Institut für Theoretische Physik
Freie Universität Berlin, Arnimallee 14, 1000 Berlin 33, Germany
Joint Institut for Nuclear Research, Dubna, SU 141980, Russia
H. Kleinert and A.M. Chervyakov
Mon Apr 15 10:26:55 MET DST 1996
Color-electric flux tubes between quarks have a thickness of the order of the dimensionally transmuted coupling constant. They should therefore display a finite resistance to a change in extrinsic curvature. To account for this property, a higher-derivative stiffness term was added to the Nambu-Goto (NG) action of an infinitely thin string [1,2]. The resulting Polyakov-Kleinert (PK) string seemed to render a more relevant description of gluon forces in QCD than the former NG string. Particularly attractive was the fact that a PK model containing only a stiffness term is asymptotically free at short distances [1,2] and can generate the string tension spontaneously [3]. For other properties see [4].
Unfortunately, however, the PK model has also serious
consistency problems.
First of all, there exists
an unphysical ghost pole in the propagator
[2]. The pole is generated
by the second derivatives with respect to the string
coordinates in the action.
Second, the PK model shares with all higher-derivative
theories
the lack of a lowest-energy state
[5,6,7,8].
Third, the squared
string tension of the PK model has
an unphysical imaginary part at high
temperatures or low 
, even though this model
unlike the
NG string
displays the correct power
behavior 
for 
consistent with asymptotic freedom
(except, however, for an imaginary factor i) [9].
It is the purpose of this letter to propose a new string model in which the problems of the PK string are absent while the attractive features are preserved. This new model has a negative extrinsic curvature stiffness. It was inspired by properties of magnetic flux tubes in type-II superconductors and analogous properties of Nielsen-Olesen vortices in relativistic gauge models. In the London limit, these have a nontrivial energy spectrum, from which several calculations have deduced a negative stiffness [10,11,12]. By analogy, we hypothesize that the stiffness QCD strings is also negative. The action we propose for a description of such a string is
where 
with 
are the string coordinates in a d-dimensional
euclidean spacetime parametrized by 
,
and 
is the induced
metric on the
world surface of the
string with 
being its inverse and 
.
Covariant differentiation with respect to 
is denoted by 
,
and 
is Laplace-Beltrami operator. The physics of this model
is governed by two mass scales M and
, and a
dimensionless constant c.
Due to nonzero radius of thickness the interaction between surface elements of the string has a nonlocal character, just like in a magnetic flux tube. In momentum space, the quadratic part of the action (1) gives rise to the free propagator
For small k, this has an expansion
with the mass parameter 
.
The stiffness 
is the inverse coefficient of 
in the action and has the negative
value
.
The propagator contains only a single pole at 
.
This is to be contrasted with
the PK model
where
corresponds to the positive stiffness
and
contains an
unphysical pole with negative residue at 
.
The inverse propagator (2)
is plotted in Fig. 1 for sample parameters
and c. The
fluctuations with large momenta are more violent than in the PK
string with the opposite
curvature stiffness, but less violent than in
a NG string with reduced tension 
,
which is eventually approached.
The above string is inspired by the properties
of
a vortex line in a type-II superconductor [13],
which
has a diameter of the order of
the penetration depth
governing
the parameter
of the model.
The inverse mass 
corresponds to the coherence length
which goes to zero in the London limit.
In QCD, the parameters of the model have the
following physical origin.
As long as there are no
quarks, QCD gives rise to
a string with only one length scale
which enters into both the string tension
and the thickness of the flux tube.
Thus it determines the two parameters
M and 
with some ratio
between them, which
is a consequence
of the detailed dynamics of the gluon system.
As quarks are included,
this ratio is modified
and an additional length scale arises
at which the
vacuum polarization of the gluon propagator
becomes important
and changes the stiffness
properties of the flux tube.
This length scale is proportional
to some average over
the inverse quark masses,
and
accounted for by
the model parameter
.
For practical calculations,
it is convenient
to approximate (2) by
(3),
and cut all
momentum integrals off at 
.
To have a physical penetration depth, we must take 
to be smaller than
the cutoff 
, or
1<c<2.
The action associated with the approximate Green function (3) is
Figure:
The inverse propagators
of the model (1) (solid curve) and of the approximation
(5)
(dashed lines) for 
and 
. The dotted curve shows the
behavior in a PK string
with the opposite value of the curvature stiffness, the dash-dotted
curve corresponds to a NG string with
reduced tension 
. We see that the large-k
fluctuations are much
more violent than
in the PK string, but less violent than
in a NG string, which is eventually approached.
We shall now derive the high-temperature limit of the total
string tension, which can be done exactly in
the limit 
.
As usual in this limit,
we make the
metric field 
an independent fluctuating field,
and force it to be equal
to the induced metric 
by means of
Lagrange multipliers 
, adding to (5)
a term
It is straight-forward to calculate the partition function
of the string at a finite temperature T anchored to two static quarks
separated by a
large distance R. The world sheet
of the string
is periodic in the imaginary-time direction with a
period 
. Choosing
coordinates with
running from 0 to R,
and 
from 0 to 
, we parametrize
the world sheet as
with 
describing the transverse fluctuations
which occur quadratically in the extended action
.
These can be integrated out producing a trace log with a factor d-2
which in the limit
suppresses the fluctuations
of 
and 
,
so that the saddle point approximation provides us with
an exact partition function.
At the saddle point, we may
assume diagonal forms for
the metric and the Lagrange multipliers:
with the constants 
and 
.
The resulting effective action
=
+
consists of a mean-field term
which also appears in NG and PK string models, and a one-loop contribution
Here
the temporal
components 
are summed over all thermal Matsubara frequencies
.
The effective string tension is defined by
, where 
is the free energy per unit length
evaluated at the extremal values of 
and 
.
At low temperatures,
and the one-loop term (9) is
very small compared to the mean-field term (8),
so that the
extremum of 
lies at
, 
, where
. At high temperatures, on the other hand,
and the one-loop term (9) becomes
essential. In this limit, it can easily be evaluated analytically.
First, however, let us make
an instructive
observation:
If we assume for a moment that the cutoff 
be artificially small,
so that
, we can neglect
the denominator of the first term in Eq. (9) and see
that
the reduced one-loop action (9)
takes the
NG form.
By the same calculation as in
Ref. pa:NGstring
we then find at high temperatures
the squared free energy
per unit length
This is in bad qualitative disagreement with the behavior derived from QCD in the limit of large-N by Polchinski [15]:
In contrast to this, the present
model has the correct high-temperature behavior. Since the
cutoff 
is, of course, much
larger than 
,
the negative curvature stiffness
can take effect.
In the high-temperature limit 
,
the first term in the logarithm (9)
is small compared to the second-derivative term,
and one-loop
action (9) reduces to
The first term comes only from zero Matsubara frequency,
in the large 
-limit,
the second contains the
full spectral sum in (9), but approximated by the
leading second-derivative term.
Adding to (12)
the mean-field term (8),
the resulting
high-temperature
effective
action 
has a saddle point at
where 
and 
are determined from
In terms of 
, the associated squared free
energy per unit length reads simply
The deconfinement temperature 
is determined by
the vanishing of 
, which yields
Here 
is the total
string tension of at infinite beta, and
the parameter 
is related to the stiffness 
as follows
The parameter 
is an unique real positive root
of the polynomial equation of 9th order which
exists for a reasonable range of the mass parameter
. Numerical studies
show that the deconfinement temperature (16)
depends only weakly on the stiffness parameter 
and lies in our model always around
.
This is somewhat larger
than the estimates
derived from rigid strings [4].
Equations
(13) and (14)
determine
via a polynomial of 14th order. In the above
range of the mass parameter, 
,
there
are
real roots for all 
.
At high-temperatures,
these have the asymptotic behavior
Inserted into (13), they yield the diagonal elements of the metric. The resulting high-temperature behavior of (15) is, therefore,
This this agrees in sign and 
-behavior
with the QCD result (11), if we neglect
the
weak 
-dependence
of the running coupling constant 
in (11).
Note that the
two strings are not immediately
comparable since
the
result (11) was obtained in the 
-limit
where the QCD string
becomes
infinitely thin and has therefore
no
curvature stiffness.
At large but finite N, however,
it does acquires a finite
thickness, as we know from Monte Carlo simulations,
and the behavior (11)
can be modified at most by small terms of order 
.
In this regime the agreement
between our
result (18) and the QCD result (11)
is definitely significant.
We consider this agreement as an important evidence for a negative stiffness of hadronic strings, which thus behave very similar to vortex lines in type-II superconductors. Our conclusion should be verified in lattice gauge simulations.
Acknowledgments
The authors thank Dr. V.V. Nesterenko for
his collaboration at an early stage of this work.
One of us (A.C.) acknowledges support from the German
Academic Exchange Program and from
the Russian Foundation for Fundamental Research
(grant 93-02-3972). He
is grateful to Prof. H. Kleinert
and his
group for their kind hospitality during
his stay at the Freie
Universität Berlin.
