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Eur. Phys. J. C (2020) 80:215

https://doi.org/10.1140/epjc/s10052-020-7760-x

Regular Article - Theoretical Physics

Dimensional schemes for cross sections at NNLO

C. Gnendiger

1,a

, A. Signer

1,2

Paul Scherrer Institut, 5232 Villigen, PSI, Switzerland

Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

Received: 13 January 2020 / Accepted: 19 February 2020

Abstract So far, the use of different variants of dimen-

sional regularization has been investigated extensively for

two-loop virtual corrections. We extend these studies to real

corrections that are also required for a complete compu-

tation of physical cross sections at next-to-next-to-leading

order. As a case study we consider two-jet production in

electron-positron annihilation and describe how to compute

the various parts separately in different schemes. In particu-

lar, we verify that using dimensional reduction the double-

real corrections are obtained simply by integrating the four-

dimensional matrix element over the phase space. In addition,

we conﬁrm that the cross section is regularization-scheme

independent.

Contents

1 Introduction ......................

2 Dimensional schemes .................

3 The process e

−

→ γ

∗

→ q ¯q in DRED .......

3.1 Tree-level contribution ..............

3.2 Virtual corrections ................

3.3 Real corrections ..................

3.4 Real-virtual corrections ..............

3.5 Combination of the contributions .........

3.6 Contributions from -scalar photons .......

4 NNLO corrections in FDH and HV ..........

5 Conclusions ......................

References .........................

1 Introduction

Beyond leading order, physical cross sections are usually

computed as sums of several terms that are individually diver-

gent. These divergences stem from the ultraviolet (UV) and

e-mail: Christoph.Gnendiger@psi.ch (corresponding author)

the infrared (IR) regions of momentum integrals. In most

applications such divergences are dealt with by using dimen-

sional regularization, i. e. by working in d = 4 − 2 dimen-

sions. This renders intermediate expressions well-deﬁned

with divergences manifest as 1/

poles. While it is manda-

tory to treat integration momenta in d dimensions, there

is quite some freedom on how to treat other quantities in

such computations. Hence, in practice there is a variety

of dimensional schemes that can be used, the most com-

mon being conventional dimensional regularization (cdr),

the ’t Hooft–Veltman scheme (hv)[1], the four-dimensional

helicity scheme (fdh)[2], and dimensional reduction (dred)

[3]. For an overview and a discussion of the basic prop-

erties of these schemes we refer to [4] and references

therein.

As it might be advantageous to use different regulariza-

tion schemes for different parts of the calculation, the rela-

tions between the various schemes have to be understood.

Starting this program, transition rules for UV-renormalized

virtual amplitudes at next-to-leading order (NLO) have ﬁrst

been worked out in [5] for massless QCD and were then gen-

eralized to the massive case [6]. The regularization-scheme

independence of cross sections at NLO is discussed in [7] and

a recipe on how to compute consistently the various ingre-

dients (virtual, real, and initial-state collinear counterterm)

for hadronic collisions is given in [8,9]. The key observa-

tion is that so-called -scalars have to be introduced and to

be considered as independent from d-dimensional gluons.

Going beyond NLO, a lot of work has been done to under-

stand the UV renormalization [10–14] as well as the vir-

tual two-loop contributions [15–20] in schemes other than

cdr. The current status can be summarized as follows: for all

dimensional schemes mentioned above, UV renormalization

and virtual corrections are understood at least up to next-

to-next-to leading order (NNLO), while the computation of

real corrections is only fully understood at NLO. A ﬁrst step

towards NNLO for the latter has been made in [21] where

the hv scheme is used for real corrections. It is the purpose

0123456789().: V,-vol

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215 Page 2 of 12 Eur. Phys. J. C (2020) 80:215

of the present paper to make further progress on the scheme

dependence of real contributions and to discuss the calcula-

tion of double-real and real-virtual corrections at NNLO in

different schemes. In particular, we will focus on dred as

this is the most general dimensional scheme usually consid-

ered.

To this end, we consider NNLO QCD corrections to

the process e

−

→ 2 jets and compute the double-

virtual, double-real, and real-virtual contributions separately

in dred. This is an extension of the corresponding com-

putation in cdr [22,23]. Moreover, we perform the com-

putations also in hv and fdh and show that the physical

cross section is regularization-scheme independent. The pro-

cess at hand has also been considered in non-dimensional

regularization schemes, i. e. schemes that keep the inte-

gration momenta in strictly four dimensions. In fact, the

fermionic contributions have been computed recently [24]

using ’four-dimensional regularization’ (fdr)[25,26]. Sim-

ilar to ’implicit regularization’ [27–29] and ’loop regular-

ization’ [30,31], fdr is a four-dimensional framework to

compute higher-order corrections. Another approach that

is being investigated is to use loop-tree duality to deal

with IR singularities at the integrand level [32,33]. While

these are interesting developments, they typically require

that the full computation is performed in the correspond-

ing scheme. It will be very difﬁcult to combine partial results

obtained in dimensional schemes with computations in non-

dimensional schemes. Hence, in this work we focus on the

former.

We start in Sect. 2 with a brief recapitulation of the

most important aspects of the dimensional schemes before

we consider QCD corrections to the process e

−

→

∗

→ q ¯q in Sect. 3. This section contains the main

results of the paper including a description on how to

compute double-virtual, double-real, and real-virtual cor-

rections in dred. We also show that the total cross sec-

tion is scheme independent, as required. The particular

role of the -scalars is investigated in Sect. 3.6.The

computation of the cross section in fdh is discussed in

Sect. 4 together with the hv scheme, before we conclude

in Sect. 5.

2 Dimensional schemes

As mentioned before, an efﬁcient way to regularize UV and

IR divergences at the same time is to formally shift the dimen-

sion of loop and phase-space integrations from (strictly) four

d ≡ 4 − 2 , (2.1)

with an arbitrary regularization parameter . In this way,

divergent integrals are parametrized in terms of 1/

poles.

Although not strictly necessary, it is usually advantageous

to also modify the dimensionality of other algebraic objects.

The most commonly used approach in this respect is cdr,

where all Lorentz indices are considered in quasi d dimen-

sions. Indicating the dimension by a subscript, we therefore

write

cdr : k

[d]

,γ

[d]

, g

μν

[d]

, A

[d]

,... (2.2)

for loop momenta, γ matrices, metric tensors, vector ﬁelds,

etc. From a conceptual point of view this approach is the sim-

plest realization of dimensional regularization in the sense

that all dimensionful quantities are treated on the same foot-

ing. As a consequence, it is for example sufﬁcient to impose

one single (modiﬁed) Lorentz algebra. However, it is impor-

tant to realize that this formal simplicity does not automat-

ically guarantee that cdr is also the best choice regarding

computational efﬁciency.

A second realization of dimensional regularization is

dred where all dimensionful quantities except for loop

momenta are treated in quasi d

dimensions with

≡ d + n



. (2.3)

The value of d

does not necessarily have to be ﬁxed as long

as the limit n



→ 0 is implied at the end of the calculation.

Usually, however, it is taken to be d

= 4, resulting in n



2. The dimensionful quantities from above are accordingly

written as

dred : k

[d]

,γ

]

, g

μν

]

, A

]

,... . (2.4)

One important aspect of dred is that (in contrast to what the

name of the scheme suggests) the underlying vector space

is ’bigger’ than the one of cdr, as indicated by (2.3). Thus,

in spite of d

= 4, for consistency the vector space of dred

is in fact inﬁnite-dimensional [34]. Therefore, it is always

possible to split quasi d

-dimensional quantities into a quasi

d-dimensional ’cdr part’ and an evanescent remainder, e. g.

]

= A

[d]

+ A



]

(2.5)

for vector ﬁelds. The ﬁeld A



]

is often referred to as -

scalar. In the case of fdh and hv we also need strictly four-

dimensional quantities such as γ

[4]

, g

μν

[4]

, and A

[4]

,asdis-

cussed in [4].

The application of the schemes mentioned above to the

computation of two-loop virtual corrections is well under-

stood and leads to universal scheme dependences which can

be described in terms of scheme-dependent IR anomalous

dimensions. Since physical cross sections must not depend

123

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