Published for SISSA by Springer
Received: October 30, 2012 Accepted: February 15, 2013
Published: March 13, 2013
S. Caron-Huota,c and Yu-tin Huanga,b,d
aSchool of Natural Sciences, Institute for Advanced Study,
Princeton, NJ 08540, U.S.A.
bDepartment of Physics and Astronomy, UCLA,
Los Angeles, CA 90095-1547, U.S.A.
cNiels Bohr International Academy and Discovery Center, The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
dMichigan Center for Theoretical Physics, Randall Laboratory of Physics, University of Michigan, Ann Arbor, MI 48109, U.S.A.
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: In this paper we present the rst analytic computation of the six-point two-loop amplitude of ABJM theory. We show that the two-loop amplitude consists of corrections proportional to two distinct local Yangian invariants which can be identied as the tree-and the one-loop amplitude respectively. The two-loop correction proportional to the tree-amplitude is identical to the one-loop BDS result of N = 4 SYM plus an additional
remainder function, while the correction proportional to the one-loop amplitude is nite. Both the remainder and the nite correction are dual conformal invariant, which implies that the two-loop dual conformal anomaly equation for ABJM is again identical to that of one-loop N = 4 super Yang-Mills, as was rst observed at four-point. We discuss the
theory on the Higgs branch, showing that its amplitudes are infrared nite, but equal, in the small mass limit, to those obtained in dimensional regularization.
Keywords: Supersymmetric gauge theory, Scattering Amplitudes, Chern-Simons Theories, Field Theories in Lower Dimensions
c
The two-loop six-point amplitude in ABJM theory
JHEP03(2013)075
[circlecopyrt] SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP03(2013)075
Web End =10.1007/JHEP03(2013)075
Contents
1 Introduction 2
2 Conventions 5
3 One-loop integrand and amplitude 63.1 Leading singularity and the one-loop integrand 73.2 The one-loop amplitude 113.3 Analytic properties of the one-loop amplitude 11
4 The two-loop six-point integrand 144.1 Integrand basis 154.2 Constraints from vanishing three point sub amplitudes 174.3 Constraints from one-loop leading singularity 184.4 Collinear divergences and the ABJM two-loop integrand 21
5 Interlude: infrared regularization using the Higgs mechanism 22
6 Computation of two-loop integrals 256.1 A Feynman parametrization trick 266.2 Divergent double-triangles 276.3 A dual conformal integral: Icritter 28
6.4 Another divergent integral: I2mheven 29
6.5 The integral Icrab 31
6.6 Parity odd box-triangles 31
7 The six-point two-loop amplitude amplitude of ABJM 32
8 Conclusions 34
A Identities 36
B ABJM theory on the Higgs branch 37
C Integrals using the mass regularization 38
D Integrals using dimensional regularization 39
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JHEP03(2013)075
1 Introduction
Amidst the shadow of tremendous progress in N = 4 super Yang-Mills (SYM4) amplitudes,
three-dimensional Chern-Simons matter (CSM) theory has recently enjoyed a quiet surge of interest. This reects an interesting dual aspect of the latter: on the one hand it is a close cousin to SYM theory in four-dimensions and thus provides a fruitful arena to apply the methods that was developed there-in. On the other, while scattering amplitudes of SYM4 theory, both perturbative and non-perturbative, are closely related to string theory scattering amplitudes, such relations for CSM theory are either obscure or in some cases simply absent as the proper correspondence is with M-theory instead. The latter is intriguing in that it implies that certain novel properties that is shared between the scattering amplitudes of both Yang-Mills and CSM may in fact have a deeper purely eld theoretical origin.
A prominent example is the N = 6 theory constructed by Aharony, Bergman, Ja eris
and Maldacena (ABJM) [1]. Being dual to type IIA string theory in AdS4 [notdef] CP3 back
ground, it is very similar to SYM4 in terms of providing an exact AdS/CFT pair. This similarity inspired the discovery of many common features between the two theories such as the presence of a hidden Yangian symmetry [2] (or equivalently dual superconformal symmetry [36]) of the tree- and planar loop-amplitudes [7],1 as well as the realization that the leading singularities of both theories are encoded by the residues of a contour integral over Grassmaniann manifolds [16, 17].
In many aspects, ABJM amplitudes are simpler than its four-dimensional relative. This simplicity is already reected in the fact that only even legged amplitudes are nontrivial [18]. Furthermore, all one-loop amplitudes consist solely of rational functions [19 22] (multiplied by , in a natural normalization) while the two-loop amplitudes are of transcendentality-two-functions [23, 24]. This should be compared to transcendentality -two- and four-functions for one and two-loop amplitudes respectively in SYM4. As all
one-loop amplitudes can be conveniently expressed in terms of a basis of massive triangle integrals, whose coe cients can be directly computed via recursion relations [25], the one-loop amplitude for ABJM theory with arbitrary multiplicity is e ectively solved.
On the other hand some properties of CSM theory, while shared with YMs theory, demand an alternative explanation other than the stringy origin currently available for the latter. Consider the color-kinematic duality [26], which leads to non-trivial amplitude relations for YMs and relates the amplitudes of the gauge theory to that of the corresponding gravity theory to all order in perturbation theory [27, 28]. For CSM, it was shown that similar duality, although based on three-algebra [2931], is also present for the N = 8 [32] and
N < 8 [33] theory. While the relations implied by the duality in YMs can be traced back
to monodromy relations of string amplitudes [34], such correspondence does not exist for CSM theory since the amplitudes are not directly related to any open string amplitudes in a at back-ground. As the color-kinematic identity allows one to obtain the amplitudes of
1At this stage it is unclear what role, if any, AdS/CFT plays in the existence of these symmetries, as explicit attempts at proving self-T-duality [8, 9] on the string or supergravity side have encounter technical di culties and have not been fully carried out [1013], except in the pp-wave limit [14]. For a recent review on these issues see [15].
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gravity-matter theory from that of CSM theory,2 the fact that gravity amplitudes can be extracted from close string amplitudes, render the role of string theory even more mysterious.
In this paper our main focus is the loop amplitudes of ABJM theory, in particular, the six-point one- and two-loop amplitude, in the planar (t Hooft) limit. It was shown in ref [23, 24] that the two-loop four-point amplitude has the same functional dependence as that of the one-loop four-point SYM4 amplitude. This equivalence was latter shown to persist to all orders in [epsilon1] expansion [35]. As the four-point amplitude can be uniquely determined by the dual conformal anomaly equation [3638], this results states that the anomaly equation for both ABJM and SYM4, up to four-points, are identical. However, taking into account the fact that the theory is conformal, the simplicity of four-point kinematics and the transcendental requirement of the nite function, this result might be deemed accidental (although not for the all order correspondence). At six-point, it is nontrivial that the anomaly equations should match. Furthermore, the anomaly equations xes the result only up to homogenous terms and six-point is the rst place where nontrivial invariant remainder functions might appear. Thus the six-point computation is an important piece of data to clarify these issues.
As ABJM theory consists of matter elds transforming under the bi-fundamental representation of the gauge group SU(N)k[notdef]SU(N)k, the amplitude has a denite parity
under the exchange of Chern-Simons level k $ k. More precisely and L-loop amplitude
is weighted by a factor of (4/k)L+1, and hence (odd-)even-loop amplitudes are parity (even)odd. Assuming that parity is non-anomalous, in order for odd-loops to give an acceptable contribution it must compensate for its opposite parity. Since the exchanging of k $ k can be translated into the exchange of the gauge group, this implies that a
non-vanishing odd-loop all scalar-amplitude must pickup a minus when cyclicly shifted by
one-site. This is indeed the case.
We construct the six-point integrand using leading singularity methods. Since it was shown in ref. [7] that there is only one pair of leading singularity at six-points, it is straightforward to construct the integrand by choosing an integral basis consisting of integrands with uniform leading singularities. At one-loop there are two types, the one-loop box and massive triangle integrals with loop momentum dependent and independent numerators respectively. There are two distinct combinations of the leading singularity pair, the di erence and the sum. The former is simply the tree amplitude while the latter is the conjugate tree-amplitude, with even and odd sites now belonging to the conjugate multiplet, denoted as Atree6,shifted since the identication of multiplets are shifted by one site. We nd that those
two objects do appear in integrand.
In ref [2022], the one-loop amplitude was given solely in terms of triangle integrals, proportional to Atree6,shifted. This is valid up to order O([epsilon1]) as the box-integrals integrate to zero at O(1). Having a result at one-loop that is valid to all orders in [epsilon1] will be extremely
important for the construction of the two-loop amplitude.3 The integrated result is pro-2Both pure Chern-Simons and gravity in three-dimensions are topological.
3This was already seen for the four-point amplitude [23] where the one-loop result vanishes up to O([epsilon1]),
yet it has a nontrivial box integrand. This integrand later becomes the seed of the two-loop integrand. The relevance of the O([epsilon1]) pieces can also be understand from unitarity cuts, where such terms might combine
with collinear singularity of the tree amplitudes to give non-trivial two loop contribution.
3
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portional to a step function, which as we will see nicely captures the non-trivial topology of 3d massless kinematics. More precisely, massless kinematics in three-dimensions can be parameterized by points on S1. For color ordered amplitudes, distinct kinematic congu-rations can be categorized by a winding number which can be unambiguously dened. The sign function then simply distinguishes the congurations with even or odd winding number, for a given kinematic channel.
With the one-loop integrand in hand, we construct the two-loop amplitude by simply requiring that on the maximal cut of one of the sub-loops, one obtains the full one-loop integrand. This xes the integrand up to possible double triangles, which are further xed by soft-collinear constraints. We compute the integrals using both dimensional reduction regularization as well as mass regularization. This mass regulator can actually be given a physical interpretation in terms of moving to the Coulomb branch of the theory and giving the scalars a vev, similar to that used for SYM4 [39]. Interestingly, while the result for the individual integrals di er between the two schemes, they give, up to additive constant, identical results when combined into the nal physical amplitude.
Using ve-dimensional embedding formalism, the tree amplitude is multiplied by ve-dimensional parity even integrals, while the conjugate tree-amplitude is multiplied by parity odd integrals. Introducing the cross-ratios (only two of the these are algebraically independent)
u1 = (1 [notdef] 3)(4 [notdef] 6)
(1 [notdef] 4)(3 [notdef] 6)
the two-loop amplitude is given as:
A2-loop6 = [parenleftbigg]
3
Xi=1
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, u2 = (2 [notdef] 4)(5 [notdef] 1)
(2 [notdef] 5)(4 [notdef] 1)
, u3 = (3 [notdef] 5)(6 [notdef] 2)
(3 [notdef] 6)(5 [notdef] 2)
, (1.1)
BDS6 + R6[bracketrightbigg]+ Atree6,shifted4i
[bracketleftbigg]log u2u3 log [vector]1 + cyclic [notdef] 2[bracketrightbigg]
, (1.2)
where BDS6 is the one-loop MHV amplitude for N = 4 SYM [40, 41], with proper rescaling
of the regulator to account for the fact this is at two-loops, and the remainder function R6 is given as
R6 = 22 +
N k
2 Atree6 2
Li2(1 ui) +12 log ui log ui+1 + (arccos pui)2[bracketrightbigg]
.
The [vector]i are little-group-odd cross-ratios dened in (7.4); we warn the reader that these variables may require some care when analytically continuing to Minkowski kinematics. An alternative form of the amplitude with explicit dependence on conventional invariants is given in eq. (7.3).
The presence of the BDS result demonstrates that infrared divergence and the dual conformal anomaly equation of the two-loop ABJM theory is identical are that of one-loop SYM4. Furthermore, similar to SYM4, using the mass regulator we show how the anomaly equation can be converted into a statement of exact dual conformal symmetry in higher dimensions, with the mass playing the role of the extra dimension.
This paper is organized as follows: in section (2) we lay out some basic conventions, while in section (3) we begin with the discussion of general one-loop dual conformal integrand and its integration in the embedding formalism. We then explicitly construct the
4
one-loop six-point integrand and well as the integrated result. We end with a more detailed discussion of the properties of the one-loop amplitude in terms of the topological properties of three-dimensional kinematics. In section (4), we employ leading singularity methods and soft-collinear constraints to x the two-loop integrand. In section (5) we briey discuss two regularization schemes, dimensional reduction regularization and higgs mass regulation, with special emphasis on the latter. In section (6) we will use mass regularization to explicitly compute the integrals. In section (7) we combine the integrated expressions and give the complete six-point two-loop amplitude. We give a brief conclusion and discussion for future directions in section (8).
2 Conventions
Since we will be interested in planar amplitudes, it is useful to dene the dual coordinates
xi+1 xi = pi. (2.1)
Special interest in the xi coordinates resides in the fact that planar amplitudes in ABJM theories are invariant under the so-called dual conformal transformations, which act as conformal transformations of the xi. To make the action of this symmetry simplest, and at the same time trivialize several operations which occur when doing loop computations, we will systematically use the so-called embedding formalism [4245] (for more recent discussion, see [46]).
The idea is to uplift three-dimensional xis to (projectively identied) null ve-vectors
yi := (xi, 1, x2i) (2.2)
such that inverse propagators become the (2,3)-signature inner product
(i [notdef] j) := yi [notdef] yj := (xi xj)2. (2.3)
The group of conformal transformations SO(2,3) of three-dimensional Minkowski spacetime is then realized linearly as the transformations of the yi which preserve this inner product.
It was shown in ref. [7] that the tree-level amplitude and loop-level integrand in ABJM inverts homogeneously under dual conformal inversion:
I [An] =
n
Yi=1
JHEP03(2013)075
qx2iAn . (2.4)
Due to the fact that at weak coupling the theory only has N = 6 supersymmetry, the
on-shell states are organized into two di erent multiplets:
( ) = 4 + I I + 12[epsilon1]IJK I JK +
1 3![epsilon1]IJK I J K 4,
( ) = 4 + I
I + 12[epsilon1]IJK I J
K + 1
3![epsilon1]IJK I J K
4, (2.5)
5
where I are Grassmann variables in the fundamental of U(3)2SU(4). The kinematic
information are encoded in terms of SL(2,R) spinors , with
sij = [angbracketleft]ij[angbracketright]2, [angbracketleft]ij[angbracketright] := i j[epsilon1] (2.6)
where sij = x2i,i+2 when j = i + 1. Note that x2ij is positive when the corresponding momentum is spacelike, while [angbracketleft]ij[angbracketright]2 is negative in that case. For more detailed discussion
of the on-shell variables ( i, Ii) see ref. [2]. In this paper, we will use the convention where the barred multiplet sits on the odd sites. The four-point amplitude is given as [2]:
A4(1234) = 4 k
3(P )
Q3I=1 2(QI)
h12[angbracketright][angbracketleft]23[angbracketright]
with 2(QI) :=
X1
i<j 4
Ii[angbracketleft]ij[angbracketright] Ij. (2.7)
3 One-loop integrand and amplitude
Dual conformal symmetry restricts the integral basis to be constructed of SO(2,3) invariant projective integrals. At one-loop, this restricts the n-point amplitude to be expanded on the basis of scalar triangles with appropriate numerators, as well as scalar box integrals with numerator constructed from the ve-dimensional Levi-Cevita tensor:
Ibox(i, j, k, l) =
Za
[epsilon1](a, i, j, k, l)(a [notdef] i)(a [notdef] j)(a [notdef] k)(a [notdef] l)
. (3.1)
The integral in eq. (3.1) is analogous to the four-dimensional pentagon integral described in ref. [47] and it integrates to zero up to order [epsilon1] in dimension regularization [23]. To demonstrate how dimensional regularization is employed in the embedding formalism, we explicitly demonstrate this result in the following.
We rst note that eq. (3.1) can be rewritten using Feynman parametrization as
Ibox(i, j, k, l) =
dF [epsilon1](i, j, k, l, @Y )
JHEP03(2013)075
Za [3] (a [notdef] Y )3
(3.2)
where dF :=
Q4i=1 d i (1
Pi i) and Y := 1yi + 2yj + 3yk + 4yl. We now focus on the inner integral, which for the purpose of dimensional regularization, we dene in
D-dimensions:
I0 = [3]
Za1(a [notdef] Y )3:= [3]
[integraldisplay]
dD+2a (a2)
i(2)DVol(GL(1))1(a [notdef] Y )3(a [notdef] I)D3. (3.3)
Let us illuminate this denition of
Ra by comparing it with (2.2). First, the GL(1) symmetry can be gauge-xed by setting the next-to-last component of a to 1, at the price of a unit
Jacobian. Then the (a2) factor forces the last component of a to equal x2, thus reducing
Ra to the usual loop integration[integraltext]
dDx
i(2)D . Finally, the factor of i is removed by the Wick rotation from Minkowski to Euclidean space.
The key feature away from D = 3 is the factor (a[notdef]I) where yI := (
[vector]0D, 0, 1) is the innity
point. This signals the breaking of dual conformal symmetry, and is required to maintain
6
the projective nature of the integrand (the GL(1) invariance) for arbitrary D. This feature remains clearly visible when switching to the easily-obtained integrated expression:
I0 =
Plugging this into eq. (3.2), we nd that the box integral gives:
Ibox(i, j, k, l) =
[integraldisplay]
dF (4)
3 D2 (4 )
D 2
1(I [notdef] Y )D3(12Y 2)3
D2. (3.4)
3 D2[bracketrightbig]
[epsilon1](i, j, k, l, I)(I [notdef] Y )D2(12Y 2)3
D2. (3.5)
The rst term vanishes due to the fact that Y is a linear combination of the four external coordinates, while the second term is at least O([epsilon1]) with D = 3 2[epsilon1].
As the one loop box integral vanishes, dual conformal symmetry implies that the amplitude, up to O([epsilon1]), can be solely expressed in terms of scalar triangles. However as
discussed in the introduction, for the purpose of constructing the two-loop integrand it will be extremely useful (and actually essential) to have a one-loop integrand valid beyond O([epsilon1]).
In the following, we will derive the full one-loop six-point integrand that includes both the scalar triangle and the tensor box integrals. We note that the form of the amplitude in terms of scalar triangles were given in [20, 21].
3.1 Leading singularity and the one-loop integrand
At six-point there are three possible box integrals, the one mass box, two-mass-easy and two-mass-hard box integrals.4 Using the ve term Schouten identity of the ve-dimensional Levi-Cevita tensor one nds the following linear identity for the box integrals:
Ibox(1, 3, 4, 6) = Ibox(3, 4, 5, 6) + Ibox(4, 5, 6, 1) + Ibox(1, 3, 5, 6) + Ibox(1, 3, 4, 5) . (3.6)
Thus the two-mass-easy integral can be expressed in terms of linear combinations of the two-mass-hard and one mass integrals. We will use the later two as the basis for box integrals. The relative coe cient of the box integrals can be easily xed by requiring that the two particle cuts which factorize the amplitude into two ve-point tree amplitudes, must vanish. Cutting in the x214-channel, shown in gure 1, this requires four box integrals to come in the following combination:
Ibox(3, 4, 5, 1) + Ibox(1, 2, 3, 4) Ibox(4, 5, 6, 1) Ibox(6, 1, 2, 4) . (3.7)
Using the Schouten identity, one can show that this combination is actually invariant under cyclic permutation by one site up to overall sign. This extra sign will be important as we will discuss shortly.
The other allowed scalar integrals are the massive triangles. Their coe cients along with that of the boxes can be xed by the two triple-cuts C1,2 (and their conjugate C 1,2),
4Here we borrow the nomenclature of four-dimensional box integrals to denote the propagator structure.
7
4 D2[bracketrightbig]
[epsilon1](i, j, k, l, Y )(I [notdef] Y )D3(12Y 2)4
D2+ (D 3)
D
2
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x2
x1 x4
x1
x5
x3
x6
xa xa xa xa x4
x5
x4
x1
x2
x4
x1
0
x3
x6
Figure 1. The particular combination of tensor box integrals in eq. (3.7) combines to give vanishing two particle cut x2a1 = x2a4 = 0. This cut must vanish as the amplitude factorizes a ve-point tree amplitude, which vanishes.
where the subscripts correspond to the the two distinct maximal cut, indicated as channel(1) (2) in gure 2. Explicitly they are given by:
C1 =
[integraldisplay]
3
YI=1d Il1d Il2d Il3A4(1, 2, l2, l1)A4(3, 4, l3, l2)A4(5, 6, l1, l3)
C2 =
[integraldisplay]
3
YI=1d Il1d Il2d Il3A4(l1, 2, 3, l2)A4(l2, 4, 5, l3)A4(l3, 6, 1, l1)
We note that there is always an ambiguity in distinguishing C1 versus C 1, since they arise
from the two solutions of a quadratic equation. However, two convention-independent combinations always exist. One is the average of the two cuts C1 + C 1 and the other is
the average of the leading singularities, LS1 + LS 1 = (C1 C 1)/[4 det(l1, l2, l3)(C1)], e.g.,
the numerators weighted by the Jacobian. Independence of the second combination follows from sign ip of the Jacobian on the two solutions, det(l1, l2, l3)(C1) = det(l1, l2, l3)(C 1).
The leading singularities have the following analytic form [7]:
LS1 = 3(P ) 6(Q)
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Q3I=1( +I)2c+25c+41c+63, LS 1 = LS1(+ ! ) . (3.8)
The functions c[notdef]ij and [notdef]I are dened as
c[notdef]ij := [angbracketleft]
i[notdef]p135[notdef]j[angbracketright] i[angbracketleft]i + 2, i 2[angbracketright][angbracketleft]j 2, j + 2[angbracketright] p2135
, [notdef]I := ([epsilon1]ij
k[angbracketleft]
i, j[angbracketright]
I
k [notdef] i[epsilon1]lmn[angbracketleft]l, m[angbracketright] In) p2135
,
where in the denition of [notdef]I, the (un-barred)barred indices indicate (odd)even labels. One can conveniently x the convention of C1 and C 1 as:
C1 := 2[angbracketleft]12[angbracketright][angbracketleft]34[angbracketright][angbracketleft]56[angbracketright]LS1, C 1 = C1(+ ! ) . (3.9) As one can check, the two combinations LS1 + LS 1 and C1 + C 1 both have the correct little
group weights for an amplitude.
The leading singularities of ABJM have a dual presentation as the residues of an integral over orthogonal Grassmanian [17]. As discussed in ref. [7] at n = 2k-point there are (k2)(k3)/2 number of integration variables in the orthogonal Grassmanian. This implies
that at six-point, there are no integrals to be done and one has a unique leading singularity from the Grassmanian (plus its complex conjugate due to the orthogonal condition). This
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p1 p2
p2 p3
x2 x4
x6
x1 x3
x5
l1
l2
l1
l2
p6
p3
p4
p4
p5
p1
l3
l3
p5
p6
Channel(1) Channel(2)
Figure 2. The two maximal cuts at one-loop six-point.
implies that the second maximal cut C2 and C 2 must be related to C1 and C 1. Indeed one
can check thatC1 + C 1 h12[angbracketright][angbracketleft]34[angbracketright][angbracketleft]56[angbracketright]
= C2 + C 2
h23[angbracketright][angbracketleft]45[angbracketright][angbracketleft]61[angbracketright]
= 2iAtree6,shifted , (3.10)
where we have further identied the combination as the tree amplitude rotated by one,
Atree6,shifted(123456) := Atree(234561). Note that all objects in this equation have the same little group weights (odd under reversal of the even s) so the identication makes sense.
A remarkable feature of 6-point kinematics is that the expressions for C1 are explicit
in terms of angle brackets, that is they contain no square roots. This reects the fact that at six-points the cut solutions can be expressed explicitly in terms of angle brackets. Let us see this explicitly. At the same time, this will make apparent the following connection between the leading singularities and the BCFW form of the six-point tree amplitude,
Atree6 = LS1 + LS 1 = C1 C 1
2[angbracketleft]12[angbracketright][angbracketleft]34[angbracketright][angbracketleft]56[angbracketright]
= LS2 + LS 2 , (3.11)
in line with the original BCF logic [48] and as explained recently in [22]. The main point is that the on-shell condition l21 = l22 = 0 in channel (1) of gure 2 indicates that the loop momentum spinors can be parameterized as
l1 = 1 sin + 2 cos , l2 = i( 1 cos 2 sin ) . (3.12)
This is precisely the BCFW parameterization discussed in [7]. On the double-cut there are three poles as a function of cos , whose residues are respectively LS1, LS 1, and Atree6.
(The latter is located at cos = 0 and a computation of its residue is detailed in subsection (4.3), as part of our determination of the two-loop integrand.) The desired relation then follows from the fact that the three residues must sum up to zero by Cauchys theorem. For completeness, we record here the explicit solution which corresponds to C1
sin = ic+45/
q(c+36)2 (c+45)2 and cos = c+36/[radicalBig](c+36)2 (c+45)2. (3.13)
From eq. (3.9), one also sees that Ci has a non-uniform weight under conformal inversion:
I [C1] = C1
Q6 i=1
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q(x2i)x21x23x25, I [C2] = C2
Q6 i=1
q(x2i)x22x24x26. (3.14)
9
We are now ready to use the maximal cut to completely x the integrand. Two types of integrals contribute to the cut in channel (1) in gure 2, the massive triangles as well as the two-mass-hard box integrals. As there are two solutions for the maximal cut in channel (1), giving di erent cut results C1 and C 1, the massive triangle by itself cannot
simultaneously reproduce both. This implies the need for the box integrals. On the cut
the box integrals give:
Ibox(3, 4, 5, 1) [vextendsingle][vextendsingle][vextendsingle][vextendsingle]
= p2[angbracketleft]12[angbracketright][angbracketleft]34[angbracketright][angbracketleft]56[angbracketright] , (3.15)
where [notdef]C1 indicates the maximal cut it is evaluated on. A simple way to verify these formu
las, up to a common sign, is to compare their square with the square of [epsilon1](a, 3, 4, 5, 1)/(a[notdef]4)
on the cut, using the identity
[epsilon1](i1, . . . , i5)[epsilon1](j1, . . . , j5) := det
(ii [notdef] jj)
Ibox(3, 4, 5, 1) + Ibox(1, 2, 3, 4) Ibox(4, 5, 6, 1) Ibox(6, 1, 2, 4)
[bracketrightbigg]
+ C1 + C 12 2 Itri(1, 3, 5) + C2
+ C 2
2 Itri(2, 4, 6) .
44[angbracketright] component of the amplitude, from the explicit form of C1,2
in eqs. (3.8)(3.10), one sees that under a cyclic shift:
C1(
44
44 44)
[vextendsingle][vextendsingle]
44) . (3.18)
These additional signs are important for a non-vanishing one-loop amplitude as we now discuss. In ABJM, the tree and even-loop six-point amplitudes are parity even under k !
k, while odd-loops are parity odd. As parity is believed to be non-anomalous, this naively forbids non-trivial corrections from odd-loops unless these are odd under parity. Due to the change from k ! k, we are really exchanging the two gauge group U(N)k[notdef]U(N)k, and
thus resulting in a cyclic shift in the identication of the barred and unbarred-multiplet. Thus if the one-loop amplitude picks up a minus sign under the cyclic shift, this will compensate for the parity odd nature, and gives an acceptable one-loop correction. This aspect of the one-loop amplitude has been discussed previously in refs. [1921].
10
= p2[angbracketleft]12[angbracketright][angbracketleft]34[angbracketright][angbracketleft]56[angbracketright], Ibox(3, 4, 5, 1)[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
C 1
C1
, (3.16)
which in fact denes our normalization of the Levi-Cevita tensor. The sign can be computed by a judicious use of eq. (A.2).
Since the one-mass box must combine with the two-mass-hard box in the combination given in eq. (3.7), this xes the nal integrand that reproduces the correct maximal cut to be (stripping a loop factor 4N/k):
A1-loop6 = Atree6p
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(3.17)
Using eqs. (3.11) and (3.15), one can see that all maximal cut are correctly reproduced.
An important feature of the integrand in eq. (3.17) is that it picks up a minus sign under a cyclic shift of the all scalar component amplitude by one-site. For the box integrals, this is a consequence of the linear combination dictated by the vanishing two-particle cut in eq. (3.7). For the triangles, this is a consequence of their coe cients: if one considers the all scalar [angbracketleft]
44
44
i!i+1 = C 2(
44
44
3.2 The one-loop amplitude
The box integrals integrate to zero, thus the one-loop amplitude, at order O([epsilon1]0) is given
solely by the massive triangles:5
A1-loop6 =
The fact that (i [notdef] i+2) = [angbracketleft]ii+1[angbracketright]2 motivates the following denition [21]:
sgnc[angbracketleft]ij[angbracketright] := [angbracketleft]
i
2 Atree6,shifted (sgnc[angbracketleft]12[angbracketright]sgnc[angbracketleft]34[angbracketright]sgnc[angbracketleft]56[angbracketright] + sgnc[angbracketleft]23[angbracketright]sgnc[angbracketleft]45[angbracketright]sgnc[angbracketleft]61[angbracketright]) .
(3.20)
Thus the one-loop amplitude is proportional to the tree-amplitude shifted by one-site multiplied by a step function. This result has been obtained previously in [1921].
In closing, we note that at six-point there are only two distinct Yangian invariant, the sum and the di erence of the leading singularity and its conjugate. Interestingly, both combinations are local quantities, with the di erence appearing as the tree-amplitude, while the sum appears as the one-loop amplitude. From eq. (3.8) this property is rather obscure, however due to the following non-trivial identity, equivalent to eq. (A.4),
hi[notdef]j + k[notdef]l[angbracketright]2 (pi + pj + pk + pl)2[angbracketleft]jk[angbracketright]2 = (pi + pj + pk)2(pj + pk + pl)2 (3.21) one nds for example
c+41c41 = p2345
p2135
p(i j)
p(i k) p(j k)].6The locality of the leading singularities at six-point has been recently understood as a special property of the orthogonal Grassmaniann [49].
11
N k
(C1 + C 1) 4
p(1 [notdef] 3) p(5 [notdef] 3)
p(1 [notdef] 5)+ (C2 + C 2) 4 p(2 [notdef] 4) p(4 [notdef] 6)
p(6 [notdef] 2)
[parenrightBigg]
p[angbracketleft]ij[angbracketright]2 i[epsilon1]= [notdef]1. (3.19)
Using eq. (3.10) the one loop six-point ABJM amplitude can thus be rewritten as
A1-loop6 =[parenleftbigg]
N k
ij[angbracketright]
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[parenrightbigg]
. (3.22)
Thus the denominators of the leading singularities are in fact local propagators.6 However, only the sum of the leading singularities has the correct little group weights to appear in an amplitude. The di erence does not, unless it is multiplied by sign functions, which explains why it can appear only at loop level.
3.3 Analytic properties of the one-loop amplitude
The one-loop result (3.20) displays some remarkable properties which are worth spending some time on. In particular, step functions are rarely seen in loop amplitudes, so we need to understand well why they are allowed to appear in three space-time dimensions.
First, we would like to give some topological interpretation to the region where the amplitude is nonzero. In Minkowski space as null momenta can be parameterized as pi =
5The basic integral with massless internal lines, which follows easily from (3.4) with D = 3, is
Ra1 (ai)(aj)(ak) = 1/[8
Ei(1, sin i, cos i), the kinematic conguration of the scattering can be projected to a set of points on S1. The rst thing to notice is that the invariant [angbracketleft]ij[angbracketright] ips sign whenever the
points i, j on S1 crosses each other, as was also noted in [1921]. This is easy to see by writing the invariants in terms of coordinates on S1:
hij[angbracketright] = 2 sin
2 12[parenrightbigg] [radicalbig]E
i + i[epsilon1]
pEj + i[epsilon1] . (3.23)
Thus the function changes sign whenever point j crosses point i on S1. It is thus natural to divide the phase space into chambers depending on the ordering of the angles of the particles; the one-loop amplitude is locally constant in each of these chambers.
Given that the product of sign functions changes sign whenever two angles cross, the angular dependence can be given a simple topological interpretation in terms of a winding number counting the number of angle crossings compared to the color ordering. This can be dened as follows: if multiples of 2 are added to angles such that they are strictly increasing, 0 < i+1 i < 2, i = 1 . . . 7, then w := ( 7 1)/(2). Then one can show sgnc[angbracketleft]12[angbracketright]sgnc[angbracketleft]34[angbracketright]sgnc[angbracketleft]56[angbracketright]
sgnc[angbracketleft]23[angbracketright]sgnc[angbracketleft]45[angbracketright]sgnc[angbracketleft]61[angbracketright]
= (1)w(1)k. (3.24)
The second factor (1)k has a kinematical origin and originates from the factors
p(i [notdef] j) i[epsilon1] which can be real or imaginary depending on whether the given channel is space-like or time-like, respectively. The number k then simply equals the number of positive-energy timelike two-particle channels. We see that the 1-loop amplitude is a highly intricate function of the kinematical conguration.
As discussed in ref. [21], the fact that the one-loop amplitude is a step function can be readily understood from superconformal anomaly equations. Using free representation for the OSp(6[notdef]4) superconformal generators, it was shown that acting on the one-loop
six-point amplitude with the linear generators, one must obtain an anomalous term that is proportional to ([angbracketleft]ij[angbracketright]), i.e. it has support on regions where two external legs become
collinear. As the generators are linear, single derivatives in the on-shell variables, this implies that the amplitude must be proportional to step functions, or equivalently, sign functions.
However, we are rather disturbed by the notion of an amplitude vanishing in an open set but nonzero elsewhere this would seem to clash with the amplitude being an analytic function of the external momenta. In the rest of this section, we will propose that the step functions behavior are not actually incompatible with analyticity of the amplitude, but are likely only an artifact of xed-order perturbation theory.
It is useful to rst ask what would happen if we added small masses to the internal propagators still keeping the external lines massless. This could arise naturally by giving a vacuum expectation value to some of the scalars of the theory as discussed in section (5). In that case, the sign function singularity would split into two threshold singularities at = [notdef] with =
JHEP03(2013)075
p4m2/E1E2. Schematically,
sgn( 2 1) ! F ( 2 1) (3.25)
12
(a) (b)
Figure 3. (a) The amplitude F in the presence of a small mass. It can be continued from < 0 to > 0 through a narrow window of size , which shrinks to zero size in the massless limit. (b)
The advocated behavior in the massless setup, at a small but nite value of the coupling. A branch cut covers the whole imaginary axis but the discontinuity across it tends to zero at the origin.
where F ( 2 1) is an analytic function with an analytic window of width 2 around the
origin.7 This amplitude is plotted in the complex plane in gure 3. We see that as long as m [negationslash]= 0 there exists a small window of width around the origin along which the amplitude
can be rightfully continued.
We also see clearly why such behavior is possible in three space-time dimensions but not in higher dimensions. In three dimensions the physical (real) phase space for a set of massless particles splits into chambers which are separated by singular, collinear congurations. To analytically continue from one chamber to the next one must avoid the singularity, since the amplitude is not required to be analytic around that point. But attempts to avoid the singularity by passing through the complex plane may fail: the singularity can be surrounded by cuts.
At the massless point but at the nonperturbative level, we expect an analogous situation but with a nonperturbatively small window of width e#
k
N . Indeed, in a theory where soft and collinear quanta are copiously produced, as is ABJM, we nd it unlikely for a sharp feature such as a sign function to remain unwashed. Rather, the backreaction of the radiation on the ongoing hard quanta should smear the small angle behaviour. In perturbation theory this would become visible through large logarithms N/k log 1/ , which would have to be resumed at small angles. Indeed such logarithms will come out of our two-loop computation. Thus a more faithful model for the small angle behavior at small
7The precise form of F can be worked out from the following exact expression for the internally massive loop integral, writing = x2ijx2ikx2jk + m2(2x2ijx2ik + 2x2ijx2jk + 2x2ikx2ik x4ij x4ik x4jk)
1/2:
1[(a i) + m2][(a j) + m2][(a k) + [notdef]2]= 18i logi + m(x2ij + x2ik + x2jk + 8m2)
i + m(x2ij + x2ik + x2jk + 8m2) . (3.26)
In the collinear regime x2ij m2, this exhibits on the rst sheet a pair of logarithmic branch points at
the threshold x2ij + 4m2 = 0. However, on the second sheet there is also a square-root branch point at
x2ij = m2 (x
2
2 jk)
2
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Za
2
jk . The latter could be visible with physical Minkowski space kinematics, depending on whether the xik and xjk channels are time-like or not.
13
x2
ikx
xi+1
xi
xa
xi xa
xi+1
xi1
xi1
Figure 4. The triple cut of consecutive massless corners corresponds to soft exchange between the two external lines. In dual space, this correspond to the loop region xa approaching xi.
but nite coupling should be a function of the sort
sgn( 2 1) !
2 1 (( 2 1)2)
1
2
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#N/k (3.27)
which can be happily continued from the left region to the right region. It would be very interesting to investigate the small-angle behavior quantitatively and conrm that the discontinuity across the cut goes to zero as 2 1 ! 0.
4 The two-loop six-point integrand
We shall now proceed to determine the two-loop six-point integrand from a variety of on-shell constraints. In ABJM theory we get a large number of constraints just from the fact that there are no 3- and 5-point on-shell amplitudes. This gives a large number of cuts on which the integrand must vanish. In addition, there are some very simple non-vanishing triple-cuts associated with soft gluon exchanges which can be used to x the remaining freedom.
Our rst goal in this section is thus to determine the two-loop hexagon integrand using just the following constraints:
0. The integrand is dual conformal invariant.
1. Cuts isolating a ve-point amplitude must vanish.
2. Cuts isolating a three-point vertex must vanish.
3. Triple-cuts of consecutive massless corners, as shown in gure 4, correspond to soft gluon exchange and must reduce to the 1-loop integrand.
4. Absence of non-factorizable collinear divergences.
As an example, we now show that by simply using steps 2 and 3, one completely xes the four-point two-loop integrand to be that constructed in ref. [23]. This also illustrate the importance of obtaining the one-loop amplitude beyond O([epsilon1]). The one-loop four-point
integrand is given by
Atree4p
2
[epsilon1](a1234)(a [notdef] 1)(a [notdef] 2)(a [notdef] 3)(a [notdef] 4)
, (4.1)
14
x2
xa
x3
x1
xb xb
x2 xa
x3
x1
Figure 5. The triple cut of the double box integral. As xa approaches x2 on the cut condition, one should obtain the one-loop integrand given in eq. (4.1).
where the unpleasant-looking factor of p2 is due to our normalization of the ve-dimensional [epsilon1]-symbol as discussed around eq. (3.16). Now consider the triple cut of a double-box integral in gure 5. On the cut, xa approaches x2 and in this limit one should recover eq. (4.1).
With a little thought one sees that the following double-box numerator does the job:
Atree4
[epsilon1](a123 )(b341 )
2 (a [notdef] 1)(a [notdef] 2)(a [notdef] 3)(a [notdef] b)(b [notdef] 3)(b [notdef] 4)(b [notdef] 1)
where [epsilon1](a, i, j, k, )[epsilon1](b, l, m, n, ) := [epsilon1](a, i, j, k, [notdef])[epsilon1](b, l, m, n, [notdef]). The detailed behavior of
such numerator under the cut condition will be discussed in subsection (4.3). This however, is not complete as one sees that there is a non-trivial contribution to the cut (a[notdef]3) = (a[notdef]b) =
(b [notdef] 3) = 0. This separates out a three-point tree amplitude and hence must vanish. On
this cut, using (3.16) and setting ya = yb to restrict to an easy subcase, the double box gives a nontrivial contribution
Atree42
.
One can see that the above also satisfy requirement (1) and is equivalent to that of [23].
4.1 Integrand basis
We begin by constructing the most general algebraic basis of dual-conformal integrals at two loops. In three dimensions, the most general two-loop integral is a double-box
Iijk;lmn2box[(v1 [notdef] a)(v2 [notdef] b)] := [integraldisplay]a,b
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(4.2)
(1 [notdef] 3)2
(a [notdef] 1)(b [notdef] 1)
. (4.3)
One can easily see that this contribution can be cancelled by a double triangle integral. Thus combining requirements (2) and (3) uniquely xes the two-loop four-point integrand to be:
A2-loop4 = Atree42 [integraldisplay]a,b [bracketleftbigg]
[epsilon1](a123 )(b341 ) + (a [notdef] 2)(b [notdef] 4)(1 [notdef] 3)2
(a [notdef] 1)(a [notdef] 2)(a [notdef] 3)(a [notdef] b)(b [notdef] 3)(b [notdef] 4)(b [notdef] 1)
+ (s $ t)
[bracketrightbigg]
(v1 [notdef] a)(v2 [notdef] b)
(a [notdef] i)(a [notdef] j)(a [notdef] k)(a [notdef] b)(b [notdef] l)(b [notdef] m)(b [notdef] n)where v1 and v2 are some 5-vectors. Note that the presence of the numerator is required by dual conformal invariance, as the integrand must have scaling weight -3 with respect to
15
x1 x1 x1 x1
Figure 6. Possible box integrals that have non-trivial two-particle cut which would correspond to factorization channel that factorize the amplitude into a product of 5-pt amplitudes. The contributions to the cut from each box integral are distinct, leading to the conclusion that they will not appear.
both a and b.8 Numerators v1 proportional to yi, yj or yk are reducible, which would leave a-priori 2 distinct numerators on each side. However, at 6 points constraint 2 above is very powerful as it requires the numerator to have zeros on any double cut isolating a massless external leg. For dual conformal invariant integrals, this restricts the numerators to be of the [epsilon1]-type
Iijk;lmn2box[[epsilon1](a, i, j, k, )[epsilon1](b, l, m, n, )] or Iijk;kli2box[[epsilon1](a, i, j, k, b)] ,
where the second possibility is allowed only when (k [notdef] l) and (j [notdef] l) are both nonvanishing.
Note that this latter parity-odd double-box integral has excessive weight on (i, k, l). At six-point, this can be naturally absorbed by the extra weights of C1+C 1 shown in eq. (3.14).
Six-point double-box integrals with a three-legged massive corner, and some with two legged massive corners, will have two-particle cuts that factories into a product of ve-point amplitudes as shown in gure 6. Since ve-point amplitudes vanish to all order in [epsilon1], the contributions of these double-box integrals must cancel out on such cuts or they are not allowed in the integral basis. It is straight forward to see that the contributions are distinct and cannot cancel. Thus by imposing conditions 1 and 2 on one-loop subdiagrams the allowed parity even double box integrals are restricted to
I2mheven(i) :=
Za,b
[epsilon1](a, i, i + 1, i + 2, )[epsilon1](b, i + 2, i 2, i, )(a [notdef] i)(a [notdef] i + 1)(a [notdef] i + 2)(a [notdef] b)(b [notdef] i + 2)(b [notdef] i 2)(b [notdef] i)
Icrab(i) :=
Za,b
[epsilon1](a, i, i + 1, i + 2, )[epsilon1](b, i 2, i 1, i, )(a [notdef] i)(a [notdef] i + 1)(a [notdef] i + 2)(a [notdef] b)(b [notdef] i 2)(b [notdef] i 1)(b [notdef] i)
Icritter(i) :=
Za,b
[epsilon1](a, i, i + 1, i + 2, )[epsilon1](b, i + 3, i + 4, i + 5, )(a [notdef] i)(a [notdef] i + 1)(a [notdef] i + 2)(a [notdef] b)(b [notdef] i + 3)(b [notdef] i + 4)(b [notdef] i + 5)
I2mhodd(i) :=
Za,b
[epsilon1](a, i, i + 1, i + 2, b)(a [notdef] i)(a [notdef] i + 1)(a [notdef] i + 2)(a [notdef] b)(b [notdef] i + 2)(b [notdef] i 2)(b [notdef] i)(4.5)
8The absence of pentagon-boxes or more complicated topologies can be easily proved as follows. A pentagon would need a numerator quadratic in a. Lets assume the ve external propagators involving a are a1, . . . a4 and ab. Then we can expand the numerator in terms of products (a v1)(a v2) where the
(a vi) are chosen lie in the following basis
(a 1), (a 2), (a 3), (a 4) and [epsilon1](a1234). (4.4) All numerators in this basis trivially cancel some propagator, except for ([epsilon1](a1234))2, which would appear to be irreducible. However, this can be reduced using the Gram identity (3.16) together with a2 = 0.
16
JHEP03(2013)075
where the subscript even and odd denotes the two-mass hard integrals with parity-even and -odd numerators.
The same conditions also leave box-triangle integrals
Iijk;lmbox;tri[[epsilon1](a, i, j, k, l)] :=Za,b
[epsilon1](a, i, j, k, l)(a [notdef] i)(a [notdef] j)(a [notdef] k)(a [notdef] b)(b [notdef] l)(b [notdef] m)
. (4.6)
Other choices for the numerator here, such as the other natural choice [epsilon1](a, i, j, k, m), would be related by a Schouten identity plus double triangle integrals. The box-triangles again have excessive weight which would imply that they should come with factors of C1 + C 1.
We will see that they indeed arise in this way.Finally, the conditions applied so far leave only three double-triangle integrals, namely
Ii,i+2;i+2,i2tri :=Za,b
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(i [notdef] i + 2)2(a [notdef] i)(a [notdef] i + 2)(a [notdef] b)(b [notdef] i)(b [notdef] i + 2)
Ii,i+2;i2,i2tri :=
Za,b(i [notdef] i + 2)(i [notdef] i 2)(a [notdef] i)(a [notdef] i + 2)(a [notdef] b)(b [notdef] i 2)(b [notdef] i)
Ii,i+2;i3,i12tri :=
Za,b(i [notdef] i + 2)(i 1 [notdef] i 3)(a [notdef] i)(a [notdef] i + 2)(a [notdef] b)(b [notdef] i 3)(b [notdef] i 1)
. (4.7)
We now nish to implement constraint 2, the vanishing of all three-point sub amplitudes.
4.2 Constraints from vanishing three point sub amplitudes
We consider the cut (a[notdef]3) = (a[notdef]b) = (b[notdef]3) = 0 which separates out a three-point amplitude
and thus must vanish. Two types of double boxes contribute to such cut, I2mh and Icrab, and they contribute:
(1)
1
3
2
a b
5
!
(1 [notdef] 3)(b [notdef] 2)[(a [notdef] 1)(3 [notdef] 5) (a [notdef] 5)(3 [notdef] 1)]
(a [notdef] 2)(a [notdef] 1)(b [notdef] 1)(b [notdef] 5)
(2)
3
2
a b 4
1
5
!
(b [notdef] 2)(1 [notdef] 3)(a [notdef] 4)(3 [notdef] 5)
(a [notdef] 1)(a [notdef] 2)(b [notdef] 5)(b [notdef] 4)
(3)
5
4
a b
1
3
!
(b [notdef] 4)(3 [notdef] 5)[(a [notdef] 5)(3 [notdef] 1) (a [notdef] 1)(3 [notdef] 5)]
(a [notdef] 4)(a [notdef] 5)(b [notdef] 5)(b [notdef] 1)
17
(4)
where weve indicated the non-vanishing remainder on the cut. This was obtained by using eq. (3.16) to reduce the [epsilon1]-symbols to dot products and dropping terms which vanish on the cut. This can be simplied further when we take into account that the general solution to the cut is parametrized by ya,b = y3 + a,bv, where v is any null ve-vector such that
v[notdef]3 = 0. Physically, on the cut the two loop momenta are collinear with each other. Then
one nds that the following combination of double-box and double-triangle integrals vanish on the cut and are thus allowed
I2mheven(1) + I1,3;3,12tri I1,3;3,52tri I1,3;5,12tri Icrab(1) + I1,3;3,52tri
Icritter, Ii,i+2;i1,i32tri, Ii,i+1,i+2;i1,i3box;tri and I2mhodd .
In addition the integral Ii,i+1,i+2;i+2,i2box;tri is immediately ruled out. In the following, we will use constraint 3 to x the relevant coe cient of the double box integrals.
4.3 Constraints from one-loop leading singularity
The particular cut we will be interested in is the maximal cut of one of the sub loops with adjacent massless legs. This cut corresponds to a kinematic conguration where there is a soft-exchange between the two external legs, as can be deduced from eq. (3.12) using the fact that the (a [notdef] 2) only gives a pole at cos ! 0. In terms of dual regions, this correspond
to when the loop region ya ! yi as illustrated in gure 4.
More specically, we compute the leading singularity (a [notdef] 1) = (a [notdef] 2) = (a [notdef] 3) = 0 of
[epsilon1](a, 1, 2, 3, )
(a [notdef] 1)(a [notdef] 2)(a [notdef] 3)(a [notdef] i)
Normally there are two solutions to such a cut constraint, but let us verify explicitly that here there is only one solution ya = y2 as claimed. To do so we expand a over a natural basis, such as a = a1y1 + y2 + a3y3 + a4y4 + a[epsilon1]y[epsilon1] where y[epsilon1] := [epsilon1](1, 2, 3, 4, ). Imposing the
two cuts (a [notdef] 1) = (a [notdef] 3) = 0 gives that a1 = 0 and a3/a4 = (1 [notdef] 4)/(1 [notdef] 3), and thus a3 / a2[epsilon1]
due to the y2 = 0 constraint. Then (a [notdef] 2) a2[epsilon1] so the only solution is a[epsilon1] = 0.
Let us work out the details. Normalizing the leading singularities in a convenient way
F (a)(a [notdef] i)(a [notdef] j)(a [notdef] k)
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
residue i,j,k
Za = 4 [integraldisplay]
d5a (a2)vol(GL(1)) = N
[integraldisplay]
da1da3da4da[epsilon1] (a2)
18
3
a b
1
2
5
! 0 ,
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.
:= 4
Za ((a [notdef] i) ((a [notdef] j)) ((a [notdef] k))F (a), (4.8)
we have here
4
where N = p2[epsilon1](y1, y2, y3, y4, y[epsilon1]) = p2y2
[epsilon1] . After taking the rst two cuts and evaluating the Jacobian from the -functions we get
Za
((a [notdef] 1)) ((a [notdef] 3))(a [notdef] 2)= N
y2[epsilon1](1 [notdef] 3)2(2 [notdef] 4)
[integraldisplay]
da[epsilon1]
a2[epsilon1] (4.9)
where a([epsilon1]) = y2 + a[epsilon1]y[epsilon1] + a2[epsilon1]
(this could be mapped to the BCFW parametrization (3.12), since this solves the same cut constraints). Multiplying by [epsilon1](a, 1, 2, 3, )/(a [notdef] i) and taking the residue at a[epsilon1] = 0, we immediately get the box leading
singularity
[epsilon1](a, 1, 2, 3, )
(a [notdef] 1)(a [notdef] 2)(a [notdef] 3)(a [notdef] i)
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
(1[notdef]3) y3 y4
(2 [notdef] )(2 [notdef] i)
[bracketrightbigg]
. (4.10)
Note that although the numerator vanishes on the cut solution ya = y2, reecting the absence of three-point vertices in this theory, a nonvanishing residue remains due to the double pole in (4.9). The residue reects the the exchange of a zero-momentum Chern-Simons eld.
This physical origin implies that these leading singularities are universal, and must reduce to the lower-loop integrand with the loop variable ya omitted. Thus, with a normalization easily xed from the 1-loop integrand,
A[lscript]loopn[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
= A([lscript]1)loopn. (4.11)
Similarly, but with an opposite sign due to k ! k,
A[lscript]loopn[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
residue 2,3,4
y2[epsilon1] 2(2[notdef]4)
[bracketleftbig]
[bracketrightbig]
(1[notdef]4)
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residue 1,2,3
= p2
residue 1,2,3
= A([lscript]1)loopn. (4.12)
These relations can easily be veried to hold for the one-loop integrand (3.17), where the right-hand side reduces to the tree amplitude. However, these relations must hold at any loop order. In a sense they are analogous to the so-called rung rule [50].
At two loops, this requires to see the one-loop integrand emerge on the cut, i.e. eq. (3.17). Indeed one nds that the various pieces of the one-loop integrand do appear from the double-box and the box-triangle integrands. More specically, for the cut (a [notdef] 1) = (a [notdef] 2) = (a [notdef] 3) = 0, omitting the common p2 factor, we nd the following
contributions:
I2mheven(1) ! Ibox(3, 5, 1, 2)
Icrab(1) ! Ibox(5, 6, 1, 2), Icrab(3) ! Ibox(3, 4, 5, 2)
Icritter(1) ! Ibox(4, 5, 6, 2),
I1,2,3;4,6box;tri[[epsilon1](a, 1, 2, 3, i)] ! (2 [notdef] i)Itri(2, 4, 6), I2mhodd(1) ! Itri(1, 3, 5) . (4.13)
As one can see, all one-loop integrals appearing in eq. (3.17) are present. Thus the constraint of reproducing the one-loop integrand on the one-loop maximal cut, combined with
19
3
3
2
a b
5
2
a b 4
1 6
1
1
5
tree
( C2 + C2 )
6
2
3
3
a b
4
2
a b
5
6
2
5
JHEP03(2013)075
1
3
a b
4
3
* * a
1
2
( C1 + C1 )
2
b
5
2 2 2 2
1 6
Figure 7. The terms that contribute to the leading singularity (a [notdef] 1) = (a [notdef] 2) = (a [notdef] 3) = 0, which
has to reduce to the one-loop integrand. The blue lines indicate the one-loop propagators that remain and uncancelled after the cut. The term in the bottom of each diagram is the numerator factor. One can see that the combination is precisely the one-loop answer.
previous results derived from constraint 2, xes the two-loop integrand to be the following combination:
Atree6
2 I2mheven(1) + Icrab(1) Icritter(1) + I1,3;3,12tri I1,3;5,12tri + cyclic[bracketrightbigg]
+C1 + C 1
2p2 [bracketleftbigg]
I2mhodd(1)
I4,5,6;1,3box;tri[[epsilon1](a, 4, 5, 6, 1)]
(1 [notdef] 5)
+ cyclic [notdef] 2
[bracketrightbigg]
+C2 + C 2
2p2 [bracketleftbigg]
I2mhodd(2) + I1,2,3;4,6box;tri[[epsilon1](a, 1, 2, 3, 6)]/(2 [notdef] 6) + cyclic [notdef] 2[bracketrightbigg]
+
6
Xi=1 iIi,i+2;i3,i12tri
where cyclic [notdef] 2 implies cyclic by two sites and C1,2, C 1,2 are dened as before. The presence
of the one-loop integrand on the cut (a [notdef] 1) = (a [notdef] 2) = (a [notdef] 3) = 0 are shown in gure 7.
We see that at this point, the only remaining freedom is the triangle integrals Ii,i+2;i3,i12tri. However, as we will now see these integrals are badly collinear divergent and so they are constrained by other physical considerations.
20
4.4 Collinear divergences and the ABJM two-loop integrand
As was demonstrated in [5153] (in the context of planar N = 4), the exponentiation of
divergences leads to constraints which can be formulated in a very simple way at the level of the integrand, e.g., before even performing any integral. We will now formulate similar constraints in ABJM theory, but these will have a somewhat di erent avor due to the absence of one-loop divergences.
In ABJM theory, the twist-two anomalous dimensions which control the collinear and soft-collinear divergences begin at order (k/N)2, e.g. two-loops. Thus the divergences at two-loop are the leading ones and must be proportional to the tree amplitude in a specic way. Some qualitative constraints can be deduced in a simple way as follows: we note that soft divergences can be computed by replacing the external states by Wilson lines. Just this fact imposes two simple constraints. First, the coe cient of proportionality of the 1/[epsilon1]2 divergence must be a pure number, e.g. independent of the kinematics (ultimately, 6 times the so-called cusp anomalous dimension). Second, kinematic dependence of the subleading 1/[epsilon1] divergence, which can arise from soft wide-angle radiation but not collinear radiation (and hence is controlled by the Wilson lines) can only be of the simple dipole invariants of the form [1/[epsilon1]] log x
2i,i+1
[notdef]2IR . We will call divergences of these forms factorizable. These are rather general constraints that any physically acceptable amplitude must possess and we will see that they impose nontrivial constraints on the integrand.
We will consider the collinear divergence from the region collinear to momentum p3.
To have a divergence we need both loop momenta to be collinear, thanks to the special [epsilon1]-numerators, so we consider the limit
ya ! y3 + ay4, yb ! y3 + by4. (4.14)
A rst requirement is that the integrals proportional to the parity-odd structure, e.g.
Ci + C i, be nite. For the box-triangles, we nd the following combination is free of
divergences:
(C1 + C 1)[parenleftbigg]
I4,5,6;1,3box;tri[[epsilon1](a, 4, 5, 6, 1)]
(1 [notdef] 5)
JHEP03(2013)075
[parenrightbigg]
(C2 + C 2)[parenleftbigg]
I1,2,3;4,6box;tri[[epsilon1](a, 1, 2, 3, 6)]
(2 [notdef] 6)
.
To see that this combination is indeed nite in the collinear region, note that in the limit eq. (4.14), factoring out the divergent factors 1/(a [notdef] 3)(a [notdef] b)(b [notdef] 4) one has
(C1 + C 1)
[epsilon1](3, 4, 5, 6, 1)(5 [notdef] 1)(a [notdef] 1)(3 [notdef] 5)(b [notdef] 6)
(C2 + C 2)
[epsilon1](4, 1, 2, 3, 6)(2 [notdef] 6)(a [notdef] 1)(4 [notdef] 2)(b [notdef] 6)
.
where weve symmetrized in (a $ b). The above combination vanishes thanks to the
following identity:
C1 + C 1 C2 + C 2
=
[epsilon1](6, 1, 2, 3, 4)(3 [notdef] 5)(5 [notdef] 1)
[epsilon1](3, 4, 5, 6, 1)(6 [notdef] 2)(2 [notdef] 4)
. (4.15)
This identity is proven in appendix (A). Thus we conclude that the parity-odd part of the integrand in eq. (4.14) is already complete, provided that the box-triangle numerators are chosen as there.
21
We now turn to the parity-even sector. We need to study the divergences of the yet-unconstrained integral I1,3;4,62tri in more detail. Integrating out the remaining variables around the limit (4.14) one easily obtains the divergent contribution from the collinear region
Za,b
1(a [notdef] 3)(a [notdef] b)(b [notdef] 4)(a [notdef] 1)(b [notdef] 6) /log [notdef]2
Z0<a<b<1
dadb
pa(b a)(a [notdef] 1)(b [notdef] 6)
where a and b are as in (4.14). Such a divergence violates factorizability in two ways: it depends on y1 through (a [notdef] 1) and on y6 through (b [notdef] 6). This leads, for instance, to
dependence on the cross-ratio u1. A quick look at (4.14) reveals that the only other integral with potentially similar dependence on a,b is Icritter. However the divergence
cancels exactly, pre-integration, in the combination
Icritter(1) + I1,3;4,62tri .
Thus we nally arrive at the complete integrand for two-loops six-point amplitude in ABJM theory:
A2-loop6 = [parenleftbigg]
4N k
2 Atree6 2
I2mheven(1)+Icrab(1)Icritter(1)+I1,3;3,12triI1,3;5,12triI1,3;4,62tri+cyclic[bracketrightbigg]
+ C1 + C 1
JHEP03(2013)075
2p2 [bracketleftbigg]
I2mhodd(1)
I4,5,6;1,3box;tri[[epsilon1](a, 4, 5, 6, 1)]
(1 [notdef] 5)
+ cyclic [notdef] 2
[bracketrightbigg][parenleftBigg][bracketrightbigg] (4.16)
5 Interlude: infrared regularization using the Higgs mechanism
The two-loop amplitude is infrared divergent and must be regulated in some way. For inspiration we can look at the four-dimensional sibling of ABJM, N = 4 SYM. In that
theory there exists a canonical and self-contained infrared regularization, associated to giving small vacuum expectation values to the scalar elds of the theory [39]. The elds running in loops then acquire masses through the Higgs mechanism, rendering the loop integrations nite.
Does a similar regularization exist in ABJM theory? As was shown in the original paper [1], this theory has a moduli space (C4/Zk)N where N characterizes the SU(N)[notdef]
SU(N) gauge group and k is the level. It is described simply by diagonal vacuum expectation values (vevs) for the 4 scalar elds, [angbracketleft]A[angbracketright] = diag(vAi) (with corresponding vevs for
the conjugate elds, [angbracketleft]
+ C2 + C 2
2p2 [bracketleftbigg]
I2mhodd(2) +
I1,2,3;4,6box;tri[[epsilon1](a, 1, 2, 3, 6)]
(2 [notdef] 6)
+ cyclic [notdef] 2
A[angbracketright] = diag((vAi))). The Zk identications will play no role in what
follows, although we will be able to see that our formulas are invariant under it.
The rst question to address is what is the spectrum of the theory at a given point on the moduli space. While we have not found the general answer to this question in the literature, this can easily be answered in perturbation theory in the usual way by studying the linearized action for uctuations around the vacuum. (Due to the amount
22
m3
m5
m1 m1
Figure 8. Pattern of masses for the Higgsed theory following [39]. The loop propagators in the interior of the graph remain massless while those at the boundary, represented in bold, acquire a mass. External states can be chosen to remain massless or not, depending on whether the mi are equal or not.
of supersymmetry, it is plausible that the resulting spectrum is valid for all values of the coupling, although this will not be important for us.) To be safe we have computed the linearized action for both scalar, fermion and gauge eld uctuations, and conrmed that the spectra are related by supersymmetry as required. These computations are reproduced in appendix (B). From the linearized action it is then possible to nd the poles in the propagators and read o the spectrum.
The result is very simple. We nd that the diagonal elds remain massless, while the o -diagonal elds stretching between i and j acquire the mass squared
m2ij = (vi[notdef]vi + vj[notdef]vj)2 4vi[notdef]vjvj[notdef]vi. (5.1)
Note that this vanishes when vi = vj, as expected.9 Furthermore, when m2ij is nonzero, the computation in appendix demonstrates that the corresponding components of the gluon propagator lack a pole at zero momentum. Thus all modes acquire a mass. This is in contrast with the mass-deformed supersymmetric CSm amplitudes discussed in [18].
Following ref. [39], this can be used to regulate planar amplitudes. The idea is to split SU(N) as SU(M)[notdef] SU(N-M) with M N and turn on vevs only within the smaller SU(M),
restricting attention to external states within that SU(M). Many variants are possible. For instance if the vev preserves the SU(M) symmetry all external states remain massless. Or a generic vev can break SU(M) down to U(1)M, rendering the external states massive according to eq. (5.1).
However, as long as none of the SU(M) vevs vanishes, all outermost propagators in a Feynman diagram will be massive as depicted in gure 8. This ensures the niteness of the corresponding integral. (At least for all integrals that have been considered in the literature so far.)
For the purpose of regularization we will restrict to the simplest setup, taking all nonzero vevs to be aligned in the SU(4) directions: vAi = A1vi. Then the mass formula reduces to
where mi := [notdef]vi[notdef]2. We see that the ABJM masses behave exactly like the extra-dimensional
coordinates in N = 4 SYM discussed in [39]!
9More generally, this vanishes whenever the (Zk)N-invariant combination vi vi vj vj vanishes.
23
m2
m6
JHEP03(2013)075
m2ij = (mi mj)2 (aligned vevs). (5.2)
In the remainder of this section, we discuss how to implement this regulator in a simple way within the embedding formalism. This will be applied to numerous examples in the next section.
Following the extra-dimensional interpretation of the masses it is natural to enlarge the external ve-vectors yi to six-vectors
y(6)i = (xi, 1, x2i + m2i, mi) (5.3)
with a inner product dened such that (i [notdef] i)(6) = 0 for vectors of this form. Then one can
verify that (i [notdef] j)(6) = (xi xj)2 + (mi mj)2, automatically generating the correct internal masses provided that the 6-dimensional product is used in propagators. Furthermore, the on-shell constraints are simply (i [notdef] i+1)(6) = 0. Regarding loop integrations, we set
the extra-dimensional component of loop variables to zero, a = (x, 1, x2, 0), e.g. the loop variables remain 5-dimensional. Then all propagators come out correctly.
Since only the ve-dimensional components of vectors yi couple to the loop variables, it is immediate that all Feynman parametrization formulas in section (3) go through unchanged. One must simply continue to use the ve-dimensional inner product (i [notdef] j) := (xi xj)2 + m2i + m2j in them. The ve-dimensional inner product of the
external ys obeys the following identity
(i [notdef] i+1)2 (i [notdef] i)(i+1 [notdef] i+1) = 0 (5.4)
which can be seen to be equivalent to the on-shell relation (5.2) for the external states.
A simple consequence of this procedure is that as long as integrands written in terms of the y(6) are SO(2,3)-covariant, resulting amplitude will be invariant under the modied dual conformal generator
K[notdef]An = 0, where K[notdef] =
n
Xi=1
JHEP03(2013)075
x2i @
@x[notdef]i 2x[notdef]ixi[notdef]
@
@xi 2x[notdef]imi
@
@mi x[notdef]i[bracketrightbigg]
. (5.5)
This equation is essentially trivial by assumption, and will remain true as long as the integrals are indeed rendered nite by the regularization.
An important question is whether the SO(2,3) symmetry is an actual property of the ABJM integrand even for nite values of the masses. We expect this to be the case, although we cannot prove it. The logic is that the dual conformal symmetry SO(2,3) is associated with integrability, which we do not expect to vanish into thin air just because one moves away from the origin of moduli space. Indeed, physically, the Higgs branch can be explored by considering amplitudes at the origin of moduli space but with soft scalars added, as was demonstrated in the context of tree amplitudes in refs. [54, 55]. By exploring such a construction in three dimensions, it might even be possible to establish whether the dual conformal symmetry of tree amplitudes, hence presumably of loop integrands by unitarity, holds away from the origin of moduli space.10
10Dual conformal symmetry of maximal super-Yang-Mills at nite values of the masses can also be established by considering the symmetry as a property of the higher dimensional parent theory [56, 57].
24
In the present paper we work only to lowest order in the masses, e.g. we keep only the logarithmic dependence on them. At that level the SO(2,3) symmetry is more or less tautological as it is the same as the existing dual conformal symmetry. Thus to logarithmic accuracy in the masses the validity of (5.5) is already guaranteed by existing results.
Let us elaborate on eq. (5.5). A consequence of it together with the on-shell condition (5.4) is that the dependence on the individual mi can be determined simply from consideration of conformal weights. For instance, suppose an amplitude is known in the case that all internal masses are equal and all external masses vanish. Then, the most general aligned case with internal masses mi (and thus generic external masses) can be obtained (to the same order in the small mass expansion) through the simple substitution
x2ij
[notdef]2IR !
The point is that there are no ratios of the masses invariant under (5.5). Therefore, with no loss of generality, the Higgs regulator to logarithmic accuracy (and perhaps more generally) can be summarized by the simple rule
yi ! yi + [notdef]2IRyI, e.g. ([vector]xi, 1, x2i) ! ([vector]xi, 1, x2i + [notdef]2IR) (5.6)
for each external region momenta, where we only need to keep track of the ve-dimensional components of the yi, and where the massless external momenta remain undeformed. This recipe is the main result of this section.
Alternatively, infrared divergences can be regulated using dimensional regularization. Due to the presence of Levi-Cevita tensors in Chern-Simons theory, dimensional regularization has always been used with a great deal of caution. However, as one can use tensor algebra in three-dimensions to covert the Levi-Cevita tensors into Lorentz invariant scalar dot products and analytically continue to D = 3 2[epsilon1], this is more similar to dimensional
reduction regularization, commonly applied to supersymmetric theories. This regularization scheme has been shown to be gauge invariant up to three-loops for Chern-Simons like theories in ref. [58], and has also been applied to Wilson-loop computations in refs. [60, 61] establishing duality with the amplitude result. We will demonstrate below that the individual dimensionally regulated integrals di er from the mass regulated result functionally. However, when combined into the physical amplitude, the two regulated results agree.
6 Computation of two-loop integrals
In this technical section we describe our computation of the two-loop integrals relevant for the two-loop hexagon. Although we feel that some of the tricks employed here can nd application elsewhere, the reader not interested in these details can safely skip to the next section.
Certain integrals, or combinations of integrals, are absolutely convergent and can be computed directly in D = 3. Examples are I2mhodd, Icritter(1) + I1,3;4,62tri, or a certain combination of the odd box-triangles described below. It is very convenient to treat these
25
JHEP03(2013)075
x2ij
mimj .
combinations separately since they can be evaluated without regularization. Furthermore they automatically give rise to functions of the cross-ratios ui. On the other hands, IR divergent integrals cannot be avoided. In this section we will use the Higgs regulator described in the previous section.
We will describe the various steps in our integration method, starting from the steps common to all integrals.
6.1 A Feynman parametrization trick
There is a particular version of Feynman parameterization which is particularly e ective for our calculations. It is inspired by a formula obtained in [62] using intuition from Mellin space techniques, but can be derived very simply by a judicious change of variable in standard Feynman parameter space as was demonstrated in ref. [63].
We illustrate it in detail in the case of the double triangle I1,3;3,12tri, the other integrals are entirely similar. The rst step starting from the denition (4.7) is the usual Feynman (Schwinger) trick
I1,3;3,12tri = (3)2 [integraldisplay]
1
0
JHEP03(2013)075
[d1a1a3]
vol(GL(1))
[d1b1b3]
vol(GL(1))
Za,b(1[notdef]3)2(a [notdef] A)2(a [notdef] b)(b [notdef] B)2. (6.1)
where A =
Pi=1,3 aiyi and B = Pi=1,3 biyi.
A word about the notation. The 1/vol(GL(1)) symbol means to break the projective invariance (ai, bi) ! (ai, bi) by inserting any factor which integrates to 1 on GL(1) orbits.
Standard choices include (
Pai 1), which give the Feynman parameter measure dF in section (3), or (a1 1) which give rise to Schwinger parameters. The choice of a gauge-
xing function will play no role in what follows, and for all practical purposes it can be gleefully ignored it until the nal step.
The loop integrals over a, b can all be done using only the one-loop integral (3.4). Since the Higgs regulator already renders the integrals nite, we set D = 3 immediately to obtain (to avoid cluttering the formulas in this section, we will strip a factor 1/(4) for each loop)
(3)Za1(a[notdef]A)3 !14(12A [notdef] A)3/2. (6.2)
By repeatedly using this formula and its corollary valid for b2 = 0
Za1(a [notdef] A)2(a [notdef] b)= (3)
[integraldisplay]
10 df [integraldisplay]a1(a [notdef] (A + fb))3= 121
q12 A [notdef] A(A [notdef] b)
, (6.3)
we derive the following, key formula:
Za,b
1(a [notdef] A)2(a [notdef] b)(b [notdef] B)2= 12 [integraldisplay]b
1
q12 A [notdef] A(A [notdef] b)(b [notdef] B)2
= 18 [integraldisplay]
1
0
de
3/2
q12 A [notdef] A
1
2 (B + eA) [notdef] (B + eA)
1
0
=
[integraldisplay]
dc 4pc
[integraldisplay]
1
0 de
1
c12A [notdef] A + 12(eA + B) [notdef] (eA + B)
2 .
(6.4)
26
The key idea here is the introduction of the new Feynman parameter c in the last step, as done in [63]. Although it could be removed immediately, it will prove advantageous to leave it untouched until the nal stage. For example, this will allow us to postpone dealing with square roots until the very end.
Upon substituting (6.4) into (6.1), one notes that the variable e is charged under both GL(1) symmetries. Therefore, it is allowed to gauge-x one of them by setting e = 1, which e ectively locks the two GL(1) together. This will always be the case: the variable e is always removable in this way. Thus we have
I1,3;3,12tri = [integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d3a1a3b1b3]
vol(GL(1))
(1 [notdef] 3)2
(1 + c)12A [notdef] A + A [notdef] B + 12B [notdef] B
2 . (6.5)
As mentioned, this is similar to the formula for the double-box obtained in [62].
So far all we have done is rewrite the standard Feynman parameter integral in some specic form. As we will now see, in all cases the variables ai, bi can be integrated out rather straightforwardly, and will generate some logarithms or dilogarithms to be integrated over c. The c integration in the nal step then poses no particular di culty.
6.2 Divergent double-triangles
Let us see carry out the remainder of this procedure for I1,3;3,12tri starting from (6.5).
Notice that we havent said anything about the regularization yet. This is because everything is fully accounted for by the rules (5.6). According to it, we simply have to take (i [notdef] j) ! x2ij + 2[notdef]2IR.
After evaluating the dot products and doing a simple rescaling of the integration variables, the double-triangle is thus easily seen to depend only on the ratio [epsilon1] := [notdef]
2IR
x213 :
JHEP03(2013)075
I1,3;3,12tri
= [integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d3a1a3b1b3]
vol(GL(1))
1
(a1+b1)(a3+b3)+ca1a3+[epsilon1] ((a1+a3+b1+b3)2+c(a1+a3)2)
2 .
We are interested in the small mass limit [epsilon1] 1. The only sensitivity to [epsilon1] comes from
the two regions where a1, b1 [epsilon1] or a3, b3 [epsilon1]; these correspond physically to collinear
congurations. However, everywhere else we can ignore [epsilon1]. Consequently, let us parametrize the variables as
a1 = 1, b1 = x, a3 = a, b3 = ay, [d3a1a3b1b3]
vol(GL(1)) = adadxdysuch that the dangerous regions are a = 0 and a = 1. Since the region a > 1 contributes
the same as a < 1, by symmetry, we need only consider the former and multiply it by 2. Furthermore, in that region, we can neglect a in the terms proportional to [epsilon1] since they are only needed when a ! 0. Thus
I1,3;3,12tri = 2 [integraldisplay]
1
0
1
dc 4pc
[integraldisplay]
0 ada [integraldisplay]
1
0
dxdy(a((1 + x)(1 + y) + c) + [epsilon1]((1 + x)2 + c))2
1
0 dxdy
=
[integraldisplay]
dc 4pc
[integraldisplay]
log (1+x)(1+y)+c[epsilon1]((1+x)2+c) 1 (1 + x)(1 + y) + c
2 = log
4[notdef]2IR x213
+ O([notdef]IR). (6.6)
(This was most readily done by evaluating the integrals in the following order: y, x and c.)
27
The second type of double-triangle I1,3;3,52tri is entirely similar. The general formula (6.4) then gives directly, after a simple rescaling of the variables,
I1,3;3,52tri =[integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d3a1a3b3b5]
vol(GL(1))
[notdef]
1
(a1+b5)(a3+b3) + a1b5 + ca1a3 + [epsilon1][prime] ((a1+a3+b3+b5)2 + c(a1+a3)2)
2
with [epsilon1][prime] = [notdef]
2IRx215x213x235 . The dangerous region is the collinear region a1 ! 0 and b5 ! 0, so up to
power corrections in [epsilon1][prime] we can drop a1 and b5 in the terms multiplying [epsilon1][prime]. These can then be easily integrated out, leaving:
I1,3;3,52tri =
JHEP03(2013)075
[integraldisplay]
1
0
dc 4pc
[integraldisplay]
1
0 db3
log (1+b3)(1+b3+c)(1+b3)2+c log [epsilon1][prime] (1 + b3)(1 + b3 + c)
= 1
1
2 log
4[notdef]2IRx215 x213x235
+ O([notdef]IR). (6.7)
6.3 A dual conformal integral: Icritter
As our next example, we turn to the integral Icritter(1). This integral is collinear diver
gent, but it becomes absolutely convergent after combining it with I1,3;4,62tri as explained in section (4). Therefore, we will only consider the sum
[notdef]critter(1) := Icritter(1) + I1,3;4,62tri
= 4
[integraldisplay]
1
0
[d2a1a2a3]
vol(GL(1))
[d2b4b5b6]
vol(GL(1))
Za,b
[epsilon1](a, 1, 2, 3, )[epsilon1](b, 4, 5, 6, ) + (1 [notdef] 3)(4 [notdef] 6)(a [notdef] 2)(b [notdef] 5) (a [notdef] A)3(a [notdef] b)(b [notdef] B)3
.
(6.8)
Note that we have combined the two integrals into a common Feynman parameter integral, by inserting the inverse propagators (a.2)(b.5) into the numerator of the double-triangle. This allows us to immediately set the regulating masses to zero, since we are dealing with an absolutely convergent integral.
To apply the formula (6.4), we use the familiar fact that numerators turn into derivatives in Feynman parameter space. Thus for instance for the rst term in (6.8)
4[epsilon1](a, 1, 2, 3, )[epsilon1](b, 4, 5, 6, )
(a [notdef] A)3(a [notdef] b)(b [notdef] B)3
= [epsilon1](@A, 1, 2, 3, )[epsilon1](@B, 4, 5, 6, )
1(a [notdef] A)2(a [notdef] b)(b [notdef] B)2
.
Thus, after setting e = 1 in (6.4) to remove one of the GL(1) symmetries as done previously, we obtain
[notdef]critter(1) =
[integraldisplay]
1
0
1
0
dc 4pc
[integraldisplay]
[d5a1a2a3b4b5b6]
vol(GL(1))
[epsilon1](@A, 1, 2, 3, )[epsilon1](@B, 4, 5, 6, )
+(1 [notdef] 3)(4 [notdef] 6)(2 [notdef] @A)(5 [notdef] @B)
[parenrightbigg]
1
(c + 1)12A [notdef] A + A [notdef] B + 12B [notdef] B
2 . (6.9)
28
To proceed from here, we simply integrate over the variables ai, bi one at a time. This can be done in an essentially automated way using the method described in detail in a four-dimensional context in [63]. The idea is that at each stage the integral can be decomposed into a rational factor which takes the form dx/(x xi)n with n 1, times logarithms
or polylogarithms with arguments that are rational functions of x. Such integrals can be performed, at the level of the symbol, in a completely automated way. After this is done, we integrate the symbol and obtain the c-integrand as described in [63].
We applied this method, doing the integrals in the order a2, b5, a1, b6 and a3, to obtain the symbol of a function to be integrated over c. After a step of integration by parts in c to remove degree-three components, we obtained the symbol of a degree-two function, which could easily promoted to a function
[notdef]critter(1) = 2
[integraldisplay]
1
0
JHEP03(2013)075
dc 4pc
2
3 Li2 1u1(c+1)
Li2(1u2)Li2(1u3)log u2 log u3 c+1
=
1
2Li2(1 u2)
1
2Li2(1 u3)
1
2 log u2 log u3 (arccos pu1)2 +
2
3 . (6.10)
Here all non-constant terms come out of the symbol computation, while the 2/3 term
is a beyond-the-symbol ambiguity. We have xed it by an analytic computation at the symmetrical point u1 = u2 = u3 = 1, where the integral simplies dramatically. Assuming the principle of maximal transcendentality for this integral, this is the only possible ambiguity. As a cross-check, we have veried that this result agrees with a direct numerical evaluation of eq. (6.9), to 6 digit numerical accuracy at several random kinematical points with Euclidean kinematics, which we take to conrm our assumptions.11
6.4 Another divergent integral: I2mheven
We now consider a somewhat more nontrivial divergent integral,
I2mheven(1) =
[integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d5a1a2a3b3b5b1]
vol(GL(1))
([epsilon1](@A, 1, 2, 3, )[epsilon1](@B, 3, 5, 1, )) (c + 1)12A [notdef] A + A [notdef] B + 12B [notdef] B
2 (6.11)
where the derivative operators are understood to act on the rational function underneath, to avoid an unnecessary lengthening of the formula.
This integral requires regularization, and as in the rest of this section we use the Higgs regularization described in section (5). The procedure has the following precise meaning here. In both the numerator and denominator, we use the shifted ve-vectors dened in eq. (5.6), so the formula amounts to (i [notdef] j) ! (xi xj)2 + 2[notdef]2IR.
A rst observation is that in all divergent regions b5 ! 0. Thus we can drop b5
from terms multiplying the mass in the denominator, which allows us to integrate out b5
11Numerics with this level of accuracy can be easily obtained starting directly from (6.9) and performing the a2, b5 and c integrals analytically, which are readily done using computer algebra software such as Mathematica. The remaining 3-fold numerical integration poses no particular problem.
29
explicitly:12
I2mheven(1)
= [integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d4a1a2a3b1b3]
vol(GL(1))
a2(2 [notdef] 5) 2((A + B) [notdef] 5) (2 [notdef] 5)/(1 [notdef] 3)/((A + B) [notdef] 5)2 (1 + c)a1a3 + a1b3 + a3b1 + b1b3 +
[notdef]2IR (1[notdef]3) X
2
where X := (
Pa +
Pb)2 + c( Pa)2.
To proceed further, we need a small bit of physical intuition about this integral. It has collinear divergences in the region a3, b3 ! 0 (both loop momenta collinear to p1)
and in the region a1, b1 ! 0 (both loop momenta collinear to p2). In addition, there are
soft-collinear divergences where these two regions meet. Thus a reasonable strategy is to subtract something which has the same divergent behavior as [notdef]2IR ! 0 but which is simpler
to integrate. A good candidate is
I2mheven[prime](1) := I2mheven(1) X ! a22(1 + c) [parenrightbig]
(6.12)
since this remains nite and has identical soft and soft-collinear regions. But thanks to the simplied denominator, this can be integrated more easily. Indeed after a shift a1+b1 ! b1, a3 + b3 ! b3 together with a simple rescaling of the variables, it can be seen to depend
only on a single parameter [epsilon1] := 4[notdef]
2IRx215x235 x213x435 :
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I2mheven[prime](1) = [integraldisplay]
1
0
dc 4pc
Za1<b1a3<b3 [d4a1a2a3b1b3] vol(GL(1))
a2 + 2b1 + 2b3(a2 + b1 + b3)2(b1b3 + a1a3c + [epsilon1]a22(1 + c))2
= 2
14 log2 [epsilon1] + O([epsilon1]). (6.13)
From this point we omit further details on the computation of integrals, as they proceed using the same strategy as in previous examples. It remains to correct for the error introduced by eq. (6.12) in the hard collinear regions. At xed a2, a1, b1 1 one can see that the region a3, b3 ! 0 produces a logarithm whose cuto depends on X. The error is given
by the change in the logarithmic cuto
Icoll(y) :=
[integraldisplay]
1
0
72
12
dc 4pc
Za1<b1[d2a1a2b1]vol(GL(1))
y(a2y + 2b1) log (a2+b1) 2+c(a1+a2)2 a22(1+c) b1(b1 + a1c)(a2y + b1)2
= 2
6 Li2(1 y) (6.14)
so that
I2mheven(1) = I2mheven[prime](1) + Icoll(x225/x215) + Icoll(x225/x235). (6.15)
This gives the result quoted in appendix (C).
12Strictly speaking the numerator derived from eq. (6.11) contains terms proportional to [notdef]2IR. However, due to the special properties of the [epsilon1]-symbol numerators, one can see that these terms only give rise to power-suppressed contributions. That is, they are never accompanied by compensating 1/[notdef]2IR power infrared divergences which would render them relevant. We have veried that the same is true also for the integral
Icrab considered below.
30
6.5 The integral Icrab
The nal divergent integral we have to compute is
Icrab(1) =
[integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d5a1a2a3b5b6b1]
vol(GL(1))
([epsilon1](@A, 1, 2, 3, )[epsilon1](@B, 5, 6, 1, )) (c + 1)12A [notdef] A + A [notdef] B + 12B [notdef] B
2 . (6.16)
Its evaluation is extremely similar to that in the previous subsection. The regions which diverge as [notdef]2IR ! 0 are the p6-collinear and p1-collinear regions, and their intersection, the
soft-collinear region A, B ! y1. Therefore, if we denote the [notdef]2IR-containing terms in the
denominator by [notdef]2IRX, we see that we can neglect a3 and b5 in X:
X = (a1 + a2 + b6 + b1)2 + c(a1 + a2)2.
Then we proceed as in the previous example: we replace the integral by the simpler one
Icrab[prime](1) := Icrab(1) X ! (a1 + b1)2 + ca21
, (6.17)
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which we have been able to evaluate as (setting [epsilon1]1 := 4[notdef]
2IRx235 x213x215 )
1
2Li2(1 1/u3).
The error introduced by (6.17) is by construction localized to the hard collinear regions and turns out to be given by the same eq. (6.14): Icrab(1) = Icrab[prime](1) + Icoll(x215/x225) +
Icoll(x213/x236). Collecting the terms gives the result recorded in appendix (C).
6.6 Parity odd box-triangles
As shown in section (4), parity odd box-triangles appear in the six-point amplitude only in absolutely-convergent combinations of the form
Ioddbox;tri(1) := (1 [notdef] 4)(3 [notdef] 6) [bracketleftBigg]
Icrab[prime](1) = 1
2
12 +
1
2(1 + log u3) log [epsilon1]1
14 log2 [epsilon1]1 +
(2 [notdef] 4)I1,2,3;4,6box;tri[[epsilon1](a, 1, 2, 3, 6)][epsilon1](1, 2, 3, 4, 6) +
(3 [notdef] 5)I4,5,6;1,3box;tri[[epsilon1](a, 4, 5, 6, 1)] [epsilon1](3, 4, 5, 6, 1)
[bracketrightBigg]
.
(6.18)
The Feynman parametrization formula (6.4) reads in this case
Ioddbox;tri(1) =[integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d5a1a2a3b4b5b6]
vol(GL(1))
[epsilon1](@A, 1, 2, 3, 6)
[epsilon1](1, 2, 3, 4, 6) (2 [notdef] 4)(5 [notdef] @B)
+ (2 [notdef] @A)(3 [notdef] 5)
[epsilon1](@B, 4, 5, 6, 1)
[epsilon1](4, 5, 6, 1, 3)
[parenrightbigg]
(1 [notdef] 4)(3 [notdef] 6)
(c + 1)12A2 + A [notdef] B + 12B2
2 . where we have combined the two integrals under a common Feynman parameter integral sign. This may now be evaluated directly without regularization. One can see that the integrations over a2, b5 do not produce any transcendental functions, which suggests to do them rst. In the process, the [epsilon1]-symbols neatly cancel out:
Ioddbox;tri(1) =
[integraldisplay]
1
0
dc 4pc
[integraldisplay]
[d3a1a3b4b6]
vol(GL(1))
[parenleftbigg]
a3(3 [notdef] 5)
a1(1 [notdef] 5) + a3(3 [notdef] 5)
b4(2 [notdef] 4)
b4(2 [notdef] 4) + b6(2 [notdef] 6)
[parenrightbigg]
(1 [notdef] 4)(3 [notdef] 6)
[notdef] (c + 1)a1a3(1 [notdef] 3) + a1b4(1 [notdef] 4) + a3b6(3 [notdef] 6) + b4b6(4 [notdef] 6)
2 .
31
The three integrations over ai, bi remain elementary, and we obtain a pleasingly simple result
Ioddbox;tri(1) = [integraldisplay]
1
0
dc 4pc
log u2u3 log(u1(c+1)) 1 u1(c+1)
= log u3u2 arccos(pu
1)
2 pu1(1 u1)
. (6.19)
7 The six-point two-loop amplitude amplitude of ABJM
We now construct the nal integrated result. We rst consider the parity even part,i.e. terms in eq. (4.16) proportional to Atree
2 . We begin by summing all the divergent
integrals, or more specically,
P6i=1(I2mheven(i) + Icrab(i) + Ii,i+2;i+2,i2tri Ii,i+2;i+2,i22tri), using the formulas recorded in appendix (C). Pleasingly, we nd that all terms of non-uniform transcendentality have canceled in the sum! The nal uniform transcendental result is given by:
1
2
"Li2(1ui)+log ui logx2i1,i+2 [notdef]2IR[bracketrightBigg]
:= BDS6 2. (7.1)
As we see the above result is nothing but the BDS Ansatz [40, 41] in the Higgs regulator!. That is, up to the constant term and the substitution [notdef]2IR ! 4[notdef]2IR! The evaluation of the parity even terms will be complete upon adding the dual conformal integrals
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6 "log2
x2i,i+3 x2i+1,i+3
!log2[parenleftBigg]
x2i,i+3[notdef]2IR x2i,i+2x2i+1,i+3[parenrightBigg][bracketrightBigg]+3
Xi=1
P3i=1(Icritter(i) + Ii,i+2;i+3,i12tri).
As a cross-check on our evaluation of the integrals, we have evaluated the above com
bination of integrals using dimensional regularization, which has been successfully implemented in obtaining the two-loop four-point result [23, 24]. While we nd that the results for individual integrals di er functionally, we nd perfect agreement for the combination just considered, up to an expected scheme-dependent constant. This constant is given in (D.3).
We next consider the parity odd part, i.e. terms in eq. (4.16) proportional to the sum of one-loop maximal cuts. The I2mhodd integrals integrate to zero at order O([epsilon1]), thus we have:
C1
+ C 1 2p2
I4,5,6;1,3box;tri[[epsilon1](a, 4, 5, 6, 1)]
(1 [notdef] 5)
+ C2 + C 2
2p2
I1,2,3;4,6box;tri[[epsilon1](a, 1, 2, 3, 4)]
(2 [notdef] 4)
+ cyclic [notdef] 2.
Using the identity (A.1) together with the denition (6.18), this can be expressed in terms of the dual conformal invariant nite integral
C1
+ C 1 2p2 [bracketleftbigg]
[epsilon1](3, 4, 5, 6, 1)(1 [notdef] 4)(3 [notdef] 6)(1 [notdef] 5)(3 [notdef] 5)
Ioddbox;tri(1) + cyclic [notdef] 2[bracketrightbigg]
= C1 + C 1
2p2 [bracketleftBigg]
[epsilon1](3, 4, 5, 6, 1)(1 [notdef] 4)(3 [notdef] 6)(1 [notdef] 5)(3 [notdef] 5)
log u2u3 arccos(pu1)
2 pu1(1 u1)+ cyclic [notdef] 2
[bracketrightBigg]
. (7.2)
Due to Yangian invariance, the coe cients of the transcendental functions must be expressible in terms of the leading singularities LS1 and LS 1 dened in section (3.1). This
32
can be veried thanks to the remarkable identity (A.5), together with (3.10):
(C1 + C 1)[epsilon1](3, 4, 5, 6, 1)
2p2(1 [notdef] 4)(3 [notdef] 6)(1 [notdef] 5)(3 [notdef] 5) pu1(1 u1) = Atree6,shiftedsgnc[angbracketleft]12[angbracketright]sgnc[angbracketleft]45[angbracketright][parenleftBigg][parenleftBigg][angbracketleft]34[angbracketright][angbracketleft]46[angbracketright]
+ [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]
[radicalBig][parenleftBigg]
h34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]
2 .
Thus combining everything, we nd the two-loop six-point amplitude of ABJM to be
A2-loop6 = [parenleftbigg]
N k
BDS6 + R6[bracketrightbigg]+ Atree6,shifted2 [notdef]
sgnc([angbracketleft]12[angbracketright])sgnc([angbracketleft]45[angbracketright]) [angbracketleft]
34[angbracketright][angbracketleft]46[angbracketright]+[angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]
2 Atree6 2
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u2u3 arccos(pu
1) + cyclic [notdef] 2
[radicalBig][parenleftBigg]
h34[angbracketright][angbracketleft]46[angbracketright]+[angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]
2 log
[bracketrightbigg](7.3)
where the remainder function R6 is given as
R6 = 22 +
3
Xi=1
Li2(1 ui) +12 log ui log ui+1 + (arccos pui)2[bracketrightbigg]
.
We like to stress that the remainder function, up to an additive constant, is given entirely by the dual-conformal nite integral[notdef]critter(i) for the parity even structure (and by Ioddbox;tri(i)
for the parity-odd structure). This is in contrast with N = 4 SYM, where the remainder
function is mixed with BDS and spread across a number of divergent integrals.
The part proportional to Atree6,shifted can be written in a more compact way if we assume
certain restrictions on the kinematics. We will assume so-called Euclidean kinematics, e.g. all non-vanishing invariants (i [notdef] j) are spacelike. (This is a nonempty region even for real
Minkowski momenta.) In that case, it is correct to naively rewrite the original expression in terms of angle-brackets:
arccos(pu1)
pu1(1 u1):= 1 2i
log
pu1+ip1u1 pu1ip1u1 [parenrightBig]
pu1(1 u1) !
(1 [notdef] 4)(3 [notdef] 6) log [vector]1
2i[angbracketleft]12[angbracketright][angbracketleft]45[angbracketright] [angbracketleft]34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]46[angbracketright]
[parenrightbig]
where
[vector]1 := [angbracketleft]12[angbracketright][angbracketleft]45[angbracketright] + i([angbracketleft]34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]) h12[angbracketright][angbracketleft]45[angbracketright] i([angbracketleft]34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright])
. (7.4)
Indeed, in that region, u1 > 0 and the rst expression is always real and positive. This is also the case for the second expression, as can be seen from the fact that [angbracketleft]12[angbracketright] and [angbracketleft]45[angbracketright] are
real, while ([angbracketleft]34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]) is either real or smaller in magnitude than [angbracketleft]12[angbracketright][angbracketleft]45[angbracketright]. Note
that, dening cross-ratios [vector]2 ([vector]3) from cyclic shifts by minus 2 (plus 2) of this expression, it can be shown that
[vector]1[vector]2[vector]3 = 1 (7.5)
Finally we would like to note that the function [vector]i also appears in the one-loop six-point bosonic Wilson-loop as eq. (5.4) in [59]. The vanishing of the one-loop Wilson-loop is simply due to the above identity.
33
This allows us, in these kinematics, to simplify the answer to
A2-loop6 = [parenleftbigg]
N k
BDS6 + R6[bracketrightbigg]+ Atree6,shifted4i
[bracketleftbigg]log u2u3 log [vector]1 + cyclic [notdef] 2[bracketrightbigg]
. (7.6)
While strictly derived from eq. (7.3) in Euclidean kinematics, we expect this expression to be valid in other kinematic regions for a suitable analytic continuation of the variables [vector]i. Note that as [vector]i ! [vector]1i under the Z2 little group transformation of any external leg,
the little group weight of log [vector]1 is exactly what is needed to compensate the little group mismatch of Atree6,shifted.
8 Conclusions
In this paper, we construct the two-loop six-point amplitude of ABJM theory. The result can be separated into a two-loop correction proportional to the tree amplitude, and a correction proportional to the shifted tree amplitude, which are distinct Yangian invariants. The correction proportional to the rst is infrared divergent and we use mass regularization. The result shows that the infrared divergence is identical to that of N = 4 super Yang-
Mills and is thus completely captured by the BDS result. This establishes that the dual conformal anomaly equation is identical between the one-loop SYM4 and two-loop ABJM, which was rst observed at four-points and we conjecture will persists to all points. The correction multiplying the shifted tree amplitude is completely nite.
As a comparison, we also computed the divergent integrals using dimensional reduction. We nd that the individual integrals give di erent functional answer between the two regularization schemes. However, when combined into the amplitude, they give the same result up to a physically expected constant.
We nd in addition to the BDS result a nonzero (dual-conformal invariant) remainder function. This implies that the six-point ABJM amplitude cannot be dual to a bosonic Wilson-loop, which only captures the BDS part [61].13 This does not rule out a possible duality with a suitable supersymmetric Wilson loop, however. The reason is that, if SYM4 is to be of any guidance [6466], the correct Wilson loop dual for amplitudes with n 6
particles should reproduce, at lowest order in the coupling, the n-point tree amplitude.14
Since no candidate Wilson loop with this property, or even just the correct quantum numbers, are presently available in the literature, we nd it hard to say anything conclusive about the duality. Our results demonstrate that the dual conformal symmetry persists at the quantum level up to an anomaly which is identical to that of a Wilson loop. We interpret this as strong evidence for the existence of a dual Wilson loop which remains to be constructed.
We list a number of open questions for future work. A rst one concerns the status of the dual conformal symmetry away from the origin of moduli space, e.g. in the Higgsed theory. As demonstrated in section (5), to lowest order in the masses (logarithmic accuracy), the Higgsed theory enjoys an exact dual conformal symmetry under which the
13Note that the 1-loop Wilson loops has been shown to vanish numerically [60] and analytically [59].
14At least up to 3(P ) 6(Q) and a purely bosonic factor, akin to the Parke-Taylor denominator in SYM4.
34
2 Atree6 2
JHEP03(2013)075
masses transform in a nontrivial way. It is not clear whether this symmetry extends all the way into the moduli space; for one thing, the origin of the symmetry is mysterious and the original string theory argument in [39] does not apply in ABJM due to di culties with the T-duality. As discussed in the main text, a key step here would be to settle this question for the tree amplitudes.
We note that a 3-loop computation of the 4-point and 6-point amplitude in ABJM would probably be feasible with the same techniques, although a more sustained e ort would be required. For instance, we expect only degree-3 transcendental functions in the result. Furthermore, the only divergences should be double-logarithms multiplying the 1-loop amplitude. Given the absence of overlapping divergences, the integration technology developed in section (6) might thus plausibly be su cient.
An interesting property of our remainder function R6 is that it does not vanish in collinear limits, contrary to the case in SYM4. In fact, it even diverges logarithmically in the simple collinear limit (six point goes to ve), this even though the ve-point amplitude is zero. This does not violate any physical principle, since the Atree6 and Atree6,shifted prefactors do not have any pole in this limit. In the absence of a pole, there is no need for the amplitude to factorize into a product of lower-point amplitudes. In other words, the leading term in the collinear limit in ABJM is similar to subleading, power-suppressed terms in the collinear limits in D = 4. The factorization theory for these terms is more complicated, and in fact it has only been worked out recently in the dual Wilson loop language [67]. It would be very interesting to work out the general structure of this limit, using eld theory arguments, as this should place strong constraints on the amplitudes. In subsection (3.3), for instance, we have conjectured from analyticity of the scattering amplitudes that a certain discontinuity of the amplitude should vanish in the collinear limit, but we have no idea how this could be established.
Also interesting are the double-collinear limit (six point goes to four), or factorization limits (p2123 goes to zero, but momenta p1,2,3 do not become collinear). Since it was not clear to the authors what kind of eld theory predictions are available for these limits, we did not discussed them on our 6-point result. However, it is possible that this could shed further light on our result itself, for instance by giving a physical interpretation for the relative signs between di erent terms. These limits may also yield some interesting constraints on the higher-point amplitudes.
Another interesting direction for future work concerns the rest of the Yangian algebra at loop level. As one easily sees from [3], the Yangian algebra in ABJM is generated by the bosonic dual conformal symmetry together with the (ordinary) superconformal symmetry. Since the former is presently conjectured to become anomaly-free to all loops after dividing by the BDS Ansatz, the crux is the superconformal anomaly. By analogy with SYM4, the properly understood symmetry at the quantum level should uniquely determine the amplitudes, providing for an e cient way to compute them. Our two-loop result (7.6) should thus provide an important data point to understand the quantum symmetries of ABJM, perhaps combining the 1-loop ABJM analysis in [21] with the all-loop SYM4 analysis in [68].
35
JHEP03(2013)075
Acknowledgments
We would like to thank Johannes Henn for many enlightening discussions. Y-t would like to thank N. Arkani-Hamed for invitation as visiting member at the Institute for Advanced Study at Princeton, where this work was initiated. SCH gratefully acknowledges support from the Marvin L. Goldberger Membership and from the National Science Foundation under grant PHY-0969448. This work was supported in part by the US Department of Energy under contract DEFG03 91ER40662.
A Identities
In this appendix, we aim to prove a series of identities used in the text. First consider the following identity:
C1 + C 1 C2 + C 2
=
JHEP03(2013)075
[epsilon1](6, 1, 2, 3, 4)(3 [notdef] 5)(5 [notdef] 1)
[epsilon1](3, 4, 5, 6, 1)(6 [notdef] 2)(2 [notdef] 4)
. (A.1)
The strategy is to express the ve-dimensional [epsilon1] symbol in terms of angle brackets. To do so, we start from the denition of the [epsilon1]-symbol as a determinant and use the manifest translation invariance of the formula to set x2 = 0. In doing so, we must remember to normalize the determinant such that [epsilon1](i, j, k, l, m)2 agrees with the Gram determinant formula (3.16), since this is the convention used in the main text; this requires an extra factor of 2ip2. Thus
[epsilon1](6, 1, 2, 3, 4) := 2ip2 det(y6, y1, y2, y3, y4) = 2ip2 det 0
B
@
[vector]p6[vector]p1 [vector]p1 0 [vector]p2 [vector]p2+[vector]p31 1 1 1 1
[angbracketleft]61[angbracketright]2 0 0 0 [angbracketleft]23[angbracketright]2
1
C
A
.
where the rst three rows are real in Minkowski signature. This determinant can now be evaluated in terms of three-dimensional ones, which in turn give two-brackets: det([vector]pi, [vector]pj, [vector]pk) := 12[angbracketleft]ij[angbracketright][angbracketleft]jk[angbracketright][angbracketleft]ik[angbracketright]. This way we obtain
[epsilon1](6, 1, 2, 3, 4) = ip2[angbracketleft]61[angbracketright][angbracketleft]12[angbracketright][angbracketleft]23[angbracketright] [angbracketleft]31[angbracketright][angbracketleft]16[angbracketright] + [angbracketleft]32[angbracketright][angbracketleft]26[angbracketright]
. (A.2)
Performing a similar computation for [epsilon1](3, 4, 5, 6, 1) and using that (3 [notdef] 5) = [angbracketleft]34[angbracketright]2 etc., we
thus nd[epsilon1](6, 1, 2, 3, 4)(3 [notdef] 5)(5 [notdef] 1)
[epsilon1](3, 4, 5, 6, 1)(6 [notdef] 2)(2 [notdef] 4)
= [angbracketleft]12[angbracketright][angbracketleft]34[angbracketright][angbracketleft]56[angbracketright] h23[angbracketright][angbracketleft]45[angbracketright][angbracketleft]61[angbracketright]
[angbracketleft]31[angbracketright][angbracketleft]16[angbracketright] + [angbracketleft]32[angbracketright][angbracketleft]26[angbracketright] h34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]
. (A.3)
Using momentum conservation, the parenthesis can be shown to equal 1, proving the
desired formula using (3.10).
Another remarkable algebraic identity is
x214x236 x213x246 = ([angbracketleft]34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright])2 (A.4) which one might call a Dirac matrix trace identity, and follows from squaring eq. (A.2) and using that the square should give the Gram determinant. Using this identity we have that
h12[angbracketright][angbracketleft]34[angbracketright][angbracketleft]56[angbracketright][epsilon1](3, 4, 5, 6, 1)
(1 [notdef] 4)(3 [notdef] 6)(1 [notdef] 5)(3 [notdef] 5) pu1(1 u1)= ip2[angbracketleft]12[angbracketright][angbracketleft]45[angbracketright] [angbracketleft]34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]
2 (A.5)
q[angbracketleft]12[angbracketright]2[angbracketleft]45[angbracketright]2 [angbracketleft]34[angbracketright][angbracketleft]46[angbracketright] + [angbracketleft]35[angbracketright][angbracketleft]56[angbracketright]
which was used around eq. (7.2).
36
B ABJM theory on the Higgs branch
The action of ABJM takes the form (see for instance [69] for an explicit component form):
L =
k4 Lkin + L4 + L6
. (B.1)
To describe the spectrum of the theory on the Higgs branch, we begin by describing the fermion mass matrix. The interactions of the fermions can be written, following [69] but as can also be veried directly by comparing against various components of the four-point amplitude (2.7),
L4 = Tr[ A BC
D] Tr[ A
DC B]
[parenrightbig]
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(2 AC DB AB DC) +[epsilon1]ABCDTr[A BC D] + [epsilon1]ABCDTr[ A
B C
D]. (B.2)
As already mentioned in the main text, the moduli space (C4/Zk)N of this theory is characterized by diagonal vacuum expectation values for the scalar elds. Let us denote the elds above (below) the diagonal with a plus (minus) superscript, so that ( [notdef]A) =
A .
As one can easily see from the action (B.2), the diagonal fermions remain massless while
+B and
B+ mix with each other. Upon inserting a diagonal vev for the scalars, the mass term thus takes the form ( A, A)Mf( +B,
B+)T where Mf is the 8[notdef]8 Hermitian matrix
Mf = 2(xx yy)BA BA(x[notdef]x y[notdef]y) 2[epsilon1]ABCDxCyD
2[epsilon1]ABCD yC xD 2(yy xx)AB AB(y[notdef]y x[notdef]x) [parenrightBigg]
.
In this appendix, (x, y)A := (vi, vj)A will denote the diagonal vevs coupled to o -diagonal components under consideration. As one can verify M2f = 18m2 with
m2 = (x[notdef]x + y[notdef]y)2 4x[notdef]yy[notdef]x,
showing that all 8 o -diagonal fermions acquire the same mass.
The scalar potential was described in detail in ref. [1],
L6 = Tr[A
[AB
C]C
B]
1 3Tr[A
[AB
BC
C]]
where here the square bracket means antisymmetrization in the indices. Again one can see that the diagonal uctuations remain massless while o -diagonal ones A+ and
+A mix
with each other. It follows that the mass term takes the form (
A, A)M2s( B+,
+B)T ,
and a computation gives the 8 [notdef] 8 Hermitian matrix as
M2s = m2 (18 P8) (B.3)
where P8 = xT yT
yT xT [parenrightBigg] [notdef][parenleftBigg][parenleftBigg]
x[notdef]x + y[notdef]y 2x[notdef]y
2y[notdef]xy x[notdef]x + y[notdef]y
![notdef]
x y y x
!/m2 is an orthogonal pro-
jector onto the two would-be Goldstone bosons ( +,
+) (x, y) and ( +,
+)
(y, x). We conclude that six of the eight scalars acquire the same mass squared as the
37
fermions, while the remaining two acquire no mass, although they are soon to be eaten by the gauge elds through the Higgs mechanism.
Finally, we consider the gauge elds, which acquire mass terms through the scalar kinetic term TrD[notdef]D[notdef]
. First we discuss the o -diagonal components. As one can see again the elds A+1,2 from the two gauge groups mix with each other, so the mass term is characterized by a 2[notdef]2 Hermitian matrix (A1, A2)Mg(A+1, A+2). However, in this case the
kinetic term is also characterized by a nontrivial matrix d(A1, A2) ^ Kg(A+1, A+2). These two matrices are
Kg = 1 0 0 1
Fortunately, to obtain the propagator it is not necessary to diagonalize these two matrices simultaneously as pointed out in [70, 71] this may not even be possible in general. In the present case, one can verify that (KgMg)2 = m212 and this su ces in order to write down the propagator in a simple way. To see this, let us rst add a gauge-xing term to the action (@[notdef]A[notdef] v[notdef]( ))Kg(@[notdef]A[notdef]+ v[notdef]( +))/, designed to remove the mixing
between the gauge bosons and the scalar elds, where is some arbitrary scale. Then the two unphysical scalars acquire masses squared m, and a short computation gives the
gluon propagator as
h(A[notdef]1, A[notdef]2)(p) A +1
A +2
38
and Mg = x[notdef]x + y[notdef]y 2x[notdef]y
2y[notdef]x x[notdef]x + y[notdef]y
!.
JHEP03(2013)075
![angbracketright] /
[epsilon1][notdef] pKg + [notdef] KgMgKg
p2 + (KgMg)2 + p[notdef]p
Kg + KgMgKg 2p2 p2+(KgMg)2 p4 + 2(KgMg)2 .
In particular, this formula shows that there are no singularities at zero momentum provided m2 [negationslash]= 0, as required in the main text.
Finally, we discuss the diagonal gauge elds. Since only the combination (A1 A2) receives a mass term in this case, we have that Mg / (1, 1) (1, 1)T which is e ectively
nilpotent: (KgMg)2 = 0. As the above propagator shows, even though the mass matrix is nonzero, no massive states appear in the spectrum (as required by supersymmetry). This situation has been discussed in detail in [70, 71].15
C Integrals using the mass regularization
Here we summarize the results obtained in section (6) for the integrals dened in eqs. (4.5), multiplied by 162, evaluated using a small internal mass to regulate infrared divergences
15Note that in refs. [70, 71] it was further shown that the eld (A1 A2) can be integrated out in a
systematic expansion in 1/m, yielding a Yang-Mills term kF 2/m for the remaining gauge eld plus other
terms. But since for us m |vi|2 is an infrared scale, not an ultraviolet scale, such a (in any case not
strictly necessary) procedure would be inappropriate in our context.
as dened in section (5).
Icrab(1) = 1 +
2
4 +
1
2(1 + log u3) log
4[notdef]2IRx235 x213x215
14 log2
4[notdef]2IRx235 x213x215
Li2(1 x213/x236) Li2(1 x215/x225) +
1
2Li2(1 1/u3),
I2mheven(1) = 2
2
4
14 log2
4[notdef]2IRx215x235 x213x425
Li2(1 x225/x215) Li2(1 x225/x235),
I1,3;3,12tri = log
4[notdef]2IR x213
,
I1,3;3,52tri = 1
1
2 log
4[notdef]2IRx215 x213x235
,
I2mhodd(1) = 0. (C.1)
In addition, we have the following two absolutely-convergent integrals:
Icritter(1) + I1,3;4,62tri =
1
2Li2(1 u2)
1
2Li2(1 u3)
1
2 log u2 log u3 (arccos(pu1))2 +
2 3
(C.2)
JHEP03(2013)075
and (see eq. (6.18) for the denition)
Ioddbox;tri(1) =
log u3u2 arccos(pu1)
2 pu1(1 u1)
. (C.3)
For completeness, we nd for the four-point double-box using the same regularization:
I1,2,3;3,4,12box[[epsilon1](a, 1, 2, 3, )[epsilon1](b, 3, 4, 1, )] =
2
3 + log
4[notdef]2IR x213
log 4[notdef]2IR x213
log 4[notdef]2IR x224
. (C.4)
D Integrals using dimensional regularization
In this appendix, we present the integrated result of infrared divergent integrals using dimensional regularization. Here all integrals are again multiplied by 162. After obtaining the integrals in terms of Feynman parameters, we integrate by converting the integrand into Mellin-Barnes representation, and implement the Mathemtica package MB.m [72] to obtain the result up to O([epsilon1]). The result is expressed in terms of zero-, one- and two-dimensional
integrals in Mellin space. The one and two-dimensional integrals are analytically evaluated by performing sum over residues. That such sum can be carried out analytically, is simply due to the fact that the two-loop amplitude should be of transcendental two functions.
39
The two mass hard integral gives:
I2mheven(1) = [integraldisplay]
[epsilon1](a, 1, 2, 3, )[epsilon1](b, 3, 5, 1 )
(a [notdef] 1)(a [notdef] 2)(a [notdef] 3)(a [notdef] b)(b [notdef] 3)(b [notdef] 5)(b [notdef] 1)
=
e 2[epsilon1]
(4)2[epsilon1]
(2)2[epsilon1](3(x213)2[epsilon1] (x215)2[epsilon1] (x235)2[epsilon1] + 2(x225)2[epsilon1]) 16[epsilon1]2
(2)2[epsilon1]
16[epsilon1]2
(x213)(x215)(x235) (x225)2
2[epsilon1]
[bracketrightbigg]
1 8
[bracketleftbigg]
log2(x213/x215) log2(x213/x235) + 2 log2(x213/x225)
+2 log2(x215/x225) 2 log2(x235/x225) + 3 log2(x215/x235)[bracketrightbigg]
arcsin(
qx215/x225) + arcsin2( qx215/x225) arcsin([radicalBig]
x235/x225) + arcsin2(
qx235/x225)
+12Li2(1 x215/x225) +
1
2Li2(1 x235/x225)
112
24
log2 2
4 + 2 + 22 a (D.1) where a 0.3594267177020808. The crab integral gives:
Icrab4(1) = [integraldisplay]
[epsilon1](a, 1, 2, 3, )[epsilon1](b, 5, 6, 1, )
(a [notdef] 1)(a [notdef] 2)(a [notdef] 3)(a [notdef] b)(b [notdef] 5)(b [notdef] 6)(b [notdef] 1)
=
JHEP03(2013)075
e 2[epsilon1]
(4)2[epsilon1]
(x213)2[epsilon1] + (x215)2[epsilon1] + (x226)2[epsilon1] (x225)2[epsilon1] (x236)2[epsilon1] 8[epsilon1]2
+(x235)2[epsilon1] (x213)2[epsilon1] (x215)2[epsilon1]
4[epsilon1]
[bracketrightbigg]
+14[bracketleftbigg]
log2(x213/x215) log2(x213/x225) log2(x215/x236) log2(x215/x225) log2(x213/x236)[bracketrightbigg] +14[bracketleftbigg]
log2(x226/x213) + log2(x226/x215) + log2(x225/x235) + log2(x236/x235) log2(x235/x226)[bracketrightbigg] + arcsin( qx215/x225) arcsin2([radicalBig]
x215/x225) + arcsin( qx213/x236) arcsin2([radicalBig] x213/x236)
1
2
12 log2 (u3) + Li2(1 x215/x225) + Li2(1 x213/x236) + Li2 (1 u3) [bracketrightbigg]
248 1 2
(D.2)
Notice the presence of arcsin functions with non-conformal cross-ratios as arguments. Such function did not appear in the mass regulated result and marks a stark distinction between the two regularizations. The ArcSin functions always come in the combination
arcsin(pm) arcsin2(pm)/ .
This particular combination is necessary for the integral to remain real.
For completeness, we list the double triangle result:
I1,3;3,12tri =
[integraldisplay]
(1 [notdef] 3)2
(a [notdef] 1)(a [notdef] 3)(a [notdef] b)(b [notdef] 1)(b [notdef] 3)
=
e x213 4
2[epsilon1] 12[epsilon1] + 1 ,
I1,3;5,12tri =
[integraldisplay]
(1 [notdef] 3)(1 [notdef] 5)
(a [notdef] 1)(a [notdef] 3)(a [notdef] b)(b [notdef] 5)(b [notdef] 1)
=
e x213x215 4 x235
2[epsilon1] 14[epsilon1] 1 2 .
40
While the integrated results appears to be regularization scheme dependent (compare with eqs. (C.1)), when combined into amplitudes they give identical result up to additive constants. In particular, the arcsin functions completely cancel. Considering the sum of infrared divergent integrals in eq. (7.1) one obtains:
6
Xi=1I2mheven(i) + Icrab(i) + Ii,i+2;i+2,i2tri Ii,i+2;i+4,i2tri
=
[bracketleftbigg]
6
Xi=1
[parenleftbigg]
e 2[epsilon1]
(8)2[epsilon1]
(x2i,i+2)2[epsilon1]
(2[epsilon1])2 log
x2ii+2 x2ii+3
log x2i+1i+3 x2ii+3
+ 14 log2
xii+3
xi+1i+4
JHEP03(2013)075
1
2Li2
1
x2ii+4x2i+1i+3 x2ii+3x2i+1i+4[parenrightbigg] [parenrightbigg]
2
238 12 a
[parenrightbigg] [bracketrightbigg]
= BDS6([epsilon1] ! 2[epsilon1]) 2 [parenleftbigg]
31
8 12 a
[parenrightbigg]
(D.3)
where BDS6 is the one-loop six-point MHV amplitude of N = 4 sYM [41] with [epsilon1] replaced by
2[epsilon1], reecting the two-loop nature of the result. Thus the dimensionally regulated infrared divergent integrals combine to give the BDS answer, just as in the mass regulated result.
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SISSA, Trieste, Italy 2013
Abstract
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)
In this paper we present the first analytic computation of the six-point two-loop amplitude of ABJM theory. We show that the two-loop amplitude consists of corrections proportional to two distinct local Yangian invariants which can be identified as the tree- and the one-loop amplitude respectively. The two-loop correction proportional to the tree-amplitude is identical to the one-loop BDS result of ... SYM plus an additional remainder function, while the correction proportional to the one-loop amplitude is finite. Both the remainder and the finite correction are dual conformal invariant, which implies that the two-loop dual conformal anomaly equation for ABJM is again identical to that of one-loop ... super Yang-Mills, as was first observed at four-point. We discuss the theory on the Higgs branch, showing that its amplitudes are infrared finite, but equal, in the small mass limit, to those obtained in dimensional regularization.
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