Perturbative correlation functions of null Wilson loops and local operators

0.16 MB·English

by  Luis F. Alday

DCPT-12/31
Perturbative correlation functions
2 of null Wilson loops and local operators
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0
2

l
u Luis F. Aldaya1, Paul Heslopb2 and Jakub Sikorowskic3
J

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1 a Mathematical Institute, University of Oxford, Oxford OX1 3LB, U.K.

] b Department of Mathematical Sciences, Durham University, Durham DH1
h
t 3LE, U.K.
-
p
c Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
e
h Oxford OX1 3NP, U.K.
[

1
v
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1 Abstract
3
4 We consider the correlation function of a null Wilson loop with four
.
7 edgesandalocaloperatorinplanarMSYM.Byapplyingtheinsertion
0
procedure, developed for correlation functions of local operators, we
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1 give an integral representation for the result at one and two loops.
: We compute explicitly the one loop result and show that the two loop
v
i result is finite.
X
r
a
[e-mail]
[e-mail]
[e-mail]
1 Introduction
Correlation functions of gauge invariant local operators are the natural ob-
servables of any conformal field theory. Over the last few years, there has
been rapid progress in the understanding/computation of correlations func-
tions of = 4 SYM, see for instance [1–3], and now explicit results, that
N
would be impossible to obtain by standard Feynman diagram techniques, are
available.
Given an n point correlation function (x )... (x ) an interesting
1 n
− hO O i
limit to consider is the one where consecutive (after choosing a specific or-
dering) distances became null x2 0, at equal rate. It was argued in [4]
i,i+1 →
that in such a limit one obtains
(x )... (x )
lim hO 1 O n i = Wn [ ] (1)
x2i,i+1→0 hO(x1)...O(xn)itree h adj C i
where Wn [ ] is a Wilson loop in the adjoint representation, over the null
adj C
polygonal path , with cusps at x . This relation is quite general and does
i
C
not require the theory to be planar. If we focus on a planar theory, as we
will do in this paper, then Wn [ ] = Wn [ ] 2, the square of a Wilson
h adj C i h fund C i
loop in the standard fundamental representation.
One can also consider a generalization of the above limit, in which all
distances but one became null. It was argued in [5], see also [6,7], that in
this limit one obtains
(x )... (x ) (y) Wn [ ] (y)
lim hO 1 O n O i = h adj C O i . (2)
x2i,i+1→0 hO(x1)...O(xn)i hWandj[C]]i
On the right hand side we obtain the correlation function of a null Wilson
loop with a local operator. This is a very interesting class of objects, in par-
ticular, they interpolate between a Wilson loop and a correlation function,
and they are finite, since UV divergences in the numerator and denomina-
tor cancel out. The planar limit of a Wilson loop with operator insertions
was discussed in detail in [6], where it was shown that Wn [ ] (y)
h adj C O i →
2 Wn [ ] Wn [ ] (y) . Hence, in the planar limit
h fund C ih fund C O i
(x )... (x ) (y) Wn [ ] (y)
lim hO 1 O n O i = 2h fund C O i (3)
x2i,i+1→0 hO(x1)...O(xn)i hWfnund[C]]i
2
In this paper we will focus on the simplest case, where the polygonal null
Wilson loop has four edges, i.e. n = 4. In this case conformal symmetry
implies:
W4(x ,x ,x ,x ) (y) x x 2
1 2 3 4 13 24
h O i = | | F(ζ) , (4)
W4(x ,x ,x ,x ) 4 y x 2
h 1 2 3 4 i i=1| − i|
where ζ is the cross-ratio that can be constructed out of the location of the
Q
local operator y and the location of the cusps x :
i
y x 2 y x 2x2
ζ = | − 2| | − 4| 13 . (5)
y x 2 y x 2x2
| − 1| | − 3| 24
Hence F(ζ) is a function of a single variable ζ, in addition to the coupling
constant a = g2N. From the definition of the cross-ratio and cyclic symmetry
4π2
of the location of the cusps, we expect F(ζ) to have “crossing” symmetry:
F(ζ) = F(1/ζ) (6)
For the case of = , the operator that couples to the dilaton (i.e. the
dil
O O
= 4 action), this function was computed in [5] at leading order both in
N
the weak and strong coupling expansions1
a
F(ζ)= +..., a 1 (7)
−4π2 ≪
ζ √a
F(ζ)= ( 2(ζ 1)+(ζ +1)logζ) +...,a 1 (8)
(ζ 1)3 − − 2π2 ≫
−
we can see that both expressions satisfy the crossing symmetry (6). The aim
of the present paper is to compute F(ζ) to higher orders in perturbation
theory.
A related quantity, namely the four-point correlation function of the
stress-tensor multiplet, has been extensively studied in the past as well as
more recently and has now been explicitly computed at the integrand level to
6loops[1,2,8,9]. Thismultiplet, inparticular, contains thechiral Lagrangian
1In [5] the strong coupling result was found to be F(ζ) = c ζ ( 2(ζ 1)+(ζ +
3(ζ−1)3 − −
1)logζ)√λ, with λ = 4π2a. In the appendix we show that c = 3/(4π3) in order for (10)
to be satisfied. As forthe weakcoupling result,one is to setcˆ =1/2in [5], andfurther
dil
multiply by 1/4, since [5] used a non-standard convention for traces in the fundamental
representation.
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of = 4 SYM. Computations of the correlator have made extensive use of
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the method of Langrangian insertions. This method relies on the observa-
tion that derivatives with respect to the coupling constant of any correlation
function can be expressed in terms of a correlation function involving an
additional insertion of the = 4 SYM action. For instance,
N
∂
a (x )... (x ) = d4x (x )... (x ) (x ) . (9)
1 4 5 1 4 N=4 5
∂ahO O i hO O L i
Z
This method is very powerful: by successive differentiation with respect to
the coupling, it allows one to express the ℓ loop correction for the four-point
−
correlation in terms of the integrated tree-level correlation function with ℓ
additional insertions of the = 4 SYM Lagrangian.
N
From the discussion above it is clear that a particular limit of those inte-
grands will produce the integrands for (x )... (x ) (x ) in the par-
1 4 N=4 5
hO O L i
ticular null limit we are interested in. This will give integrand expressions
for loop corrections to W4 (x ) .
N=4 5
h L i
In the next section we start by writing down those integral expressions.
Then we compute the one-loop correction to F(ζ) (proportional to a2) and
show that the two-loop correction (proportional to a3) is finite. This is to be
expected, but it is far from obvious from the integral expressions, since each
integral diverges as 1 in dimensional-regularization.
ǫ4
Before proceeding, let us finish with a brief comment. The insertion
procedure in particular implies an integral constraint on F(ζ), namely
F(ζ)
x2 x2 d4y = a∂ log W4 (10)
13 24 4 y x a h i
Z i=1| − i|
One can check that this equation is indeed satisfied by the leading results at
Q
weak and at strong coupling, and we do so in the appendix.
2 Explicit results
2.1 General expressions and one-loop result
Following [1,11,12] we introduce
4
∞
(x )... (x ) =G = aℓG(ℓ)(1,2,3,4)
hO 1 O 4 i 4 4
ℓ=0
X
∞
(x )... (x ) (x ) =1/4 d4ρ G = 1/4 aℓ+1 d4ρ G(ℓ)(1,2,3,4,5)
hO 1 O 4 L 5 i 5 5;1 5 5;1
Z ℓ=0 Z
X
here ρ is a Grassmann variable, is the lowest component of the stress-
O
tensor multiplet and is the component proportional to ρ4. We define the
L
’tHooft coupling constant a = g2N/(4π2). The object we want to compute
is then simply given by
(x )... (x ) (x ) d4ρ G
1 4 5 5 5;1
hO O L i = (11)
(x )... (x ) 4G
hO 1 O 4 i R 4
(ℓ) (ℓ)
Expressions for G and G (in terms of certain functions to be defined bel-
4 5;1
low), can be found in [1]. In general, those depend on the insertion points,
together with certain auxiliary harmonic variables y . In the null limit con-
i
sidered in this paper, however, the dependence on the harmonic variables
factors out, and goes away when taking the ratio (11). In the null limit we
obtain
1 2x2 x2
G(ℓ)(1,2,3,4) = 13 24 G(0) d4x ...d4x f(ℓ)(x ,...,x ) (12)
4 ℓ!( 4π2)ℓ 4 5 4+ℓ 1 4+ℓ
− Z
8 x2 x2
d4ρ G(ℓ) = 13 24 G(0) d4x ...d4x f(ℓ+1)(x ,...,x )
5 5;1 ℓ!( 4π2)ℓ+1 4 6 5+ℓ 1 5+ℓ
Z − Z
(13)
which is consistent with the insertion formula
∂
a G = 1/4 d4x d4ρ G . (14)
4 5 5 5;1
∂a
Z Z
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Finallywealsoneedexpressions forthef functions. Thesehavearemarkably
simple form [1]. At 1,2,3 loops these are given by2
1
f(1)(x ,...,x ) = ,
1 5 x2 x2 x2 x2
15 25 35 45
1 x2 x2 x2
f(2)(x ,...,x ) = 48 σ∈S6 σ(1)σ(2) σ(3)σ(4) σ(5)σ(6) (15)
1 6 (x2 x2 x2 x2 )(x2 x2 x2 x2 )x2
P15 25 35 45 16 26 36 46 56
1 x4 x2 x2 x2 x2 x2
f(3)(x ,...,x ) = 20 σ∈S7 σ1σ2 σ3σ4 σ4σ5 σ5σ6 σ6σ7 σ7σ3 .
1 7 (x2 x2 x2 x2 )(x2 x2 x2 x2 )(x2 x2 x2 x2 )(x2 x2 x2 )
15 25 35P45 16 26 36 46 17 27 37 47 56 57 67
These functionssatisfycertainsymmetries. Uponmultiplicationbytheprod-
uct of all external kinematic invariants (x2 x2 x2 x2 x2 x2 ) and for generic
12 13 14 23 24 34
(non-null-separated) points, these functions are completely symmetric under
interchange of any two points and can be written as P(ℓ)(x1,...,x4+ℓ), where P(ℓ)
Q x2
1≤i<j≤4+ℓ ij
is a homogeneous polynomial in x2 of uniform weight (ℓ 1) at each point.
ij − −
These properties hold at all loops in perturbation theory [1]. When taking
the null limit the functions f(ℓ) will have fewer terms, but some symmetries
will be lost.
Let us now consider the ratio (11) order by order in perturbation theory
d4ρ G G(0) G(1) G(0)G(1)
5 5;1 = d4ρ a 5;1 + a2 5;1 5;1 4
R G4 Z 5( "G(40)# "G(40) − G(40)G(40)#
G(2) G(1) G(1) G(0)G(2) G(0) G(1) 2
+a3 5;1 5;1 4 5;1 4 + 5;1 4 +...
"G(0) − G(0) G(0) − G(0)G(0) G(0) G(0)! # )
4 4 4 4 4 4 4
(16)
Hence, at leading order in perturbation theory (proportional to a) we find
W4 (0) a d4ρ G(0) ax2 x2
h Li = 5 5;1 = 13 24 f(1)(x ,...,x )
(cid:18) hW4i (cid:19) 8R G(40) (−4π2) × 1 5
a x2 x2
= 13 24 , (17)
( 4π2)x2 x2 x2 x2
− 15 25 35 45
2Note that the functions f(ℓ) are multiplied by the overallfactor (x2 x2 x2 x2 x2 x2 )
12 13 14 23 24 34
compared to the definition in [1].
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which precisely agrees with the leading order result found in [5]. At next
order we find
W4 (1) a2
h Li = x2 x2 d4x f(2)(x ,...,x ,x )
W4 ( 4π2)2 × 13 24 × 6 1 5 6
"
(cid:18) h i (cid:19) − Z
2x2 x2 f(1)(x ,...,x ) d4x f(1)(x ,...,x ,x )
− 13 24 1 5 6 1 4 6
#
Z
(18)
In the light-like limit the numerator of f(2) becomes simply
1
x2 x2 x2
48 σ(1)σ(2) σ(3)σ(4) σ(5)σ(6)
σX∈S6
= x2 x2 x2 +x2 x2 x2 +x2 x2 x2 +x2 x2 x2 +x2 x2 x2 (19)
13 24 56 15 36 24 25 46 13 35 16 24 45 26 13
When integrating over x in (18) we recognize two kinds of contributions
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1 x2 x2
F(1,2,3,4) = d4x 13 24 (20)
−4π2 6x2 x2 x2 x2
Z 16 26 36 46
1 x2 x2
F(1,2,3,5) = d4x 13 25 (21)
−4π2 6x2 x2 x2 x2
Z 16 26 36 56
The first is the conformal massless box function, while the second is the two
mass hard box function (since x and x are not null).3 To be more precise,
51 53
we have
W4 (1) a2 x2 x2
h Li = 13 24 (22)
W4 ( 4π2) × x2 x2 x2 x2
(cid:18) h i (cid:19) − 15 25 35 45
F(1,2,3,5)+F(4,1,2,5)+F(3,4,1,5)+F(2,3,4,5) F(1,2,3,4) .
× −
(cid:16) (cid:17)
The first (f(2)) term in (18) contributes a similar expression with all coeffi-
cients +1 whereas the second term in (18) subtracts a term proportional to
2F(1,2,3,4) thus swapping the sign of the last term.
3Theseintegralsareofcourseinfrareddivergentandneedregularisation. Thecombina-
tion of these integrals we consider below will be finite howeverand so we do not specify a
regulator. Inpractisewewillusedimensionalregularisation(wherethex’sareinterpreted
as dual momenta).
7
The explicit expression for the box functions can be found for instance
in [14,15], where dimensional regularization is used. Even though each box
function is divergent, the above combination is finite. Furthermore this com-
bination is dual conformally invariant (see for example (2.23,2.22) of [13] for
the divergences and conformal variation of the box functions in dimensional
regularization). Plugging the analytic expressions for the box functions and
expanding up to finite terms we obtain
W4 (1) a2 x2 x2 1
h Li = 13 24 log2ζ +π2 (23)
W4 ( 4π2) × x2 x2 x2 x2 × −4
(cid:18) h i (cid:19) − 15 25 35 45 (cid:18) (cid:19)
(cid:0) (cid:1)
a2 1
F(ζ) = log2ζ +π2 (24)
( 4π2) −4
− (cid:18) (cid:19)
(cid:0) (cid:1)
This result has homogeneous degree of transcendentality and the correct
symmetry F(ζ) = F(1/ζ).
2.2 Two-loop result
At O(a3) we have
W4 (2)
h Li
W4
(cid:18) h i (cid:19)
1a3x2 x2
= 13 24 d4x d4x
2( 4π2)3 × 6 7×
− Z
f(3)(x ,...,x ,x ) 4x2 x2 f(2)(x ,...,x )f(1)(x ,...,x ,x )
× 1 6 7 − 13 24 1 6 1 4 7
"
2x2 x2 f(1)(x ,...,x )f(2)(x ,...,x ,x ,x )
− 13 24 1 5 1 4 6 7
+8(x2 x2 )2f(1)(x ,...,x )f(1)(x ,...,x ,x )f(1)(x ,...,x ,x ) (25)
13 24 1 5 1 4 6 1 4 7
#
The integrals which arise from this are a 2-mass pentabox, 2-mass (2 types)
and massless double boxes, and products of massless and 2 mass boxes. All
of these are illustrated in the figures.
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2 3 2
1 3 2 5 1 3
4 1 5
1x2 x4 1x4 x2 1x2 x4
4 13 24 2 13 25 2 13 25
I I I
1 2 3
Figure 1: All contributing double box integrals at 2 loops with the corre-
sponding numerator.
2
3
7
1
4
5
x2 x2 x2 x2
17 24 35 25
I
4
Figure 2: The 2 mass pentabox which contributes at 2 loops with the corre-
sponding numerator.
More specifically we have
f(3)(x ,...,x ,x )
d4x d4x 1 6 7 = I +I +I +I +I +I
6 7 f(1)(x ,...,x ) 1 2 3 4 6 7
1 5
Z 16Xperms(cid:16) (cid:17)
x2 x2 x2 x2 1
13 24 13 24 d4x d4x f(2)(x ,...,x )f(1)(x ,...,x ,x ) = I + I
f(1)(x ,...,x ) 6 7 1 6 1 4 7 5 2 6
1 5
Z 16Xperms(cid:16) (cid:17)
x2 x2 d4x d4x f(2)(x ,...,x ,x ,x ) = I +I
13 24 6 7 1 4 6 7 1 5
Z 16Xperms(cid:16) (cid:17)
(x2 x2 )2 d4x d4x f(1)(x ,...,x ,x )f(1)(x ,...,x ,x ) = I (26)
13 24 6 7 1 4 6 1 4 7 5
Z 16perms
X
where the sum over 16 permutations indicates that we must sum over 16
permutations generated by cycling the external points x ,x ,x ,x , parity
1 2 3 4
(x x , x x ) together with swapping the internal coordinates x ,x .
1 4 2 3 6 7
↔ ↔
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2 2
1 3 1 3
×
4 4
1 x4 x4
16 13 24
I
5
2 2
1 3 5 3
×
4 4
1x2 x4 x2
2 13 24 35
I
6
2 2
1 5 5 3
×
4 4
1x2 x4 x2
4 15 24 35
I
7
Figure 3: The 3 types of products of boxes which contribute at two loops
with the corresponding numerator.
These permutations will not always produce a different integrand (for ex-
ample I is completely symmetric under all such permutations). We have
5
divided by the corresponding symmetry factor in the definition of the inte-
gral (see figures).
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