Questionable and unquestionable in the perturbation theory of non-Abelian models

28 Pages·1998·0.55 MB·English

MPI{PhT/96{121
BUTP/96{27
Questionable and unquestionable in the
perturbation theory of non-Abelian models(cid:3)
y
Ferenc Niedermayer
Institute for Theoretical Physics
University of Bern
Sidlerstrasse 5, CH-3012 Bern, Switzerland
Max Niedermaier and Peter Weisz
Max-Planck-Institut fu¨r Physik
F¨ohringer Ring 6, D-80805 Mu¨nchen, Germany
January 1998
Abstract
We show, by explicit computation, that bare lattice perturbation the-
oryinthetwo-dimensionalO(n)nonlinear(cid:27) models withsuperinstan-
tonboundaryconditionsisdivergentinthe limitofanin(cid:12)nitenumber
ofpointsj(cid:3)j. ThisistheanalogueofDavid’sstatementthatrenormal-
ized perturbation theory of these models is infrared divergent in the
limit where the physical size of the box tends to in(cid:12)nity. We also give
arguments which support the validity of the bare perturbative expan-
sionof short-distancequantities obtainedby takingthe limitj(cid:3)j!1
(cid:3)Work supported in part by Schweizerischer Nationalfonds
yOn leave from the Institute of Theoretical Physics, Eo¨tvo¨s University, Budapest,
Hungary
term by term in the theory with more conventional boundary condi-
tions such as Dirichlet, periodic, and free.
2
1 Introduction
Quantum chromodynamics (QCD) is the presently favored candidate the-
ory for strong interactions. However, to establish its esteemed status the
theory must be able to reproduce the low-lying spectrum and low-energy
S-matrix elements to a respectable precision. To accomplish this we require
a nonperturbative de(cid:12)nition of the theory { the most promising of which
is lattice regularization. On the other hand, many aspects of high energy
phenomena involving hadrons (e.g., jets, deep inelastic scattering, etc.) are
described successfully using renormalized perturbation theory (PT); the ra-
tionale being the expected property of asymptotic freedom, i.e., that the
amplitudes can be expressed as a series in a coupling which depends on the
energy of the process and goes to zero as the energy goes to in(cid:12)nity. The
conventionalwisdomisthat\inprinciple"onecouldjustifythisuseofPTby
showing that the coe(cid:14)cients of the conventional perturbative series provide
the true asymptotic expansion of the full theory de(cid:12)ned nonperturbatively
via the lattice regularization. Con(cid:12)rming this proclaimed status of PT is
also essential in aiming at a criterion where to truncate it. Without such
a criterion PT produces sequences of predictions, one for each order where
one decides to truncate it, and a priori none of them might be close to the
true answer. It must be emphasized, however, that the usual arguments for
the above scenario are far o(cid:11) a proof and Patrascioiu and Seiler repeatedly
emphasized that also alternative scenarios can be imagined [1, 2].
In full QCD a rigorous understanding of the status of PT is likely not
to become amenable in the near future, so that it is sensible to address the
question in a simpler context, e.g., the O(n) nonlinear (cid:27) models in two di-
mensions. These models are also perturbatively asymptotically free and are
thought to describe a multiplet of stable massive particles, the mass scale
being nonperturbative in the coupling constant. In the lattice formulation
one usuallystarts witha (cid:12)nitenumberof lattice pointsj(cid:3)jand thestandard
discretization (2.1). The conventional picture is that the critical point at
which the continuum limit for n > 2 should betaken is (cid:12) = 1. Patrascioiu
c
and Seiler [1], on the other hand, conjecture that there is a critical (cid:12) < 1
c
(for all n > 1)beyondwhich thetheoryis inamassless phase. Itmightthen
happen that although the continuum limit obtained by approaching (cid:12) from
c
below has a mass gap and describes the expected low-energy properties, the
theory thus obtained is not asymptotically free at high energies. In par-
3
ticular, conventional perturbation theory in the in(cid:12)nite volume limit would
then be wrong for 2-d models with a non-Abelian global symmetry. Their
arguments arebased onresults frompercolation theory[2]andwith analogy
to the case of the Abelian model n = 2 (for which it is rigorously proven
that there is a phase transition at (cid:12)nite (cid:12) [3]). Numerical investigations can
give useful hints but obviously cannot establish which scenario is correct.
Concerning the status of perturbation theory it is known that for a (cid:12)xed
numberoflatticepointsj(cid:3)j (cid:24) L2andforalargeclassofboundaryconditions
(BC’s) observables have a well-de(cid:12)ned perturbative expansion in the bare
coupling (cid:12)−1. We shall use the term \short-distance observable" to denote
O(n){invariant correlation functions of some local (cid:12)eld, where (in the bulk
ofthepaper)allargumentsare(cid:12)xedlattice sites, adistanceO(L)away from
the boundary. The problems now arise with the limit L ! 1. Let us (cid:12)rst
consider the following question:
Q1: Is the asymptotic expansion of short-distance observables for (cid:12)xed L
and (cid:12) ! 1 uniform in L? Equivalently, can for such quantities the
limits (cid:12) !1 and L! 1 be exchanged in bare lattice PT?
A major (cid:12)nding of [4] is that the answer to Q1 actually depends on the
choice of boundary conditions. The reasoning can be illustrated with the
computation of the energy density E at the center of the lattice. Although
E does not have a continuum limit relevant to the physics, by application of
the Mermin-Wagner theorem [5] it should be independent of the boundary
conditionsinthelimitofanin(cid:12)nitenumberoflatticepoints. Patrascioiuand
Seiler invented boundary conditions (which they coined superinstanton (SI)
boundaryconditions; c.f.sect.2)andcomputedtheone-loopcoe(cid:14)cientofE.
They showed that it has a (cid:12)nite limit as L!1 but the result was di(cid:11)erent
from that with more conventional BC’s such as Dirichlet. Furthermore they
claim a similar result holds for the renormalization group (cid:12)-functions. The
conclusion is that for at least one of the BC’s involved the answer to Q1 is
negative. The authors interpret their result as casting doubt on the validity
of standard PT for all BC’s. They claim that the correct perturbative
treatment in the limit of in(cid:12)nite volume must include expansions around
the so-called superinstanton gas (con(cid:12)gurations whose energy tends to zero
as L!1; see sect. 3).
4
We believe that such conclusions are too strong and we set out to clarify
some of the points raised. In the next section we show that the SI BC are
\sick" in the sense that the limits in Q1 cannot be interchanged. We do
this byshowing that the perturbativecoe(cid:14)cients of E with SIBC diverge at
two-loop order in the limit L! 1.1 We have not considered the particular
(cid:12)-functions de(cid:12)ned in [4] but we believe a similar result will be found. This
infrared divergence of the PT with SI BC was already suspected by David
[6] but as his argument was formulated in the framework of (renormalized)
continuum PT his claims were open to a counterattack [7] and the issue
remained unsettled. Our result can be viewed as the lattice analogue of
David’s statement.
Having identi(cid:12)ed the SI BC as \sick", the next question is whether the
conventional BC are \healthy". Consider the bare lattice PT expansion
of some short-distance observable on a (cid:12)nite lattice j(cid:3)j (cid:24) L2 with various
BC’s. Let c(cid:11)(x ;:::;x ;L); r (cid:21) 1bethecoe(cid:14)cient of (cid:12)−r in an asymptotic
r 1 n
expansion with BC of type (cid:11), where the sites x ;:::;x are (cid:12)xed and a
1 n
distance O(L) away from the boundary.
Q2: Which BC’s (cid:11) can be anticipated to give (cid:12)nite and coinciding answers
fortheL!1limitsoftheirperturbativecoe(cid:14)cientsc(cid:11)(x ;:::;x ;L)?
r 1 n
Q3: Dothein(cid:12)nitevolumeshort-distanceobservablesadmitanasymptotic
expansionin(cid:12)−1? If\yes", cantheircoe(cid:14)cientsc1(x ;:::;x ); r (cid:21) 1
r 1 n
be obtained from PT via limL!1c(cid:11)r(x1;:::;xn;L) = c1r (x1;:::;xn),
where (cid:11) is one of the BC meeting the conditions in Q2?
Insection3weshallconsiderthesedeeperquestionsraisedin[1,4]andargue
in favor of the following picture: (i) All BC’s involving only ferromagnetic
couplings and leaving the interior spins unconstrained meet the condition in
Q2 { provided free and Dirichlet BC meet it. (ii) Assuming the latter also
the answer to both parts of Q3 is a(cid:14)rmative. We do not have a rigorous
theorem but our considerations may constitute a strategy for a future proof
thereof. Section 4 is devoted to some further discussion.
1We have veri(cid:12)ed, however, that if we compute quantities well away from the bound-
aries and the centerthen thesame results are obtained as for conventional BC.
5
2 IR divergence of PT with superinstanton BC
We consider a two-dimensional square lattice (cid:3) = f(x ;x );−L=2 + 1 (cid:20)
1 2
x (cid:20) L=2−1g, L a positive even integer. The set of points surrounding the
k
square, fjx j = L=2; jx j(cid:20) L=2g[fjx j (cid:20) L=2; jx j = L=2g will be referred
1 2 1 2
to as the boundary @(cid:3) of (cid:3). The O(n) spin (cid:12)eld S de(cid:12)ned on (cid:3)[@(cid:3) is
x
an n-component (cid:12)eld of unit length S2 = 1. We restrict attention to the
x
standard lattice action
X
A(S) = (cid:12) (1−S (cid:1)S ): (2.1)
x x+(cid:22)
x;(cid:22)
For (cid:12)xed L the perturbative expansion of the two-point function is
X
C(cid:11)(x;y;(cid:12);L) (cid:17) hS (cid:1)S i = 1− (cid:12)−rc(cid:11)(x;y;L) ; (2.2)
x y r
r(cid:21)1
where the superscript (cid:11) indicates the dependence on the boundary condi-
tions under consideration. In this section we will only consider two sorts of
BC’s, the Dirichlet boundary condition Sa = (cid:14) for x 2 @(cid:3) and the so-
x an
called superinstanton (SI) boundary conditions [4]. The latter are Dirichlet
BC’s with the additional constraint that the (cid:12)eld at some point z is also
0
held (cid:12)xed parallel to the (cid:12)elds at the boundary.
The purposeof this section is to show, by explicit computation, that PT
with SIBC’s is infrared(IR) divergent at third order, while PT with Dirich-
let or periodic BC is IR (cid:12)nite at the same order. For completeness, and to
introduce the notations, we (cid:12)rst reconsider the (cid:12)rst and second orders in
some detail. Bare perturbation theory can be performed by parametriz-
ing the (cid:12)elds by2 Sxa = (cid:12)−12(cid:25)xa; a = 1;:::;N where N = n − 1, and
Sxn = (1 − (cid:12)−1~(cid:25)x2)12. The measure (cid:5)x2(cid:3)dSPx(cid:14)(S2x − 1) is then replaced by
exp(−Ameasure)(cid:5)x2(cid:3)d~(cid:25)x with Ameasure = 12 xln(1−(cid:12)−1~(cid:25)x2). TheFeynman
diagrams contributing to the two-point function are the same for the two
BC’s, only the expression for the free propagator di(cid:11)ers. We denote the free
Dirichlet propagator by D(x;y) and the free SI propagator by G(x;y). One
has
D(x;z )D(z ;y)
G(x;y) = D(x;y)− 0 0 : (2.3)
D(z ;z )
0 0
2Thisisnotindispensable,aparametrizationindependentde(cid:12)nitioncouldbegivenvia
the Schwinger-Dyson equations.
6
In the following we will take the point z to be the origin z = (0;0). Also
0 0
we denote by z the point (1;0). One easily sees that each c(cid:11)(x;y;L) is a
1 r
polynomial in N = n−1 of degree r or less, and we write
Xr
c(cid:11) = c(cid:11) Nj: (2.4)
r r;j
j=1
2.1 First order
In (cid:12)rst order we have, for the Dirichlet BC,
h i
1
cDir(x;y) = N D(x;x)+D(y;y)−2D(x;y) : (2.5)
1 2
As mentioned above, for the SI BC one simply replaces D by G. For the
special case of the energy expectation value we have
(cid:16) (cid:17)
1 1
cDir(z ;z )= N − 4 ; (2.6)
1 0 1 4 2 0
where
4 = D(z ;z )−D(z ;z ):
0 0 0 1 1
cDir approaches its asymptotic value with power corrections since 4 =
1 0
O(1=L2) for L!1. On the other hand
1
cSI(z ;z ) =N G(z ;z ) (2.7)
1 0 1 2 1 1
approaches the same asymptotic value extremely slowly with corrections
1=lnL since
1 1
G(z ;z )= −4 − Z−1 ; (2.8)
1 1 0
2 16
with
Z (cid:17) D(z ;z )(cid:24) (2(cid:25))−1lnL; for L!1: (2.9)
0 0
2.2 Second order
At the next order there are three contributions
X3
c(cid:11)(x;y) = c(cid:11) (r)(x;y) (2.10)
2 2
r=1
7
Figure 1: One-loop diagrams contributing to the spin-spin correlation func-
tion
correspondingto the diagrams depicted in Fig. 1 (diagram 2comes from the
measure part of the action). One has
h i
1 2
cDir (1)(x;y)= N2 D(x;x)−D(y;y)
2 8
h i
1
+ N D(x;x)2+D(y;y)2−2D(x;y)2 ; (2.11a)
4
X
cDir (2)(x;y)= 1N De(x;y;i)2 ; (2.11b)
2 2
i
X
cDir (3)(x;y)=−1N2 [D(i;i)−D(j;j)][De(x;y;i)2 −De(x;y;j)2]
2 4
hi;ji
X n
1
− N De(x;y;i)[D(i;i)De(x;y;i)−D(i;j)De(x;y;j)]
2
hi;ji
o
+(i $j) ; (2.11c)
where the sums in the last equation are restricted to the case when i and j
are nearest neighbors and we have introduced the notation
De(x;y;i) = D(x;i)−D(y;i):
Letusjustinspectthecoe(cid:14)cient ofN2. Herediagram 2does notcontribute
and for diagram 1 one has simply
1
cSI (1)(z ;z )−cDir (1)(z ;z ) = [G(z ;z )2−42] (2.12)
2;2 0 1 2;2 0 1 8 1 1 0
which by eq. (2.8) tends to 1=32 as L!1. For diagram 3 we (cid:12)nd
1
cSI (3)(z ;z )−cDir (3)(z ;z ) = 4 +t Z−1+t Z−2+t Z−3 ; (2.13)
2;2 0 1 2;2 0 1 16 0 1 2 3
8
with
X
t = 1 De(z ;z ;i)2E(i) ;
1 0 1
4
i6=z0
X
1
t = − D(z ;i)2F(i) ;
2 0
64
i6=z0
X
1
t = D(z ;i)2E(i) ; (2.14)
3 0
64
i6=z0
where
Xh i
E(i) = D(i;z )2−D(j;z )2 ;
0 0
j=hii
Xh i
F(i) = D(i;i)−D(j;j) : (2.15)
j=hii
Here the sums are taken over sites j which are nearest neighbors to i. The
decomposition intheformofeq.(2.13) ismadeinordertobeabletoextract
the asymptotic behavior reliably. Our ansatz (which we have not proven
analytically) is that the functions t have an expansion of the form
i
XRi
t (L) (cid:24) t(r)[(2(cid:25))−1lnL]r +O(1=L); (2.16)
i i
r=0
with R (cid:12)nite. To obtain the leading behavior we have computed t (L) over
i i
a large range of L (up to L = 220, using extended precision arithmetics)
and taken logarithmic derivatives with respect to L. Our (cid:12)ndings are that
R = R = 0 ; (2.17)
1 2
and hence the contributions in eq. (2.13) involving t ;t vanish in the limit
1 2
L! 1. However,
1
R = 3; t(3) = − : (2.18)
3 3 96
Onecaninfactunderstandtheresultfort analytically inthefollowingway.
3
Far from the origin and the boundary, i.e., for 1 (cid:28) jij (cid:28) L the Dirichlet
propagator is well approximated by its continuum behavior
L
D(z ;i) (cid:24) (2(cid:25))−1ln ; (2.19)
0 jij
9
and so
E(i) (cid:24) −(cid:50) (D(z ;i)2)(cid:24) −2(2(cid:25))−2jij−2; (2.20)
(i) 0
where (cid:50) is the laplacian in the variable i. Thus for t we have
(i) 3
X(cid:16) (cid:17)
t (cid:24) − 1 (2(cid:25))−4 1 L 2ln2 L
3 32 L2 jij jij
i6=z0
Z
1 jxj=1 1
(cid:24) − (2(cid:25))−4 d2xjxj−2ln2jxj = − (2(cid:25))−3ln3L: (2.21)
32 jxj>1=L 96
This is the same result as was guessed in [4]. These authors were quite
fortunate to get the correct result because initially they just evaluated the
fullfunctioncSI (3) numericallyandthenmadearathernaiveextrapolation.
2;2
We stress that the analysis has to be done extremely carefully on the lines
outlinedabovetoproperlytreattheappearanceofpolynomialsoflogarithms
in the denominators.3
The (cid:12)nal result of this subsection is that
1
lim [cSI (z ;z )−cDir(z ;z )] = (2.22)
L!1 2;2 0 1 2;2 0 1 48
(we also know limL!1cD2;i2r(z0;z1) = 0). The Mermin-Wagner theorem im-
plies that the perturbative expansion of the two-point function is indepen-
dent of the boundary conditions in the in(cid:12)nite volume limit. Accepting this
fact we mustconclude thatthe interchange of the limits(cid:12) !1andL!1
is not permissible for at least one of the boundary conditions involved. We
now go on to show that this is indeed the case for the SI BC because one
encounters an infrared divergence.
2.3 Leading N contribution to the third order
For the purpose of showing an IR divergence at some order r, for generic n,
it is su(cid:14)cient to show that one of the coe(cid:14)cients of this polynomial in N
is IR divergent. We claim that this is the case for the coe(cid:14)cient of N3 in
c(cid:11)(x;y;L) with the SI BC.
3
3Indeed,cSI (3) isnotmonotonicinLandhasaminimumatalargevalueofL(cid:24)120.
2;2
10