The principle of maximal transcendentality and the four-loop collinear anomalous dimension
The principle of maximal transcendentality and the four-loop collinear anomalous dimension
Lance J. Dixon 0 1
0 Stanford , CA 94309 , U.S.A
1 SLAC National Accelerator Laboratory, Stanford University
We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar N = 4 super-Yang-Mills theory from recent QCD results, obtaining Gˆ(4) = −300ζ7 − 256ζ2ζ5 − 384ζ3ζ4. This value agrees with a previous numerical result to 0 within 0.2%. It also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension through four loops.
Scattering Amplitudes; Supersymmetric Gauge Theory; 1/N Expansion
Dedicated to the memory of Lev Lipatov.
1 Introduction 2 3
Review of infrared structure of planar N = 4 SYM
2] has played a key conceptual role, especially in the planar limit of a large number of
colors where the theory becomes integrable [3–8] and its amplitudes become dual to Wilson
loops [9–15]. So much is known about the analytic structure of scattering amplitudes in
planar N = 4 SYM that the first amplitude with non-trivial kinematic dependence, the
six-point amplitude, can be bootstrapped to at least five loops [16–22].
Ironically, the finite parts of the six-point amplitude (the remainder function and ratio
function), which are polylogarithmic functions of three variables, are now known to higher
loop orders than is the dimensionally-regulated infrared-divergent prefactor — the BDS
ansatz  — even though the latter depends only on four constants per loop order. One
of these constants, the (light-like) cusp anomalous dimension [24, 25] is known to all orders
in planar N = 4 SYM, thanks to integrability . The cusp anomalous dimension controls
the double pole in ǫ in the logarithm of the dimensionally regularized BDS ansatz. The
single pole is controlled by the “collinear” anomalous dimension. In planar N = 4 SYM,
it is known analytically through three loops [23, 26], and it was computed numerically at
four loops a decade ago . The nonplanar contribution to the four-loop collinear
anomalous dimension was computed numerically very recently [28, 29]. The collinear anomalous
dimension also enters the Regge trajectory for forward scattering [10, 30, 31]. An eikonal
(Wilson line) version of it enters both the threshold soft anomalous dimension for
threshold resummation [32–35] and the rapidity anomalous dimension for transverse momentum
The BDS ansatz also contains two finite constants at each loop order, one for the four
point amplitude and one for the five-point amplitude. One of these constants is known
analytically at three loops , but the other is only known numerically at this loop order .
The purpose of this paper is to provide an analytical value for one of the four constants
in question at four loops, namely the collinear anomalous dimension in planar N = 4
SYM. We do so by leveraging two recent four-loop computations in QCD in the large Nc
limit [40, 41], as well as the principle of maximal transcendentality [42–45]. This principle
states that for suitable quantities, such as the BFKL and DGLAP kernels, the result in
N = 4 SYM can be obtained from that in QCD by converting the fermion representation
– 1 –
from the fundamental (quarks) to the adjoint (gluinos) and then keeping only the functions
that have the highest transcendental weight. In momentum space (x-space) these functions
are typically iterated integrals, and the weight is the number of iterations; in Mellin-moment
space, it corresponds to the number of sums in the nested sums . Here we will only need
the notion of weight for ordinary Riemann zeta values, ζn ≡ ζ(n), for which the weight is
n. Also, the weight is additive for products, and rational numbers have weight zero.
The complete set of observables to which this principle can be applied is still unclear. Besides anomalous dimensions, it has also been successfully applied to form factors, matrix elements of gauge-invariant operators with two or three external partons [47–51], and to certain configurations of semi-infinite Wilson lines [34, 35]. However, it does not hold for
scattering amplitudes with four or five external gluons, even at one loop , in the sense
that there are maximally transcendental parts of the QCD one-loop amplitudes which have
different rational prefactors from the corresponding N = 4 SYM amplitudes.
Here we will apply the maximal transcendentality principle to the collinear anomalous
dimension. This quantity depends on the method of regularization. We will compute its
value in dimensional regularization — in fact, in a supersymmetric version of dimensional
regularization such as dimensional reduction. The collinear anomalous dimension has also
been computed using the so-called massive, or Higgs, regularization [53–57]. The
Higgsregulated result begins to differ from the dimensionally-regularized value starting at three
loops , the last value for which it is known. A dual conformal regulator for infrared
divergences has also been defined ; however, the multi-loop values of the collinear
anomalous dimension in this scheme are still under investigation .
One might think that the collinear anomalous dimension in planar N = 4 SYM could
simply be read off from the leading transcendental terms in the large-Nc quark collinear
anomalous dimension . However, the full-color expression for this quantity is a
polynomial in the adjoint and fundamental quadratic Casimirs, CA and CF . In the large-Nc limit,
CA → Nc while CF → Nc/2. In order to apply the principle of maximal transcendentality
at large Nc, we should first set CF → CA; that is, CA → Nc and CF → CA → Nc, not Nc/2.
There is not enough information left in the large-Nc limit to make the correct replacement.
However, if we can first convert the collinear anomalous dimension to an appropriate
eikonal (Wilson line) quantity, then we can make the correct replacement. In a conformal
theory, this “eikonal bypass” only requires  knowledge of the virtual anomalous
dimension, the coefficient of δ(1 − x) in the DGLAP kernel. The virtual anomalous dimension
in large-Nc QCD was computed recently at four loops , and we can make use of its
leading transcendental part to do the conversion. Once we have the eikonal quantity, we
use its non-abelian exponentiation property [61, 62], which means that it is “maximally
non-abelian”. That is, at any loop order, it can contain only one quadratic Casimir for the
representation of the Wilson line; the remaining group theory factors must all be CA. (A
subtlety that arises when quadratic Casimir scaling does not hold is addressed in section 3.)
This information suffices to allow us to apply the principle of maximal transcendentality
and extract the eikonal quantity in planar N = 4 SYM. Then we use the virtual anomalous
dimension in planar N = 4 SYM, which has been computed to all orders using
integrability [63–65], to convert back to the non-eikonal collinear anomalous dimension.
– 2 –
This paper is organized as follows. In section 2 we briefly review the infrared structure of scattering amplitudes, form factors and Wilson loops in planar N = 4 SYM. In section 3 we carry out the computation and then conclude.
Review of infrared structure of planar N
= 4 SYM
In this section we give a very brief review of the infrared structure of scattering
amplitudes, form factors and Wilson loops in planar N = 4 SYM. In general, multi-loop n-point
amplitudes can be factorized into soft, collinear and hard virtual contributions, where soft
gluon exchange can connect any of the n hard external legs [66, 67]. This factorization
adjoint particles becomes the product of n “wedges”, each equivalent to the square root
of a Sudakov form factor for producing two adjoint particles . The infrared behavior
of the Sudakov form factor was studied using factorization and renormalization group
evolution, beginning in the 1970s [70–75]. Besides the β function (which of course vanishes
in N = 4 SYM), the only quantities that enter are the (light-like) cusp anomalous dimension
γK [24, 25] and an integration constant for a function G(q2), which we will refer to as the
collinear anomalous dimension and denote by G0 [69, 75].
We consider gauge group SU(Nc) and adopt the “integrability” notation for the large
Nc coupling constant,
(Note that another normalization is often used for the cusp anomalous dimension, γK =
2Γcusp.) The cusp anomalous dimension is known to all orders, thanks to integrability .
The first four terms in its perturbative expansion are:
γplanar N =4 = 8 g2 − 16 ζ2 g4 + 176 ζ4 g6 − 1752 ζ6 + 64 (ζ3)2 g8 .
We give the previously-known three-loop result for G0(g) below, in eq. (3.5).
2 ≡ Nc (4π)2
= CA 4π
where αs = gY2M/(4π), λ = NcgY2M is the ’t Hooft coupling, and a was used e.g. in ref. .
The quadratic Casimir in the adjoint representation is CA = Nc, while in the fundamental
representation it is CF = (Nc2 − 1)/(2Nc).
We expand the cusp and collinear anomalous dimensions in terms of g2:
γK (g) =
X g2LγˆK(L) ,
X g2LGˆ0(L) .
– 3 –
In a non-conformal theory, when the differential equation for the Sudakov form factor
is integrated up, infrared poles are obtained that involve integrals over functions of the
running coupling in D = 4 − 2ǫ dimensions. Because planar N = 4 SYM is conformally
invariant, the integrals can be performed analytically. One obtains for the color-ordered
n-point scattering amplitude An :
where si,i+1 = (ki + ki+1)2.
The form factor F (Q2) for producing two adjoint particles corresponds to setting n = 2
in this formula, in which case the two wedges have the same kinematics,
ln F (Q2) = −
Wilson loops for light-like n-gons Cn contain ultraviolet poles rather than infrared ones.
These poles have a very similar form (with ǫ → −ǫ due to their ultraviolet nature) :
ln WCn = −
L2 ǫ2 γˆ(L) − 2L ǫ Gˆ0(L,e)ik
where xi2,i+2 = (xi − xi+2)2 are invariant distances between the corners of the polygons xµi .
The amplitude-Wilson loop duality makes the identification (xi − xi+2)
2 = (ki + ki+1)2.
While the leading double poles in Wilson loops are governed by the same quantity as in
amplitudes, namely γK , a different quantity appears in the subleading poles, G0, eik, whose
expansion is defined by
G0, eik(g) = X g2LGˆ0(L,e)ik .
G0 = G0, eik + 2 B .
instead of G0.
The relation between G0 and G0, eik was explored in ref. , where it was shown that
for a conformal theory, they obey a particularly simple relation,
(Empirical evidence for this kind of relation was given in refs. [32, 76].) Here B, sometimes
called Bδ or the virtual anomalous dimension, is the coefficient of the first subleading term
in the limit as x → 1 of the DGLAP kernel for parton i to split to parton i:
2(1 − x)+
+ Bi δ(1 − x) + . . . .
In a general theory, B = Bi depends on the type of parton i (also the leading, cusp, term
in eq. (2.10) depends on the color representation of parton i), but in N = 4 SYM B is the
same for all partons, by supersymmetry.
– 4 –
In planar N = 4 SYM, thanks to dual conformal symmetry, the gluon Regge trajectory
governing the forward limit of the four-point amplitude can be computed from the cusp
and collinear anomalous dimensions [10, 30, 31]. The result is 
∂ ln s
ln A4(s, t)
4 γK ln
Hence our four-loop result for G0 will also provide ωR(−t) to the same order.
In ref. , the quark form factor was computed to four loops in the large Nc limit of QCD,
and the cusp and collinear anomalous dimensions for large-Nc QCD were determined from
it. In ref.  it was proposed that the N = 4 super-Yang-Mills results for the
twisttwo anomalous dimensions (which includes the cusp anomalous dimension, but not the
collinear anomalous dimension) could be extracted from the leading transcendental terms
in the QCD result by setting CF → CA. Through three loops, where full-color QCD results
are known, the same extraction procedure also works for the collinear anomalous dimension.
Unfortunately, as mentioned in the introduction, the large Nc limit corresponds to CF =
Nc2 − 1
The factor of 1/2 means that the CF → CA replacement can’t be deduced in general
from the large Nc limit. However, there is a workaround, the eikonal bypass discussed
in the introduction, which involves converting the non-eikonal quark collinear anomalous
dimension to an eikonal (Wilson line) quantity , with the help of the recent four-loop
result for the DGLAP kernels in the large Nc limit of QCD . In particular, we need
the coefficient of δ(1 − x) in this result, the virtual anomalous dimension. We will see that
th CF → CA replacement can be performed for the eikonal quantity we have constructed.
Afterwards, one can use the virtual anomalous dimension for planar N = 4 SYM [63–65]
to convert back to the non-eikonal collinear anomalous dimension. We will find an analytic
expression that is quite close to the numerical result .
Through four loops, the leading transcendental part of the leading-color quark collinear
anomalous dimension is [40, 77, 78]
γq|L.C.L.T. = 7 ζ3 g4 − 68 ζ5 +
3 ζ2ζ3 g6 + 705 ζ7 + 144 ζ2 ζ5 + 164 ζ3 ζ4 g8 . (3.2)
– 5 –
136CA2 + 120CF CA − 240CF2 ζ5 +
88 CA2 + 16CF CA − 32CF2 ζ2ζ3
CA αs !
CA αs !
Letting CF → CA, the N = 4 SYM result, for a gluon or gluino in the adjoint representation
γN =4 = 2 ζ3
− 16 ζ5 +
+ . . . .
In refs. [23, 69], the collinear anomalous dimension G0 was evaluated to two and three
loops in planar N = 4 SYM, although to this order there are no subleading color terms.
GN =4 = −4 ζ3
+ 32 ζ5 +
the coefficient of the leading 1/(1 − x)+ term is the cusp anomalous dimension . The
next-to-leading term as x → 1 is the coefficient of δ(1 − x), sometimes called the virtual
anomalous dimension, or Bδ, or just B. The large-Nc, leading transcendentality terms in
B for quarks are given by [41, 80]:
Bq|L.C.L.T. = 20 ζ5 g6 − 280 ζ7 + 40 ζ2 ζ5 − 16 ζ3 ζ4 g8 .
Through three loops, we also know the full group-theoretical decomposition :
Bq|L.T. = CF −12(CA − 2CF )ζ3
40 CA2 + 120 CF CA − 240CF2 ζ5 + 16CF (CA − 2CF )ζ2ζ3
Letting CF → CA, the N = 4 SYM result, for a gluon or gluino, is :
CA αs !
BN =4 = 12 ζ3
− 80 ζ5 + 16 ζ2ζ3
CA αs !
loops, using eqs. (3.3) and (3.7):
G0, eik,F = CF
−28 CA ζ3
+ . . . .
We see that all the CF terms cancel, except for the overall one. This result reflects
nonabelian exponentiation for this type of Wilson line [61, 62]. These results agree with the
leading transcendental part of the results for fLq in ref. .
The threshold soft anomalous dimension γ
s defined in refs. [34, 35] (called γW in
ref. ) is the same as G0, eik,F up to a conventional minus sign, γ
s = −G0, eik,F , and
eq. (3.10) agrees with the leading transcendental part of the QCD result in refs. [34, 35].
The rapidity anomalous dimension γr, which enters the SCET description of transverse
momentum resummation, has also been computed to three loops . The result agrees
with the threshold soft anomalous dimension, up to terms that are proportional to
coefficients of the QCD beta function. This result was explained in ref.  by mapping the
appropriate configurations of Wilson lines for the two computations into each other using a
conformal transformation. Hence we will obtain the four-loop values of both the threshold
soft and rapidity anomalous dimensions in planar N = 4 SYM from
γs, planar N =4 = γr, planar N =4 = −G0p,leainkar N =4 .
In planar N = 4 SYM, the natural Wilson line is in the adjoint representation, not the
fundamental. In the large Nc limit, this collinear anomalous dimension can be obtained
from eq. (3.10) simply by multiplying by an overall factor of 2, since CA = 2CF in the large
Nc limit. What about at four loops? At this order, quadratic Casimir scaling might be
violated. That is, inspecting the color factors of all the Feynman diagrams that contribute
at this order, we see that G0, eik,F might contain — besides CF times a polynomial in CA
— a color factor of
(See e.g. eq. (2.14) of ref. .) If so, the corresponding term in the case of an adjoint
Wilson line would have the same numerical coefficient multiplying
(Nc2 − 1)(Nc2 + 6)
– 7 –
However, the latter factor is precisely twice the former factor in the large Nc limit, which
is the same factor as for the conversion CF → CA in this limit. Given that there are
no CF terms in G0, eik,F except for the overall CF , G0, eik in the large Nc limit of N = 4
SYM can be extracted from the leading transcendality terms of the corresponding eikonal
quantity in the large Nc limit of QCD. (The beta-function correction terms to eq. (2.9) for
a non-conformal theory are also subleading in transcendentality.)
In summary, the eikonal collinear anomalous dimension for planar N = 4 SYM can be
obtained from the large-Nc QCD results for γq and Bq through four loops, using
G0p,leainkar N =4 = 2 −2γq|L.C.L.T. − 2Bq|L.C.L.T. .
Inserting eqs. (3.2) and (3.6), we obtain
G0p,leainkar N =4 = −28 ζ3 g4 + 192 ζ5 +
ζ2ζ3 g6 − 1700 ζ7 + 416 ζ2 ζ5 + 720 ζ3 ζ4 g8 .
= −4 ζ3 g4 +
ζ2ζ3 g6 − 300 ζ7 + 256 ζ2 ζ5 + 384 ζ3 ζ4 g8 .
The virtual anomalous dimension in planar N = 4 SYM is known to all orders from
Bplanar N =4 = 12 ζ3 g4 − 80 ζ5 + 16 ζ2ζ3 g6 + 700 ζ7 + 80 ζ2 ζ5 + 168 ζ3 ζ4 g8 + . . . .
We set L = 2 in eq. (3.16) of ref. , and multiply by −1/2 to account for the different
The non-eikonal collinear anomalous dimension in planar N = 4 SYM is then:
Gplanar N =4 = G0p,leainkar N =4 + 2Bplanar N =4
– 8 –
The numerical value of the four-loop coefficient is − 1238.7477172547735332918988 . . . which can be compared with the number from ref. : − 1240.9(3).
The two results are within about 0.2%, although they are not within the error budget of
0.3 reported in ref. . It would be very nice to check the analysis in this paper with an
improved numerical value.
The first order at which GN =4 can have a subleading-color term is four loops. Recently
this term has been computed numerically [28, 29],
G0(4,N),PN =4 = −384 × (−17.98 ± 3.25)
Could one try to get an analytic value for this quantity using the methods in this paper?
One issue is that the principle of maximal transcendentality has not really been tested
yet for cases where there is a subleading-color contribution to N = 4 SYM, but one could
try nevertheless. The good news is that the simple relation (2.9) continues to hold at
subleading color — whereas in a non-conformal theory it would receive additional corrections
depending on the infrared-finite part of a form factor . The bad news is that there are
not yet analytic values for the subleading-color terms in QCD at four loops, for either γq or
Bq. (Approximate numerical values are available for Bq .) Once they become available,
it will be possible to compute eq. (3.20) analytically, if it is not already known by then. In
fact, the eikonal bypass of using eq. (2.9) should become unnecessary at that point, once
the full color dependence of the QCD result for γq is known.
In summary, in this paper we obtained an analytical value (3.17) for the four-loop collinear anomalous dimension in planar N = 4 SYM, which also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension at this order. We hope that this additional data point will inspire those versed in integrability
methods to try to compute this quantity to all loop orders!
I am grateful to Benjamin Basso, Tomasz Lukowski, Mark Spradlin, Matthias Staudacher
and HuaXing Zhu for useful discussions, and to HuaXing Zhu for very helpful comments
on the manuscript. This research was supported by the US Department of Energy under
contract DE-AC02-76SF00515, and by the Munich Institute for Astro- and Particle Physics
(MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”.
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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