The allloop conjecture for integrands of reggeon amplitudes in \( \mathcal{N}=4 \) SYM
JHE
The allloop conjecture for integrands of reggeon amplitudes in
0 Skobeltsyn Institute of Nuclear Physics
1 Institutskiy Pereulok str. , Dolgoprudny , Russia
2 JoliotCurie str. , Dubna , Russia
3 The Center for Fundamental and Applied Research
4 A.E. Bolshov
5 National Research Nuclear University , MEPhI
6 Moscow Institute of Physics and Technology, State University
7 Joint Institute for Nuclear Research
8 Leninskiye Gory str. , Moscow , Russia
9 Kashira Highway , Moscow , Russia
10 Sushchevskaya str. , Moscow , Russia
11 Bolshaya Cheremushkinskaya str. , Moscow , Russia
In this paper we present the allloop conjecture for integrands of Wilson line form factors, also known as reggeon amplitudes, in N = 4 SYM. In particular we present a new gluing operation in momentum twistor space used to obtain reggeon treelevel amplitudes and loop integrands starting from corresponding expressions for onshell amplitudes. The introduced gluing procedure is used to derive the BCFW recursions both for treelevel reggeon amplitudes and their loop integrands. In addition we provide predictions for the reggeon loop integrands written in the basis of local integrals. As a check of the correctness of the gluing operation at loop level we derive the expression for LO BFKL kernel in
Scattering Amplitudes; Extended Supersymmetry; Perturbative QCD

4 SYM
HJEP06(218)9
N = 4 SYM.
1 Introduction 2
Form factors of Wilson lines and gluing operation
Form factors of Wilson lines operators
Gluing operator: transforming onshell amplitudes into Wilson line form
factors
HJEP06(218)9
Gluing operation in momentum twistor space
Loop integrands
Gluing operation and local integrands
LO BFKL and gluing operation
5.3.1
5.3.2
Virtual part of LO BFKL
Real part of LO BFKL
6
Conclusion
A Gluing operation and Grassmannians
BCFW for integrands of Wilson lines form factors and correlation functions
1
Introduction
In the last two decades the tremendous progress in understanding of the structure of
Smatrix (amplitudes) in gauge theories in various dimensions has been achieved. The most
prominent examples of such progress are various results for the amplitudes in N = 4 SYM
theory. See for review [1, 2]. These results were near to impossible to obtain without
plethora of new ideas and approaches to the perturbative computations in gauge theories.
These new ideas and approaches mostly exploit analytical properties of amplitudes rather
then rely on standard textbook Feynman diagram technique.
It is important to note that these analyticity based approaches appear to be effective
not only for computations of the amplitudes but for form factors and correlation functions
of local and nonlocal operators in N = 4 SYM and other gauge theories as well [3–12]. So
for many important results for the amplitudes in N = 4 SYM their analogs for the form
factors and correlation functions were found [3, 4, 6, 8–20]. First of all, new variables such
as helicity spinors and momentum twistors appear also useful for the description of the
– 1 –
form factors and correlation functions [3, 4, 10]. At tree level various recurrence relations
(BCFW, CSW e t.c.) were constructed for the form factors of some local [3, 4, 6, 7, 21] as
well as nonlocal [22] operators and various closed solutions for such recurrence relations
were obtained [3, 4, 6, 7, 10–12, 19, 20, 23]. Ultimately for the form factors of operators
from stress tensor supermultiplet [8] as well as for Wilson line operators [19] the
representation in terms of integral over Grassmannian was discovered.1 Also in the case of the
Wilson line operators such representation was generalized to the form factors with
arbitrary number of Wilson line operator insertions as well as correlation functions [20]. Dual
description for such objects in terms of twistor string theories was investigated and in this
context different CHY like representations for form factors were obtained [14, 15, 24]. In
of external particles [3, 13, 16, 31–41] and connection between form factors and integrable
systems [8, 42] was discussed. Interesting results [43, 44] also should be mentioned.
The ultimate goal for such investigations, similar to the amplitude case, would be the
evaluation, in some closed form, of all factors and correlation functions off all possible
operators in N = 4 SYM at arbitrary value of coupling constant.
In this note we are going to continue to work in this direction and consider the
possibility of constructing recurrence relations for the loop integrands of the Wilson line form
factors in N = 4 SYM theory.
Wilson lines are nonlocal gauge invariant operators and are interesting objects not
only from pure theoretical but also from phenomenological point of view. They appear, for
example, in the description of reggeon amplitudes in the framework of Lipatov’s effective
QCD lagrangian2 [22, 46–56], within the context of kT or highenergy factorization [57–60]
as well as in the study of processes at multiRegge kinematics. The Wilson line operators
play the role of sources for the reggeized gluons, while their form factors are directly related
to amplitudes with reggeized gluons in such framework. The results in this field to a large
extent originate from long lasting efforts of St.Peterburg and Novosibirsk groups in the
investigation of asymptotic behavior of QFT scattering amplitudes at high energies (Regge
limit), which can be tracked back in time to early works [61] of Gribov. These results, in
particular, include resummation of leading high energy logarithms (αs ln s)n to all orders
in strong coupling constant (LLA resummation) in QCD, which eventually resulted in the
discovery of BalitskyFadinKuraevLipatov (BFKL) equation [62–66] governing the LLA
high energy asymptotic behavior of QCD scattering amplitudes. Today BFKL equation is
known at nexttoleadinglogarithmicapproximation (NLLA) [67–69]. The current article
can be also considered as an effort in this direction, namely towards NNLLA BFKL in
the context of N = 4 SYM. More accurately, the results of this article can be considered
as a solution to the problem of the reduction of individual Feynman diagrams to a set of
1So in this sense these objects at tree level are known for arbitrary number of external legs.
2We also want to mention recent work [45], where Wilson lines arise in the process of offshell analytic
continuation of lightfront quantized YangMills action.
– 2 –
master integrals in BFKL computations. As in general amplitudes multiloop calculations,
in BFKL calculations there are basically two steps in getting the final result. The first one
is the reduction of contributing individual Feynman diagrams to a finite set of so called
master integrals and the second one is the evaluation of these master integrals themselves.
In this paper we discuss only the first part of this problem, which is the easiest one.
This article is organized as follows. In section 2 we remind the reader the definition of
the Wilson line form factors and correlation functions as well as give the definition of so
called gluing operator Aˆi−1i considered in our previous papers [24]. This operator allows
one to convert the onshell amplitudes into the Wilson line form factors and will be heavily
used throughout the paper. In section 3 we discuss how the BCFW recurrence relations for
the Wilson line form factors are constructed with the use of helicity spinor variables used
to describe kinematical data. After that we show how the mentioned BCFW recursion
can be derived from the BCFW recursion for onshell amplitudes by means of application
of the gluing operators. Section 4 contains the derivation of the gluing operator in the
case when kinematical data are encoded by momentum twistor variables. In section 5
we remind the reader the necessary facts about BCFW recursion for integrands of the
onshell amplitudes in momentum twistor space. After that we show how applying the
gluing operator one can formulate similar recurrence relation for the Wilson line form
factor integrands as well. We also show how one can directly transform the integrands of
onshell amplitudes into the integrands of the Wilson line form factors on the examples
of local onshell integrands. After that we perform simple but interesting self consistency
check of our considerations. Namely starting from our results for tree and one loop level
Wilson line form factors we correctly reproduce LO BFKL kernel. Appendix A contains the
derivation of the Grassmannian integral representation for the reggeon amplitudes starting
from corresponding representation for the onshell amplitudes.
2
2.1
Form factors of Wilson lines and gluing operation
Form factors of Wilson lines operators
To describe the form factors of Wilson line operators we will use the definition in [22]:
Wpc(k) =
Z
d4xeix·kTr
1 c
πg
t P exp
√
ig Z ∞
2 −∞
ds p · Ab(x + sp)tb
(2.1)
where tc is SU(Nc) generator,3 k (k2 6= 0) is the offshell reggeized gluon momentum and
p is its direction or polarization vector, such that p2 = 0 and p · k = 0. The polarization
vector and momentum of the reggeized gluon are related to each other through the so called
kT decomposition of the latter:
μ
kμ = xpμ + kT ,
x ∈ [0, 1] .
(2.2)
3The color generators are normalized as Tr(tatb) = δab.
– 3 –
HJEP06(218)9
It is convenient to parametrize such decomposition by an auxiliary lightcone fourvector
qμ, so that
kTμ (q) = kμ − x(q)pμ
with
x(q) =
and q2 = 0.
q · k
q · p
write down the latter in the basis of two “polarization” vectors4 [46]:
Noting that the transverse momentum kTμ is orthogonal to both pμ and qμ vectors, we may
kTμ (q) = − 2 [pq]
κ hpγμq]
− 2
κ∗ hqγμp]
hqpi
with
κ = hqk/p] , κ∗ = hpk/q] .
hqpi
[pq]
It is easy to check, that k2 = −κκ∗ and both κ and κ∗ variables are independent of auxiliary
fourvector qμ [46]. Also, it turns out convenient to use spinor helicity decomposition of the
lightcone fourvector q as q = ξi[ξ. Wpc(k) is nonlocal gauge invariant operator and plays
the role of source for the reggeized gluon [22, 70], so the form factors of such operators, or
offshell gauge invariant scattering amplitudes in our terminology, are closely related to the
reggeon scattering amplitudes, and we will use words offshell gauge invariant scattering
amplitudes, reggeon amplitudes and Wilson line form factors hereafter as synonyms.
Both usual and color ordered reggeon amplitudes with n reggeized and m usual
onshell gluons could be then written in terms of the form factors with multiple Wilson line
insertions as [22]:
n
Y
j=1
n
Y
j=1
(2.3)
(2.4)
Here asterisk denotes an offshell gluon and p, k, c are its direction, momentum and color
index. Next h{ki, ǫi, ci}im=1 = Nm
i=1hki, εi, ci and hki, εi, ci denotes an onshell gluon state
with momentum ki, polarization vector εi− or εi+ and color index ci, pj is the direction
of the j’th (j = 1, . . . , n) offshell gluon and kj is its offshell momentum. To simplify
things, here we are dealing with color ordered amplitudes only. The usual amplitudes are
then obtained using their color decomposition, see [19, 71]. For example, the color ordered
amplitude with one reggeon and two onshell gluons with opposite helicity at tree level is
given by the following expression:
A2+1(1−, 2+, g3∗) =
∗
δ4(λ1 λ˜1 + λ2λ˜2 + k3)
κ
∗
3
hp3 1i4
.
When dealing with N = 4 SYM we may also consider other onshell states from N = 4
supermultiplet. The easiest way to do it is to consider color ordered superamplitudes
defined on N = 4 onshell momentum superspace [72, 73]:
Am+n Ω1, . . . , Ωm, gm∗+1, . . . , gn∗+m
∗
= hΩ1 . . . Ωm
Wpm+j (km+j )0i,
(2.7)
Am+n 1±, . . . , m±, gm∗+1, . . . , gn∗+m
∗
= h{ki, ǫi, ci}im=1
Wpcmm++jj (km+j )0i .
(2.5)
sometime abuse spinor helicity formalism notations and write hqγμp]/2 ≡ p]hq, λq ≡ hq and λ˜q ≡ [p.
– 4 –
where hΩ1Ω2 . . . Ωm ≡ Nim=1h0Ωi and Ωi (i = 1, . . . , m) denotes an N = 4 onshell chiral
superfield [73]:
Ω = g
+ + η˜AψA +
1
2! η˜Aη˜BφAB +
1
1
3! η˜Aη˜Bη˜C ǫABCDψ¯D +
4! η˜Aη˜Bη˜C η˜DǫABCDg−.
(2.8)
¯
Here, g+, g− are creation/annihilation operators of gluons with +1 and −1 helicities, ψA,
ψA stand for creation/annihilation operators of four Weyl spinors with negative helicity
−1/2 and four Weyl spinors with positive helicity correspondingly, while φAB denote
creation/annihilation operators for six scalars (antisymmetric in the SU(4)R Rsymmetry
indices AB). The A∗m+n Ω1, . . . , gn∗+m superamplitude is then the function of the
following kinematic5 and Grassmann variables
Ak,m+n Ω1, . . . , gm∗+n = Ak,m+n {λi, λ˜i, η˜i}im=1; {ki, λp,i, λ˜p,i}i=m+1 .
∗ ∗ m+n
(2.9)
and encodes in addition to the amplitudes with gluons also amplitudes with other onshell
states similar to the case of usual onshell superamplitudes [1]. Here, additional index6 k
in A∗k,m+n denotes the total degree of A∗k,m+n in Grassmann variables ηi, which is given by
4k − 4n. For example the supersymmetrised (in onshell states) version of (2.6) is given by:
δ4(λ1λ˜1 + λ2λ˜2 + k3) δ4 (η˜1hp3 1i + η˜2hp3 2i) .
κ
∗
3
Then it was conjectured that the following relation holds at least at tree level:
Ak,m+n Ω1, . . . , gm∗+n = hV1, . . . VmVmge+n.1, . . . , Vmge+n.niworldsheet fields.
∗
Here h. . .i means average with respect to string worldsheet fields. This conjecture was
successfully verified at the level of Grassmannian integral representations for the whole
5We used helicity spinor decomposition of onshell particles momenta.
6We hope there will be no confusion with momentum labels.
– 5 –
Here we have k = 2, m = 2 and n = 1. We also for simplicity will often drop ∂4/∂η˜4pi
projectors in further considerations.
2.2
Gluing operator: transforming onshell amplitudes into Wilson line form
In [24, 74] it was conjectured that one can compute the form factors of Wilson line operators
by means of the four dimensional ambitwistor string theory [75]. In an addition to the
standard vertex operators V and Ve, which describe Ωi onshell states in N = 4 SYM field
theory, one can introduce, so called, generalised vertex operators V
Vj
gen.
∼
Z
A2,2+1(Ωj , Ωj+1, g∗)
∗
Y
i=j,j+1
d2λid2λ˜i
Vi Vol[GL(
1
)]
gen. [24]:
d4η˜i.
A2,2+1(Ω1, Ω2, g3∗) =
∗
factors
(2.10)
(2.11)
(2.12)
tree level Smatrix [24, 74] and on several particular examples [24] with fixed number of
external states. Effectively the evaluation of the string theory correlation function in (2.12)
can be reduced to the action of some integral operator Aˆ on the onshell amplitudes. In
the case of one Wilson line operator insertion the relation between onshell amplitude and
the Wilson line form factor looks like:
An+1 = Aˆn+1,n+2[An+2] ,
∗
(2.13)
where An+2 is the usual onshell superamplitude with n + 2 onshell external states and
the gluing integral operator Aˆn+1,n+2 acts on the kinematical variables associated with the
The action of Aˆn+1,n+2 on any function f of variables {λi, λ˜i, η˜i}in=+12 is formally given by
This expression can be simplified. Performing integration over λ˜n+1, λ˜n+2, η˜n+1 and η˜n+2
variables [24] in (2.14) we get
Aˆn+1,n+2[f ] = hpn+1ξn+1i Z dβ1
∗
κn+1
dβ2
1
where ∗ denotes substitutions {λi, λ˜i, ηi}i=n+1 7→ {λi(β), λ˜i(β), η˜i(β)}in=+n2+1 with
n+2
λn+1(β) = λn+1 +β2λn+2 ,
λ˜n+1(β) = β1λ˜n+1 +
β2
(1+β1) λ˜n+2 , η˜n+1(β) = −β1η˜n+1 ,
β1β2
(1+β1) λn+1 , λ˜n+2(β) = −β1λ˜n+2 −β1β2λ˜n+1 ,
η˜n+2(β) = β1β2η˜n+1 ,
A3,3+1(1−, 2+, 3−, g4∗) = δ4
∗
3
i=1
X λiλ˜i + k4
! 1
[2p4]4
κ4 [12][23][3p4][p41]
– 6 –
λn+2(β) = λn+2 +
and
where [19, 46]
λn+1 = λp, λ˜n+1 = hξk , η˜n = η˜p; λn+2 = λξ, λ˜n+2 = hpk , η˜n+2 = 0.
hξpi
hξpi
All other variables left unshifted.
The integration with respect to β1,2 will be understood as a residue form [76] and will
be evaluated by means of the composite residue in points resβ2=0 ◦ resβ1=−1. For example,
one can obtain [24] the Wilson line form factor A∗3,3+1(1−, 2+, 3−, g4∗) from 5 point NMHV
onshell amplitude A∗3,5(1−, 2+, 3−, 4−, 5+):
A3,3+1(1−, 2+, 3−, g4∗) = Aˆ45[A3∗,5(1−, 2+, 3−, 4−, 5+)],
∗
(2.18)
(2.16)
(2.17)
=
Z dα
α
where
There is another way of representing the action of gluing operator. One can note
that (2.14) is in fact equivalent to the action of a pair of consecutive BCFW bridge
operators, in terminology of [77], on the f function weighted with an inverse soft factor. Namely,
function f of the arguments {λi, λ˜i, η˜i}in=+12, 1 ≤ i, j ≤ n + 2 according to:
if one [77] defines [i, ji BCFW shift operator as Br(i, j) (see figure 1) which acts on the
Br(i, i + 1) hf . . . , λi, λ˜iη˜i, . . . , λj , λ˜j , η˜j , . . .
i
=
ˆ
f . . . , λi, λ˜i, η˜ˆi, . . . , λˆj , λ˜j , η˜j , . . .
Z dα
α
f . . . , λi, λ˜i − αλ˜j , η˜i − αη˜j , . . . , λj + αλi, λ˜j , η˜j , . . . ,
then one can see that the following relation holds:
Aˆn+1,n+2[ f ] = Br(n + 1, n + 2) ◦ Br(n + 2, n + 1) S−1(1, n + 2, n + 1) f ,
Several Wilson line operator insertions correspond to the consecutive action of several
gluing operators. For example A∗3,0+3(g1∗, g2∗, g3∗) can be obtained [24] from 6 point NMHV
amplitude A3,6(1−2+3−4+5−6+):
A3,0+3(g1∗, g2∗, g3∗) = (Aˆ12 ◦ Aˆ34 ◦ Aˆ56)[A3,6(1−2+3−4+5−6+)],
∗
where A∗3,0+3 is given by (P′ is the permutation operator which shifts all spinor and
momenta labels by +1 mod 3.):
A3,0+3(g1∗, g2∗, g3∗) = δ4(k1 + k2 + k3) 1 + P′ + P′2 f,
∗
and function the f depends on {λi, λ˜i, η˜i}in=+12 arguments.
Note also that since Br(i, j) operators act naturally on onshell diagrams [77] one can
easily consider the action of Aˆn+1,n+2 operator on the topcell diagram corresponding to
the Ak,n+2 tree level onshell amplitude. The topcell for Ak,n+2 in its turn can be
represented as the integral over Grassmannian Lkn+2 [77] (here let’s ignore integration contour
for a moment):
dk×n+2C δk×2(C · λ˜)δk×4(C · η˜)δ(n+2−k)×2(C⊥ · λ)
Vol[GL(k)] (1 · · · k)(2 · · · k + 1) · · · (n + 2 · · · k − 1)
.
Then one can see that the following relation also holds:
h k
i
Aˆn+1,n+2 Ln+2 = Ωn+1 ,
k
– 7 –
i
where Ωkn+2 is the Grassmannian integral representation for the offshell amplitude A∗k,n+1,
with the Wilson line insertion positioned after the onshell state with number n [19], if the
appropriate integration contour is chosen for Ωkn+2:
7
HJEP06(218)9
with
and
k
Ωn+2 =
Z dk×(n+2)C′
Vol[GL(k)]
δk×2 (C′ · λ˜) δk×4 (C′ · η˜) δ(n+2−k)×2 C′⊥ · λ
(1 · · · k) · · · (n + 1 · · · k − 2)(n + 2 1 · · · k − 1)
,
Reg. = hξn+1pn+1i (n + 2 1 · · · k − 1)
,
∗
κn+1
(n + 1 1 · · · k − 1)
λi = λi,
λ˜i = λ˜i,
η˜i = η˜i,
i = 1, . . . n,
i = 1, . . . n,
i = 1, . . . n,
λn+1 = λpn+1 ,
λ˜n+1 = hξn+1kn+1 ,
hξn+1 pn+1i
η˜n+1 = η˜pn+1 ,
λn+2 = ξn+1
λ˜n+2 = − hξn+1 pn+1i
hpn+1kn+1 ,
η˜n+2 = 0.
The action of several Aˆi,i+1 operators can be considered among the same lines and the
result reproduces Grassmannian representation of the form factors with multiple Wilson
line operator insertion obtained in [20].
At the end of this section let us make the following comment. Both onshell and
offshell amplitudes (Wilson line form factors) can be represented by means of the BCFW
recursion relations. But due to different analytical properties (Wilson line form factors
will have additional type of poles corresponding to the Wilson line propagators [46]) the
recursion for onshell and offshell amplitudes looks rather different. However, from the
examples similar to ones considered above (namely (2.18) and (2.18)) one can note that
7One can think of this as alternative derivation of the results of appendix A of [24]. See also appendix
of the current article for notation explanation.
(2.27)
(2.28)
(2.29)
– 8 –
not only gluing operator maps onshell amplitudes to offshell ones but one can choose
representation for the onshell amplitude in terms of the BCFW recursion in such a way
that each BCFW term from onshell amplitude will be mapped onetoone to the terms from
the BCFW recursion for the offshell amplitudes. So a natural question to ask is whether it
is possible to derive the BCFW recursion for the Wilson line form factors from the BCFW
recursion for the onshell amplitudes. We will address this question in the next section.
BCFW recursion for Wilson line form factors
Offshell BCFW from analyticity
First let us remind the reader the main results of [46] and comment on supersymmetric
extension of the offshell BCFW recursion. The offshell BCFW recursion for the reggeon
amplitudes with an arbitrary number of offshell reggeized gluons was worked out in [46].
Similar to the BCFW recursion [78, 79] for the onshell amplitudes it is based on the
observation, that a contour integral of an analytical function f vanishing at infinity equals
where the sum is over all poles of f and resif (z) is a residue of f at pole zi. Using this,
one can relate the offshell amplitude to the sum over contributions of its factorisation
channels, which in turn can be represented as the offshell amplitudes with smaller number
of external states. In the original onshell BCFW recursion the zdependence of scattering
amplitude is obtained by a zdependent shift of particle’s momenta. Similarly, the offshell
gluon BCFW recursion of [46] is formulated using a shift of momenta for two external
gluons i and j with a vector
so that
1
2
eμ =
hpiγμpj ],
pi · e = pj · e = e · e = 0,
kˆiμ(z) ≡ kiμ + zeμ = xi(pj )piμ −
kˆjμ(z) ≡ kj − zeμ = xj (pi)pjμ − 2
μ
κj hpj γμpi]
[pj pi]
−
κi − [pipj ]z hpiγμpj ]
2
This shift does not violate momentum conservation and we still have pi · kˆi(z) = 0 and
pj · kˆj (z) = 0. We would like to note, that the overall effect of shifting momenta is that the
values of κi and κj∗ shift, while κi∗ and κj stay unshifted. In the onshell limit the above
to zero, that is
and the integration contour expands to infinity. Taking the above integral by residues
I dz f (z)
2πi z
iμ + ki+1 + · · · + kjμ and h is an internal onshell gluon helicity or a summation
μ
index over all onshell states in the Nair onshell supermultiplet in the supersymmetric
case discussed later. Here and below we use the convention that double lines may stand
both for offshell and onshell gluons. The coil crossed with a line correspond to the offshell
gluons (Wilson line operator insertion). The thick solid lines stand for onshell particles.
The offshell coil lines can be bent apart to form a single eikonal quark lines [22, 46].
According to this kjμ in kiμ,j can be either offshell or onshell depending on the context.
Let’s now discuss each type of the terms encountered in (3.6) in more details. The
Ai,h terms are usual onshell BCFW terms, which correspond to the zpoles at which
denominator of internal gluon (and also fermion or scalar) propagator kˆ12,i(z) vanishes:
shift corresponds to the usual [i, ji BCFW shift. Note also, that we could have chosen
another shift vector eμ = 12 hpj γμpi] and shift κi∗ and κj instead. The offshell amplitudes
we consider in this paper do also have a correct large z (z → ∞) behavior [46], so that we
should not worry about boundary terms at infinity.
The sum over the poles (3.2) for zdependent offshell gluon scattering amplitude is
given by the following graphical representation8 [46]:
(3.6)
This is standard BCFW onshell condition for physical states of N
= 4 SYM
supermultiplet.
The Bi term is a new one and is unique to the BCFW recursion for the offshell
amplitudes. It originates from the situation when the denominators of eikonal propagators
coming from Wilson line expansion vanish, that is
8We are considering the color ordered scattering amplitudes and without loss of generality may use shift
of two adjacent legs 1 and n.
kˆ12,i(z) = 0.
pi · kˆi,n(z) = 0
and piμ is the direction of the Wilson line associated with the offshell gluon. It is important
to understand that condition pi · kˆi,n(z) = 0 fixes only the direction of momentum flowing
through the Wilson line kˆi,n.
The offshell momenta kˆL = ki−1 + . . . + kˆ1 and kˆR =
ki + . . . + kˆn, which belongs to the offshell amplitudes in eq. (3.7), term Bi, are different
(kˆL = −kˆR), but satisfy the same condition pi · kˆL/R = 0. We also want to stress that
this term is present only if i labels an offshell external gluon. In addition, let us note
that κi∗ factors in the pair of offshell amplitudes, which contribute to this term, are given
explicitly by the following expressions:
[piqi]
[piqi]
κi,L = hpikˆ1 + . . . + ki−1qi]
∗
, κi∗,R = hpikˆn + . . . + ki+1qi]
,
(3.10)
where κi∗,L and κi∗,R belong to the offshell amplitudes positioned to the left and to the right
The C term is only present if the gluon number 1 is offshell. It is also unique to the
BCFW recursion for the offshell amplitudes. It appears due to vanishing of the external
momentum square
Similarly, the D term is due to vanishing of the external momentum square kˆn2(z). It turns
out that both these contributions could be calculated in terms of the same BCFW term
with the offshell gluons 1 or n exchanged for the onshell ones. The helicity of the onshell
12 hpj γμpi]) used. We refer the reader to [46] for further details and examples.
gluons depends on the type of the term (C or D) and the shift vector eμ ( 12 hpiγμpj ] or
The use of shifts involving only onshell legs also allows one to perform the
supersymmetrization of the offshell BCFW recursion introduced in [46]. Indeed, it is easy to see,
that the supersymmetric shifts of momenta and corresponding Grassmann variables are
given by the onshell BCFW [i, ji supershifts:9
ˆi] = 1] + zj],
ˆji = ji − zii,
ηˆAi = ηAi + zηAj.
(3.12)
No other spinors or Grassmann variables shift.
3.2
Offshell BCFW from gluing operation
The aim of this section is to derive the offshell recursion relations described above from
the BCFW recursion for the onshell amplitudes by means of the gluing operator. Before
proceeding with general derivation let us consider a simple example first: we will take
the BCFW recursion for the onshell 6point NMHV amplitude A3,6(1−, 2+, 3−, 4+, 5−, 6+)
and transform it into the three point offshell amplitude A∗0+3(g1∗, g2∗, g3∗) considered in [46].
This offshell amplitude in its turn also can be obtained from the offshell BCFW recursion,
when external momenta 1 and 3 are shifted. Contributions corresponding to this shift are
given in figure 2, and the sum of these three terms is given by (2.21).
9These shifts respect both momentum and supermomentum conservation.
external momenta. The first and the third terms labeled A) and C) are type C and D contributions,
while the second, labeled B), term is type B contribution of the offshell BCFW recursion (3.6).
representation of A3,6(1−, 2+, 3−, 4+, 5−, 6+) amplitude.
So let’s consider A3,6(1−, 2+, 3−, 4+, 5−, 6+) amplitude represented via the standard
BCFW shift [1−, 6+i. The amplitude is then given, once again, by three terms (see figure 3)
A3,6(1−, 2+, 3−, 4+, 5−, 6+) = A + B + C,
where
A = A2,5(3−, 4+, 5−, ˆ6+, Pˆ+) 2 A2,3(1ˆ−, 2+, −Pˆ−),
B = A2,4(4+, 5−, ˆ6+, Pˆ−) 2 A2,4(1ˆ−, 2+, 3−, −Pˆ+),
C = A1,3(5−, ˆ6+, Pˆ+) 2 A3,5(1ˆ−, 2+, 3−, 4+, −Pˆ−).
1
q5,6
1
q1,3
1
Here qa,b = Pib=a qi and qi denote onshell particle momenta.
Now we are going to consider the action of our gluing operators on A3,6 onshell
amplitude, which will convert all pairs of i−, (i + 1)+ gluons into Wilson line operator
insertions (reggeized gluons). As we have discussed in previous sections to do this we have
to take into account the action of the following combination of the gluing operations on A3,6:
A0+3(g1∗, g2∗, g3∗) = Aˆ12 ◦ Aˆ34 ◦ Aˆ56[A3,6(1−, 2+, 3−, 4+, 5−, 6+)].
(3.17)
Let’s consider each contribution in details.
We will start with A contribution first:
Aˆ12 ◦ Aˆ34 ◦ Aˆ56[A] = Aˆ34 ◦ Aˆ56[A2,5(6ˆ+, 5−, 4+, 3−, Pˆ−)]Aˆ12
2 A2,3(1ˆ−, 2+, −Pˆ+) .
It turns out, that both the value of the BCFW shift parameter z as well as shifted spinors
are regular after we made ∗ substitutions (see (2.15) and (2.2)) corresponding to gluing
operations and took limits β2 → 0, β1 → −1. Let us introduce the following notation for
the spinors entering kT decomposition of momenta of three reggeized gluons, each spinor
will be labeled by corresponding gluing operator:
The original value of the onshell z parameter (z = [[6221]] ) transformed after the action of
Aˆ12 into
and helicity spinor decomposition of momentum Pˆ is given now by
where we used kT decomposition of the first reggeized gluon g1∗ momentum k1:
z =
[21]
Then, it is easy to see that
Aˆ34 ◦ Aˆ56[A2,5(3−, 4+, 5−, ˆ6+, Pˆ+)] = A1+2(g2∗, gˆ3∗, Pˆ+),
∗
where gˆ3∗ denotes reggeized gluon g3∗ with momentum shifted as
For the term
we have
kˆ3 = k3 +
κ1
[p1 p3] p1i[p3.
Aˆ12
2 A2,3(1ˆ−, 2+, −Pˆ−) = resβ1=−1 ◦ resβ2=0[ωA],
#
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
ωA = −
=
1
hp1 ξ1i
κ
∗
1
1
1
k2
1 h12ih2Pˆi
h1Pˆi3 !
dβ1 ∧ dβ2 + less singular terms.
where
z =
where helicity decomposition of momentum Pˆ is given by
Aˆ12 ◦ Aˆ34 ◦ Aˆ56[C] =
A1+2(gˆ1∗, g2∗, Pˆ−) ,
∗
This is precisely the D term from the offshell BCFW recursion [46] for A∗0+3(g1∗, g2∗, g3∗)
amplitude.
Now let us turn to B contribution to A3,6 (see figure 2). The value of z parameter in
this case transforms under the action of the gluing operator as
Taking residues and combining everything together we finally get for A term
which is precisely the C term from the offshell BCFW recursion [46] for A∗0+3(g1∗, g2∗, g3∗)
reggeon amplitude (see (3.6)). The C term can be analysed similarly. In this case we get
HJEP06(218)9
where
ωB = hp2 ξ2i
∗
κ
2
Evaluating corresponding residues we get
Aˆ34[B] =
1
h5p2i4
1
κ1
1
κ3
1
1
q1,3
hˆ1 3i4
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
Here it is convenient to consider first the action of Aˆ34. In this case the value of Pˆ
momentum is given by
β2
2 hξ2 p2i
Pˆ = qˆ1 + q2 +
(1 + β1) κ∗ p2i[p2 + less singular terms,
before residues evaluation. So for the whole B term after Aˆ34 action we have
Aˆ34[B] = Aˆ34 A2,4(4+, 5−, ˆ6+, Pˆ−) 2 A2,4(1ˆ−, 2+, 3−, −Pˆ+)
"
#
= resβ1=−1 ◦ resβ2=0[ωB]
h5 Pˆi4
1
2
h4 5ih5 6ˆih6ˆ PˆihPˆ 4i q1,3 hˆ1 2ih2 3ih3 PˆihPˆ 1i
!
∗ β12β2
β1β2
1 dβ1 ∧ dβ2
. (3.34)
κˆ2,L h6ˆ5ih5p2ihp26ˆi × hp2k1 + k2p2] × κˆ∗2,R h12ih2p2ihp21i
∗
1
h1p2i4
(3.35)
The action of Aˆ12 ◦ Aˆ56 can be evaluated in similar fashion and finally we arrive at
Aˆ12 ◦ Aˆ56 ◦ Aˆ34[B] = A0+2(g2∗, gˆ3∗)
∗
1
which is exactly B term in the offshell BCFW recursion [46] for A∗0+3(g1∗, g2∗, g3∗) amplitude.
So we see the pattern here: the contributions C and D from the offshell BCFW
recursion for A∗0+3(g1∗, g2∗, g3∗) amplitude are reproduced from the onshell BCFW recursion
for A3,6(1−, 2+, 3−, 4+, 5−, 6+) when gluing operator is acting on three point MHV3 or
MHV3 subamplitudes (with degenerate kinematics), while B contribution is reproduced
by the action of gluing operator on both sides of the BCFW bridge, see figure 3.
These observations can be immediately generalized to the situation with arbitrary
onshell amplitude. One can obtain the offshell BCFW recursion for A∗0+n(g1∗, . . . , gn∗)
with the shift of the offshell momenta k1 and kn from the BCFW recursion for
An,2n(1−, 2+, . . . , (2n)+) onshell amplitude represented by [1−, (2n)+i shift. Indeed, the
terms C and D are reproduced when gluing operator acts on MHV3 or MHV3 onshell
amplitudes. Repeating the steps identical to the previous discussion (see (3.28)) we get:
κˆ2,L = hp2q5 + qˆ6ξ] , κˆ2,R = hp2q2 + qˆ1ξ] .
∗ ∗
[p2ξ]
[p2ξ]
"
Pˆi =
"
where
where
with
(3.36)
(3.37)
(3.38)
(3.39)
#
(3.40)
(3.41)
(3.42)
Aˆ2n−1 2n ◦ . . . ◦ Aˆ12 An−1,2n−1 3−, . . . , (d2n)+, Pˆ+
1
q1,2
#
=
1
κ1
A1+(n−1)(g2∗, . . . , gˆn∗, Pˆ+),
∗
Similarly for MHV3 we have:
1
x(pn)p1i − hpnp1i
κ
∗
pni ,
Pˆ] = p1].
Aˆ2n−1 2n ◦ . . . ◦ Aˆ12 A1,3 (2n − 1)−, (d2n) , −Pˆ+
+
1
2
q2n−1,2n
An,2n−1(1ˆ−, . . . , Pˆ−)
1
=
κ∗ A1+(n−1)(gˆ1∗, . . . , gn∗−1, Pˆ−),
∗
n
x(p1)pn] − [pnp1] p1] ,
κ1
Pˆi = pni.
When gluing operator Aˆii+1 acts on legs separated by the BCFW bridge the B type
contribution is reproduced. In this case the onshell BCFW shift z is replaced by
z = hpik1 + . . . + kipi]
,
hp1 pii[pi pn]
shell [1−, n+i recurcion in general case. A) diagrams will give A type terms of the offshell BCFW
recurcion, B) type diagrams will give B type terms, while C) diagrams will give C and D type terms
of the offshell BCFW recurcion.
and the factor 1/q12,i is replaced by
(3.43)
HJEP06(218)9
1
hpik1 + . . . + kipi]
times βi factors. The explicit proof that such contribution in general case gives us the
B term can be done by induction and can be sketched as follows: one can decompose
each Aki,ni onshell amplitude in individual BCFW terms via onshell diagram
representation [77] into combination of MHV3 and MHV3 vertexes (onshell diagrams). The action
of the gluing operator on such onshell diagrams was considered in the previous section.
After that one have to reassemble A∗2,2+1, MHV3 and MHV3 amplitudes together. As the
result one can obtain that:
"
. . . ◦ Aˆii+1 ◦ . . . Ak1,n1 (i + 1)+, . . . , (d2n)+, Pˆ−
2 Ak2,n2 (1ˆ−, . . . , i−, −Pˆ+)
1
q1,i
= A0∗+i(gˆ1∗, . . . , gi∗)
1
hpik1 + . . . + kipi] A0+(n−i)(gi∗, . . . , gˆn∗).
∗
#
(3.44)
Also, presented above relation is implicitly guaranteed by the Grassmannian integral
representation of the onshell and offshell amplitudes. It was shown [24, 74] that the latter
could be easily related with each other by means of the same gluing operations (see (2.26)
and appendix A) which imply similar relation for the individual residues of the topforms.
We see that the gluing operator transforms ordinary 1/P 2 propagator type poles of the
onshell amplitudes into eikonal ones, when external legs, on which the gluing operator
acts, are separated by the BCFW bridge (see figure 4 B). The value of the BCFW shift
parameter z is adjusted accordingly to match the offshell BCFW recursion term B.
All other contributions (see figure 4) reproduce A type terms from [46], which also
can be shown by induction. Indeed it is easy to see that the pole factor remains of the
same 1/P 2 type in this case, which corresponds to the propogators of the onshell states
of N = 4 SYM, and the value of z is adjusted to match A term of the offshell BCFW.
The fact that one can transform each individual term in the BCFW recursion for the
onshell amplitudes into terms of the BCFW recursion for the Wilson line form factors using
gluing operators Aˆi,i+1 in fact is not (very)surprising and in some sense trivial. Indeed, as
was mentioned before, if the relation (2.26) holds on the level of the Grassmannian integrals
(topcell diagrams) then it likely will hold for the individual residues (boundaries of
topcells) as well.10 And it is also natural, that in the case of Ωkn+2, and its generalisations to
multiple Wilson line insertions, the residues of Ωkn+2 can be identified with the individual
BCFW terms of recursion for the offshell amplitudes in full analogy with the onshell case.
However observation that one can transform the BCFW recursion for the onshell
amplitudes into the BCFW recursion for the Wilson line form factors at tree level opens
up exiting possibility. It is known that one can formulate the BCFW recursion not only
for the tree level onshell amplitudes in N = 4 SYM theory but for the loop integrands as
well [77, 80]. If we can transform onshell BCFW recursion into offshell one at tree level,
then what about recursion for the loop integrands ?
The next sections will be dedicated to discussion of this question. Namely we will
investigate what will happen if we apply the analog of the gluing operators to the BCFW
recursion for N = 4 SYM loop integrands of the onshell amplitudes. Our ultimate goal
is to present arguments that using the gluing operator one can transform the integrands
of the onshell amplitudes into the integrands of the Wilson line form factors at any given
number of loops and external states.
4
Gluing operation in momentum twistor space
The integrands of the onshell amplitudes in the planar limit are naturally formulated using
momentum super twistors variables, in particular this is the case for theis BCFW recursion
representation [80]. So, to proceed with our main goal we need to fprmulate our gluing
operation in momentum twistor variables as well.
To do this, let us recall how the momentum twistor variables are introduced. We start
with so called zone super variables or dual super coordinates yi and ϑi which are related
with onshell momenta and its supersymmetric counterpart as [1, 81]:
qi = λiλ˜i = yi+1 − yi,
λiηi = ϑi+1 − ϑi.
(4.1)
The introduction of dual super coordinates helps to trivialize the conservation of super
momentum [1, 81] and figure 5 shows the momentum conservation geometrically for the
case of n = 4 onshell and one offshell momenta as an example. There we have a contour
in the dual space formed by onshell particles momenta together with two auxiliary onshell
momenta 5 and 6 used to describe offshell momentum.
The momentum super twistor variables Zi = (λi, µ i, ηi) [81] are then defined through
the following incidence relations
µ i = λiyi = λiyi+1,
η˜i = λiϑi = λiϑi+1.
(4.2)
10This can be explicitly seen for some particular case considering integration contours for tree level
3 3
amplitudes Ln+2 and Ωn+2, and probably can be easily generalised for the case of arbitrary number of
Wilson line insertions and arbitrary value of k [24]. Here however we avoided considerations of integration
contours completely.
onshell legs. In contrast to the case of the onshell amplitudes, the n onshell momenta do not
add up to zero but to the offshell gluon momentum k: q1 + . . . + q4 = k, which in its turn can be
decomposed as a pair of auxiliary onshell momenta k = q5 + q6.
The bosonic part of Zi will be labeled as Zi = (λi, µ i). Inverting presented above relations
δi−1 j hi i + 1i + δij hi + 1 i − 1i + δi+1 j hi − 1 ii
hi − 1 iihi i + 1i
.
(4.3)
Here λ˜ ≡ (λ˜1 · · · λ˜n), η˜ ≡ (η˜1 · · · η˜n) and it is assumed that Pin=1 qi = 0. The transition
from momentum twistors to helicity spinors could be performed with a formula like:11
µ = Q˜ · λ˜ ,
η = Q˜ · η˜ ,
˜
Qij =
(hj ii if 1 < j < i
0
otherwise
(4.4)
Note, that momentum super twistors trivialize both onshell condition qi2 = 0 and
mentioned above conservation of super momentum.
To construct the gluing operator acting in momentum twistor space let us recall that
the initial gluing operator in helicity spinor variables can be represented as an action of
two consecutive BCFW bridges times some regulator12 factor (2.23). The BCFW bridge
operators can also be defined in momentum twistor space using special version of the
onshell diagrams [84]. The action of [i, i + 1i the BCFW shift bridge operator br(ˆi, i + 1) in
momentum twistor representation on the function Y of {Zi}in=1 variables is given by [80, 84]:
Y ′(Z1, . . . , Zn) = br(ˆi, i + 1) [Y (Z1, . . . , Zn)] ≡
Y (Z1, . . . , Zˆi, . . . Zn),
(4.5)
Z dc
c
cZi+1. We also do not require Y and Y ′ to be Yangian invariants.
where Y , Y ′ are both functions of n momentum super twistors variables and Zˆi = Zi +
11The matrix Q˜ij is a formal inverse of singular map Qij, see [82, 83] for details.
12We call inverse soft factor (2.24) regulator because it makes soft holomorphic limit with respect to one
of the auxiliary onshell momenta, which encodes offshell one, regular [19, 46].
require that
sumed):
As new onshell diagrams in momentum twistor space are no longer built from ordinary
MHV3 and MHV3 vertexes (amplitudes), then in the definition of the gluing operator we
will have, in principle, to change the form of regulator factor. So, to construct the gluing
operator in momentum twistors we will consider the following ansatz:
Aˆim−.t1w,iistor [. . .] = N br(ˆi, i + 1) ◦ br(i [+1, i) [M . . .]
with two unknown rational functions of helicity spinors λi (first components of
momentum twistors) — measure M and normalization coefficient N . To fix N , M functions we
Aˆnm+.t1w,nis+to2r hLkn+2i = ωnk+2 ,
where (k is N(k−2)MHV degree and the use of an appropriate integration contour is
ask
ωn+2 =
Reg. =
1
,
,
is the momentum twistor Grassmannian integral representation for the ratio of amplitudes
with one Wilson line operator insertion A∗k,n+1/A∗2,n+1 and Ln+2 is the Grassmannian
k
representation for Ak,n+2/Ak=2,n+2 onshell amplitude ratio:
That is our gluing operation should transform the Grassmannian integral representation
of onthe shell amplitudes into corresponding Grassmannian integral representation for the
offshell amplitudes. From this requirement we get
(4.6)
(4.7)
(4.8)
(4.10)
(4.11)
HJEP06(218)9
where S is the usual soft factor
M = N −1 = S(i + 1, i, i − 1) ,
S(i + 1, i, i − 1) =
∗
κi−1hi − 1 i + 1i
hi i + 1ihi − 1 ii
.
Computation details can be found in appendix A.
So, finally, we have the following expression for the gluing operation in momentum
twistor space
Aˆim−.t1w,iistor [. . .] = S(i + 1, i, i − 1)−1 br(ˆi, i + 1) ◦ br(i [+1, i) [S(i + 1, i, i − 1) . . .] . (4.12)
It may be, at first glance, surprising that here in momentum twistor space we used [i, i + 1i
BCFW shift and not [i − 1, ii as for the gluing operation in the helicity spinors
representation. In fact, mentioned before the two BCFW shifts are equivalent, see, for example,
Aˆ5,6 P64 =
Aˆ5,6 P64 =
1
1
1 + hhpp55ξ15ii hh13344556ii [13456] +
1 + hhpp55ξ15ii hh12233556ii [12356] + [12345],
A∗2,4+1
A∗3,4+1 (Ω1, . . . , Ω4, g5∗) ,
Aˆ3,4 ◦ Aˆ5,6 P64 = c35[12345] + c36[12356] + c46[13456],
Aˆ3,4 ◦ Aˆ5,6 P64 =
A∗2,2+2
A∗3,2+2 (Ω1, Ω2, g3∗, g4∗) ,
1
1
1
1
1 + hhpp33pξ34iihh11223354ii , c36 =
1 + hhpp44ξ14ii hh12233556ii , c46 =
1 + hhpp33pξ34ii hh11334566ii 1 + hhpp44ξ14ii hh13344556ii .
These results are in complete agreement with previously obtained results from the offshell
BCFW [46] and Grassmannian integral representation [19, 20].
In general the onshell ratio function Pk,n+2 = Pn+2
4(k−2)(Z1, . . . , Zn+2) can be found
for fixed n and k via the solution of the onshell BCFW recursion in momentum twistor
and14
with
c35 =
Pk,n(Z1, . . . , Zn) = Pk,n−1(Z1, . . . , Zn−1)
+ X[j − 1, j, n − 1, n, 1]Pk1,n+2−j (ZIj , Zj , Zj+1, . . . , Zˆnj )Pk2,j (ZIj , Z1, Z2, . . . , Zj−1) ,
where15
Znj = (n − 1, n) ∩ (1, j − 1, j), ZˆIj = (j − 1, j) ∩ (1, n − 1, n), k1 + k2 + 1 = k. We
will make more comments about the structure of this recursion relation in the next section.
From practical point of view the easiest way to compute Wilson line form factor with f
onshell states and m Wilson line operator insertions is to solve (4.27) for n = f + 2m and
then apply m gluing operators via (4.16) rule.
Now, when we have the definition of the gluing operator Aˆii+1 in momentum twistor
space and some practice with the tree level answers we are ready to consider loop integrands.
5
Loop integrands
The natural way to define planar loop integrands unambiguously is to use momentum
twistors or dual coordinates. The loop integrand IkL,n for onshell Lloop amplitude AkL,n
13It is assumed that the momentum super twistors Z5 and Z6 are sent to corresponding offshell kinematics
related to offshell momenta of g5 reggeized gluon.
∗
reggeized gluons g3 and g4 .
∗
∗
15(i, j) ∩ (k, p, m) ≡ Zihjkpmi + Zjhikpmi.
14We again assume corresponding offshell kinematics for momentum super twistors Z3Z6 describing
in this language is defined as16
Z
L
reg m=1
A(kL,n)/A(20,n) =
Y d4lmIk(L,n)(Z1, . . . , Zn; l1, . . . , lL) ,
functions
n−2
j=2
where momentum super twistors Z1, . . . , Zn describe kinematics of external particles and
reg stands for regularization needed by loop integrals. Here Ik,n is a rational function of
both loop integration and external kinematical variables. Moreover, Ik,n is cyclic in
external momentum super twistors. It is also assumed that loop integrand is completely
symmetrized in loop variables l1, . . . , lL.
Rewriting the latter in terms of bitwistors
(lm ≡ (AmBm) ≡ (AB)m) the loop integration measure takes the form [80]:
where we dropped out factors hλA λBi = hZAZBI∞i as the integrands in N = 4 SYM
are always dual conformal invariant. Here I∞ denotes infinity bitwistor [81]. The
integral over the line (AB) is given by the integrals over the points ZA, ZB modulo GL(
2
)
transformations leaving them on the same line.
BCFW for integrands of Wilson lines form factors and correlation
Now let us see what modifications occur to the onshell integrand BCFW recursion in the
offshell case. The looplevel BCFW for onshell amplitudes in N = 4 SYM was worked
out in detail in [80] (see also [85, 86] for situation with less SUSY) and the result for
Zˆn = Zn + wZn−1 shift reads
Ik(L,n) = Ik(L,n)−1(Z1, . . . , Zn−1)
+X[j −1, j, n−1, n, 1]Ik(L1,1n)+2−j (ZIj , Zj , Zj+1, . . . , Zˆnj )Ik(L2,2j)(ZIj , Z1, Z2, . . . , Zj−1)
+
Z d44ZAd44ZB Z
Vol[GL(
2
)]
GL(
2
)
[A, B, n−1, n, 1]Ik(L+−1,1n)+2(Z1, Z2, . . . , ZˆnAB , ZA, ZB) ,
where Zˆnj = (n−1, n)∩(1, j−1, j), ZIj = (j−1, j)∩(1, n−1, n), ZˆnAB = (n−1, n)∩(A, B, 1)
and k1+k2+1 = k. The RGL(
2
) integral is defined as follows. First we set ZA → ZA+αZB ≡
ZA′ and ZB → ZB + βZA ≡ ZB′, which is equivalent to moving points ZA and ZB without
changing the line they span.
Then we calculate composite residue in α, β such that
hA′, 1, n − 1, ni → 0 and hB′, 1, n − 1, ni → 0, what is equivalent to taking points A′, B′ to
lie on the plane h1, n − 1, ni:
Z
GL(
2
)
Z
dα
Z
hA′,1,n−1,ni→0
hB′,1,n−1,ni→0
dβ (1 − αβ)2 .
the Jacobian factor (1 − αβ)2 makes poles in α, β simple.
Taking the residue as above is equivalent to setting ZA′, ZB′ to (A, B) ∩ (1, n − 1, n) and
16Here by dividing on MHV amplitude we mean that we are factoring out h12i . . . hn1i product and
dropping momentum conservation delta function.
(5.1)
(5.2)
(5.3)
(5.4)
scalar integrals for two loop n = 4 example. Red arrows indicate propagators which we are cutting
when evaluating residues. Term C) is actually absent in N = 4 SYM case as well as A).
HJEP06(218)9
Next, let us make some comments about the origin of different terms in (5.3). The
first two terms, namely
n−2
j=2
Ik(L,n)−1(Z1, . . . , Zn−1) + X[j − 1, j, n − 1, n, 1]Ik(L1,1n)+2−j (ZIj , . . . , Zˆnj )Ik(L2,2j)(ZIj , . . . , Zj−1)
originate from the poles in the BCFW shift parameter w coming from propagators which
does not contain loop momentum dependence:
hi − 1in − 1nˆ(w)i = 0.
that is from propagators connecting loop integrals, see figure 7 A. These contributions are
identical both at tree and loop level. The term containing GL(
2
) integration
Z
GL(
2
)
[A, B, n − 1, n, 1]Ik(L+−1,1n)+2(Z1, Z2, . . . , ZˆnAB , ZA, ZB)
is present only at the loop level. It originates from the poles in the BCFW shift parameter
w coming from propagators containing loop momenta [80], see figure 7 B. At L loop level for
n point amplitude the residue at such pole corresponds to the so called forward limit of L−1
loop n + 2 point amplitude. Indeed, if we consider L loop integrand17 of some amplitude
at the pole 1/lL2 corresponding to L’th loop integration we will get (see figure 8)
In(L)({p1, . . . , pn}, l1, . . . , lL), where {p1, . . . , pn} are external momenta and consider residue
ReslL2=0 In(L)
∼ In(L+−21)({p1, . . . , pn, −lL, lL}, l1, . . . , lL−1).
In momentum twistor space residue can be evaluated as follows. For simplicity let’s
consider L = 1 example to make formulas more readable. The generalization for general
L is trivial. The npoint amplitude integrand is the function of the following variables
17Here we assume some specific “appropriate” choice of loop momenta. The corresponding ambiguity in
the choice of loop momenta can be removed [80] if one considers dual (or momentum twistor) variables and
planar limit, which we are interested in.
(5.5)
(5.6)
(5.7)
(5.8)
A L−1
=
l
A L−1
Red arrow indicates which propagator we are cutting.
which propagator we are cutting.
In(
1
)({Z1 . . . , Zn}, ZA, ZB). The residue at the point (we consider Zˆn = Zn + wZn−1 shift
and take residue with respect to w parameter)
is given by:
where
hAB1nˆ(w)i = 0
ReshAB1nˆi=0 In(
1
)
∼ Atnr+ee2(Z1, . . . , Zˆn, ZˆB, ZˆB),
ˆ
Zˆn = (n − 1, n) ∩ (A, B, 1),
ZB = (A, B) ∩ (n − 1, n, 1).
(5.9)
(5.10)
(5.11)
This is analog of (5.8) in momentum twistor space, see also figure 9 and 10. The first
expression for Zˆn solves hAB1nˆi = 0. The second expression for ZˆB is the consequence of the
first one and the forward limit. See [1] for detailed derivation and discussion. The
expression (5.10) in this limit could be obtained from the expression for Atnr+ee2(Z1, . . . , Zˆn, ZA, ZˆB)
at general kinematics18 by introducing GL(
2
) integration with [A, B, n−1, n, 1] weight (5.7).
Now let’s see how similar to (5.3) the recurrence relation for Wilson line form factors
can be constructed. Let’s consider integrand Ik∗,(nL+) 1 of A∗k(,nL+)1(Ω∗1, . . . , Ωn, gn∗+1) Wilson line
18General in a sense that there are no collinear twistors in contrast to (5.10).
form factor. As one will try to reconstruct it via Zˆi = Zi+wZi−1 shift he/she will encounter
two types of contributions. The first type will be given by the residues with respect to
propagators which does not contain loop momentum dependence. These can be considered
along the same lines as in sections 3 and 4. The second type of contribution is the residues
with respect to propagator poles with loop momentum dependence. Now in contrast to the
case of onshell amplitudes we have two types of propagator poles. Ordinary 1/l2 poles and
eikonal ones 1/hplp]. To simplify discussion let’s consider oneloop case. Generalization
to higher loops can be easily done by induction. The residue evaluation with respect to
1/l2 poles is identical to the onshell amplitudes case and is given by forward limit of tree
level Wilson line form factor with n + 2 onshell legs (k as usual is offshell momentum
with direction p and {q1, . . . , qn} are onshell momenta):
∼ A(∗n(t+re2e))+1({q1, . . . , qn, −lL, lL}, {p, k}).
(5.12)
The terms which include eikonal propagator pole residue are a little more complicated.
Surprisingly, here similar to the onshell case we also have forward like limit. For example,
consider Wilson line form factor at one loop level In∗(
1
)({q1, . . . , qn}, {k, p}, l). Here once
again {q1, . . . , qn} are onshell momenta and k is offshell momentum with direction p.
Using decomposition (4.13) we can decompose our offshell momentum into pair of
onshell momenta k′ = pikξi/hpξi, k′′ = ξikpi/hξpi and formally write this integrand as
In∗(
1
)({q1, . . . , qn, k′′, k′}, l). Considering residue for the pole 1/hplp] we enforce on loop
momentum l condition hplp] = 0, l2 6= 0 (see also (3.9) and discussion there). This results
can set l′ and k′ collinear to each other:
where l′ = pilξ′i/hξ′pi, l′′ = ξ′ilpi/hξ′pi. Now using the freedom in the choice of ξ′i one
(5.13)
(5.14)
(5.15)
(5.16)
(l′)μ = −(k′)μ/hξ′pi
up to scalar factor hξ′pi. This resembles the onshell forward limit kinematics of (5.8)
for n + 4 point offshell amplitude. So presumably the residue with respect to Wilson line
propagators can, in principle, be evaluated in momentum twistor space along the same lines
as (5.10) and (5.11). Consideration of this eikonal residue type, however, can be avoided
entirely if one will choose BCFW shift in such a way that w parameter will not appear in
eikonal propagators at all.
To see this let’s consider once again oneloop case, that is we take the solution of (5.3)
for n external particles Ik(L,n=1) and apply gluing operator Aˆn−1n to it
Ik∗,((Ln=−12))+1 = Aˆn−1n hIk(L,n=1)i .
We will assume that the tree level form factors and onshell amplitudes are related as
Aˆn−1n[Ak,n(Ω1 . . . , Ωn)] = Ak,(n−2)+1(Ω1 . . . , Ωn−2, gn∗−1).
∗
x n + 1 x
1
x n + 2
x
2
Z
n
Z
n
Z n−1
Z
1
Z
B
Z n + 2
gluing xn+1 with x1 while keeping xn fixed in such a way that x21n = 0. In momentum twistor space
this equivalent to gluing Zn+1 and Zn+2 with ZˆB. The same is also true for their supersymmetric
What we are going to show now is that Ik∗,((Ln=−12))+1 will have appropriate factorization
properties19 for one loop Wilson line form factor and that it can be obtained from recurrence
relation similar to (5.3), where only poles of (5.12) type will contribute. I.e. there always
will be possibility to choose the BCFW shift in such a way that only 1/P 2 type poles will
contribute to recursion.
To show this let us consider all possible BCFW shifts in Ik∗,((Ln=−12))+1. But first let us
note that in the case under consideration the pair of axillary momentum twistor
variables Zn and Zn−1 is used to encode information about offshell momentum k according
to (4.13). So, the only possible propagators which contain loop momentum and which
will be affected by gluing operator Aˆn−1n are given by hABn − 1ni and hABn1i. More
accurately, only hABn − 1ni will be transformed into eikonal propagator hABn − 1n∗i since
hABn∗1i = hABn1i. Equivalently one can note that due to the cyclical symmetry the only
possible eikonal propagator with loop momentum dependence in A∗k(,(Ln=−12))+1 will depend on
ZA, ZB, Zn−1, Zn, Z1 momentum twistors.
Now let’s return to the shifts. If we shift Zi as Zˆi = Zi + wZi−1 for i = 1, . . . , n − 2,
then the shift parameter w will not affect the eikonal propagator and the corresponding
residues with respect to w can be evaluated according to (5.12), so that the result will be
given by the forward limit of the tree level Wilson line form factor with n onshell states.
These is precisely the desired factorization property. For this form factor we also know
that the relation (5.16) holds. So we see that in such cases the gluing operation indeed
transforms solutions of (5.3) into Wilson line form factors similar to tree level.
As for the shifts involving Zn−1 and Zn, we can always choose to shift Zˆn = Zn +
wZn−1, so that the w parameter drops out of hABn − 1n∗i bracket and will remain only in
hABnˆ1i bracket, which is again not affected by the action of Aˆn−1n gluing operator. This
gives us
ReshABnˆ1i=0 Aˆn−1n[In(
1
)] ∼ Aˆn−1n[Atnr+ee2] = A(n)+1(Z1, . . . , Zn−1, Zˆn∗, ZˆB, ZˆB),
∗
(5.17)
19That is the corresponding residue will be given by forward limit of tree level Wilson line form factor
with n + 2 onshell states.
where
Zˆn∗ = (n − 1, n∗) ∩ (A, B, 1),
Zn∗ = Zn + hpξi
hp1i Z1.
Here we see that Zˆn∗ solves hABnˆ∗1i = hABnˆ1i = 0 — the same condition as in the case
of the onshell amplitudes. So once again we have appropriate factorization properties and
we also see that the gluing operation indeed transforms solutions of (5.3) into the Wilson
Equivalently using the same arguments as above one can show that in A∗k(,(Ln=−12))+1 in
pair Zn−1, Zn one can always choose to shift Zˆn = Zn + wZn−1 so that w will drop
out from eikonal propagator. That is for all Zˆi, which describe both onshell and
offshell momenta, one can choose such shifts that will not affect eikonal propagators with
loop momentum dependance and the corresponding recurrence relations will contain only
contribution of (5.5) and (5.12) type.
This considerations can be easily generalized by induction to arbitrary loop level and
ˆ
to arbitrary number of gluing operators applied. So we may conclude that application of
Ai−1i to (5.3) will likely result in a valid recursion relation for loop integrands of Wilson line
form factors (offshell amplitudes) similar to tree level case. For example if we chose i = n,
to match our previous considerations, we will get recurrence relation for the integrand
Ik∗,((Ln)−2)+1 of Wilson line form factor when operator is inserted after onshell state with
Ik∗,((Ln)−2)+1 = Ik(L,n)−1(Z1, . . . , Zn−1)
+ X[j − 1, j, n − 1, n∗, 1]Ik(L1,1n)+2−j (ZIj , Zj , Zj+1, . . . , Zˆn∗j )Ik(L2,2j)(ZIj , Z1, Z2, . . . , Zj−1)
(5.18)
(5.19)
[A, B, n − 1, n∗, 1]Ik(L+−1,1n)+2(Z1, Z2, . . . , Zˆn∗AB , ZA, ZB) ,
where Zˆnj = (n − 1, n∗) ∩ (1, j − 1, j), ZIj = (j − 1, j) ∩ (1, n − 1, n), ZˆnAB = (n − 1, n∗) ∩
(A, B, 1) and k1 + k2 + 1 = k. Zn∗ is given by (5.18). As before, to encode offshell momenta
we use twistor variables with numbers n − 1 and n. p and ξ are lightcone vectors entering
kT decomposition of this offshell momentum k. Spinors pi and ξi are obtained from
corresponding vectors.
One can also skip the solution of this new recursion and apply Aˆi−1i directly to the
solutions of onshell recursion relation (5.3), that is to the onshell integrands, similar to
the tree level case (4.16). In the next section we will consider such action using local form
of integrands instead of nonlocal form produced directly by BCFW recursion.
At the end of this section we want to make the following note: in general forward limits
may not be well defined [80], because on the level of integrands one may encounter
contributions from tadpoles and bubble type integrals on external onshell legs (see figure 7 C as
an example). However, such contributions are absent in N = 4 SYM onshell amplitudes
due to the enhanced SUSY cancellations [1, 80]. Their analogs are also absent for the
Wilson line form factors (offshell reggeon amplitudes) — there are no tadpoles diagrams
involving closed Wilson line propagators and bubbles on external Wilson line are also equal
to 0 on integrand level (see Feynman rules in [46]).
Gluing operation and local integrands
Now, following our discussion in the previous subsection we conclude that the integrands
for the planar offshell Lloop amplitudes could be obtained from the corresponding
onshell integrands by means of the same gluing procedure as was used by us at tree level.
Namely, for reggeon amplitude with n reggeized gluons (Wilson line operator insertions)
and no onshell states Ik∗,(0L+)n(g1∗, . . . , gn∗) we should have:
Ik∗,(0L+)n = Aˆ2n−1 2n ◦ . . . ◦ Aˆ12hIk,2n
i
= Ik(L,2)n
h1 2i
Z1, Z2 − h1 3i Z3, . . . , Z2n−1, Z2n −
h2n − 1 2ni
h2n − 11i
Z1 .
(5.20)
Here it is assumed that Ik∗,(0L+)n is normalized by A∗2,(00+)n similar to the definition of
onshell integrands (5.1). The loop integrands for reggeon amplitudes (Wilson line form
factors) with onshell states can be obtained from (5.20) by removing necessary number of
ˆ
Ai−1i operators.
The loop integrands produced by the BCFW recursion are nonlocal in general [80].
However, it is still possible to rewrite the integrands in a manifestly local form.20 Moreover,
one may choose as a basis the set of chiral integrals with unit leading singularities [80, 87].
The leading singularities are generally defined as the residues of a complex,
multidimensional integrals of integrands in question over C4L, where L is the loop order. The
computation of residues for the integrands expressed in momentum twistors is then ultimately
related to the classic Schubert problem in the enumerative geometry of CP3 [87]. When
the residues of integral associated to at least one of its Schubert problems are not the same
then the integral is called chiral. In the case when the integral has at most one nonzero
residue for the solutions to each Schubert problem then the integral is called completely
chiral. If all nonvanishing residues are the same up to a sign then it is possible to
normalize them, so that all residues are ±1 or 0. The integrals with this property are called pure
integrals or integrals with unit leading singularities.
The application of the gluing operation to the onshell integrands written in the local
form follows the general rule (5.20). Let’s see some particular examples. At oneloop for
MHV npoint integrand we have21 [80, 87]:
I2(1,n) =
X
hAB(i − 1 i i + 1) ∩ (j − 1 j j + 1)ihXiji
i<j hAB XihAB i − 1 iihAB i i + 1ihAB j − 1 jihAB j j + 1i
.
(5.21)
20This procedure spoils the Yangianinvariance of each term in the onshell case however.
21(i − 1 i i + 1) ∩ (j − 1 j j + 1) ≡ Zi−1Zihi + 1j − 1jj + 1i + ZiZi+1hi − 1j − 1jj + 1i + Zi−1Zi+1hij − 1jj + 1i.
I2∗,((1n)−2)+1 =
X
where Zn∗ is given by:
responds to eikonal propagators with shifted twistor. Wavy line corresponds to numerator of the
form hAB(ij)W i, where (ij)W = (i − 1 i i + 1) ∩ (j − 1 j j + 1).
This expressions is cyclic invariant and sum in the above expression is independent from
X, but contains spurious poles hAB Xi term by term. If we choose X = (k k + 1) then all
poles are manifestly physical but cyclic invariance will be lost. To obtain corresponding
expression I∗(
1
)
2,(n−2)+1 for the amplitude with one offshell leg in place of two last onshell
legs A∗2,((1n)−2)+1(Ω1, . . . , Ωn−2, gn∗−1) we just shift momentum super twistor Zn. Also it is
convenient to choose X = (n − 1n):
hAB(i − 1 i i + 1) ∩ (j − 1 j j + 1)ihn − 1n∗iji
i<j hAB n − 1 n∗ihAB i − 1 iihAB i i + 1ihAB j − 1 jihAB j j + 1i
Zn∗ = Zn − hp1i
hpξi Z1.
, (5.22)
(5.23)
See figure 11 A. Legs n−1 and n describe offshell momentum, so that p and ξ are lightcone
vectors entering kT decomposition of this momentum k.
Next, taking the expression for the integrand of 2loop 4point MHV onshell
amplitude [80, 87]:
I2(
2,4
) =
h2341ih3412ih4123i
hAB41ihAB12ihAB23ihCD23ihCD34ihCD41ihABCDi
+cyclic, no repeat (5.24)
and applying Aˆ3,4 gluing operation we get for the integrand of A∗2,(22+)1(Ω1, Ω2, g3∗) (See
h3412ih4123ih1234i
(5.25)
1
2
2
1
2
Vertical red line represents cuts of corresponding propagators. Grey blobs are onshell amplitudes
with k = 2, 3. Dark grey blobs are Wilson line form factors with k = 2, 3.
HJEP06(218)9
where Z4∗ is given by:
Z4∗ = Z4 − hp1i
hpξi Z1.
As always we assume offshell kinematics for legs 3 and 4, so that p and ξ are lightcone
vectors entering kT decomposition of the offshell gluon momentum k. Note also that this
result is consistent with two and three particle unitarity cuts. See figure 12.
The introduced gluing operation also allows us easily obtained expressions for
integrands of offshell remainder functions starting from their onshell counterparts. Indeed,
starting from integrand for 1loop onshell remainder function
R(k1,n) = Ik,n − Pn
(
1
)
4(k−2)I2(1,n)
and applying gluing operation Aˆn−1,n we may obtain the expression for offshell remainder
function with one offshell leg in place of two last onshell legs
Rk,(n−2)+1 = Ik∗,((1n)−2)+1 − Aˆn−1,n Pn
∗(
1
)
That is, for example taking integrand for R3∗,(61) onshell remainder function written in terms
of chiral octagons [87]:
R3∗,(61) =
1
2
1
where
and (see figure 13)
([1, 2, 3, 4, 5]+[1, 2, 3, 5, 6∗]+[1, 2, 3, 6∗, 4])I8(1, 3, 4, 6∗)+ [1, 2, 3, 4, 6∗]I8odd(1, 3, 4, 6∗)
1
− 6 ([1, 3, 4, 5, 6∗]−[1, 2, 3, 4, 5])I8odd(
1, 3, 4, 5
)+ 6 ([1, 2, 4, 5, 6∗]+[1, 3, 4, 5, 6∗])I8odd(1, 4, 5, 6∗) ,
1
6
I8odd(i, j, k, l) ≡ I8(i, j, k, l) − I8(j, k, l, i)
I8(i, j, k, l) =
hABijihAB(j − 1 j j + 1) ∩ (k − 1 k k + 1)i
hABi − 1 iihABi i + 1ihABj − 1 jihABj j + 1i
×
hABklihAB(l − 1 l l + 1) ∩ (i − 1 i i + 1)i
hABk − 1 kihABk k + 1ihABl − 1 lihABll + 1i
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
(5.31)
k
j
numerator of the form hABiji.
As before Z6∗ is defined as
Z6∗ = Z6 − hp1i Z1 ,
hpξi
and we again assume offshell kinematics for legs 5 and 6 with p and ξ denoting lightcone
vectors entering kT decomposition of reggeized gluon momentum.
Now we would like to show one simple but interesting test both for our tree and loop
level constructions (4.16), (5.20) and obtain the expression for LO BFKL kernel with gluing
operation.
LO BFKL and gluing operation
at large center of mass energy √
s and fixed momentum transfer √
Within BFKL approach [62–66] amplitudes of scattering of some quantum states A + B →
A′ + B′, which can be partons in hadron, hadrons themselves, high energy electrons etc.,
−t, s ≫ t can be
represented as
AAB
A′B′ = hΦA′AeαsN ln(s/s0) KBFKL ΦB′Bi ,
where the so called impact factors hΦA′A and ΦB′Bi are process dependent functions and
describe the transitions A → A′ and B → B′. This scattering, in the mentioned above
regime, can be described via interaction with special quasiparticles — so called reggeized
gluons. BFKL kernel KBFKL describes the self interaction of these reggeized gluons. s0 is
some process related energy scale. See for example [70] for detailed discussion.
Let us now calculate the LO kernel of BFKL equation in N = 4 SYM with the use of
our gluing operation. At LO order it is given by two contribution so called real and virtual
one. Consider virtual contribution first (also see figure 14 A).
5.3.1
Virtual part of LO BFKL
To compute virtual contribution to the LO BFKL we need the Regge trajectory. The latter
could be conveniently extracted from the oneloop correlation function of two Wilson lines
playing the role of sources for reggeized gluons [70]. Namely, we have to compute the
(5.32)
(5.33)
following offshell amplitude:
h0Wp1 (k)Wp2 (−k)0i = A2,0+2(g1∗, g2∗) = Aˆ12 ◦ Aˆ34 A2,4(1−, 2+, 3−, 4+) .
∗
where β1,(
1
), β2,(
1
) parameters correspond to Aˆ12 gluing operation and those with (
2
)
subscripts to Aˆ34. Evaluating
β2,(i) = 0 we get
substitutions and taking composite residues at β1,(i) = −1,
HJEP06(218)9
A∗2,(00+)2 = −
hpκ1∗pκ2∗i2 .
1 2
Now we should recall that the Wilson lines were used here to describe scattering of two fast
moving particles at high energy.23 This restricts further our kinematics, so that p1 ·p2 = s/2
(s is the usual Mandelstam variable) and momentum transfer between two particles is
restricted by two orthogonality conditions k · p1 = k · p2 = 0. The latter two conditions
allow us to write down transverse momentum transfer as
k = c1λp1 λ˜p2 + c2λp2 λ˜p1 ,
so that t ≡ k2 = c1c2s and24
Then for A∗2,(00+)2 amplitude we have
κ∗1κ∗2 =
c2hp1 p2i [p1 p2] c1hp2 p1i [p2 p1] = −c1c2hp1 p2i2 = − s hp1 p2i2.
t
[p1 p2] [p2 p1]
Now let us turn to the integrand of the corresponding oneloop amplitude. The latter is
given for n = 4 by (5.21):
A∗2,(00+)2 =
t
I2∗,(01+) 2 =
h1234i2
h12∗ABih23ABih34∗ABih41ABi
Z2∗ = Z2 − hp1p2i
hp1ξ1i Z3, Z4∗ = Z4 − hp2p1i
hp2ξ2i Z1.
,
1
(5.34)
(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
with
This expression can be rewritten in spinor helicity variables as:
22The gluing details are similar to those considered in sections 2 and 3.
23For the introduction to corresponding description see [70].
24It is convenient here to chose ξ1 = p2 and ξ2 = p1.
I2∗,(01+) 2 =
t2hp1 p2i2
1
=
st
4κ∗1κ∗2 l2(l + k)2 l · p1 l · p2
4 l2(l + k)2 l · p1 l · p2
(5.42)
The same result can also be obtained within helicity spinor picture
where we arrange loop momenta as:
I2∗,(01+) 2 = Aˆ12 ◦ Aˆ34 hI2(
1,4
)(1−, 2+, 3−, 4+)i ,
I2(
1,4
)(1−, 2+, 3−, 4+) = l2(l + q2)2(l + q1 + q2)2(l − q3)2
.
(q1 + q2)2(q2 + q3)2
In the expression above qi, (i = 1, . . . , 4) are momenta of external gluons and l is loop
momentum.
In LO BFKL regime we are interested in leading logarithmic approximation (LLA) to
highenergy scattering amplitude. The latter could be obtained using Sudakov
decomposition of loop integration momentum and retaining only logarithmic in Mandelstam invariant
s contribution. That is
2
dDl =
dα dβ dD−2l⊥,
l = αp1 + βp2 + l⊥,
pi · l⊥ = 0,
k · pi = 0,
and we are interested in the following regime (here m is some problem related mass scale):
(5.43)
(5.44)
(5.45)
(5.46)
(5.47)
(5.48)
(5.49)
(5.50)
(5.51)
(5.52)
(5.53)
(5.54)
Then
Now taking residue in β at 0 and integrating over α from m2/s to 1 we get
1 ≫ α ≫ β ∼
≪ 1.
p2 · l = αs/2 + p1 · l⊥ = αs/2
p1 · l = βs/2 + p2 · l⊥ = βs/2
l2 = αβs/2 − l⊥
2
(l + k)2 = αβs/2 − (l⊥ + k⊥)2.
– 34 –
Z
1
(2π)4 l2(l + k)2(p1l)(p2l)
Z
2(2π)4
1
4π3s log
dα
dβ
dD−2l⊥
αs/2 βs/2 [αβs/2 − l⊥2][αβs/2 − (l⊥ + k⊥)2]
m2
Z
A∗2,(00++21) = A∗2,(00+)2
g
2
1 − 16π3 log
m2
Z
k2 d2l⊥
⊥
l⊥2(l⊥ + k⊥)2
This expression tells us that in LLA approximation25 with account for color factor (CA = N
for SU(N ) gauge group) for reggeized gluon propagator we get
h0Wp1 (k)Wp2 (−k)0i LLA ∼ k2
1
⊥
s
m2
ω(t) = −αsN
Z d2+εl⊥
k
2
N αs 2(k⊥2)ε/2
ε
is the famous LO BFKL Regge trajectory, which at LO is the same in QCD and N = 4
SYM. See for example [88–90]. Using this result for virtual part of BFKL kernel we can
write [70]:
− αsN KBVFKL = − 12 δ(
2
)(k − k′) (ω(k⊥) + ω(k⊥ − r⊥)) .
Here r is the momentum transfer for A → A′ scattering r = pA′ − pA.
In conclusion we would like also to note the following interesting fact. In N = 4
SYM the four point onshell amplitude A2,4 has a remarkable property27 of being Regge
exact, i.e. the contribution of the gluon Regge trajectory to the amplitude (c(t) is the gluon
A2,4(s, t) = c(t)2
+ subleading terms in   ,
t
coincides with the exact expression for A2,4(s, t) as a function of arbitrary s and t.
5.3.2
Real part of LO BFKL
Now let’s consider real contribution. This contribution is given by the integrated product of
two, so called, Lipatov’s Lμ RRP vertexes [70], see figure 14 B. To compute this contribution
we may note, that Lipatov’s RRP Lμ vertex (tree level reggeonreggeonparticle amplitude)
is related to reggeon amplitudes A∗2,2+1(g1∗, g2∗, 3+) and A∗3,2+1(g1∗, g2∗, 3−) as
Lμ(k, k′) = (k′ + k)μ + n
μ
k
2
k′− − k
+ n
k+ − k′− ,
A2∗,2+1(g1∗, g2∗, 3+) = δ4(k − k′ − q3)A2∗,2+1(k, −k′, −q3) = δ4(k − k′ − q3) 3
A3∗,2+1(g1∗, g2∗, 3−) = δ4(k − k′ − q3)A3∗,2+1(k, −k′, −q3) = δ4(k − k′ − q3) 3
ǫμ,+Lμ(k, k′)
ǫμ,−Lμ(k, k′)
which in their turns could be obtained with two our gluing operations applied to 5point
onand n± are normalized light like directions for reggeized gluons
shell amplitude A2,5. Here k, k′ are reggeized gluons g1∗ and g1∗ momenta with k −k′ −q3 = 0
n
− = √ ,
2p1
s
n
+ = 2√p2 , (n−n+) = 2, (kn±) ≡ k±,
and ǫ3± are polarization vectors of onshell gluon with momentum −q3. It is assumed that
in the definitions of A∗2,2+1(g1∗, g2∗, 3+) and A∗3,2+1(g1∗, g2∗, 3−) amplitudes one has to take in
26Here we introduced dimensional regularization of otherwise divergent integral.
27See the discussion in [91].
(5.55)
(5.56)
(5.57)
(5.58)
(5.59)
(5.60)
B
B’
B
B’
A)
B)
N = 4 SYM. At large center of mass energy √
s and fixed momentum transfer √
totical behaviour of an amplitude is given by its imaginary part [50, 70]: AAB
Grey squares represents impact factors, wavy lines represents gluon propagators, vertical red line
represents cuts of corresponding propagators and impact factors. Diagrams of type A) gives
congive contribution to KBRFKL and their total sum is equivalent to evaluation of (5.62).
tribution to KBVFKL and their total sum is equivalent to evaluation of (5.53). Diagrams of type B)
A B
′ ′
s≫1 ∼ Im[AAABB ].
kT decomposition of k and k′ momenta direction vectors as p1 = n− and p2 = n+. We
have also defined functions A∗2,2+1 and A∗3,2+1 which are given by corresponding Wilson
line form factors stripped from momentum conservation delta functions:
A∗2,2+1(k1, k2, q3) =
A∗3,2+1(k1, k2, q3) =
[n−n+]3
κ∗1κ∗2 h3n+ihn−3i
κ1κ2 [3n+][n−3]
.
,
Performing Sudakov decomposition28 of reggeized gluon momentum the contribution
of real radiation to BFKL kernel takes the form [70]:
KBRFKL(k⊥, k⊥′, r) ln
m2 =
s Z
2
dαkdβk′ Lμ(k, k′)Lμ(r − k, r − k )
′ δ(αkβk′ s + (k⊥ − k⊥′)2)
′
k⊥2(r⊥ − k⊥′)2
Note that factor Lμ(k, k′)Lμ(r − k, r − k′) = gμν Lμ(k, k′)Lν (r − k, r − k′) can be
rewritten purely in terms of Wilson line form factors.
Namely using gauge invariance of
A∗2,2+1(g1∗, g2∗, 3+) and A∗3,2+1(g1∗, g2∗, 3−) we can replace
28See for example [70] for details.
gμν 7→
X ǫ(μi)ǫ(νi),
i=±
(5.61)
.
(5.62)
(5.63)
HJEP06(218)9
so that (r − k ≡ m and r − k′ ≡ m′)
Lμ(k, k′)Lμ(m, m′)
= A∗2,2+1(k, −k′, −q3)A2∗,2+1(m, −m′, −q3)
+ A∗3,2+1(k, −k′, −q3)A3∗,2+1(m, −m′, −q3).
(5.64)
r
2
r
2
αk is performed over the interval [ ms2 , 1]. This way we get
Also note that in this case no other particles besides gluons from N = 4 SYM supermultiplet
give contribution to real radiation. This happens due to the Rcharge conservation.
The integral over βk′ is taken with the help of δfunction, while the integration over
KBRFKL(k⊥, k⊥′, r) = − k⊥′2(r⊥ − k⊥′)2 + ′
⊥
⊥
k⊥2(k⊥ − k⊥′)2 +
k
2
(r⊥ − k⊥)2
(r⊥ − k⊥′)2(k⊥ − k⊥′)2
. (5.65)
Altogether with account for the Regge trajectories contributions we recover LO expression
for BFKL kernel KBFKL = KBRFKL + KBVFKL [70]:
KBFKL(k⊥, k⊥′, r) = − k⊥′2(r⊥ −k⊥′)2
⊥
+ ′
k
2
(r⊥ −k⊥)2
(r⊥ −k⊥′)2(k⊥ −k⊥′)2
− 12 δ(
2
)(k −k )
′ Z d2l⊥
4π2
k
2
(k⊥ −r⊥)2
(l⊥ −r⊥)2(k⊥ −l⊥)2
. (5.66)
6
In this paper we considered the derivation of the BCFW recurrence relation for the Wilson
line form factors and correlation functions (offshell reggeon amplitudes) both at tree and
at integrand level. We have shown that starting from the BCFW recursion for onshell
amplitudes and using so called “gluing operator” one can obtain recursion relations for the
Wilson line form factors. The latter is true both at tree and integrand level in helicity
spinor and momentum twistor representations. The gluing operation also allows one easily
convert known local integrands of the onshell amplitudes into integrands of the Wilson
line form factors. These results are condensed in formulas (4.16), (4.17) and (5.19), (5.20)
for tree and loop level correspondingly. We have verified our considerations by reproducing
LO BFKL kernel. We also made some predictions regarding the structure of the integrands
of Wilson line form factors at higher loops/large number of external states.
As far as we can understand our construction is not limited to the Wilson line operators
only. Indeed, using [24] similar gluing operator the form factors of stress tensor operator
supermultiplet could be constructed. Also, presumably, analogs of gluing operator for all
other type of local operators in N = 4 SYM theory should exist. The only real obstacle
in this direction is that at the level of integrands for local single trace operators we should
account for nonplanar contributions. So the notion of “integrand” in this case is somewhat
obscure at first sight. Nevertheless we think that one can still introduce integrands in setup
similar to considerations in [92], where nonplanar contributions to the onshell amplitudes
were considered in momentum twistor variables.
We hope that the presented results will be interesting and useful for people both from
N = 4 SYM “amplitudology” and BFKL/reggeon physics communities.
Acknowledgments
This work was supported by RFBR grants # 170200872, # 160200943, contract #
02.A03.21.0003 from 27.08.2013 with Russian Ministry of Science and Education and
HeisenbergLandau program. The work of L.V. Bork was supported by the grant #
17133251 of the “Basis” foundation for theoretical physics.
A
Gluing operation and Grassmannians
Let’s see how the use of the gluing operation in momentum twistors could easily reproduce
known Grassmannian integral representations for the treelevel offshell amplitudes [19, 20].
We start with the Grassmannian integral representation for the onshell amplitudes in
momentum twistors:
Lk,n+2 =
Ak,n+2 =
Z d(k−2)×(n+2)D
δ4(k−2)4(k−2)(D · Z)
Vol[GL(k − 2)] (1 . . . k − 2) . . . (n + 2 . . . k − 3)
,
here (also similar notations are used in (2.26))
δ4(k−2)4(k−2)(D · Z) =
k−2
Y δ44
a=1
X DaiZi ,
and (i1, . . . , ik−2) is minor constructed from columns of D matrix with numbers i1, . . . , ik−2.
Applying to this expression the gluing operation Aˆn+1,n+2 amounts to the following shifts
of momentum super twistors:
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
so that
where
′
Z1 → (1 + α1α2)Z1 + α1Zn+2 ≡ Z1 ,
′
Zn+2 → Zn+2 + α2Z1 ≡ Zn+2 ,
D1′ = (1 + α1α2)D1 + α2Dn+2 ,
′
Dn+2 = α1D1 + Dn+2
′ ′
D1 = D1 − α2Dn+2 ,
Dn+2 = −α1D1′ + (1 + α1α2)Dn′+2 .
transformation from D’s to D′’s is then given by
All other momentum super twistors are unshifted and we have Di′ = Di. The inverse
With these transformations it is easy to write down transformation rules for minors. For
example, we have
(1 . . . k−2) → (1 . . . k−2)′ −α2(n+2 2 . . . k−2)′
(n−k+5 . . . n+2) → −α1(n−k+5 . . . n+1 1)′ +(1+α1α2)(n−k+5 . . . n+2)′
Finally performing transition in the Grassmannian integral from D’s to D′’s and taking
residues at α1 = 0 and α2 = − hn+1 1i
hn+1 n+2i = − hpn+1 1i
hpn+1 ξn+1i we get
A∗k,n+1 =
A∗2,n+1
Z d(k−2)×(n+2)D′
1
(1 ... k−2)′
Γ Vol[GL(k−2)] 1+ hpn+1 ξn+1i (n+2 2 ... k−2)′ (1 . . . k−2)′ . . . (n+2 . . . k−3)′
δ4(k−2)4(k−2)(D′ ·Z)
,
(A.12)
Similarly applying several gluing operations we recover formula [20].
Open Access.
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