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JHEP12(2018)077

Published for SISSA by Springer

Received: April 30, 2018

Revised: November 30, 2018

Accepted: December 1, 2018

Published: December 13, 2018

Tr(F

) supersymmetric form factors and maximal

transcendentality. Part II. 0 < N < 4 super

Yang-Mills

Andreas Brandhuber,

Martyna Kostaci´nska,

Brenda Penante

and

Gabriele Travaglini

Centre for Research in String Theory, School of Physics and Astronomy,

Queen Mary University of London,

Mile End Road, London E1 4NS, U.K.

CERN Theory Division,

1211 Geneva 23, Switzerland

E-mail: a.brandhuber@qmul.ac.uk, m.m.kostacinska@qmul.ac.uk,

b.penante@cern.ch, g.travaglini@qmul.ac.uk

Abstract: The study of form factors has many phenomenologically interesting applica-

tions, one of which is Higgs plus gluon amplitudes in QCD. Through eﬀective ﬁeld theory

techniques these are related to form factors of various operators of increasing classical di-

mension. In this paper we extend our analysis of the ﬁrst ﬁnite top-mass correction, arising

from the operator Tr(F

), from N = 4 super Yang-Mills to theories with N < 4, for the

case of three gluons and up to two loops. We conﬁrm our earlier result that the maximally

transcendental part of the associated Catani remainder is universal and equal to that of

the form factor of a protected trilinear operator in the maximally supersymmetric theory.

The terms with lower transcendentality deviate from the N = 4 answer by a surprisingly

small set of terms involving for example ζ

, ζ

and simple powers of logarithms, for which

we provide explicit expressions.

Keywords: Eﬀective Field Theories, Scattering Amplitudes, Supersymmetric Gauge The-

ory

ArXiv ePrint: 1804.05828

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP12(2018)077

JHEP12(2018)077

top quarks is replaced by a set of local interactions in an expansion in 1/m

where m

the top mass. This expansion has the form [3, 4]

eﬀ

i=1

+ O





, (1.1)

where O

, i = 1, . . . , 4 are dimension-7 operators made of gluon ﬁeld strengths and covariant

derivatives, and O

= H Tr(F

are the matching coeﬃcients and are proportional

to 1/v, where v is the Higgs ﬁeld vacuum expectation value. Due to the equations of

motion, in pure Yang-Mills one can eliminate two of the four operators in the sum [4, 5],

and the remaining two operators can be chosen to be H Tr(F

) and H Tr



νρ



One is then led to the study of the form factors of the two operators

Tr(F

) = Tr(F

ASD

) + Tr(F

) ∝ O

+ O

, O

∝ Tr



νρ



, (1.2)

where ASD stands for the anti-selfdual part of the gluon ﬁeld strength (which is the only

part contributing at two loops for our external state).

Our main goal is to identify some universal structures in the expressions of such form

factors, in particular across diﬀerent classes of operators and for various amounts of super-

symmetry. Several hints of this universality have already been found in related investiga-

tions. In particular, in [6] it was found that the form factor remainder for the half-BPS

bilinear scalar operator Tr(X

) in N =4 super Yang-Mills (SYM) captures the maximally

transcendental part of the remainder computed in pure Yang-Mills of the operator Tr(F

)

with a state of three gluons [7].

In turn, these particular form factors compute the leading-

order Higgs plus gluon amplitudes in the 1/m

expansion, related to the term O

in (1.1).

This surprising coincidence was the motivation for the study begun in [8–10] of form

factors of operators containing three scalar ﬁelds in N = 4 SYM. In particular, it was

found in [10] that at two loops, the minimal form factor for the non-protected operator

Tr(X[Y, Z]) has the same maximally transcendental part of the minimal form factor re-

mainder of the protected operator Tr(X

). The Tr(X[Y, Z]) operator (or more precisely a

certain admixture of it with a fermion bilinear) is a descendant of the simplest non-protected

operator, namely the Konishi. While the form of universality we alluded to earlier is across

diﬀerent theories, this new appearance is across diﬀerent types of operators. Other purely

transcendental terms of decreasing transcendentality three to zero (which we will refer to

as “pure” terms) were found in the remainder for Tr(X[Y, Z]), and unexpected connec-

tions of these terms to certain spin-chain remainder densities in the SU(2) sector [11] were

identiﬁed. This was quite surprising since the operator Tr(X[Y, Z]) belongs to a diﬀerent

sector, namely the SU(2|3) sector [12].

The calculation of [10] was a stepping stone for the computations of the form factors of

the operator O

in N =4 SYM in [1]. More precisely, in that paper two diﬀerent operators

were considered: O

and a particular supersymmetric completion thereof denoted by O

belonging to the Konishi supermultiplet, whose MHV form factors have recently been

computed [13]. It was found in [1] that the maximally transcendental part of these form

Here X denotes one of the three complex scalar ﬁelds of the N =4 theory.

– 2 –

JHEP12(2018)077

factors with an external state of three gluons is one and the same across theories with

any amount of supersymmetry, including pure Yang-Mills (or QCD), and also identical for

and O

. These form factors in turn describe the ﬁrst subleading corrections to Higgs

+ many gluons in the 1/m

expansion. Unlike the case of the operator Tr(X[Y, Z]), the

remainders for O

and O

show a new feature in that they are accompanied by “non-pure”

terms, i.e. terms of transcendentality degree ranging from three to zero which are further

multiplied by ratios of kinematic invariants. Interesting relations across terms with varying

degree of transcendentality were observed in [2] as a consequence of requiring the absence

of unphysical singularities in soft/collinear limits.

In this paper we quantify these ﬁndings by providing explicit expressions for the re-

mainder functions in N =2, 1 SYM, both for the component operator O

and for its super-

symmetric version O

, whose form factors can be simply obtained by a truncation [14] of

the result of [13] (we note in passing that we will never need to know the explicit expression

of the operator O

, only of its MHV super form factors).

An important disclaimer is in order here. Throughout our calculations we use four-

dimensional expressions of amplitudes and form factors as input in the unitarity cuts. As

mentioned in [2], there are examples in N = 4 SYM where it has explicitly been observed

that four-dimensional cuts are suﬃcient for computing ﬁnite remainders, namely for four-

[15], ﬁve- [16] and six-point [17] two-loop remainders of MHV amplitudes. This happens

because of the absence of so-called µ

-terms (that can only be detected by performing cuts

in D dimensions) at four points, and because of remarkable cancellations in the ﬁve- and

six-point cases which occur thanks to the particular deﬁnition of the remainder function. To

the best of our knowledge, no such examples exist with N <4 supersymmetry. We cannot

a priori exclude the presence of such µ

-terms, and the potential modiﬁcations to the

ﬁnite remainder function they could induce, however we do mention that our result passes

several consistency checks. These include reproducing the correct infrared and ultraviolet

divergences, and soft/collinear factorisation at two loops. Furthermore, we observe that the

relevant one-loop form factor used throughout this paper as obtained from four-dimensional

cuts is also correct in D dimensions [18], i.e. its expression has no additional µ

-terms. This

quantity plays a twofold rˆole, in that it enters cuts of two-loop form factors, and is also

used in the deﬁnition of our two-loop remainders.

The results of our investigation can be summarised as follows:

1. The maximally transcendental part of the form factors of the operators O

and O

the same as that of the half-BPS operator Tr(X

) in the N = 4 SYM theory, regardless

of the amount of supersymmetry (including N = 0) [1]. The latter statement was

conﬁrmed by a recent explicit computation in [19].

2. The non-pure terms of our remainders are identical to those computed in the maxi-

mally supersymmetric theory.

3. The only diﬀerences arise in the pure terms at transcendentality below four, and

are limited to a very restricted type of terms involving ζ

, ζ

and simple powers

– 3 –

JHEP12(2018)077

of logarithms (after disentangling the mixing). The results of our calculations are

collected in tables 4 and 5.

The rest of the paper is organised as follows. In section 2 we brieﬂy discuss the opera-

tors studied in this paper and their tree-level form factors, while in section 3 we summarise

the one-loop calculation. In section 4 we move on to calculate the two-loop minimal form

factors in theories with less than maximal supersymmetry. In section 5 we compute the

Catani two-loop form factor remainder functions in N =2, 1 SYM. We conclude in section 6

with a discussion of our results, their implications, and a number of consistency checks.

2 Operators and tree-level form factors in N = 1, 2, 4

As explained in detail in [1, 2], a central point of our discussion consists of appropriately

translating the operator O

∝ Tr(F

ASD

) to a supersymmetric completion O

+ O(g).

In [1] we have identiﬁed O

for the case of N = 4 SYM as a Supersymmetric descendant

of the Konishi, generated by acting with tree-level supercharges on the lowest-dimensional

operator in the multiplet. Notably, the Component operator O

is contained within O

The key point to make here is that similar supersymmetric completions of O

can be

obtained in N = 2, 1 SYM by an appropriate truncation [14]. We will see shortly that for

the concrete calculations in this paper, we will only need O

for N = 2 SYM.

We now review some of the ingredients of the calculations. For both operators, the tree-

level minimal form factor with the external state of three positive-helicity gluons is given by

(0)

, 2

, 3

; q) = −[12][23][31] . (2.1)

Next, we recall the tree-level MHV super form factors [20] of the full Konishi multiplet in

N =4 SYM have been constructed and expressed in a compact formula in [13],

h1, 2, . . . , n|K(θ,

θ)|0i

(0)

MHV

l=1

[l|

θθ|li+η

hθli

h12i · · · hn1i

i≤j<k≤l

(2−δ

)(2−δ

)

ABCD

ˆη

hjkihlii ,

(2.2)

where ˆη

= η

+ 2[

] and η

are the usual on-shell superspace coordinates labelling

the external on-shell states. The θ

and

A ˙α

label the components of the Konishi super-

multiplet. MHV form factors of O

are obtained by setting θ =

θ = 0, while the form

factors of O

are obtained by setting

θ = 0 and extracting the θ

-term:

(0)

,MHV

(1, 2, . . . , n; q) =

144

(8)

(

i=1

)

h12i · · · hn1i

i≤j<k≤l

(2−δ

)(2−δ

)

ABCD

hjkihlii .

(2.3)

More details on the form of the operator O

can be found in section 2.2 of [2] and in

particular a number of examples of four-point tree-level form factors relevant to unitarity

cuts below are given in (2.13)-(2.20) of [2], describing the diﬀerences between O

and O

– 4 –

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