Four-Point Functions with a Twist

0.73 MB·

by  Till Bargheer

DESY 16-248
Four-Point Functions with a Twist
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Till Bargheer

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a DESY Theory Group, DESY Hamburg,
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Notkestraße 85, D-22603 Hamburg, Germany
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[email protected]

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[ Abstract

1 We study the OPE of correlation functions of local operators in planar
v
N = 4 super Yang–Mills theory. The considered operators have an ex-
4
2 plicit spacetime dependence that is defined by twisting the translation
4
generators with certain R-symmetry generators. We restrict to opera-
4
0 tors that carry a small number of excitations above the twisted BMN
.
1 vacuum. The OPE limit of the four-point correlator is dominated by
0
internal states with few magnons on top of the vacuum. The twisting
7
1 directly couples all spacetime dependence of the correlator to these
: magnons. We analyze the OPE in detail, and single out the extremal
v
i states that have to cancel all double-trace contributions.
X
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Contents
1 Introduction 1
2 Twisted States 2
3 Two-Point and Three-Point Correlators 6
4 Four-Point Correlators 10
5 OPE Analysis 12
6 Conclusions 18
A Details on Tensors 21
1 Introduction
Solving planar N = 4 super Yang–Mills theory (sYM) at any value of its coupling constant con-
tinues to be an important goal in mathematical physics. This most symmetric four-dimensional
gauge theory serves as a key towards understanding general properties of interacting gauge
theory, AdS/CFT duality, and quantum gravity. In probing the theory at ever deeper levels, we
have witnessed extraordinary progress due to the emergence of integrability [1]. By now, the
spectrum of single-trace operators is virtually solved, for any value of the coupling constant [2].
In the past few years, also three-point functions of local operators, as well as scattering ampli-
tudes have come into focus. For both types of observables, essential proposals for finite-coupling
descriptions have been made [3,4], and highly non-trivial implications could be derived from
these proposals [5]. At this point, it appears not unlikely that all correlation functions will
eventually be computable at any value of the coupling, at least in the planar limit.
In this paper, we focus on four-point correlation functions of local gauge-invariant operators.
While two- and three-point functions in principle determine all higher-point functions via the
conformal operator product expansion, actually resumming the expansion to recover explicit
higher-point functions is very difficult in practice. Conversely, four-point (and higher-point)
functions contain a wealth of information on structure constants and scaling dimensions, and
are thus very interesting physical objects in their own right. It therefore remains desirable to
devise efficient methods for computing higher-point correlators directly. Here, our primary goal
will be to find the right language and variables for making use of the integrable structure for
general correlation functions.
A key point of the integrability-based solution to the spectral problem is the organization of
local operators in terms of excitations above a “vacuum” BPS operator tr[ZJ], where Z is a
complex scalar. The spectrum of anomalous scaling dimensions is then encoded in the eigenstates
and eigenvalues of the dilatation generator, which are governed by the Beisert scattering matrix
among the set of excitations (scalars, fermions, and covariant derivatives) above the tr[ZJ]
vacuum [6]. In particular, the scattering matrix is, up to its overall phase, completely constrained
by the psu(2|2)2 subalgebra of the superconformal symmetry that preserves the vacuum.
For higher-point functions, it is again desirable to consider operators that preserve as much
symmetry as possible. The familiar BPS operators tr[ZJ] suggest themselves. But when
1
considering more than two insertions of such operators, most of the symmetry is broken, because
the psu(2|2)2 symmetries of the individual insertions are not aligned. The situation can be
improved by considering slightly generalized BPS operators, which are related to tr[ZJ] by an
internal rotation. Namely, consider tr[φJ], where φ = X ·Φ and X ∈ C6, X2 = 0 specifies a
complex, lightlike direction in the internal space of scalar fields ΦI, I = 1,...,6. By judiciously
choosing the vector X as a function of the spacetime coordinate, a substantial part of the
superconformal symmetry can be preserved. This has been noted before [7,8]. Namely, after
choosing a vector X at the spacetime origin x = 0, define X(x) throughout spacetime via a
ˇ
twisted translation generator P ∼ P−R that is a combination of the conventional spacetime
translationPandaninternalrotationR. Thistwistedtranslationisassociatedtoawholetwisted
conformal symmetry algebra so(2,4), which is a diagonal combination of the original conformal
c
algebra so(2,4) and the internal so(6) R-symmetry. The twisted fields were first considered in
the context of topological twistings of N = 4 super Yang–Mills theory [7]. Later, correlation
functions of single-trace operators of a single twisted complex scalar φ were considered [8], and
found to (i) have trivial spacetime dependence, and (ii) be protected from quantum corrections.
These properties make such operators good candidates for an integrability-based description in
terms of excitations on top of vacuum operators tr[φ...φ]. In fact, the hexagon form factor
proposal for three-point functions [4] relies on an excitation picture around vacuum operators of
this type.
In this paper, we consider four-point functions of twisted single-trace operators with few
excitations at the one-loop level. For the time being, the analysis is restricted to zero-momentum
excitations. Whilethismeansthatthefouroperatorsarestillhalf-BPS,theirfour-pointfunctions
are not protected. We extract the one-loop correlation functions of such operators from [9],
and analyze their OPE decomposition. In the double-coincidence limit |x |,|x | (cid:28) |x |, the
12 34 23
one-loop correlator displays a logarithmic singularity, which is generated by the anomalous
dimensions of the internal states in the OPE. We show how this leading term is captured by
states with a minimal number of excitations on top of the twisted vacuum. For “extremal” OPE
contributions (which stem from internal states with the maximal R-charge compatible with the
external operators), it is known that there is a 1/N2 mixing with double-trace operators. We
c
find that the contribution of double-trace operators can be neglected by simply projecting out
specific extremal states in the OPE.
Note: While this work was being completed, I learned of the very interesting parallel paper [10],
which also discusses the OPE of four BPS operators from an integrability perspective. While
this work considers operators of any charge, [10] focuses on operators with large charges and
the relation to the hexagon form factor approach.
2 Twisted States
We start by reviewing essential parts of [7], [8], and [11]. Consider a general complex linear
combination X ·Φ = X ΦI of the six real scalar fields ΦI. The propagator for two such fields
I
equals (neglecting color factors)
D E X1 ·X2
X ·Φ(x )X ·Φ(x ) = . (2.1)
1 1 2 2 (2π)2x2
12
Correlation functions of single-trace operators are difficult to compute in general, and have
a complicated dependence on the spacetime coordinates as well as the coupling constant,
unless they obey extra relations due to supersymmetry. A prominent case are half-BPS chiral
2
primary operators tr[(X ·Φ)J] with X2 = 0. Every such operator preserves 24 supersymmetries.
Therefore any two such operators share 16 supersymmetries, and any three such operators
share 8 supersymmetries. This is the reason why two-point and three-point functions of such
half-BPS operators are protected from quantum corrections. The same is generically not true
for correlation functions of four or more such operators, which generically do not share any
supersymmetry. However, making the judicious, spacetime-dependent choice
√
(cid:16) (cid:17)
X = Xφ(xµ) ≡ 2 ixµ, 1(1+x2), i(1−x2) , (2.2)
2 2
φ(x) ≡ Xφ(x)·Φ(x) (2.3)
has two important consequences. Firstly, Xφ(x )·Xφ(x ) = x2 , therefore the free propagator
1 2 12
of two fields φ becomes constant,
1
hφ(x )φ(x )i = . (2.4)
1 2 (2π)2
Hence, in the free field theory, correlation functions of any number of operators
Q(x) ≡ tr[φ(x)J] (2.5)
are completely spacetime-independent and just evaluate to constants. Secondly, all correlation
functions
D E
Q(x )...Q(x ) (2.6)
1 n
are protected from quantum corrections, for any number of operators Q(x). The reason is that
all operators Q(x) share two universal supercharges Q±, and the perturbative action of N = 4
sYM is Q±-exact [7,8]. We will refer to the operators (2.5) as vacuum operators.
Thefieldφ(x)canbedefinedviaatranslationthatistwistedwithanappropriateR-symmetry
generator. In fact, the field (2.3) was first considered in the context of topological twists of
N = 4 sYM [7]. The twisted translation is associated with a whole twisted conformal symmetry
algebra so(2,4), which is a diagonal combination of the original conformal algebra so(2,4) and
c
the internal so(6) R-symmetry. Denoting the generators of the original conformal symmetry by
{P ,K ,L ,D}, the twisted conformal symmetry generators read1
µ µ µν
ˆ ˆ
P = P +R , L = L +R ,
µ µ µ+ µν µν µν
ˆ ˆ
K = K +R , D = D+R. (2.7)
µ µ µ−
Here, the R-symmetry algebra has been split into an so(4) with generators R and a remainder
µν √
with generators R ≡ R ±iR , and R ≡ −iR . Starting with φ(0) = (Φ5 +iΦ6)/ 2 ≡ Z
µ± µ6 µ5 56
at the origin, φ(x) throughout spacetime is obtained by
φ(x) ≡ exp(xµ(P −R ))·Z(0) = exp(−xµR )·Z(x). (2.8)
µ µ+ µ+
The splitting is chosen such that R annihilates φ(0) = Z at the origin, and R measures the
µ−
R-charge along the direction of Z,
R·Z = −Z, R·Z¯ = +Z¯, [R,R ] = ±R . (2.9)
µ± µ±
1A similar twisting was used in [12] to construct a chiral algebra of operators living in a two-dimensional
hyperplane of the four-dimensional Minkowski spacetime. The construction of [12] involves passing to the
cohomology of a suitable supercharge, upon which all operators transform trivially under a twisted conformal
algebra. On the contrary, here we study fully four-dimensional operators that transform non-trivially under the
twisted conformal symmetry.
3
Together with φ, the remainder of the six scalars ΦI naturally organizes into a multiplet
{φ,V ,B} of the twisted conformal symmetry, with
µ
V ≡ XVµ ·Φ, XVµ ≡ √1 (∂ Xφ) = (ie ,x ,−ix ), (2.10)
µ µ µ µ µ
2
B ≡ XB ·Φ, XB ≡ √1 (∂2Xφ) = (0,0,0,0,1,−i). (2.11)
4 2
Just as φ(x), the fields V (x) and B(x) throughout spacetime are obtained by the same twisted
µ
translation (2.8) applied to V (0) and B(0) at the origin. The latter reduce to
µ
√
¯
V (0) = iΦ , B(0) = Φ −iΦ = 2Z. (2.12)
µ µ 5 6
The twisted translation is defined such that, when acting on the twisted fields φ, V , and B, the
µ
generators of the twisted conformal algebra so(2,4) take the standard realization
c
ˆ
P = ∂ ,
µ µ
Lˆ = (x ∂ −x ∂ )+Σˆ ,
µν µ ν ν µ µν
Dˆ = xµ∂ +∆ˆ,
µ
Kˆ = 2x x·∂ −x2∂ −2x ∆ˆ+2xνΣ +κˆ , (2.13)
µ µ µ µ µν µ
where the derivatives only act on the fields ΦI, and not on the explicit coordinates xµ in the
ˆ ˆ
definition of the fields φ, V , and B. The generators ∆, Σ and κˆ specify the transformation
µ µν µ
properties of the fields at the origin,
ˆ ˆ
[∆,φ] = 0, [Σ ,φ] = 0, [κˆ ,φ] = 0,
µν µ
√
ˆ ˆ
[∆,V ] = V , [Σ ,V ] = η V −η V , [κˆ ,V ] = −η 2φ,
ρ ρ µν ρ ρµ ν ρν µ µ ρ µρ
ˆ ˆ
[∆,B] = 2B, [Σ ,B] = 0, [κˆ ,B] = −2V . (2.14)
µν µ ρ
That is φ is a scalar with dimension zero, V is a vector with dimension one, and B is a scalar
µ
with dimension two under the twisted conformal symmetry. More generally, by construction,
the twisted scaling dimension of any operator is the sum of the untwisted scaling dimenion and
ˆ
the R-charge in the Z direction. What is nonstandard is that the special conformal generator K
acts non-trivially at the origin, even though the fields V and B are not conformal descendants.2
ρ
ˆ
Due to the non-trivial action of κˆ, the representation of the little group (with generators Σ ,
µν
ˆ
∆, and κˆ ) is not further reducible, even though φ, V , and B have different scaling dimensions.
µ µ
This affects the conformal Ward identities for correlation functions, and hence correlators of
twisted fields generically do not have the standard form (that follows from the Ward identities
for κˆ = 0). In particular, operators with different twisted dimensions may have non-vanishing
µ
two-point functions.
The twisting is explained in detail in [7,8]. Here, we only note that under the twisted
conformal symmetry, the fermion fields organize into vectors ψ(1), ψ˜(1), two-form fields χ±,(2),
µ µ µν
and scalars η(2), η˜(2), where the superscript numbers denote the twisted scaling dimensions. The
gauge fields are R-symmetry singlets and thus not affected by the twisting. The twisted matter
fields {φ,V ,B,ψ ,ψ˜ ,χ± ,η,η˜} transform covariantly under fifteen combinations Q , S , Q ,
µ µ µ µν µ µ µν
and Q of the fermionic generators of psu(2,2|4). Commuting the latter with the singlets Q±
D
that preserve the vacuum field φ yields the twisted conformal algebra,
[Q±,Q ] = Pˆ , [Q±,Q ] = Lˆ ,
µ µ µν µν
[Q±,S ] = Kˆ , [Q±,Q ] = Dˆ . (2.15)
µ µ D
2In the classification of representations of the conformal algebra by Mack and Salam [13], the representa-
tion (2.14) belongs to class Ib (κ 6=0 but nilpotent).
µ
4
ˆ ˆ ˆ ˆ
The fermionic generators {Q ,S ,Q ,Q } and their bosonic counterparts {P ,K ,L ,D}
µ µ µν D µ µ µν
combine into the strange superalgebra q(4) [14].
Besides the vacuum operators Q, we will also consider operators with zero-momentum
excitations
J−2
QJ = tr[V φJ−1], QJ = Xtr[V φjV φJ−2−j], ... , (2.16)
µ µ (µν) (µ ν)
j=1
and two-magnon operators
1 "1 J √ #
OJ = √ Xcos(p(j +3/2))tr[V φjVµφJ−j]+ 2cos(p/2)tr[BφJ+1] ,
p J +3 2 µ
j=0
1 J+2
OJ,VV = √ Xcos(p(j +1/2))tr[V φjV φJ+2−j],
p,(µν) J +3 (µ ν)
j=0
1 1 J+2
OJ,∂∂ = √ Xcos(p(j +1/2))tr[φ φjφ,µφJ+2−j]+... ,
p J +34 ,µ
j=0
1 "1 J 1 #
OJ,∂∂ = √ Xcos(p(j +3/2))tr[φ φjφ φJ−j]+ cos(p/2)tr[φ φJ+1] ,
p,(µν) J +3 2 ,(µ ,ν) 2 ,(µν)
j=0
−i " 1 J √ #
OJ,V∂,1 = √ √ Xcos(p(j +3/2))tr[V φjφ φJ−j]+ 2cos(p/2)tr[V φJ+1]+... ,
p,µν J +3 2 µ ,ν µ,ν
j=0
−i 1 J+2
OJ,V∂,2 = √ √ Xcos(p(j +1/2))tr[V φjφ φJ+2−j]+... . (2.17)
p,µν J +3 2 µ ,ν
j=0
of charge J ≥ 0. Here, φ = Xφ ·D Φ and V = XVµ ·D Φ are the twisted combinations of
,µ µ µ,ν ν
covariant derivative fields D ΦI. These two-magnon states are simply twisted cousins of the
µ
two-magnon states listed in Appendix B of [11]. They all belong to a superconformal multiplet
whose primary is the operator OJ. The two magnons carry opposite momenta p and −p, with
p
2πn
p ∈ , 1 ≤ n ≤ bJ+2c. (2.18)
J +3 2
Since the twist amounts to a (position-dependent) R-symmetry rotation, and the one-loop
dilatation operator commutes with the R-symmetry, all eigenstates remain eigenstates, with
unmodified eigenvalues, and all states with different eigenvalues remain orthogonal.3 In other
words, the anomalous dimension is unaffected by the twisting. The two-magnon states therefore
have twisted dimensions
1
∆ˆ(p) = 2+λ∆ (p)+O(λ2), ∆ (p) = sin2(p/2), (2.19)
1 1 π2
with the familiar value ∆ of the anomalous dimension. The ellipses in (2.17) stand for terms
1
withfermionfieldsandnon-linearcorrectionswhichdonotcontributetothefreeOPEcoefficients.
Instead of the charge J, we will sometimes refer to the “length” L, which equals the number of
fields within a single-trace operator. For the two-magnon states above, the relation between
charge and length are:
OJ,OJ,∂∂ ,OJ,V∂,1 : L = J +2, OJ,VV ,OJ,∂∂,OJ,V∂,2 : L = J +4. (2.20)
p p,(µν) p,µν p,(µν) p p,µν
3For states with identical anomalous dimension, orthogonality may not be preserved by the twisting. In fact,
states with scalar zero-momentum excitations V or B, which we will consider below, are not orthogonal to the
µ
vacuum operators Q, even though they are orthogonal before twisting.
5
3 Two-Point and Three-Point Correlators
In the following, we will compute two-point and three-point correlators of vacuum operators,
operators with zero-momentum excitations, and two-magnon operators, at leading order in the
Yang–Mills coupling g . The three-point correlators yield the structure constants that will
YM
later be needed for the OPE analysis of four-point correlators. Here and in everything that
follows, we omit all gauge group factors as well as all factors of 1/(2π)2 from propagators (2.4).
Keeping this in mind, the classical contractions for the twisted fields are:
η
µν
hφ(x )φ(x )i = 1, hV (x )V (x )i = − , hB(x )B(x )i = 0,
1 2 µ 1 ν 2 x2 1 2
12
√ x √ 1
12,µ
hφ(x )V (x )i = − 2 , hφ(x )B(x )i = − 2 , hV (x )B(x )i = 0. (3.1)
1 µ 2 x2 1 2 x2 µ 1 2
12 12
The contractions for twisted fields with twisted versions of derivative fields similarly follow by
adding the twisting factor X ·X to the untwisted propagators, for instance
1 2
D E x12,µ
(X ·Φ(x ))(X ·D Φ(x )) = 2 (X ·X ). (3.2)
1 1 2 µ 2 x2 1 2
12
Two-Point Functions. For non-zero momenta, the two-magnon states (2.17) are annihilated
by the special conformal generator κˆ (2.14). They are thus proper conformal primaries of the
µ
twisted conformal algebra. Hence their two-point functions take the standard form as dictated
by conformal symmetry:4
DOJ OJ E = DOJ,∂∂OJ,∂∂E = δp,q
p,1 q,2 p,1 q,2 x4
12
DOJ,VV OJ,VV E = DOJ,∂∂ OJ,∂∂ E = δp,qI I
p,(µν),1 q,(ρσ),2 p,(µν),1 q,(ρσ),2 x4 12,ρ(µ 12,ν)σ
12
DOJ,V∂,1OJ,V∂,1E = DOJ,V∂,2OJ,V∂,2E = δp,qI I (3.3)
p,µν,1 q,ρσ,2 p,µν,1 q,ρσ,2 x4 12,µρ 12,νσ
12
Here,
2x x
α β
I ≡ I (x ), I (x) = η − (3.4)
ij,αβ αβ ij αβ αβ x2
is the inversion tensor, and parentheses denote traceless symmetrization with unit weight:
x ≡ 1x + 1x −η xρ. (3.5)
(µν) 2 µν 2 νµ µν ρ
All other two-point functions among two-magnon states with non-zero momenta vanish.
Two Vacua. By explicit computation, one finds that the three-point functions of two-magnon
states (with non-zero momentum) and two vacuum states of the same weight are:
D E D E x2
OJ QkQk = OJ,∂∂QkQk = CJk2 23 ,
p,1 2 3 p,1 2 3 p x2 x2
12 13
D E D E
OJ,VV QkQk = OJ,∂∂ QkQk = CJ2k2Y Y ,
p,(µν),1 2 3 p,(µν),1 2 3 p (µ,12,13 ν),12,13
D E D E
OJ,V∂,1QkQk = OJ,V∂,2QkQk = CJ2ik2Y Y , (3.6)
p,µν,1 2 3 p,µν,1 2 3 p µ,12,13 ν,12,13
4The two-point functions for the operators OJ,V∂,1 and OJ,V∂,2 get non-trivial contributions from fermion
termsnotdisplayedin(2.17). ForOJ,V∂,1, reducingtothebosonictermsgives(cid:0)1/2+(1+cos(p))/(J+3)(cid:1)times
p
the full two-point function. For OJ,V∂,2, reducing to the bosonic terms gives 1/2 of the full two-point function.
6
with
xα yα
Y (x,y) = − , Y = Y (x ,x ). (3.7)
α x2 y2 α,ij,kl α ij kl
In this case, all three operators are proper conformal primaries (they are annihilated by κˆ ), and
µ
hence these correlation functions are of the standard form as dictated by conformal symmetry.
Here, the coefficient depending on the charge and momentum is
1 −e−ip/2
CJ = √ . (3.8)
p J +3(1+ei(J+2)p/2)2
The two-magnon states (2.17) are all mutually orthogonal as well as orthogonal to the vacuum
states and all their descendants. Hence, the coefficients of the two-magnon states in the OPE of
two vacuum operators can simply be extracted from the three- and two-point functions,
DO (x )O (x )O (x )E −x−2−→−x→3 C1 (x )DO (x )O (x )E+... , (3.9)
1 1 2 2 3 3 23 23 1 1 1 2
where the ellipsis stands for higher-order terms in x that stem from descendants of O . The
23 1
resulting OPE coefficients for two vacuum operators read
CJ,p = CJ,p,∂∂ = CJk2x2 ,
k;k k;k p 12
CJ,p,VV,(µν) = CJ,p,∂∂,(µν) = CJ2k2x(µxν),
k;k k;k p 12 12
CJ,p,V∂,1,µν = CJ,p,V∂,2,µν = CJ2ik2xµ xν . (3.10)
k;k k;k p 12 12
One Excitation. Zero-momentum one-excitation operators transform non-trivially under κˆ ,
µ
hence they are not proper conformal primaries, and their correlation functions will not be of the
familiar form as dictated by conformal symmetry. By direct computation, one finds the following
three-point functions of two-magnon states (with non-zero momentum), a one-excitation BPS
state Qk and a vacuum state Qk:
ρ
D E √ x2 (cid:16) (cid:17)
OJ Qk Qk = CJ 2k 23 1Y −Y ,
p,1 ρ,2 3 p x2 x2 2 ρ,21,23 ρ,21,23
12 13
D E √ 2 !
OJ,VV Qk Qk = CJ 2k Y Y Y − I Y ,
p,(µν),1 ρ,2 3 p ρ,21,23 (µ,12,13 ν),12,13 x2 12,ρ(µ ν),12,13
12
D E √ x2
OJ,∂∂Qk Qk = CJ 2k 23 1Y ,
p,1 ρ,2 3 p x2 x2 2 ρ,21,23
√ 12 13
D E
OJ,∂∂ Qk Qk = CJ 2kY Y Y ,
p,(µν),1 ρ,2 3 p ρ,21,23 (µ,12,13 ν),12,13
D E √ 1 !
OJ,V∂,1Qk Qk = CJ 2ik Y Y Y − I Y ,
p,µν,1 ρ,2 3 p ρ,21,23 µ,12,13 ν,12,13 x2 12,ρµ ν,12,13
12
D E √ 1 !
OJ,V∂,2Qk Qk = CJ 2ik Y Y Y − I Y . (3.11)
p,µν,1 ρ,2 3 p ρ,21,23 µ,12,13 ν,12,13 x2 12,ρµ ν,12,13
12
Here,
(cid:18) xρ (cid:19)
Y = LY +2k ik . (3.12)
ρ,ij,ik ρ,ij,ik x2
ik
The resulting OPE coefficients are
√
CJ,p = CJ 2k(2k −L+2)1x ,
k,ρ;k p 2 12,ρ
√ (cid:18)(2k −L)x(µxν)x (cid:19)
CJ,p,VV,(µν) = CJ 2k 12 12 12,ρ +2x(µην) ,
k,ρ;k p x2 12 ρ
12
7
√
CJ,p,∂∂ = CJ 2k(2k −L)1x ,
k,ρ;k p 2 12,ρ
√ x(µxν)x
CJ,p,∂∂,(µν) = CJ 2k(2k −L) 12 12 12,ρ ,
k,ρ;k p x2
12
√ (cid:18)(2k −L)xµ x (cid:19)
CJ,p,V∂,1,µν = CJ 2ikxν 12 12,ρ +ηµ ,
k,ρ;k p 12 x2 ρ
12
√ (cid:18)(2k −L)xµ x (cid:19)
CJ,p,V∂,2,µν = CJ 2ikxν 12 12,ρ +ηµ . (3.13)
k,ρ;k p 12 x2 ρ
12
Two Excitations. Thethree-pointfunctionsoftwo-magnonstates(withnon-zeromomentum)
and two one-excitation BPS states read:
D E x2 " 2k −L !
OJ Qk Qk = CJ 23 1Y Y − I
p,1 ρ,2 σ,3 p x2 x2 2 ρ,21,23 σ,31,32 2x2 23,ρσ
12 13 23
I I µ #
12,ρµ 13 σ
−Y Y −Y Y − ,
ρ,21,23 σ,31,32 ρ,21,23 σ,31,32 x2
23
" !
D E 2k −L
OJ,VV Qk Qk = CJ Y Y − I Y Y
p,(µν),1 ρ,2 σ,3 p ρ,21,23 σ,31,32 x2 23,ρσ (µ,12,13 ν),12,13
23
#
2 2 2I I
12,ρ(µ 13,ν)σ
−Y I Y − I Y Y − ,
ρ,21,23x2 13,σ(µ ν),13,12 x2 12,ρ(µ ν),12,13 σ,31,32 x2 x2
13 12 12 13
D E x2 2k −L !
OJ,∂∂Qk Qk = CJ 23 1Y Y − I ,
p,1 ρ,2 σ,3 p x2 x2 2 ρ,21,23 σ,31,32 2x2 23,ρσ
12 13 23
!
D E 2k −L
OJ,∂∂ Qk Qk = CJ Y Y − I Y Y ,
p,(µν),1 ρ,2 σ,3 p ρ,21,23 σ,31,32 x2 23,ρσ (µ,12,13 ν),12,13
23
" !
D E 2k −L
OJ,V∂,1Qk Qk = CJi Y Y − I Y Y
p,µν,1 ρ,2 σ,3 p ρ,21,23 σ,31,32 x2 23,ρσ µ,12,13 ν,12,13
23
#
1 1
−Y I Y − I Y Y ,
ρ,21,23x2 13,σµ ν,13,12 x2 12,ρµ ν,12,13 σ,31,32
13 12
" !
D E 2k −L
OJ,V∂,2Qk Qk = CJi Y Y − I Y Y
p,µν,1 ρ,2 σ,3 p ρ,21,23 σ,31,32 x2 23,ρσ µ,12,13 ν,12,13
23
#
1 1
−Y I Y − I Y Y . (3.14)
ρ,21,23x2 13,σµ ν,13,12 x2 12,ρµ ν,12,13 σ,31,32
13 12
The resulting OPE coefficients are
(2k −L+2)(cid:18) 2x x (cid:19)
CJ,p = −CJ I + 12,ρ 12,σ ,
k,ρ;k,σ p 2 ρσ,12 x2
12
x(µ (cid:16) (cid:17) ! !
CJ,p,VV,(µν) = −CJ (2k −L) 12 2 x ην) +x ην) +xν)I +2η(µην) ,
k,ρ;k,σ p x2 12,σ ρ 12,ρ σ 12 ρσ,12 ρ σ
12
(2k −L)
CJ,p,∂∂ = −CJ I ,
k,ρ;k,σ p 2 ρσ,12
x(µxν)
CJ,p,∂∂,(µν) = −CJ(2k −L) 12 12I ,
k,ρ;k,σ p x2 ρσ,12
12
8
1
0
p
π/2 π
Figure 1: The two branches of the OPE coefficient function CJ, here displayed for
p
J = 42. Onthelowerbranch,pisamultipleof4π/(J+3),andthereforeCJ ∼ (J+3)−1/2.
p
On the upper branch, CJ becomes large for small p, due to its small denominator. For
p
the smallest two values p = 2π{1,3}/(J +3) on the upper branch, the coefficient is off
the chart at CJ ≈ {30.6, 3.41}.
p
xν (cid:18)(cid:16) (cid:17) (cid:19)
CJ,p,V∂,1,µν = −CJi(2k −L) 12 x ηµ +x ηµ +xµ I ,
k,ρ;k,σ p x2 12,σ ρ 12,ρ σ 12 ρσ,12
12
xν (cid:18)(cid:16) (cid:17) (cid:19)
CJ,p,V∂,2,µν = −CJi(2k −L) 12 x ηµ +x ηµ +xµ I , (3.15)
k,ρ;k,σ p x2 12,σ ρ 12,ρ σ 12 ρσ,12
12
where
(2k −L)xρxσ
Iρσ(x) ≡ Iρσ(x)+ . (3.16)
x2
Even though this will not be needed in the subsequent analysis, we note that for two zero-
momentum excitations on one of the two external half-BPS states, the OPE coefficient of the
two-magnon operator OJ reads
p
x x
CJ,p = CJk(2k −L+2)(2k −L) 12,(ρ 12,σ) . (3.17)
k,(ρσ);k p x2
12
The Coefficient Function. Using the on-shell condition (2.18), the OPE coefficent func-
tion (3.8) can be rewritten as
−1
CJ = √ . (3.18)
p (cid:16) (cid:17)
2 J +3 cos(p/2)+eip(J+3)/2
It has a few interesting properties. First of all, because p is a multiple of 2π/(J + 3), the
term eip(J+3)/2 alternates between −1 and +1 when p is varied. The value +1 is assumed
when p is a multiple of 4π/(J +3). For fixed J, the coefficient CJ therefore has two branches
p
(see Figure 1). Since p takes values between 0 and π, the coefficient is small on the “+” branch,
√ √
CJ ≈ −1/4 J +3. On the “−” branch, CJ becomes large when p is small, CJ ≈ 4/ J +3p2.
p,+ p p,−
In fact, for large values of J, the smallest momentum p = 2π/(J +3) is strongly dominating
(see Figure 2). The reason for the alternating behavior of CJ is that the coefficient of the “local”
p
two-excitation operator
tr[V ,φ,...,φ,V ,φ,...,φ] (3.19)
µ ν
in the OPE of two vacuum (or zero-momentum) operators has a part that is independent of
the distance between the two excitations, and a part that is proportional to that distance. In
9