#### Regge meets collinear in strongly-coupled \( \mathcal{N}=4 \) super Yang-Mills

Received: November
meets collinear in strongly-coupled
Martin Sprenger 0
Open Access 0
c The Authors. 0
0 Institut fur Theoretische Physik, Eidgenossische Technische Hochschule Zurich
We revisit the calculation of the six-gluon remainder function in planar N = 4 super Yang-Mills theory from the strong coupling TBA in the multi-Regge limit and identify an in nite set of kinematically subleading terms. These new terms can be compared to the strong coupling limit of the nite-coupling expressions for the impact factor and the BFKL eigenvalue proposed by Basso et al. in [1], which were obtained from an analytic continuation of the Wilson loop OPE. After comparing the results order by order in those subleading terms, we show that it is possible to precisely map both formalisms onto each other. A similar calculation can be carried out for the seven-gluon amplitude, the result of which shows that the central emission vertex does not become trivial at strong coupling.
Scattering Amplitudes; AdS-CFT Correspondence; Supersymmetric gauge
1 Introduction 2
The 3 ! 3{amplitude at strong coupling
Six-point amplitude from the TBA
Multi-Regge kinematics
Calculation of the remainder function R3!3
R3!3 in the Mandelstam region
R3!3 in the Mandelstam region and subleading kinematics
Subleading kinematics from the OPE
Extracting the subleading pieces
Mapping TBA $ BFKL OPE
Subleading kinematics for the 2 ! 5{amplitude
A Derivation of the impact factor at strong coupling
A.1 u^; v^ > 1
A.2 u^; v^ < 1
Introduction
The high-energy behavior of scattering amplitudes in planar N
= 4 super Yang-Mills
number of loops to
nite coupling still seems di cult. One might therefore hope that
for full kinematics.
Indeed, in a series of papers it was shown that the six-point remainder function has a
15], before a
nite-coupling proposal was put forward in [1].
Expanding the dispersion integral at weak coupling leads to an expansion of the
reof this paper.
not the case is another point of this paper.
Another special kinematic con guration is the collinear limit, which is governed by
the Wilson loop OPE [33{36]. This expansion takes the form of a
ux-tube spanned
expressions for the BFKL eigenvalue and the impact factor from
nite-coupling expressions
governing the energies and momenta of certain
ux-tube excitations. Remarkably, the
and the strong coupling result of [14, 15] and pass all tests.
In this paper, we identify kinematically subleading terms at strong coupling which
allow a more detailed check of the
nite-coupling expressions with the result of the TBA
potential nite-coupling expression for the central emission vertex.
This paper is organized as follows. In section 2 we brie y review the calculation of
the 3 ! 3{amplitude at strong coupling from the TBA. We then show how to obtain
2 ! 5{amplitude in section 4 before concluding in section 5. The technical details for the
appendix A.
The 3 ! 3{amplitude at strong coupling
In this section, we brie y review the calculation of the 3 ! 3{amplitude at strong coupling
Six-point amplitude from the TBA
remainder function,1 which depends only on the three dual-conformal cross ratios
u1 =
u2 =
u3 =
only arise in contributions which are subleading in p .
1Note that at strong coupling, there is no distinction between the MHV and the NMHV case. Di erences
xi, with xi+N
normalization, because the prefactor in eq. (2.1) is subleading in p . The remainder
xi and xi;j := xi
xj . The amplitude is only
xed up to an overall
function is given by several terms,
The simplest piece of eq. (2.3) is
which is directly given in terms of the cross ratios
and reads
R6 :=
(Afree + Aper +
Afree = jmj Z
Aper =
u1 =
to the remainder function are given by
d cosh log
1 + Ye1( )
1 + Ye3( )
1 + Ye2( )
The cross ratios (2.2) can be calculated from the Yea-functions through the relations
u2 =
u3 =
Those Yea-functions satisfy a set of TBA-like equations,
log Yea( ) =
ma cosh
where the parameters are given by
and the integration kernels read
m1 = m3 = jmj; m2 = p
C1 =
C3 = C; C2 = 0
in terms of the two functions
Kaa0 ( ) = B@K2( ) 2K1( ) K2( )C
A
K1( ) K2( ) K1( )
K1( ) =
K2( ) =
a dependence on the parameter ' arises. For completeness, let us mention that the
Yeafunctions satisfy a recursion relation,
Yea( ) =
This relation allows us to easily construct the Yea-functions far away from the real axis,
tion kernels.
All of the above has a nice generalization to the general n-gluon case, but the
expressions become more complex. We therefore refer the reader to [23] for details.
Multi-Regge kinematics
Regge limit of the 3 ! 3{amplitude. As described in [58], this limit is characterized by the
following behavior of the cross ratios,
reduced cross ratios
remain constant. This di ers from the 2 ! 4{amplitude only in that the large cross ratio
from the exact relation
C = cosh 1
= cosh 1
1 + u~2 + u~3
2 u1u~2u~3
which can be derived from the recursion relation (2.11) as well as the exact relation
than minus one, which leads to
with Ce being real. For comparison, in the 2 ! 4{case C is purely imaginary. This, however,
the analysis of [59] for the 2 ! 4{case, and use the relations between the Yea-functions and
= e2C ;
C = i + Ce;
limit. The result is that the limit
jmj ! 1;
C const:
which behave as " ! 0 and w ! const: in the multi-Regge limit, we nd the following
parametrization of the cross ratios in terms of the TBA parameters
u1 = 1
+ 2 cosh C ;
u2 = "w;
u3 =
with corrections of O("2). This parametrization nicely shows the behavior (2.12) once we
take into account eq. (2.16).
Calculation of the remainder function R3!3
of the remainder function (2.3). The two contributions
and Aper are easily computed.
Simply plugging in the parametrization (2.18) and expanding in " we obtain
Aper =
+ O (" log ") ;
are exponentially suppressed in jmj. Indeed, a careful analysis shows that
and is therefore negligible in the limit " ! 0, see [14] for details. Summing up all
contributions, we nd that the remainder function is a constant
Afree = O(" log ")
R3!3 =
This constant, however, comes solely from the Li2-part of
and cancels with a similar term
there is a kinematic region of the 3 ! 3{amplitude, in which a Regge cut is known to
continuation in the cross ratios as
= 0 : : : ;
to nd a non-trivial remainder function.
and refer the reader to those references for details.
As explained in the last section, we want to perform an analytic continuation in the
cross ratios. This is trivial for the
-contribution to the remainder function (2.4), as it
is subtle for the following reason. For any given Yea-function there are locations in the
complex -plane where Yea( ) =
1. The location of these points, of course, depends on
the TBA parameters. Hence these points will move in the complex
-plane during the
Yea-function they are associated to, i.e. we have
Yea(ea;i) =
1 for i = 1; : : : ; na:
TBA equations as
Kaa0 ( ) =:
@ log Saa0 ( ):
S1( ) = i
S2( ) =
log Ye a0( ) =
m0a cosh
0) log 1 + Ye a00 ( 0)
a0 i=1
X sign(Im(ea;i)) log Saa0 (
The quantities Saa0 ( ) appearing in eq. (2.24) are related to the kernels via
For the basic kernels K1( ) and K2( ) they explicitly read
contributions to the remainder function using
A0per =
A0free = jmj0 Z
d cosh log
1 + Ye 10( )
1 + Ye 30( )
1 + Ye 20( )
a i=1
X sign(Im(ea;i)) sinh ea;i:
of the residue terms in both the TBA equations and the A0free-contribution. To obtain an
explicit result for the 3 ! 3{remainder function for the continuation (2.22), all we need to
do is gure out how many crossing solutions there are for the three Yea-functions and what
their locations ea;i at the end of the continuation are.
The key di culty in those calculations is to gure out which path the TBA parameters
jmj and C have to follow for a given path in terms of the cross ratios.
Basically, it
the equations Yea( ) =
1 numerically to see whether any of those solutions cross the real
proceed with the analytic continuation.
the continuation we go to the multi-Regge regime jmj0 ! 1, where we can neglect the
the continuation, the Yea-functions can be evaluated at the locations of the crossed solutions
which by de nition yields
locations ea;i. Therefore, our
though it involves numerical intermediate steps.
nal result for the remainder function will be exact, even
Following this algorithm for the path (2.22) for the cross ratios, we
solutions of the equation Ye1( ) =
1 cross the real axis,2 as shown in
nd that two
gure 1. No
solutions of the other Yea-functions cross the real axis. We therefore have two crossing
solutions, which we will call
in the following. Furthermore, solving the Bethe ansatz
2The fact that we nd crossing solutions for Ye1 is related to our choice of Re (Ce) > 0 in our numerical
analysis. Choosing Re (Ce) < 0 would lead to crossing solutions in Ye3, which, however, gives rise to the
same result for the remainder function.
a0 j=1
−1
−1
−0.5
1 during the analytic continuation (2.22). The
w = 1 and C = i
We switch colors from blue to red when two of the solutions cross
the real axis. The convergence of the endpoint position of the crossing solutions against
clearly visible.
equations (2.29) we nd that
is exactly the same as in the 2 ! 4{case [15].
We have now assembled all necessary
ingredients to calculate the remainder function in the Mandelstam region.
R3!3 in the Mandelstam region and subleading kinematics
considered in those references.
We begin from the modi ed TBA equations valid in the Mandelstam region. Since we
send jmj0 ! 1 at the end of the continuation, we can neglect all integrals and nd
log Ye a0( ) =
m0a cosh
Ca0 + log
From the Ye a0-functions we can then calculate the cross ratios at the endpoint of the
continuation through the relations (2.10), from which we nd
u02 =
cos '0 + sin '0
u03 =
2 + cos '0 + sin '0
; (2.32)
tions of O("0 2). Using our choice of path (2.22) we then demand that
u02 =
u03 =
eters " and w. Using '0 = tan 1
logarithm in eq. (2.35) is positive. For Aper, we use eq. (2.34) and nd4
A0per =
log2 w0 = Aper
3Note that for any given numerical value of " there is, of course, an exponent N such that log N " and
when indeed all corrections of the form log n " are smaller than the corrections of the form "n.
4Note that due to the quadratic term log2 "0 in Aper we always have to expand the parameters "0, w0 in
eq. (2.34) one order higher than the order we want to compute the remainder function to.
"0 =
w0 = w
2p2. These are the subleading kinematic corrections we are after in this
paper. In the previous analysis, only the leading terms (i.e. without any factors of log n ")
" ! 0, since both the integrals we neglect in the TBA equations as well as the higher
order terms neglected in eq. (2.32) are of the form O("n) and therefore much smaller than
-contribution to the remainder function is easily evaluated, since it is a function
tion (2.22) to nd
0 =
+ 2 cosh C
llooggw"00 , these equations can be solved order by order
Lastly, the A0free-contribution can be calculated via eq. (2.28) once we specify the number
and endpoints of the crossing solutions. After neglecting the integrals, we obtain
A0free =
2 jmj0 = p
2 log "0 1 +
= p
Note again that as described in section 2.3 the integrals in A0free are of O(" log ") and
all contribution to nd the remainder function
2 ) log " +
+ 2 cosh C
coe cients from this approach.
Subleading kinematics from the OPE
As mentioned in the introduction, a conjecture for a
nite-coupling expression for the
m= 1
where the kinematic variables are related to the cross ratios via
log u2u3;
= log w:
variables , for the TBA variables and nd
in eq. (3.1) agrees with the phase of the TBA result (2.39). The nite
factor (or BFKL measure, both expressions will be used interchangeably)
given in [1]. Here, we only spell out their form in the strong coupling limit p
derived in [1] and read
!( ) SC =
( ) SC =
2 2 cosh
2 2 sinh
1 + i 2 sinh
i 2 sinh
need the behavior of the impact factor
BFKL(u) at strong coupling. This quantity is not
related object, the OPE measure of the small fermion.5
Given the technical nature of
which reads
5We would like to point out that the result (3.5), while unpublished, was already derived by Benjamin
Basso and thank him for sharing the nal expression (3.5) with us.
we can evaluate the integral by means
log BFKL( ) SC =
cosh(2 1) cosh(2 2) cosh( 1
which holds in the region u^ < 1.
Extracting the subleading pieces
Since all quantities in eqs. (3.4), (3.5) scale like p
of a saddle point approximation, i.e. we rst solve the equation
0 = @ log BFKL( ) + i log w @
log " @ !( )
for the saddle point 0 and then obtain the remainder function as
elog BFKL( 0)+i log w ( 0) log " !( 0):
not know which m-mode is dominant. We therefore write
in eq. (3.7). Similar to the
TBA-case, we can expand the result in orders of lo1g " . For example, it is easy to see that
the leading order result is given by
0 = 0 + O(log 1 ")
which leads to
2 log(1+
2 2 log(1+
2 ) + log2(1+
We have performed this calculation up to ten orders in log n " and found perfect matching
of the BFKL OPE and the TBA results, which strongly supports the conjectured
in the next section that this is indeed the case.
Mapping TBA $ BFKL OPE
remainder function |
rst the kinematic aspect of nding the saddle point for the OPE
remainder function on these solutions.
It is therefore natural to expect that the saddle point equation (3.6) can be mapped
to the equation determining '0 (2.33). To see this, we start from the de nition of '0,
log w =
tan '0 log " + h5('0);
'0 = tan 1
and use eqs. (2.32), (2.33), (2.18) to rewrite this in the form
i log w g1('0)
log " g2('0) + g3('0) = 0;
perfectly, once we identify
0 = i'0:
Similarly, it should then be possible to map the expressions for the remainder
functhe TBA side can be written as
in terms of the parameters jmj0 and jmj. Keeping in mind the identi cation (3.12) as well
to rewrite the TBA remainder function (3.13) in the form
e p2 (i log w h1('0) log " h2('0)+h3('0)):
freedom in rearranging terms due to the relation
which we do not spell out explicitly. We then obtain
e 2 (i log w(h1('0)+h4('0)) log "(h2('0) ih4('0) tan '0)+(h3('0) ih4('0)h5('0))); (3.16)
two formalisms are identical at strong coupling.
Subleading kinematics for the 2 ! 5{amplitude
u11 = 1
u12 = 1
+ 2 cosh C2 ; u21 = "2w2; u31 =
+ 2 cosh C1 ; u22 = "1w1; u32 =
in the multi-Regge limit where "i ! 0, the wi are real and constant, and Ci are purely
imaginary and constant. Corrections to the cross ratios in eq. (4.1) are of O("2).
Furthermore, there is another, dependent cross ratios u~, which behaves as 1
analytic continuation
with all other cross ratios held
xed. Subtleties in probing this region, usually denoted as
, from the TBA are discussed in [27], but do not play a role here. In this region,
the all-loop remainder function is expected to be of the form [25]
= i
!( 2;n2)jz2j2i 2 ( 2; n2) sub + : : : ;
by zi = w3 i
1 eC3 i . While the BFKL eigenvalue !( ; n) and the impact factors ( ; n) in
investigated in [27] with the result that
where again e2 =
= 7;
Note that the phase ~7;
in eq. (4.5) already slightly di ers from the predicted valued
in [24, 25] by an additional piece. This di erence, however, could well arise from the
not trivial at strong coupling and links the two integrations.
parametrize the remainder function as
k1= 1 k2= 1
ck1;k2 (w1; w2) log k1 "1 log k2 "2:
In this notation, the leading terms of eq. (4.4) correspond to the terms c 1;0, c0; 1 and
c0;0. Some of the lowest subleading terms read6
c1;0(w1; w2) =
log w1 + const: ; (4.7)
c0;1(w1; w2) = c1;0 (1=w2; 1=w1)
via target-projectile symmetry and
c1;1(w1; w2) =
6 log w1 log w2 + 6 log 1 + p
2 2 log w2 + const:
6We provide the full form of the rst four orders of subleading terms in the le 7pt subleading.m
attached to the arXiv submission of this paper.
While their explicit form is not particularly simple, it is very interesting that we
subleading terms of the form log 1 "1 log 1 "2. These terms couple the two triplets of cross
coupling, assuming that the form (4.3) holds.
Conclusions
can actually be precisely mapped onto each other at strong coupling.
in the Mandelstam region P7;
and found that there are subleading terms which couple
an open question for future investigations.
Acknowledgments
Science Foundation through the NCCR SwissMAP.
Derivation of the impact factor at strong coupling
at strong coupling, eq. (3.5). We are interested in the limit g ! 1, while keeping the
rescaled rapidity u^ = 2ug
xed and the mode number m of O(1). To derive the impact
as derived in [1],
g2(x[+m]x[ m]
x[+m]x[+m]
eA+2fB(3F)KL;m(u) 2fB(4F)KL;m(u);
The constant A is given by
x[ m] = x u i
x(u) = (u + pu2
A = 2
J0(2gt)2
and the source terms
M = (1 + K) 1;
Qij = ij( 1)i+1i (A.4)
BFKL =
Z 1 dt Jj(2gt) et jeven + ( 1)jet jodd
cos(ut)e mt=2
J0(2gt) ;
sin(ut)e mt=2;
f B(4F)KL;m(u) are de ned as
f B(3F)KL;m(u) = 2eBmFKL(u) Q M
f B(4F)KL;m(u) = 2 BmFKL(u) Q M
BFKL(u) and
for the functions f B(3F)KL;m(u), f (4)
expanding at strong coupling, we easily see that
The strong coupling limit g ! 1 can be readily performed for all parts of eq. (A.1) except
BFKL;m(u), which are more involved. Rescaling u ! u^ and
nd that the leading term of the prefactor eq. (A.1) is of order g0. Since we
drop the prefactor from now on. For the constant A we make the substitution t ! 2tg and
expand at strong coupling to obtain
= x(u^)jSC ;
AjSC =
where we used that cusp = 2
f B(3F)KL;m(u), f B(4F)KL;m(u). The de nition of these functions given in eq. (A.6) is well-suited
at strong coupling. Let us now turn to the functions
to nite size, since Kij
entries are of the same order Kij
gi+j at weak coupling. At strong coupling, however, all matrix
g and only become numerically smaller as i; j grow.
string [37]. Indeed, at strong coupling we have
BFKL(u^) SC =
2 Jj (t)
where SF stands for small fermion, see appendix B of [37]. Similarly we have that
SC =
SC = 4g ejSF(u^) SC :
We can therefore use the strong coupling expansion of the functions fS(3F;4) for the small
drop the subscript SC from now on.
We start from a slight generalization of the functions f (3;4), namely
f B(3F)KL(u^; v^) := 2 BFKL(u^) Q
f B(4F)KL(u^; v^) := 2 BFKL(u^) Q
BFKL(v^);
BFKL(v^);
and equivalently for the small fermion functions fS(3F;4)(u^; v^). We will determine these
functions and take the limit v^ ! u^ in the end. As in the case of the BFKL eigenvalue !(u^)
both examine in the following.
of the function fS(3F)(u^; v^),7
fS(3F)(u^; v^) =
7For the calculations in the small fermion case, we use the notation of [60] for all quantities with the
to leading order at strong coupling. In eq. (A.14), the function W +( ; u^) is de ned as
where P denotes the principal value. To carry out the integrals, we use the relation
W +( ; u^) :=
2 Z 1
cos( k)
d sin(v^ ) cos(k ) =
where the function ef ;u^( ) is shown to be given by
ef ;u^( ) =
W +( ; u^) +
u^ + 1 4
using the identity
for jpj > 1, see e.g. [60], we obtain the result
fS(3F)(u^; v^) =
32g u^ + v^
u^ + 1 4
u^ + 1 4
We then obtain8
f B(4F)KL(u^; v^) = 2
BFKL(u^) Q
BFKL(1) =
= 16g2
Taking the limit v^ ! u^ then gives the expression f B(4F)KL(u^) needed for the BFKL measure.
8In the second step we assume that summation and integration commute, which is supported by
numerSimilarly the function fS(4F)(u^; v^) has an integral representation as
fS(4F)(u^; v^) =
1 Z 1 d
cos(v^ ) +f;u^( );
where the function +f;u^( ) is given by
+f;u^( ) =
4g 4 u^ + 1
W ( ; u^) +
W ( ; u^)
to leading order at strong coupling. Furthermore, the function W ( ; u^) is de ned as
W ( ; u^) :=
Performing the integrals as before one obtains the result
fS(4F)(u^; v^) =
32g u^ + v^
u^ + 1 4 v^
To integrate this expression to the BFKL case we use the boundary values
BFKL(1) =
BFKL(u^) =
( (u^) 4gu^) ;
sion of eq. (3.4) for the region u^ > 1 and is given by
( ) =
sinh + i
2u^ + 2
1 is obtained by noting that the
measure at strong coupling is symmetric under u^ $
see [1]. We then nd
log (u^) =
+ ( (u^) 4gu^) 4g
f B(3F)KL(u^; v^) = 16g2
d 2 fS(4F)( 1; 2) + ( (v^) 4gv^)
Putting all contributions together, we obtain the full measure for u^ > 1 as
sinh(2 1) sinh(2 2) cosh( 1
slightly di erent form and is given by
in the main text, as the saddle point turns out to be close to u0
0. This region is
ef ;u^( ) =
W +( ; u^) +
1 + u^ 4
now all integrals are principal value integrals. Partitioning principal values as
where jpj < 1, we obtain the result
fS(3F)(u^; v^) =
64g u^ + v^
and performing the integrals using the identity
f B(4F)KL(u^; v^) = 2
= 16g2
To integrate to the BFKL case, we again use relation eq. (A.19) and nd
f B(4F)KL(u^) =
= 12 tanh 1(u^).
cosh(2 1) cosh(2 2) cosh( 1
cosh( 1 + 2)
In the same way, we obtain the result for f B(3F)KL(u^). We begin with the modi cations
for the small fermion function fS(4F)(u^; v^), where the only change from eq. (A.21) is in +f;u^,
which now reads
+f;u^( ) =
W ( ; u^) +
1 + u^ 4
Going through the same steps as before we obtain the result
fS(4F)(u^;v^) =
64g u^ + v^
This can be integrated to the BFKL case using the boundary condition e
BFKL(0) = 0
f B(4F)KL(u^; v^) = 2 BFKL(u^) Q M
BFKL(v^)
= 2
1 Z Z
2 0 0 cosh(2 1) cosh(2 2) cosh( 1
cosh( 1 + 2)
leading order at strong coupling is given by
0 0 cosh(2 1) cosh(2 2) cosh( 1
cosh( 1 + 2)
cosh(2 1) cosh(2 2) cosh( 1
cosh( 1 + 2)
cosh(2 1) cosh(2 2) cosh( 1
eq. (3.5). We provide a plot of the strong coupling measure in gure 2.
f B(3F)KL(u^) =
log BFKL(u^)
blue when ju^j becomes bigger than one, i.e. when the description switches between eq. (A.39) and
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