Correlation functions in the Non Perturbative Renormalization Group and
  field expansion
Correlation functions in the Non Perturbative Renormalization
Group and field expansion
Diego Guerra,∗ Ramón Méndez-Galain,† and Nicolás Wschebor‡
Instituto de F́ısica, Facultad de Ingenieŕıa, Univ. de la República,
J.H.y Reissig 565, 11000 Montevideo, Uruguay
(Dated: November 4, 2018)
Abstract
The usual procedure of including a finite number of vertices in Non Perturbative Renormalization
Group equations in order to obtain n-point correlation functions at finite momenta is analyzed.
This is done by exploiting a general method recently introduced which includes simultaneously all
vertices although approximating their momentum dependence. The study is performed using the
self-energy of the tridimensional scalar model at criticality. At least in this example, low order
truncations miss quantities as the critical exponent η by as much as 60%. However, if one goes to
high order truncations the procedure seems to converge rapidly.
PACS numbers: 03.75.Fi,05.30.Jp
∗Electronic address: dguerra@fing.edu.uy
†Electronic address: mendezg@fing.edu.uy
‡Electronic address: nicws@fing.edu.uy
http://arxiv.org/abs/0704.0258v2
mailto:dguerra@fing.edu.uy
mailto:mendezg@fing.edu.uy
mailto:nicws@fing.edu.uy
I. INTRODUCTION
In nearly all fields in physics, there are systems having a large number of strongly cor-
related constituents. These cannot be treated with usual perturbative methods. Phase
transitions and critical phenomena, disordered systems, strongly correlated electrons, quan-
tum chromodynamics at large distances, are just a few examples which demand a general
and efficient method to treat non-perturbative situations. In problems as those just quoted,
the calculation of correlation functions of the configuration variables is, in general, a very
complicated task.
The non perturbative renormalization group (NPRG) [1, 2, 3, 4, 5] has proven to be a
powerful tool to achieve this goal. It presents itself as an infinite hierarchy of flow equations
relating sequentially the various n-point functions. It has been successfully applied in many
different problems, either in condensed matter, particle or nuclear physics (for reviews, see
e.g. [6, 7, 8]; a pedagogical introduction can be found in [9]). In most of these problems
however, one is interested in observables dominated by long wavelength modes. In these
cases, it is then possible to approximately close the infinite hierarchy of NPRG equations
performing an expansion in the number of derivatives of the field. This approximation
scheme is known as the derivative expansion (DE) [10]. The price to pay is that the n-point
functions can be calculated only at small external momenta, i.e. smaller than the smallest
mass on the problem (vanishing momenta in the case of critical phenomena).
In many other physical problems however, this is not enough: the full knowledge of the
momentum dependence of correlation functions is needed in order to calculate quantities of
physical interest (e.g. to get the spectrum of excitations, the shape of a Fermi surface, the
scattering matrix, etc.). There have been many attempts to solve the infinite system of flow
equations at finite momenta; most of them are based on various forms of an early proposal
by Weinberg [11]. Although some of these attempts [12, 13, 14, 15] introduce sophisticated
ansatz for the unknown correlation functions appearing in a given flow equation, most efforts
simply ignore high order vertices. In all these works, only low order vertices are taken into
account: usual calculations do not even include the complete flow of the 3- and 4-point
functions. Moreover, it is not possible a priori to gauge the quality of such approximations
schemes.
Recently, an alternative general method to get n-point functions at any finite momenta
within the NPRG has been proposed [16]. It has many similarities with DE. First, it is
an approximation scheme that can be systematically improved. Second, the scheme yields
a closed set of flow equations including simultaneously an infinite number of vertices; one
thus goes far beyond schemes including a small number of vertices, as those quoted in
the previous paragraph. Moreover, it has been proven [16] that in their corresponding
limits, both perturvative and DE results are recovered; this remains valid at each order of
the respective expansion. Finally, in the large-N limit of O(N) models, the leading-order
(LO) of the approximation scheme becomes exact for all n-point functions. (The expression
“leading order” means the first step in the approximation scheme; it does not refer to an
expansion in a small parameter which usually does not exist in these kind of problems.).
In [17], the method has been applied, in its leading order, to the calculation of the self-
energy of the scalar model, at criticality. That is, we have fine-tuned the bare mass of the
model in order for the correlation length to be infinite, and then we have studied the full
range of momenta, from the high momenta Gaussian regime to the low momenta scaling
one. At this order of the approximation scheme the self-energy is expected to include all
one loop contributions and to achieve DE at next-to-leading order (NLO) precision, in the
corresponding limit [16]. The numerical solution found in [17] verifies these properties.
Moreover, the function has the expected physical properties in all momenta regime. First,
it presents the correct scaling behavior in the infrared limit. The model reproduces critical
exponent η with a level of precision comparable to the DE at NLO. Moreover, contrarily to
DE, the anomalous power-law behavior can be read directly from the momentum dependence
of the 2-point function. Second, it shows the expected logarithmic shape of the perturbative
regime even though the coefficient in front of the logarithm, which is a 2-loops quantity,
is only reproduced with an error of 8%. In order to check the quality of the solution in
the intermediate momentum region, a quantity sensitive to this crossover sector has been
calculated: one gets a result almost within the error bars of both Monte-Carlo and resumed
7-loops perturbative calculations. Please observe that this quantity is extremely difficult to
calculate: even these sophisticated methods give an error of the order of 10%.
Another interesting similarity between DE and the method presented in [16] is that, as a
price to pay in order to close the equations including an infinite number of vertices, one has
to study the problem in an external constant field. Accordingly, one ends up with partial
differential equations which may be difficult to solve. A useful approximation scheme, widely
used in DE calculations, is to perform, on top of the expansion in derivatives of the field,
an extra expansion in powers of the field (see e.g. [7]), in the spirit of Weinberg proposal.
During the last 10 years, this strategy has been widely used [18, 19, 20, 21, 22, 23, 24, 25];
in many studied situations this expansion seems to converge (generally oscillating) [19, 20,
21, 23, 24, 25], while in many others it does not [22, 26]. In d = 2, the field expansion has
been explored with no indication of convergence for critical exponents, even going to high
orders [27].
In this work we shall explore this procedure of expansion in powers of the field, in the
framework of the calculation scheme presented in [16]. More precisely, we shall make a field
expansion on top of the already approximated 2-point function flow equation solved in [17].
Then we shall compare results with and without field expansion. In doing so, we have two
goals. First, we shall study the apparent convergence of this procedure. This comparison is
essential if one hopes to apply the scheme described in [16] to situations more complicated
than that considered in [17]. For example, within DE scheme, when trying to go to higher
orders or when considering more involved models, the expansion in powers of the field on top
of the corresponding approximate flow equations is sometimes the only practical strategy to
solve them [8, 23, 28]. The second and more important goal is the following: as we shall see
in section III, truncation in powers of the field is equivalent to ignoring high order vertices
in the flow equations. Thus, the comparison presented here can help to estimate the quality
of the calculations made so far to get n-point functions at finite momenta neglecting high
order vertices.
The article is organized as follows. In the next section we describe the basics ingredients
of both the NPRG and the approximation scheme introduced in [16]. We also present the
results obtained in [17], when this scheme is used to find the 2-point function of the scalar
model. In section III, we apply the expansion in the field at various orders and compare
these results with those found in [17]. Finally, we present the conclusions of the study.
II. GENERAL CONSIDERATIONS
Let us consider a scalar field theory with the classical action
(∂µϕ(x))
ϕ2(x) +
ϕ4(x)
. (1)
Here, r and u are the microscopic mass and coupling, respectively.
The NPRG builds a family of effective actions, Γκ[φ] (where φ(x) = 〈ϕ(x)〉J is the ex-
pectation value of the field in presence of an external source J(x)), in which the magnitude
of long wavelength fluctuations are controlled by an infrared regulator depending on a con-
tinuous parameter κ. One can write for Γκ[φ] an exact flow equation [5, 14, 21, 29]:
∂κΓκ[φ] =
(2π)d
∂κRκ(q
Γ(2)κ +Rκ
, (2)
where Γ
κ is the second functional derivative of Γκ with respect to φ(x), and Rκ denotes
a family of “cut-off functions” depending on κ: Rκ(q) behaves like κ
2 when q ≪ κ and it
vanishes rapidly when q ≫ κ [19, 20]. The effective action Γκ[φ] interpolates between the
classical action obtained for κ = Λ (where Λ−1 is the microscopic length scale), and the full
effective action obtained when κ → 0, i.e., when all fluctuations are taken into account (see
e.g. [7]).
By differentiating eq. (2) with respect to φ(x), and then letting the field be constant, one
gets the flow equation for the n-point function Γ
κ in a constant background field φ. For
example, for the 2-point function one gets:
κ (p;φ) =
(2π)d
∂κRk(q)
Gκ(q;φ)Γ
κ (p, q,−p− q;φ)
×Gκ(q + p;φ)Γ
κ (−p, p+ q,−q;φ)Gκ(q;φ)
Gκ(q;φ)Γ
κ (p,−p, q,−q;φ)Gκ(q;φ)
, (3)
where
G−1κ (q;φ) ≡ Γ
κ (q,−q;φ) +Rκ(q
2), (4)
and we used the definition
(2π)d δ(d)
Γ(n)κ (p1, . . . , pn;φ) =
ddx1 . . .
ddxne
δφ(x1) . . . δφ(xn)
φ(x)≡φ
The flow equation for a given n-point function involves the n+1 and n+2 point functions
(see, e.g., eq. (3)), so that the flow equations for all correlation functions constitute an infinite
hierarchy of coupled equations.
In [16], a general method to solve this infinite hierarchy was proposed. It exploits the
smoothness of the regularized n-point functions, and the fact that the loop momentum q in
the right hand side of the flow equations (such as eq. (2) or eq. (3)) is limited to q <∼ κ due
to the presence of ∂κRκ(q). The leading order of the method presented in [16] thus consists
in setting
Γ(n)κ (p1, p2, ..., pn−1 + q, pn − q) ∼ Γ
κ (p1, p2, ..., pn−1, pn) (6)
in the r.h.s. of the flow equations. After making this approximation, some momenta in some
of the n-point functions vanish, and their expressions can then be obtained as derivatives of
m-point functions (m < n) with respect to a constant background field.
Specifically, in the flow equation for the 2-point function, eq. (3), after setting q = 0 in
the vertices of the r.h.s., the 3- and 4-point functions will contain one and two vanishing
momenta, respectively. These can be related to the following derivatives of the 2-point
function:
Γ(3)κ (p,−p, 0;φ) =
κ (p,−p;φ)
, Γ(4)κ (p,−p, 0, 0;φ) =
κ (p,−p;φ)
. (7)
One then gets a closed equation for Γ
κ (p;φ):
2;φ) = J
d (p, κ;φ)
κ (p,−p;φ)
d (κ;φ)
κ (p,−p;φ)
, (8)
where
d (p; κ;φ) ≡
(2π)d
κ∂κRκ(q
2)Gκ(p+ q;φ)G
(n−1)
κ (q;φ), (9)
d (κ;φ) ≡
(2π)d
κ∂κRκ(q
2)Gnκ(q;φ). (10)
In fact, in order to preserve the relation
Γ(2)κ (p = 0;φ) =
, (11)
Vκ(φ) = Γκ[φ(x) ≡ φ]/Vol being the effective potential, it is better to make the approxima-
tion (6) (followed by (7)) in the flow equation for Σκ(p;φ) defined as
Σκ(p;φ) = Γ
κ (p;φ)− p
2 − Γ(2)κ (p = 0;φ). (12)
The 2-point function is then obtained from Γ(2)(p;φ) = ∂2Vκ(φ)/∂φ
2 + p2 +Σκ(p;φ), which
demands the simultaneous solution of the flow equations for Vκ(φ) and Σκ(p;φ).
As shown in [17], even if the complete solution of these equations is a priori complicated,
a simple, and still accurate, way of solving them consists in assuming in the various integrals
G−1κ (q;φ) ≃ Zκq
2 + ∂2Vκ(φ)/∂φ
2 +Rκ(q
2), (13)
where Zκ ≡ Zκ(φ = 0), with Zκ(φ) ≡ 1+∂Σκ(p;φ)/∂p
2|p=0. This approximation is consistent
with an improved version of the Local Potential Approximation (LPA, the first order of the
DE), which includes explicitly a field renormalization factor Zκ [7]. Doing so, the “p = 0”
sector decouples from the p 6= 0 one. Here, by “p = 0” we mean the sector describing vertices
and derivative of vertices at zero momenta, i.e., flow equations for Vκ and Zκ. Moreover, it
is useful to use the regulator [20]
2) = Zκ(κ
2 − q2) Θ(κ2 − q2), (14)
which allows the functions J
d (p; κ;φ) and I
d (κ;φ) to be calculated analytically. The
corresponding expressions can be found in [17]. In fact, all quantities are functions of
ρ ≡ φ2/2. The problem is then reduced to the solution of the three flow equations for Vκ(ρ)
and Zκ(ρ), for the p = 0 sector, and for Σκ(p; ρ), in the p 6= 0 one. As only the “effective
mass”
m2κ(ρ) ≡
∂2Vκ(φ)
∂Vκ(ρ)
∂2Vκ(ρ)
(and its derivatives with respect to ρ) enters in the p 6= 0 sector, it is more convenient to
work with the flow equation for m2κ(ρ) instead of that for Vκ(ρ) itself. The non-trivial fact is
that by differentiating twice the flow equation for Vκ(ρ) w.r.t φ, one gets a closed equation
for m2κ(ρ).
In order to make explicit the fixed point in the κ → 0 limit, it is necessary to work with
dimensionless variables:
µκ(ρ̃) ≡ Z
−2 m2κ(ρ) , χκ(ρ̃) ≡ Z
κ Zκ(ρ) , ρ̃ ≡ K
d Zκ κ
2−d ρ , (16)
which, in the critical case, have a finite limit when κ → 0. Above, Kd is a constant
conveniently taken as K−1d ≡ d 2
d−1 πd/2 Γ(d/2) (e.g., K3 = 1/(6π
2)). In the p 6= 0 sector,
the dimensionful variable p in the self-energy flow equation makes Σκ(p; ρ̃) reach a finite value
when κ → 0. As discussed in [17], the inclusion of the flow equation for the renormalization
factor Zκ(ρ̃) is essential in order to preserve the correct scaling behavior of Γ
(2)(p; ρ̃) in the
infrared limit. Doing so, in the critical case, the function Γ(2)(p; ρ̃)/(Zκκ
2) has to reach a
fixed point expression depending on ρ̃ and p/κ, when κ, p ≪ u and ρ̃ ∼ 1.
Putting all together, in d = 3, the three flow equations that have to be solved are
κ∂κµκ(ρ̃) = −(2 − ηκ)µκ(ρ̃) + (1 + ηκ)ρ̃µ
κ(ρ̃)−
µ′κ(ρ̃) + 2ρ̃µ
κ(ρ̃)
(1 + µκ(ρ̃))2
4ρ̃µ′κ(ρ̃)
(1 + µκ(ρ̃))3
κ∂κχκ(ρ̃) = ηκχκ(ρ̃) + (1 + ηκ)ρ̃χ
κ(ρ̃)− 2ρ̃
µ′2κ (ρ̃)
(1+µκ(ρ̃))4
1− ηκ
8ρ̃χ′κ(ρ̃)
µ′κ(ρ̃)
(1+µκ(ρ̃))3
χ′κ(ρ̃)+2ρ̃χ
κ(ρ̃)
(1+µκ(ρ̃))2
, (18)
together with
χ′κ(0)
χ′κ(0)/5 + (1 + µκ(0))
, κ∂κZκ = −ηκZκ , (19)
for the p = 0 sector, and
κ∂κΣκ(p, ρ̃) = (1 + ηκ)ρ̃Σ
κ(p, ρ̃) +
2ρ̃µ′2κ (ρ̃)κ
(1+µκ(ρ̃))2
fκ(p̃, ρ̃)−
2(1−ηκ/5)
(1+µκ(ρ̃))2
2ρ̃fκ(p̃,ρ̃)
(1+µκ(ρ̃))2
2µ′κ(ρ̃)Σ
κ(p, ρ̃) +
Σ′2κ (p,ρ̃)
(1−ηκ/5)
(1+µκ(ρ̃))2
(Σ′κ(p, ρ̃) + 2ρ̃Σ
κ(p, ρ̃)) (20)
for the p 6= 0 one. In these equations, the prime means ∂ρ̃ and we used the explicit expression
for I
3 = 2K3κ
5−2nZ1−nκ (1−ηκ/5)/(1 + µκ(ρ̃))
n. In eq. (20), we introduced the dimensionless
expression fκ defined as J
3 (p; κ; ρ) ≡ K3κ
−1Z−2κ /(1 + µκ(ρ̃))
2 × fκ(p̃; ρ̃), with p̃ ≡ p/κ.
In [17], this strategy is used to get the 2-point function of the scalar model at criticality
and zero external field (i.e., Σ(p = 0, ρ = 0) = 0), in d = 3. As recalled above, the function
thus obtained has the correct shape, either in the scaling, perturbative and intermediate
momenta regimes.
III. EXPANSION IN POWERS OF THE FIELD
In this section, we shall compare the solution obtained in [17] using the procedure de-
scribed above, with the solution of the same three flow equations expanded in powers of
ρ̃ and truncating up to a given order. Before doing so, let us first consider only the flow
equation for the potential or, equivalently, that for the effective mass, i.e., eq. (17), with
Zκ ≡ 1 (ηκ ≡ 0). This corresponds to the pure LPA sector and it is thus independent of
the scheme presented in [16]. In d = 3, its expansion in powers of the field has been widely
studied during the last ten years, using various regulators [22, 24]. Recently, another inter-
esting truncation scheme has also been considered in [30] showing much better convergence
properties. However, here we shall consider the simpler expansion in powers of the fields;
as shall be seen bellow this is the field expansion that can be compared to usual truncation
in the number of vertices. It has been shown [25] that, using the regulator we consider here
(see eq. (14)), this expansion seems to converge. This result follows when expanding both
around finite and zero external field, although faster in the first case. In [25] the convergence
in this situation has been discussed studying the critical exponent ν. In order to strengthen
this conclusion, as a first step in our study, we have analyzed the effect of the expansion on
the function µκ(ρ̃):
µκ(ρ̃) =
µ(n)κ ρ̃
n. (21)
More precisely, we shall gauge the impact of truncating this sum on the fixed point values of
the coefficients µ
κ , which are proportional to vertices at zero momenta and zero external
field. This study is motivated by the fact that these µ
κ shall appear in the Σκ(p; ρ̃) flow
equation, eq. (20), when the later shall be expanded around ρ̃ = 0. Results are shown
in Figure 1. The four plots present the fixed point value for the first 4 couplings, µ
n = 0, · · · , 3. For each coupling, we present the result which follows by solving the complete
LPA equation, eq. (17), together with the result obtained with the equation expanded in
powers of ρ̃. For example, when going only up to the first order (i.e., neglecting all µ
κ with
n ≥ 2), the corresponding equations for µ
κ and µ
κ , are:
κ = (ηκ − 2)µ
(1− ηκ/5)µ
(1 + µ
κ = (2ηκ − 1)µ
6(1− ηκ/5)(µ
(1 + µ
. (23)
which have to be solved simultaneously. (In fact, if solving just the LPA, ηκ = 0; nevertheless,
we have kept ηκ in eqs. (22)-(23) for a later use of these equations). When going to the second
order, eq. (23) acquires a new term and a new flow equation, that for µ
κ , appears; and so
on. According to Figure 1, an apparent convergence shows up. In all cases one observes
1 2 3 4 5 6 7 8 9
order
-0.18
-0.15
-0.12
-0.09
4 5 6 7 8 9
-0.189
-0.1875
-0.186
4 5 6 7 8 9
0.246
0.2475
0.249
1 2 3 4 5 6 7 8 9
order
0.135
0.225
6 7 8 9
0.093
0.0945
0.096
2 3 4 5 6 7 8 9
order
0.075
3 4 5 6 7 8 9
order
0.0280
0.0315
0.0350
0.0385
FIG. 1: First four dimensionless fixed point couplings at zero momenta and zero external field:
results obtained by truncating the flow equation, as a function of the order; the corresponding
value for the complete equation is represented by the dotted-line.
that: 1) there seems to be an oscillating convergence, 2) the value of µ(i) is found with about
1% error truncating at order i+ 3.
Let us now turn to the study of the flow equation for the 2-point function coming from
the scheme proposed in [16]. As the effective potential (or the effective mass), Γ
κ (p; ρ) can
also be expanded in powers of the external field:
Γ(2)κ (p; ρ) =
(2n)!
Γ(2n+2)κ (p,−p, 0, 0, · · · , 0; ρ)|ρ=0 ρ
n, (24)
because
Γ(m+2)κ (p,−p, 0, 0, · · · , 0; ρ) =
κ (p;φ)
3,5×10
3,6×10
2,0×10
2,5×10
FIG. 2: Comparison of the self-energy when expanding only the flow equations for the self-energy
Σκ(p; ρ̃) and its derivative Zκ(ρ̃) (strategy I): truncation is made at first (double dotted-dashed),
second (dotted-dashed), third (dashed) and fourth (dotted) order; the complete solution is given
by the straight line. In the figure, u = 5.9210−4Λ.
and we used that, at zero field, all odd vertex functions vanish. Equation (24) makes clear
the point stated above: once approximation (6) is performed, truncating the expansion
in powers of the external field is equivalent to neglecting high order vertices. Moreover,
eqs. (24) and (25) show that the procedure proposed in [16] indeed includes all vertices,
although approximately.
We have now all the ingredients to discuss the main goal of this paper: the analysis of
the expansion of the three flow equations for µκ(ρ̃), Zκ(ρ̃) and Σκ(p; ρ̃), eqs. (17-20), around
ρ̃ = 0. In doing so, one can write:
Σκ(p, ρ̃) =
Σ(n)κ (p) ρ̃
n. (26)
χκ(ρ̃) =
χ(n)κ ρ̃
n. (27)
together with eq. (21). For example, when going to the first order, the six equations that
have to be solved are:
κ (p) = −
(1− ηκ/5)Σ
κ (p)
(1 + µ
κ (p) = (1 + ηκ)Σ
κ (p) +
(1 + µ
fκ(p̃, 0)−
2(1− ηκ/5)
(1 + µ
2fκ(p̃, 0)
(1 + µ
2µ(1)κ Σ
κ (p) +
κ (p)
2(1− ηκ/5)µ
κ (p)
(1 + µ
, (29)
which correspond to the expansion of eq. (20),
κ = ηκχ
(1− ηκ/5)χ
(1 + µ
κ = (1 + 2ηκ)χ
(1 + µ
κ (1− ηκ/5)
(1 + µ
, (31)
which correspond to the expansion of eq. (18), together with eqs. (22) and (23).
In fact, it is possible to perform two kinds of expansion. First, in order to isolate the
effect of the field expansion just in the flow equations provided by the scheme presented in
[16], we shall expand only the flow equations for Σκ(p; ρ̃) and its derivative at zero momenta
Zκ(ρ̃), eqs. (20) and (18), solving exactly the differential flow equation for µκ(ρ̃), eq. (17).
For example, at first order, one should solve simultaneously eqs. (28-29), (30-31), and (17).
This is called “strategy I”. Second, to consider all the effects, we shall make the expansion
in the three flow equations. For example, at first order, one should solve simultaneously
eqs. (28-29), (30-31), and (22-23). We call this “strategy II”. Notice that, as explained in
[17], in order to get the correct scaling behavior it is mandatory to treat the equations for
Zκ(ρ̃) and Σκ(p; ρ̃) with the same approximations; it is then not possible to solve one of
them completely while expanding the other one.
Figure 2 presents the self-energy one gets truncating up to fourth order, following strategy
I; it is also shown the function obtained in [17] (from now on, the latter function, obtained
by solving the 3 differential equations, eqs. (17), (18) and (20), shall be called the “complete
solution”). Figure 3 presents the same results when following strategy II. These Figures
show that, in both strategies of expansion, by truncating at first order one already gets a
function with the correct shape in all momenta regimes.
3,5×10
3,6×10
1,9×10
2,0×10
FIG. 3: Comparison of the self-energy when expanding the three flow equations (strategy II):
truncation is made at first (double dotted-dashed), second (dotted-dashed), third (dashed) and
fourth (dotted) order; the complete solution is given by the straight line. In the figure, u =
5.9210−4Λ.
In order to make a quantitative evaluation of the approximate solution obtained doing
the expansion, we have calculated different numbers describing the physical properties of
the self-energy. First, as can be seen in both figures above, all solutions have, in the infrared
(p ≪ u), the potential behavior characterizing the scaling regime: Σ(p) + p2 ∼ p2−η, where
η is the anomalous dimension. We have checked that, at each order and in both strategies,
the resulting self-energy does have scaling, and we extracted the corresponding value of
η. In fact, this can be done in two different ways: either using the κ–dependence of Zκ
(η = − limκ→0 κ∂κ logZκ) or the p–dependence of Σ(p) stated above. We checked that those
two values always coincide, within numerical uncertainties. Figure 4 presents the relative
error for η, at each order, when compared with the value following from the complete
solution. One observes: 1) in both strategies of expansion there is an apparent convergence,
which is oscillatory; 2) the solution from strategy I reaches faster the correct result; 3) when
following strategy I, already with a second order truncation the error is about 3% and it
drops to less that 1% at the third order. Nevertheless, due to the mixed characteristic of
3 4 5 6 7 8
1 2 3 4 5 6 7 8
order
FIG. 4: Relative error (measured in percent) for the anomalous dimension, with respect to the
value coming from the complete solution, as a function of the truncation order. Full line: expanding
only the flow equations for the self-energy Σκ(p; ρ) and its derivative Zκ(ρ) (strategy I); dashed
line: expanding all flow equations (strategy II).
strategy I, when using this strategy at high order numerical problems arise: indeed, this
task demands the numerical evaluation of high order derivatives of µκ(ρ̃), to be used in the
various flow equations obtained when expanding that of Σκ(p; ρ̃). If high precision in the
result is required, strategy II is then numerically preferable.
It is important to observe here that the procedure which can be compared to the usual
truncation including a finite number of vertices is strategy II. Moreover, the inclusion of
high order vertices without performing any other approximation is difficult; for example,
the complete inclusion of the 6-point vertex has never been done. Accordingly, as can be
seen in fig. 4, when including only up to the 4-point vertex, as it usually done, the error in
η can be as large as 60%.
A second number to assess the quality of the approximate solution is the critical exponent
ν. In order to calculate it, we extract the renormalized dimensionful mass from m2R =
κ2µκ(ρ̃ = 0) and we relate it to the microscopic one by
m2R(κ = 0) ∝ (m
R(κ = Λ)−m
R,crit(κ = Λ))
2ν , (32)
where mR,crit is the critical renormalized mass. With the complete solution one gets ν =
3 4 5 6 7 8
1 2 3 4 5 6 7 8
order
FIG. 5: Relative error (measured in percent) for the critical exponent ν, with respect to the value
coming from the complete solution, as a function of the truncation order. Full line: expanding
only the flow equations for the self-energy Σκ(p; ρ) and its derivative Zκ(ρ) (strategy I); dashed
line: expanding all flow equations (strategy II).
0.647, to be compared with the best accepted value [31]: ν = 0.6304 ± 0.0013. Figure 5
presents the relative error of the value of ν extracted from the expansion. Once again, one
observes that the convergence is much faster when following strategy I, i.e., when considering
the effect of the expansion only on the self-energy equation.
The large momenta regime (p ≫ u) of the self-energy can be calculated using perturbation
theory, yielding the well known logarithmic shape: Σ(p) ∼ A log(p/B), where A and B
are constants. For the complete solution presented in [17] one can prove analytically that
A = u2/9π4, which is only 8% away from the exact result A = u2/96π2 (please observe that
this coefficient is given by a 2-loop diagram for the self energy which is only approximatively
included at this order). The proof of this analytical result remains valid when performing
the field expansion, at any order and within both strategies. We have checked that our
numerical solution always has the correct shape, with A = u2/9π4. This is due to the fact
that already the first order in the expansion of Σκ(p; ρ̃) around ρ̃ = 0 contains the same
2-loop diagrams contributing to the complete solution.
In order to study the quality of the self-energy in the intermediate momenta regime, we
2 3 4
1 2 3 4
order
FIG. 6: Relative error (measured in percent) for ∆〈φ2〉, with respect to the value coming from
the complete solution, as a function of the truncation order. Full line: expanding only the flow
equations for the self-energy Σκ(p; ρ) and its derivative Zκ(ρ) (strategy I); dashed line: expanding
all flow equations (strategy II).
have calculated a quantity which is very sensitive to this cross-over region:
∆〈φ2〉 =
(2π)3
p2 + Σ(p)
. (33)
(the integrand is non zero only in the region 10−3 <∼ p/u <∼ 10, see for example [15]). This
quantity received recently much attention because it has been shown [32] that for a scalar
model with O(N) symmetry, in d = 3 and N = 2, it determines the shift of the critical
temperature of the weakly repulsive Bose gas. It has then been widely evaluated by many
methods, for different values of N , in particular, for N = 1. With the numerical solution
found in [17], one gets a number almost within the error bars of the best accepted results
available in the literature, using lattice and 7 loops resumed perturbative calculations. Please
observe that these errors are as large as 10%, which is an indication that this quantity is
particularly difficult to calculate. In Figure 6 we plot the relative error in ∆〈φ2〉, at each
order of the expansion, when compared with the complete solution result found in [17]. One
can appreciate that 1) for both expansion strategies there is an apparent convergence, which
is also oscillatory; 2) in both strategies, already with a second order truncation the error is
about 1%.
IV. SUMMARY AND CONCLUSIONS
In this article, the inclusion of a finite number of vertices in NPRG flow equations is
analyzed. An unsolved difficulty of this usual strategy (originally proposed by Weinberg)
is the estimation of the error introduced at a given step. Moreover, without performing
further approximations, it is very hard to reach high orders of the procedure. The study of
its convergence is thus a difficult task. In the present work we analyse this problem using a
different approximation scheme [16]: instead of considering a finite number of vertices, this
procedure includes all of them, although approximately. Within this context, it is possible
to estimate the error of the Weinberg approximation, order by order. To do so one can
perform, on top of the approximation presented in [16], the usual truncation in the number
of vertices. The analysis has been done in the particular case of the 2-point function of
the scalar field theory in d = 3 at criticality. It has been shown [17] that, at least in this
case, the procedure proposed in [16] yields very precise results. Another interesting outcome
of the present work follows from the fact that, within the approximation [16], truncation
in the number of vertices is equivalent to an expansion in powers of a constant external
field. The latter is usually employed in the DE context in order to deal with complicated
situations. The analysis of the present paper generalizes this expansion procedure when non
zero external momentum are involved.
The calculation of the 2-point function demands the study of both the p = 0 and the
p 6= 0 sectors. While the first one is given by the well studied DE flow equations, the latter
follows from the approximation scheme introduced in [16] to calculate the flow of Σκ(p; ρ).
We used two different strategies to perform the field expansion, both of them around zero
external field: either expanding only the flow equation for the self-energy (and its derivative)
(strategy I), or both the effective potential and the self-energy (and its derivative) flow
equations (strategy II). We have studied the convergence of various quantities measuring
physical properties of the self-energy in all momenta regimes: the critical exponents η and
ν of the infrared regime, the coefficient of the ultraviolet logarithm, and ∆〈φ2〉 which is
dominated by the crossover momenta regime.
As stated in section III, the strategy that can be compared to the usual truncation which
includes a finite number of vertices is strategy II. For example, including completely the
4-point vertex as it is usually done (i.e., in the language of field expansion, going only up to
the first order of the expansion), when describing the deep infrared regime one could make
errors as big as 60% in the critical exponent η (see figure 4). If one wants results with less
that 5% error for this quantity, the inclusion of up to 8-point vertices (i.e., going up to third
order) is necessary.
However, when going to higher orders in the field expansion, the series for all considered
quantities seem to converge rapidly, within both strategies. The convergence is faster when
using strategy I, i.e., when making the expansion only for the approximate flow equation
resulting from the method presented in [16]. For example, using strategy I, a third order
truncation introduces a relative error smaller that 1% for all studied quantities; while using
strategy II, in order to reach the same error one needs 6th order for η, 4th order for ν and
2nd order for ∆〈φ2〉. Nevertheless, due to numerical difficulties, if trying to go to high order
expansions, it is preferable to use strategy II, i.e., expanding also the effective potential flow
equation.
It is difficult to assess the generality of these results on the use of field expansion on top of
the strategy proposed in [16]. Of course, there are situations where expanding in an external
field is not a priori convenient. One can mention as a first example, situations where there
is a physical external field (as in a broken phase or when an external source for the field is
considered). A second example is two-dimensional systems where even in the DE, the field
expansion does not seem to converge. Nevertheless, the short study presented in the present
paper allows to consider field expansion on top of the approximation proposed in [16] as a
possible strategy to deal with many involved models, as for example QCD.
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http://arxiv.org/abs/hep-th/0611306
	Introduction
	General considerations
	Expansion in powers of the field
	Summary and Conclusions
	References
