Mean field quantization of effective string
Revised: May
eld quantization of e ective string
Yuri Makeenko 0 1
0 B. Cheremushkinskaya 25 , Moscow, 117218 Russia
1 Institute of Theoretical and Experimental Physics
I describe the recently proposed quantization of bosonic string about the meaneld ground state, paying special attention to the di erences from the usual quantization about the classical vacuum which turns out to be unstable for d > 2. In particular, the string susceptibility index str is 1 in the usual perturbation theory, but equals 1/2 in the mean eld approximation that applies for 2 < d < 26. I show that the total central charge equals zero in the mean eld approximation and argue that uctuations about the mean eld do not spoil conformal invariance.
Long strings; Conformal Field Models in String Theory; Bosonic Strings

Mean
1 Introduction
2 The mean eld ground state
3 Instability of the classical vacuum
4 Stability of the mean eld vacuum
5 The string susceptibility index
6 Fluctuations about the mean eld
7 Computation of the central charge
8 \Semiclassical" correction to the mean eld
8.1
8.2
Correction to Tzz
Correction to Se
8.3 Remark on the universality
9 Discussion
A.1 T a
a
A.2 Tzz
A Leadingorder explicit computations
B \Semiclassical" corrections
B.1 Contribution to Tzz
B.2 Contribution to Se
C Mathematica programs
C.1 Program for computing diagrams in gure 3 that results in (8.9)
C.2 Program for computing diagrams in gure 4 that results in (8.12)
1
Introduction
Strings or more generally twodimensional random surfaces have wide applications in
physics: from biological membranes to QCD. However, a nonperturbative theory of
quantum strings, which goes back to 1980's, makes sense only if the dimension of target space
{ 1 {
d < 2, where the results both of dynamical triangulation [1{3] and of conformal eld
theory [4{6] are consistent and agree. For d > 2 the scaling limit of dynamically
triangulated random surfaces is particlelike1 rather than stringlike because only the lowest mass
scales but the string tension does not scale and tends to in nity in the scaling limit. [8]
Analogously, the conformal eld theory approach does not lead to sensible results for
2 < d < 26. [4{6] The conclusion was that quantum string does not exist
nonperturbatively for 2 < d < 26, while it beautifully works for d < 2. However, we understand that
strings do exist as physical objects in d = 4 spacetime dimensions.
A potential way out was to adopt the viewpoint that string is not a fundamental object
but is rather formed by more fundamental degrees of freedom. This philosophy perfectly
applies to QCD string,2 where these are uxes of the gauge eld. String description makes
sense only for the distances larger than the con nement scale. For shorter distances the
quarkgluon degrees of freedom are more relevant due to asymptotic freedom. This picture
is well justi ed both by experiment and by lattice simulations. The string tachyon which
is a shortdistance phenomenon does not show up in the QCD spectrum.
A breakthrough along this line is due to the \e ective string" philosophy [10], which
works perturbatively order by order in the inverse string length (for recent advances see [11{
15]). Then string quantization is consistent even below the critical dimension (d = 26 for
the relativistic bosonic string) and a few leading orders reproduce [10, 16, 17] the
AlvarezArvis groundstate energy [18{20]. In this Paper I shall pay much attention to this issue.
In the recent series of papers [21{24] it has been understood why lattice string
formulations resulted in the particlelike continuum limit. A nonperturbative mean eld solution of
the NambuGoto string showed that the usual classical vacuum about which string is
quantized is unstable for d > 2, while another nonperturbative vacuum is stable for 2 < d < 26,
like it happens in the wellknown example of the twodimensional O(N ) sigmamodel. For
the true ground state the value of the metric at the string worldsheet ( ab =
ab in the
conformal gauge) becames in nite in the scaling limit. For this reason an in nite amount
of stringy modes (which is /
=a2 with a being a UV cuto ) can be reached even at
the distances of order a. That was in contrast to the usual continuum limit in quantum
eld theory, where the amount of degrees of freedom can be in nite only if the correlation
length is in nite. The discovered phenomenon is speci c to theories with di eomorphism
invariance and was called the Lilliputian continuum limit.
The task of this Paper is to analyze properties of the mean eld vacuum which plays
the role of a \classical" string ground state and \quantum"
uctuations about it. I put
here quotes to emphasize this state is a genuine nonperturbative quantum state from the
viewpoint of the usual semiclassical expansion in 0 about the classical vacuum. We thus
perform a resummation of this expansion with the leading order given by the sum of
bubblelike diagrams. An analogy with the twodimensional O(N ) sigmamodel at large N can
be again instructive. I shall pay special attention to a comparison with the
KnizhnikPolyakovZamolodchikov (KPZ){DavidDistlerKawai (DDK) results [4{6] for the
parturbative vacuum, which is applicable for d < 2.
1A detailed description can be found in the book [7].
2For a brief introduction see e.g. [9].
{ 2 {
where the consistency is explicitly demonstrated to a few lower orders of the perturbative
expansion [10, 16, 17]. I then analyze a \semiclassical" correction to the mean eld
approximation and show that it does not spoil conformal invariance in spite of logarithmic
infrared divergences caused by the propagator of a massless eld, which cancel in the sum
of diagrams.
This Paper is organized as follows. In sections 2, 3, 4 I review the results [21{23] which
form a background for further investigations. Section 5 is devoted to the computation of the
mean eld value of the string susceptibility index str = 1=2 and its comparison to the
perturbative value str = 1. In section 6 I formulate a general procedure for expanding about
the mean eld and describe PauliVillars' regulation for computing the energymomentum
tensor and its trace anomaly, which does not rely on approximating the involved
determinants by (the exponential of) the conformal anomaly. It section 7 I compute the total
central charge of the system in the mean eld approximation and show that it vanishes for
2 < d < 26. Section 8 is devoted to the \semiclassical" expansion about the mean
eld.
I show that logarithmic infrared divergences which might spoil conformal invariance are
mutually canceled. The results obtained and tasks for the future are discussed in section 9.
Some explicit computations are presented in appendices A, B by using the Mathematica
programs from appendix C.
2
The mean eld ground state
We consider a closed bosonic string in target space with one compacti ed dimension of
circumference . The string wraps once around this compact dimension and propagates
through the distance L. The string worldsheet has thus topology of a cylinder. There is
no tachyon for such a string con guration, if
is larger than a certain value to guarantee
that the classical energy of the string dominates over the energy of zeropoint uctuations.
The NambuGoto string action is given by the area of the surface embedded in target
space. It is highly nonlinear in the embeddingspace coordinate X . To make it quadratic
in X , we rewrite it, introducing a Lagrange multiplier ab and an independent intrinsic
metric ab, as3
S = K0
d2! pdet +
K0 Z
2
where K0 stands for the bare string tension. The equivalence of the two formulations can be
proven by path integrating over the functions ab(!) and ab(!) which take on imaginary
and real values, respectively.
It is convenient to choose the worldsheet coordinates !1 and !2 inside an !L
!
rectangle in the parameter space. Then the classical solution Xcl minimizing the action (2.1)
3We denote det = det ab and det = det ab.
ab) ;
(2.1)
{ 3 {
linearly depends on !
The classical value of ab coincides with the classical induced metrics
L2
!L2 ; !2
2 !
which becomes diagonal for
The classical value of ab reads
! =
L
!L:
calb = calbpdet cl
and simpli es to calb = ab if eq. (2.4) is satis ed.
We apply the pathintegral quantization to account for quantum
uctuations of the
Xq . We thus obtain the action, governing the elds ab and ab,
X elds by splitting X
= Xcl + Xq and then performing the Gaussian path integral over
Z
1
O = pdet
K0 Z
2
Sind = K0
d2! pdet +
d
2
ab) +
tr log( O);
The operator O reproduces the usual twodimensional Laplacian
righthand side of eq. (2.5). Its determinant is to be computed with the Dirichlet boundary
condition imposed. Quantum observables are determined by the path integral over
and ab with the action (2.8), which runs as is already mentioned over imaginary
and real ab(!). The action (2.6) is often called the induced (or emergent ) action to be
distinguished from the e ective action which is usually associated with slowly varying elds
for ab given by the
in the lowmomentum limit.
It is convenient to x the conformal gauge when ab =
the log of the determinant of the ghost operator [25]
ab, so that pdet
= . Then
Ogh ba =
a
b
1
2
( ba log )
is to be added to the induced action (2.6) [or (4.11) below]. The operator (2.7) acts on
twodimensional vector functions whose one component obeys the Dirichlet boundary condition
and the other obeys the Robin boundary condition [26, 27]. The subtleties associated with
the boundary conditions will be inessential both for the matter and ghost determinants for
when only the bulk terms survive.
We shall describe in section 6 how to accurately compute the determinants using
the PauliVillars regularization but let us assume for a moment that
what is needed for the mean eld approximation.
an exercise in string theory courses with the result
Se =
K0
guarantees that ab and
return to this issue soon.
for L
. Here
2 cuts o eigenvalues of the operators involved. The rst and second
terms on the righthand side are classical contributions, while the sign of the third term is
negative for d=
> 2 to comply with positive entropy. Technically, it comes as the product
of the eigenvalues divided by
, where every multiplier is less than 1. The last term is
to nd the mean eld con guration which
describes the string ground state. The di erence from the classical ground state (2.2),
(2.3), (2.5) is that we now minimize the action, taking into account the determinants
coming from X
and ghosts, while the classical (perturbative) ground state minimizes the
classical action. Additionally, similarly to the classical case we have to minimize (2.8)
over the ratio ! =!L which plays the role of the modular parameter of the cylinder. This
ab are diagonal as is required by the conformal gauge. We shall
The minimum of (2.8) is remarkable simple [21{23]
(d
3K0
2)
:
s
2
{ 5 {
The value of the action (2.8) at the minimum (2.9) is
The meaning of the above minimization procedure is clear: we have constructed a
saddlepoint approximation to the path integral, which takes into account an in nite set of
diagrams of perturbation theory about the classical vacuum. This approach is quite similar
to that4 in the twodimensional O(N ) sigmamodel, where one sums up bubble diagrams
of the 1=N expansion by introducing the Lagrange multiplier u to resolve the constraint
~n2 = 1. After integration over the elds ~n one obtains an induced action as a functional
4See e.g. the book [28].
=
=
!
=
1
2
+
2
!L
L
s
2
2K0
!2
2
s
+
(d 2)
6K0
1
4
r
1 +
uctuations of u about this mean eld vacuum have to be included, but they are small
even at N = 3 because, roughly speaking, there is only one u while the induced action is of
order N , i.e. large as is needed for a saddle point. Alternatively, the perturbative vacuum
~ncl = (1; 0; : : : ; 0) possesses an O(N
1) symmetry rather than the O(N ) symmetry as the
saddlepoint vacuum does and the elds ~n
uctuate strongly. For our case the number of
elds X
in the sigma model (2.1) is d, so the saddlepoint is justi ed by K0
d !
At nite d the saddlepoint solution (2.9) is associated with the mean eld approximation.
The minimization of the action (2.8) over ! =!L can be now understood as follows.
In the mean eld approximation we consider the action to be large, doing all integrals by
the saddle point, including the integral over the modular parameter, which is present for
the cylinder topology.
A few comments concerning the solution (2.9) are in order:
Equation (2.9a) is wellde ned if the bare string tension K0 > K given by
K
=
d
1 + pd2
2d
2
The classical vacuum (2.2), (2.3), (2.5) is recovered by (2.9) as K0 ! 1, while the
expansion in 1=K0 makes sense of the semiclassical (perturbative) expansion about
this vacuum. The usual oneloop results are recovered to order 1=K0.
The larged groundstate energy [
18
],5 where an analytic regularization was used, are
recovered by eq. (2.10) for
2 = 0. Analogously, the groundstate energy obtained
by the old canonical quantization [
19
] is reproduced by our mean eld
approximation. This is not surprising because
uctuations of
are ignored in the canonical
quantization.
squared.
Equation (2.10) is wellde ned for
larger than p (d
2)=3K0
1= , but
becomes imaginary otherwise. The singularity was linked [18{20] to the tachyon mass
The metric (2.9b) becomes in nite when K0 ! K given by eq. (2.11). This is crucial
for constructing the scaling limit.
At the classical level cl coincides with the induced metric as is displayed in (2.3). In
the mean eld approximation it is superseded by
(2.12)
where the average is understood in the sense of the path integral over X . Equation (2.12)
follows from the minimization of the e ective action over
approximation
coincides with the averaged induced metric.
ab. Thus, in the mean eld
5The original computation [
18
] used the NambuGoto string. How the same result can be obtained for
the Polyakov string is shown in [29].
{ 6 {
Instability of the classical vacuum
The usual semiclassical (or oneloop) correction to the classical groundstate energy due to
zeropoint uctuations [30] is described in textbooks. The sum of the two reads
To make the bulk part of (3.1) nite, it is usually introduced the renormalized string
S1l =
K0
(d
2
2) 2 L
(d
2) L
in the mean eld approximation for !L = L and ! =
1pK0. Inverting eq. (3.6), we
(j) =
1
2
+ r
1 + j + K0
2
1 + j + K0
! 1. Then it is assumed that it works order by order of the
perturbative expansion about the classical vacuum, so that KR can be made nite by ne
We see however from eq. (2.10) how it may not be case. The righthand side of
eq. (2.10) never vanishes with changing K0. The point of view on eq. (3.1) should be that
for d > 2 the oneloop correction simply lowers the energy of the classical ground state
which therefore may be unstable.
As we show in the next section, the action (2.8) indeed increases if we add a constant
imaginary addition
to . However, the sum of the two linear in
terms in eq. (2.8)
vanishes for
given by eq. (2.9a), so the action does not depend on
at the minimum.
This reminds a valley in the problem of spontaneous symmetry breaking.
To investigate it, we proceed in the standard way, adding to the action the source term
{ 7 {
and de ning the eld
Ssrc =
K0 Z
2
d2! jab
ab
ab(j) =
2
K0 jab log Z:
Minimizing the action with the source term added for constant jab = j ab, we nd [23]
1
2
(j) =
1 + j +
2
K0
+
s
1
4
1 + j +
like in the studies of symmetry breaking in quantum
eld theory.
In the mean eld approximation we then obtain
1
K0L
2
K0
s
2d 2
K0
1):
(3.9)
1 and the potential (3.9) decreases
with increasing
because the second term on the righthand side has the negative sign,
demonstrating an instability of the classical vacuum. If K0 > K given by eq. (2.11), the
potential (3.9) linearly increases with
for large
and thus has a (stable) minimum at
(0) =
+
1
2
2
r
which is the same as (2.9b) for ! =
1=pK0. Near the minimum we have
"
( ) =
1 +
The coe cient in front of the quadratic term is positive for K0 > K
which explicitly
demonstrates the (global) stability of the mean eld minimum (2.9b).
The situation is di erent for d < 2, where quantum corrections increase the vacuum
energy. For this reason the classical vacuum is energetically favorable to the mean eld
one. It is explicitly seen for d < 0 from eq. (3.9) where
1 has to be negative. The
function ( ) then increases with decreasing
near
= 1 and the mean eld solution is a
maximum, not a minimum.
The conclusion of this section is that the classical vacuum is not stable for d > 2 where
the mean eld vacuum is energetically favorable. This reminds spontaneous generation of
in quantum
eld theory. The situation is opposite for d < 2, where the classical vacuum
has lower energy than the mean eld vacuum.
{ 8 {
Stability of the mean eld vacuum
Let us now consider stability of the mean eld vacuum under wavy uctuations, when
(!) =
+
(!);
and
.
The divergent part of the e ective action reads [23]
For constant ab =
this reproduces the divergent part of eq. (2.8) above.
Expanding to quadratic order in uctuations
q
det( ab +
ab) =
+
1
2
aa
1
8
2 =
1
1 aa
2
2pdet
+
2
;
where
S2 =
Z
d2p
(2 )2
A
d2!
2
K0
d 2 Z
The rst term on the righthand side of eq. (4.4) plays a very important role for
dynamics of quadratic
uctuations. Because the path integral over
imaginary axis, i.e.
ab is pure imaginary, the rst term is always positive. Moreover, its
ab goes parallel to
exponential plays the role of a (functional) deltafunction as
The same is true for a constant part of
ab.
! 1, forcing
ab =
ab.
For the e ective action to the second order in uctuations we then nd the following
(p) ( p)
+ 2A
(p) ( p)
+ A
(p) ( p)
2
;
Here c is a regularizationdependent constant.
A
A
A
2
=
=
=
(26
96
1
2
K0
d 2
{ 9 {
In the scaling limit, where [21{23]
as
! 1 keeping the renormalized string tension KR
xed, we have
K0 ! K +
KR2
1 +
r
1
2
d
!
;
so only A
value
diverges as
2
. Therefore, typical
1=
so that ab is localized at the
This is quite similar to what is shown in the book [31] for the
uctuations about the
classical vacuum. Thus only
uctuates.
case the eld ab(!) is localized at the value
Equation (4.9) holds in the conformal gauge, where abpdet
=
ab. In the general
where
is constant for the worldsheet parametrization in use.
We can therefore rewrite the righthand side of eq. (2.1) in the scaling limit as
which reproduces the Polyakov string formulation [25] for
= 1. As shown in [
21, 22
] the
action (4.11) is consistent only for a certain value of
which is regularizationdependent.
One has
= 1 for the zetafunction regularization but
< 1 for the propertime
regularization or the PauliVillars regularization.
A subtlety with the computation of the determinants in the conformal gauge is that
X
and
do not interact in the action (4.11) since
in the conformal gauge ab = g^ab . Here g^ab is a ducial metric which we can set g^ab = ab
without loss of generality.
larization
But the dependence of the determinants on
appears because the worldsheet
reguab =
ab:
ab =
abpdet ;
" =
1
2pdet
=
1
2
depends on
owing to di eomorphism invariance. For smooth the determinants are given
by the usual conformal anomaly [25]. An advantage of using the PauliVillars regularization
in the conformal gauge is that the implicit dependence on the metric becomes explicit as
is described in section 6.
Integrating over the matter and ghost elds, we arrive for g^ab = ab to the induced
Sind =
where the ghost operator is displayed in eq. (2.7). Evaluating the determinants, we nd
which for
= 1 reproduces the usual result.
We see from eq. (4.15) (as well as from eq. (4.5) with
= 0) that the action, describing
uctuations of the metric, is positive only for d < 26 and becomes negative if d > 26. Thus,
as far as the local stability of the action under wavy uctuations is concern, it is the same
about the mean eld vacuum as about the usual classical vacuum. This instability is
probably linked to the presence of negativenorm states for d > 26 [32, 33].
)
Z
d2!
a
reg
1
2
tr log
+
The string susceptibility index
A very important characteristics of the string dynamics is the string susceptibility index
str which characterizes the string entropy and is determined from the preexponential in
the number of surfaces of xed area A by
e F (A)
Z
d2z
/
A!1 A str 2 eCA;
where C is a nonuniversal constant. F (A) on the lefthand side has the meaning of the
Helmholtz free energy of a canonical ensemble at xed area A. Introducing the Lagrange
multiplier, we rewrite (5.1) as
Z
d2z
A
=
dj ej(R d2z
A) ;
where the integral over j runs parallel to the imaginary axis. This j is the same as
introduced in section 3 except for the integral over j.
Let us rst consider the integrand. The saddlepoint solution is given by eq. (3.6).
Then the integrand in (5.2) has an extremum at j(A) given by eq. (3.7) with
substituted
by A=Amin, Amin = L . Expanding about the extremum, we nd [24]
jA
K0Amin
(A) =
s
2d 2 s
+
K0
r 2K0
Amin
A
Amin
A
Amin
A
Amin
1
1
A
Amin
3=2
( j)2 :
1 +
2
K0
The integral over j = j
j(A) goes along the imaginary axis and thus converges. For
F (A) we obtain
F (A) = p2d 2K0pA(A
Amin)
A(K0 + 2) + log [A(A
Amin)] + const:
(5.4)
According to the de nition (5.1) of the string susceptibility index, we expect
F (A) = regular + (2
str) log
3
4
A
Amin
for A
Amin. Comparing with (5.4), this determines str = 1=2. It can be shown [24] that
the oneloop correction contributes only to the regular part of F (A) and does not change
the singular part that gives str = 1=2. This value can be exact because it is linked only
to the emergence of the squareroot singularity which is not changed by higher orders.
We are to compare the mean eld result for str with the oneloop computation of (5.1)
about the classical vacuum which is almost trivially done by changing
! A and gives
(5.1) / A 1 e(d 2) 2A=2
resulting in str = 1. We can compare it with the formula of the d !
1 expansion [34]
generalized to an arbitrary genus in [
35, 36
]. Since we deal with the worldsheet having
topology of a cylinder which has two boundaries, its Euler character equals 0 like for a
torus. This explains why there is no ddependence of str. We have got str = 1 rather
than str = 2 as in [
35, 36
] because we deal with an open rather than a closed string.
The discrepancy between the obtained mean eld value str = 1=2 and the perturbative
value str = 1 is due to the fact that the vacua are di erent. The former applies for
2 < d < 26, while the latter applies for d < 2.
6
Fluctuations about the mean eld
The instability of the e ective action for d > 26 implies that we cannot straightforwardly
make a systematic 1=d expansion as d ! +1. This is in contrast to the d !
1 limit
which comes along with the usual perturbative expansion because the vacuum is then just
classical. The usual semiclassical expansion as d !
1 cannot be extended to d > 2
because the vacuum states are di erent for d < 2 and d > 2.
To go beyond the mean eld for 2 < d < 26, we de ne the partition function
Z[h] =
Z
D e Sind=h
(5.5)
(5.6)
(6.1)
with Sind given by eq. (4.14). Here we have introduced an additional parameter h to control
the \semiclassical" expansion about the mean
eld which plays the role of a \classical"
vacuum. This procedure makes sense of the change d ! d=h for the number of the X elds
and simultaneously 2 ! 2=h for the number of the ghost elds. Diagrammatically, this
mean eld corresponds to summing up bubbles of both matter and ghosts. The mean eld
approximation is associated with h ! 0, while the expansion about the mean eld goes
so that
is convergent. Here M
equations by
For the propertime regularization we have instead
det( O) det( O + 2M 2)
det( O + M 2)2
;
det( O) reg
! 1 is the regulator mass which is related to
in the above
2 =
1
are then proportional to hl. In reality h = 1
but we can expect that the actual expansion parameter is 6h=(26
d) like in the usual
semiclassical expansion as d !
1. Then the expansion can make sense for d = 4.
The action in eq. (6.1) is given by (4.14). For the PauliVillars regularization the
determinants are regularized by the ratio of massless to massive determinants [24]
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
#
:
(6.8)
(6.9)
the type
to get
The path integral over the regulator elds generates the propagator
XM (k)XM ( k) =
K0( k2 + M 2 )
We have added in (6.2) the ratio of the determinants for the masses p
2M and M to
cancel the logarithmic divergence at small , because the Seeley expansion
D
! e O !
E
=
4
1
1
pdet
+
R
24
+ : : :
starts with the term 1= . This is speci c to the twodimensional case.
The massive determinants in eq. (6.2) can also be represented as path integrals of
det
d=2
Z
DXM e 2
over the elds XM (!) with normal statistics or YM (!) with ghost statistics and the double
number of components. We can explicitly add these regulator elds to the action (4.12)
)
Z
d2!
+
K0 Z
2
d2!
"
and the triple vertex of the
XM XM interaction
( p)XM (k + p)XM ( k) truncated =
K0M 2
The latter vanishes for M = 0 as it should owing to conformal invariance, but explicitly
breaks it at nonzero M . Notice that path integration over all matter elds (both X
and
the regulators) runs with a simple nonregularized measure. This makes it very convenient
to derive (regularized) Noether's currents and to calculate their anomalies.
An instructive exercise is how to compute the usual anomaly in the trace of the
energymomentum tensor
2
[Taa]mat = 4 K0 41
+
0
1
13
HJEP07(218)4
M 2(YM(i))2 + 2M 2Xp22M A5 :
(6.11)
Averaging (6.11) over the regulator elds, we obtain the diagrams depicted in gure 1,
where the solid line corresponds to the propagator of the regulator elds Xp2M or YM
while the wavy line corresponds to
. We have explicitly in momentum space
d Z
h
d2k
(2 )2
2M 2
k2 + 2M 2
2M 2
k2 + M 2
d
h
=
M 2 log 2;
(6.12)
Figure 1b = 2
d Z
h
d2k
(2 )2
4M 4
( k2 + 2M 2 )( (k
p)2 + 2M 2 )
2M 4
( k2 + M 2 )( (k
p)2 + M 2 )
d
d
Figure 1a = 2
reproducing eq. (6.4), and
where
and for brevity we denoted
det
a
=
det
(6.13)
(6.14)
(6.15)
(6.16)
:
(6.17)
G(p) = 12 B
0
4m4 arctanh
ppp2+8m2
p2+4m2
2m4 arctanh
ppp2+4m2 1
p2+2m2
p(p2 + 8m2)
p(p2 + 4m2)
C
A
p m 2
= p
m2 = M 2 :
Ogh b
a
det
The e ect of the diagram in gure 1c and the next orders is to complete the result to
scalar curvature R as is discussed in appendix A. Adding all diagrams and using eq. (6.4),
we obtain for the contribution from matter
h[Taa]mati = 4
K0(1
)
2
+
d
12
R:
It still remains to compute the contribution of the ghost determinant which we also
regularize by the PauliVillars regularization
d
2
det
Ogh ba + 2M 2 a
b
Ogh ba + M 2 a 2
b
The computation of the contribution from ghosts is pretty much similar to the one [25{27]
for the perturbative vacuum and adding it with (6.16) we obtain for the trace of the total
energymomentum tensor (matter plus ghosts)
hTaai
D [Taa]mat + [Taa]gh
E = 4
K0(1
)
d
2
1
2 +
d
26
12
R;
(6.18)
which is the same as =
acting on (4.15). The average in this formula is over the matter
and ghost elds but not over
which plays the role of an external eld.
For
given by eq. (2.9a) the divergent term vanishes, so we reproduce the usual
conformal anomaly. The reason is that we have essentially made a oneloop calculation for
the Polyakovlike action (4.12) with a constant
ducial metric ^ab =
ab and the result
coincides with the one about the classical vacuum because of the background independence.
7
Computation of the central charge
If ab is considered as a classical background metric, only matter and ghosts contribute
to the central charge of the Virasoro algebra which equals d 26 like in eq. (6.18). Then
the conformal anomaly vanishes only in d = 26 (the critical dimension) which reproduces
the result of the old canonical quantization. We shall now see how this is modi ed when
quantum
uctuations of ab are taken into account in the mean eld approximation.
For this purpose let us compute the correlator of the two zzcomponents of the
energymomentum tensor
Tzz
T (z) = Tmat(z) + Tgh(z):
Classically, the X eld does not interact, as is already pointed out, with the metric
in the conformal gauge because of conformal invariance. Like in the previous section we
shall make use of the PauliVillars regularization, where Tmat(z) explicitly depends on the
regularizing elds as
(7.1)
(7.2)
Tmat(z) = 2 K0
The diagrams contributing to the correlator hT (z)T (0)i in the mean eld
approximation are depicted in gure 2, where the solid line corresponds to the propagator of the eld
X (and its regulators Xp2M and YM ) or the ghosts (and their regulators), while the wavy
line corresponds to the propagator of
HJEP07(218)4
h ( k) (k)i =
To each closed line there is associated a factor of (d
26)=h coming from summation over
the matter and ghosts like in eq. (6.18).
The diagram in gure 2a (which have a combinatorial factor of 2) gives the usual result
hT (z)T (0)ia) =
d
26
2hz4
associated with the central charges of free elds: d for matter and 26 for ghosts, whose
di erence vanishes only in the critical dimension d = 26. Only massless elds contribute
to the most singular as z ! 0 part of the correlator shown in eq. (7.4) via the propagator
Xq (z)Xq (0) =
log(zz):
1
The diagram in gure 2b is usually associated with the next order of the perturbative
expansion about the classical vacuum because it has two loops, but in the mean eld
approximation it has to be considered together with the diagram in
gure 2a since both
are of the same order in h. We shall return soon to the discussion of this issue. Every of
the two closed loops in the diagram in gure 2b involves the momentumspace integral
2
where we have absorbed the ratio = into M 2 for simplicity. The power counting predicts
a quadratically divergent term like M 2 zz in the integral (7.6), but it vanishes in the
conformal gauge.
Each of the two closed lines is associated ether with matter (the factor of d) of ghosts
(the fector of 26). Multiplying the contribution of the two loops by the propagator (7.3),
we nd for the diagram in gure 2b
hT (z)T (0)ib) =
(d
26)
12h
(d
26) 6
12h
(26
d)
12h
z4 =
d
26
2hz4 :
(7.4)
(7.5)
=
p
2
z
12
;
(7.6)
(7.7)
Notice this result is pure anomalous: it comes entirely from the regulator elds but M has
canceled. Both diagrams in gure 2 give a \classical" (i.e. saddlepoint) contribution from
the viewpoint of the mean eld. Adding (7.4) and (7.7), we obtain zero value of the total
central charge in the mean eld approximation.
The fact that the total central charge of the bosonic string is always zero in the
meaneld approximation, independently on the number of the targetspace dimensions d, is
remarkable. Thus it always reminds the string in the critical dimension d = 26.
A very similar situation occurs in the PolchinskiStrominger approach [10] to the e
ective string theory, where the AlvarezArvis groundstate energy (same as (2.10) for
= 1)
was obtained from the requirement of vanishing the central charge at large
to order
1= [10], 1= 3 [16] and 1= 5 [17]. The mean eld approximation we used apparently sums
up bubble graphs to all orders in 1= and explicitly results in the AlvarezArvis formula.
8
\Semiclassical" correction to the mean
eld
Let us consider a \semiclassical" correction to the mean eld approximation which comes
from averaging over uctuations of
about .
Integrating over the matter and ghost elds (including their regulators), we obtain the
following induced action for the eld
to quadratic order in
:
Se =
Z
)
Si(n2d) =
(26
d) Z
96 h
d2p
(2 )2
( p)G(p) (p)
with G(p) given by eq. (6.14).
This is not the end of the story because there are diagrams with three, four, etc.
's
in (4.14), whose contributions we denote as Si(n3d), Si(n4d), etc. As is explicitly demonstrated
in appendix A, it is convenient to introduce instead of
another variable ' by
(z) =
e'(z);
(z) =
e'(z)
1
and to expand in '. Then the terms higher than quadratic order in ' are mutually canceled
in the sum
Sind = Si(n2d) + Si(n3d) + : : : =
(26
d) Z
d2p
(2 )2 '( p)G(p)'(p) + O('3)
large, pi2
M 2 . There is no reason to expect the cancellation in this case.
in the IR limit where all variables pi's obey pipj
M 2 , so the induced action (8.3)
reproduces the usual e ective action for smooth '(z). However, we consider below explicitly
the case of four ''s, where two momenta are small, pi2
M 2 , but two other momenta are
The e ective action describes \slow"
uctuations of ' with p2
M 2 and emerges
after averaging over \fast" uctuations with p2
M 2 . The quadratic part of the e ective
action gets then contribution from averaging higher terms in the induced action (which are
generically nonlocal), so we write it in the spirit of DDK (a good review is [
37
]) as
(8.1)
(8.2)
(8.3)
(8.4)
a)
d)
b)
e)
c)
HJEP07(218)4
with a certain constant b2. The di erence between the induced action (8.3) and the e ective
action (8.4) will show up when virtual momenta of the propagator h'( p)'(p)i in diagrams,
which emerge after averaging over ', are large: p2 & M 2 . Hence the higher order in '
terms in eq. (8.3) may and will, as we see below, then play an important role. The reason
why they survive is, roughly speaking, a quadratic divergence of the involved integrals.
These terms are however subordinated in h because h'( p)'(p)i / h owing to eq. (7.3).
It is instructive to give yet another explanation why the higher terms can emerge. Let
us consider Tzz given by eqs. (7.1) and average (7.2) over the regulator elds with ' playing
again the role of an external eld. The result is given by the diagrams in
gure 1 whose
analytic expressions are listed in eqs. (A.7){(A.11) of appedix A, where it is explicitly shown
the cancellation of higher than quadratic terms when all momenta squared of external lines
are small (i.e. pi2
M 2 ). I do not see again any reason to expect such a cancellation for
momenta of the order of M 2 , so counterparts of the higher terms in eq. (8.3) may emerge.
The result of the averaging over the regulators will not be yet the energymomentum
(pseudo)tensor because the averages in the pathintegral language are associated with T
products in the operator language. To obtain a genuine Tzz, we have to normal order the
operators ' which produces additional terms like the diagrams in gure 3 coming from
normal ordering in '4. We thus write
T z'z =
1
2
1
+ O('3)
again in the spirit of DDK.
nonlinear
One more source of the nonlinearity is the wellknown fact that the norm of ' is
jj 'jj2 =
Z
d2z (z)( '(z))2:
We can adopt the philosophy of DDK and replace the path integral over ' with the
non(8.5)
(8.6)
linear norm (8.6) by the path integral over the eld '0 with a linear one
by introducing the Jacobian for the transformation from ' to '0. It has again the form of
(the exponential of) the action (8.3) and simply changes its coe cients. We shall therefore
replace in eq. (8.3)
26
d
6h
) b2
0
1
;
1
b
0
The di erence between this b20 and b2 in eq. (8.4) comes to order h from the diagrams with
one propagator which are computed below.
My last comment before proceeding with the computations is that the propagator
h'( p)'(p)i behaves as 1=p2 for small p2, so one might expect therefore logarithmic IR
divergences, associated with this behavior, which would spoil conformal invariance. However,
the lowmomentum e ective action is quadratic in the variable ' as is already mentioned
(and demonstrated by explicit computations in appendix A), so the divergences are
expected to cancel each other because the induced action coincides with the e ective action
in the IR domain.
We shall see in explicit computations this is indeed the case. The
remaining contribution to be calculated will come from virtual momenta squared of the
order of the cuto : k2
M 2 . I believe this is a heuristic proof of the theorem about the
cancellation of the IR divergences.
8.1
Correction to Tzz
The diagrams of the next to the leading order in h which describe \quantum" corrections
to the mean eld approximation for Tzz are depicted in gure 3. Their analytic expressions
are listed in eqs. (B.1){(B.5) of appendix B.
Every individual diagram has an IR divergence coming from the '' propagator, but
it has indeed canceled in the sum as anticipated. Actually the cancellation happens for the
sum a) + 2b) 2c) because d) = 12 e) so only the diagrams in gure 3a, 3b and 3c contribute
resulting in
It is instructive to present the result in the DDK form
Multiplying (8.9) by the normalizations of the propagator (7.3) and of the integrals and
summing with the leadingorder diagrams in
gure 1, we obtain for the coe cient Q
in eq. (8.5)
where b20 is de ned in eq. (8.8).
Q =
q0
b
2
0
13
6
+ O(h);
(8.7)
(8.8)
HJEP07(218)4
(8.9)
(8.10)
(8.11)
a)
d)
b)
e)
Z 1 d
0
An analogous direct computation of the quadratic in ' term in eq. (8.5) is a bit more
tedious and involves 12 diagrams: 7 of which are new, while the contribution of the sum of
remaining 5 diagrams is like 12 Q@2'2. The integrals involve two external momenta, which
complicates their computation.
The diagrams of the next to the leading order in h which describe \quantum" corrections
to the mean eld approximation for Se are depicted in gure 4. Their analytic expressions
are listed in eqs. (B.12){(B.17) of appendix B. Every individual diagram has again an IR
divergence coming from the '' propagator, but it has canceled in the sum as anticipated.
Actually the cancellation happens for the sum a) + 2b)
4c) + f ) because d) = e); so only
the diagrams in gure 4a, 4b, 4c and 4f contribute. Accounting for combinatorial factors,
we obtain
Multiplying (8.12) by the normalization of the propagator (7.3) and of the integrals,
accounting for ghosts and summing with the mean eld result, we obtain for the coupling
constant in the e ective action (8.4)
1
b2 =
1
b
2
0
5 + O(h):
8.3
Remark on the universality
In the above computations of Tzz and Se we substituted G(p) in eq. (8.3) by p2 because
otherwise the computation is hopeless. A question arises as to whether this a ects the
results because characteristic virtual momenta squared in the diagrams are
M 2 .
It is possible to verify that by changing the regularization procedure (6.3) to
ization (8.14) involves N PauliVillars regulators with masses pnM (n = 1; : : : ; N ) which
complicates the computation. It can be shown however that the results (8.11), (8.13) do
not change which is an argument in favor of their universality.
9
Discussion
The main result of this Paper is that a quantization of the e ective string about the
mean eld ground state works in 2 < d < 26. The mean eld approximation corresponds
to conformal eld theory with the central charge vanishing for any d, resulting in the
AlvarezArvis groundstate energy and complimenting the PolchinskiStrominger approach.
A \semiclassical" expansion about the mean eld can be treated adopting the philosophy
of DDK.
The di erence from DDK is that our ' is massless as a consequence of the minimization
at the mean eld saddle point. The massless ' is thus a consequence of the nonperturbative
mean eld ground state for 2 < d < 26 in contrast to the usual perturbative one for
d < 2. This may lead to infrared logarithms which would spoil conformal invariance when
accounting for uctuations about the mean eld, but we argued they have to cancel because
the lowmomentum (or e ective) action is quadratic in '. This cancellation is explicitly
shown to the lowest order of the \semiclassical" expansion about the mean eld. Thus, we
expect that conformal invariance should be maintained order by order of the expansion.
The explicit computation shows the (induced) action governing
uctuations about
the mean eld is however not quadratic in ', while only its lowmomentum limit  the
e ective action  is quadratic. The reason for that is, roughly speaking, quadratic
divergences of the involved integrals. Using the PauliVillars regularization, I have shown
how to systematically treat the induced action (4.14) and to deal with these higher
order terms without assuming that ' is smooth and the determinants are approximated by
the conformal anomaly. Their emergence may in uence the results and deserve further
investigation.
The most interesting question is what would be the spectrum of the NambuGoto string
beyond the mean eld approximation. In particular, whether the universal correction to the
AlvarezArvis spectrum at the order 1= 5 (see [12] and references therein) is reproduced
in the \semiclassical" expansion about the mean
eld at one loop. This issue will be
considered elsewhere.
Acknowledgments
I am grateful to Jan Ambj rn for sharing his insight into Strings.
A
Leadingorder explicit computations
To the leading order in h we consider
(or ') as an external eld over which we shall
average to next orders.
The contribution of diagrams in gure 1 to Taa from the regulators read
2M 2
a) =
b) =
c) =
d) =
e) =
Z
Z
Z
Z
Z
d2k
d2k
d2k
d2k
(2 )2
p
p
2M 8
60
p
[(k
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
HJEP07(218)4
M=!1 2p2 + 3q2 + 3r2 + 2t2 + 3pq + 2pr + pt + 4qr + 2qt + 3rt
:
p
32M 10
p
2M 10
q
q
p
r
q
r
Multiplying each wavy line by
, passing to the coordinate space and summing up
the contributions of the diagrams in gure 1 with these of ghosts, we nd
hTaai =
=
26
26
12h
(4
12h
d
d
+
'
2
)
6
where we have used (8.2) and expanded in '. We have thus reproduced eq. (6.18) to this
order.
In eq. (A.6) we simply subtracted 26 from d to account for the ghost contribution
because the contribution of diagrams which emerge from ghost and the regulators of ghosts
are identical for the mean eld and perturbative vacua just as it is for the matter elds
and regulators. The same applies below for Tzz.
(k2 + M 2)
= 0;
pz)
p
q
p
q
=
26
12h
d
d
3 2
+ O('5);
16M 8kz(kz
qz
q
qz
rz
p
tz)
tz)
q
p
r
q
r
M=!1 2pz2 + 3qz2 + 3rz2 + 2tz2 + 6pzqz + 8pzrz + 10pztz + 7qzrz + 8qztz + 6rztz :
(A.11)
Using (8.2), we analogously to eq. (A.6) obtain for Tzz
a) =
b) =
c) =
d) =
e) =
Z
Z
Z
Z
Z
d2k
d2k
d2k
d2k
d2k
M=!1 3pz2 + 4qz2 + 3rz2 + 9pzqz + 12pzrz + 9qzrz
240
(A.7)
(A.8)
(A.9)
(A.10)
(A.12)
The analogous contribution of diagrams in gure 1 to Tzz read
i.e. the free energymomentum tensor to this order.
The reason why I presented in this appendix the explicit computations of Taa and Tzz
is to emphasize that numerical factors are most important to get the freetheory results.
The cancellation would no longer take place if these factors were changed due to induced
interactions, as we shall immediately see in the next appendix.
;
pz
qz
p
rz
'
+ 3
\Semiclassical" corrections
Contribution to Tzz
The contributions of diagrams in gure 3 to Tzz involve
a) =
b) =
c) =
d) =
e) =
(k
q)2[(k
;
pz)
pz)
q)2[(k
p)2 + 2M 2](k2 + 2M 2)2(q2 + 2M 2)
pz)
q)2[(k
p)2 + 2M 2](k2 + 2M 2)(q2 + 2M 2)
pz)
pz)
pz)
p)2 + 2M 2](k2 + 2M 2)2
q2[(k
pz)
pz)
p)2 + 2M 2](k2 + 2M 2)
q2[(k
For the computation of integrals it is convenient to multiply a generic integral
by the projector
Z
d2k
(2 )2 ka(kb
pb)f (k2; p2; kp) = f1(p2)gab + f2(p2)papb
P ab = 2
papb
p2
gab
Z
d2k
(2 )2
Z
2
(2 )2 kz(kz
kp
k
2 f (k2; p2; kp) = f2(p2)p2:
pz)f (k2; p2; kp) = f2(p2)pz2:
This trick is implemented in the Mathematica program of appendix C, where the integrals
are computed by rst integrating over the two relative angles and then by the two absolute
values of the virtual momenta.
ing for combinatorial factors, we obtain
Performing the computation by the Mathematica program in appendix C and
account2c)
d) +
e) =
1
2
13pz2
288
:
(B.1)
(B.2)
(B.3)
;
(B.4)
: (B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
Notice that d) = 12 e) so only the diagrams in gure 3a, 3b and 3c contribute. The infrared
divergence coming from the '' propagator has indeed canceled in the sum, as anticipated.
Multiplying (B.10) by the normalization of the propagator (7.3) and of the integrals
and summing the diagrams in gure 1 and gure 3, we obtain
1
B.2
Contribution to Se
The contributions of diagrams in gure 4 to Se involve
q)2[(k
2M 8
;
q)2[(k
p)2 + 2M 2](k2 + 2M 2)2(q2 + 2M 2)
q)2[(k
p)2 + 2M 2](q2 + 2M 2)
(k
q)2[(k
p)2 + M 2](q2 + M 2)
Performing the computation by the Mathematica program of appendix C and
accounting for combinatorial factors, we obtain
4c)
d) + e) + f ) =
5p2
48
:
Notice that again d) = e) so only the diagrams in gure 4a, 4b, 4c and 4f contribute. The
infrared divergence coming from the '' propagator has indeed canceled in the sum, as
anticipated.
a) =
b) =
c) =
d) =
e) =
f ) =
2M 6
8M 6
4M 4
4M 4
2M 4
16M 8
8M 6
16M 8
:
(k
q)2[(k
p)2 + M 2](k2 + M 2)2(q2 + M 2)
q)2[(k
p)2 + 2M 2](k2 + 2M 2)(q2 + 2M 2)
(k
q)2[(k
p)2 + M 2](k2 + M 2)(q2 + M 2)
p)2 + 2M 2](k2 + 2M 2)2
q2[(k
p)2 + 2M 2](k2 + 2M 2)
q2[(k
2M 6
2M 4
(B.11)
(B.12)
(B.13)
(B.14)
;
(B.15)
;
(B.16)
(B.17)
(B.18)
Multiplying (B.18) by the normalization of the propagator (7.3) and of the integrals,
accounting for ghosts and summing with the mean eld result, we obtain
Se =
(B.19)
C Mathematica programs
C.1
Program for computing diagrams in gure 3 that results in (8.9)
(* Computation of T_zz by integration over angles and x=k^2, y=q^2 *)
HJEP07(218)4
x =.
y =.
M =.
al = 1
ii1 = Normal[
ii2 = Normal[
M = 1
Resa1 = ResaM
M =.
ii1 = Normal[
ii2 = Normal[
M = Sqrt[2]
Resb2 = ResbM
Normal[Series[(2 k^2 Cos[bet]^2  al k p Cos[bet]  k^2) M^6/(k^2 + q^2
2 k q Cos[the])/(k^2 + al^2 p^2  2 al k p Cos[bet] + M^2)/(q^2 + al^2 p^2
2 al p q Cos[bet + the] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]];
ixyan = Integrate[%, {bet, 0, 2 Pi}]
iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {x > y > 0}]
Series[Integrate[iixlay, {y, 0, x  mu^2},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]
iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {y > x > 0}]
Series[Integrate[iiylax, {y, x + mu^2, Infinity},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]
ResaM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity},
Assumptions > {M > mu > 0}]
tResa = FullSimplify[(Resa2  2 Resa1)/16/Pi^2]
Normal[Series[(2 k^2 Cos[bet]^2  al k p Cos[bet]
k^2) M^6/(k^2 + q^2  2 k q Cos[the])/(k^2 + al^2 p^2
2 al k p Cos[bet] + M^2)/(k^2 + M^2)^2/(q^2 + M^2), {p, 0, 2}]];
ixyan = Integrate[%, {bet, 0, 2 Pi}]
iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {x > y > 0}]
Series[Integrate[iixlay, {y, 0, x  mu^2},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]
iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {y > x > 0}]
Series[Integrate[iiylax, {y, x + mu^2, Infinity},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]
Assumptions > {M > mu > 0}]
tResb = FullSimplify[(Resb2  2 Resb1)/16/Pi^2]
Normal[Series[(2 k^2 Cos[bet]^2  al k p Cos[bet]
k^2) M^4/(k^2 + q^2  2 k q Cos[the])/(k^2 + al^2 p^2
2 al k p Cos[bet] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]];
ixyan = Integrate[%, {bet, 0, 2 Pi}]
iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {x > y > 0}]
Series[Integrate[iixlay, {y, 0, x  mu^2},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]
iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {y > x > 0}]
Series[Integrate[iiylax, {y, x + mu^2, Infinity},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]
RescM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity},
Assumptions > {M > mu > 0}]
tResc = FullSimplify[(Resc2  2 Resc1)/16/Pi^2]
FullSimplify[tResa + 2 tResb  2 tResc, Assumptions > {mu > 0}]
(* gives 13 p^2/288 *)
C.2
Program for computing diagrams in gure 4 that results in (8.12)
(* Computation of S_eff by integration over angles and x=k^2, y=q^2 *)
M = 1
Resc1 = RescM
M =.
x =.
y =.
M =.
k = Sqrt[x]
q = Sqrt[y]
al = 1
Normal[Series[M^8/(k^2 + q^2  2 k q Cos[the])/(k^2 + al^2 p^2
2 al k p Cos[bet] + M^2)/(q^2 + al^2 p^2
2 al p q Cos[bet + the] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]];
ixyan = Integrate[%, {bet, 0, 2 Pi}];
iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {x > y > 0}]
ii1 = Normal[Series[Integrate[iixlay, {y, 0, x  mu^2},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {y > x > 0}]
ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
ResaM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity},
Assumptions > {M > mu > 0}]
M = Sqrt[2];
Resa2 = ResaM;
M = 1;
M =.
SResa = FullSimplify[(Resa2  2 Resa1)/16/Pi^2]
Normal[Series[M^8/(k^2 + q^2  2 k q Cos[the])/(k^2 + al^2 p^2
2 al k p Cos[bet] + M^2)/(k^2 + M^2)^2/(q^2 + M^2), {p, 0, 2}]];
ixyan = Integrate[%, {bet, 0, 2 Pi}];
ii1 = Normal[Series[Integrate[iixlay, {y, 0, x  mu^2},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {y > x > 0}]
ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
ResbM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity},
Assumptions > {M > mu > 0}]
M = Sqrt[2];
Resb2 = ResbM;
M = 1;
Resb1 = ResbM;
M =.
SResb = FullSimplify[(Resb2  2 Resb1)/16/Pi^2]
Normal[Series[M^6/(k^2 + q^2  2 k q Cos[the])/(k^2 + al^2 p^2
2 al k p Cos[bet] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]];
ixyan = Integrate[%, {bet, 0, 2 Pi}];
iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {x > y > 0}]
ii1 = Normal[Series[Integrate[iixlay, {y, 0, x  mu^2},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {y > x > 0}]
ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
RescM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity},
Assumptions > {M > mu > 0}]
M = Sqrt[2];
Resc2 = RescM;
M = 1;
Resc1 = RescM;
M =.
SResc = FullSimplify[(Resc2  2 Resc1)/16/Pi^2]
Normal[Series[M^4/(k^2 + q^2  2 k q Cos[the])/(k^2 + al^2 p^2
2 al k p Cos[bet] + M^2)/(q^2 + M^2), {p, 0, 2}]];
ixyan = Integrate[%, {bet, 0, 2 Pi}];
iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {x > y > 0}]
ii1 = Normal[Series[Integrate[iixlay, {y, 0, x  mu^2},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions > {y > x > 0}]
ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity},
Assumptions > {M^2 > x > mu^2 > 0}], {mu, 0, 0}]];
ResfM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity},
Assumptions > {M > mu > 0}]
Resf2 = ResfM;
SResf = FullSimplify[(Resf2  2 Resf1)/16/Pi^2]
FullSimplify[SResa + 2 SResb  4 SResc + SResf,
Assumptions > {mu > 0}]
FullSimplify[% 48 Pi/(2 Pi)^2, Assumptions > {mu > 0}]
(* gives 5 p^2/48 *)
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Phys. Lett. A 3 (1988) 1651 [INSPIRE].
321 (1989) 509 [INSPIRE].
[1] V.A. Kazakov, A.A. Migdal and I.K. Kostov, Critical Properties of Randomly Triangulated
Planar Random Surfaces, Phys. Lett. B 157 (1985) 295 [INSPIRE].
[2] F. David, Planar Diagrams, TwoDimensional Lattice Gravity and Surface Models, Nucl.
Phys. B 257 (1985) 45 [INSPIRE].
[3] J. Ambj rn, B. Durhuus and J. Frohlich, Diseases of Triangulated Random Surface Models
and Possible Cures, Nucl. Phys. B 257 (1985) 433 [INSPIRE].
Gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].
[4] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2D Quantum
[5] F. David, Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge, Mod.
[6] J. Distler and H. Kawai, Conformal Field Theory and 2D Quantum Gravity, Nucl. Phys. B
[7] J. Ambj rn, B. Durhuus and T. Jonsson, Quantum geometry. A statistical eld theory
approach, Cambridge University Press, Cambridge U.K. (1997).
[8] J. Ambj rn and B. Durhuus, Regularized bosonic strings need extrinsic curvature, Phys. Lett.
B 188 (1987) 253 [INSPIRE].
[9] Y. Makeenko, QCD String as an E ective String, in proceedings of the Low dimensional
World Scienti c (2012), pp. 211{222 [arXiv:1206.0922] [INSPIRE].
[10] J. Polchinski and A. Strominger, E ective string theory, Phys. Rev. Lett. 67 (1991) 1681
118 [arXiv:1302.6257] [INSPIRE].
[11] S. Dubovsky, R. Flauger and V. Gorbenko, E ective String Theory Revisited, JHEP 09
[12] O. Aharony and Z. Komargodski, The E ective Theory of Long Strings, JHEP 05 (2013)
[13] S. Dubovsky, R. Flauger and V. Gorbenko, Flux Tube Spectra from Approximate Integrability
at Low Energies, J. Exp. Theor. Phys. 120 (2015) 399 [arXiv:1404.0037] [INSPIRE].
[INSPIRE].
[INSPIRE].
[14] S. Hellerman, S. Maeda, J. Maltz and I. Swanson, E ective String Theory Simpli ed, JHEP
09 (2014) 183 [arXiv:1405.6197] [INSPIRE].
Phys. A 31 (2016) 1643001 [arXiv:1603.06969] [INSPIRE].
[15] B.B. Brandt and M. Meineri, E ective string description of con ning ux tubes, Int. J. Mod.
[16] J.M. Drummond, Universal subleading spectrum of e ective string theory, hepth/0411017
[17] O. Aharony, M. Field and N. Klingho er, The e ective string spectrum in the orthogonal
gauge, JHEP 04 (2012) 048 [arXiv:1111.5757] [INSPIRE].
HJEP07(218)4
142 [arXiv:1601.00540] [INSPIRE].
93 (2016) 066007 [arXiv:1510.03390] [INSPIRE].
Lett. B 770 (2017) 352 [arXiv:1703.05382] [INSPIRE].
[22] J. Ambj rn and Y. Makeenko, Scaling behavior of regularized bosonic strings, Phys. Rev. D
[23] J. Ambj rn and Y. Makeenko, Stability of the nonperturbative bosonic string vacuum, Phys.
[24] J. Ambj rn and Y. Makeenko, The use of PauliVillars regularization in string theory, Int. J.
Mod. Phys. A 32 (2017) 1750187 [arXiv:1709.00995] [INSPIRE].
[25] A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B 103 (1981) 207
[26] B. Durhuus, P. Olesen and J.L. Petersen, Polyakov's Quantized String With Boundary
Terms, Nucl. Phys. B 198 (1982) 157 [INSPIRE].
Geometry, Nucl. Phys. B 216 (1983) 125 [INSPIRE].
Cambridge U.K. (2002), pp. 208{210.
[27] O. Alvarez, Theory of Strings with Boundaries: Fluctuations, Topology and Quantum
[28] Y. Makeenko, Methods of contemporary gauge theory, Cambridge University Press,
(1987), pp. 173{174.
Rev. D 6 (1972) 1655 [INSPIRE].
[29] Y. Makeenko, An interplay between static potential and Reggeon trajectory for QCD string,
Phys. Lett. B 699 (2011) 199 [arXiv:1103.2269] [INSPIRE].
[30] L. Brink and H.B. Nielsen, A Simple Physical Interpretation of the Critical Dimension of
SpaceTime in Dual Models, Phys. Lett. B 45 (1973) 332 [INSPIRE].
[31] A.M. Polyakov, Gauge elds and strings, Harwood Academic Publishers, Reading U.K.
[32] R.C. Brower, Spectrum generating algebra and no ghost theorem for the dual model, Phys.
[33] P. Goddard and C.B. Thorn, Compatibility of the Dual Pomeron with Unitarity and the
Absence of Ghosts in the Dual Resonance Model, Phys. Lett. B 40 (1972) 235 [INSPIRE].
[34] A.B. Zamolodchikov, On the entropy of random surfaces, Phys. Lett. B 117 (1982) 87
[INSPIRE].
(2008) and online pdf version at http://qft.itp.ac.ru/ZZ.pdf.
[18] O. Alvarez , The Static Potential in String Models, Phys. Rev. D 24 ( 1981 ) 440 [INSPIRE].
[19] J.F. Arvis , The Exact qq Potential in Nambu String Theory, Phys . Lett. B 127 ( 1983 ) 106
[20] P. Olesen , Strings and QCD , Phys. Lett. B 160 ( 1985 ) 144 [INSPIRE].
[21] J. Ambj rn and Y. Makeenko , String theory as a Lilliputian world , Phys. Lett. B 756 ( 2016 )
[35] S. Chaudhuri , H. Kawai and S.H.H. Tye , Path Integral Formulation of Closed Strings, Phys. Rev. D 36 ( 1987 ) 1148 [INSPIRE] . Phys. Lett. B 187 ( 1987 ) 149 [INSPIRE].
[36] I.K. Kostov and A. Krzywicki , On the Entropy of Random Surfaces With Arbitrary Genus,
[37] A. Zamolodchikov and A. Zamolodchikov, Lectures on Liouville theory and matrix models,