![](https://csdnimg.cn/release/download_crawler_static/12287940/bgc.jpg)
We conclude this section with a comment on the regularisation of MPLs. We have seen
that G(a
1
, . . . , a
n
; z) diverges whenever z = a
1
. Hence, when evaluating HPLs at z = ±1
the result may be divergent. It is easy to see that all divergencies are logarithmic, and it is
useful to introduce a formal quantity lim
z→1
G(1; z) = −ζ
1
which acts as a regulator and
allows one to keep track of divergences. Indeed, all divergencies show up as powers of ζ
1
,
and the dependence on the regulator ζ
1
must cancel for finite quantities. The regulator ζ
1
is represented by HPL[{1},1] in Hpl and PolyLogTools.
There is, however, a subtle point about the regularisation discussed in the previous
paragraph. As usual, there is an ambiguity in how to introduce the regulator, related
to shifting divergent quantities by finite terms. In practice, one would like to choose a
regulator that preserves as many of the algebraic structures as possible. MPLs are equipped
with both a shuffle and a stuffle algebra structure. However, it is not possible to preserve
both structures at the same time, leading to two different ‘schemes’ to regulate MPLs and
MZVs [73], referred to as shuffle regularisation and stuffle regularisation respectively. For
example, using the shuffle product on MPLs, we obtain
ζ
2
1
= lim
z→1
G(1; z)
2
= lim
z→1
2 G(1, 1; z) , (5.3)
so that the shuffle-regulated value of G(1, 1; z) at z = 1 is given by
1
2
ζ
2
1
. Instead, using the
fact that G(1; z) = −Li
1
(z) as well as the stuffle product, we find
ζ
2
1
= lim
z→1
G(1; z)
2
= lim
z→1
Li
1
(z)
2
= lim
z→1
2Li
1,1
(z, z) + Li
2
(z
2
)
= ζ
2
+ lim
z→1
2 G(1, 1; z) ,
(5.4)
so that the stuffle-regulated value of G(1, 1; z) at z = 1 is given by
1
2
ζ
2
1
−
1
2
ζ
2
. We see that
the shuffle- and stuffle-regulated values are different. Note that any finite quantity must
be independent of the regularisation scheme chosen to compute it. However, care is needed
that the same scheme is applied consistently throughout a computation!
By default, PolyLogTools uses shuffle regularisation to regulate the G and H func-
tions, because in applications it is often more convenient to preserve the shuffle alge-
bra structure (which is more closely related to the definition of MPLs as iterated inte-
grals). It is possible to switch to stuffle regularisation by changing the value of the variable
PLT$ShuffleRegularisation from True to False. We point out that, unlike PolyLog-
Tools, the Hpl package uses stuffle regularisation to define the values of MZVs and HPLs
at z = ±1. Care is thus needed if the output and functions from this package are used in
conjunction with PolyLogTools.
6 The Hopf algebra of MPLs
An important property of MPLs is that they can be equipped with a coproduct turning
them into a Hopf algebra. The Hopf algebra of MPLs has led to the development of
novel techniques to deal with Feynman integrals that evaluate to MPLs, and it has seen a
multitude of applications over the last few years. In particular, it allows one to control more
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