We compute the hierarchical $\phi^4$-trajectory in terms of perturbation
theory in a running coupling. In the three dimensional case we resolve a
singularity due to resonance of power counting factors in terms of logarithms
of the running coupling. Numerical data is presented and the limits of validity
explored. We also compute moving eigenvalues and eigenvectors on the trajectory
as well as their fusion rules.Comment: 24 pages, 9 pictures included, uuencoded compressed postscript fil

Rolf, J.

Wieczerkowski, C.

English

arXiv

We compute the hierarchical \phi^4-trajectory in terms of perturbation theory in a running coupling. In the three dimensional case we resolve a singularity due to resonance of power counting factors in terms of logarithms of the running coupling. Numerical data is presented and the limits of validity explored. We also compute moving eigenvalues and eigenvectors on the trajectory as well as their fusion rules

Rolf, J

Wieczerkowski, C

CERN Document Server

The Hierarchical 
4
- Trajectory by Perturbation
Theory in a Running Coupling and its Logarithm
J. Rolf
1
and C. Wieczerkowski
2
Institut fur Theoretische Physik I, Universitat Munster,
Wilhelm-Klemm-Strae 9, D-48149 Munster,
1
rolfj@@planck.uni-muenster.de,
2
wieczer@@yukawa.uni-muenster.de
Abstract
We compute the hierarchical 
4
-trajectory in terms of perturbation
theory in a running coupling. In the three dimensional case we resolve
a singularity due to resonance of power counting factors in terms of
logarithms of the running coupling. Numerical data is presented and
the limits of validity explored. We also compute moving eigenvalues
and eigenvectors on the trajectory as well as their fusion rules.
1 Introduction
In the block spin renormalization scheme of Wilson [?, ?] renormalized theo-
ries come as renormalized trajectories of eective actions. Departing from a
bare action the renormalized trajectory is reached by an innite iteration of
block spin transformations. For this limit to exist the bare couplings have to
be tuned as the number of block spin transformations is increased. Consider
an asymptotically free model at weak coupling. There the point is to keep
couplings under control which increase by value under a block spin transfor-
mation. Such couplings are called relevant. In weakly coupled models they
can be identied by naive power counting. This renormalization scheme has
been beautifully implemented both within and beyond perturbation theory.
Let us mention the work of Polchinski [?], Gawedzki and Kupiainen [?] ,
Gallavotti [?], and Rivasseau [?] as a guide to the extensive literature. The
underlying picture of an ultraviolet asymptotically free model is to think of
the renormalized trajectory as unstable manifold of a trivial xed point. Al-
though this picture has been in mind behind block spin renormalization since
the very beginning [?] it has not been formalized yet to an approach free of a
bare action. This paper is a contribution to ll this gap. It extends the anal-
ysis begun in [?] and [?] in the context of renormalization group improved
actions for the two dimensional O(N)-invariant nonlinear -model. Here we
will work it out for the 
4
-trajectory in the hierarchical approximation. The
hierarchical model was invented by Dyson [?] and Baker [?] and has enjoyed
the attention of Bleher and Sinai [?], Collet and Eckmann [?], Koch and Wit-
twer [?], Felder [?], and Pordt [?], to mention a few. The 
4
-trajectory will
be dened as a curve which departs the trivial xed point in the 
4
-direction.
Technically we perform a renormalized perturbation expansion in a running
coupling. In the three dimensional case we perform a perturbation expan-
sion in a running coupling and its logarithm. The dynamical principle which
proves to be strong enough to determine the trajectory at least in perturba-
tion theory is stability under the renormalization group. With stability we
mean that the trajectory is left invariant under a transformation as a set in
theory space. Recall that a renormalized action always comes together with
a sequence of descendents generated by further block spin transformations.
Even in the case of a discrete transformation this sequence will prove to con-
sist of points on a continuous curve in theory space which is stable under the
renormalization group. It is the computation of this curve we address. The
result is an iterative form of renormalized perturbation theory. Its closest
1
relatives in the literature are the renormalized tree expansions of Gallavotti
and collaborators [?, ?, ?]. A pedagocial exposure can be found in [?]. Our
expansion is however free of divergencies piled up in standard perturbation
theory by innitely iterated renormalization group transformations from the
very beginning. Surprisingly we do not need to treat relevant and irrelevant
couplings on a dierent footing. It will involve neither bare couplings nor
renormalization conditions in the original sense. A renormalization group
transformation in our approach translates to a transformation of the run-
ning coupling according to some -function. We will consider in particular
a choice of coordinate whose associated -function is exactly linear. This
idea has also appeared in [?] and references therein. Renormalized perturba-
tion theory furthermore will surprise us with a sequence of discrete poles at
special rational dimensions. These poles will be traced back to certain reso-
nance conditions on the scaling dimensions of powers of elds. In particular
the case of three dimensions will be shown to be resonant. We will resolve the
associated singularity by a double expansion in both the running coupling
and its logarithm. The expansion will then be extended to the computation
of moving eigenvectors in the sense of [?] on the renormalized trajectory and
their fusion rules. Finally we perform a numerical test of our renormalized
actions. As expected they prove to work well in a small eld region. The
extension of our program to full models is under way. A prototype with
momentum space regularization has been developed in [?].
2 Hierarchical renormalization group
The hierarchical renormalization group in the form advocated by Gawedzki
and Kupiainen [?] is a theory of the non linear transformation
RZ( ) =

Z
d

() Z(L
1 
D
2
 + )

L
D
m(1)
on some space of Boltzmann factors Z(). In the scalar theory  is a single
real eld variable.
d

() = (2)
 
1
2
exp

 

2
2

dm(2)
is the Gaussian measure on R with mean zero and covariance . The pa-
rameters of (1) are the Euclidean dimension D and the block scale L. The
2
subspace of even Boltzmann factors Z( ) = Z() is stable under (1). We
will restrict our attention to this subspace. Let the potential be given by
Z() = exp( V ()). The transformation for the potential is
RV ( ) =  L
D
log

Z
d

() exp

 V (L
1 
D
2
 + )


:m(3)
The below analysis will be done in terms of the potential. The method will be
perturbation theory. The question of stability bounds will not be addressed.
Regarding mathematical aspects of (1) and (3) we refer to the work of Collet
and Eckmann [?], Gawedzki and Kupiainen [?] and of Koch and Wittwer
[?].
3 The trivial xed point
(3) has a trivial xed point V

() = 0. This xed point is the hierarchical
massless free eld. The linearization of (3) at this trivial xed point is given
by
L
V

RO( ) = L
D
Z
d

() O

L
1 
D
2
 + 

:m(4)
This linearization is diagonalizable. The eigenvectors are normal ordered
products
: 
n
:
0
=
@
n
@j
n




j=0
exp
 
j 
j
2
0
2
!
m(5)
with normal ordering covariance 0 =

1   L
2 D

 1
. The normal ordering
covariance has been chosen in order to be invariant with respect to integration
with d

. Its singularity at D = 2 is an infrared singularity of the hierarchical
massless free eld in two dimensions. The eigenvalues are

n
= L
D+n
(
1 
D
2
)
:m(6)
The eigenvalue of : 
4
:
0
is 
4
= L
4 D
. The eigenvector : 
4
:
0
is therefore
relevant for D < 4, marginal for D = 4, and irrelevant for D > 4 dimen-
sions. Perturbation theory can be used to compute corrections to (4) in a
neighbourhood of V

().
3
4 The 
4
- trajectory
Let us dene a curve V (; g) in the space of potentials parametrized by a
local coordinate g. We call it the 
4
- trajetory. We expand the potential
V (; g) =
1
X
n=0
V
2n
(g) : 
2n
:
0
m(7)
in the base of eigenvectors (5). A natural coordinate in a vicinity of V

()
is the 
4
- coupling dened by V
4
(g) = g. Let us use it for a moment. Let
the 
4
- trajectory then be the curve V (; g) dened by the following two
conditions:
1) V (; g) is stable under R. Then there exists a function (g) such that
RV (; g) = V (; (g)) :m(8)
The function (g) is of course coordinate dependent. With the 
4
-
coupling as coordinate it is called  - function.
2) V (; g) visits the trivial xed point V

() at g = 0. The tangent to
V (; g) at V

() is given by
@
@g




g=0
V (; g) = : 
4
:
0
m(9)
This condition is equivalent with V
4
(g) = g + O(g
2
) together with
V
2n
(g) = O(g
2
), n 6= 2.
The 
4
- trajectory is the object of principal interest in massless 
4
- theory
at weak coupling.
5 Perturbation theory
The 
4
- trajectory can be computed by perturbation theory in g as solution
to (8) and (9). Potentials on the 
4
- trajectory are said to scale. A potential
V (; g) is said to scale to order s in g if there exists a function
(g) = 
(s)
(g) +O(g
s+1
) ;

(s)
(g) =
s
X
r=1
b
r
g
r
;m(10)
4
such that
V (; g) = V
(s)
(; g) +O(g
s+1
) ;
RV
(s)
(; g) = V
(s)
(; (g)) +O(g
s+1
) ;m(11)
and
V
(1)
(; g) = g : 
4
:
0
:m(12)
The scheme is to compute 
(s+1)
(g) and V
(s+1)
(; g) given 
(s)
(g) and
V
(s)
(; g) to some order s. Let us explain it in some detail at the case of
D = 4 dimensions, block scale L = 2, and covariance  = 1. Then the normal
ordering covariance is 0 =
4
3
. Computing a block spin transformation (3),
we speak of V () as bare potential and of RV () as eective potential. The
point of departure is (12). Anticipating the terms generated in RV
(1)
(; g)
to second order in g we make the ansatz
V
(2)
(; g) = c
0
g
2
+ c
2
g
2
: 
2
: +g : 
4
: +c
6
g
2
: 
6
: :m(13)
The coecients are determined by the condition that (11) be fullled to
second order. (13) is mapped to
RV
(2)
(; g(g
0
)) = (16c
0
 
5440
9
)g
0
2
+ (4c
2
  448)g
0
2
: 
2
:
+g
0
: 
4
: +(
c
6
4
  2)g
0
2
: 
6
: +O(g
0
3
) :m(14)
Here the eective coupling dened as the coecient of : 
4
: in the eective
potential is given by
g
0
(g) = g   60g
2
+O(g
3
) :m(15)
Comparing the eective potential as a function of the eective coupling with
the bare potential as a function of the bare coupling we conclude that
c
0
=
1088
27
; c
2
=
448
3
; c
6
=  
8
3
m(16)
on the 
4
- trajectory. The coecients of the  - function (10) to this order
are
b
1
= 1 ; b
2
=  60 :m(17)
5
It follows that g is marginally irrelevant in four dimensions. This completes
the rst step. It is iterated in the obvious manner. The general form of the
order s approximation is
V
(s)
(; g) =
s+1
X
n=0
c
(s)
2n
(g) : 
2n
: ;
c
(s)
2n
(g) =
s
X
r=2
c
2n;r
g
r
; n 6 1 ;
c
(s)
4
(g) = g ;
c
(s)
2n
(g) =
s
X
r=n 1
c
2n;r
g
r
; n > 3 :m(18)
It includes all normal ordered products generated in the eective potential by
(3) from (12) to order s in g. The iteration proceeds as above with the order
s + 1 ansatz of the form (18). The condition (11) yields a system of linear
equations for the order s + 1 coecients. (To highest order the coecients
have no other choice.) This system has a unique solution: the 
4
- trajectory.
Note that the coecient b
s+1
of the  - function is already determined by
V
(s)
(; g). For instance (15) does not contain any of the coecients in (13).
The expansion can be computed to higher orders using computer algebra.
To third order we nd
c
(3)
0
(g) =
1088
27
g
2
 
54784
27
g
3
;
c
(3)
2
(g) =
448
3
g
2
 
497408
27
g
3
;
c
(3)
6
(g) =  
8
3
g
2
+ 352g
3
;
c
(3)
8
(g) =
32
3
g
3
m(19)
together with

(3)
(g) = g   60g
2
+ 8880g
3
:m(20)
Let us remark that the perturbation coecients (10) and (11) come with
alternating signs. The coecients show a frightening increase in absolute
value with the order in g. The full series is not expected to converge. Note
that the coecients look better when g is replaced by g=4!.
6


The Hierarchical $\phi^4$ - Trajectory by Perturbation Theory in a Running Coupling and its Logarithm

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The Hierarchical ϕ4\phi^4ϕ4 - Trajectory by Perturbation Theory in a Running Coupling and its Logarithm

Abstract

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The Hierarchical $\phi^4$ - Trajectory by Perturbation Theory in a Running Coupling and its Logarithm