The Wilson exact renormalization group equation and the anomalous dimension parameter
The non-linear way the anomalous dimension parameter has been introduced in the historic first version of the exact renormalization group equation is compared to current practice. A simple expression for the exactly marginal redundant operator proceeds from this non-linearity, whereas in the linear...
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irk-123456789-1208102017-06-14T03:05:47Z The Wilson exact renormalization group equation and the anomalous dimension parameter Bervillier, C. The non-linear way the anomalous dimension parameter has been introduced in the historic first version of the exact renormalization group equation is compared to current practice. A simple expression for the exactly marginal redundant operator proceeds from this non-linearity, whereas in the linear case, first order differential equations must be solved to get it. The role of this operator in the construction of the flow equation is highlighted. Параметр аномальної вимiрностi, введений в нелiнiйний спосiб у першiй iсторичнiй версiї рiвняння точної ренормалiзацiйної групи, порiвнюється з сучасною методикою. Простий вираз для точно граничного (маргiнального) надлишкового оператора слiдує з цiєї нелiнiйностi, тодi як для того, щоб отримати цей результат у лiнiйному випадку необхiдно розв’язати диференцiальнi рiвняння першого порядку. Висвiтлено роль цього оператора в побудовi рiвняння потоку. 2013 Article The Wilson exact renormalization group equation and the anomalous dimension parameter / C. Bervillier // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23003:1-10. — Бібліогр.: 41 назв. — англ. 1607-324X PACS: 05.10.Cc, 11.10.Gh, 64.60.ae DOI:10.5488/CMP.16.23003 arXiv:1304.4131 http://dspace.nbuv.gov.ua/handle/123456789/120810 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The non-linear way the anomalous dimension parameter has been introduced in the historic first version of the exact renormalization group equation is compared to current practice. A simple expression for the exactly marginal redundant operator proceeds from this non-linearity, whereas in the linear case, first order differential equations must be solved to get it. The role of this operator in the construction of the flow equation is highlighted. |
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Bervillier, C. The Wilson exact renormalization group equation and the anomalous dimension parameter Condensed Matter Physics |
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Bervillier, C. |
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Bervillier, C. |
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The Wilson exact renormalization group equation and the anomalous dimension parameter |
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The Wilson exact renormalization group equation and the anomalous dimension parameter |
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The Wilson exact renormalization group equation and the anomalous dimension parameter |
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The Wilson exact renormalization group equation and the anomalous dimension parameter |
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The Wilson exact renormalization group equation and the anomalous dimension parameter |
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wilson exact renormalization group equation and the anomalous dimension parameter |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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The Wilson exact renormalization group equation and the anomalous dimension parameter / C. Bervillier // Condensed Matter Physics. — 2013. — Т. 16, № 2. — С. 23003:1-10. — Бібліогр.: 41 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bervillierc thewilsonexactrenormalizationgroupequationandtheanomalousdimensionparameter AT bervillierc wilsonexactrenormalizationgroupequationandtheanomalousdimensionparameter |
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2025-07-08T18:36:52Z |
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2025-07-08T18:36:52Z |
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1837104956275949568 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 2, 23003: 1–10
DOI: 10.5488/CMP.16.23003
http://www.icmp.lviv.ua/journal
The Wilson exact renormalization group equation
and the anomalous dimension parameter
C. Bervillier∗
Laboratoire de Mathématiques et Physique Théorique, UMR 7350 (CNRS), Fédération Denis Poisson,
Université François Rabelais, Parc de Grandmont, 37200 Tours, France
Received December 14, 2012, in final form March 8, 2013
The non-linear way the anomalous dimension parameter has been introduced in the historic first version of
the exact renormalization group equation is compared to current practice. A simple expression for the exactly
marginal redundant operator proceeds from this non-linearity, whereas in the linear case, first order differ-
ential equations must be solved to get it. The role of this operator in the construction of the flow equation is
highlighted.
Key words: exact renormalization group equation, anomalous dimension
PACS: 05.10.Cc, 11.10.Gh, 64.60.ae
1. Introduction
The Wilson renormalization group (RG) ideas [1–3] have formalized and clarified the notions of scal-
ing and of universality attached to critical behavior. As several modern subjects in physics, critical phe-
nomena cannot be studied by pure perturbative methods. The development of nonperturbative methods
is thus very important. As is well demonstrated by the work of M. Kozlovskii (see, e.g., [4–8]), the collec-
tive variables method [9], close to the early ideas of Kadanoff [10], is one of the RG-like nonperturbative
methods allowing one to investigate the critical point, starting with a microscopic Hamiltonian — thus
allowing one to calculate nonuniversal quantities such as the critical temperature [4]. The subject of the
present article is another expression of the nonperturbative RG framework, called the exact RG equation
(ERGE) — see in [1] section 11, equations (11.14, 11.15), and for reviews see, e.g., [11–19]. Though one
could also calculate nonuniversal quantities with an ERGE (see, e.g., [20–22]), our purpose is presently
limited to the close vicinity of a RG fixed point with a view to discuss the particular issue of the way one
may account for the anomalous dimension of the field (the critical exponent η) within an ERGE.
Among an infinite variety of equivalent ERGE, only those based on an effective cutoff function asso-
ciated with a bilinear kinetic term are currently used [23]. Despite this limited number of usual variants
(a priori close to each other), they show differences sufficient to make it sometimes difficult to clearly
display (or even understand) their relationships. Of course, several authors have addressed this issue in
general [24–30] but, in contrast to [31–34], the anomalous dimension parameter η was not included in
their considerations. For example, it is known that theWilson historic first version [1] is made equivalent
to the Polchinski version [35], provided there is a choice of an exponential cutoff function and after a spe-
cific field redefinition [28, 31], but the parameter η was not explicitly considered in this relation except
in [31].
Recently Osborn and Twigg [33] and Rosten [32, 34] have independently established the relation be-
tween a version of the Polchinski ERGE — “modified” to include the anomalous dimension [36] — and
a “fixed-point equation” for the effective average action [28]. This relation appears to be rather compli-
cated: it “involves solving some first order differential equations” [33]. O’Dwyer and Osborn [37] had
∗E-mail: claude.bervillier@lmpt.univ-tours.fr
© C. Bervillier, 2013 23003-1
http://dx.doi.org/10.5488/CMP.16.23003
http://www.icmp.lviv.ua/journal
C. Bervillier
previously encountered similar first order differential equations in the process of constructing the (ex-
actly) marginal, redundant “operator”1 O (EMRO) associated with the change of normalization of the field
by a constant factor but they did not discuss the effect of the way η had been accounted for in the ERGE
on this complexity. Yet, it is a well known fact that the EMRO of the Wilson version [1] takes on a simple
form— e.g., see equation (24) of [12] or equation (2.35) of [38]. The object of the present article is to com-
pare the main ways of introducing η in an ERGE encountered in the literature. We show, in particular,
that the simplicity of the form of the EMRO is maintained in the Wilson version of the ERGE extended to
an arbitrary cutoff function [31]. The role of the EMRO in the construction of an ERGE is also highlighted.
A larger discussion is left to an upcoming detailed publication [39].
2. The RG steps
2.1. Reminder
Basically, a RG transformation of a cutoffed action S[φ,ℓ] of a scalar field φ involves three steps.
Having chosen a fixed arbitrary momentum scale of reference Λ0, and having defined the RG-scale
parameter ℓ=Λ/Λ0 É 1, these three steps are:
Step 1: an integration of the high momentum components of the field generating an effective action
with a reduced cutoff Λ′
= (1−dℓ/ℓ)Λ.
Step 2: a rescaling of the momenta back to the initial value of the cutoff:
∣
∣q
∣
∣→ (1−dℓ/ℓ)
∣
∣q
∣
∣. This step
is accounted for by a simple dimensional analysis such as defining a dimensionless momentum
as q̃ = q/Λ and a dimensionless field φ̃q̃ =Λ
−d̄φφq [see also equation (7)].
Step 3: a renormalization of the field φ̃q̃ −→ ζ(ℓ−dℓ) φ̃q̃ , with ζ(ℓ) = ℓ̟ and ̟ related to η, e.g., see
equation (19). This step is required to keep constant (i.e., independent of ℓ) one term of S in
order to set up the system of scale of reference.
Having defined the RG-time t as:
t =− lnℓ=− ln
(
Λ
Λ0
)
, (1)
the combined effect of the above three steps on the action S may be expressed as:
Ṡ[φ̃,e−t ]=Gtra (S)+Gdil
(
S,dφ
)
+Gren (S) , (2)
in which:
Ṡ[φ̃,e−t ] ≡
d
dt
S[φ̃,e−t ]
∣
∣
∣
∣
φ̃
, (3)
Gdil
(
S,dφ
)
=
∫
p̃
(
p̃ ·
∂φ̃p̃
∂p̃
− d̄φφ̃p̃
)
δS
δφ̃p̃
. (4)
The expression of Gtra (S) in (2), depends on the way the above step 1 is realized. In any case, for a
complete action S, it corresponds to a (quasi local) field redefinition which leaves the partition function
unchanged and, consequently, must have the form [2, 3]:
Gtra (S) = Ξ [S,Ψ] , (5)
Ξ [S,Ψ] =
∫
q̃
(
Ψq̃
δS
δφ̃q̃
−
δΨq̃
δφ̃q̃
)
, (6)
in which Ψq̃ is some functional of φ̃.
1 Actually O is not an operator but represents a direction in the space of actions {S}.
23003-2
Wilson’s ERGE and anomalous dimension
As for the expression of Gren (S), it is the object of the present article to discuss its various forms
encountered in the literature. It is related to the expression of d̄φ in (4); if dφ is the dimension of φ(x),
d̄φ is that of its Fourier transformed φq , if D is the spatial dimension, they are related through:
d̄φ = dφ−D . (7)
In the following we use two kinds of dφ: a “classical” family d (c)
φ
:
d (c)
φ
=
D
2
−n0 , (8)
in which2 n0 is related to the behavior of the cutoff function at small momenta [see equations (16, 17)
where n0 = 0 or n0 = 1] , and the fixed-point-value dimension:
d (a)
φ
=
D −2+η
2
, (9)
in which η is the usual critical exponent.
2.2. Main ways of introducing η
Though it gave the name to the theory, step 3 is currently skipped in the modern constructions of an
ERGE. A perfect example of this fact is the original Polchinski ERGE [35] a version of which — including
η as proposed in [36] — has been called the “modified” Polchinski version [32, 34] or the Polchinski
version “extended to include the parameter η” [33]. In modern approaches, the renormalization step
3 is currently avoided and η is merely included in the rescaling step 2 via some ad hoc “anomalous”
dimensional analysis [28, 40]. There is nothing wrong with such a procedure: the large freedom that is
offered in the construction of an ERGE makes it well acceptable. It remains no less true that it is fair to
ask whether or not there are important consequences of doing this or that.
Essentially, two ways of introducing η are encountered in the literature:
1. via the explicit renormalisation step 3 associated with Gren (S). In that case the dimension dφ in
Gdil
(
S, dφ
)
is classical, given by (8). This is the original and most general procedure but it is not
currently used.
2. via some “anomalous” dimensional analysis as proposed, e.g., by Morris [28]. In that case Gren (S)≡
0 and dφ is given by (9). In fact, this procedure, or some equivalent variant, is generally used in the
current versions of the ERGE (see also, e.g., Berges et al [13], Ball et al [36] etc. . . . ). It is a correct
procedure in case of the proximity of a fixed point.
In addition, one may observe that the effective contribution of η within the ERGE is either purely
linear or not (w.r.t. S). For example, the procedure of Ball et al [36] is linear whereas the Wilson [1],
Morris [28] andWetterich [13] procedures are non-linear. These differences, although well allowed, have
some consequences in the relations between the main versions of the ERGE. In the following sections we
examine the non-linear Wilson procedure in greater detail than previously done in [31] and compare it
with the linear and non-linear procedures of, respectively, Ball et al [36] and Morris [28].
3. The extended Wilson ERGE
3.1. Presentation
Let us consider the extended Wilson ERGE (i.e., the flow equation of S with an arbitrary cutoff func-
tion), as obtained in [31]. For convenience, we adopt the notations of Osborn and Twigg [33] and write
2The value of d (c)
φ
depends onwhich term is chosen to set the standard of the scale of reference. Usually one refers to the kinetic
term and d
(c)
φ
=
D−2
2 (n0 = 1). Wilson has chosen a pure quadratic term and, for him d
(c)
φ
=
D
2 (n0 = 0).
23003-3
C. Bervillier
the flow equation under the form of (2) with (notice the change in Gdil):
Gtra (S) =
∫
q̃
[
G
(
q̃2
)
(
δ2S
δφ̃q̃δφ̃−q̃
−
δS
δφ̃q̃
δS
δφ̃−q̃
)
+H
(
q̃2
)
φ̃q̃
δS
δφ̃q̃
]
, (10)
Gren (S) = ̟O(S, P̃ ) , (11)
Gdil
(
S,dφ
)
= Gdil(S,d (c)
φ
) , (12)
where:
O(S, P̃ ) =
∫
q̃
[
P̃
(
q̃2
)
(
δ2S
δφ̃q̃δφ̃−q̃
−
δS
δφ̃q̃
δS
δφ̃−q̃
)
+ φ̃q̃
δS
δφ̃q̃
]
, (13)
G
(
q̃2
)
= −q̃2P̃
(
q̃2
) K ′
(
q̃2
)
K
(
q̃2
) , (14)
H
(
q̃2
)
= −2q̃2 K ′
(
q̃2
)
K
(
q̃2
) . (15)
Compared to [31], the cutoff function P
(
q2,Λ
)
has been given the (regular) dimension dim[P ] =−2n0
so that:
P (q2,ℓΛ0) = Λ
−2n0
0 ℓ2̟−2n0 P̃
(
q̃2
)
, (16)
P̃
(
q̃2
)
=
K
(
q̃2
)
(
q̃2
)n0
, (17)
where K is dimensionless — this is to permit comparison with the current uses where n0 = 1 whereas
n0 = 0 in [31].
It is useful to recall the following points:
• the general form of an ERGE involves an additive field independent term which will be sytemati-
cally neglected in the following;
• the establishment of the extended Wilson ERGE is based on the relation of the complete action S to
a partial action Sint by extracting a quadratic form involving the arbitrary cutoff function P :
S
[
φ
]
=
1
2
∫
q
φq P−1(q2,ℓΛ0)φ−q +Sint
[
φ
]
; (18)
• in its original version [35] Polchinski expresses the flow of Sint under a change of ℓ— instead of S
for the Wilson ERGE;
• the factorized ℓ2̟-term in (16) was not part of Polchinski’s assumptions. This kind of ℓ-dependency
in front of the cutoff function is a convenient artefact to introduce η non-linearly in the ERGE. This
is not unusual since Morris [28] already used it when he gave the cutoff function an anomalous
dimension (see section 4.2). We shall show that the recourse to the EMRO enables us to get rid of
this artefact;
• in order to specify the nature of the renormalization step, one must make reference to the right
power law behavior at large distances of the critical two point correlation function and this implies
that [31]:
̟= 1−n0 −
η
2
. (19)
[In the case of anomalous dimensional analysis, this step is not required since the field has a priori
been given the “right” fixed point dimension d (a)
φ
defined by (9)];
23003-4
Wilson’s ERGE and anomalous dimension
• it is easy to verify that for n0 = 0 and ̟ given by (19), (10)–(17) gives the historic first version of the
ERGE [1] with the choice P̃ (q̃2) = e−2q̃2
, and the redefinition of the field [31]:
φ̃q̃ →
√
P̃ (q̃2) φ̃q̃ ; (20)
• the writing of equations (10)–(15) suggests that O(S, P̃ ) plays the role of the redundant “operator”
O (S,1) which is associated with an infinitesimal renormalization of the field by a constant factor
[38, 41]. It is, actually, via O (S,1) that η was introduced in the historic first version3.
Instead of assuming the presence of the factor ℓ2̟ in (16), we could as well have implemented step 3
using O (S,1) in place of O(S, P̃ ) in (11), but then this would have destroyed the currently admitted equiv-
alence [23, 28] [under the change (20)] between Polchinski’s and Wilson’s ERGE. This is merely because,
given a fixed point S∗ [characterized by Ṡ∗
= 0], O(S∗,1) is not an EMRO for the extended Wilson ERGE
whereas O(S∗, P̃ ) is. It is the object of the next section to discuss this issue.
3.2. The exactly marginal redundant “operator”
In this section, we apply the procedure described in appendix D of [37] to the complete action S (see
also [19]) and we demonstrate that O(S, P̃ ) defined by (13) with S = S∗, corresponds to an EMRO for the
ERGE given by (2), (10)–(12), and that it may be used to implement the RG step 3.
It is easy to see that O(S, P̃ ) is a redundant “operator” since it may be written under the form of the
r.h.s. of (6) with:
Ψ
(P̃)
q̃
= φ̃q̃ − P̃
(
q̃2
) δS
δφ̃−q̃
. (21)
We then proceed in two stages, showing successively that:
Stage 1: whatever the functions G
(
q̃2
)
and H
(
q̃2
)
:
(a) there exists a function B
(
q2
)
— solution of a first order differential equation— such that
O (S∗,B) is exactly marginal for the ERGE without renormalization (i.e., with Gren (S) ≡ 0
and an arbitrary dimension parameter dφ),
(b) O (S∗,B) is again exactly marginal for the ERGE translated by αO (S,B) [i.e., with
Gren (S)=αO (S,B)] with α an arbitrary constant.
Stage 2: B
(
q2
)
identifies with P̃
(
q̃2
)
in the case of (14), (15).
Let us consider the ERGE linearized about a fixed point S∗ and its associated eigenvalue equation for
some eigenfunctional Θi [φ̃]:
D̂Θi =λiΘi . (22)
For the ERGE given by (2), (10)–(12), without the renormalisation part Gren (S) and for arbitrary dφ,
3Via a scale dependent factor ̟(t ) in front of O (S,1) instead of a constant ̟ as in (11), ̟(t ) coincides with ̟ in the vicinity of a
fixed point.
23003-5
C. Bervillier
we have:
D̂ = D̂1 + D̂2 + D̂3 + D̂4 ,
D̂1 =
∫
p̃
[
p̃ ·
∂φ̃p̃
∂p̃
− d̄φ φ̃p̃
]
δ
δφ̃p̃
,
D̂2 =
∫
p̃
G
(
p̃2
) δ2
δφ̃p̃δφ̃−p̃
,
D̂3 =
∫
p̃
H
(
p̃2
)
φ̃p̃
δ
δφ̃p̃
,
D̂4 = −2
∫
p̃
G
(
p̃2
) δS∗
δφ̃p̃
δ
δφ̃−p̃
.
Considering a general redundant “operator” Ξ [S,Ψ] as defined by (6), it is not very complicated to
verify that the general property [2, 3] of the redundant “operators” to form a closed subspace under the
flow in the vicinity of a fixed-point takes on the form:
D̂Ξ[S∗,Ψ]=Ξ[S∗,D̂tΨ] , (23)
with:
D̂tΨq̃ [φ̃]=
(
D̂+ d̄φ−G
(
q̃2
)
− q̃ ·
∂
∂q̃
)
Ψq̃ [φ̃] . (24)
Following the procedure of [37], we try to build up the EMRO [i.e., a solution Θ0 of (22) with λ0 = 0]
from equations (23), (24) with a function Ψ
(B )
q̃
[φ̃] similar to (21) and find that B (x) must be the solution
of the following first order differential equation (with initial value):
G (x)−̟g B (x)−H (x)B (x)− xB ′ (x) = 0, (25)
B (0) =
G (0)
̟g
, (26)
in which:
̟g =−
dφ+ d̄φ
2
. (27)
Assuming an explicit solution of B (x) [see equations (31), (32)] this closes stage 1a.
The quantity O (S,B) , similar to (13), is thus redundant and exactly marginal at the fixed point S∗. Let
us consider a new ERGE with:
Gren (S) =αO (S,B) . (28)
We easily see that this modification amounts to merely having performed the following translations
in the flow equation:
G (x) −→ G (x)+αB , (29)
̟g −→ ̟g +α , (30)
which keep equation (25) unchanged. Consequently, stage 1b is also verified. It is clear that this would
not be the case using O (S,1) in (28).
The solution of the differential equations (25), (26) is:
B (x) =
(
1
x
)̟g 1
C0 (x)
x
∫
0
u̟g −1G(u)C0 (u)du , if ̟g Ê 0 , (31)
C0 (x) = exp
x
∫
0
H(u)
u
du
, if H(0) = 0. (32)
23003-6
Wilson’s ERGE and anomalous dimension
Using an integration by parts, one may verify that B (x), as given by (31), satisfies the initial condition
(26), provided that ̟g Ê 0.
In the case where the functions G and H are defined by (14), (15), equation (31) simplifies to give:
B (x) ≡ P̃ (x) , (33)
so that with ̟g =̟, as given by (19), the quantity ̟O(S, P̃ ) corresponds to the effective non-linear real-
ization of the field renormalization adapted to the ERGE under consideration (QED).
In order to discuss the linear implementation of η, it is useful to first consider the relation back to the
original Polchinski version of the ERGE.
4. From Wilson to Polchinski
In this section we scrupulously look at the relation between the Wilson and the Polchinski ERGE
considering successively the linear (Ball et al [36]) and non-linear (Wilson [1, 31], Morris [28]) ways of
introducing η.
The Polchinski equation for Sint [35] is originally limited to the implementation of the RG step 1 via
the variation of Sint under the change of Λ introduced by a bilinear term in S [as shown in (18)]. It reads:
ℓ
∂
∂ℓ
Sint
∣
∣
∣
∣
φ
=−
1
2
∫
q
ℓ
∂
∂ℓ
P
(
q2,ℓΛ0
)
∣
∣
∣
∣
q
[
δ2Sint
δφqδφ−q
−
δSint
δφq
δSint
δφ−q
]
. (34)
From this incomplete expression of the RG flow, the shift to S is easily implemented via the following
functional derivatives of (18):
δSint
δφq
=
δS
δφq
−P−1φ−q , (35)
δ2Sint
δφqδφ−q
=
δ2S
δφqδφ−q
−P−1 , (36)
from which we get (up to an additive field-independent term):
ℓ
∂
∂ℓ
S
[
φ
]
∣
∣
∣
∣
φ
=−
1
2
∫
q
ℓ
∂
∂ℓ
P
(
q2,ℓΛ0
)
∣
∣
∣
∣
q
[
δ2S
δφqδφ−q
−
δS
δφq
δS
δφ−q
+2P−1
(
q2,ℓΛ0
)
φq
δS
δφq
]
. (37)
Then, depending on the procedure chosen for introducing η, the implementation of the RG steps 2
and 3 may be easier to get a hand on according to whether one considers (34) or (37). In the linear case
of Ball et al [36] this does not matter, however.
4.1. Linear introduction of η
The cumulative account from Λ0 to Λ = ℓΛ0 which has been conveniently included as an explicit
ℓ-factor within the cutoff function [as displayed in (16)] is not necessary to the construction of an ERGE .
Actually, only the reduction of the degrees of freedom in the infinitesimal range
[
Λ,
(
1− dℓ
ℓ
)
Λ
]
is obliga-
tory. In other terms: the renormalization step 3 may be implemented linearly only. This is the procedure
adopted by Ball et al [36] when they introduced η. However, there is a supplementary freedom.
The role of the renormalization step 3 is to compensate the modification of one term of S induced by
the implementation of the RG step 1. Instead of explicitly renormalizing the field, one may as well view
this modification as an integral part of φ so that it may be absorbed in the rescaling step 2 by assuming
that φ has the (anomalous) dimension (9)— notice that this implies n0 = 1. Ball et al [36] have chosen that
possibility. For both S and Sint, RG steps 2 and 3 are then implemented by a mere dimensional analysis
that readily provides the final flow equations. In terms of t defined in (1) we get:
23003-7
C. Bervillier
• the flow equation for S under the form (2) with Gren (S) = 0, Gdil
(
S,dφ
)
= Gdil(S,d (a)
φ
) and Gtra (S)
given by (10), (14), (15);
• the flow equation for Sint as:
Ṡint = Gtra (Sint)+Gdil(Sint,d (a)
φ
) , (38)
Gtra (Sint) =
∫
q̃
[
G
(
q̃2
)
(
δ2Sint
δφ̃q̃δφ̃−q̃
−
δSint
δφ̃q̃
δSint
δφ̃−q̃
)]
(39)
with G
(
q̃2
)
given by (14).
However, since the anomalous dimension attributed to φ is not compensated by an anomalous di-
mension of the cutoff function in the bilinear term of S, the relation between the two flow equations is
altered and we have:
Ṡ = Ṡint +̟
∫
q̃
φ̃q̃ P̃−1
(
q̃2
)
φ̃−q̃ , (40)
with ̟ given by (19) in which n0 = 1. This implies that the two flow equations do not have equivalent
fixed points.
Equation (40) has induced the notion of “modified” Polchinski flow equation [32–34] which refers
to the flow equation for S (expressed in terms of Sint) rather than to equations (38), (39) as it would be
normally. It is worth to underline, however, that the latter flow equation for Sint is perfectly valid and
may be studied for its own sake.
Another consequence of the linear introduction of η is the complicated expression of the EMRO. In-
deed, the linear introduction of η is not compatible with the translation (29), (30) and the solution of (25),
(26) must actually be considered with ̟g = 1−η/2 so that, for the choice (14), (15) and with P̃ (q̃2) given
by (16) in which ̟ is formally set equal to 0, we get, provided η< 2:
B (x) =
(
1
x
)1−η/2
[K (x)]2
x
∫
0
u−η/2 K ′ (u)
[K (u)]2
du , (41)
which cannot be reduced to the simple form of (33), for arbitrary K . The result (41) is equivalent to that
obtained in appendix D of [37] for the EMRO constructed in terms of Sint.
4.2. Non-linear introduction of η
In this section we compare two non-linear ways of introducing η: the Morris version [28] which is
based on an “anomalous” dimensional analysis and the Wilson version extended to an arbitrary cutoff
described in section 3. We show that the two versions are formally very close to each other but yield
different RG flow equations.
In [28], Morris has a priori given the field φ the dimension d (a)
φ
[given by (9)] with an anomalous part
that is compensated by a cutoff function anomalously dimensioned. Applied to our present matter, the
direct consequence (the easiest to grasp) of the Morris procedure is the modification of (40) into:
Ṡ = Ṡint , (42)
that expresses the interesting property of the flow equations for S and Sint to have equivalent fixed points.
It is important to notice that this property is also true with the Wilson version extended to an arbitrary
cutoff function since, by construction, the renormalization of the field exactly compensates the extra ℓ-
dependency of the cutoff function displayed in (16).
To get the Morris version of the flow equations for S and Sint, one must first come back to the deriva-
tion of the Polchinski-like equation (34) where the derivative w.r.t. ℓ is performed at a fixed dimensioned
field (RG step 1). This time, P
(
q2,ℓΛ0
)
is not given by (16) but, for dimensional reason, by
23003-8
Wilson’s ERGE and anomalous dimension
P (q2,ℓΛ0) =Λ
−2̟
0 ℓ−2̟P̃
(
q̃2
)
, (43)
with ̟ given by (19) in which n0 is equal to zero to avoid singularities at
∣
∣q
∣
∣ = 0. Consequently, after
the rescaling step 2, it comes a flow equation for Sint of the form (38), (39) but with an effective G
(
q̃2
)
translated by the constant term −̟ compared to (14) so that this flow equation may finally be written
under the form:
Ṡint =Gtra (Sint)+Gdil(Sint,d (c)
φ
)−̟O(Sint, P̃ ) , (44)
in which O(Sint, P̃ ) is the expression in terms of Sint of the redundant “operator” obtained from (13) using
(35), (36) (up to a field independent term):
O(Sint, P̃ ) =
∫
q̃
[
P̃
(
q̃2
)
(
δ2Sint
δφ̃q̃δφ̃−q̃
−
δSint
δφ̃q̃
δSint
δφ̃−q̃
)
− φ̃q̃
δSint
δφ̃q̃
]
. (45)
The Morris-like flow equation for S is then readily obtained under the form:
Ṡ =Gtra (S)+Gdil(S,d (c)
φ
)−Gren (S) , (46)
in which the three terms are defined by equations (10)–(12). Notice the negative sign in front of Gren (S)
which is opposite to that of Wilson as described in section (3) — one would observe the same change
of sign in front of ̟ in (44) compared to its Wilson-like version. This difference is due to the fact that
some aspects have formally been reversed. On the one hand, (Wilson) one lets the coefficients of S vary
under the change of scale and then renormalizes the field. On the other hand, the “renormalization step”
is anticipated and included within the cutoff function because the (useful) variation of the coefficients
of S has been arbitrarily incorporated in the scaling property of the field. Notice that in both cases the
procedure amounts to keeping the same term of the action constant.
The two versions provide well allowed forms of ERGE, but it is worth underlining the unusual expres-
sion of (46) that may have consequences in practical calculations. The discussion of that issue— as well as
of the consequences of other considerations presented in this article— is left to another publication [39].
5. Summary and conclusion
Three different ways of introducing the anomalous dimension parameter η in an ERGE have been
considered explicitly. The non-linear procedure of the historic first version [1] (extended to an arbitrary
cutoff function in [31]) has been compared to the linear and non-linear versions associated with the
“anomalous” dimensional analysis procedure of respectively the “modified” Polchinski version [32–34,
36] and theMorris version [28]. Their differences in essence have been emphasized, as well as the reasons
why the non-linear versions should provide simpler calculational frameworks. The role of the exactly
marginal redundant operator in the construction of an ERGE has also been underlined.
Acknowledgements
It is with great pleasure that I dedicate this article to M. Kozlovskii on his sixtieth anniversary and in
remembrance of the times when we organized some symposium in the early 1990s.
I thank H. Osborn for useful remarks.
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Рiвняння точної ренормалiзацiйної групи Вiльсона
i параметр аномальної вимiрностi
К. Бервiльє
Лабораторiя математики i теоретичної фiзики, UMR 7350 (CNRS), Унiверситет Франсуа Рабле,
37200 Тур, Францiя
Параметр аномальної вимiрностi, введений в нелiнiйний спосiб у першiй iсторичнiй версiї рiвняння то-
чної ренормалiзацiйної групи, порiвнюється з сучасною методикою. Простий вираз для точно гранично-
го (маргiнального) надлишкового оператора слiдує з цiєї нелiнiйностi, тодi як для того, щоб отримати цей
результат у лiнiйному випадку необхiдно розв’язати диференцiальнi рiвняння першого порядку. Висвi-
тлено роль цього оператора в побудовi рiвняння потоку.
Ключовi слова: рiвняння точної ренормалiзацiйної групи, аномальна вимiрнiсть
23003-10
http://dx.doi.org/10.1002/pssb.2221970221
http://dx.doi.org/10.1103/PhysRevB.73.174406
http://dx.doi.org/10.1088/1751-8113/43/49/495001
http://dx.doi.org/10.5488/CMP.14.43002
http://dx.doi.org/10.1142/S0217979200000923
http://dx.doi.org/10.1016/S0370-1573(00)00137-X
http://dx.doi.org/10.1016/S0370-1573(01)00098-9
http://dx.doi.org/10.2478/BF02475552
http://dx.doi.org/10.1103/PhysRevB.69.134413
http://dx.doi.org/10.1016/j.aop.2007.01.007
http://dx.doi.org/10.1007/978-3-642-05094-7
http://dx.doi.org/10.1016/j.physrep.2011.12.003
http://dx.doi.org/10.1016/S0550-3213(99)00545-3
http://dx.doi.org/10.1103/PhysRevLett.84.5208
http://dx.doi.org/10.1103/PhysRevE.82.041128
http://dx.doi.org/10.1088/1126-6708/2000/11/004
http://dx.doi.org/10.1016/0370-2693(94)90767-6
http://dx.doi.org/10.1142/S0217751X94000972
http://dx.doi.org/10.1016/0550-3213(95)00541-2
http://arxiv.org/abs/hep-th/0002231
http://dx.doi.org/10.1016/0370-2693(94)90767-6
http://dx.doi.org/10.1143/PTPS.131.395
http://dx.doi.org/10.1007/BF01555911
http://dx.doi.org/10.1016/j.physleta.2004.09.037
http://dx.doi.org/10.1088/1751-8113/44/19/195401
http://dx.doi.org/10.1016/j.aop.2011.10.011
http://arxiv.org/abs/1106.2544
http://dx.doi.org/10.1016/0550-3213(84)90287-6
http://dx.doi.org/10.1016/0370-2693(95)00025-G
http://dx.doi.org/10.1016/j.aop.2007.10.005
http://dx.doi.org/10.1016/0003-4916(85)90341-0
http://arxiv.org/abs/hep-th/9801124
http://dx.doi.org/10.1103/PhysRevB.11.3431
Introduction
The RG steps
Reminder
Main ways of introducing
The extended Wilson ERGE
Presentation
The exactly marginal redundant “operator”
From Wilson to Polchinski
Linear introduction of
Non-linear introduction of
Summary and conclusion
|