#### Leading CFT constraints on multi-critical models in d > 2

Received: March
Leading CFT constraints on multi-critical models in d
Alessandro Codello 1 2
Mahmoud Safari 1 2
Gian Paolo Vacca 1 2
Omar Zanusso 1 2
W Symmetry 1 2
Open Access 1 2
c The Authors. 1 2
0 -Origins, University of Southern Denmark
1 via Irnerio 46 , 40126 Bologna , Italy
2 Campusvej 55 , 5230 Odense M , Denmark
We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial m below their upper critical dimensions dc = m2m2 , and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d = 2, while for odd m the theories are non-unitary and start at m = 3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators k and of some families of structure constants in either the coupling's or the -expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling's expansion.
cDipartimento di Fisica e Astronomia; Universita di Bologna
Contents
1 Introduction
Schwinger-Dyson consistency and CFT
2n-theory in d = d2n
Anomalous dimensions
Warm-up: 1
Climbing up: 2
The general case: k
Structure constants
Structure constants C1;2k;2l 1
Structure constants C1;1;2k
Critical coupling g( )
Collecting the results: even potentials
2n+1-theory in d = d2n+1
Anomalous dimensions
Structure constants
Structure constants C1;k;l
Structure constants C1;1;2k
The special case of C111 for n > 1
Critical coupling g( ) for n = 1
Collecting the results: odd potentials
Case n > 1
Conclusions A Free theory B Action of the Laplacian
Introduction
of the standard methods of quantum
eld theory (QFT) are used, including perturbation
when critical properties are under investigation.
We will be interested in generalizing this idea to theories governed by the general
potential. In a Ginzburg-Landau description their action is
moment we shall simply ignore them.
g is canonically dimensionless
S[ ] =
protected by a Z2 parity (
the 2n e ective potential describes a statistical system with a phase-transition that can be
gk of (1.1) do not play a signi cant role in tuning the action to criticality, therefore for the
The upper critical dimension of (1.1) is de ned as the dimension d at which the coupling
dm =
mension the statistical
uctuations are weak and the physics of (1.1) is Gaussian and
with an anomalous dimension. In the latter case a consistent expansion for the
crit1We follow the convention that universality classes such as Ising's are denoted with typeset font,
therefore the spin
1 Ising model at criticality is only one speci c realization of the Ising universality class and the two should not generally be confused. The paper will deal with universality classes to a greater extent.
d = dm
and that for n
d2n of (1.2)!
quantities in the form of a Taylor series in .
The most important critical exponents of all the aforementioned special cases (Ising,
let us point out that for n
3 the leading contributions arise from multiloop computations,
4 the divergences are subtracted as poles of the fractional dimensions
Another interesting property is that the even models are known to interpolate in
the sequence of minimal non-unitary multi-critical theories M(2; m + 2) studied in [13].
eld theories for any dimension 2
dm. The straightforward question that we will
determining the critical properties of (1.1)?
The paper is organized as follows: in section 2 we brie y summarize the main features
the CFT correlators.
Schwinger-Dyson consistency and CFT
2In the non unitary models, e.g. Lee-Yang universality class, the critical coupling and some structure
m-theory
S[ ] =
ing (2.1) with (1.1). In (2.1) we introduced a reference (mass) scale
which makes the
almost marginal coupling g dimensionless for any d. The presence of the mass scale
example, all the n di erent phases of a
2n theory coexist.
Before diving more deeply into some technical details, it is worth noting that, with the
More generally, after the displacement by
all the theories will live in the arbitrarily
real dimension d = dm
. Theories living in continuous dimensions have already been
realized for any value of the dimension d.3
The key idea of [1] is that all the CFT data of (2.1) must interpolate with that of
the Gaussian theory in the limit
dimensions for the eld
and the composite operators
in d dimensions. Let the canonical dimension of
! 0. We set some notation by de ning the scaling
m of an interacting scalar theory
1 = m
m =
and the scaling dimensions of
and k be respectively
k = k
The -terms represent the corrections from the canonical scaling dimensions
and therefore must be proportional to some power of g or
to ensure consistency of the
Gaussian limit.
3Scale invariance seems to imply conformal invariance for several physically interesting critical models,
supported at theoretical level [21].
The Schwinger-Dyson equations (SDE) generalize the notion of equations of motion
zero. In practice, for any state of the CFT and for any list of operators Oi the relation
(x) O1(y) O2(z) : : :
= 0
order one can use the relation
h2x (x)O1(y)O2(z) : : :i =
m 1(x)O1(y)O2(z) : : :i
teracting CFT the operator
1 are primaries, while the operator
m 1 is a descendant.4 In other words, the interacting CFT enjoys one less independent
operator, that is
m 1, and a recombination of the conformal multiplets must take place.
In particular, the scaling dimensions of
m 1 must be constrained
m 1 =
=)
m 1 = 1 + (m
sides of the SDE. It is possible to nd a basis Oa of scalar primary operators with scaling
a whose two point correlators are diagonal
k with k 6= m
hOa(x)Ob(y)i =
factors which can in principle be set to one. However, for the moment, we will
more constrained by conformal symmetry and reads
hOa(x)Ob(y)Oc(z)i =
are completely and uniquely speci ed by providing the scaling dimensions
a and the
reasons paramount target of any computation.
Our goal is to extract the leading informations for a part of the conformal data of all
4A descendant operator in d > 2 is the derivative of a primary operator, which is annihilated by the
of the d = 2 case.
theories at an interacting xed point in dm
with also conformal bootstrap techniques.
2n-theory in d = d2n
dimensions as a power series in , eventually
This section is dedicated to the investigation of the even potentials
2n in d = d2n
that corresponds to the Ising universality class, the upper critical dimensions are
d2n =
= 4 ; 3 ; ; ;
In the limit n ! 1 the critical dimensions tend to two, that is the dimensionality for which
serves as a testing ground for the entire method.
In the rst part of this section, we will kickstart the computation by obtaining the
anomalous dimension for the eld
by using a constraint which comes from the consistency
of the two point function (2.7) with the SDE (2.5) in the limit
! 0. Then we will repeat
scaling dimensions of all the composite operators k
. We will see that 2, which is related
on those that are not present at zeroth order in
and are thus generated at quantum level.
In the third part we will exploit the fact that the scaling dimension of the 2n 1 descendant
operator can be computed in two di erent ways and use it to
nd a critical value for the
perturbation theory of [10]. All the results are summarized at the end of the section.
Anomalous dimensions
eld 1 and of the composite operators k with k
2. By LO we will generally mean leading order in g and in . Only when an explicit relation g( ) will be available (as for even potentials in section 3.3), leading order will mean leading order in . { 6 {
We start with a simple analysis of the two point function that will directly uncover
a precise leading order relation between 1 and the coupling g. The determination of 2
requires the analysis of three point function h
2i and is a bit more involved for n > 2.
Finally we shall be able to obtain the anomalous dimensions k with k
n from the
k k+1i. In these rst computations we will proceed step by step in order to
correlators as detailed in appendix A.
Warm-up: 1
Let us consider in d dimensions the propagator of the interacting theory
h (x) (y)i =
in eq. (A.2). Thus we will make the replacement C ! c everywhere from now on.
On applying rst the SDE in one point one shows that 1 is at least of order g2. Then
applying the SDE also to the second point gives the leading expression for 1 in terms of
and recalling
1 =
+ 1 gives
2x h (x) (y)i = 2x x
= c
tions 4( + 1) 1 ! 4 2n 1 in the numerator and 1 +
+ 1 ! 1 + 2n in the denominator,
expression using the SDE one nds
h2x (x) (y)i =
2n 1(x) (y)i = O(g2) :
1 is at least of order g2. To obtain another useful relation one acts with a
2x2y h (x) (y)i = 2x2y x
= c
2 1(2 1 +2)(2 1 2 )(2 1 +2 2 )
16( + 1)( + 1 + 1) 1(1 + 1) ! 16 2n( 2n + 1) 1 in the numerator and 2 1 + 4 ! 2 2n + 4
two point function of eq. (A.3) of appendix A gives instead
h2x (x)2y (y)i =
(2n 1)!
2n 1(x) 2n 1(y)i L=O
anomalous dimension
Using the fact that
we nd the explicit formula
g2 + O(g3) :
c =
1 =
h2x (x) (y) 2(z)i =
2n 1(x) (y) 2(z)i L=O g
which agrees with the perturbative result [10].
Climbing up: 2
where it appears is
To determine 2 we need to consider the three point functions. The simplest correlator
h (x) (y) 2(z)i =
Laplacian can be easily obtained from eq. (B.3) given in appendix B by setting
2 = 2 1
2 and 2 =
3 =
2 = 2 + 2
From this expression we easily determine the leading order contributions
2x h (x) (y) 2(z)i L=O
should match the one obtained by applying the SDE
1 =
following expression
2 =
n = 2 :
In order to nd the leading value of 2 in the general case n > 2 we act with the second
one nds (we skip the intermediate steps)
which we should compare with the leading order result obtained applying the SDE,
2n 1(x) 2n 1(y) 2(z)i L=O
so that by comparison we obtain
2 1 =
16(n 2)(2n 1)!2
Using the explicit expression for 1 given in eq. (3.9) we nd
2 = 8
(n 2)(2n)!
This quantity has not been reported in the perturbative results given in [10].
The general case: k
To determine k at rst we could think to consider h
the free theory whenever k > 2. To investigate all k
ki, but this correlator is zero in
2 we instead consider the following
three point function
h (x) k(y) k+1(z)i =
k 6= 2n
subsection 3.3.
1. Indeed for k = 2n 2; 2n 1 other terms are present. Nevertheless, if one restrict the analysis to the lowest order, these extra terms which are subleading can
from the general expression (A.8) and reads
C1fr;kee;k+1 = (k +1)! ck+1 :
The main recursion relation can then be derived for k
n 1 applying a Laplacian in x
the same reasoning of the previous subsections, we nd the following LO expression
On the other hand using the SDE one gets
h2x (x) k(y) k+1(z)i =
C2frnee 1;k;k+1 =
k!(k +1)!(2n 1)!
(k n+1)!(n
Vice versa when k
n 2 the free correlator is zero and the the full correlator in eq. (3.21)
is at least of order O(g2). The expression obtained from the SDE in eq. (3.21) has a leading
the rst term in eq. (3.20) is negligible and that k+1
k = O(g). Then by comparing
eqs. (3.20) and (3.21) one nds the recurrence relation
k =
(n 2)! n! (k n+1)!
O(g) ceases to exists for k
2 and is substituted by some relation involving O(g2)
solve the recurrence relation to obtain
n 1. With this condition one can
k =
we correctly reproduce eq. (3.13).
The above relation says that for k
1 the leading contribution to the anomalous
next to leading order O( 2), and for k
in the next subsection.
ourselves to h
for this to be nonzero is
family of even universality classes.
In order to get some information from the three point functions using the
SchwingerDyson equations we need to have one of the elds to appear with power one. The h
are already explored and give information on the scaling dimensions
i. In the following
we therefore concentrate on the rest of these correlation functions.
Structure constants C1;2k;2l 1
The remaining correlation functions consist of h
lj 6= 1. These vanish in the
these are at least proportional to the coupling or smaller. Now if h
in the free theory it implies that C1kl are at least of order O(g2) and to nd their value at
leading order we need to know h
2n 1 k li beyond free theory. Therefore we will not be
2n 1 k li not to vanish in the free theory we must have the following conditions.
1 is odd, either k or l must be even while the other must be odd, so we restrict
1; n > 1. As previously discussed, the condition
These are equivalent to k +l
otherwise we will be back to the case h
for k; l satisfying the conditions
k k+1i which is already studied. In summary,
k 6= 0 or 1 ;
we can nd the leading order (O(g)) structure constants C1;2k;2l 1. One can use the SDE
h x (x) 2k(y) 2l 1(z)i =
x to the correlation function h (x) 2k(y) 2l 1(z)i one nds
L=O C1;2k;2l 1 x
l + 1)(d2n
One readily sees, using the relation
2n 1 =
1 + 2, that the denominators in the above
two expressions are equal. Comparing the coe cients we nd
C1;2k;2l 1 =
2n 1;2k;2l 1
1)! 4(k
C2frnee 1;2k;2l 1 =
(n+l k 1)!(k +n l)!(k +l n)!
and c is the normalization of the free propagator given in eq. (A.2).
Structure constants C1;1;2k
O(g) for
other coe cients of the form C1;1;2k with k in the range 2
1, which turn out
to be of order O(g2). These can be extracted from the analysis of the family of correlators
considered in the previous subsection
h (x) (y) 2k(z)i =
where k > 1. Clearly the coe cients C1;1;2k for k > 1 vanish in the free theory. We proceed
so that we obtain
h2x (x)2y (y) 2k(z)i =
C1;1;2k =
(n 1)4Cfree
2n 1;2n 1;2k
1)!2 16k(k 1)(k
For higher values of k one needs to know the correlation function h
range of validity for this formula is therefore 2
1 and k 6= n
1; n. As mentioned
before, 1; n
1; n were excluded from the possible values k can take in this subsection, and
1; n provide a
di erent way to compute 2(n 1); 2n, which can be shown to be consistent with the results
of the previous subsections.
Critical coupling g( )
1) , only if we knew
the anomalous dimension 2n 1. The general formula for the anomalous dimension k was
excluded there because the correlation function h
k 2ni in these cases would involve a
(2n 1)!
g (n 1) 2y h (x) (y) 2n(z)i
one notices that
h (x) (y)2n 1 2n(z)i =
(2n 1)!
(2n 1)!
LO (2n 1)!
C1;1;2n 4
The structure constant on the r.h.s. evaluated in the free theory is nonzero for k
C2frnee 1;2n 1;2k =
(2k)!(2n 1)!2
k!2(2n k 1)!
C1;1;2k =
(2k)!(n 1)4 c2n+k 1
16k(k 1)(k
n)(k n+1)k!2(2n k 1)!
since 2 1 +2 d = 2 1 = O(g2), and
C1;1;2n =
(2n 1)! n(n
1)(d2n 2)2
on the the scaling dimension of the descendant operator 2n 1 from the equation of motion,
(n 1) + O(g2) = 2n 1 =
which gives the linear relation
2(n 1) (2n
a non primary operator,
h (x) (y)2n 1 2n(z)i L=O
We notice that the expressions (3.43) and (3.45), for 1 and k, are in agreement with the
results obtained in [10] with a perturbative computation.
+ O( 2) =
is linear in , it is possible to uniquely determine
g =
1) g + (n 1)
which shows that the non trivial xed point of the CFT is IR attractive (g > 0).
Collecting the results: even potentials
Anomalous dimensions.
The anomalous dimensions k for 1
1 are found to
be of O(g2) but only the rst two, 1 and 2, are determined at leading order. The rest
are of O(g) and their leading values, together with 1 and 2 are summarized here
2 = 8
k = 2(n 1)
(n 1)3(n+1) n!6
(2n)! (k n)!
One may write a generating function for the anomalous dimensions of all these
multiat O( ), which gives k for any k
n in eq. (3.45), can be written as
F (even)(x; y; ) = ex (pxy sinh pxy
2 cosh pxy) + O( 2) ;
k(n; ) =
@yn @xk
so that one has
We nd
C1;2k;2l 1 =
C116 =
C114 =
+ O( 2) ; (3.48)
k 6= 0; 1, and
(2k)!(2l 1)!
l + 1) (n+l k 1)!(k +n l)!(k +l n)!
within the limits of eq. (3.26), namely k + l
C1;1;2k =
k(k 1)(k n)(k n+1) (2n)!2 k!2(2n k 1)!
for k 6= n
1; n and 2
1. In this scheme all the factors are absent. We note, however, that for comparison with results obtained from perturbation theory other normalizations may prove more convenient.
set of leading order structure constants that we have found we report all the ones of O( 2)
and only a few of the in nite sequence of order O( ). For the Ising universality class:
C114 =
C125 =
C136 = 20 ;
C116 =
For the Tricritical universality class:
C114 =
And nally for the Tetracritical universality class:
C118 =
C136 =
C127 =
C125 =
; C1;1;10 =
; C1;1;12 =
; C1;1;14 =
C116 =
C118 =
C125 = 6 ;
C136 = 54 ;
C1;1;10 =
555984 2
odd potentials 2n+1 for n a natural number n
1 which arise as particular cases of (2.1)
P T -symmetry [23{25]. On a general action S[ ] as in (1.1) P T -symmetry acts as
d2n+1 = 2 +
= 6 ;
which similarly to d2n tend to two in the limit n ! 1. In a Ginzburg-Landau description
fact these models seem to be non-unitary for all d
The well-known upper critical dimension of the Lee-Yang universality class is six.
< 1 in a physically interesting scenario. The models with odd potentials are much
quantities in the process.
As for the content of this section, it will mostly follow the development of section 3,
C1;1;1. In the third part we will show that the possibility to x the coupling to its critical
value as a function of
only occurs for the Lee-Yang universality class. All the results will
be summarized in the nal part of this section.
Anomalous dimensions
1 and the coupling g by acting with two Laplacians on the propagator and using the
SDE, which now gives the operatorial relation
2 so that 2n is a descendant of .
Taking into account that the results of section 3 must be shifted as n ! n + 12 , so that
1 = c2ondd 1 ((22nn+11))2! 3g22
+ O(g3) =
g2 + O(g3) ;
codd is obtained from (A.2) after the shift n ! n + 12 ,
codd =
correlator h
we can directly infer
i when all the operators are primary. Therefore from expression (3.17)
2 = c2ondd 1 ((22nn + 3)(2n
3)(2n + 1)! 16
+ O(g3) =
(2n + 3)(2n
3)(2n + 1)!
g2 + O(g3) ;
2n = 1 +
dimensions. From the study of the correlator of primary operators
Unfortunately we are not able to
nd a closed expression for the other anomalous
h (x) k(y) k+1(z)i =
one which involves the descendant operator '2n
2n(x) k(y) k+1(z)i =
Acting on (4.7) with a Laplacian in x and keeping only leading order terms gives
h2x (x) k(y) k+1(z)i L=O 2C1fr;kee;k+1
2n k k+1i is zero in the free theory we
corrections are expressed in terms of integer powers of g we conclude that k+1
least of order O(g2) and thus
k = O(g2) ;
Structure constants
potentials, one can consider the correlation functions h
2ki with the action
correlation function h
i with the action of a triple Laplacian also gives some leading
order information on the structure constants. Below, we consider each case in turn.
Structure constants C1;k;l
for the correlator h
2n k li to acquire a contribution in the free theory is di erent. Here
k; l have to be either both even or both odd. Furthermore they must satisfy
(2n)!k!l! codd
; (4.12)
This is equivalent to k+l
free theory is
2n(x) k(y) l(z)i L=O
2n. In this case the correlator h
Let us therefore consider for k; l 6= 2n
h (x) k(y) l(z)i =
This tells that the correlator involving the descendant operator 2n gets three contributions
depends on the C1;k;l, the scaling dimensions
2n =
k, l and the dimension d.
In the following we shall restrict to few considerations based on this relation. Acting
2n(x) k(y) l(z)i =
one nds
4 C1;k;l
2x h (x) k(y) l(z)i = C1;k;l 1
(d 2)(1+k l) + 2( 1 + k
(d 2)(1+l k) + 2( 1 + l
[(d 2)(k +l 1) + 2( k + l
2)(1+k l) + 2( 1 + k
+ : : : (4.16)
the same coordinate dependence of the expression in eq. (4.12), so that
n and gives
C1;k;k =
k!2(2n 1)2
Structure constants C1;1;2k
Let us nally consider the correlator h
2k . Again, the analysis in this case follows
function obtained using the SDE twice
2n(x) 2n(y) 2k(z)i L=O
This gives the structure constants
26k(k 1)(4(k n)2
1) k!2(2n
C2frne;e2n;2k
: (4.20)
nonzero if k
2n, therefore the range of validity of this equation is 2
2n. For k = n
the correlation function under study h
2ki involves a descendent operator and therefore
this includes several terms as can be seen by writing
Using eq. (B.3) of appendix B, the leading term in this expression can be shown to be
h (x) (y) 2n(z)i =
2z h (x) (y) (z)i =
It turns out that the coe cient of this leading term which we can now call C1;1;2n satis es
structure constant C111 which we compute in the next subsection.
The special case of C111 for n > 1
Let us now consider the action of a triple Laplacian on h
i for n > 1, which lies outside
box operator three times one nds the following leading contribution
the SDE has been used three times
2n(x) 2n(y) 2n(z)i L=O
C2frne;e2n;2n
Comparing the two, we obtain the following expression of order O(g3) for the structure
C111 =
LO C2frne;e2n;2n (2n 1)6
(2n)!3 28n(n 1)
g3 =
28n(n 1)n!3
dimension 1. Following [15], one may evaluate at leading order
2x2y2z h (x) (y) (z)i L=O 32(
On the other hand, in this case C111 is already known, because eq. (4.18) is still valid
for k = 1 and gives5 C111 =
corresponding equation found from the SDE
8 32C111
8 codd =
we nd the relation
Lee-Yang universality class
g at the xed point is proportional to p
5This is also in agreement with the OPE coe cient found in [16].
which shows again that the interacting xed point is IR attractive.
Collecting the results: odd potentials
stants in the normalization obtained by rescaling the elds
the propagator to unity.
cod1d=2 which normalizes
1 in terms of , and
nally using the relation (4.6), which links the anomalous scaling of
the descendant operator 2 to the one of
summary, for the Lee-Yang universality class we get
one obtains the leading -dependence of 2. In
g2 =
1 =
2 =
g2 =
g =
, we nd that
Moreover, the fact that 1 + k
k = O( ) :
p2=3
+ O( ) ;
In fact one can restrict to l
repetition. Some of these structure constants are listed as follows
C122 =
; C111 =
; C113 =
; C133 =
: (4.35)
C114 =
Case n > 1
constraints. It is not possible to
xed point g( ) so the results are expressed in
terms of the coupling g, which always appears through the combination g condd
1=2, with codd
given in eq. (4.4).
We start from the anomalous dimensions. The leading order constraints give
xed point coupling g( ) and therefore we expect both negative 1 and 2 (which is instead
1 =
2 =
(2n+3)(2n 1)3 (codd2 g)2
(2n 3)(2n+1)!
+ O(g3) ;
+ O(g3) ;
2 = 2
(2n+3)(2n 1)
+ O(g) :
k = O(g2) ;
2n = 1 +
from which we can deduce a well determined leading order result for their ratio
While for k > 2, all one can get is
Furthermore, from the relation between the scaling dimension of
and 2n one nds
C1;1;2k =
C111 =
26k(k 1)(4(k n)2
1) k!2(2n
28n(n 1)n!3 (codd2 g)3 + O(g4) :
For the structure constants we have, at the leading order approximation:
+ O(g2) ;
Conclusions
value d = dm
. What renders our analysis unique is that for most values of m, the upper
The sequence of models for m even enjoys Z2 parity and encodes the scale invariant
independent computation based on perturbation theory [22].
The extent of our results di ers between even and odd models, and the strength of the
2n, for which we could obtain the anomalous dimensions 1 and 2 and k n, two entire
families of structure constant C1;2k;2l 1 and C1;1;2k, as well as a relation between
critical coupling g( ). In section 4 we studied the odd potentials 2n+1, for which we could
determine 1 and 2 together with the structure constants C1;k;l, C1;1;2k and C1;1;1. Only
for the cubic potential 3, corresponding to the Lee-Yang universality class, we could
to re-express all critical quantities in terms of 1, which yields some simpli cation. All
Our analysis is very encouraging in that it can be considered as a rst step in the
bridge from criticality in dimension d
2 to the well known minimal models in CFT
perturbation theory [41, 42], which may prove useful in this direction.
A special comment must be made on unitarity of the spectrum. In fact, the -expansion
Furthermore, almost all the
m potentials have a purely rational upper critical dimension.
the presence of negative norm states should be investigated.
The possible non-unitarity of the spectrum should be distinguished from the
nonall odd models we would like to point out that the quintic model
5 has upper critical
dimension dc = 130 > 3, implying that
model further in the future [30].
Note added.
After the completion of this work we became aware of the two works [43, 44]
here. In particular the leading anomalous dimensions
k for k > n. Moreover
coincides with our eq. (3.48), once the composite operators k are rescaled by pk! in order
nd (the square of) a family of leading OPE coe cients (see eq. (4.36) of [44]) which
family of O( 2) structure constants C1;1;2k that we have reported in eq. (3.49).
Free theory
A.C. and O.Z. are grateful to INFN Bologna for hospitality and support.
one can rescale the elds to obtain two point functions normalized to one.
We nally consider a generic three point correlator of the form
n1, n2 and n3 connected by l12, l23 and l31 propagators, in cyclic order respectively. One
ni = lij + lki () lij =
(ni + nj
i 6= j 6= k :
The correlator is non zero when there exists a solution such that lij are non negative
integers (lij
0). Then the number of all possible con gurations (contractions) is given by
Here Sdm is the area of the dm-dimensional sphere. A generic two point correlator for the
k is given by
h (x) (y)i f=ree
c =
k(x) l(y)i f=ree
Nn1;n2;n3 =
n1! n2! n3!
l12! l23! l31!
so that, with the above normalization, the explicit form of the correlator is given by
Cnfr1e;en2;n3 =
n1! n2! n3!
n1+n2 n3 ! n2+n3 n1 ! n3+n1 n2 !
( +2)( + 2
d)( +4 d)
correlators and nd some lengthy expressions. The action of one Laplacian 2x is:
Action of the Laplacian
the simple relations @x jx y
one rst derives
The action of two Laplacians 2x2y is:
1 2( 1 + 3 +2 d)( 1 + 2
1 2(2 + 1)( 1 + 3 +2 d)
2 3( 1 + 2 +2 d)( 1 + 3 +2 d)
1 3( 1 + 2 +2 d)( 1 + 3
1(2 + 1) (2 2 3 + ( 1 +2 d)( 1 + 2 + 3 +4 d))
1 3(2 + 1)( 1 + 2 +2 d)
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