=0 aainstitutetext: CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
bbinstitutetext: School of Fundamental Physics and Mathematical Sciences,
Hangzhou Institute for Advanced Study & ICTP-AP, UCAS, Hangzhou 310024, China
ccinstitutetext: Peng Huanwu Center for Fundamental Theory, Hefei, Anhui
230026, China ddinstitutetext: Niels Bohr International Academy, Niels Bohr
Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark

# A nice two-loop next-to-next-to-MHV amplitude in ${\cal N}=4$ super-Yang-
Mills

Song He a Zhenjie Li d Chi Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a scalar component of the 8-point next-to-next-to-maximally-helicity-
violating (N2MHV) amplitude at two-loop level in ${\cal N}=4$ super-Yang-Mills
theory; it has a leading singularity proportional to the inverse of the four-
mass-box square root and receives contributions from only two types of non-
trivial integrals with one-loop infrared (IR) divergences. We compute such
two-loop 8-point integrals by taking (double-)collinear limits of certain
finite, dual-conformal-invariant integrals, and they nicely give the IR-safe
ratio function after subtracting divergences. As the first genuine two-loop
N2MHV amplitude computed explicitly, we find remarkable structures in its
symbol and alphabet: similar to the next-to-MHV (NMHV) case, there are still
$9$ algebraic letters associated with the square root, and the latter also
becomes a letter for the first time; unlike the NMHV case, such algebraic
letters appear at either one or all of the second, third and last entry, and
the part with three odd letters is particularly simple.

##  1 Introduction

Scattering amplitudes are central objects in fundamental physics: they are
crucial for connecting theory to experiments in particle accelerators such as
Large Hadron Collider, and they play a central role in discovering new
structures of Quantum Field Theory (QFT). In particular, tremendous progress
has been made for planar ${\cal N}=4$ supersymmetric Yang-Mills theory (SYM);
not only have hidden mathematical structures for all-loop integrands been
unraveled ArkaniHamed:2010kv ; Arkani-Hamed:2012zlh ; Arkani-Hamed:2013jha ,
but amplitudes have also been determined to impressively high loops, for
$n=6,7$ Dixon:2011pw ; Dixon:2014xca ; Dixon:2014iba ; Drummond:2014ffa ;
Dixon:2015iva ; Caron-Huot:2016owq ; Dixon:2016nkn ; Drummond:2018caf ; Caron-
Huot:2019vjl ; Caron-Huot:2019bsq ; Dixon:2020cnr  and beyond
Anastasiou:2009kna ; CaronHuot:2011ky ; Zhang:2019vnm ; He:2020vob . A
remarkable duality between Maximally Helicity Violating (MHV) scattering
amplitudes and null polygonal Wilson loops (WL) of the theory was discovered
at both strong Alday:2007hr ; Alday:2007he ; Alday:2009yn  and weak coupling
Brandhuber:2007yx ; Drummond:2007aua ; Drummond:2007cf ; Drummond:2007bm ;
Drummond:2008aq ; Bern:2008ap ; later it was established that super-amplitudes
(after stripping off MHV tree prefactor) are dual to supersymmetric WL
CaronHuot:2010ek ; Mason:2010yk , and quite a lot of what we have learned
about amplitudes are from this duality picture. For example, one can compute
amplitudes at any value of the coupling around collinear limits Basso:2013vsa
based on integrability Beisert:2010jr  and operator product expansions (OPE)
of WL Alday:2010ku . From such a dual WL picture, one can derive the powerful
$\bar{Q}$ anomaly equation CaronHuot:2011kk , which has been the driving force
for computing multi-loop amplitudes, including two-loop MHV to all
multiplicities Caron-Huot:2011zgw , $n\leq 9$ NMHV CaronHuot:2011kk ;
Zhang:2019vnm ; He:2020vob  and even three-loop MHV for $n\leq 8$ Li:2021bwg .

As shown in CaronHuot:2011kk , given lower-loop amplitudes with higher $n,k$,
the ${\bar{Q}}$ anomaly equation can be used to uniquely determine MHV and
NMHV amplitudes, up to “constants” which are in the kernel of $\bar{Q}$
operator: any function that satisfies $\bar{Q}F_{n,k}=0$ must be a constant
for $k=0$ or constant times R-invariants for $k=1$ CaronHuot:2011kk . However,
this is no longer true for $k\geq 2$: there exist transcendental functions in
the kernel of ${\bar{Q}}$ for N2MHV amplitudes and beyond. In principle such
amplitudes can be determined by exploiting anomaly equations for level-one
generators such as $Q^{(1)}$, which can be obtained from spacetime parity-
conjugation of the $\bar{Q}$ equation, but it is a formidable task already for
the simplest case with $n=8,k=2$. It remains an important open question to
determine, or constrain as much as possible, multi-loop amplitudes with $k\geq
2$ using ${\bar{Q}}$ and $Q^{(1)}$ anomaly equations; in particular, what do
these equations say about amplitudes which evaluate to functions beyond
multiple polylogarithms (MPL) such as $n=10,k=3$ case Caron-Huot:2012awx ;
Bourjaily:2017bsb ; Kristensson:2021ani ?

In this paper, we will take the more traditional approach, namely directly
computing Feynman loop integrals contributing to (certain components of)
amplitudes with $k\geq 2$, which has proved very successful for e.g. two-loop
MHV and NMHV amplitudes. The main motivation is to obtain more data about
scattering amplitudes and Feynman integrals in ${\cal N}=4$ SYM, which in turn
allows us to discover more mathematical structures about their symbology and
physics related to the ${\bar{Q}}$ anomaly equation etc.. Our main interests
are two-loop N2MHV amplitudes, where the integrands in four-dimensional space
are known from generalized unitarity Bourjaily:2015jna ; at least for $n=8$,
all such integrals are expected to evaluate to MPLs, which can in principle be
done by using direct integration method CaronHuot:2011kk  if infrared
divergences are suitably regulated. We will not compute the entire two-loop
8-point N2MHV super-amplitude through this method, mainly for two reasons: (1)
there are too many integrals with rather complicated kinematic dependence with
as many as $9$ dual-conformal-invariant cross-ratios; additional technical
difficulties are caused by regulating divergent integrals; (2) we expect that
most components of N2MHV amplitudes are in a sense similar to those of
NMHV/MHV amplitudes, since they receive contributions from same types of
integrals and there are certain similarities regarding their symbology; thus
we will focus on what we consider as “genuinely new” components of N2MHV
amplitudes, which exhibit new structures invisible at NMHV/MHV level.

Therefore, we will focus on a class of particularly nice components of N2MHV
amplitudes, which vanish at tree level and remain finite at one-loop level.
The (algebraic, Grassmann-valued) coefficients of these integrals, which are
leading singularities or Yangian invariants, are well understood: for $k=2$
only one special class of them are non-rational functions with square roots,
which first appear in one-loop amplitudes as coefficients of four-mass box
integrals. Certain scalar components isolate such a non-rational Yangian
invariant ArkaniHamed:2009dn ; Arkani-Hamed:2012zlh , thus e.g. at one loop it
is simply given by a finite four-mass box integral which contains the same
square root. For $n=8$, there are exactly two such scalar components related
to each other by a cyclic rotation, and e.g. one of them,

$\mathcal{A}(\varphi_{12},\varphi_{12},\varphi_{13},\varphi_{13},\varphi_{34},\varphi_{34},\varphi_{24},\varphi_{24})\>,$

which depends on the square root $\Delta_{1,3,5,7}$ is given by the four-mass
box integral with dual points $x_{1},x_{3},x_{5},x_{7}$, which will be denoted
as $\Delta$ when there is no ambiguity. At two-loop level, as we will see
shortly, such a component receives contributions from three integrals (up to
cyclic rotations), a penta-box and a double-box, both of which has only one-
loop divergence, and the well-known finite box ladder integral Usyukina:1993ch
; the finite ratio function can be obtained by subtracting one-loop MHV
amplitudes multiplied by the one-loop component (which is given by the four-
mass box function). We will use the dual conformal invariant (DCI)
regularization Bourjaily:2013mma  since these integrands are defined strictly
in four-dimensional space: it turns out the $n=8$ penta-box and double-box can
be obtained from double-collinear limits of $n=10$ finite DCI integrals, and
the DCI regularization allows us to extract these divergent integrals from the
latter: by sending a dual point to infinity for the $n=10$ penta-box integral,
it is nicely reduced to some two-loop master integrals with four masses which
have been recently computed very recently using differential equations
He:2022ctv ; the $n=10$ double-box evaluates to elliptic multiple
polylogarithms Kristensson:2021ani , but we will see that it suffices to take
a double-collinear limit for its one-dimensional integral representation,
which gives the $n=8$ double-box as MPLs. After subtracting the DCI-regulated
one-loop 8-point MHV amplitude multiplied by the four-mass box function, we
confirm the cancellation of divergences and end up with a finite ratio
function.

Having computed these cutting-edge examples for the two-loop DCI-regulated
integrals contributing to this N2MHV component, we will then study the
structures of the resulting symbol and the alphabet, especially in comparison
with earlier results of $n=8$ NMHV and MHV amplitudes Caron-Huot:2011zgw ;
Zhang:2019vnm ; Li:2021bwg . We find that the alphabet of this component (and
its cyclic rotation) stays very similar to the NMHV one (MHV case is absent of
any square root), which consists of $180$ rational letters and $9+9$ algebraic
letters (associated with square root $\Delta$ and its cyclic rotation)
Zhang:2019vnm  (the appearance of these algebraic letters is studied from many
different directions He:2020uhb ; Mago:2020kmp ; Arkani-Hamed:2019rds ;
Drummond:2019cxm ; Drummond:2019qjk ; Drummond:2020kqg ; Henke:2019hve ;
Henke:2021ity ; He:2021eec ; He:2021esx ; He:2021non ; Yang:2022gko ); in
particular there are still $9$ independent algebraic letters which are odd
under the flip $\Delta\to-\Delta$ (all rational letters are even), and there
is exactly one new letter for this N2MHV component, _namely $\Delta$ itself_.
Furthermore, recall that NMHV amplitudes or other components of N2MHV ones
must be even under the flip, thus at two loops they contain none or two odd
(algebraic) letters, which can only be in the second and third entry
Zhang:2019vnm ; He:2020vob  (some individual Feynman integrals also have this
property He:2020uxy ; He:2020lcu ); in particular this means that the last
entry only consists of even (rational) letters, as predicted by $\bar{Q}$
equation. On the contrary, this special component of N2MHV must be odd, and at
two loops it contains either one or three odd letters, in the second, third
and last entries. Among other things, we find remarkably simple results for
the part containing three odd letters.

##  2 A nice component of 8-point N2MHV: two-loop ratio function

A nice representation for two-loop integrands of $n$-point, NkMHV super-
amplitude in ${\cal N}=4$ SYM has been obtained using generalized unitarity
Bourjaily:2015jna . Schematically, the amplitudes in planar ${\cal N}=4$ SYM
take the form

$\mathcal{A}_{n,k}=\sum\text{Leading Singularity}\times\text{Scalar Integral}$
(1)

where leading singularities are Yangian invariants: they are completely fixed
by super conformal symmetries and dual super conformal symmetries and hence
independent of loop level Arkani-Hamed:2012zlh .

For our purpose, it is convenient to work with the _dual_ superspace
coordinates $(x,\theta)$,

$x_{i+1}^{\alpha\dot{\alpha}}-x_{i}^{\alpha\dot{\alpha}}=p_{i}^{\mu}\sigma_{\mu}^{\alpha\dot{\alpha}}=\lambda_{i}^{\alpha}\tilde{\lambda}_{i}^{\dot{\alpha}}\>,\qquad\theta_{i+1}^{\alpha
A}-\theta_{i}^{\alpha A}=\lambda_{i}^{\alpha}\eta_{i}^{A},$ (2)

as well as the following shorthand notation for the Lorentz invariants,

$x_{i,j}^{2}=(x_{i}-x_{j})^{2}=(p_{i}+p_{i+1}+\cdots+p_{j-1})^{2}.$

To linearly realize the dual superconformal symmetries and have nice
expressions for Yangian invariants, it is convenient to introduce (super)
momentum twistor variables Hodges:2009hk ,

$\mathcal{Z}_{i}=(Z_{i}^{a}|\chi_{i}^{A}):=(\lambda_{i}^{\alpha},x_{i}^{\alpha\dot{\alpha}}\lambda_{i\alpha}|\theta_{i}^{\alpha
A}\lambda_{i\alpha})\>.$ (3)

In terms of super momentum twistors, we further define the basic
$\mathrm{SL}(4)$-invariant $\langle
ijkl\rangle:=\epsilon_{abcd}Z_{i}^{a}Z_{j}^{b}Z_{k}^{c}Z_{l}^{d}$ (or the
Plücker coordinates of $\mathrm{Gr}(4,n)$), and the basic $R$ invariant
Drummond:2008vq ; Mason:2009qx

$[i\,j\,k\,l\,m]:=\frac{\delta^{0|4}(\chi_{i}^{A}\langle
jklm\rangle+\text{cyclic})}{\langle ijkl\rangle\langle jklm\rangle\langle
klmi\rangle\langle lmij\rangle\langle mijk\rangle}\>.$ (4)

The genuine N2MHV amplitudes in planar $\mathcal{N}=4$ SYM first appear
starting from $n=8$, which is sufficient for our purpose. As we have
mentioned, we are mostly interested in (two cyclically related) component
amplitudes, which are proportional to Yangian invariants with square roots;
such a scalar component reads:

$\displaystyle\mathcal{A}(\varphi_{12},\varphi_{12},\varphi_{13},\varphi_{13},\varphi_{34},\varphi_{34},\varphi_{24},\varphi_{24})$
$\displaystyle=\int\mathrm{d}\chi_{1}^{1}\mathrm{d}\chi_{2}^{1}\mathrm{d}\chi_{3}^{2}\mathrm{d}\chi_{4}^{2}\chi_{5}^{3}\mathrm{d}\chi_{6}^{3}\mathrm{d}\chi_{7}^{4}\mathrm{d}\chi_{8}^{4}\>{\cal
A}_{8,2}$
$\displaystyle=\sum_{L=0}^{\infty}(g^{2})^{L}\mathbf{A}_{8,2}^{(L)}\,,$ (5)

where $\mathbf{A}_{8,2}^{(L)}$ denotes the $L$-loop contribution to this
component amplitude. At tree level, this component amplitude vanishes,
$\mathbf{A}_{8,2}^{(0)}=0$, the one-loop contribution $\mathbf{A}_{8,2}^{(1)}$
thus is finite. Moreover, $\mathbf{A}_{8,2}^{(1)}$ receives contribution only
from one term in eq.(1),

$\sum_{\pm}\frac{(1-u-v\pm\Delta)}{2\Delta}[\alpha_{\pm},4,5,6,7][\gamma_{\pm},8,1,2,3]\int\frac{-\Delta\,x_{1,5}^{2}x_{3,7}^{2}\>\mathrm{d}^{4}x_{0}}{x_{1,0}^{2}x_{3,0}^{2}x_{5,0}^{2}x_{7,0}^{2}}$
(6)

where $\alpha$ and $\delta$ are two solutions of the Schubert problem

$\alpha=(a{-}1\,a)\cap(d\,d{-}1\,\gamma)\>,\qquad\gamma=(c{-}1\,c)\cap(b\,b{-}1,\alpha)\>,$
(7)

and

$u=\frac{x_{1,3}^{2}x_{5,7}^{2}}{x_{1,5}^{2}x_{3,7}^{2}}\>,\quad
v=\frac{x_{1,7}^{2}x_{3,5}^{2}}{x_{1,5}^{2}x_{3,7}^{2}}\>,\quad\Delta=\sqrt{(1-u-v)^{2}-4uv}\>.$
(8)

By integrating Grassmannian variables, we have the scalar component
$\mathbf{A}_{8,2}^{(1)}$,

$\displaystyle\mathbf{A}_{8,2}^{(1)}$ $\displaystyle=I^{(1)}_{\rm
bl}:=\int\frac{\mathrm{d}^{4}x_{0}}{x_{01}^{2}x_{03}^{2}x_{05}^{2}x_{07}^{2}}$
$\displaystyle=\frac{2}{x_{1,5}^{2}x_{3,7}^{2}\Delta}\biggl{(}\text{Li}_{2}(z)-\text{Li}_{2}(\bar{z})+\frac{1}{2}\log(z\bar{z})\log\left(\frac{1-z}{1-\bar{z}}\right)\biggr{)},$
(9)

where the box integral has been denoted as the one-loop instance of well-known
box ladders Usyukina:1993ch , $I^{(1)}_{\rm bl}$, and
$z,\bar{z},1-z,1-\bar{z}$, which contain the square root $\Delta$, are defined
by

$z\bar{z}=u,\quad(1-z)(1-\bar{z})=v.$

$x_{5}$$x_{8}$$x_{0^{\prime}}$$x_{7}$$x_{0}\>$$x_{1}$$x_{3}$

$x_{3}$$x_{7}$$x_{8}$$x_{0^{\prime}}$$x_{6}$$x_{5}$$x_{0}$$x_{1}$

$x_{3}$$x_{7}$$x_{0^{\prime}}$$x_{5}$$x_{0}\>\>$$x_{1}$

Figure 1: Three Feynman diagrams as well as their dual diagrams that
contribute to the two-loop N2MHV component amplitude $\mathbf{A}^{(2)}_{8,2}$:
the penta-box diagram, the double-box diagram, and the box-ladder diagram. In
the dual diagrams, the line connecting to two dual points, $x_{a}$ and
$x_{b}$, represents a factor $x_{a,b}^{2}$ in the denominator, while the
dotted line represents a factor in the numerator.

At two-loop level, by repeating the procedure as in the one-loop case, one can
find only three types of Feynman integrals that contribute to this component
amplitude: up to cyclic rotations, we have a penta-box integral with $5$ dual
points, $I_{\rm pb}(x_{1},x_{3},x_{5},x_{7},x_{8})$, a double-box integral
with $6$ dual points, $I_{\rm db}(x_{1},x_{3},x_{5},x_{6},x_{7},x_{8})$, and
the double box integral with $4$ dual points $I_{\rm
bl}^{(2)}(x_{1},x_{3},x_{5},x_{7})$ ($L=2$ instance of the box ladder $I_{\rm
bl}^{(L)}$) Usyukina:1993ch :

$\mathbf{A}_{8,2}^{(2)}=\left(2I_{\rm
pb}(x_{1},x_{3},x_{5},x_{7},x_{8})+2I_{\rm
db}(x_{1},x_{3},x_{5},x_{6},x_{7},x_{8})+I_{\rm
bl}^{(2)}(x_{1},x_{3},x_{5},x_{7})\right)+3\leavevmode\nobreak\ {\rm cyclic}$
(10)

where these three types of Feynman integrals read

$\displaystyle I_{\rm pb}(x_{1},x_{3},x_{5},x_{7},x_{8})$
$\displaystyle=\int\frac{x_{8,0}^{2}x_{1,7}^{2}\>\mathrm{d}^{4}x_{0}\mathrm{d}^{4}x_{0^{\prime}}}{x_{1,0}^{2}x_{3,0}^{2}x_{5,0}^{2}x_{7,0}^{2}x_{0,0^{\prime}}^{2}x_{7,0^{\prime}}^{2}x_{8,0^{\prime}}^{2}x_{1,0^{\prime}}^{2}}\>,$
(11) $\displaystyle I_{\rm db}(x_{1},x_{3},x_{5},x_{6},x_{7},x_{8})$
$\displaystyle=\int\frac{x_{6,8}^{2}\>\mathrm{d}^{4}x_{0}\mathrm{d}^{4}x_{0^{\prime}}}{x_{1,0}^{2}x_{3,0}^{2}x_{5,0}^{2}x_{0,0^{\prime}}^{2}x_{6,0^{\prime}}^{2}x_{7,0^{\prime}}^{2}x_{8,0^{\prime}}^{2}}\>,$
(12) $\displaystyle I_{\rm bl}^{(2)}(x_{1},x_{3},x_{5},x_{7})$
$\displaystyle=\int\frac{x_{1,5}^{2}\>\mathrm{d}^{4}x_{0}\mathrm{d}^{4}x_{0^{\prime}}}{x_{1,0}^{2}x_{3,0}^{2}x_{5,0}^{2}x_{0,0^{\prime}}^{2}x_{5,0^{\prime}}^{2}x_{7,0^{\prime}}^{2}x_{1,0^{\prime}}^{2}}\>,$
(13)

the factor ‘2’ in front of $I_{\rm pb}$ and $I_{\rm db}$ is due to their
symmetry under the exchange $x_{0}\leftrightarrow x_{0^{\prime}}$, ‘$+3$
cyclic’ means taking $x_{i}\to x_{i{+}2}$ for $i=1,\cdots,8$ which gives a
cyclic orbit of length $4$. As remarked in the introduction and shown in the
one-loop case eq.(9), this component amplitude $\mathbf{A}_{8,2}^{(2)}$ as
well as its three ingredient integrals are pure polylogarithms divided by the
square root

$x_{1,5}^{2}x_{3,7}^{2}\Delta=\sqrt{(x_{1,5}^{2}x_{3,7}^{2}-x_{1,3}^{2}x_{5,7}^{2}-x_{3,5}^{2}x_{1,7}^{2})^{2}-4x_{1,3}^{2}x_{3,5}^{2}x_{5,7}^{2}x_{1,7}^{2}}\>,$

we thus introduce the normalized amplitudes and Feynman integrals indicated by
a hat:

$\widehat{\mathbf{A}}_{8,2}=x_{1,5}^{2}x_{3,7}^{2}\Delta\times\mathbf{A}_{8,2}\>,\quad\widehat{I}_{\bullet}=x_{1,5}^{2}x_{3,7}^{2}\Delta\times
I_{\bullet}\>.$ (14)

This component suffers from infrared divergence: both the penta-box integral
$I_{\rm pb}$ and double-box integral $I_{\rm db}$ has one-loop divergence from
the box with two massless legs ($x_{0^{\prime}}$ in the above equations).
However, it is well known that the infrared divergence is captured by the MHV
amplitudes (also known as BDS ansatz) Bern:2005iz , then it is clear that the
so-called ratio functions
$\mathcal{R}_{n,k}=\mathcal{A}_{n,k}/\mathcal{A}_{n,0}$ are finite and
independent of the regularization scheme. For our case, the corresponding
component of $\mathcal{R}_{8,2}$ is slightly simpler and reads

$\mathbf{R}_{8,2}^{(2)}=\mathbf{A}^{(2)}_{8,2}-\mathbf{A}^{(1)}_{8,2}\times\mathcal{A}_{8,0}^{(1)}$
(15)

due to the vanishing $\mathbf{A}_{8,2}^{(0)}$. As above, we introduce the
normalized (pure) ratio function $\widehat{\mathbf{R}}_{8,2}$. The infrared
structure of $\mathbf{A}_{8,2}^{(2)}$ now is manifest: it is captured by the
one-loop 8-point MHV amplitude $\mathcal{A}_{8,0}^{(1)}$. Note that this
component ratio function is the same as the BDS-normalized amplitude
$(\mathcal{A}_{n,k}/\mathcal{A}_{\text{BDS}})^{(2)}$ at two-loop order.

The general regularization for the Feynman integrals in Gauge theories is
dimensional regularization which will spoil the dual conformal symmetries of
$\mathcal{N}=4$ SYM, we instead use the so-called _dual-conformal
regularization_ Bourjaily:2013mma , which will be elaborated in the following
section.

##  3 The computation: DCI-regulated penta-box and double-box integrals and
the subtraction

The infrared divergences of the penta-box integral and the double-box integral
arise from massless legs of the box sub-integral (on the right): for the
penta-box, $x_{7,8}^{2}$ and $x_{8,1}^{2}$ vanish, and so do $x_{6,7}^{2}$ and
$x_{7,8}^{2}$ for the double-box. To regulate these divergences, the most
straightforward way is to assign a small mass $\mu$ to these massless legs,
then take the limit $\mu\to 0$, which is also known as the Higgs
regularization Alday:2009zm ; Henn:2010ir . In the dual space, this
corresponds to a shift of dual points parameterized by the dimensional scale
$\mu$ which breaks the dual conformal symmetries. The dual conformal
regularization Bourjaily:2013mma ; Bourjaily:2019vby  instead shifts each dual
point (for $a=1,2,\cdots,n$) by

$x_{a}\to\widehat{x}_{a}=x_{a}+\epsilon(x_{a+1}-x_{a})\frac{x_{a-2,a}^{2}}{x_{a-2,a+1}^{2}}\,$
(16)

which is parameterized by a dimensionless parameter $\epsilon$. With this
regularization, the one-loop $n$-point MHV amplitudes Bourjaily:2013mma ;
Bourjaily:2019vby  read

$\mathcal{A}_{n,0}^{(1)}=n\log^{2}(\epsilon)+\log(\epsilon)\,\Biggl{(}\sum_{a=1}^{n}\log\frac{x_{a,a+2}^{2}x_{a+3,a+5}^{2}}{x_{a,a+3}^{2}x_{a+2,a+5}^{2}}\Biggr{)}+n\zeta_{2}+F_{n,\text{DCI}}^{(1)}+O(\epsilon)$
(17)

where the finite weight-$2$ functions read

$F_{n,\text{DCI}}^{(1)}=\left[\sum_{b=4}^{\lfloor
n/2\rfloor+1}\mathrm{Li}_{2}(1-u_{1,b})+\tfrac{1}{2}\log u_{1,b}\log
v_{1,b}\right]+\underset{\footnotesize\text{(delete
duplicates)}}{\text{cyclic}}\>,$ (18)

with dual conformal cross-ratios

$u_{1,b}=\frac{x_{3,b}^{2}x_{2,b+1}^{2}}{x_{2,b}^{2}x_{3,b+1}^{2}}\>,\qquad
v_{1,b}=\frac{x_{2,n}^{2}x_{1,3}^{2}x_{b-1,b+1}^{2}x_{b,b+2}^{2}}{x_{3,n}^{2}x_{1,b}^{2}x_{b-1,b+2}^{2}x_{2,b+1}^{2}}\>.$
(19)

Similarly, one can compute one-loop $n$-point NkMHV amplitudes
$\mathcal{A}_{n,k}^{(1)}$ with this regularization, and obtain the finite
ration function at one-loop level:
$\mathcal{R}_{n,k}^{(1)}:=\mathcal{A}_{n,k}^{(1)}-\mathcal{A}_{n,k}^{(0)}\mathcal{A}_{n,0}^{(1)}$,
where $\log^{2}\epsilon$ and $\log\epsilon$ terms cancel nicely Caron-
Huot:2013vda .

Moving to our two loop computation, in eq.(15) we also need to regulate the
penta-box integral and the double-box integral with one-loop divergences.
Their results in dual conformal regularization can be obtained as special
double-collinear limits of their finite counterparts with $n=10$, which are
already known from the literature Kristensson:2021ani ; He:2022ctv . In the
rest of this section, we will elaborate the procedure of taking such collinear
limits.

### The double-box integral

The finite counterpart of the 8-point double-box integral is the 10-point
elliptic double-box integral Bourjaily:2017bsb ; Kristensson:2021ani :

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(20)

which is finite and depends on 7 cross-ratios

$\displaystyle
u_{1}=\frac{x_{3,6}^{2}x_{8,10}^{2}}{x_{3,8}^{2}x_{6,10}^{2}}\>,\qquad
u_{2}=\frac{x_{1,3}^{2}x_{5,8}^{2}}{x_{1,5}^{2}x_{3,8}^{2}}\>,\qquad
v_{1}=\frac{x_{6,8}^{2}x_{3,10}^{2}}{x_{3,8}^{2}x_{6,10}^{2}}\>,\qquad
v_{2}=\frac{x_{3,5}^{2}x_{1,8}^{2}}{x_{1,3}^{2}x_{5,8}^{2}}\>,\qquad$
$\displaystyle
u_{3}=\frac{x_{1,6}^{2}x_{8,10}^{2}}{x_{1,8}^{2}x_{6,10}^{2}}\>,\qquad
u_{4}=\frac{x_{6,8}^{2}x_{5,10}^{2}}{x_{5,8}^{2}x_{6,10}^{2}}\>,\qquad
u_{5}=\frac{x_{1,5}^{2}x_{6,10}^{2}}{x_{1,6}^{2}x_{5,10}^{2}}\>.$ (21)

One can easily see that this integral reduces to
$x_{1,5}^{2}x_{3,7}^{2}I_{\text{db}}$ in the collinear limit $x_{10}\to
x_{8}$, $x_{8}\to x_{7}$ at the integrand level. The elliptic double-box
integral has been evaluated as elliptic multiple polylogarithms in
Kristensson:2021ani  where its symbol is also given. However, a better start
point turns out to be its one-fold integral representation given in
Kristensson:2021ani

$I_{\text{ell-
db}}=\int_{0}^{\infty}\frac{\mathrm{d}x}{y}\mathcal{G}_{3}(x,y)\>,$ (22)

where $\mathcal{G}_{3}(x,y)$ is a known MPL of weight 3, which depends on 7
cross-rations, as well as $\\{x,y\\}$ given by the elliptic curve

$\displaystyle
y^{2}=\biggl{(}\frac{v_{1}}{u_{4}}\bigl{(}(1{-}u_{4})(x{+}1{-}v_{2}){-}u_{1}{+}u_{3}v_{2}\bigr{)}{+}h_{1}{+}h_{2}\biggr{)}^{2}{-}4h_{1}h_{2},$
(23)

with

$\displaystyle h_{1}$
$\displaystyle=\frac{u_{2}u_{4}}{v_{1}}\bigl{(}x^{2}+(1{-}u_{1}{+}v_{1})x+v_{1}\bigr{)},$
(24) $\displaystyle h_{2}$
$\displaystyle=\Bigl{(}x{+}\frac{v_{1}}{u_{4}}\Bigr{)}\Bigl{(}(1{+}x{-}u_{1})\Bigl{(}\frac{u_{2}u_{4}}{v_{1}}-1\Bigr{)}+(1{-}u_{3})v_{2}\Bigr{)}.$

Now let us turn to the 8-point double box. With the DCI regulator eq.(16), the
7 cross-ratios in eq.(21) behave like

$\displaystyle
u_{1}\to\epsilon\frac{x_{1,7}^{2}x_{3,6}^{2}}{x_{1,6}^{2}x_{3,7}^{2}}\>,\quad
u_{2}\to\frac{x_{1,3}^{2}x_{5,7}^{2}}{x_{1,5}^{2}x_{3,7}^{2}}=u\>,\quad
v_{1}\to\epsilon\frac{x_{3,8}^{2}x_{5,7}^{2}}{x_{3,7}^{2}x_{5,8}^{2}}\>,\quad
v_{2}\to\frac{x_{3,5}^{2}x_{1,7}^{2}}{x_{1,5}^{2}x_{3,7}^{2}}=v\>,$
$\displaystyle u_{3}\to\epsilon\>,\quad u_{4}\to\epsilon\>,\quad
u_{5}\to\frac{x_{1,5}^{2}x_{6,8}^{2}}{x_{1,6}^{2}x_{5,8}^{2}}\>,$ (25)

as $\epsilon\to 0$. Correspondingly, the elliptic curve eq.(23) degenerates to

$y^{2}=\Bigl{(}x^{2}+x(1+u-v)x+u\Bigr{)}^{2}=\Bigl{(}(x+z)(x+\bar{z})\Bigr{)}^{2}\>,$
(26)

and $\mathcal{G}_{3}(x,y)$ in eq.(22) simplifies dramatically as

$\mathcal{G}_{3}(x,y)=\log^{2}(\epsilon)\log\frac{(1+x)(x+u)}{xv}+\log(\epsilon)\>\mathcal{G}_{2}^{\text{fin}}+\mathcal{G}_{3}^{\text{fin}}+O(\epsilon)\>,$
(27)

where $\mathcal{G}_{2,3}^{\text{fin}}$ are MPLs of weight 2 and 3,
respectively.

A nice observation is that for the $\log^{2}\epsilon$ term, the integral
$\int\log\frac{(1+x)(x+u)}{xv}\frac{\mathrm{d}x}{(x+z)(x+\bar{z})}$ indeed
evaluates to the four-mass box integral
$\frac{1}{\Delta}\widehat{\mathbf{A}}_{8,2}^{(1)}$ in eq.(9). This already
provides some preliminary evidence that IR divergences will be cancelled in
(15) to give a finite result. It is straightforward to evaluate the remaining
integrals
$\int\mathcal{G}_{2,3}^{\text{fin}}\frac{\mathrm{d}x}{(x+z)(x+\bar{z})}$ with,
say HyperInt or PolyLogTools Panzer:2014caa ; Duhr:2019tlz . Here we only
record the _symbol_ Goncharov:2010jf  of its cyclic image
$\widehat{I}_{\text{db}}(x_{5},x_{7},x_{1},x_{2},x_{3},x_{4})$, which can be
obtained much easier by the algorithm provided in the Appendix A of Caron-
Huot:2011dec , in the ancillary file n82Lnnmhv_scale_component.txt as
variables puredb$log2e, puredb$loge and puredb$finite.

### The penta-box integral

Since $I_{\rm pb}$ can be obtained as double-collinear limits of $n=10$ penta-
box integrals, in principle we can compute the latter similarly in (DCI)
Feynman-parameterized form, but the computation turns out to be rather
tedious. Fortunately, we find that they belong to the family of integrals
recently computed in He:2022ctv  by canonical differential equations: the DCI
penta-box integral can be identified as a non-DCI double box integral by
sending $x_{5}$ to $\infty$, see figure 2. This so-called non-DCI limit gives
us a bijection between the kinematics of these two Feynman integrals,

$m_{1}^{2}\leftrightarrow x_{3,7}^{2},\,m_{2}^{2}\leftrightarrow
x_{1,3}^{2},\,m_{3}^{2}\leftrightarrow x_{8,1}^{2},\,m_{4}^{2}\leftrightarrow
x_{7,8}^{2},\,s\leftrightarrow x_{1,7}^{2},\,t\leftrightarrow
x_{3,8}^{2},\,N=x_{0,8}^{2},$ (28)

and for external Mandelstam variables, namely $x_{i,j}^{2}$ with $i,j\neq
0,0^{\prime}$, we further write them in terms of momentum-twistor variables

$x_{i,j}^{2}=\frac{\langle i{-}1\ i\ j{-}1\ j\rangle}{\langle i{-}1\ i\
I_{\infty}\rangle\langle j{-}1\ j\
I_{\infty}\rangle}\leftrightarrow\frac{\langle i{-}1\ i\ j{-}1\
j\rangle}{\langle i{-}1\ i\ 4\ 5\rangle\langle j{-}1\ j\ 4\ 5\rangle},$ (29)

where the infinite bi-twistor $I_{\infty}$ is given by $(45)$ since we have
taken $x_{5}\to\infty$.

$x_{7}$$x_{8}$$x_{1}$$x_{3}$$x_{5}$$N$

$\xrightarrow{x_{5}\to\infty}$
$p_{1}$$p_{2}$$p_{3}$$p_{4}$$N$$D_{1}$$D_{2}$$D_{3}$$D_{4}$$D_{5}$$D_{6}$$D_{7}$

Figure 2: The penta-box integral of interest and the corresponding non-DCI
double-box integral. Here $N$ represents a factor $x_{0,8}^{2}$ in the
numerator of both integrals.

The non-DCI double-box integral $I_{\text{nDCI-db}}$ is denoted in He:2022ctv
as $G_{1,1,1,1,1,1,1,-1}$, which can be expressed as linear combination of the
following three master integrals

$\displaystyle I_{5}=r_{9}G_{0,1,1,1,1,1,1,0},\quad
I_{6}=r_{8}G_{1,1,0,1,1,1,1,0}$ $\displaystyle
I_{3}=r_{2}(-m_{3}^{2}G_{0,1,1,1,1,1,1}-m_{4}^{2}G_{1,1,0,1,1,1,1}+sG_{1,1,1,1,1,1,1,-1}),$

where we have used (see the figure 2 above for definition of $D_{i}$’s):

$G_{\alpha_{1},\dots,\alpha_{7},\alpha_{8}}:=\int\frac{{\rm d}^{D}\ell_{1}{\rm
d}^{D}\ell_{2}}{(i\pi^{D/2})^{2}}\frac{1}{D_{1}^{\alpha_{1}}\cdots
D_{7}^{\alpha_{7}}N^{\alpha_{8}}}.$

Therefore, $I_{\text{nDCI-db}}=G_{1,1,1,1,1,1,1,-1}$ can be solved as

$I_{\text{nDCI-
db}}=\frac{1}{r_{2}s}I_{3}+\frac{m_{3}^{2}}{r_{9}s}I_{5}+\frac{m_{4}^{2}}{r_{8}s}I_{6},$
(30)

where $I_{3}$, $I_{5}$ and $I_{6}$ are three pure weight-4 MPL functions, and
$r_{2},r_{8},r_{9}$ are three different square roots of kinematics variables
given by

$\displaystyle r_{2}^{2}=(s-m_{1}^{2}-m_{2}^{2})^{2}-4m_{1}^{2}m_{2}^{2},$
$\displaystyle
r_{8}^{2}=((m_{1}^{2}-m_{2}^{2})m_{4}^{2}+s(t-m_{1}^{2}))^{2}-4sm_{1}^{2}m_{3}^{2}m_{4}^{2},$
$\displaystyle
r_{9}^{2}=((m_{2}^{2}-m_{1}^{2})m_{3}^{2}+s(t-m_{2}^{2}))^{2}-4sm_{2}^{2}m_{3}^{2}m_{4}^{2}.$

Applying the bijection eq.(28) and then taking double-collinear limits with
DCI regulator leads to

$m_{3}^{2}\leftrightarrow
x_{8,1}^{2}=\frac{x_{7,1}^{2}x_{8,2}^{2}}{x_{7,2}^{2}}\epsilon,\quad
m_{4}^{2}\leftrightarrow
x_{7,8}^{2}=\frac{x_{6,8}^{2}x_{7,1}^{2}}{x_{6,1}^{2}}\epsilon,$ (31)

we then get the full result of penta-box integral $I_{\text{pb}}$ in the DCI
regularization.

Under the collinear limits eq.(31), only the square root $r_{2}$ survives:

$r_{2}\to x_{3,7}^{2}\Delta,\quad r_{8}\to
x_{1,7}^{2}(x_{3,8}^{2}-x_{3,7}^{2}),\quad r_{9}\to
x_{1,7}^{2}(x_{3,8}^{2}-x_{1,3}^{2}),$ (32)

and this implies that the divergent penta-box integral shares the same overall
factor, the square root $(x_{1,5}^{2}x_{3,7}^{2}\Delta)^{-1}$, with the
divergent DCI double-box integral. Since the coefficients of the last two
terms in eq.(30) vanish, we here only need to consider the collinear limit of
$I_{3}$. By taking the collinear limit of its letters and keeping the leading
terms of $\epsilon$ in the symbol, we find that $\widehat{I}_{\text{pb}}$ has
the expected IR divergence structure and takes the form

$\widehat{I}_{\text{pb}}=\widehat{\mathbf{A}}_{8,2}^{(1)}\log^{2}(\epsilon)+I_{\text{pb}}^{(-1)}\log(\epsilon)+I_{\text{pb}}^{(0)},$
(33)

where $I_{\text{pb}}^{(-1)}$ and $I_{\text{pb}}^{(0)}$ are two finite, pure
MPL functions of weight $3$ and $4$ respectively. In the ancillary file
n82Lnnmhv_scale_component.txt, the three terms of
$\mathcal{S}(\widehat{I}_{\text{pb}}(x_{3},x_{5},x_{7},x_{1},x_{2}))$ are
stored in three variables purepb$log2e, purepb$loge and purepb$finite.

### The subtraction

Now we move to the subtraction in (15) which should give an IR finite ratio
function. The cancellation of leading divergences, i.e. the
$\log^{2}(\epsilon)$ part, is easy to see. Since the $\log^{2}(\epsilon)$
divergence of both penta-box and double-box integrals is given by the pure
function $\widehat{\mathbf{A}}_{8,2}^{(1)}$, by summing over their cyclic
images (see eq.(17)) in the ratio function, we see that $\log^{2}(\epsilon)$
divergence is cancelled perfectly.

What is much more non-trivial, is the cancellation of sub-leading
$\log(\epsilon)$ divergences in the ratio function. Note that the
$\log(\epsilon)$ part of both double-box and penta-box integrals is given by
complicated weight-$3$ MPL functions, which certainly look much more
complicated than the subtraction term in (15). We find it totally remarkable
that in the cyclic sum, all these complicated weight-3 MPL functions combine
into precisely the $\log(\epsilon)$ part of the eq.(17), namely simple
logarithmic functions times $\widehat{\mathbf{A}}_{8,2}^{(1)}$. This provides
a very non-trivial consistency check of our computations for both DCI-
regulated integrals, and such a precise cancellation of all IR divergences
produces a finite result for the ratio function. We record its symbol in
scalecomponent in the ancillary file n82Lnnmhv_scale_component.txt, and we
also provide its alphabet as the variable alphabet for convenience.

##  4 The symbol and alphabet of the ratio function

In this section, we discuss the symbol and alphabet of the ratio function,
${\widehat{\bf R}}^{(2)}_{8,2}$. The full symbol is too lengthy to be present
here, and we record it in an ancillary file. Here we briefly comment on
certain structures we find in the symbol of 8-point N2MHV, especially compared
with those for NMHV/MHV cases.

A new feature of this component of N2MHV amplitudes, as opposed to other
components of NMHV/MHV amplitudes, is that the symbol is odd with respect to
the flip $\Delta\to-\Delta$. This is due to the fact that its leading
singularity is proportional to (the inverse of) $\Delta$, thus, similar to the
one-loop component (four-mass box function) but for the first time for two-
loop amplitudes, each term of the symbol (a “word”) contains odd number of odd
letters, which are algebraic (non-rational) functions of momentum twistors.
Recall that for one-loop case, the symbol reads

${\cal
S}[\widehat{I}_{\text{bl}}^{(1)}(x_{1},x_{3},x_{5},x_{7})]=u\otimes\frac{z}{\bar{z}}-v\otimes\frac{1-z}{1-\bar{z}}\,,$
(34)

which is odd under $\Delta\to-\Delta$ since it contains the odd letters
$z/\bar{z},(1-z)/(1-\bar{z})$ in the second entry, e.g.
$\log(z/\bar{z})\to-\log(z/\bar{z})$ under the flip.

Similarly at two-loop level, since the first entry must be even letters, there
is either one such odd letter at the second, third and fourth entry, or all
these three entries are odd. In particular, this is the first example of
${\cal N}=4$ SYM amplitudes at two-loop level where the last entry can be odd
letters. Recall that for the two-loop 8-point NMHV amplitude we find $9$
multiplicatively independent algebraic letters that are odd in
$\Delta\to-\Delta$) associated with this square root (and $9$ more related by
a cyclic rotation $i\to i{+}1$). It is nice that this component of the two-
loop 8-point N2MHV amplitude still contains exactly this $9$-dimensional space
of odd letters, and we denote them as

$\\{\chi(a_{1}),\chi(a_{2})\\}|_{i\to i+2k}\text{ for $k=0,1,2,3$ and
}\frac{\chi(0)}{\chi(1)},$ (35)

where $\chi(a):=(z-a)/(\bar{z}-a)$ and

$a_{1}=\frac{\langle 4567\rangle\langle 1258\rangle}{\langle
2567\rangle\langle 1458\rangle},\quad a_{2}=\frac{\langle 4567\rangle\langle
1358\rangle}{\langle 3567\rangle\langle 1458\rangle}.$

Our result strongly supports the conjecture that altogether the $9+9$ letters
constitute all algebraic (odd) letters for $n=8$ amplitudes for all helicity
sectors.

Next we move to even letters, which are rational functions of momentum
twistors. We note that again almost all of them (together with those for the
other component, related by $i\to i{+}1$) have been seen in the alphabet of
$n=8$ NMHV amplitudes: they are $92$ polynomials of Plücker coordinates, which
can form $92-8=84$ DCI combinations. There are $56$ Plücker coordinates
($4$-brackets): under cyclic rotations $i\to i{+}2$ they form $13$ length-$4$
orbits and $2$ length-$2$ ones, with $15$ seeds chosen as

$\displaystyle\langle 1234\rangle,\langle 1235\rangle,\langle
1236\rangle,\langle 1237\rangle,\langle 1238\rangle,\langle
1245\rangle,\langle 1248\rangle,\underline{\langle 1256\rangle},\langle
1258\rangle,\langle 1267\rangle,\langle 1268\rangle,$ (36)
$\displaystyle\langle 1358\rangle,\langle 1368\rangle,\langle
1468\rangle,\underline{\langle 1458\rangle}\,,$

where two brackets with underline generate length-$2$ orbits under $i\to
i{+}2$ rotations. The remaining $36$ letters are quadratic in Plücker
coordinates, which form $9$ length-$4$ orbits with seeds

$\displaystyle\langle 1(23)(45)(67)\rangle,\langle
1(23)(45)(78)\rangle,\langle 1(28)(34)(67)\rangle,\langle
1(28)(45)(67)\rangle,$ (37) $\displaystyle\langle 2(13)(45)(67)\rangle,\langle
2(13)(45)(78)\rangle,\langle 2(18)(34)(67)\rangle,\langle
2(18)(45)(67)\rangle,\langle 812(67)\cap(345)\rangle\,,$

where we have defined

$\displaystyle\langle ijk(ab)\cap(cde)\rangle:=\langle ijka\rangle\langle
bcde\rangle-\langle ijkb\rangle\langle acde\rangle$ (38) $\displaystyle\langle
i(ab)(cd)(ef)\rangle:=\langle iacd\rangle\langle ibef\rangle-\langle
iaef\rangle\langle ibcd\rangle.$ (39)

In addition to these letters which have appeared in NMHV/MHV amplitudes, we
find exactly one new letter, which is nothing but the inverse of leading
singularity, $\Delta$!

Let us briefly comment on how the letters appear in the symbol. As indicated
by physical discontinuity conditions Gaiotto:2011dt , we find exactly $20$
letters of the form $\langle i\,i{+}1\,j\,j{+}1\rangle$ which appear in the
first entry; the frozen ones $\langle i\,i{+}1\,i{+}2\,i{+}3\rangle$ (for
$i=1,\cdots,8$, treated as $2$ length-$4$ orbits above), and $3$ length-$4$
orbits generated by seeds $\langle 1245\rangle$, $\langle 1256\rangle$ and
$\langle 1267\rangle$. The second entry respects Steinmann relations, and we
find that all first two entries are consistent with the general prediction of
He:2021mme : they are either $\log\log$ terms respecting Steinmann relations,
or the symbol of one-loop box functions i.e.
$I^{(1)}(x_{1},x_{3},x_{5},x_{7})$ and finite part of its lower-mass
degenerations

$\operatorname{Li}_{2}\biggl{(}1-\frac{x_{a,c}^{2}x_{a-1,b}^{2}}{x_{a-1,c}^{2}x_{a,b}^{2}}\biggr{)}\>,$

with $a+1<b<c<a-2$ in a cyclic order. The third entry contains almost all
letters in the alphabet, except for $4+4$ quadratic letters generated by
$\langle 1(28)(34)(67)\rangle$ and $\langle 2(13)(45)(78)\rangle$ which have
appeared in the second entry. Finally, we find the following letters for the
last entry: the $9$ algebraic (odd) letters of eq.(35), $\Delta$, $4+4$
quadratic ones generated by $\langle 1(23)(45)(67)\rangle$ and $\langle
2(81)(45)(67)\rangle$, and the following $34$ Plücker coordinates: the frozen
ones $\langle i\,i{+}1\,i{+}2\,i{+}3\rangle$, $6$ length-$4$ orbits with seeds
(all of the form $\langle\bar{i}j\rangle:=\langle i-1\,i\,i+1\,j\rangle$,
originated from MHV amplitude in the subtraction term):

$\langle 1235\rangle,\langle 1236\rangle,\langle 1237\rangle,\langle
1248\rangle,\langle 1258\rangle,\langle 1268\rangle\,.$ (40)

and a length-$2$ orbit $\\{\langle 1458\rangle$, $\langle 2367\rangle\\}$.

We have observed more structures for terms in the symbol involving odd
letters. In particular, those terms with $3$ odd letters are very simple: the
first-two entries can only be the symbol of the four-mass box, eq.(34), and we
record the last-two (odd) entries: in principle we have $9\times 9$ terms but
the result is much more compact, which we record as

$\sum_{i,j=1}^{9}c_{ij}v_{i}\otimes v_{j}\>,$

where

$(c_{ij})=\frac{1}{4}\begin{pmatrix}6&1&0&1&-4&-2&0&2&-2\\\
1&6&1&0&2&-4&-2&0&-2\\\ 0&1&6&1&0&2&-4&-2&-2\\\ 1&0&1&6&-2&0&2&-4&-2\\\
-4&2&0&-2&8&-2&0&-2&1\\\ -2&-4&2&0&-2&8&-2&0&1\\\ 0&-2&-4&2&0&-2&8&-2&1\\\
2&0&-2&-4&-2&0&-2&8&1\\\ -2&-2&-2&-2&1&1&1&1&6\\\ \end{pmatrix}$

and

$\\{v_{1},\cdots,v_{9}\\}=\biggl{\\{}\chi(a_{1}),\chi(a_{1})|_{i\to
i+2},\cdots,\chi(a_{2}),\chi(a_{2})|_{i\to
i+2},\cdots,\frac{\chi(0)}{\chi(1)}\biggr{\\}}.$

Alternatively, if we denote $\chi_{i}:=v_{i}$ and
$\chi^{\prime}_{i}:=v_{i{+}4}$ for $i=1,\cdots,4$ and
$\chi^{\prime\prime}:=v_{9}$, we can write this symmetric, weight-$2$ word as
the symbol of

$\displaystyle G=$
$\displaystyle\frac{1}{4}\sum_{i=1}^{4}\left(3\log^{2}\chi_{i}+\log\chi_{i}\log\chi_{i{+}1}+4\log^{2}\chi^{\prime}_{i}-2\log\chi^{\prime}_{i}\log\chi^{\prime}_{i{+}1}+3\log^{2}\chi^{\prime\prime}-\log\chi^{\prime\prime}\log\biggl{(}\frac{\chi_{i}^{2}}{\chi^{\prime}_{i}}\biggr{)}\right)$
$\displaystyle-\sum_{i=1}^{4}\left(\log\chi_{i}\log\chi^{\prime}_{i}+\frac{1}{2}\log\chi_{i}\log\frac{\chi^{\prime}_{i{+}1}}{\chi^{\prime}_{i{-}1}}\right)\,,$
(41)

where the symmetry under the cyclic rotation $i\to i+2$ is manifest. The part
of the complete symbol which contains $3$ odd letters can be put in a
remarkably compact form:

${\cal S}[\widehat{{\bf R}}^{(2)}_{8,2}]={\cal
S}[\widehat{I}_{\text{bl}}^{(1)}]\otimes{\cal S}[G]+\text{terms with one odd
letter}\,,$ (42)

where the remaining terms have one odd letter in either second, third or last
entry. These terms seem to be significantly more complicated. For example, for
terms where only the second entry is odd ($\chi(0)$ or $\chi(1)$), the first
two entries must again form the symbol of $\widehat{I}_{\text{bl}}^{(1)}$,
while the last two entries consist of even letters, including the new letter
$\Delta$.

##  5 Conclusion and Outlook

In this paper, we have computed a particularly nice component of two-loop
N2MHV amplitudes, whose leading singularity is proportional to the inverse of
the four-mass square root, which makes it distinct from other N2MHV
components. The component receives contributions from only two new integrals,
which we have computed using the DCI regularization and the finite ratio
function is obtained by subtracting IR divergences. This is the first two-loop
amplitudes beyond MHV and NMHV cases that have been computed, and the
resulting symbol has several interesting features; compared to NMHV case it
contains exactly one new letter, which is _the square root itself_ , and
either one or all of the last three entries must be algebraic (odd) letters.
We have found a remarkable simplicity for the latter case.

There are several directions worth further investigations. Two immediate
generalizations are (1) computing all components of the two-loop 8-point N2MHV
amplitude, which should be straightforward but rather tedious; (2) computing
such special components for $n$-point N2MHV amplitudes, e.g. $9$ such
components each with a square root for $n=9$ case; our techniques should be
applicable but several new cases of penta-box integrals etc. are needed. It
would be interesting to look for more structures in the symbol and alphabet of
such special components for $n=8$ and higher, similar to what we have
considered for NMHV and MHV cases; it would be nice to further study
mathematical structures in the symbol of our two-loop integrals (see
He:2021eec ; He:2021esx ; He:2021mme ; Yang:2022gko ; He:2022tph  and
references therein). Another important question is how to uplift these symbols
to functions and directly bootstrap them. Now that we have some control of
$n=8$ MHV, NMHV and these components of N2MHV amplitudes, it would also be
intriguing to compare them in more details.

Relatively, it would be highly desirable to study what do such structures,
especially the appearance of algebraic letters in the last entries, mean for
the ${\bar{Q}}$ and $Q^{(1)}$ anomaly equations, and eventually use them for
computing amplitudes with $k\geq 2$. In this regard, it would be nice to apply
these anomaly equations for certain components rather than for the complete
super-amplitude (related anomaly equations have been applied to other theories
and even individual integrals in Chicherin:2022gky ; Corcoran:2021gda ). We
have also demonstrated how DCI-regulated integrals are useful for computing
IR-safe quantities for loop amplitudes with higher multiplicities. It would be
interesting to push this direction further, including the computation of non-
planar SYM amplitudes and those in supergravity theories Bourjaily:2019iqr ;
Bourjaily:2019gqu , as well as amplitudes in other theories (such as ABJM, see
Caron-Huot:2012sos ; He:2022toappear ).

###### Acknowledgements.

We thank Rourou Ma, Zihao Wu, Qinglin Yang and Yang Zhang for collaborations
on related projects. SH is supported by National Natural Science Foundation of
China under Grant No. 11935013, 11947301, 12047502, 12047503 and the Key
Research Program of the Chinese Academy of Sciences, Grant NO. XDPB15. CZ was
supported in part by the ERC starting grant 757978 and grant 00025445 from the
Villum Fonden.

##  Appendix A Review of multiple polylogarithm and its symbol

Many amplitudes and Feynman integrals in QFT can be described by multiple
polylogarithms. In this appendix, we briefly review some basic facts about
multiple polylogarithms.

Multiple polylogarithms are generalizations of classical polylogarithms, such
as $\log$, $\operatorname{Li}_{2}$, $\operatorname{Li}_{3}$, $\dots$, which
are defined by goncharov2005galois

$G(a_{1},\ldots,a_{n};z):=\int_{0}^{z}\mathrm{d}\log(t-a_{1})\,G(a_{2},\ldots,a_{n};t),$
(43)

with $G(;z):=1$ and exceptional cases

$G(\underbrace{0,\ldots,0}_{k};z):=\frac{1}{k!}(\log z)^{k},$

where $n$ is called the weight of $G(a_{1},\ldots,a_{n};z)$. For example,

$G(0;x)=\log x,\quad G(1;x)=\log(1-x),\quad
G(0,1;x)=-\operatorname{Li}_{2}(x),\quad\dots.$

It is straightforward to see that the total differential of eq.(43) satisfies

$\mathrm{d}G(a_{1},\dots,a_{n};z)=\sum_{i=1}^{n}G(a_{1},\dots,\hat{a}_{i},\dots,a_{n};z)\,\mathrm{d}\log\frac{a_{i}-a_{i{-}1}}{a_{i}-a_{i{+}1}},$
(44)

where $a_{i}$ is deleted in the $i$-th summand with boundary cases $a_{0}:=z$
and $a_{n{+}1}:=0$.

There is a well-known Hopf algebra structure Goncharov:2005sla  on multiple
polylogarithms, which has led to the notion of symbol Goncharov:2010jf ;
Duhr:2011zq . For any weight-$n$ multiple polylogarithm $G^{(n)}$ whose
differential reads

$\mathrm{d}G^{(n)}=\sum_{i}G^{(n-1)}_{i}\mathrm{d}\log x_{i},$

where $\\{G^{(n-1)}_{i}\\}$ are polylogarithms of lower weight $(n{-}1)$, its
symbol $\mathcal{S}(G^{(n)})$ is recursively defined by

$\mathcal{S}(G^{(n)}):=\sum_{i}\mathcal{S}(G^{(n-1)}_{i})\otimes x_{i}.$

For example,

$\mathcal{S}(\log(x))=x,\quad\mathcal{S}(\operatorname{Li}_{2}(x))=-\,(1-x)\otimes
x.$

The entries of a symbol are called its letters, and the collection of all
letters is called its alphabet.

For our calculation, it’s more convenient to perform integrations on the
symbol level directly based on the following rules CaronHuot:2011kk : Suppose
we have an integral

$\int_{a}^{b}{\rm d}\log(t+c)\,(F(t)\otimes w(t)),$

where $F(t)\otimes w(t)$ is a linear reducible symbol in $t$, i.e. its entries
are products of powers of linear polynomials in $t$, and $w(t)$ is the last
entry. The total differential of this integral is the sum of the following two
parts:

  1. (1)

the contribution from endpoints:

${\rm d}\log(t+c)\Bigl{(}F(t)\otimes
w(t)\Bigr{)}\Bigr{|}_{t=a}^{t=b}\Rightarrow\Bigl{(}F(t)\otimes
w(t)\otimes(t+c)\Bigl{)}\Bigr{|}_{t=a}^{t=b},$ (45)

  2. (2)

contributions from the last entry: for a term where $w(t)$ is a constant,

$\left(\int_{a}^{b}{\rm d}\log(t+c)\,F(t)\right){\rm d}\log
w\Rightarrow\left(\int_{a}^{b}{\rm d}\log(t+c)\,F(t)\right)\otimes w,$ (46)

and for a term where $w(t)=t+d$,

$\left(\int_{a}^{b}{\rm d}\log\frac{t+c}{t+d}\,F(t)\right){\rm
d}\log(c-d)\Rightarrow\left(\int_{a}^{b}{\rm
d}\log\frac{t+c}{t+d}\,F(t)\right)\otimes(c-d).$ (47)

##  Appendix B Alphabet of 3-loop MHV octagon and 2-loop NMHV octagon

The symbol alphabet for the three-loop MHV BDS-normalized octagon
$(\mathcal{A}_{8,0}/\mathcal{A}_{8}^{\text{BDS}})^{(3)}$ consists of $204$
multiplicative-independent rational letters and 18 independent DCI algebraic
letters Li:2021bwg . The 204 rational letters are organized as follows

  * •

$\binom{8}{4}-2=68$ : all $\langle abcd\rangle$ except $\langle 1357\rangle$
and $\langle 2468\rangle$;

  * •

1 cyclic class of $\langle 12(345)\cap(678)\rangle$;

  * •

7 cyclic classes of $\langle 1(ij)(kl)(mn)\rangle$ with $2\leq i<j<k<l<m<n\leq
8$;
5 cyclic classes of $\langle 1(28)(kl)(mn)\rangle$ with $2<k<l<m<n<8$;

  * •

5 cyclic classes of
$\langle\bar{2}\cap\bar{4}\cap(568)\cap\bar{8}\rangle,\langle\bar{2}\cap\bar{4}\cap\bar{6}\cap(681)\rangle,\langle(127)\cap(235)\cap\bar{5}\cap\bar{7}\rangle,\langle(127)\cap\bar{3}\cap(356)\cap\bar{7}\rangle,\langle\bar{2}\cap(278)\cap(346)\cap\bar{6}\rangle.$

Here we introduce the notations $\bar{a}=(a{-}1,a,a{+}1)$ and

$\displaystyle\langle ab(cde)\cap(fgh)\rangle:=\langle abde\rangle\langle
cfgh\rangle+\langle abec\rangle\langle dfgh\rangle+\langle abcd\rangle\langle
efgh\rangle,$
$\displaystyle\langle(a_{1}b_{1}c_{1})\cap(a_{2}b_{2}c_{2})\cap(a_{3}b_{3}c_{3})\cap(a_{4}b_{4}c_{4})\rangle:=\langle(a_{1}b_{1}c_{1})\cap(a_{2}b_{2}c_{2}),(a_{3}b_{3}c_{3})\cap(a_{4}b_{4}c_{4})\rangle.$

For the two-loop NMHV BDS-normalized octagon Zhang:2019vnm , there are only
180 rational letters, the following 24 rational letters do not appear,

$\text{cyclic images of }\langle
1(23)(46)(78)\rangle,\>\langle\bar{2}\cap\bar{4}\cap(568)\cap\bar{8}\rangle\text{
and }\langle\bar{2}\cap\bar{4}\cap\bar{6}\cap(681)\rangle.$

The algebraic letters are the same for the 3-loop MHV octagon and the two-loop
NMHV octagon, which are cyclic images of eq.(35) by taking $i\to i+1$.

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