Volume 97B, number 2
PHYSICS LETTERS
1 December 1980
SELF-CONSISTENT AREA LAW IN QCD
Yu. M. MAKEENKO
Institute for Theoreticaland ExperimentalPhysics,Moscow, USSR
and
A.A. MIGDAL
LandauInstitute for TheoreticalPhysics,Moscow, USSR
Received 2 September 1980
The exact equation for the loop average in multicolor QCD is reduced to a bootstrap form. Its iterations yield a new
manifestly gauge invariant perturbation theory in the loop space, reproducing asymptotic freedom. For large loops, the
area law appears to be a self-consistent solution.
Everybody believes that quarks are confined in
QCD. The problem is to understand how are they confined.
To answer this question, we should derive some
dynamical equations for QCD which lead to quark
confinement in the infrared domain. In the ultraviolet
one, the same equations should reproduce the perturbative QCD.
A somewhat similar infrared problem was solved
in the theory of phase transitions some time ago. There
the scaling law arose as a sum of infrared divergencies.
In order to justify the scaling law dynamically, the
bootstrap equations were derived. In contrast with
equations o f motion, the bootstrap equations were
closed in the infrared domain, so that the scaling law
arose as a self-consistent solution.
In the case o f QCD, we expect the area law for
the Wilson loop average
W(C)= g2 (Tr P exp ~A d.xtl ----+ exp[-°Amin(C)]
c
a~ge c
(1)
where Amin(C ) is the minimal area enclosed by C
and o stands for the string tension.
In this note, we derive an analogue of the bootstrap
equations for W(C). The areas law, indeed, appears to
be self-consistent.
The set o f exact equations for the loop function-
als ,1 was derived in our papers [ 2 - 5 ] . In the limit of
large N, corresponding to the WKB approximation
in the loop space, these equations reduce to the "classical" equation
6 W(C) _
D~ 6a.u(x )
f
Cxx
dy v 8(x - y ) W(Cxy)W(Cyx), (2)
x
with the initial condition W(O) = X = g2N. Here 8v
and O/D6uv(x ) stand for path and area variation,
respectively (see refs. [ 2 - 5 ] for more details).
This equation as it stands is not suitable for checking the area law. For a simple loop without self-intersections, one may verify that the l.h.s, indeed vanishes ,2 due to properties of the minimal surface. However,
this is true for any string tension, o. Moreover, the
same " p r o o f ' can be applied to the abelian theory as
well, so one cannot take it seriously.
t l The fruitful idea of writing down a functional equation
satisfied by W(C) for infinitesimal variation of C was proposed in ref. [1].
4:2 For a simple loop, the r.h.s, can be easily reproduced multiplying the ansatz (1) by the abelian factor
exp(- 1~,ffdx
dyD(x-y))-+exp[-pl(C)].
The divergent coefficient 0 depends on the regularization
prescription.
253
Volume 97B, number 2
PHYSICS LETTERS
1 December 1980
As is discussed in detail in refs. [3-6], all the dynamical information about QCD is hidden in the current
Jv(Cxx)-- f dy~ 6(x- y)W(Cxy)W(Cyx),
(3)
-
j*4<->
9
Fig. 2. The tadpole frame graph, which should be added to
the graph of fig. 1 to preserve gauge invariance.
Cxx
which displays itself for self-intersecting loops. In the
boostrap version of the dynamical equations, this information will be manifest.
The bootstrap equation expresses the area derivative as an infinite series of path integrals of the product
of W-functionals. The systematic method which generates the successive terms is sketched below.
The first term reads
8w(c)
E fay.
8o,
v(x) _ fdaZ Fxz
T
z)
(4)
X W(CyxFxy)W(CxyPyx).
Here Fxz = Fz-x1 is the contour beginning at x and ending at z. The path integration in d-dimensional euclidean space is defined as
x(s)=z
...
~--"
I f ds f
FX2
s
C/)x(~-)exp(-lf "2dr)
X
x(O)=x
0
The part y ~ Fxz, rzx yields the frame graph of fig.
2. The frame graph is "glassed" with the W-functionals, like the skeleton graph in bootstrap approach was
"dressed" by the Green functions and vertices. The
arrow stands for the velocity ~(0) in eq. (4), while the
dot stands for the d y , integration over the corresponding contour.
It is useful to associate the diagram of fig. 1 with
the whole integral in eq. (4), keeping in mind the diagrams of fig. 2. The mnemonic rule is that the dot
never should cross the cut. When it comes to the point
x, it continues along the same solid line (with the coefficient 2 - d).
The compact operator form for eq. (4) reads
8 W(C) _
60lay(X)
....
(5)
~),O-2(Ju + [0X, 0v] O - 2 J x) - U +~ P + O(W3)
(7)
In eq. (4), the inverse operator was represented as
the path integral
The y integration goes along the total loop
-(xlO-2lz)=~pexp(fzd~C~)'rxz
T
Fzx
Cxx
Fxz
The symbol ~ implies antisymmetrization.
Different parts of the y integral produce terms,
which are treated separately within the standard perturbation theory. The gauge dependence of these
terms arises in our approach as freedom of adding
derivatives prior to splitting the total loop integral into
parts.
The part y ~ Cxx corresponds after integration
dC/z 6 (y z) to the frame graph depicted in fig. 1.
~...~~.
•
'~
~
WJ
_
/~ <->
Fig. 1. Frame graph of lower order.
254
(8)
The ordered exponential added, after the Feynman
disentangling, the finite wires Fxz, Pzx between the
points x and z, so that one obtained Jv(FzxCxxFxz).
The next terms in the bootstrap equation can most
easily be found in operator form, iterating eq. (2)
together with the Bianchi identity
euvxo0)/6 W(C)/6 a o x (x) = 0.
(9)
The detailed calculations will be published in an extended paper.
The answer for the next term can be represented
as the sum of four frame graphs shown in fig. 3. Again
the tadpole graphs should be added, so that the dots
Yl, Y2 move along the total closed loops preserving
the planar configuration. The arrow ~ stands for the
difference of two velocities. In the first graph there
are three independent paths Fxz, Pzyl, I'zy2 and hence
there are three path integrations. In the second graph
Volume 978, number 2
PHYSICS LETTERS
l%is law follows from the stand~d analysis of per-
turbative QCD.
Xnour case, one may wenormalize the bootstrap
equation as it stands. The domain of small paths, E
< si <pMm2,produces the counter terms which renor*
m&e the contribution of the basic domain g-z
s
< ~0in the frame graphs. The net effect is the substie
tution of e by pV2, X by Xfland
<
Fig. 3. Frame graphs of the next order.
WC; XI= exp(-PIClWJC;
The solution of the bootstrap equation by iterations,
starting from W= h, yields a ma~fest~y gauge invariant
~agram technique in the loop space In order to recover the ordinary Faddeev-Popov graphs, the arisirrg
gaussian path integrals should be performed using the
identities
A,> I
where ICl is the length of the loop, C, and p - e-1 is
is the known renormaIizatio~ constant [6,7]. it is implied that y is much less than the conAnement scale,
so that ~rturbat~ve QCD holds,
Let us now leave the domain of perturbat~ve QCD
and try to obtain some new results in the infrared domain. It is clear from our pianar topology that the
area Iaw will be self-consistent, since the m~~ima1surface is additive.
One step forward can be made without much effort: Consider the string ansatz
~tC)-rZa~~~exp(-olSl-plCl).
FX_Y
that the tadpole graphs display themselves in
higher orders, when the &on line starting from the
loop of the tadpole connects it with the original loop,
C. The role of the tadpole graphs is to preserve the
gauge invariance of the bootstrap equation. The naive
procedure of summing the Fadd~ev-Popov graphs inside the windows would spoil the gauge invariance.
The general boostrap term contains some planar
~on~guratio~ of paths with the W-f~~~~~~~~~wrresending to each window.
Everywhere the lower bound, E, for the proper
times, s;, in the path integrals should be kept finite.
This provides the ultravio&3 cutoff of our theory.
The bare coupling constant, X, should vary with E, at
E + 0, according to the renormalization group law
(141
WI
Here IS t is the area of an arbitrary surface, S, enclosed
by C When the ansatz is substituted in the r.h.s. of the
bootstrap equation, the arising sums can be transformed as follows:
Notice,
w
The ~eonletry is shown in fig, 4. ltt is ~~~portant that the
area is an additive functional of the surface.
Xv
Fig. 4.
Geometry used in eq. (IQ.
Volume 97B, number 2
PHYSICS LETTERS
Several comments should be made concerning relation (16). It is implied that the surfaces are without
handles - a handle connecting two surfaces in the
1.h.s. would spoil the identity. As for the self-intersections, those should be allowed - otherwise there would
be correlation between two surfaces in the 1.h.s. On the
other hand, the path, 7, in the r.h.s, should avoid itself
at the surface. Otherwise the decomposition of S(C)
into two pieces would not be unique. Also there would
be double counting of the self-intersections of the
image I'xy in physical space, if we allow self-intersections of the original 7xy at the surface.
So, the r.h.s, of the boostrap equation reduces to
the similar ansatz
Z 2 ~ e-o,s,-o'c' friZz ~
8S=C
× f
e-2plTI
7xz
dYe
(17)
z) + O(z3),
TcS
whe re
5s(X, x ' )
= ~ (2)(x -
x')/v~
We are grateful to S.B. Khokhlachev for valuable
discussions and to E.F. Kasatkina for kind hospitality
at Nicolina Gora, where this paper was written.
(19)
OS=C
Here tuu = 8 IS I/Souv is the tangent plane tensor of the
surface and tu(S)is the tangent vector of C.
We observe that the same integral over surfaces is
present in both sides of equation. As for the integrand,
in general, it is different in the 1.h.s. and in the r.h.s.
However, for sufficiently large and smooth surfaces
(which dominate at large smooth loops), the surface
paths in the r.h.s, are relatively small, so that the path
integrals depend on the local properties of the surface.
There are only two local tensors of appropriate rank
and dimension - the same as in eq. (19). The comparison of coefficients yields the relation between o, 0
and Z in eq. (15).
Up to now, the preexponential factor, Z, was quite
arbitrary. As is done usually in the bootstrap approach,
it should be fixed by joining, at intermediate loops,
256
the self-consistent asymptotics (1)with the perturbatire solution respecting the initial condition. We expect that infrared properties of 4-dimensional QCD
will be important at this matching.
Let us compare our mechanism of confinement
with that of the lattice gauge theories. In our equations, we did not utilize the global properties of the
SU(N) group, which are responsible for the area law
in lattice gauge theories. In particular, the topological
excitations, corresponding to the center Z N ~ U(1) of
SU(N), were not taken into account. These exictations modify the equation of motion [8].
lnstead of that we used the proper time regularization [4], which makes perturbative QCD manifest.
As a consequence, the string tension in our theory is
related to the gauge coupling constant, whereas in the
Z N theory o f confinement this was an independent
parameter.
So, our mechanism of confinement is closer in spirit
to the 't Hooft condensation of the planar graphs [9],
rather than to the Wilson string of the strong coupling
expansion.
(18)
is the surface &function.
As for the 1.h.s. of the bootstrap equation, it yields
e -°lSl-olCI (-atuu +ptu dtu/dS ) .
1 December 1980
References
[1] J. Gervais and A. Neveu, Phys. Lett. 80B (1979) 255;
Y. Nambu, Phys. Lett. 80B (1979) 372;
A.M. Polyakov, Phys. Lett. 82B (1979) 247.
[2] Yu.Mu. Makeenko and A.A. Migdal, Phys. Lett. 88B
(1979) 135.
[3] Yu.M, Makeenko and A.A. Migdal, Moscow preprint
ITEP-23 (1980); Yad. Fiz. 32 (1980) 838.
[4] A.A. Migdal, Ann. Phys. 126 (1980) 279.
[5] Yu.M. Makeenko, Moscow preprint ITEP-141 (1979);
Yad. Fiz. 32 (1980) No. 6.
[6] A.M. Polyakov, Nucl. Phys. B164 (1980) 171.
[7] V.N. Dotsenko and S.N. Vergeles, Chernogolovka preprint 1980-1 (1979).
[8] S.B. Khokhlachev and Yu.M. Makeenko, Moscow preprint 1TEP-126 (1979).
[9] G. 't Hooft, Nucl. Phys. B72 (1974) 461.