Lepage and Mackenzie have shown that tadpole renormalization and systematic
improvement of lattice perturbation theory can lead to much improved numerical
results in lattice gauge theory. It is shown that lattice perturbation theory
using the Cayley parametrization of unitary matrices gives a simple analytical
approach to tadpole renormalization, and that the Cayley parametrization gives
lattice gauge potentials gauge transformations close to the continuum form. For
example, at the lowest order in perturbation theory, for SU(3) lattice gauge
theory, at $\beta=6,$ the `tadpole renormalized' coupling $\tilde g^2 = {4\over
3} g^2,$ to be compared to the non-perturbative numerical value $\tilde g^2 =
1.7 g^2.$Comment: Plain TeX, 8 page

A. Hasenfratz

G. P. Lepage

K. Symanzik

M. Alford

R. Dashen

Vipul Periwal

English

arXiv

Lepage and Mackenzie have shown that tadpole renormalization and systematic improvement of lattice perturbation theory can lead to much improved numerical results in lattice gauge theory. It is shown that lattice perturbation theory using the Cayley parametrization of unitary matrices gives a simple analytical approach to tadpole renormalization, and that the Cayley parametrization gives lattice gauge potentials gauge transformations close to the continuum form. For example, at the lowest order in perturbation theory, for SU(3) lattice gauge theory, at \beta=6, the `tadpole renormalized' coupling \tilde g^2 = {4\over 3} g^2, to be compared to the non-perturbative numerical value \tilde g^2 = 1.7 g^2

Periwal, V

CERN Document Server

PUPT-1549jhep-lat/9507007
Improving lattice perturbation theory
Vipul Periwal
y
Department of Physics
Princeton University
Princeton, New Jersey 08544-0708
Lepage and Mackenzie have shown that tadpole renormalization and systematic improve-
ment of lattice perturbation theory can lead to much improved numerical results in lattice
gauge theory. It is shown that lattice perturbation theory using the Cayley parametriza-
tion of unitary matrices gives a simple analytical approach to tadpole renormalization,
and that the Cayley parametrization gives lattice gauge potentials gauge transformations
close to the continuum form. For example, at the lowest order in perturbation theory, for
SU(3) lattice gauge theory, at  = 6; the `tadpole renormalized' coupling ~g
2
=
4
3
g
2
; to be
compared to the non-perturbative numerical value ~g
2
= 1:7g
2
:
y
vipul@puhep1.princeton.edu
1. Introduction
Lepage and Mackenzie[1] have made considerable advances in numerical simulations
of lattice gauge theories by improving the lattice gauge theory action in two ways:
1. The action is improved in the sense of Symanzik's[2] program.
2. The action is improved by `tadpole renormalization'|a procedure by which the gauge
potential associated to each link variable is given not by the usual correspondence
U

$ exp(igaA

)
but by
U

$ u
0
(1 + igaA

):
It is observed by Lepage and Mackenzie[1] (and collaborators[4]) that these two improve-
ments together lead to a much better convergence to a continuum limit, with orders of
magnitude less computational eort.
Given these remarkable results, it is important to understand exactly what is analyt-
ically involved in these improvements, especially for tadpole renormalization, which is not
as well-understood as the Symanzik-type improvements. One might hope that a better
understanding of the continuum limit would be useful as well for understanding how to
link the strong coupling expansion and continuum perturbation theory[5], for example.
That the Lepage-Mackenzie prescription needs some explication can be seen, for in-
stance, by considering the gauge transformation properties of the above correspondence.
A prime requirement for gauge invariance is UU
y
= 1; which is not consistent with
U

$ u
0
(1 + igaA

) unless u
 2
0
= 1 + (ga)
2
A
2

: The intuition is that u
0
incorporates
short distance uctuations of the link variable, leaving A

as the continuum gauge eld
at long distances. However, it is obvious that the exponential parametrization that is
manifestly unitary cannot be factorized in any simple manner for non-Abelian gauge the-
ories, so as to lead to the required form. One is lead, then, to ask: Given that tadpole
renormalization does appear to work, what is the systematic analytical approximation that
underlies it? Clearly, from the nomenclature, one hopes to resum some subsets of graphs
in lattice perturbation theory, so as to make the rest of the lattice perturbation theory
closer to continuum perturbation theory. Consistent resummations of selected subsets of
graphs in continuum relativistic gauge theories are rare. On the lattice, of course, one has
a great deal more freedom, and reorganizing the perturbation theory to make the approach
to the continuum simpler is a good thing to consider. The basic idea, then, is that the
reorganized lattice perturbation theory should be closer to continuum perturbation theory.
I show, in the present work, that the classical Cayley parametrization of group ele-
ments has two things to recommend it over the exponential parametrization in common
use:
I. This parametrization leads to gauge transformations for the lattice gauge potentials
associated with links, that are `closer' to continuum gauge transformations, than are
the gauge transformations for the potentials obtained via the exponential map.
II. In this parametrization, it is natural to reorganize the lattice perturbation theory in a
manner that eliminates certain tadpoles, and renormalizes the nave lattice coupling
constant in the way posited by Lepage and Mackenzie. I calculate the background
eld eective action in this parametrization after a suitable resummation, and show
that some care is needed in interpreting the resulting expansion parameter.
2. Gauge transformations
The Cayley parametrization of a unitary matrix is
U 
1 + igA=2
1  igA=2
; A Hermitian:
This parametrization is valid so long as det(U + 1) 6= 0; so we may use it in lattice
perturbation theory, when we expect U  1: So at the most simplistic level, one could
arrive at a relation close to the Lepage-Mackenzie form by expanding
U

=
1 + igaA

=2
1  igaA

=2
=
1
1  c
2

1 ++
2
+ : : :
  
1 + igaA

+(1  c
2
)

;
with   (c
2
  g
2
a
2
A
2

=4)=(c
2
  1): This expansion is now consistent with unitary link
variables. The series expansion will converge the best when inserted in correlation functions
if c
2
=  hg
2
a
2
A
2

i: The higher terms in the series will then involve insertions of powers of
(c
2
  g
2
a
2
A
2

=4); and self-contractions (tadpoles) will be cancelled. The higher terms are,
of course, crucial for unitary link variables. This particular form of resumming tadpoles
is likely not a good idea, since it is manifestly not gauge invariant. The same idea can,
however, be implemented at the level of the lattice gauge theory action, as will be shown
later in this paper.
Let us turn now to the question of gauge invariance of the gauge potential A dened
by the Cayley parametrization. It will be useful to contrast the lattice gauge invariance of
this parametrization with that of the usual exponential parametrization, so I rst recall the
standard result. (I have rescaled A to absorb g for this part of the discussion.) If U

(x) 
exp(iaA

(x)); then gauge transformations map U

to
~
U

(x)  (x)U

(x)
y
(x + e

): Let
(x) = 1 + i"(x); then one has
U

(x)U
y

(x) =  iaD

"(x);
with the lattice covariant derivative dened by
D

"(x) 
1
a
 
U

(x)"(x + e

)U
y

(x)   "(x)

:
We are interested in the change in A

; induced by such a change in U

; so we expand both
sides in powers of A; to nd
iA

(x)  
a
2
fA

; A

(x)g  
ia
2
6
 
fA

(x); A

(x)
2
g+A

(x)A

(x)A

(x)

+ : : :
=  i

~
D

"(x) +
ia
2
fA

(x);

"(x)gB  
a
2
[A

(x)
2
; "(x)] + : : :

;
where 

"(x)  ("(x + e

)   "(x)) =a; and
~
D

"(x) 
1
a
("(x + e

)  "(x)) +
i
2
[A

(x); "(x + e

) + "(x)]:
It follows then that the gauge transformations of the gauge potential on the lattice, de-
ned by the exponential parametrization, dier from the continuum form of the gauge
transformations at O(a): Explicitly, if A

=  
~
D

"(x) + a$; then
$ =  
1
4
fA

; [A

; "(x + e

) + "(x)]g +
1
2
[A

(x)
2
; "(x)] +O(a):
The same calculation for the Cayley parametrization is much simpler, and leads in a
straightforward manner to
A

(x) =  
~
D

"(x)  
a
2
4
A

(x)

"(x)A

(x);
an exact result as opposed to the innite series one obtains in the exponential parametriza-
tion. Thus the Cayley parametrization is closer to the continuum in the sense that gauge
transformations of the Cayley gauge potential dier only at O(a
2
); from  
~
D

"(x); whereas
gauge transformation for the exponential parametrization dier at O(a); with contribu-
tions at all higher powers of a: Of course,  
~
D

"(x) itself diers from the continuum form
at O(a):
One may wonder if it is possible to improve the lattice denition of gauge potential
even further, to make the deviation from the continuum even higher order in a: It is a
simple matter to show that parametrizations of the form U

= 1 + f(iaA)=1 + f( iaA);
with f a real polynomial, do not accomplish this.
3. Tadpole resummation in the action
Consider now using the Cayley parametrization in the Wilson action,
S
W
=
1
g
2
X
plaquettes
<tr [1  U

(x)U

(x + e

)U

(x + e

)U

(x)] 
1
g
2
X
plaquettes
<tr [1 W

] :
A particularly interesting question to address is the following: The renormalization group
scale parameter for weak coupling lattice perturbation theory, 
L
; is very small[6,3] com-
pared to the continuum value, computed with e.g. Pauli-Villars,

L

PV
 0:02; for SU(1):
The fact that this ratio is small due to tadpole diagrams was noted in the original calcu-
lation of Dashen and Gross[3]|this is one of the motivations for tadpole renormalization
according to Lepage and Mackenzie. Therefore, a good test of the ecacy of Cayley's
parametrization in lattice perturbation theory is to compute the same background eld
one-loop eective action taking into account a resummation of tadpoles.
In the background eld approach, we parametrize the link variables as
U

(x)  C(

)U
0

(x);
where C(x)  (1 + iax=2)=(1   iax=2);  are the uctuations in the gauge potential, and
U
0
is the background conguration of the link variables, supposed to be weak (i.e. close
to the identity) and varying only over long distances. The lattice covariant derivatives D
0

are now dened with parallel transport by U
0
: It is a standard exercise to show that
trW

= trC(X

)W
0

;with W
0

 U
0

(x)U
0

(x + e

)U
0

(x + e

)U
0

(x);
and
X

a
 D
0



 D
0



+
i
2
 
2[

; 

]  a[D
0



; 

] + a[D
0



; 

]  a
2
[D
0



;D
0



]

;
which we divide into two parts, X

=a  2(E

+ iB

): We write 2W
0

 G

+ iH

;
with G = 2 +O(F
2
); H = O(F ); and note that X
2
= 4a
2
(E
2
 B
2
+ ifE;Bg): Further,
4<trW

= tr
"
1
1 + a
2
X
2

=4


1  i
X

a
2

2
(G

  iH

)
+

1 + i
X

a
2

2
(G

+ iH

)

#
;
so for a one-loop computation we need only
2<trW

= tr

1
1 + a
2
E
2

 
(1  a
2
E
2

)G

  4iaB

H



:
Note that at this order in the weak-coupling background eld calculation we do not need
to consider the lattice functional Haar measure.
Expanding S
W
about U
0
; we have
S
W
=
1
g
2
X
plaquettes
<tr
 
1 W
0


+<tr
 
W
0

 W


;
which gives
S
W
=
1
g
2
X
plaquettes
<tr
 
1 W
0


+ tr
"
a
2
E
2

1 + a
2
E
2

G+ i
aB

H

1 + a
2
E
2

#
:
Dene E

= (a
2
E
2

  C)=(1 + C): with C a number that we shall x momentarily. In
terms of E

;
S
W
=
1
g
2
 
1 C
1 +C

X
plaquettes
<tr
 
1 W
0


+
1
g
2
(1 + C)
X
plaquettes
tr
h
2a
2
E
2

1 + E

+
E

1 + E

(G

  2)   i
aB

H

1 + E

i
:
Observe that the coupling constant g
2
in front of the classical action has been renormalized
to g
2
(1 + C)=(1  C):
We now wish to x C: To this end, rescale the uctuation elds by a factor of g
p
1 + C:
The part of S
W
governing uctuations becomes
S() =
X
plaquettes
tr
h
2a
2
E
2

1 + g
2
b
E

+
b
E

1 + g
2
b
E

1
1 + C
(G

  2)   ia
B

H

1 + g
2
b
E

i
;
with
b
E

= a
2
E
2

 
C
g
2
(1 + C)
:
It follows therefore that at this order, we should choose
C =
g
2
ha
2
E
2

i
1  g
2
ha
2
E
2

i
;
which implies that
1  C
1 + C
= 1  2g
2
ha
2
E
2

i:
The gauge xing terms are as usual, with the gauge parameter dependent on C for Feynman
gauge, but this is not a problem.
It should be noted that there are two types of contributions to the one-loop renormal-
ization of the lattice coupling constant, those that have continuum analogues and those
that are specic to lattice gauge theory. Ideally, one would want to incorporate all the lat-
tice specic renormalizations into a resummed action|the resummation proposed above
does not include all the lattice specic renormalizations, it only includes the tadpole dia-
gram that provides the bulk of the renormalization. Within the present calculation, it does
not seem possible to resum the other type of lattice specic renormalizations, essentially
because they are not tadpole diagrams. It may be possible to eliminate such contributions
if one starts with an action dierent from the Wilson action.
When  = 6; for SU(3), this tells us that the coupling constant in front of the classical
action is
~g
2
=
4
3
g
2
;
at this order in perturbation theory. This should be compared with the non-perturbative
value found numerically by Lepage and Mackenzie,
~g
2
 1:7g
2
:
It would appear then that the use of the Cayley parametrization combined with our resum-
mation of the perturbation expansion is a decent approximation to the numerical renor-
malization, since this value of  corresponds to g
2
= 1: Note that we have ignored terms of
order g
2

2
from the measure[6] for example, and we have not included any Symanzik-type
improvements in the action|nevertheless, we still obtained a lowest order result that is
about half of the numerical non-perturbative value.
Two more comments are in order. Firstly, perturbation theory is still organized in
powers of g
2
since higher terms in the uctuation eld appear in the combination g
2
b
E :
Of course, if one calculated at higher order, one would also need to change the denition
of C to be consistent. Secondly, the tadpole contribution to 
L
=
PV
does not appear
since G

  2 couples to
b
E ; and h
b
Ei = 0: The one-loop renormalization of the tree action
is still non-trivial since we have only resummed some of the graphs contributing to the
determinant. In particular, C should be adjusted order by order in the perturbative
expansion in order to eliminate the largest contribution from diagrams specic to lattice
gauge theory.
4. Conclusions
It is plausible, on the basis of the simple calculation presented here, that combining the
Cayley parametrization with Symanzik-type improvements will lead to a systematic lat-
tice perturbation theory that is much closer to the continuum and to the non-perturbative
numerical simulations. It seems unlikely that lattice perturbation theory, even if it is refor-
mulated as described above, will become a practical tool for calculations in gauge theories.
The interest in understanding the structure of lattice perturbation theory is in being able
to match numerical results with continuum results. As I hope is evident from the preceding
discussion, there is freedom in the denition of the lattice perturbation expansion that can
be used to render such a comparison more transparent. Deviations from perturbation the-
ory can then be understood as either lattice artifacts, or genuine non-perturbative eects.
The numerical phenomenology of genuine non-perturbative physics would be useful in
constructing analytical approximations to gauge theory dynamics. While such systematic
study does not have the theoretical allure of direct attempts at phenomenological models
of low-energy dynamics in gauge theories, keep in mind the words of Simone Weil|`We
must prefer real hell to an imaginary paradise.'
I am grateful to Mark Alford for comments on the manuscript.
References
1. G.P. Lepage and P.B. Mackenzie, Phys. Rev. D48 (1993) 2250
2. K. Symanzik, Nucl. Phys. B226 (1983) 187
3. R. Dashen and D.J. Gross, Phys. Rev. D23 (1981) 2340
4. M. Alford, W. Dimm, G.P. Lepage, G. Hockney, and P.B. Mackenzie, QCD on coarse
lattices, in the Proceedings of Lattice '94, Bielefeld, Germany.
5. G.P. Lepage, seminar at Princeton University, April 1995
6. A. Hasenfratz and P. Hasenfratz, Phys. Lett. 93B (1980) 165


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