#### Mellin space bootstrap for global symmetry

HJE
Mellin space bootstrap for global symmetry
Parijat Dey 0 1
Apratim Kaviraj 0 1
Aninda Sinha 0 1
0 C.V. Raman Avenue , Bangalore 560012 , India
1 Centre for High Energy Physics, Indian Institute of Science
We apply analytic conformal bootstrap ideas in Mellin space to conformal eld theories with O(N ) symmetry and cubic anisotropy. We write down the conditions arising from the consistency between the operator product expansion and crossing symmetry in Mellin space. We solve the constraint equations to compute the anomalous dimension and the OPE coe cients of all operators quadratic in the elds in the epsilon expansion. We reproduce known results and derive new results up to O( 3). For the O(N ) case, we also study the large N limit in general dimensions and reproduce known results at the leading order in 1=N .
Conformal Field Theory; 1/N Expansion; AdS-CFT Correspondence
1 Introduction and summary of results
1.1
Summary of the results 1.1.1 1.1.2 1.1.3
-expansion from constraint equations
-expansion for higher spin exchange
3.2 Simpli cation under -expansion s-channel t-channel Total amplitude
5 Comparison with known results
Pade approximations
Large spin analysis
6 Cubic anisotropy
6.1 Solutions
7 Discussion
A Essential formulas
A.1 The normalization A.2
Mack polynomials
A.3 Continuous Hahn polynomials
A.4 t-channel integral
B Obtaining the cT from symmetry
C Obtaining the large N corrections
{ i {
1
Introduction and summary of results
Wilson's renormalization group approach to understanding critical phenomena [1] has led to
it does not make use of the enhanced conformal symmetry at the
xed point. In the
1970s, [2{5] initiated the study of the conformal bootstrap approach in understanding
critical phenomena. Unfortunately, the resulting equations proved very di cult to solve
and not much progress was made. The work of [6] in the 1980s made remarkable progress
in understanding 2d CFTs. It would take another two decades before progress was made,
starting with the work of [7] which made use of the development in understanding conformal
blocks in [8, 9] in the bootstrap program in higher dimensions [10{47].
In the modern formalism of the conformal bootstrap, building on the work of [7], one
expands a four point function in a conformal eld theory in terms of the conformal blocks
of one of the channels (direct channel). Then one imposes crossing symmetry in the next
step. This is a nontrivial constraint and forms the starting point for the powerful
numerical approach to constraining conformal eld theories. Analytic progress, with this as
the starting point, has been limited [28, 29, 35{40]. In cases with weakly broken higher
spin symmetry, some progress has been made in understanding the leading order anomalous
dimensions [48{50] for lower spin operators as well | however, it is not clear how to
systematize this approach to get subleading orders. The double light cone limit of the bootstrap
equations in the works of [51{54] gives a systematic approach for the large spin limit. For
low spin cases, the methods of [53] allow a resummation to nite values of the spin, including
spin zero but the issue is subtle.1 It is worth exploring other methods which do not require
a resummation. In [55] it was shown how to make use of conformal symmetry of three point
functions to get the leading order (in epsilon) anomalous dimension of a large class of scalar
operators (see also [56{61]). This approach depended indirectly on the equations of motion
that follows from a lagrangian and leads to the question: how does one recover these results
using the bootstrap and go further? The modern incarnation of the bootstrap can be used
to gain some insight into the epsilon expansion using numerics [62] but is not very e cient
in getting analytic results. Hence it is desirable to seek a di erent starting point.
In [5], Polyakov considered a version of the conformal bootstrap that made use of
crossing symmetric blocks from the beginning. Thus while crossing symmetry was in-built,
1In some cases one can add solutions consistent with crossing and with
nite support in the spin. In
consistency with the operator product expansion, for instance in the direct channel, was
not guaranteed. There are spurious poles in the expansion which need to be cancelled.
Demanding this consistency leads to an in nite number of constraints. This approach lay
dormant for a long time. In [63], this approach was revisited and it was pointed out that
it could be made to work at the next nontrivial order in the epsilon expansion. In [64,
65], it was realized that the full power of [5] could be harnessed in Mellin space [66{77]
where the systematics of non-zero spin exchange was both conceptually and calculationally
simpler. Quite remarkably, the epsilon expansion results at three loops in the Feynman
diagram approach were reproduced leading to agreement with existing results for anomalous
dimensions as well as new results for OPE coe cients which have never been calculated,
barring the stress tensor and conserved current exchanges (which are known upto two loop
order in the epsilon expansion for the Ising case).
The reason why [64, 65] worked so e ciently relied on two key ingredients. First, the
direct channel expression for the leading spurious pole naturally leads to an expansion in
terms of a convenient orthonormal basis in terms of the continuous Hahn polynomials [78,
79]. Second, the crossed channels got contributions from only one scalar operator upto the
rst two or three (depending on the spin in the s-channel) subleading orders in epsilon.
We will demonstrate that this approach works for the O(N ) case as well in the epsilon
expansion. This will lead to reproducing known three loop results as well as new results
for the OPE coe cients for various operators. Another reason for looking at the O(N ) case
is that 1=N in the large-N limit provides another expansion parameter for a xed spacetime
dimension d and it is natural to ask what happens in this case. There is a large body of
work using a bootstrap type approach and conformal symmetry to understand this very
important case [80{86] which ties up with the AdS/CFT correspondence. As we will show
that the single operator contribution in the crossed channel holds only upto leading order in
1=N , enabling us to easily extract the leading 1=N terms. To go beyond these orders, will
require a careful study of the systematics of all the spurious poles, not just the leading one,
and also some mixed correlators. This will be taken up in the near future in a separate work.
Another important case of N -scalars that we will consider is the theory with cubic
anisotropy [87]. In the space of couplings there are four xed points | the Gaussian
xed
point corresponding to the free theory, the Ising xed point corresponding to N decoupled
4 theories, the O(N ) xed point arising from an interaction g1(PN
xed point corresponding to a continuum theory with the interaction g1(PN
i=1 i i)2 and the cubic
g2 PiN=1( i)
4 | for this last case the discrete symmetry i $
j; i !
i=1 i i)2 +
i is preserved.
For a certain N < Nc, the O(N ) xed point is the stable xed point while for N > Nc the
cubic
xed point is the stable xed point. The value of Nc that follows from an epsilon
expansion analysis is less than 3. To our knowledge, this value has not been determined
using the modern numerical bootstrap and our analysis may be a useful starting point
to address the same. The N = 3 case is relevant for ferromagnets. We will set up the
equations for this problem and derive anomalous dimensions and OPE coe cients for
operators quadratic in the eld. While the anomalous dimension of the singlet scalar and
the fundamental scalar are known to ve loop order [87], many of the results we will quote
appear to be unknown in the literature (to the best of our knowledge).
{ 2 {
Assumptions.
The essential assumptions that we will make in order to solve the
bootstrap conditions, in addition to there being a Z2 symmetry in all cases are:
1. There is a unique conserved stress tensor and a conserved spin-1 current. However,
the conservation of spin-1 current does not hold for cubic anisotropy.
2. In the -expansion the OPE coe cients of higher order operators like ( i i)2 begin at
O( ). This is expected from the free theory, the only nontrivial bit in this assumption
is that it begins at O( ) rather than say O( 1=2).
We summarize below the ndings of the paper. We use the colour code of blue to indicate
We nd the anomalous dimensions and OPE coe cients (squared) of operators for the
critical O(N ) model in d = 4
at the Wilson-Fisher zed point, for general N . The
results are obtained as an expansion in . The table below summarizes the operators, and
the equations showing the corresponding corrections. The anomalous dimensions below
agree with literature [88{92] while the OPE coe cients are all new results.2;3
operator O
dimension
O
OPE coe cient C
O
CO
(i j)
(3.16)
(3.19)
metrized traceless combination of ` derivatives.
2The symmetrization and antisymmetrization brackets are de ned as A(ab) = (Aab + Aba)=2 and A[ab] =
3We show only a schematic form of the operators. For example i@` i indicates a primary with
sym{ 3 {
Nij k k
(4.5)
These results are in agreement with the results from [48, 86].
We also consider the special case of a broken O(N ) symmetry with an interacting term like
into a diagonal part and an o -diagonal part. We have obtained the anomalous dimensions
and OPE coe cients for a few operators, which are summarized below. The generalized
ijkl notation is introduced in (6.2).
. In this model the symmetric traceless operators break
(i j)
ijkl k l
ijkl k l
(6.13)
(6.14)
predictions and also a large spin analysis to compare with known results. In section 6 we
show how to modify our equations for a di erent kind of symmetry, which is the cubic
anisotropic case. There are four appendices which give the essesntial formulas, alternate
methods to verify our ndings and some results which were too big for the main text.
2
Mellin space bootstrap for O(N )
We begin by reviewing the analysis of [64, 65] and extend the ideas to theories with O(N )
symmetry. For the O(N ) case the spectrum contains operators that behave di erently
{ 4 {
from one another under O(N ) transformations. A two point OPE can have the following
operator content in the spectrum,
i
j
f1; S; T(ij); A[ij]g ;
where S denotes O(N ) singlets of even spin, T denotes O(N ) symmetric traceless tensors of
even spin, and A denotes O(N ) antisymmetric tensors of odd spin. These operators do not
mix with each other under O(N ) rotations, and give rise to di erent symmetry structures
in a four point OPE. More discussion for O(N ) models and their symmetries can be found
A generic four point function of the fundamentals of the O(N ) in the s-channel can be
The + denotes even spins, and the
denotes odd spins. Each sector has di erent C ;`-s
corresponding to exchanges in that sector. The sums run over primary operators of
dimension
and spin ` and C ;` is the square of the OPE coe cient of the operator carrying
and spin `. We will sometimes loosely refer to C ;` as the OPE coe cient.
Following the analysis of [64, 65] we will write the four point function in the basis of
Witten diagrams, as follows
A(u; v) =
X c ;`(W (s;)`(u; v) + W (t;)`(u; v) + W (u;`)(u; v)) :
;`
Here the constants c ;` are related to the OPE coe cients C ;` via a normalization factor
de ned in (A.1). We can write the Mellin representation of a Witten diagram (for identical
external scalars) as follows,
W (s;)`(u; v) =
Z
ds dt
s)2 M (s;)`(s; t)
where M (s;)`(s; t) is the Mellin amplitude of W (s;)`(u; v) given by,
(s;)`( ) (s;`)(s)P (;s`)(s; t)
4We will choose the normalization of blocks such that in small u and 1
v limit g ;`(u; v)
u
( `)=2(1
v)`.
{ 5 {
Here we have 2 = (h +
`)=2,
2 = (h
`)=2 and h = d2 . The Mack polynomials
h;`(s; t) are polynomials of degree ` in s and t. Their form is shown explicitly in
For the four point function written in (2.2) carrying the O(N ) indices, the sum of
Witten diagrams in a certain channel can be decomposed according to singlet, traceless
symmetric and anti-symmetric operator exchanges. The s-channel can be written as,
X c ;`W (s;)`(u; v) =
;`
Z
ds dt
where i stands for S; T; A for singlet, symmetric traceless and antisymmetric operators
and
P (s)
appendix A.2.
with,
respectively.
2.2
dsdt
{ 6 {
(s;)`( ) =
2 i((
h)2
(s;`)(s) =
The t-channel can be done in a similar manner by replacing x2 $ x4 , i2 $ i4 and
u $ v. After this interchange we can bring the integral into the form (2.5), by relabelling
! t. Then the Witten diagrams in the t-channel can be written in
M i;(t)(s;t)=
d ci ;`0 M (t;)`0 (s;t)=
(t);`0 ( ) (t)
;`0 (t)P (;t`)0 (s
and P (;t`)(s; t) is obtained from P (;s`)(s; t) by interchanging s $ t. For identical scalars,
(t);`0 ( ) is the same as (s;)`0 ( ).
Similarly in u-channel we can write,
Xc ;`0 W (u)
;`0 (u;v)=
;`0
dsdt
In u-channel the Mack polynomial P (;u`)(s; t) is obtained from P (;s`)(s; t) by transforming s !
s t and t ! t. Once again for identical scalars, we have (u)
;`0 ( ) is the same as (s;)`0 ( ).
2.3
Total amplitude
The total crossing symmetric amplitude is given by,
These poles correspond to operators present in the OPE (there are also shadow poles
occuring at (d
`) + 2n but they can be eliminated by an appropriate choice of
contour). They come from the Mellin amplitude M ;` when we look at the simple pole at
=
(
h) and the -functions in
(s;`)(s). These poles then reproduce the u(
`)=2+n
dependence that one expects in the OPE. We call these the physical poles.
; t) :
(2.15)
(2.16)
(2.17)
{ 7 {
Now there are also poles in s that do not correspond to operators present in the OPE.
These poles occur at,
s =
+ n where n = 0; 1; 2
+n dependence which are spurious because they typically
do not occur in the s-channel OPE.5 Since one already obtains s-channel OPE, which is
the full A(u; v), from the physical poles, these other poles are called unpysical poles, and
the spurious u-dependences as unphysical terms.
Let us look at unphysical terms, with the leading order in u. These occur at the pole
s =
s =
and the residues are simply given by the individual Mellin apmlitudes evaluated at
. They can be expanded in terms of the basis of the continuous Hahn polynomials
Q`2;s0+`. So let us write
denote physical pole contributions, and other spurious poles. The polynomials
Q`;0(t) are given in terms of the Mack polynomaials P (;s`)(s; t) as
M i;(s)(s !
M i;(t)(s !
M i;(u)(s !
X
ci ;`qi;(s) 2
;` Q`;0
+`(t) +
;`;i
X
;`;`0;i
X
;`;`0;i
ci ;`qi;(t)
;`j`0 Q`;0
ci ;`qi;(u)
;`j`0 Q`;0
2
2
+`(t) +
+`(t) +
Q`;0(t) =
(
1)`(2h
1)`
P
h;` s =
In the s-channel we have,
qi;(;s`)(s) =
41 ` (2s + `
1)` (2h
2s
`
(
1)` (h
`
s)2
:
(2.22)
Let us write this as,
qi;(;s`)(s) = q(2;;`t) + (s
)q(1;;`s) + O((s
)2)
=
41 ` (2
+ `
h)
(`
+ 2
)(` +
+ 2
2h)
+ (s
)
(`
42 ` (2
+ 2
+ `
h + 1)
)2(` +
+ 2
where i stands for S; T; A for singlet, symmetric traceless and antisymmetric operators
respectively. In the above equation the rst term in the second line, is associated with the
log term u
term u
log u , while the second term is part of the coe cient of the non-logarithmic
(we will call this term the power law). We will need to sum up the coe cients of
both log and power law terms from all the three channels and equate them to 0. For this
5Except when the operator with dimension 2
is protected in which case we will have to consider the
contribution from these operators like the way we would treat the disconnected part.
2
`
; t :
2s) (s;)`( )
(2.20)
(2.21)
(2.23)
2h)2
;
purpose, only the two terms shown in (2.23) are enough since once the log coe cients are
0, all that is left in the power law coe cient is the second term in (2.23).
The expansions in the t and u channels are possible because the continuous Hahn
polynomials Q`;0(t) are orthogonal polnomials. Their orthogonality property reads
where `(s) is de ned in (A.7). The properties of continuous Hahn polynomials are detailed
in appendix A.3 . Using this for the crossed channels we have,
HJEP07(21)9
qi;(;t`)j`0(s)=
qi;(;u`)(s)=
`1(s) Z 2dti (s+t)2 Q`2;s0+`(t)Z d ( 2 t
`1(s) Z 2dti (s+t)2 Q`2+s+;0`+(t)
Z
) ( 2 t
) (t);`( )P (;t`)0(s
;t+
)
d ( 2 t
) ( 2 t
) (`u0)( )P (;u`)0(s
So near the pole s =
the integrand would look like,
A(u;v)=
qi;(;t`)j`0(s)
qi;(;u`j)`0(s)
qi;(;2`j;`t)0 + (s
qi;(;2`j;`u0) + (s
to get,
)qi;(;1`j;`t)0 + : : :
)qi;(;1`j;`u0) + : : :
2cA;`qA(;`s)+XcA;`0(qA(;`uj)`0 +qA(;`tj)`0)
expand qi;(;t`)j`0(s) and qi;(;u`j)`0(s) around the point s =
The explicit formulas for qi;(;t`)j`0 and qi;(;t`)j`0 are given in (A.9). As given in (2.23) we can
The terms qi;(;2`;s=t=u) give the log unphysical terms. Once we set the log terms to 0, the
power law unphysical terms are given in terms of qi;(;1`;s=t=u). In the crossed channel we will
mostly need the expression for the `0 = 0 contributions which are given by,
q(2;;`tj)`0=0 =
Z
d
(t);0( ) ( ) ( )2`((
)`)2 X` ( `)q(2
`
(
)(2
+ ` 1)`
q=0
+ ` 1)q (q + ) (q + )
((
)q)2 q! (q
2k +
+ )
{ 9 {
and
q(1;;`tj)`0=0 =
Z
d
"
+ )
one can show that,
The above analysis does not include the case where the exchange operator is an identity
operator which gives the disconnected part of the four point function. This is given by,
This has a Mellin transform that can be written as,
Adis(u; v) =
Z (d2s di)t2 us vt ( t)2 (s + t)2 (
s)2
i1i2 i3i4 Md(sis) + i1i4 i2i3 Md(tis) + i1i3 i2i4 Md(uis) ;
q(t);`j`0 (s) = ( 1)` q(u;)`j`0 (s) :
(2.31)
where the Mellin amplitudes of the disconnected s, t and u channels are given by,
;
;
:
`
(2.30)
(2.32)
(2.33)
(2.34)
(2.35)
=0;`j`0=0
(2.36)
Md(sis)(s; t) =
Md(tis)(s; t) =
Md(uis)(s; t) =
( 2(
( 2(
( 2(
(s
(
s) 2( t) 2(s + t)) 1
s) 2( t) 2(s + t)) 1
s) 2( t) 2(s + t)) 1
st
)(t +
)
s
t)t
Note that, the Mellin amplitude of the identity piece is not well-de ned. However, for our
purposes it su ces to consider only the relevant poles, as in (2.34).
The equation (2.33) can be rearranged according to singlet, traceless symmetric and
antisymmetric tensor combinations, as below,
Adis(u;v)=
+
1
2
Md(tis)
Md(uis)
This integral also has spurious poles, but they only come from the simple poles of Md(tis)(s; t)
and Md(uis)(s; t). Now the spurious terms come only from t and u channel disconnected parts
and there are no log terms from them. Since we have only power law u
we will expand
the Mellin amplitude in the basis of continuous Hahn Polynomials, as shown below,
Md(tis)(s !
`
X q(t=)0;`j`0=0 +q(u=) 0;`j`0=0
X q(t=)0;`j`0=0 q(u=) 0;`j`0=0
indicate terms regular when s !
Let us now take the equations (2.27) and (2.38) and equate the coe cients of logs and
powers laws to 0. For the log terms we just put s =
in q ;`(s) in all channels, and equate
them to 0 for every value of `, and for each of the singlet, symmetric and antisymmetric
sectors. This is because since the Q2s+`(t)-s form a complete basis, each of them are
independent, and so are each of the tensor structures. So we get six equations for every `, two
corresponding to each sector. Using (2.31) the constraint equations undergo considerable
simpli cations. For even spin singlet exchange in the s-channel the constraints reduce to
the following,
X
"
q(1=;t)0;`j`0=0(s)=
and q(1=;u0);`j`0=0(s)=
(
(
`(s) 1 Z dt
`(s) 1 Z dt
s)2
s)2
)q0 ;`)js=
terms from (2.20). In terms of q(1;;`s=t=u), (2.35) reads,
Since this term is associated with the power law term, the q(1=;t=0u;`)j`0=0(s) will have to cancel
cS ;`qS(;2`j;`s0) +
2 XcS ;`0 qS(;2`j;`t0) +2 1+
N
1
N
2
N 2
`0
XcT ;`0 q ;`j`0
T (2;t)
cS ;`qS(;1`j;`s0) +
2 XcS ;`0 qS(;1`j;`t0) +2 1+
N
1
N
2
N 2
XcT ;`0 q ;`
T (1;t) +
`0
2 (1;t)
N q0;`j0
For a symmetric traceless operator exchange in s-channel, we have,
X
"
X2cS ;` qS(;2`j;`t0) +2cT ;`qT (;`2;s) +2 1
2
N
`0
XcT ;`0 q ;`j`0
T (2;t)
`0
2 XcS ;`0 qS(;1`j;`t0) +2cT ;`qT (;`1;s) +2 1
2
N
`0
XcT ;`0 qT (;`1;;`t0) +2q(1;t)
Similarly, the constraint equations for antisymmetric odd spin operator exchange in the
s-channel are given by,
In writing the above equations we have used (2.31) and the fact that antisymmetric
operators have odd spins and others have even spins.
HJEP07(21)9
3
-expansion from constraint equations
In this section we will use the above equations to get the dimensions and OPE coe cients
of operators in ( i i)2 with an O(N ) global symmetry in d = 4
Wilson-Fisher xed point. The lagrangian in this theory is given by,
dimension at the
S =
Z
ddx
2
2
+ g( i )
i 2 :
However we will not be using the explicit form of the lagrangian. Instead we will use the
following assumptions:
(3.1)
(3.2)
(3.3)
There is a conserved stress tensor.
Z2 symmetry ( i $
i) is present.
There are N identical fundamental scalars.
The OPE coe cients of higher order operators which vanish in the free theory start
at O( ) in the interacting theory so that the C ;` of these operators are O( 2).
These assumptions will be enough, to determine the dimensions and OPE coe cients,
from the equations above. Our starting point will be the conservation of stress tensor. This
means we will use `=2 = d as an input. Let us write the dimension of i as
(2) 2 + (3) 3 + O( 4). It starts with 1 becuase in the free theory dimension of the
fundamental scalar is (d
2)=2 = 1+O( ). For the stress tensor OPE coe cient we write C2h;2 =
CS(0;2) + CS(1;2) + CS(2;2) 2 + CS(3;2) 3 + O( 4). In the singlet equations (2.39) and (2.40), we have,
= 1+ (1) +
45
4
c2h;2q2Sh(;2`;=s)2 =
S
CS(0;2) 1 + 2 (1)
+ O( 2) :
Expansion of the derivative q`(=1;2s) in is given by,
c2h;2q2Sh(;1`;=s)2 =
S
45CS(0;2) +
2
3
2
2CS(0;2) + 15CS(1;2) + 30CS(0;2) E
(1)
+ O( 2) :
and
C
(1)
S;2 =
11
36N
The spin-0 singlet and traceless symmetric operators have the leading contributions in
t-channel. We denote their dimensions as,
With this let us look at the singlet and traceless symmetric equations (2.39), (2.40), (2.41)
E is the Euler-gamma. The disconnected part (2.37) for ` = 0 gives,
q
(1;t)
+ q
(1;u)
=0;`=2j`0=0
=0;`=2j`0=0
=
15
2
(47 + 60 E )
(1)
4
+
+O( 2) :
The O( 2
) and O( 3
) terms are too tedious and hence not written here. The crossed channel
terms as shown below start from O( 2
) order. So solving (2.39) and (2.40) at O( ) we get,
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
1
A
(2)
2
1
A
E )
(2)
:
(3.10)
2C
(0)
T;0
1 + 2 T;0
(1)
(1)
S;0
1 + 2
(1)
T;0
1 +
(1)
S;0
1 +
(1)
S;0
1 +
(1)
T;0
1 +
(1)
S;0
(1)
T;0
C
(1)
S;0 + C
(0)
S;0
E +
(1)
S;0
C
(1)
T;0 + C
(0)
T;0
E +
(1)
T;0
(2)
(2)
:
E +
E +
+
+
C
(0)
S;0
2
C
(0)
T;0
2
(1)
S;0
2
E
(1)
T;0
2
E
(1)
and (2.42) for ` = 0. The q`=0
c
S
S ;0q
S(2;s)
S ;`=0 =
(0) (1)
S;0 S;0
c
T
T ;0q
T (2;s)
T `=0 =
(0) (1)
T;0 T;0
1 +
C
C
C
C
(0)
S;0
(0)
T;0
T (s)
1 +
The derivatives are given by,
c
S
S ;0q
S(1;s)
S ;`=0 = C
(0)
S;0 +
C
(1)
S;0 + C
(0)
S;0
c
T
T ;0q
T (1;s)
T ;`=0 = C
(0)
T;0 +
C
(1)
T;0 + C
(0)
T;0
(2)
S;0 + C
(1)
S;0
E +
(1)
S;0
(1)
2 E S;0 + 2
(2)
S;0
+ 2C
(0)
S;0 E
(2)
(2)
T;0 + C
(1)
T;0
E +
(1)
T;0
(1)
2 E T;0 + 2 T;0
(2)
+ 2C
(0)
T ;0 E
2
:
The spin 0 disconnected parts reads,
q
(1;t)
+ q
(1;u)
=0;`=0j`0=0
=0;`=0j`0=0
=
2
4(1 +
E )
t(or u)-channel for q`S=;(0t) and q`T=;(0t) only the `0 = 0 operators have the leading contributions.
This is also true for ` > 0. This nice feature is discussed in detail later in this section. So
we would also need the crosssed channels. In the
and the derivative,
The correponding terms for q`T=(t0) and its derivatives are simply given by replacing with the
traceless symmetric scalar,
and
qT (T2;;`t=)0j`0=0 = q`S=(20;t)( CS(0;0) ! CT(0;0); CS(1;0) ! CT(1;0);
S;0 ! T(1;)0 )
(1)
qT (T1;;`t=)0j`0=0 = qT (S1;;`t=)0j`0=0(CS(0;0) ! CT(0;0))
Now using the above in (2.39), (2.40), (2.41) and (2.42) to get the solutions of
CS(0;0); CT(0;0); CS(1;0), CT(1;0);
S(1;0) and T(1;)0. These solutions are listed at the end of this
subsection. But let us rst use them to obtain the crossed channel terms with ` = 2. It is only
the spin 0 operators that will contribute to the q`i(=t)2. They are given by,
cS S;0qS(S2;;`t=)2j0 =
cS S;0qS(S1;;`t=)2j0 =
352 CS(0;0)( S(1;0) 2 (1))2 2
6241 CS(0;0)( S(1;0) 2 (1))2 2
and
and
cT T ;0qT (T2;;`t=)2j0 =
cT T ;0qT (T1;;`t=)2j0 =
352 CT(0;0)( T(1;)0 2 (1))2 2 :
6241 CT(0;0)( T(1;)0 2 (1))2 2 :
This allows us to solve for (2.39) and (2.40) for ` = 2 at the O( 2) order. Thus we get (2).
With this information we go back to solving (2.39), (2.40), (2.41) and (2.42) for ` = 0 at
the order of O( 2). Then we return to ` = 2 and solve for ` = 2. That gives us
up to
the O( 3) order. Skipping the details of O( 3) let us just give the results. The dimension
of
is obtained to be,
= 1
2
+
(N + 2)
4 (N + 8)2
2
(2 + N ) ( 272 + N (N
16 (8 + N )4
56)) 3 + O( 4) :
HJEP07(21)9
(3.12)
(3.13)
(3.14)
(3.15)
For the singlet and symmetric traceless scalars we get,
The OPE coe cient of the stress tensor is given by,
CS;2 =
1
3N
11
36N
+
108N (8+N )2
+
1296N (8+N )4
514+145N +7N 2 2
41824+27968N +3462N 2 193N 3 +N 4 3
This is how far one can go with ` = 0 and ` = 2. Obtaining the anomalous dimensions
and OPE coe cients to next order of
becomes di cult since, as we discuss below, an
in nite number of operators start contributing in both the channels at the next order.
However for the terms qi;(;1`=;s)0 and qi;(;1`=;t)0;`0 at the O( 3), only one operator contributes in
each channel, even though for the analogous terms qi;(;2`=;s)0 and qi;(;2`=;t)0;`0 an in nite number
of terms contribute at O( 3). This means even though we would not be able to compute the
O( 3) ` = 0 anomalous dimensions, S(3;0) and T(3;)0, if we knew these quantites, we would be
able to compute the O( 3) OPE coe cients CS(3;0) and CT(3;0). Borrowing the O( 3) anomalous
dimensions from [88{91],
(2 + N ) 3N 3 + 96(8 + N )(22 + 5N ) (3)
5312
2672N
452N 2
10624 + 4192N + 56N 2
134N 3
5N 4
192(8 + N )(22 + 5N ) (3)
S(3;0) =
T(3;)0 =
CS(3;0) = (2+N )
CT(3;0) =
we can solve (2.39) and (2.41) for ` = 0 at O( 3) order, to get,
46N 2 +19N 3 +(8+N )(1504+N (344+N (14+N ))) (3)
27N 3 +N 4 +2(8+N )(752+N (204+7N )) (3) 4256 1728N
110N 2
2(8+N )5
:
(3.23)
3.1
-expansion for higher spin exchange
We now proceed to study the higher spin operators using the constraint equations. We use
the constraint equations (2.39){(2.44) to nd the OPE coe cient and anomalous dimension
for the spin ` singlet, symmetric traceless and antisymmetric operator exchange in the
schannel. The higher spin conformal dimension are of the order
= 2
us denote their dimensions and OPE coe cients as,
+ ` + O( 2). Let
8(8 + N )5
8(8 + N )5
2N (8+N )5
i;` = 2
+ ` + i(;2`) 2 + i;`
(3) 3 + O( 4)
Ci;` = Ci(;0`) + Ci(;1`)
+ Ci(;2`) 2 + +Ci(;3`) 3 + O( 4)
(3.17)
(3.18)
where the subscript i stands for singlet(S), symmetric traceless (T ) and antisymmetric
(A) exchange respectively. Here we use the fact that the singlet and symmetric traceless
operators exist for even spins only, and the antisymmetric ones for odd spins. Even for the
general ` cases, we nd only the spin 0 singlet and symmetric traceless scalars contributing
to the t and u channels, under the -expansion. The higher spin operators do not contribute
to the crossed channels upto O( 3). However, they will contribute to the O( 4) order. This
is discussed at the end of this section.
To
nd the unknowns we solve (2.39){(2.44), order by order. The steps are exactly
similar as chalked out for spin 2. So we skip the details and give the solutions directly.
The conformal dimensions in the three sectors are given by,
16(N +8)2`(`+1)H` 1 448(N +4)+16`(17`(`(`+2)+3) 126) + O( 4)
where H` is a harmonic number of order `.
It should be noted that for ` = 1 the anomalous dimension
A;` vanishes as it is the
conserved current. Also for ` = 2 we have vanishing anomalous dimension for singlet
representation which is the conserved stress-tensor. However, for ` = 2 symmetric traceless
operators acquire an anomalous dimension. We can compute the higher spin OPE coe
cients for any given spin ` upto O( 3). The explicit expressions for individual N and ` up
to the O( 2) are given below. Their O( 3) part have been obtained for individual `-s, and
can be automated for any value of ` | a general formula can be obtained using a di erent
method [100]. First few have been listed here.
CCSfSr;;e``e =1 + 2" (2+N ) 6+2 3 2`+2`2 +`3 H`
2(8+N )2`(1+`)2
6 5`+2`2 +`3 H2` #
+ c(S3;)` 3
CCTfTr;e;``e =1+ 2" (2(6+N )+2(1+`)( 6 N +(2+N )`(1+`))H` (1+`)( 2(6+N )+(2+N )`(1+`))H2`
2(8+N )2`(1+`)2
#
+ c(T3;)` 3
CCAfAr;e;``e =1 + 2" (2+N ) 2+2 1+2`2 +`3 H` + 2+` 2`2 `3 H2` #
2(8+N )2`(1+`)2
+ c(T3;)` 3
The O( 3) terms ci(;3`) can be computed for any given spin. They also obey a general `
formula. These are given in appendix D .
The anomalous dimension corrections are in good agreement with the known
results [88].
The OPE ccoe cient corrections are new, except for ` = 1 and 2, which
correspond to cJ and cT , which were calculated till O( 2) [97], and agree with our results.
In appendix B we give an alternative computation of cT from symmetry arguments that
agrees with our result (5.3), up to the O( 3) order.
Simpli cation under -expansion
While evaluating the equations (2.39){(2.44), we could get away with just one operator in
the s-channel and two operators (the singlet and traceless symmetric scalars) in the crossed
channels. This is because the other operators in the sum contribute from a subleading
(O( 4)) order in . Let us see how the other operators are suppressed, in in all the channels.
For a spin ` in the s-channel one can have operators with dimension
2m + m + O( 2). Such operators have the form O2m;`
a + b + c = 2m + ` derivatives, 2m derivatives are contracted and ` derivatives carry
indices. For them, the q(2;;`s) is given by,
Here C2m;` is the OPE coe cient of the operator. Since the operator is made of four -s,
it does not exist in the free theory, and in the interacting theory the generic three point
function goes as h i j O2m;`i
function, we have C2m;`
. Since the OPE coe cient is the square of the three point
2 and the whole expression above contributes at O( 4). Now,
there can be other operators too, with higher number of -s. But they contribute at a
more subleading order because their OPE coe cients are even further suppressed in .
3.2.2
In the crossed channels the simpli cations happen due to cancellation of residues of various
poles among one another, under
expansion | the discussion is similar to the one in [65].
The cancellations are such that all the operators in the crossed channels start contributing
from O( 4).
The only operators that can contribute at a more leading order are the
lowest dimension scalars. Even for these operators there are only two poles whose residues
contribute at a leading order, and all other residues cancel among one another to start
from O( 4) or more.
For `0 = 0 the expression (2.25) or (A.9) can have poles at (after substituting s =
),
= (
= (2
= (h + 2n) .
h)
h + 2n)
Here n is a positive integer. For the lowest dimension operators, i.e. those with dimension
= 2 + 0(1) + O( 2), we have the following residues cancelling each other,
Res =h+2n 2 + Res =2
h+2n = O( 4) :
(3.30)
So only the poles I and II (for n1 = 0) contribute to our computations.
Heavier scalars having dimensions of the form
2m;0 = 2 + 2m + m + O( ) contribute
only from O( 4). Tis is due to the following cancellations of residues:
Res =
h + Res =h 2 +2s+2m 2 + Res =2
Res =2
h = O( 5)
h+2m = O( 4)
For spin `0 > 0 we have the following poles in the crossed channels:
= (
= (h
+ `0 h + 2n)
1); (h
2);
; (h
2 + `0)
Here we observe two di erent cases:
Lowest dimension spin `0. These are operators of the form
`0 = 2
+`0+O( 2), whose
dimensions we computed in the paper . We observe the following cancellations for them,
Res =
h + Res =h 2+`0 + Res =2 +`0 h = O( 4)
Res =h+`0+2n + Res =2
h+`0+2n+2 = O( 4) where n = 0; 1; 2;
Res =h 1
Res =h 2
Res =h+`0 3 = O( 4) :
go like C2m;`0
cancellations,
Higher dimensional spin `0 operators. These are the operators O2m;`0 we discussed
in 3.2.1 having the dimensions
2m;` = ` + 2 + 2m + m + O( 2) . Their OPE coe cients
O( 2) or higher. Accounting for this suppression we have the following
Res =h 2+`0 + Res =2 +`0 h = O( 4)
Res =
h + Res =h+`0+2m 2 + Res =2
h+`0+2m = O( 4)
Res =h+`0+2n + Res =2
h+`0+2n+2 = O( 4) where n 6= m
Res =h 1
Res =h 2
Res =h+`0 3 = O( 4) :
(3.32)
1
(3.33)
Thus the crossed channels get contributions from only a nite number of operators. We
refer the readers to appendix F of [65] for a more detailed discussion of these simplifcations.
N
2
3
4
5
6
10
20
3 results
sults [95] for cT =cT free in d = 3 for the O(N ) model.
Large N results
Numerical results
sults [96] for cJ =cJ free in d = 3 for the O(N ) model. (Only numerical results which were precisely
computed in [96] have been presented.) For N = 2 quantum Monte Carlo [93] results quote the
value 0.917 or 0.904 [94] depending on the extrapolation scheme used.
5.2
Large spin analysis
In [65] it is shown how the s, t and u channels simplify when we consider large spin in
the s-channel. Let us assume a weakly coupled theory, which means a CFT where we
have a certain suitable small parameter g, in terms of which the anomalous dimension can
be expanded. The large spin analysis predicts the behavior of large spin operators as an
expansion in a small parameter, at large `, which matches with the prediction of [51]. In
this section we will brie y review that analysis for O(N ) models with large N . So we will
have g = 1=N , and demonstrate how it can correctly reproduce known results for large
spin dimensions in 4 < d < 6 dimension.
We will start with the correlator h i j k li. The external elds have the dimension,
=
=
d
2
2
+
:
(5.4)
In the s-channel we have the large spin operators having dimensions of the form
i;`
Here p(N 1) are given by,
p(N 1) =
The above formula gives the contributions from di erent O(N ) sectors in the t-channel.
Since this is the t channel we have a sum over i and `0. However one can see from the
large ` dependence, that only operators with small twists dominate the sum.
Here we assume the presence of a singlet scalar of dimension
S = S = 2 + O(g) = 2 +
N
1 S(1;0) + O(1=N 2)
in the spectrum. This operator is the lagrange multiplier eld present in the large N critical
theory. It becomes signi cant in 4 < d < 6 dimensions because it is the leading operator
at large ` in the t-channel. Then the sum in the t-channel goes away and we use (5.9) to
expand (5.8). Finally using the contraint equations (2.39). (2.41) and (2.43), we get,
We use (2.25) to evaluate q(t);`j`0 . Both t and
` in the integrand. With the above approximation in the integrand, if we do the t integral
rst, we will have poles at t = 2
and t = 2
. All other poles in t have residues
suppressed in ` or lie out of the contour. Similarly in the
integral we will only have poles
at
=
( i
h) (signs depend on the pole of t considered before). Other poles have residues suppressed in `, or are out of the contour. Writing contours are determined from the power of
c i;`q(2i;;t`)j`0 =
C i;`0 N i+`0;`0 2 21 +` 3`0+2
p
i e`` ` i `0 + 2i
(1 h+`0 + i)
2`0+ i 1
2
i = i
`0 +
`0, we arrive at,
+ 2i h 2 (`0 + i 1)
(5.5)
(5.6)
(5.7)
:
(5.8)
(5.9)
:
1
N
1
N
p+q
p+1!
(5.10)
(5.11)
:
Both
and `i allow expansions in N 1 as
O(N 2). From (2.23), we can use `
1 to obtain,
=
(1)=N + O(N 2) and i;` = i(;1`)=N +
i;` = d
2 + ` + i;` :
In the above equation, PS = 1=N , PT = PA = 1=2.
To evaluate the t-channel, we will use the following approximation of Q`2;s0+` [65],
Q`2;s0+`(t) ` =s;t 2`` s t (s + `)2 ( 1 + s
( t)2 ( 1 + 2s + 2`)
t + `)
:
0(N 1) + 1(N 1) log ` + 2(N 1)(log `)2 +
CS(1;0) X2 ( 1)p+q+1
N
+ O
q=0
1
N
2p!
p 2
(1) p+q
0
1
N
( 0(1)
p 1
(d
4)3;
(d
4)2;
q
1
N
(d
p
4)2 q
(d
4);
2 (1))q 2
In general dimension the second line of (5.11) is not signi cant, and we can just take q = 0.
For large N critical model we have,
Plugging this in, we get the correct 1=`2 dependence for the large spin currents in all the
sectors as given in [48] for p = 0,
CS(1;0) =
2 (2h
(h
2) sin( h)
2) (h
1)2
0 =
2h (2h
N (h
1) sin( h)
2) (h + 1)
:
:
One can also compute the leading log ` term, given by,
1 =
N 2
1 2 1+4h (h
2)(2h
1) 2 h
h 3 2(h
1)
12 sin2(h )
:
This matches with the expected log term at O(1=N 2) [52].
6
Cubic anisotropy
The 4 interaction of (3.1) can be extended to the case of cubic anisotropy, whose lagrangian
(5.12)
(5.13)
(5.14)
(6.1)
(6.2)
(6.3)
(6.4)
Here we have introduced the generalized ijkl-function. It is de ned by
2
S =
ddx
+ gijkl i j k l ;
gijkl =
( ij kl + il jk + ik jl) + g2 ijkl :
(1; when i = j = k = l;
0; otherwise :
Z
g1
3
ijkl =
Also ijkl kl = ij . The interaction term then looks like g1( i i)2 + g2 P
i i
4 . This action
breaks the O(N ) symmetry. The symmetries respected by this system are: i $
i and
We will now study our bootstrap conditions to understand what happens in the cubic
anisotropy case without referring to the lagrangian. Since O(N ) symmety is absent, we
cannot use the form (2.3). To get the operator content let us look at the two point OPE i
j .
i
j
f1; S; T(ij); Vij ; A[ij]g :
Here schematically S
diagonal operator (i.e. 0 when i 6= j) and A[ij] =
j] denotes the antisymmetric
ij k@` k is a traceless
N
operator. The traceless symmetric operator of the O(N ) case has now broken up into the
two pieces T and V which are di erent multiplets that do not mix with each other. In
are to indicate even and odd spins respectively and the C ;` appearing in
a sector corresponds to operators in that sector.
The corresponding Witten diagram expansion can be written as,
Xc ;`W (s;)`(u;v)=
Z (d2s di)t2 usvt ( t)2 (s+t)2 (
s)2 ( i1i2 i3i4)MS;(s)(s;t)
free theory, the two-point function of one operator with another in a di erent multiplet is
0, and we will take the same operator basis for the interacting theory.
Now let us take h i1 i2 i3 i4i = (x212x324)
A(u; v). We will use the above OPE to
write the conformal blocks and the associated irreducible tensor structures. We have,
A(u;v)=XC ;`( i1i2 i3i4)g ;`(u;v)
X 2XcV ;`0qV;;(`2j`;t0)(N 1)+N 2XcS ;`0qS;;(`2j`;t0) +2XcT ;`0qT;;(`2j`;t0)(N 1)
`0
+( i1i3 i2i4 + i1i4 i2i3 2 i1i2;i3i4)MT;(s)(s;t)+ i1;i2;i3;i4 N i1i2 i3i4 MV;(s)
1
+( i1i4 i2i3 i1i3 i2i4)MA;(s)(s;t) :
(6.6)
As before, the M i;(s) are given by (2.10). The t channel is obtained by changing s ! t+
,
t ! s
and 2 $ 4. The u channel is obtained by changing s !
s
t, and 2 $ 3.
The rest of the analysis is similar to the O(N ) case. We sum over all the channels and
rearrange them according to the tensor structures appearing in the s-channel (6.6). Now
we expand the Mellin amplitudes in terms of the continuous Hahn polynomials as in (2.20).
Then corresponding to i1i2 i3i4 we get the equations,
X 2XcV ;`0qV;;(`1j`;t0)(N 1)+N 2XcS ;`0qS;;(`1j`;t0) +2q(1;t)
=0;`j`0=0 +2XcT ;`0qT;;(`1j`;t0)(N 1)
S+
T+
A
`0
`0
`0
`0
`0
Corresponding to i1i3 i2i4 + i1i4 i2i3
2 i1i2i3i4 we get the equations,
X
cT ;`qT;;(`1;s) + X cS ;`0qS;;(`1j`;t0) + X cT ;`0qT;;(`1j`;t0) + q(1=;t)0;`j`0=0
`0
X
cT ;`qT;;(`2;s) + X cS ;`0qS;;(`2j`;t0) + X cT ;`0qT;(2;t)
;`j`0
`0
`0
`0
+cS ;`qS;;(`1;s)N
=0
+cS ;`qS;;(`2;s)N
=0: (6.7)
N
N
`0
`0
1 X cV ;`0qV;(1;t) = 0
1 X cV ;`0qV;(2;t) = 0
;`j`0
;`j`0
(6.8)
The part i1i2i3i4
X cV ;`qV;;(`1;s)+2X
XcS ;`0qS;(1;t) XcT ;`0qT;;(`1j`;t0)+q(1=;t0);`j`0=0+XcV ;`0qV;;(`1j`;t0) 1
;`j`0
Finally we have the antisymmetric i1i4 i2i3
i1i3 i2i4 sector equations, which are
X
cA;`qA;;(`1;s) + X cS ;`0qS;(1;t)
;`j`0
X cT ;`0qT;;(`1j`;t0) + q(1=;t)0;`j`0=0
X
cA;`qA;;(`2;s) + X cS ;`0qS;(2;t)
;`j`0
`0
`0
X cT ;`0qT;(2;t)
;`j`0
1
N
1
N
=0
=0:
(6.9)
(6.10)
`0
`0
N
N
;`j`0
;`j`0
HJEP07(21)9
In writing the above equations we have used the relation (2.31).
In this subsection we will solve the above equations to nd the anomalous dimensions and
OPE coe cients of the operators in the spectrum. Once again, we will use the conservation
of stress tensor (i.e. `=2 = 4
write it as,
) as the input. This will give the dimension of . Let us
= 1 + (1) + (2) 2 + (3) 3 + O( 4)
(6.11)
To determine
we will have to solve the above equations simultaneously order by order
in . This will require the crossed channels too. For the same reason as described in 3.2.2
only the scalar operators of lowest dimension will contribute to the t-channel till the O( 3)
order. So, in order to solve
till O( 3) we also have to know the dimensions and OPE
coe cients of the ` = 0 operators and OPE coe cient of the spin 2 operator.,
i;0 = 2 + i(;10) + i(;20) 2 + O( 3) ;
Ci;0 = Ci(;00) + Ci(;10) + Ci(;20) 2 + O( 3) ;
C2h;`=2
CS;2 = CS(0;2) + CS(1;2) + CS(2;2) 2 + CS(3;2) 3 + O( 4)
(6.12)
where i = S; T; V . There is no spin 0 antisymmetric operator. Now using this
parametrisation we solve the equations (6.7) for ` = 2 and (6.7), (6.8) and (6.9) for ` = 0 simultaneously.
= 1
2
(N
1)(2 + N ) 2
108N 2
+
(N
1) 1728N
222N 2 + 109N 3
The spin 0 dimensions and OPE coe cients are given by,
1)(19N
81N 3
(N
1)(424 + N ( 326 + 19N )) 2
162N 3
1)(106
89N + N 2
424 + N (530 + N ( 127 + 3N )) 2
81N 4
162N 3
424 + N ( 538 + N (131 + 19N )) 2
2(N
2)(N
1)(106 + 17N ) 2
and the spin 2 OPE coe cient which is given by,
CS;2=
2
3N
11
18N
486N 3
( 22+N (11+74N )) 2+ 18656+N ( 37664+N (22206+N ( 4019+902N ))) 3
:
52488N 5
The quantities and
S;0 are known in literature [87], and our results agree with them.
Now let us turn our attention to higher spin operators. Using the information obtained
above we can determine their anomalous dimensions and OPE coe cients order by order
in . Let us denote them as
i;` and Ci;`. Here i stands for S; T; V; A. Now we solve (6.7){
(6.10) and use (6.13){(6.19) order by order, to determine the above unknowns. For i =
S; T; V we have only even spins, and for i = A we have only odd spins. We obtain,
S;` =2
54N 2`(1+`)
2)(N +3) 2 +
(N 1)
5832N 4`2(1+`)2 ((` 2)`(1+`)(3+`)
168) 108) +216N 2(N +2)`(1+`)(H` 1 3H`) 3
T;` =2
+`+ (12 18N +( 1+N )(2+N )`(1+`)) 2 +
54N 2`(1+`)
1
5832N 4`2(1+`)2 109N 4`2(1+`)2
+1696( 2+`)`(1+`)(3+`) 32N (1+`)( 108+`( 636+107`(1+`)))+N 3(540+`(2952
+`(3053 331`(2+`)))) 6N 2(540+`(2656+`(1899 325`(2+`))))+216N `(1+`)
16 18N +7N 2 H 1+` +3(2 3N )N H`
3
V;` =2
54N 2`(1+`)
2 +
1
5832N 4`2(1+`)2
1696(` 2)`(1+`)(3+`) 32N `(1+`)(107`(1+`) 324)+6N 2( 108+`(1+`)( 112
+325`(1+`)))+N 4( 108+`( 168+`(373+109`(2+`))))+N 3(540 `(648+`(1843
+331`(2+`))))+216N 2`(1+`) N +N 2 6 H` 1 3( 2+N )(1+N )H`
3
( 18(N
2)+(N
1)(2+N )`(1+`)) 2 +
54N 2`(1+`)
1
5832N 4`2(1+`)2
109N 4`2(1+`)2 +1696`(1+`)
18+`+`2
6N 2(324+`(1+`)(3696 325`(1+`)))
32N (1+`)( 54+`( 1440+107`(1+`)))+N 3(540+`(2952+`(3053 331`(2+`))))
+216( 2+N )N `(1+`)((7N 4)H` 1 9N H`) 3
:
(6.24)
(6.14)
(6.15)
11
324
55
1458
11
300
1133429981
26592073200
1133429981
23932865880
1133429981
24622290000
N =2
N =3
N =10
`=2
955
17496
40379
708588
2291
45000
N =2
N =3
N =10
`=4
33071
699840
6959771
141717600
130753
3000000
`=6
30739672087
588170555280
1723185619
37352700000
`=8
45230019647
834534005760
1892198482723
33798627233280
317067803821
6439305600000
`=10
16447155548067179
285712467504710400
686900153698555567
11571354933940771200
38269562772181723
734857169508000000
`=2
11
324
187
5832
187
8100
`=4
2039
64800
136421
4082400
114437
3780000
`=6
30893
873180
57793
1496880
170857
4677750
`=8
1666423
42456960
36326611
840647808
483771131
11675664000
`=10
1133429981
26592073200
293100163357
6222545128800
130842828107
2880807930000
Note that the anomalous dimension of spin-1 current does not vanish. This is expected
since the rotational symmetry of O(N ) is no longer present, implying J is not conserved.
Now let us come to the OPE coe cients. We write them as,
Ci;`
Cif;r`ee
= 1 + ci;`
(2) 2 + ci;`
(3) 3
:
Here we de ne the free
V;`
Cfree = Cfree
O(1);` and CAfr;e`e = Cfree
O(1);`
spin in the O(1) theory, given by,
S;`
eld OPE coe cients by Cfree = Cfree
O(1);`
T;`
=N , Cfree = Cfree
O(1);`
=2,
=2 . Here Cfree
O(1);`
is the free eld OPE coe cient for any
COfr(e1e);` =
2 (` + h
`! (h
1)2
(` + 2h
1)2
(2h + 2`
3)
3)
:
(6.25)
(6.26)
The quantities ci;`
(2)
and ci;`
(3)
have been obtained in a closed form for general N and `.
However the expression is too big to present here. So we give their values for speci c N -s
and `-s. The general form can be made available on request.
N =2
N =3
N =10
955
17496
64489
1417176
29837
1215000
33071
699840
1263227303
27776649600
420441947
11907000000
11300769391
224064973440
27216198529
648336150000
320561250041821
5842334136038400
140366883740173
3005315913600000
2292070375098731839349
39111179676719806656000
506565599576907257701
10059459793395012000000
`=4
1217
45360
4066
127575
15619
472500
`=6
49807
1496880
54163
1428840
9898997
261954000
`=8
1481069
38918880
8997979
210161952
246069253
5837832000
`=10
526157011
12570798240
145754265827
3111272564400
24030761429
523783260000
`=4
13146719
342921600
121937213
2777664960
1024214479
23814000000
`=6
5729638637
124480540800
53003131961
1069401009600
1660750757513
36306824400000
`=8
866670614587
16829769116160
17350445830481
319502648064600
257502902121943
5259302848800000
`=10
97858846018004947
1755832982119856640
1138230092181377071783
19555589838359903328000
94869301081115147749
1828992689708184000000
N =2
N =3
N =10
73
3240
101
3645
709
27000
`=5
4127
136080
212491
6123600
189809
5670000
`=7
696991
19459440
24712403
612972360
14778019
378378000
(2)
To the best of our knowledge, almost none of these results have been computed before.
So apart from and
S;0 all results presented in this section are new. The new ones
also pass some consistency checks like giving free theory results at N = 1 and Ising model
results for N = 2 [87].
7
Discussion
We have analyzed the Mellin space analytic bootstrap techniques to conformal eld theories
with O(N ) symmetry. Consistency with the OPE imposes non trivial constraints on the
dimensions and the OPE coe cients of the operators appearing in the singlet, symmetric
traceless and antisymmetric representations of O(N ). By considering the leading spurious
pole s =
, we looked at the -expansion and the large-N expansion and demonstrated
that the consistency conditions lead to known results as well as new results for OPE
coe cients. We also studied the case with cubic anisotropy and obtained new results. We
list below some future directions.
It will be interesting to compare the new O( 3) results we have for the OPE coe cients
with what arises from numerical boostrap. In [64, 65] we compared the Ising case
with the spin-4 OPE result in [99] and found impressive agreement.
It will be desirable to develop and algorithm to compute systematically subleading
corrections. We have not used all the equations. There are spurious poles of the form
s =
+ n and we just considered n = 0. With a judicious choice, it should be
possible to extract a lot more information from these equations. By exploiting these
equations it should be possible [100] to extract more information about subleading
terms as well as about other higher order operators in the spectrum for which some
information is known [101{103]. It should also be possible to consider CFTs in higher
dimensions as in [104{106].
It will be important to develop numerical algorithms to solve these equations. As
pointed out in [65], it may be easier to consider expanding these equations around
some t = t0 rather than in terms of the continuous Hahn polynomials. We go from
one set to the other by taking an in nite linear combination. Hence, it is not apriori
guaranteed that convergence (as a sum over the spectrum) in one case will lead to
convergence in the other. It appears to us that expanding around a special point
in t may be more suited for numerics. While the issue about convergence as a sum
over the spectrum is solved in the conventional approach to numerics [107, 108], this
question still needs to be resolved in our approach.
It will be interesting to understand whether this approach can be extended to
logarithmic conformal eld theories [109]. For N = 0; 2; 4, a logarithmic behavior
arises in the correlation function. It is desirable to extend our analysis to physical
systems exhibiting logarithmic behavior in appropriate limits.
To make contact with AdS/CFT it will be interesting to understand the large-N
systematics in more detail. Our progress in this paper was quite modest but that
was because we concentrated on the leading spurious pole. However, our methods
should be useful for future studies along similar lines and also for extending them to
supersymmetric theories [110{112].
Acknowledgments
We especially thank Rajesh Gopakumar for numerous discussions and comments on the
manuscript. We acknowledge useful discussions with F. Alday, J. Cardy, S. Giombi, T.
Hartman, K. Jensen, J. Kaplan, I. Klebanov, P Kraus, G. Mandal, S. Minwalla, H. Osborn,
J. Penedones, D. Poland, S. Pufu, L. Rastelli, S. Rychkov, K. Sen, M. Serone, S. Wadia
and W. Witczak-Krempa. PD thanks Johns Hopkins University, SFSU, UC Berkeley and
UCLA for hospitality during the course of this work. AK thanks Princeton University,
Cornell University, Johns Hopkins University, YITP Stony Brook, Yale University, EPFL
Lausanne, SISSA and CERN for hospitality during this work. A.S. acknowledges support
from a DST SwarnaJayanti Fellowship Award DST/SJF/PSA-01/2013-14.
1) (1 h+ ) 2(`+
1) 4 `+
2
1
+ 1 + 2 2h+`
1 `
+ 3 + 4
1)
2
1 `
+ 1 + 2
1
2
A
A.1
Essential formulas
The normalization
c ;` =C ;`N ;` =C ;`
When we expand the correlator h
i in terms of Witten diagrams, we write the constant
coe cients as c ;`. These constants are related to the OPE coe cients C ;` through a
normalization N ;` which is given by,
As explained in [65] this is obtained by computing the leading power law u
from the Witten diagram and comparing with the conformal blocks.
A.2
Mack polynomials
The Mack polynomials P (;s`)(s; t), for identical external scalars, are given by [66, 70, 72]
X
P (;s`)(s; t) = g
where
X
g
2k
m
:
2
( 1) 2( 1)( 2
s)k( 2
s)k(s + t) (s + t) ( t)m
( t)` 2k m
Q
i (li)
2`(h 1)` k=0 m=0 =0
X
X
X
[ 2` ] ` 2k m ` 2k m
( 1)` k
X
=0
(` k +h 1) ` 2k
m
1)k!(`
2k)!
m
+ 3 + 4 2h+` :
2
(
(A.1)
;
(A.2)
The other notations are given by,
1 =
h+ +` ;
2
1 =
h
2
+` ;
2 =
h+
`
2
and
2 =
h
2
`
;
(A.3)
l1 = 2+` k m+
; l2 = 2+k+m
+ ; l3 = 2 +k+m;
l4 = 2 +` k m: (A.4)
A.3
Continuous Hahn polynomials
We brie y summarize the key properties of the continuous Hahn polynomials. More details
can be found in [65]. It is given by,
Q`2;s0+`(t) =
2`(s)`2
(2s + `
1)` 3F2
`; 2s + `
1; s + t
s ;
s
; 1 :
These polynomials have the orthogonality property [78],
Further we have the identity,
dt (s + t)2 ( t)2Q`2;s0+`(t)Q`0;0
2s+`0 (t) = ( 1)` `(s) `;`0 ;
`(s) =
4``!
(2s + `
1)`2 (2s + 2`
4(` + s)
1) (2s + `
1)
:
Q`2;s0+`(t) = ( 1)`Q`2;s0+`( s
t) :
(A.5)
(A.6)
(A.7)
(A.8)
HJEP07(21)9
`
q(t);`j`0 (s)= `(s) 1 XXgZ
q=0
1
2`((s)`)2
Q
i (li) (2s+` 1)`
Now one can use this identity (A.8) on the t-channel expression (2.25) and u-channel
expression (2.26), with which the two expressions become equal under the exchange t $
s
t. Hence we get the equality (2.31).
A.4
t-channel integral
`0, is given by,
The most general form of q(t);`j`0 (s) in the t-channel for an exchange of operator with spin
d c ;`0 (t);`0 ( ) 2( 1) 2( 1)(
s)m
(
s)`0 2k m
( `)q(2s+` 1)q (k+q+s+ + 2
) (k+q+s+ + 2
(q+2s+2k+ + + 2 + 2 2
+2
1 k q s
2 +
;1 k q s
2
2
2 +
;1 :
(A.9)
This general form is derived in [65]. { 33 {
The second and third on the right are three loop diagrams. The colour indices are indicated at each
vertex. The blobs denote a composite operator (T or J ) insertion.
B
Obtaining the cT from symmetry
d2 2
The central charge cT which is given by cT = (d 1)2C2h;2 can be obtained using symmetries
of the problem and known large N results | the argument for cT in this section is due to
Hugh Osborn. C2h;2 is related to the square of the three point function h i j T i. Here T
k. To obtain the three
r(z))i. For the stress
tensor we will have to contract this with kr rst. Then we will contract the whole 3-point
function with itself.
Now assume a generalised interacting term given by 214 ijkl i j k l. The Feynman
r)i at the 2-loop and 3-loop orders are shown below in
gure 1. Other diagrams go to 0 upon the action of derivatives in
Let us rescale the interaction ijkl ! 16 2 ijkl. The general term from these two
processes can be written as
r)i = O(1) + a ikmn mnjr + b ikmn mnpq pqjr + c inkm rmpq pqjn : (B.1)
The O( 0) term can be anything of the form x ij kl + y ik jl + z il jk. Now the contracting
the above with kr, we get the form,
k)i = 1 ij + 1 ikmn jkmn + 1 ikmn mnpq pqkj + 2 inkm kmpq pqjn :
This contracted with itself should give us the OPE coe cient C2h;2. Hence we can correctly
guess the form,
cT =cT;scalar = N +
ijkl ijkl +
ijkl klmn mnij :
Here cT;scalar is the central charge for N = 1 theory. The O( 0) is just N which follows
from free theory. Now for the O(N ) case we have,
(B.2)
(B.3)
(B.4)
ijkl = ( ij kl + ik jl + il jk) :
Also at the xed point in d = 4
This gives,
N + 8
+ 3
(3N + 14) 2
(N + 8)3
cT =cT;free = 1 +
cT =cT;free = 1
2 +
N
Now the large N expansion of cT =cT;free can be found in [97, 98], and it is given by,
Using this we get,
5=36 and
7=36. So we obtain,
cT =cT free = 1
5 (N + 2)
12 (N + 8)2
2
36 (N + 8)4
(N + 2)(7N 2 + 382N + 1708) 3 + O( 4) ;
which exactly matches with our result (5.3).
Even though cT was obtained this way, it is not possible to do the same for cJ | in
terms of the OPE coe cients cJ =cJfree = CAfr;e1e=CA;1. This is because although cJ is known
up to the 1=N order, it has a more complicated structure in terms of the perturbative
parameter ijkl. So instead of (B.1) we have the form,
h i j J[kr]i = a0( ik jl
il jk) + a ikmn mnjr + b ikmn mnpq pqjr + c inkm rmpq pqjn :
Here J[kr] is the spin 1 antisymmetric current. The rst term in the r.h.s. above comes
from antisymmetrization. We get the OPE coe cient of J by contracting h i j J[kr]i with
itself. This gives the form,
cJ =cJ;free = 1 +
ijkl ijkl +
ijkl klmn mnij +
iikl kmnp lmnp :
(B.10)
Since there are three undetermined coe cients ; ; to x, we would not be able to do it
from the 2 and 3 terms of 1=N expansion alone.
C
Obtaining the large N corrections
The correction to the 4-point function h i j k li has been computed exactly at the 1=N
order [113]. It is can be written in a compact way as [51, 52],
h i j j li =
ij kl
il jk
ik jl
(x13x24)
(1)h (h)
fijkl
(h
fijkl = ij klD1;1; 1; 1(u; v) + il jkD
1;1;1; 1(u; v) + ik jlD
1;1; 1;1(u; v) : (C.2)
The D functions are de ned in [70]. We can rearrange the 4-point function into singlet,
symmetric traceless and antisymmetric parts. With the overall factor of (x212x324)
, the
(B.7)
HJEP07(21)9
(B.5)
(B.6)
(B.8)
(B.9)
g ;`(u; v) = u(
v :
(C.5)
Consider the corrections C ;` = C(0) + C`(1)
` N
and
(1)
(0) + N` . We have
singlet scalar and
0 = 2h
2 + ` for all other operators. Thus we get for general `,
(0) = 2 for the
C ;`g ;`(u; v) = C(0)g2h 2+`;`(u; v) +
u
h 1(1
` log u + C(1)
+ O(u; 1
So the coe cient of uh 1(1
coe cient of the nonlog term u
v)` log u gives the anomalous dimension ` directly. The
h 1(1 v)` is associated with the OPE coe cient correction.
However all the conformal blocks C ;`~g ;`~ with `~ < ` give a contribution to the nonlog
term, which is of the form,
The last line comes from the disconnected piece. The singlet sector coe cient is simply,
HJEP07(21)9
(h
1)
u
h 1D1;1; 1; 1 :
To read o the anomalous dimension and OPE coe cient corrections, we have to identify
the above with the conformal blocks. The conformal block in the small u limit reads,
X
`~ `
2) 2 h + `
1) (2h + `
1
2)
~
`
C(1) + C(0)
~ `
`
~ (2h
2) +
h + `
1)
:
1=N correction coe cients of the latter two sectors are given by (upper sign for symmetric
traceless and lower for antisymmetric),
(h 1)n(h 1)n+m(n+m)! ( logu+ (n+1)
(n+m+1)+2 (h+2n+m)
Xuh 1
(h 1+n)
(h 1+n+m))+ (h 1)n+m2n!
( logu+2 (h+2n+m) 2 (h 1+n+m))
m
"
(1)
2 uh 1 logu+X uh 1(1 v)m (1) ( 1)m
2
m
q 1
m q+1
The above is then compared to the uh 1(1
v)` log u and uh 1(1
v)` terms from (C.3)
and (C.4) to read o the anomalous dimensions and OPE coe cients which match exactly
with our results (4.17){(4.20).
D
Higher spin OPE in -expansion
The OPE coe cients of the higher spin operators in d = 4
can be written as
C`
`
Cfree = 1 + ci;`
(2) 3 + ci;`
the O( 3) orders for the rst few spins.
Here as usual i indicates the singlet, traceless symmetric and antisymmetric sectors. In
the above formula c(S2;`)
, c(T2;)` and c(A2;)` were given in (3.26), (3.27) and (3.28). Here we give
(C.3)
(C.4)
(C.7)
(D.1)
cS;`=4 = (2+N )(405848+N (89228+989N ))
(3)
4800(8+N )4
cS;`=6 = (2+N )(27035046944+N (5902407776+47767751N ))
(3)
298821600(8+N )4
cS;`=8 = (2+N )(1002110534752+N (217772423200+1327572517N ))
(3)
10302888960(8+N )4
cS;`=10 = (2+N )(121568105958318592+N (26332733153306704+123560820979315N ))
(3)
:
1175771471212800(8+N )4
D.2
Traceless symmetric
c(T3;)`=2 = 42096+N (22504+N (2878+13N ))
216(8+N )4
c(T3;)`=4 = 59659656+N (38917600+N (5670765+34133N ))
352800(8+N )4
c(T3;)`=6 = 2919785069952+N (2004861573920+N (299681375684+1572151439N ))
16136366400(8+N )4
c(T3;)`=8 = 3637661241149760+N (2543976284881184+N (382765405683350+1644002316149N ))
18699743462400(8+N )4
c(T3;)`=10 =(1490290339762224000(8+N )4) 1(308175148604337630720+N (217302237938493487024
+N (32745609281101869405+113160657172438904N ))):
(D.3)
D.3
Antisymmetric
c(A3;)`=3 =
c(A3;)`=5 =
c(A3;)`=7 =
c(A3;)`=9 =
(2+N )(229376+N (48672+295N ))
3456(8+N )4
(2+N )(77087104+N (16575112+96553N ))
972000(8+N )4
(2+N )(144671572069952+N (31180232920640+150688639187N ))
1616027212800(8+N )4
(2+N )(5447552685503360+N (1173953131219392+4579923671359N ))
55828779552000(8+N )4
: (D.4)
The singlet O( 3) OPE coe cients are found to obey the following general ` formula,
c(S3;)` = 8(8+N )4`3(1+`)3
h 2N 2 14+`2( 25+`(3+`(3+`))) +16N ( 28
+`( 27+`(32+7`(3+`(3+`)))))+32( 56+`( 63+`(58+17`(3+`(3+`)))))
`(1+`) 16(8+N )2`(1+`)H2 1+` +
224(8+9`)+272`2(3+`(2+`)) N 2 28+`2( 39
+`(2+`)))+8N ( 56+( 1+`)`(54+7`(3+`))))H2` +2H 1+` N 2 22+`2( 19+`(2+`))
16( 88+`( 63+17`(2+`(2+`)))) 8N ( 44+`( 27+`(20+7`(2+`)))) 8(8+N )2`(1+`)H2`
+2(8+N )2`(1+`) (8 3`(1+`))H`(2) +2 6+`+`2 H2(2`) i
The traceless symmetric O( 3) OPE coe cients are found to obey the following formula,
c(T3;)` = 8(8+N )4`2(1+`)3
4(8+N )2`(1+`)2( 2(6+N )+(2+N )`(1+`))H2(2`)
2(1+`) 2(6+N )(8+N )2 +8
4+N 2 ` (1088+N (640+3N (34+N )))`2 +2(2+N )
( 272+( 56+N )N )`3 +(2+N )( 272+( 56+N )N )`4 32(4+N )(8+N )`(1+`) H2` H`
2(2+N )( 272+( 56+N )N )`3 +(2+N )( 272+( 56+N )N )`4 H2` +2(8+N )`(1+`)2
4(32+N (10+N ))+3(2+N )(8+N )`+3(2+N )(8+N )`2 H(2) :
`
The antisymmetric part is given by,
c(A3;)` =
8(8+N )4`2(1+`)3 (2+N ) (1+`) 4(8+N )2 +16(26+7N )` 368+72N +7N 2 `2
+2
272 56N +N 2 `3 +
272 56N +N 2 `4 H` +32(8+N )`(1+`)2H`2 +2(1+`)H`
2(8+N )2 +40(2+N )` 320+64N +3N 2 `2 +2 272 56N +N 2 `3 +
272 56N +N 2 `4
16(8+N )`(1+`)H`) 2 (8+N )`(1+`)2
4(6+N )+3(8+N )`+3(8+N )`2 H(2)
`
2 24`2 64 16N N 2 168` 44N ` N 2`+4N `2+2N 2`2+(8+N )2`(1+`)2 `+`2 2 H(2)
`
:
Open Access.
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
1
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