used assumptions and approximations
are clearly stated. As regards the derivation in [8], at least one essential step is obscure
and apparently contains an implicit approximation which, in very essence, is analogous to
that pointed out in our derivation. We will discuss this point in Sec. 3.
2 An elementary step of renormalization
To derive a nonperturbative RG equation for the ϕ4 model, we should start with some
elementary steps, as explained in this section.
Consider the action S[ϕ] which depends on the configuration of the order parameter
field ϕ(x) depending on coordinate x. By definition, it is related to the Hamiltonian H of
the model via S = H/T , where T is the temperature measured in energy units. In general,
ϕ(x) is an n–component vector with components ϕj(x) given in the Fourier representation
as ϕj(x) = V
k<Λ ϕj,ke
ikx, where V = Ld is the volume of the system, d is the spatial
dimensionality, and Λ is the upper cutoff of the wave vectors. We consider the action of
the Ginzburg–Landau form. For simplicity, we include only the ϕ2 and ϕ4 terms. The
action of such ϕ4 model is given by
S[ϕ] =
Θ(k)ϕj,kϕj,−k + uV
j,l,k1,k2,k3
ϕj,k1ϕj,k2ϕl,k3ϕl,−k1−k2−k3 , (1)
where Θ(k) is some function of wave vector k, e. g., Θ(k) = r0 + ck
2 like in theories of
critical phenomena [4, 5, 6, 7]. In the sums we set ϕl,k = 0 for k > Λ.
The renormalization group (RG) transformation implies the integration over ϕj,k for
some set of wave vectors with Λ′ < k < Λ, i. e., the Kadanoff’s transformation, followed
by certain rescaling procedure [4]. The action under the Kadanoff’s transformation is
changed from S[ϕ] to Stra[ϕ] according to the equation
e−Stra[ϕ] =
e−S[ϕ]
j,Λ′<k<Λ
dϕj,k . (2)
Alternatively, one often writes −Stra[ϕ]+AL
d instead of −Stra[ϕ] to separate the constant
part of the action ALd. This, however, is merely a redefinition of Stra, and for our purposes
it is suitable to use (2). Note that ϕj,k = ϕ
j,k+iϕ
j,k is a complex number and ϕj,−k = ϕ
holds (since ϕj(x) is always real), so that the integration over ϕj,k means in fact the
integration over real and imaginary parts of ϕj,k for each pair of conjugated wave vectors
k and −k.
The Kadanoff’s transformation (2) can be split in a sequence of elementary steps
S[ϕ] → Stra[ϕ] of the repeated integration given by
e−Stra[ϕ] =
e−S[ϕ]dϕ′j,qdϕ
j,q (3)
for each j and q ∈ Ω, where Ω is the subset of independent wave vectors (±q represent
one independent mode) within Λ′ < q < Λ. Thus, in the first elementary step of renor-
malization we have to insert the original action (1) into (3) and perform the integration
for one chosen j and q ∈ Ω. In an exact treatment we must take into account that the
action is already changed in the following elementary steps.
For Λ′ > Λ/3, we can use the following exact decomposition of (1)
S[ϕ] = A0 +A1ϕj,q +A
1ϕj,−q +A2ϕj,qϕj,−q +B2ϕ
j,q +B
j,−q +A4ϕ
j,−q , (4)
where
A0 = S|ϕj,±q=0 , (5)
∂ϕj,q
ϕj,±q=0
= 4uV −1
l,k1,k2
ϕj,k1ϕl,k2ϕl,−q−k1−k2 , (6)
∂ϕj,q∂ϕj,−q
ϕj,±q=0
= Θ(q) + Θ(−q) + 4uV −1
(1 + 2δlj) | ϕl,k |
2 , (7)
∂ϕ2j,q
∣∣∣∣∣
ϕj,±q=0
= 2uV −1
(1 + 2δlj)ϕl,kϕl,−2q−k , (8)
∂2ϕj,q∂2ϕj,−q
ϕj,±q=0
= 6uV −1 . (9)
Here the sums are marked by a prime to indicate that terms containing ϕj,±q are omitted.
This is simply a splitting of (1) into parts with all possible powers of ϕj,±q. The condition
Λ′ > Λ/3, as well as the existence of the upper cutoff for the wave vectors, ensures that
terms of the third power are absent in (4). Besides, the derivation is performed formally
considering all ϕl,k as independent variables.
Taking into account (4), as well as the fact that A1 = A
1+ iA
1 and B2 = B
2+ iB
2 are
complex numbers, the transformed action after the first elementary renormalization step
reads
Stra[ϕ] = A0 − ln
j,q −A
ϕ′j,q
+ ϕ′′j,q
× exp
−2B′2
ϕ′j,q
− ϕ′′j,q
+ 4B′′2ϕ
× exp
ϕ′j,q
+ ϕ′′j,q
dϕ′j,qdϕ
. (10)
Considering only the field configurations which are relevant in the thermodynamic
limit V → ∞, Eq. (10) can be simplified, omitting the terms with B2 and A4. Really,
using the coordinate representation ϕl,k = V
ϕl(x) e
−ikxdx, we can write
B2 = 2uV
(1 + 2δlj)
ϕ2l (x)e
iqx dx
− 3ϕ2j,−q
 . (11)
The quantity V −1
ϕ2l (x)e
iqx dx is an average of ϕ2l (x) over the volume with oscillating
weight factor eiqx. This quantity vanishes for relevant configurations in the thermody-
namic limit: due to the oscillations, positive and negative contributions are similar in
magnitude and cancel at V → ∞. Since 〈| ϕj,−q |
2〉 = 〈| ϕj,q |
2〉 is the Fourier transform
of the two–point correlation function, it is bounded at V → ∞ and, hence, ϕ2j,−q also
is bounded for relevant configurations giving nonvanishing contribution to the statistical
averages 〈·〉 in the thermodynamic limit. Consequently, for these configurations, A2 is
a quantity of order O(1), whereas V −1ϕ2j,−q and B2 vanish at V → ∞. Note, however,
that the term with A4 = O
in (10) cannot be neglected unless A2 is positive. One
can judge that the latter condition is satisfied for the relevant field configurations due to
existence of the thermodynamic limit for the RG flow.
Omitting the terms with B2 and A4, the integrals in (10) can be easily calculated. It
yields
Stra[ϕ] = S
′[ϕ] + ∆Seltra[ϕ] , (12)
where S′[ϕ] = A0 is the original action, where only the ±q modes of the j-th field com-
ponent are omitted, whereas ∆Seltra[ϕ] represents the elementary variation of the action
given by
∆Seltra[ϕ] = ln
| A1 |
. (13)
According to the arguments provided above, this equation is exact for the relevant field
configurations with A2 > 0 in the thermodynamic limit.
The contributions to (6) and (7) provided by modes with wave vectors k, obeying two
relations Λ − dΛ < k < Λ and k 6= ±q, are irrelevant in the thermodynamic limit at
dΛ → 0. It can be verified by the method of analysis introduced in Sec. 2. Hence, Eq. (13)
can be written as
∆Seltra[ϕ] = ln
| Ã1 |
+ δSeltra[ϕ] , (14)
where
Ã1 = P
∂ϕj,q
= 4uV −1
Λ−dΛ∑
l,k1,k2
ϕj,k1ϕl,k2ϕl,−q−k1−k2 , (15)
Ã2 = P
∂ϕj,q∂ϕj,−q
= Θ(q) + Θ(−q) + 4uV −1
Λ−dΛ∑
(1 + 2δlj) | ϕl,k |
2 , (16)
and δSeltra[ϕ] is a vanishingly small correction in the considered limit. Here the operators
P set to zero all ϕj,k within the shell Λ− dΛ < k < Λ (i. e., the derivatives are evaluated
at zero ϕj,k for k within the shell), and the upper border Λ− dΛ for sums implies that we
set ϕl,k = 0 for k > Λ− dΛ. The above replacements are meaningful, since they allow to
obtain easily the Wegner–Houghton equation, as discussed in the following section.
3 Superposition hypothesis and the
Wegner–Houghton equation
Intuitively, it could seem very reasonable that the result of integration over Fourier modes
within the shell Λ − dΛ < k < Λ at dΛ → 0 can be represented as a superposition of
elementary contributions given by (14), neglecting the irrelevant corrections δSeltra[ϕ]. We
will call this idea the superposition hypothesis.
We remind, however, that strictly exact treatment requires a sequential integration of
exp(−S[ϕ]) over a set of ϕj,q. The renormalized action changes after each such integration,
and these changes influence the following steps. A problem is to estimate the discrepancy
between the results of two methods: (1) the exact integration and (2) the superposition
approximation. Since it is necessary to perform infinitely many integration steps in the
thermodynamic limit, the problem is nontrivial and the superposition hypothesis cannot
be rigorously justified.
Nevertheless, the summation of elementary contributions in accordance with the su-
perposition hypothesis leads to the known Wegner–Houghton equation [8]. In this case
the variation of the action due to the integration over shell reads
∆Stra[ϕ] =
Λ−dΛ<q<Λ
Ã2(j,q)
| Ã1(j,q) |
Ã2(j,q)
. (17)
It is exactly consistent with Eq. (2.13) in [8]. The factor 1/2 appears, since only half of
the wave vectors represent independent modes. Here we have indicated that the quantities
Ã1 and Ã2 depend on the current j and q. They depend also on the considered field con-
figuration [ϕ]. If Ã1 and Ã2 are represented by the derivatives of S[ϕ] (see (15) and (16)),
then the equation is written exactly as in [8].
To avoid possible confusion, one has to make clear that the operators P in (15) and (16)
influence the result, as discussed further on. It means that the equation where these
operators are simply omitted, referred in the review paper [10] as the Wegner–Houghton
equation, is not really the Wegner–Houghton equation.
The derivation in [8] is somewhat different. Instead of performing only one elementary
step of integration first, the expansion of Hamiltonian in terms of all shell variables is made
there. The basic method of [8] is to show that, in the thermodynamic limit at dΛ → 0, the
expansion consists of terms containing no more than two derivatives with respect to the
field components. Moreover, it is assumed implicitly that only the diagonal terms with
k′ = −k are important finally, when performing the summation over the wave vectors
k,k′. It leads to Eq. (2.12) in [8]. The omitting of nondiagonal terms is equivalent to the
superposition assumption we discussed already. Indeed, in this and only in this case the
integration over the shell variables can be performed independently, as if the superposition
principle were hold. Hence, essentially the same approximation is used in [8] as in our
derivation, although it is not stated explicitly.
Our derivation refers to the ϕ4 model, whereas in the form with derivatives the equation
may have a more general validity, as supposed in [8]. Indeed, (14) remains correct for a
generalized model provided that higher than second order derivatives of S[ϕ] vanish for
relevant field configurations in the thermodynamic limit. It, in fact, has been assumed
and shown in [8]. Based on similar arguments we have used already, the latter assumption
can be justified for certain class of models, for which the action is represented by a linear
combination of ϕm–kind terms with wave–vector dependent weights and vanishing sum of
the wave vectors
l=1 kl = 0 related to the ϕ factors. In this case we have
Ã1(j,q) =
∂ϕj,q
− ϕj,−q
∂ϕj,qϕj,−q
, (18)
Ã2(j,q) =
∂ϕj,qϕj,−q
for the relevant configurations at V → ∞ and dΛ → 0. The second term in (18) appears
because the derivative ∂S/∂ϕj,q contains relevant terms with ϕj,−q, which have to be
removed. The influence of the operators P is seen from (15) and (18).
Here we do not include the second, i. e., the rescaling step of the RG transformation.
It, however, can be easily calculated for any given action, as described, e. g., in [4]. It is
not relevant four our further considerations.
4 The weak coupling limit
Here we consider the weak coupling limit u → 0 of the model with Θ(k) = r0 + ck
a given positive r0, i. e., in the high temperature phase. In this case ∆Stra[ϕ] can be
expanded in powers of u. It is the natural domain of validity of the perturbation theory,
and the expansion coefficients can be calculated exactly by the known methods applying
the Feynman diagram technique and the Wick’s theorem [4, 5, 11]. On the other hand, the
expansion can be performed in (17). Our aim is to compare the results of both methods
to check the correctness of (17), since the latter equation is based on assumptions.
Let us denote by ∆S̃tra[ϕ] the variation of S[ϕ] omitting the constant (independent of
the field configuration) part. Then the expansion in powers of u reads
∆S̃tra[ϕ] = ∆S1[ϕ]u+
2 [ϕ] + ∆S
2 [ϕ] + ∆S
2 [ϕ]
u2 +O
, (20)
where the expansion coefficient at u2 is split in three parts ∆S
2 [ϕ], ∆S
2 [ϕ], and ∆S
2 [ϕ]
corresponding to the ϕ2, ϕ4, and ϕ6 contributions, respectively. The contribution of order
u is related to the diagram r✐ , whereas the three second–order contributions — to the
diagrams q q❦ , ❛✦q q✦❛ , and ❛✦q q✦❛ . The diagram technique represents the
expansion of −S[ϕ] in terms of connected Feynman diagrams, where the coupled lines are
associated with the Gaussian averages. In particular, the Fourier transformed two–point
correlation function in the Gaussian approximation G0(k) = 〈ϕj,kϕj,−k〉0 = 1/[2Θ(k)]
appears due to the integration over ϕ′j,k and ϕ
j,k. It is represented as the coupling of
lines, in such a way that each line related to the wave vector k and vector–component j is
coupled with another line having the wave vector −k and the same component j. Thus, if
we integrate over ϕj,k within Λ−dΛ < k < Λ in (2), then it corresponds to the coupling of
lines in the same range of wave vectors in the diagram technique. According to the Wick’s
theorem, one has to sum over all possible couplings, which finally yields the summation
(integration) over the wave vectors obeying the constraint Λ− dΛ < k < Λ for each of the
coupled lines associated with the factors G0(k). In the n–component case, it is suitable
to represent the ϕ4 vertex as ❛✦q q✦❛ , where the same index j is associated with two
solid lines connected to one node. The above diagrams are given by the sum of all possible
couplings of the vertices ❛✦q q✦❛ , yielding the corresponding topological pictures when
the dashed lines shrink to points. In this case factor n corresponds to each closed loop
of solid lines, which comes from the summation over j. For a complete definition of the
diagram technique, one has to mention that factors −uV −1 are related to the dashed
lines, G0(k) – to the coupled solid lines, and the fields ϕj,k – to the outer uncoupled solid
lines. Besides, each diagram contains a combinatorial factor. For a diagram consisting
of m vertices ❛✦q q✦❛ , it is the number of all possible couplings of (numbered) lines,
divided by m!.
At dΛ → 0, the diagrammatic calculation for the n–component case yields
∆S1[ϕ] =
(n + 2) dΛ
Λ−dΛ∑
| ϕj,k |
2 (21)
2 [ϕ] = −4V
Λ−dΛ∑
j,l,k1,k2,k3
ϕj,k1ϕj,k2ϕl,k3ϕl,−k1−k2−k3 (22)
× [(n+ 4)Q (k1 + k2,Λ, dΛ) + 4Q (k1 + k3,Λ, dΛ)]
2 [ϕ] = −8V
Λ−dΛ∑
i,j,l,k1,k2,k3,k4,k5
ϕi,k1ϕi,k2ϕj,k3ϕj,k4ϕl,k5ϕl,−k1−k2−k3−k4−k5
×G0 (k1 + k2 + k3) F (| k1 + k2 + k3 |,Λ, dΛ) , (23)
where Kd = S(d)/(2π)
d, S(d) = 2πd/2/Γ(d/2) is the area of unit sphere in d dimensions,
Θ(Λ) is the value of Θ(k) at k = Λ, whereas F(k,Λ, dΛ) is a cutoff function which has the
value 1 within Λ− dΛ < k < Λ and zero otherwise. The quantity Q is given by
Q(k,Λ, dΛ) = V −1
Λ−dΛ<q<Λ
G0(q)G0(k− q)F(| k− q |,Λ, dΛ) . (24)
Below we will give some details of calculation of (22), which is the most important term
in our further discussion. To obtain this result, we have dechipered the ❛✦q q✦❛ diagram
as a sum of three diagrams of different topologies made of vertices ❛✦q q✦❛ , i. e.,
q q q q✐ ✦
✍ , and q
, providing the same topological picture
❛ when shrinking the dashed lines to points. Recall that any loop made of solid
lines of ❛✦q q✦❛ gives a factor n, and one needs also to compute the combinatorial
factors. For the above three diagrams, the resulting factors are 4n, 16, and 16, which
enter the prefactors of Q in (22). To obtain the correct sign, we recall that the diagram
expansion is for −S[ϕ]. The other diagrams are calculated in a similar way.
The expansion of (17) gives no contribution ∆S
2 [ϕ], and we have skipped it in the
diagrammatic calculation as an irrelevant term, which vanishes faster than ∝ dΛ at dΛ → 0
in the thermodynamic limit V → ∞. The expansion of the logarithm term in (17) yields
∆S1[ϕ] exactly consistent with (21). Similarly, ∆S
2 [ϕ] is exactly consistent with (23).
One has to remark that two propogators are involved in (24) and, therefore, the volume
of summation region with nonvanishing cut function F shrinks as (dΛ)2 for a given nonzero
wave vector k at dΛ → 0. However, there is a contribution linear in dΛ for k = 0. As a
result, a contribution proportional to dΛ appears in (22).
Note that the contributions (21) and (23) come from diagrams with only one coupled
line. The term (22) is related to the diagram with two coupled lines. The expansion
of (17) provides a different result for the corresponding part of ∆S̃tra[ϕ]:
2 [ϕ] = −
d−1 dΛ
Θ2(Λ)
Λ−dΛ∑
j,l,k1,k2
(n+ 4 + 4δjl) | ϕj,k1 |
2 | ϕl,k2 |
2 . (25)
Note that (25) comes from the ln Ã2 term in (17), and the calculation is particularly simple
in this case, since the related sum in (16) is independent of q. Eq. (25) is obtained if we
set Q(k,Λ, dΛ) → δk,0Q(0,Λ, dΛ) in (22) (in this case only the diagonal terms j = l are
relevant when summing up the contributions with Q (k1 + k3,Λ, dΛ), as it can be shown by
an analysis of relevant real–space configurations, since 〈ϕj(x)ϕl(x)〉 = 0 holds for j 6= l).
It means that a subset of terms is missing in (25), as compared to (22). The following
analysis will show that this discrepancy between (22) and (25) is important.
It is interesting to mention that (25) is obtained also by the diagrammatic perturbation
method if we first integrate out only the mode with ϕj,±q and then formally apply the
superposition hypothesis, as in the derivation of the Wegner–Houghton equation. It shows
that the discrepancy between (25) and (22) arises because in one case the superposition
hypothesis is applied, whereas in the other case it is not used.
The difference between (22) and (25) can be better seen in the coordinate representa-
tion. In this case (22) reads
2 [ϕ] = − (4n+ 16)
ϕ2(x1)R
2(x1 − x2)ϕ
2(x2) dx1dx2 (26)
ϕj(x1)ϕl(x1)R
2(x1 − x2)ϕj(x2)ϕl(x2) dx1dx2 ,
where
R(x) = V −1
G0(q)F(q,Λ, dΛ)e
iqx (27)
is the Fourier transform of G0F , and ϕ
2(x) =
l (x). In three dimensions we have
R(x) =
(2π)2Θ(Λ)
sin(Λx) (28)
for any given x at dΛ → 0 and L → ∞, where L is the linear system size. The continuum
approximation (28), however, is not correct for x ∼ L and therefore, probably, should not
be used for the evaluation of (26).
The coordinate representation of (25) is
2 [ϕ] = −
d−1 dΛ
Θ2(Λ)
(n+ 4)
ϕ2(x1)V
−1 ϕ2(x2) dx1dx2
ϕ2j (x1)V
−1 ϕ2j (x2) dx1dx2
 . (29)
Eq. (29) represents a relevant contribution at dΛ → 0, as it is proportional to dΛ. It is ob-
viously not consistent with (26). In fact, the term (29) represents a mean-field interaction,
which is proportional to 1/V and independent of the distance, whereas (26) corresponds
to another non-local interaction given by R2(x1 − x2). Hence, the Wegner–Houghton
equation (17) does not yield all correct expansion coefficients at u → 0.
5 Discussion
The results of our test, stated at the end of Sec. 4, reveal some inconsistency between
the Wegner–Houghton equation and the diagrammatic perturbation theory in the high
temperature phase at u → 0. Since this is the natural domain of validity of the per-
turbation theory, there should be no doubts that it produces correct results here, which
agree with (2). So, the results of our test point to some inconsistency between the Wegner–
Houghton equation and (2), which causes a question in which sense the Wegner–Houghton
equation is really exact. The same can be asked about the equations of Polchinski type,
since these (as it is believed) are generalizations of the Wegner–Houghton equation to
the case of smooth momentum cutoff. There is no contradiction with the tests of consis-
tency made in [13, 15], since our test is independent and quite different. According to our
derivation of the Wegner–Houghton equation and the related discussion, it turns out that
the reason of the inconsistency, likely, is the superposition approximation (defined at the
beginning of Sec. 3) used in our paper and analogous approximation implicitly used in [8].
Despite of this problem, the Wegner–Houghton equation is able to reproduce the exact
RG eigenvalue spectrum and critical exponents of the spherical model at n → ∞ [13].
This fact can be interpreted in such a way that the superposition approximation (or an
analogous approximation) is valid to derive such nonperturbative RG equations, which can
produce correct (exact) critical exponents in some limit cases, at least. From a general
point of view, it concerns the fundamental question about the relation between the form
of RG equation and the universal quantities. It has been verified in several known studies
that the universal quantities are invariant with respect to some kind of variations in the
RG equation, like changes in the shape of the momentum cutoff function. This property
is known as the reparametrisation invariance [10]. Probably, the universal quantities are
invariant also with respect to such a variation of the Wegner–Houghton equation, which
makes it exactly consistent with (2). However, this is only a hypothesis.
6 Conclusions
1. The nonperturbative Wegner–Houghton RG equation has been rederived (Secs. 2
and 3), discussing explicitly some assumptions which are used here. In particular,
our derivation assumes the superposition of small contributions provided by elemen-
tary integration steps over the short–wave fluctuation modes. We consider it as
an approximation. As discussed in Sec. 3, the original derivation by Wegner and
Houghton includes essentially the same approximation, although not stated explic-
itly.
2. According to our calculation in Sec. 4, the Wegner–Houghton equation is not com-
pletely consistent with the diagrammatic perturbation theory in the limit of small
ϕ4 coupling constant u in the high temperature phase. This fact, together with
some other important results known from literature, is discussed in Sec. 5. Apart
from critical remarks, a hypothesis has been proposed that the equations of Wegner–
Houghton type, perhaps, can give exact universal quantities.
References
[1] D. J. Amit, Field theory, the renormalization group, and critical phenomena, World
Scientific, Singapore, 1984
[2] D. Sornette, Critical Phenomena in Natural Sciences, Springer, Berlin, 2000
[3] K. G. Wilson, M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972)
[4] Shang–Keng Ma, Modern Theory of Critical Phenomena, W.A. Benjamin, Inc., New
York, 1976
[5] J. Zinn–Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press,
Oxford, 1996
[6] H. Kleinert, V. Schulte–Frohlinde, Critical properties of φ4 theories, World Scientific,
[7] J. Kaupužs, Ann. Phys. (Leipzig) 10, 299 (2001)
[8] F. Wegner, A. Houghton, Phys. Rev. A 8, 401 (1973)
[9] J. Polchinski, Nucl. Phys. B 231, 269 (1984)
[10] C. Bagnuls, C. Bervillier, Phys. Rep. 348, 91 (2001)
[11] J. Berges, N. Tetradis, C. Wetterich, Phys. Rep. 363, 223 (2002)
[12] C. Wetterich, Phys. Lett. B 301, 90 (1993)
[13] K.-I. Aoki, K. Morikava, W. Souma, J.-I. Sumi, H. Terao, Progress of Theoretical
Physics 95, 409 (1996)
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Physics 99, 451 (1998)
[15] T. R. Morris, J. F. Tighe, Journal of High Energy Physics 9908, 007 (1999)
[16] T. R. Morris, J. F. Tighe, Int. J. Mod. Phys. A 16, 2095 (2001)
[17] L. Canet, B. Delamotte, D. Mouhanna, J. Vidal, Phys. Rev. B 68, 064421 (2003)
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	1 Introduction
	2 An elementary step of renormalization
	3 Superposition hypothesis and the  Wegner–Houghton equation
	4 The weak coupling limit
	5 Discussion
	6 Conclusions

  A nonperturbative renormalization of the phi^4 model is considered. First we
integrate out only a single pair of conjugated modes with wave vectors +/- q.
Then we are looking for the RG equation which would describe the transformation
of the Hamiltonian under the integration over a shell Lambda - d Lambda < k <
Lambda, where d Lambda -> 0. We show that the known Wegner--Houghton equation
is consistent with the assumption of a simple superposition of the integration
results for +/- q. The renormalized action can be expanded in powers of the
phi^4 coupling constant u in the high temperature phase at u -> 0. We compare
the expansion coefficients with those exactly calculated by the diagrammatic
perturbative method, and find some inconsistency. It causes a question in which
sense the Wegner-Houghton equation is really exact.

Introduction
The renormalization group (RG) approach, perhaps, is the most extensively used one in
numerous studies of critical phenomena [1, 2]. Particularly, the perturbative RG approach
to the ϕ4 or Ginzburg–Landau model is widely known [3, 4, 5, 6]. However, the pertur-
bative approach suffers from some problems [7]. Therefore it is interesting to look for a
nonperturbative approach. Historically, nonperturbative RG equations have been devel-
oped in parallel to the perturbative ones. These are so called exact RG equations (ERGE).
The method of deriving such RG equations is close in spirit to the famous Wilson’s ap-
proach, where the basic idea is to integrate out the short–wave fluctuations corresponding
to the wave vectors within Λ/s < q < Λ with the upper (or ultraviolet) cutoff parameter
Λ and the renormalization scale s > 1. The oldest nonperturbative equation of this kind,
originally presented by Wegner and Houghton [8], uses the sharp momentum cutoff. Later,
a similar equation with smooth momentum cutoff has been proposed by Polchinski [9]. The
RG equations of this class are reviewed in [10].
According to the known classification [10, 11], there is another class of nonperturbative
RG equations proposed by [12] and reviewed in [11]. Some relevant discussion can be found
in [10], as well. Such equations describe the variation of an average effective action Γk[φ]
depending on the running cutoff scale k. Here φ(x) = 〈ϕ(x)〉 is the averaged order–
parameter field (for simplicity, we refer to the case of scalar field). According to [12], the
averaging is performed over volume ∼ k−d such that the fluctuation degrees of freedom
E–mail: kaupuzs@latnet.lv
http://arxiv.org/abs/0704.0142v2
with momenta q > k are effectively integrated out. In fact, the averaging over volume
∼ k−d is the usual block–spin–averaging procedure of the real–space renormalization.
At the same time, the fluctuations with q . k are suppressed by a smooth infrared
cutoff. As one can judge from [11], the existence of a deterministic relation between the
configuration of external source {J(x)} and that of the averaged order parameter {φ(x)} is
(implicitly) assumed in the nonperturbative derivation of the RG flow equation. Namely,
it is stated (see the text between (2.28) and (2.29) in [11]) that δJ(x)/δφ(y) is the inverse
of δφ(x)/δJ(y), which has certain meaning as a matrix identity. To make this point
clearer, let us consider a toy example ~J = A~φ, where ~J = (J(x1), J(x2), . . . , J(xN)) and
~φ = (φ(x1), φ(x2), . . . , φ(xN)) are N–component vectors and A is a matrix of size N ×N .
In this case ∂J(xi)/∂φ(xj) is the element Aij of matrix A, whereas ∂φ(xi)/∂J(xj) is the
element
of the inverse matrix A−1. In the continuum limit N → ∞, this toy
example corresponds to a linear dependence between {J(x)} and {φ(x)}. The calculation
of derivative always implies the linearisation around some point, so that the matrix identity
used in [11] (as a continuum limit in the above example) has a general meaning. However,
it makes sense only if there exists a deterministic relation between the configurations of
φ(x) and J(x) or, in a mathematical notation, if there exist mappings f : {J(x)} → {φ(x)}
and f−1 : {φ(x)} → {J(x)}. On the other hand, according to the block–averaging, the
values of φ(x) should be understood as the block–averages. These, of course, are not
uniquely determined by the external sources, but are fluctuating quantities. So, we are
quite sceptical about the exactness of such an approach of averaged effective action.
The integration over fluctuation degrees with momenta q > k does not alter the be-
havior of the infrared modes, directly related to the critical exponents. From this point of
view, the approach based on the equations of Wegner–Houghton and Polchinski type seems
to be more natural. These are widely believed to be the exact RG equations, although, in
view of our currently presented results, it turns out to be questionable in which sense they
are really exact. In any case, the nonperturbative RG equations cannot be solved exactly,
therefore a suitable truncation is used. The convergence of several truncation schemes and
of the derivative expansion has been widely studied in [13, 14, 15, 16, 17]. Here [17] refers
to the specific approach of [12]. A review about all the methods of approximate solution
can be found in [18].
Another problem is to test and verify the nonperturbative RG equations, comparing
the results with the known exact and rigorous solutions, as well as with the results of
the perturbation theory. In [15], the derivative expansion of the RG β–function has been
considered, showing the agreement up to the second order between the perturbative results
and those obtained from the Legendre flow equation, which also belongs to the same class
of RG equations as the Wegner–Houghton and Polchinski equations. It has been stated
in [13] that the critical exponent ν, extracted from the Wegner–Houghton equation in the
local potential approximation, agrees with the ε–expansion up to the O(ε) order, as well
as with the 1/n (1/N in the notations of [13]) expansion in the leading order. However,
looking carefully on the results of [13], one should make clear that “the leading order of
the 1/n expansion” in this case is no more than the zeroth order, whereas the expansion
coefficient at 1/n is inconsistent with that proposed by the perturbative RG calculation
at any fixed dimension d except only d = 4. The inconsistency could be understood from
the point of view that the Wegner–Houghton equation has been solved approximately.
Therefore it would be interesting to verify whether the problem is eliminated beyond the
local potential approximation. One should also take into account that the perturbative
RG theory is not rigorous and, therefore, we think that a possible inconsistency still would
not prove that something is really wrong with the nonperturbative RG equation. In any
case, it is a remarkable fact that correct RG eigenvalue spectrum and critical exponents
are obtained in the local potential approximation at n → ∞ from the Wegner–Houghton
equation [13], as well as from similar RG equations [19], in agreement with the known
exact and rigorous results for the spherical model. It shows that some solutions, being
not exact, nevertheless can lead to exact critical exponents. From this point of view, it
seems also possible that some kind of approximations, made in the derivation of an RG
equation, are not harmful for the critical exponents.
We propose a simple test of the Wegner–Houghton equation: to verify the expansion of
the renormalized action S of the ϕ4 model in powers of the coupling constant u at u → 0
in the high–temperature phase. Such a test is rigorous, in the sense that the natural
domain of validity of the perturbation theory is considered. We think that it would be
quite natural to start with such a relatively simple and straightforward test before passing
to more complicated ones, considered in [13, 15, 19]. We have made this simplest test
in our paper and have found that the Wegner–Houghton equation fails to give all correct
expansion coefficients. We have also proposed another derivation of the Wegner–Houghton
equation (Secs. 2, 3). It is helpful to clarify the origin of the mentioned inconsistency. It
is also less obscure from the point of view that the used assumptions and approximations
are clearly stated. As regards the derivation in [8], at least one essential step is obscure
and apparently contains an implicit approximation which, in very essence, is analogous to
that pointed out in our derivation. We will discuss this point in Sec. 3.
2 An elementary step of renormalization
To derive a nonperturbative RG equation for the ϕ4 model, we should start with some
elementary steps, as explained in this section.
Consider the action S[ϕ] which depends on the configuration of the order parameter
field ϕ(x) depending on coordinate x. By definition, it is related to the Hamiltonian H of
the model via S = H/T , where T is the temperature measured in energy units. In general,
ϕ(x) is an n–component vector with components ϕj(x) given in the Fourier representation
as ϕj(x) = V
k<Λ ϕj,ke
ikx, where V = Ld is the volume of the system, d is the spatial
dimensionality, and Λ is the upper cutoff of the wave vectors. We consider the action of
the Ginzburg–Landau form. For simplicity, we include only the ϕ2 and ϕ4 terms. The
action of such ϕ4 model is given by
S[ϕ] =
Θ(k)ϕj,kϕj,−k + uV
j,l,k1,k2,k3
ϕj,k1ϕj,k2ϕl,k3ϕl,−k1−k2−k3 , (1)
where Θ(k) is some function of wave vector k, e. g., Θ(k) = r0 + ck
2 like in theories of
critical phenomena [4, 5, 6, 7]. In the sums we set ϕl,k = 0 for k > Λ.
The renormalization group (RG) transformation implies the integration over ϕj,k for
some set of wave vectors with Λ′ < k < Λ, i. e., the Kadanoff’s transformation, followed
by certain rescaling procedure [4]. The action under the Kadanoff’s transformation is
changed from S[ϕ] to Stra[ϕ] according to the equation
e−Stra[ϕ] =
e−S[ϕ]
j,Λ′<k<Λ
dϕj,k . (2)
Alternatively, one often writes −Stra[ϕ]+AL
d instead of −Stra[ϕ] to separate the constant
part of the action ALd. This, however, is merely a redefinition of Stra, and for our purposes
it is suitable to use (2). Note that ϕj,k = ϕ
j,k+iϕ
j,k is a complex number and ϕj,−k = ϕ
holds (since ϕj(x) is always real), so that the integration over ϕj,k means in fact the
integration over real and imaginary parts of ϕj,k for each pair of conjugated wave vectors
k and −k.
The Kadanoff’s transformation (2) can be split in a sequence of elementary steps
S[ϕ] → Stra[ϕ] of the repeated integration given by
e−Stra[ϕ] =
e−S[ϕ]dϕ′j,qdϕ
j,q (3)
for each j and q ∈ Ω, where Ω is the subset of independent wave vectors (±q represent
one independent mode) within Λ′ < q < Λ. Thus, in the first elementary step of renor-
malization we have to insert the original action (1) into (3) and perform the integration
for one chosen j and q ∈ Ω. In an exact treatment we must take into account that the
action is already changed in the following elementary steps.
For Λ′ > Λ/3, we can use the following exact decomposition of (1)
S[ϕ] = A0 +A1ϕj,q +A
1ϕj,−q +A2ϕj,qϕj,−q +B2ϕ
j,q +B
j,−q +A4ϕ
j,−q , (4)
where
A0 = S|ϕj,±q=0 , (5)
∂ϕj,q
ϕj,±q=0
= 4uV −1
l,k1,k2
ϕj,k1ϕl,k2ϕl,−q−k1−k2 , (6)
∂ϕj,q∂ϕj,−q
ϕj,±q=0
= Θ(q) + Θ(−q) + 4uV −1
(1 + 2δlj) | ϕl,k |
2 , (7)
∂ϕ2j,q
∣∣∣∣∣
ϕj,±q=0
= 2uV −1
(1 + 2δlj)ϕl,kϕl,−2q−k , (8)
∂2ϕj,q∂2ϕj,−q
ϕj,±q=0
= 6uV −1 . (9)
Here the sums are marked by a prime to indicate that terms containing ϕj,±q are omitted.
This is simply a splitting of (1) into parts with all possible powers of ϕj,±q. The condition
Λ′ > Λ/3, as well as the existence of the upper cutoff for the wave vectors, ensures that
terms of the third power are absent in (4). Besides, the derivation is performed formally
considering all ϕl,k as independent variables.
Taking into account (4), as well as the fact that A1 = A
1+ iA
1 and B2 = B
2+ iB
2 are
complex numbers, the transformed action after the first elementary renormalization step
reads
Stra[ϕ] = A0 − ln
j,q −A
ϕ′j,q
+ ϕ′′j,q
× exp
−2B′2
ϕ′j,q
− ϕ′′j,q
+ 4B′′2ϕ
× exp
ϕ′j,q
+ ϕ′′j,q
dϕ′j,qdϕ
. (10)
Considering only the field configurations which are relevant in the thermodynamic
limit V → ∞, Eq. (10) can be simplified, omitting the terms with B2 and A4. Really,
using the coordinate representation ϕl,k = V
ϕl(x) e
−ikxdx, we can write
B2 = 2uV
(1 + 2δlj)
ϕ2l (x)e
iqx dx
− 3ϕ2j,−q
 . (11)
The quantity V −1
ϕ2l (x)e
iqx dx is an average of ϕ2l (x) over the volume with oscillating
weight factor eiqx. This quantity vanishes for relevant configurations in the thermody-
namic limit: due to the oscillations, positive and negative contributions are similar in
magnitude and cancel at V → ∞. Since 〈| ϕj,−q |
2〉 = 〈| ϕj,q |
2〉 is the Fourier transform
of the two–point correlation function, it is bounded at V → ∞ and, hence, ϕ2j,−q also
is bounded for relevant configurations giving nonvanishing contribution to the statistical
averages 〈·〉 in the thermodynamic limit. Consequently, for these configurations, A2 is
a quantity of order O(1), whereas V −1ϕ2j,−q and B2 vanish at V → ∞. Note, however,
that the term with A4 = O
in (10) cannot be neglected unless A2 is positive. One
can judge that the latter condition is satisfied for the relevant field configurations due to
existence of the thermodynamic limit for the RG flow.
Omitting the terms with B2 and A4, the integrals in (10) can be easily calculated. It
yields
Stra[ϕ] = S
′[ϕ] + ∆Seltra[ϕ] , (12)
where S′[ϕ] = A0 is the original action, where only the ±q modes of the j-th field com-
ponent are omitted, whereas ∆Seltra[ϕ] represents the elementary variation of the action
given by
∆Seltra[ϕ] = ln
| A1 |
. (13)
According to the arguments provided above, this equation is exact for the relevant field
configurations with A2 > 0 in the thermodynamic limit.
The contributions to (6) and (7) provided by modes with wave vectors k, obeying two
relations Λ − dΛ < k < Λ and k 6= ±q, are irrelevant in the thermodynamic limit at
dΛ → 0. It can be verified by the method of analysis introduced in Sec. 2. Hence, Eq. (13)
can be written as
∆Seltra[ϕ] = ln
| Ã1 |
+ δSeltra[ϕ] , (14)
where
Ã1 = P
∂ϕj,q
= 4uV −1
Λ−dΛ∑
l,k1,k2
ϕj,k1ϕl,k2ϕl,−q−k1−k2 , (15)
Ã2 = P
∂ϕj,q∂ϕj,−q
= Θ(q) + Θ(−q) + 4uV −1
Λ−dΛ∑
(1 + 2δlj) | ϕl,k |
2 , (16)
and δSeltra[ϕ] is a vanishingly small correction in the considered limit. Here the operators
P set to zero all ϕj,k within the shell Λ− dΛ < k < Λ (i. e., the derivatives are evaluated
at zero ϕj,k for k within the shell), and the upper border Λ− dΛ for sums implies that we
set ϕl,k = 0 for k > Λ− dΛ. The above replacements are meaningful, since they allow to
obtain easily the Wegner–Houghton equation, as discussed in the following section.
3 Superposition hypothesis and the
Wegner–Houghton equation
Intuitively, it could seem very reasonable that the result of integration over Fourier modes
within the shell Λ − dΛ < k < Λ at dΛ → 0 can be represented as a superposition of
elementary contributions given by (14), neglecting the irrelevant corrections δSeltra[ϕ]. We
will call this idea the superposition hypothesis.
We remind, however, that strictly exact treatment requires a sequential integration of
exp(−S[ϕ]) over a set of ϕj,q. The renormalized action changes after each such integration,
and these changes influence the following steps. A problem is to estimate the discrepancy
between the results of two methods: (1) the exact integration and (2) the superposition
approximation. Since it is necessary to perform infinitely many integration steps in the
thermodynamic limit, the problem is nontrivial and the superposition hypothesis cannot
be rigorously justified.
Nevertheless, the summation of elementary contributions in accordance with the su-
perposition hypothesis leads to the known Wegner–Houghton equation [8]. In this case
the variation of the action due to the integration over shell reads
∆Stra[ϕ] =
Λ−dΛ<q<Λ
Ã2(j,q)
| Ã1(j,q) |
Ã2(j,q)
. (17)
It is exactly consistent with Eq. (2.13) in [8]. The factor 1/2 appears, since only half of
the wave vectors represent independent modes. Here we have indicated that the quantities
Ã1 and Ã2 depend on the current j and q. They depend also on the considered field con-
figuration [ϕ]. If Ã1 and Ã2 are represented by the derivatives of S[ϕ] (see (15) and (16)),
then the equation is written exactly as in [8].
To avoid possible confusion, one has to make clear that the operators P in (15) and (16)
influence the result, as discussed further on. It means that the equation where these
operators are simply omitted, referred in the review paper [10] as the Wegner–Houghton
equation, is not really the Wegner–Houghton equation.
The derivation in [8] is somewhat different. Instead of performing only one elementary
step of integration first, the expansion of Hamiltonian in terms of all shell variables is made
there. The basic method of [8] is to show that, in the thermodynamic limit at dΛ → 0, the
expansion consists of terms containing no more than two derivatives with respect to the
field components. Moreover, it is assumed implicitly that only the diagonal terms with
k′ = −k are important finally, when performing the summation over the wave vectors
k,k′. It leads to Eq. (2.12) in [8]. The omitting of nondiagonal terms is equivalent to the
superposition assumption we discussed already. Indeed, in this and only in this case the
integration over the shell variables can be performed independently, as if the superposition
principle were hold. Hence, essentially the same approximation is used in [8] as in our
derivation, although it is not stated explicitly.
Our derivation refers to the ϕ4 model, whereas in the form with derivatives the equation
may have a more general validity, as supposed in [8]. Indeed, (14) remains correct for a
generalized model provided that higher than second order derivatives of S[ϕ] vanish for
relevant field configurations in the thermodynamic limit. It, in fact, has been assumed
and shown in [8]. Based on similar arguments we have used already, the latter assumption
can be justified for certain class of models, for which the action is represented by a linear
combination of ϕm–kind terms with wave–vector dependent weights and vanishing sum of
the wave vectors
l=1 kl = 0 related to the ϕ factors. In this case we have
Ã1(j,q) =
∂ϕj,q
− ϕj,−q
∂ϕj,qϕj,−q
, (18)
Ã2(j,q) =
∂ϕj,qϕj,−q
for the relevant configurations at V → ∞ and dΛ → 0. The second term in (18) appears
because the derivative ∂S/∂ϕj,q contains relevant terms with ϕj,−q, which have to be
removed. The influence of the operators P is seen from (15) and (18).
Here we do not include the second, i. e., the rescaling step of the RG transformation.
It, however, can be easily calculated for any given action, as described, e. g., in [4]. It is
not relevant four our further considerations.
4 The weak coupling limit
Here we consider the weak coupling limit u → 0 of the model with Θ(k) = r0 + ck
a given positive r0, i. e., in the high temperature phase. In this case ∆Stra[ϕ] can be
expanded in powers of u. It is the natural domain of validity of the perturbation theory,
and the expansion coefficients can be calculated exactly by the known methods applying
the Feynman diagram technique and the Wick’s theorem [4, 5, 11]. On the other hand, the
expansion can be performed in (17). Our aim is to compare the results of both methods
to check the correctness of (17), since the latter equation is based on assumptions.
Let us denote by ∆S̃tra[ϕ] the variation of S[ϕ] omitting the constant (independent of
the field configuration) part. Then the expansion in powers of u reads
∆S̃tra[ϕ] = ∆S1[ϕ]u+
2 [ϕ] + ∆S
2 [ϕ] + ∆S
2 [ϕ]
u2 +O
, (20)
where the expansion coefficient at u2 is split in three parts ∆S
2 [ϕ], ∆S
2 [ϕ], and ∆S
2 [ϕ]
corresponding to the ϕ2, ϕ4, and ϕ6 contributions, respectively. The contribution of order
u is related to the diagram r✐ , whereas the three second–order contributions — to the
diagrams q q❦ , ❛✦q q✦❛ , and ❛✦q q✦❛ . The diagram technique represents the
expansion of −S[ϕ] in terms of connected Feynman diagrams, where the coupled lines are
associated with the Gaussian averages. In particular, the Fourier transformed two–point
correlation function in the Gaussian approximation G0(k) = 〈ϕj,kϕj,−k〉0 = 1/[2Θ(k)]
appears due to the integration over ϕ′j,k and ϕ
j,k. It is represented as the coupling of
lines, in such a way that each line related to the wave vector k and vector–component j is
coupled with another line having the wave vector −k and the same component j. Thus, if
we integrate over ϕj,k within Λ−dΛ < k < Λ in (2), then it corresponds to the coupling of
lines in the same range of wave vectors in the diagram technique. According to the Wick’s
theorem, one has to sum over all possible couplings, which finally yields the summation
(integration) over the wave vectors obeying the constraint Λ− dΛ < k < Λ for each of the
coupled lines associated with the factors G0(k). In the n–component case, it is suitable
to represent the ϕ4 vertex as ❛✦q q✦❛ , where the same index j is associated with two
solid lines connected to one node. The above diagrams are given by the sum of all possible
couplings of the vertices ❛✦q q✦❛ , yielding the corresponding topological pictures when
the dashed lines shrink to points. In this case factor n corresponds to each closed loop
of solid lines, which comes from the summation over j. For a complete definition of the
diagram technique, one has to mention that factors −uV −1 are related to the dashed
lines, G0(k) – to the coupled solid lines, and the fields ϕj,k – to the outer uncoupled solid
lines. Besides, each diagram contains a combinatorial factor. For a diagram consisting
of m vertices ❛✦q q✦❛ , it is the number of all possible couplings of (numbered) lines,
divided by m!.
At dΛ → 0, the diagrammatic calculation for the n–component case yields
∆S1[ϕ] =
(n + 2) dΛ
Λ−dΛ∑
| ϕj,k |
2 (21)
2 [ϕ] = −4V
Λ−dΛ∑
j,l,k1,k2,k3
ϕj,k1ϕj,k2ϕl,k3ϕl,−k1−k2−k3 (22)
× [(n+ 4)Q (k1 + k2,Λ, dΛ) + 4Q (k1 + k3,Λ, dΛ)]
2 [ϕ] = −8V
Λ−dΛ∑
i,j,l,k1,k2,k3,k4,k5
ϕi,k1ϕi,k2ϕj,k3ϕj,k4ϕl,k5ϕl,−k1−k2−k3−k4−k5
×G0 (k1 + k2 + k3) F (| k1 + k2 + k3 |,Λ, dΛ) , (23)
where Kd = S(d)/(2π)
d, S(d) = 2πd/2/Γ(d/2) is the area of unit sphere in d dimensions,
Θ(Λ) is the value of Θ(k) at k = Λ, whereas F(k,Λ, dΛ) is a cutoff function which has the
value 1 within Λ− dΛ < k < Λ and zero otherwise. The quantity Q is given by
Q(k,Λ, dΛ) = V −1
Λ−dΛ<q<Λ
G0(q)G0(k− q)F(| k− q |,Λ, dΛ) . (24)
Below we will give some details of calculation of (22), which is the most important term
in our further discussion. To obtain this result, we have dechipered the ❛✦q q✦❛ diagram
as a sum of three diagrams of different topologies made of vertices ❛✦q q✦❛ , i. e.,
q q q q✐ ✦
✍ , and q
, providing the same topological picture
❛ when shrinking the dashed lines to points. Recall that any loop made of solid
lines of ❛✦q q✦❛ gives a factor n, and one needs also to compute the combinatorial
factors. For the above three diagrams, the resulting factors are 4n, 16, and 16, which
enter the prefactors of Q in (22). To obtain the correct sign, we recall that the diagram
expansion is for −S[ϕ]. The other diagrams are calculated in a similar way.
The expansion of (17) gives no contribution ∆S
2 [ϕ], and we have skipped it in the
diagrammatic calculation as an irrelevant term, which vanishes faster than ∝ dΛ at dΛ → 0
in the thermodynamic limit V → ∞. The expansion of the logarithm term in (17) yields
∆S1[ϕ] exactly consistent with (21). Similarly, ∆S
2 [ϕ] is exactly consistent with (23).
One has to remark that two propogators are involved in (24) and, therefore, the volume
of summation region with nonvanishing cut function F shrinks as (dΛ)2 for a given nonzero
wave vector k at dΛ → 0. However, there is a contribution linear in dΛ for k = 0. As a
result, a contribution proportional to dΛ appears in (22).
Note that the contributions (21) and (23) come from diagrams with only one coupled
line. The term (22) is related to the diagram with two coupled lines. The expansion
of (17) provides a different result for the corresponding part of ∆S̃tra[ϕ]:
2 [ϕ] = −
d−1 dΛ
Θ2(Λ)
Λ−dΛ∑
j,l,k1,k2
(n+ 4 + 4δjl) | ϕj,k1 |
2 | ϕl,k2 |
2 . (25)
Note that (25) comes from the ln Ã2 term in (17), and the calculation is particularly simple
in this case, since the related sum in (16) is independent of q. Eq. (25) is obtained if we
set Q(k,Λ, dΛ) → δk,0Q(0,Λ, dΛ) in (22) (in this case only the diagonal terms j = l are
relevant when summing up the contributions with Q (k1 + k3,Λ, dΛ), as it can be shown by
an analysis of relevant real–space configurations, since 〈ϕj(x)ϕl(x)〉 = 0 holds for j 6= l).
It means that a subset of terms is missing in (25), as compared to (22). The following
analysis will show that this discrepancy between (22) and (25) is important.
It is interesting to mention that (25) is obtained also by the diagrammatic perturbation
method if we first integrate out only the mode with ϕj,±q and then formally apply the
superposition hypothesis, as in the derivation of the Wegner–Houghton equation. It shows
that the discrepancy between (25) and (22) arises because in one case the superposition
hypothesis is applied, whereas in the other case it is not used.
The difference between (22) and (25) can be better seen in the coordinate representa-
tion. In this case (22) reads
2 [ϕ] = − (4n+ 16)
ϕ2(x1)R
2(x1 − x2)ϕ
2(x2) dx1dx2 (26)
ϕj(x1)ϕl(x1)R
2(x1 − x2)ϕj(x2)ϕl(x2) dx1dx2 ,
where
R(x) = V −1
G0(q)F(q,Λ, dΛ)e
iqx (27)
is the Fourier transform of G0F , and ϕ
2(x) =
l (x). In three dimensions we have
R(x) =
(2π)2Θ(Λ)
sin(Λx) (28)
for any given x at dΛ → 0 and L → ∞, where L is the linear system size. The continuum
approximation (28), however, is not correct for x ∼ L and therefore, probably, should not
be used for the evaluation of (26).
The coordinate representation of (25) is
2 [ϕ] = −
d−1 dΛ
Θ2(Λ)
(n+ 4)
ϕ2(x1)V
−1 ϕ2(x2) dx1dx2
ϕ2j (x1)V
−1 ϕ2j (x2) dx1dx2
 . (29)
Eq. (29) represents a relevant contribution at dΛ → 0, as it is proportional to dΛ. It is ob-
viously not consistent with (26). In fact, the term (29) represents a mean-field interaction,
which is proportional to 1/V and independent of the distance, whereas (26) corresponds
to another non-local interaction given by R2(x1 − x2). Hence, the Wegner–Houghton
equation (17) does not yield all correct expansion coefficients at u → 0.
5 Discussion
The results of our test, stated at the end of Sec. 4, reveal some inconsistency between
the Wegner–Houghton equation and the diagrammatic perturbation theory in the high
temperature phase at u → 0. Since this is the natural domain of validity of the per-
turbation theory, there should be no doubts that it produces correct results here, which
agree with (2). So, the results of our test point to some inconsistency between the Wegner–
Houghton equation and (2), which causes a question in which sense the Wegner–Houghton
equation is really exact. The same can be asked about the equations of Polchinski type,
since these (as it is believed) are generalizations of the Wegner–Houghton equation to
the case of smooth momentum cutoff. There is no contradiction with the tests of consis-
tency made in [13, 15], since our test is independent and quite different. According to our
derivation of the Wegner–Houghton equation and the related discussion, it turns out that
the reason of the inconsistency, likely, is the superposition approximation (defined at the
beginning of Sec. 3) used in our paper and analogous approximation implicitly used in [8].
Despite of this problem, the Wegner–Houghton equation is able to reproduce the exact
RG eigenvalue spectrum and critical exponents of the spherical model at n → ∞ [13].
This fact can be interpreted in such a way that the superposition approximation (or an
analogous approximation) is valid to derive such nonperturbative RG equations, which can
produce correct (exact) critical exponents in some limit cases, at least. From a general
point of view, it concerns the fundamental question about the relation between the form
of RG equation and the universal quantities. It has been verified in several known studies
that the universal quantities are invariant with respect to some kind of variations in the
RG equation, like changes in the shape of the momentum cutoff function. This property
is known as the reparametrisation invariance [10]. Probably, the universal quantities are
invariant also with respect to such a variation of the Wegner–Houghton equation, which
makes it exactly consistent with (2). However, this is only a hypothesis.
6 Conclusions
1. The nonperturbative Wegner–Houghton RG equation has been rederived (Secs. 2
and 3), discussing explicitly some assumptions which are used here. In particular,
our derivation assumes the superposition of small contributions provided by elemen-
tary integration steps over the short–wave fluctuation modes. We consider it as
an approximation. As discussed in Sec. 3, the original derivation by Wegner and
Houghton includes essentially the same approximation, although not stated explic-
itly.
2. According to our calculation in Sec. 4, the Wegner–Houghton equation is not com-
pletely consistent with the diagrammatic perturbation theory in the limit of small
ϕ4 coupling constant u in the high temperature phase. This fact, together with
some other important results known from literature, is discussed in Sec. 5. Apart
from critical remarks, a hypothesis has been proposed that the equations of Wegner–
Houghton type, perhaps, can give exact universal quantities.
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	1 Introduction
	2 An elementary step of renormalization
	3 Superposition hypothesis and the  Wegner–Houghton equation
	4 The weak coupling limit
	5 Discussion
	6 Conclusions
