#### To \({d}\) , or not to \({d}\) : recent developments and comparisons of regularization schemes

Eur. Phys. J. C
To d , or not to d : recent developments and comparisons of regularization schemes
C. Gnendiger 9
A. Signer 8 9
D. Stöckinger 7
A. Broggio 14
A. L. Cherchiglia 13
F. Driencourt-Mangin 12
A. R. Fazio 11
B. Hiller 16
P. Mastrolia 10 15
T. Peraro 5
R. Pittau 6
G. M. Pruna 9
G. Rodrigo 12
M. Sampaio 3
G. Sborlini 1 4 12
W. J. Torres Bobadilla 10 12 15
F. Tramontano 0 2
Y. Ulrich 8 9
A. Visconti 8 9
0 INFN, Sezione di Napoli , 80126 Naples , Italy
1 INFN, Sezione di Milano , 20133 Milan , Italy
2 Dipartimento di Fisica, Università di Napoli , 80126 Naples , Italy
3 Departamento de Fïsica , ICEX, UFMG, 30161-970 Belo Horizonte , Brazil
4 Dipartimento di Fisica, Università di Milano , 20133 Milan , Italy
5 Higgs Centre for Theoretical Physics, The University of Edinburgh , Edinburgh EH9 3FD , UK
6 Dep. de Física Teórica y del Cosmos and CAFPE, Universidad de Granada , 18071 Granada , Spain
7 Institut für Kernund Teilchenphysik, TU Dresden , 01062 Dresden , Germany
8 Physik-Institut, Universität Zürich , 8057 Zürich , Switzerland
9 Paul Scherrer Institut , 5232 Villigen, PSI , Switzerland
10 INFN, Sezione di Padova , 35131 Padua , Italy
11 Departamento de Física, Universidad Nacional de Colombia , Bogotá D.C. , Colombia
12 Insituto de Física Corpuscular, UVEG-CSIC, Universitat de València , 46980 Paterna , Spain
13 Centro de Ciências Naturais e Humanas, UFABC , 09210-170 Santo André , Brazil
14 Physik Department T31, Technische Universität München , 85748 Garching , Germany
15 Dipartimento di Fisica ed Astronomia, Università di Padova , 35131 Padua , Italy
16 CFisUC, Department of Physics, University of Coimbra , 3004-516 Coimbra , Portugal
We give an introduction to several regularization schemes that deal with ultraviolet and infrared singularities appearing in higher-order computations in quantum field theories. Comparing the computation of simple quantities in the various schemes, we point out similarities and differences between them.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 DS: dimensional schemes CDR, HV, FDH, DRED . .
2.1 Integration in d dimensions and dimensional schemes
2.2 Application example 1: electron self-energy at NLO
2.3 Application example 2: e+e− → γ ∗ → qq¯ at NLO
Virtual contributions . . . . . . . . . . . . . . . .
Real contributions . . . . . . . . . . . . . . . . .
2.4 Established properties and future developments
of DS . . . . . . . . . . . . . . . . . . . . . . .
3 FDF, SDF: four- and six-dimensional formalism . . .
3.1 FDF: four-dimensional formulation of FDH . . .
UV renormalization . . . . . . . . . . . . . . . .
5 FDR: four-dimensional regularization/renormalization
5.1 FDR and UV infinities . . . . . . . . . . . . . .
5.2 FDR and IR infinities . . . . . . . . . . . . . . .
5.3 Application example: e+e− → γ ∗ → qq¯ at NLO
Virtual contributions . . . . . . . . . . . . . . . .
Real contributions . . . . . . . . . . . . . . . . .
5.4 Established properties and future developments
of FDR . . . . . . . . . . . . . . . . . . . . . .
Correspondence between integrals in FDR and DS
Gauge invariance, unitarity, and extra integrals . .
6 FDU: four-dimensional unsubtraction . . . . . . . . .
6.1 Introduction to LTD . . . . . . . . . . . . . . . .
6.2 Momentum mapping and IR singularities . . . .
6.3 Integrand-level renormalization and self-energies
6.4 Application example: e+e− → γ ∗ → qq¯ at NLO
6.5 Further considerations and comparison with
other schemes . . . . . . . . . . . . . . . . . . .
7 Summary and outlook . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Higher-order calculations in quantum field theories usually
involve ultraviolet (UV) and/or infrared (IR) divergences
which need to be regularized at intermediate steps. Only
after renormalization and proper combination of real and
virtual corrections, a finite and regularization-scheme
independent result can be obtained. The choice of the regularization
scheme matters in several respects of conceptual and
practical relevance:
• Mathematical consistency: It must be excluded that the
calculational rules lead to internal inconsistencies such
as final expressions contradicting each other.
• Unitarity and causality: The final finite result must be
compatible with the basic quantum field theoretical
properties of unitarity and causality. In practice this
compatibility can be shown by proving the equivalence of a
given scheme with ms or bphz renormalization, which
are known to have these properties.
• Symmetries: It is desirable that symmetries like Lorentz
invariance, non-Abelian gauge invariance, or
supersymmetry are manifestly preserved by the regularization to
the largest possible extent. Symmetry breaking by the
regularization which does not correspond to anomalies
must be compensated by special, symmetry-restoring
counterterms.
• Quantum action principle: The regularized quantum
action principle is a relation between symmetries of the
regularized Lagrangian and Ward/Slavnov–Taylor
identities of regularized Green functions. If it is valid in a given
regularization scheme, the study of symmetry properties
is strongly simplified.
• Computational efficiency: The regularization scheme
should allow for efficient calculational techniques and
ideally reduce the technical complexity as much as
possible.
In recent years, the understanding of traditional
regularization schemes has further improved, and novel schemes
have been proposed and developed. The motivation for this
progress has been to broaden the conceptual basis as well as
to enable new efficient, automated analytical and numerical
calculational methods. It appears timely to present a uniform
and up-to-date description of all schemes and to collect and
compare all established properties, definitions, and
calculational procedures. This is the goal of the present report. The
covered schemes are the following:
• traditional dimensional schemes: conventional
dimensional regularization (cdr), the ‘t Hooft–Veltman scheme
(hv), the four-dimensional helicity scheme (fdh), and
dimensional reduction (dred),
• new, distinctive (re-)formulations of dimensional
schemes: the four-dimensional formulation of the fdh
scheme (fdf), the six-dimensional formalism (sdf),
• non-dimensional schemes: implicit regularization (ireg),
four-dimensional regularization/renormalization (fdr),
four-dimensional unsubtraction (fdu).
In the following we present introductions to all these
schemes. Having applications and practitioners in mind we
will perform some simple calculations to illustrate the
differences as well as common features of the schemes. In
particular, we aim to sketch the computation of the cross
section for e+e− → γ ∗ → qq¯ at next-to-leading order and
the fermion self-energy. The quantities are chosen such that
potential technical disadvantages of the traditional schemes
are exposed and the properties of novel schemes with respect
to UV and IR divergences and (sub)renormalization can be
illustrated. In a number of footnotes we will directly compare
intermediate results and features of the different schemes and
comment on their relation.
Of course, much more detailed information is available in
the literature and we refer to the references listed in the
individual sections for a more in-depth discussion. However, we
also have to warn the reader that, unfortunately, the
nomenclature and notation used in the literature is far from being
unique. This often leads to misunderstandings. In an attempt
to avoid these in the future, we have adopted a unified
description in this article. As a result, the notation and terms used
here will differ in parts from the notation used in the
specialized literature referred to. To help further with clearing out
some of the misunderstandings and elucidating the relation
between the schemes, we will conclude in Sect. 7 by giving
a list of concrete statements.
2 DS: dimensional schemes CDR, HV, FDH, DRED
2.1 Integration in d dimensions and dimensional schemes
Dimensional regularization [
1, 2
] and variants are the most
common regularization schemes for practical calculations in
gauge theories of elementary particle physics. In the
following we summarize the basic definitions common to all
dimensional schemes (ds) discussed in Sects. 2 and 3 and then
provide specific definitions for four variants of ds which differ
by the rules for the numerator algebra in analytical
expressions.
The basic idea of all ds is to regularize divergent integrals
by formally changing the dimensionality of space-time and
of momentum space. In the present report we always denote
the modified space-time dimension by d, and we set
d ≡ 4 − 2 .
Correspondingly, a four-dimensional loop integration is
replaced by a d-dimensional one,1
d4k[
4
]
(2π )4 → μ4d−sd
dd k[d] ,
(2π )d
including the scale of dimensional regularization, μds. After
this replacement, UV and IR divergent integrals lead to poles
of the form 1/ n . In Refs. [
3, 4
], it is shown that such an
operation can indeed be defined in a mathematical
consistent way and that this operation has the expected properties
such as linearity and invariance under shifts of the integration
momentum.
To define a complete regularization scheme for realistic
quantum field theories, it must be specified how to deal with
γ matrices, metric tensors, and other objects appearing in
analytical expressions. Likewise, it should be specified how
to deal with vector fields in the regularized Lagrangian. On
a basic level, two decisions need to be made,
• regularize only those parts of diagrams which can lead to
divergences, or regularize everything;
• regularize algebraic objects like metric tensors, γ
matrices, and momenta in d dimensions, or in a different
dimensionality.
1 In this section and in Sect. 3, the (quasi)dimensionality dim of an
object is indicated by a subscript [dim]. In Sects. 4–6, where loop
integrations are performed in strictly four dimensions, the subscript is
suppressed unless stated otherwise.
(2.1)
(2.2)
It turns out that there is an elegant way to unify essentially
all common variants of ds in a single framework, where
all definitions can easily be formulated and where the
differences and relations between the schemes become
transparent. This framework is based on distinguishing strictly
four-dimensional objects, formally d-dimensional objects,
and formally ds -dimensional objects.2 These objects can be
mathematically realized [
3–5
] by introducing a strictly
fourdimensional Minkowski space S[
4
] and infinite-dimensional
vector spaces QS[ds ], QS[d], QS[n ], which satisfy the
relations
QS[ds ] = QS[d] ⊕ QS[n ],
The space QS[d] is the natural domain of cdr and of
momentum integration in all considered schemes. Using
ds ≡ d + n
g[μdνs ] = g[μdν] + g[μnν ],
γ[ds ] = γ[d] + γ[μn ].
μ μ
(2.5)
Since the quantities in Eq. (2.5) do not have a
finitedimensional representation, in most of the practical
calculations only their algebraic properties are relevant,
(g[dim])μμ = dim, (g[d]g[n ])μν = 0,
{γ[dim], γ[dim]} = 2g[μdνim], {γ[d], γ[n ]} = 0,
μ ν μ ν
(2.6a)
(2.6b)
with dim ∈ {4, ds , d, n }.
Furthermore, a complete definition of the various
dimensional schemes requires one to distinguish two classes of
vector fields (VF):4
• Vector fields associated with particles in 1PI diagrams or
with soft and collinear particles in the initial/final state
are in the following called singular VF.
• All other vector fields are called regular VF.
(2.3)
(2.4)
2 In many original references, objects such as γˆ μ, γ˜ μ, γ˘ μ are
introduced with specific meanings which differ, depending on the paper,
the scheme, and the context. Hence, we avoid such short-hand
notations here and stick with a more explicit one to make the meaning of
expressions more apparent.
3 In fdh and dred, ds is usually taken to be 4, and therefore n = 2 .
4 Note that compared to Ref. [
6
] we replaced the terms ‘internal’ and
‘external’ by ‘singular’ and ‘regular’, respectively, to avoid possible
confusion in later considerations.
Since UV and IR divergences are only related to singular VF
there is some freedom in the treatment of the regular ones.
In this report, we distinguish the following four ds:
• cdr and hv are two flavours of what is commonly called
‘dimensional regularization’. They regularize algebraic
objects in d dimensions, n -dimensional objects are not
used. In cdr, all VF are regularized, in hv only singular
ones.
• fdh and dred are two flavours of what is commonly
called ‘dimensional reduction’. They regularize algebraic
objects in ds dimensions. Sometimes ds is identified as
ds ≡ 4 from the beginning, but it is possible to keep it
as a free parameter, which is set to 4 only at the end of a
calculation. In dred, all VF are regularized, in fdh only
singular ones.
The definitions of these four schemes can be essentially
reduced to the treatment of vector fields; see Table 1. This
unified formulation of the four schemes makes obvious that
a calculation in dred covers all elements of a calculation in
the other schemes.
In fdh and dred, where singular vector fields are treated
in ds dimensions, the split of Eq. (2.5) can be applied to the
regularized Lagrangian and to covariant derivatives. As an
illustration, we provide here the regularized covariant
derivatives in QED and QCD,
QED:
D[ds ]ψi = ∂[μd]ψi + i (e A[d] + ee A[n ])Qψi ,
μ μ μ
(2.7a)
QCD:
D[ds ]ψi = ∂[μd]ψi + i (gs A[μd,]a + ge A[μn,a])Tiaj ψ j .
μ
(2.7b)
It is important that the gauge-field part is not written
as a complete ds -dimensional entity but is split into
ddimensional and n -dimensional parts, and particularly with
independent couplings. Conventionally, the n -dimensional
fields are called ‘ -scalars’, the associated couplings are
called ‘evanescent couplings’. This split is strictly necessary
at the multi-loop level in non-supersymmetric theories since
the evanescent couplings are not protected by d-dimensional
Lorentz and gauge invariance and renormalize differently
compared to the corresponding gauge couplings. As an
example, we provide the (minimal) renormalization of the QED
gauge coupling and the corresponding evanescent coupling
in fdh/dred,
β = μ
βe = μ
−
2 d
dμ2
2 d
dμ2
e
4π
2
e
4π
ee
4π
ee
4π
2
2
= −
= −
2
e
4π
ee
4π
4
4
These values can be obtained e.g. from Ref. [
7
] by setting
C A → 0, NF → 2NF . It is obvious that even for ee = e, the
values of β and βe are not the same.
2.2 Application example 1: electron self-energy at NLO
To illustrate the different treatment of the Lorentz algebra
in the various ds, we consider the electron self-energy at
NLO in dred; see Fig. 1. As mentioned in the previous
section, this can be seen as the most comprehensive case of the
four considered ds. For simplicity, we use massless QED as
underlying theory. On the one hand, the Lorentz algebra can
then be evaluated by applying the split of Eq. (2.5),
−i
(1) i
dred = − {
(1)(e2) + ˜ (1)(ee2)}
d(2dπk[)dd] {e2(−d + 2) + ee2(d − ds )}
where Feynman gauge and the equality n = (ds − d) have
been used. Setting n = 0 then corresponds to the results in
cdr and hv.
On the other hand, for ee = e, the amplitude can also
be evaluated more directly by using a quasi-ds -dimensional
algebra,
−i d(1r)ed = μ4d−sd
d(2dπk[)dd] {e γ[ds ]γ[νd] γ[ρds ](g[ds])μρ }
2 μ
(k[d])ν
×
In the second line, the identity γ νd (k[d])ν = γ[νds ](k[d])ν is
[ ]
used which directly follows from the structure of the vector
spaces in Eq. (2.3).
When setting ds = 4, one obtains the result in fdh/dred.
Moreover, setting ee = e with α = e2/(4π ), it follows that
the different treatment of the algebra in Eqs. (2.9) and (2.10)
yields the same result,
α
−i d(1r)ed = i /p[d] 4π
1
As long as no distinction between gauge and evanescent
couplings is required, both approaches are therefore equivalent.
At the two-loop level, however, the different UV
renormalization of e and ee enters via the counterterm diagrams
shown on the right of Fig. 1,
−i d(2r,ecdt) = −i {δ(1)e2 ×
(1)(e2) + δ(1)ee2 × ˜ (1)(ee2)}.
(2.12)
Since no distinction between the couplings is possible when
using a quasi-ds -dimensional algebra, in this case it is
mandatory to apply the split of Eq. (2.5). Generalizing to an arbitrary
-loop calculation, the introduction and separate treatment of
-scalars has to be considered up to ( − 1) loops. Genuine
loop diagrams, on the other hand, can either be evaluated by
using the split of Eq. (2.5) or by using a quasi-ds -dimensional
Lorentz algebra. Further details regarding the UV
renormalization in the various ds can be found in Refs. [
7–11
].
2.3 Application example 2: e+e− → γ ∗ → qq¯ at NLO
Any physical observable has to be independent of the
regularization scheme. What is usually done in computing NLO
cross sections is to obtain the virtual corrections in cdr
(either directly, or first in another scheme and then
translated to cdr) and combine them with the real corrections
calculated in cdr. As shown in Ref. [
6
], it is also possible to
compute the real corrections directly in schemes other than
cdr.
We use the very simple process e+e− → γ ∗ → qq¯ with
massless quarks to illustrate the interplay between the scheme
dependence in the real and virtual corrections at NLO in
QCD. To simplify further, we average over the directions of
the incoming leptons (with momenta p and p ) and actually
consider only γ ∗ → qq¯ . This is achieved by replacing the
(regularization-scheme dependent) leptonic tensor by
Ldμsν = (i e)2Tr[ /p γ μ /pγ ν ] → 4e2 dim − 2
2(dim − 1)
× (sg[μdνim] − qμqν ) →
4e2
3 sg[μdνim],
(2.13)
where s ≡ q2 = ( p + p )2. In the first step, the average
is taken in dim dimensions. However, the prefactor will be
an overall factor of the full cross section. Hence, for this
prefactor we set dim = 4 from the beginning and the only
scheme dependence that is left in Ldμsν is in the one in g[μdνim].
The following discussion might create the impression that
schemes other than cdr are complicated to use. However, this
is simply because we will give the details of the field-theoretic
background. This results in many apparent ‘complications’
that can actually be avoided at a practical level.
Let us begin with the most straightforward case of cdr,
where the regular photon is treated in d dimensions. Here,
only the left diagram in Fig. 2 contributes. According to
Table 1, the metric tensor of the photon propagator – and
hence in Eq. (2.13) – is gμdν , the coupling at the vertices
[ ]
is the gauge coupling e. Using Eq. (2.13), we get for the
(spin summed/averaged) squared matrix element Md(0s) =
(0) (0)
Ads |Ads
where Qq = −1/3, 2/3 and Nc are the electric charge and
the colour number of the quark, respectively, and the flux
factor 1/(2s) is included.
In hv and fdh, the regular photon is kept unregularized;
the related metric tensor is therefore gμ4ν . The squared
amplitudes are then given by [ ]
M (0)
hv = Mf(d0)h = ω(0)e4(4 − 2).
In contrast to this, in dred, the regular photon is treated
in ds dimensions and thus contains a gauge-field part and
an -scalar part. It is therefore possible to decompose the
Born amplitude into the two diagrams of Fig. 2. The crucial
point is that the diagrams involve different couplings; the
left diagram is proportional to the square of the electric gauge
coupling e as in the other schemes, whereas the right diagram
is proportional to ee2. The result of the squared matrix element
in dred therefore reads
(2.14a)
(2.14b)
Md(0r)ed = Md(0r,eγd) + Md(0r,eγ˜d) = Mc(0d)r + Md(0r,eγ˜d)
= ω(0)[e4(d − 2) + ee4(n )].
(2.14c)
The appearance of a second contributions in dred is one of
those apparent complications mentioned above. In practice,
one usually sets ee = e from the beginning and computes the
two processes in a combined way like in Eq. (2.10). This is
possible since the different UV renormalizations of e and ee
are irrelevant in this case.
Using the results in Eq. (2.14) and integrating over the
phase space, we obtain the (scheme-independent) Born cross
where we separate the d-dependent two-body phase space
2( ) =
4π
s
(1 − )
(2 − 2 ) = 1 + O( ).
Virtual contributions
In a next step we consider the virtual corrections to the
(spin summed/averaged) squared matrix element, Md(1s) =
2 Re Ads |Ad(1s) . To obtain the results of the corresponding
(0)
one-loop amplitudes, we have to evaluate the diagrams shown
in Fig. 3. There are two different vector fields in the one-loop
diagrams, a virtual photon that is ‘regular’ and a virtual gluon
that is ‘singular’. According to this, the treatment of the
photon is as for the Born amplitude. For dred, this results in two
contributions, one proportional to the gauge coupling e, the
other proportional to the evanescent coupling ee. Due to the
Ward identity, only the latter coupling gets renormalized. In
the ms scheme, we obtain
(Qq ee)2
→ (Qq ee)2
1 +
+
αe
4π
C F
4 − n
2
αs
4π
.
C F
−
3
We remark that in schemes other than cdr, the ms
counterterms in general can have O(n ) terms, as discussed e.g. in
Ref. [
12
]. In dred, one therefore has to consider the (finite)
counterterm
CTdred = Md(0r,eγ˜d)C F
αs
4π
−
6
+
αe
4π
4 − n
see also Fig. 4. In the same way, when using fdh or dred,
the gluon can be split according to Eq. (2.5). Thus, in these
schemes we get terms proportional to αs = gs2/(4π ) and
terms proportional to αe = ge2/(4π ). The unrenormalized
virtual one-loop corrections are given by
(2.15)
(2.16)
(2.17)
;
(2.18)
section
σ (0) =
2( )
8π
3s
,
Mc(1d)r
= ω(1) M (0) αs
cdr π
M (1)
hv
= ω(1) M (0)
hv
Mf(d1)h
= ω(1) M (0)
fdh
Md(1r)ed = ω(1) M (0,γ )
dred
n
4
n
4
αs
π
αs
π
+
+
αe
π
αe
π
+ ω(1) M (0,γ˜ )
dred
+ O( ),
with
× 1 −
c ( ) = (4π )
In Eq. (2.19), we have dropped n terms that vanish after
setting n = 2 and taking the subsequent limit → 0.
In particular, the dred result looks awfully complicated.
However, from a practical point of view the situation is much
simpler. As discussed in the previous section, the virtual
contributions can be computed without distinguishing the
vari−
−
ous couplings and without splitting the photon or the gluon.
We can simply evaluate the algebra of the single vertex
diagram according to the scheme and perform the integration.
The only part where the split is crucial so far is to obtain
the UV counterterm, Eq. (2.18). Thus, the computation in
schemes other than cdr is not significantly more extensive.
Computing the (IR divergent) virtual cross section by
integrating the properly(!) renormalized matrix element squared
over the two-parton phase space, Eq. (2.16), we get
σc(dv)r = σ (0)
σh(vv)
= σ (0)
×
−
×
−
×
−
αs
C F
π
1 1
2 − 2
αs
C F
π
1 3
2 − 2
αs
C F
π
1 3
2 − 2
σf(dvh) = σd(vr)ed = σ (0)
2( )c ( )s−
Finally we have to face the real corrections. In cdr, the
amplitude consists of two diagrams (one of which is depicted in
Fig. 5). The matrix element squared, expressed in terms of
si j ≡ 2 pi · p j reads
Mc(0d)r(qq¯ g)
= ω(r )e4gs2(d − 2)
×
(s12 + s13)2
s13s23
+
where ω(r) = ω(0)2CF /s. In hv, the same diagrams
contribute. One might be tempted to assume that Mh(0v)(qq¯ g) can
be obtained from Eq. (2.22a) simply by setting d → 4.
However, this is incorrect. In the regime where the gluons become
collinear, they have to be treated as singular gluons. Thus, in
hv they are d-dimensional. The same is true in principle for
the soft region, but at one loop, there is no scheme dependence
in the soft singularities. This corresponds to the statement
that the cusp anomalous dimension is scheme independent at
the one-loop level [
13,14
]. Treating the gluons properly, we
obtain
2
Mh(0v)(qq¯ g) = d − 2 Mc(0d)r(qq¯ g).
In the case of fdh we get contributions ∼gs and ∼ge. Again,
the gluon has to be treated as a singular one. Hence, it is split
into a d-dimensional gluon and an -scalar, resulting in
Mf(d0)h(qq¯ g) + Mf(d0)h(qq¯ g˜) = Mh(0v)(qq¯ g)
(2.22b)
+ ω(r)e4ge2n (s13 + s23)2
.
s13s23
Finally, as illustrated in Fig. 5, in dred the matrix element
squared is formally decomposed into four parts,
Md(0r)ed(qq¯ g) = Md(0r,eγd)(qq¯ g) + Md(0r,eγd)(qq¯ g˜)
+ Md(0r,eγ˜d)(qq¯ g) + Md(0r,eγ˜d)(qq¯ g˜)
= Mc(0d)r(qq¯ g) + ee gs n
4 2
×
+
×
4ss12 + (2 − n )(s13 + s23)2
2s13s23
d − 2
2
Mf(d0)h(qq¯ g˜) + ee ge n
4 2
−4s13s23 + n (s13 + s23)2
2s13s23
.
(2.22d)
Note that if we set ee = e and ge = gs , the matrix element
in dred corresponds to the usual four-dimensional matrix
element,
Md(0r)ed(qq¯ g) geee==egs = Mc(0d)r(qq¯ g)
d=4
= ω(r)e4gs24
with yi j ≡ si j /s. This is generally true for arbitrary tree-level
amplitudes in dred, but not necessarily in any of the other
schemes. For the considered process, it happens to be true
also in fdh.
The real cross section can now be obtained in any scheme
by integrating the corresponding matrix element over the
ddimensional real phase space,
s
σd(rs) = 2(4π )3 3( ) 0
1
Using these results for the calculation of the real corrections
in the various schemes and setting ee = e, ge = gs , n = 2 ,
we obtain
×
×
σc(rd)r = σ (0) αs
π
And, at long last, we find the well-known
regularizationscheme independent physical cross section
σ (1) = σ (0) + σd(vs) + σd(rs) d→4 =
× 1 +
3CF .
αs
4π
The expressions for the virtual and the real cross sections,
Eqs. (2.21) and (2.27), have been obtained setting ee = e
and ge = gs . We reiterate that the fdh/dred
computation can be done in a much simpler way by directly
identifying these couplings from the beginning. The only place
where it is crucial to distinguish them is for the proper
UV (sub)renormalization, i.e. to obtain the counterterm in
Eq. (2.18). If we had kept the couplings apart to the very end,
the final result would have been unaffected. In other words,
terms involving the ‘unphysical’ couplings ee and ge drop
out when adding the virtual, the real, and the counterterms
contributions. For our example this can easily be verified
by using the expressions in Eqs. (2.14), (2.18), (2.19), and
(2.22).
2.4 Established properties and future developments of DS
As mentioned in the introduction, regularization schemes
should not only simplify practical calculations but also satisfy
certain basic requirements. For decades, dimensional
regularization in the two flavours cdr and hv has been the most
commonly used regularization, not only because it allows for
the use of powerful calculational techniques but also because
many all-order statements have been rigorously proved in
these schemes.
Using an infinite-dimensional vector space as domain, a
definition of the formally d-dimensional objects and
operations is given in Refs. [
3,4
]. Among the implications are
mathematical consistency and the absence of possible
ambiguities. The equivalence to bphz renormalization and the
regularized and renormalized quantum action principle is shown
in Refs. [
15,16
]. As a caveat, however, in chiral theories
these statements rely on the use of a non-anticommuting γ5
as defined e.g. in Refs. [
2,16
]. In non-chiral theories like
QCD, the quantum action principle makes it obvious that
non-Abelian gauge invariance is manifestly preserved such
that the regularized QCD Green functions automatically
satisfy the Slavnov–Taylor identities at all orders.
The situation regarding dred and fdh has been
considerably more complicated in the past. However, now these
schemes have reached a similar status as cdr and hv. After
first one- and two-loop applications of dred [
8
], the
equivalence of fdh/dred and cdr is shown in Refs. [
9,10
],
indirectly proving that these schemes are compatible with
unitarity and causality. In Ref. [5], it is shown how the spaces in
Eq. (2.3) can be defined in a rigorous way, avoiding
mathematical ambiguities and excluding the possible inconsistency
found before in Ref. [
17
]. In this way also an earlier puzzle
regarding unitarity of dred discussed in Ref. [
18
] is resolved.
The key ingredient for the solution is the introduction and
separate treatment of -scalar fields. One important
consequence of the additional scalars is the need to distinguish
gauge couplings from evanescent couplings during the
renormalization procedure, as indicated in Eq. (2.7). The relation
between unitarity and the correct renormalization of
evanescent couplings in fdh/dred has been further stressed and
exemplified with explicit calculations in Refs. [
7,11
].
Apart from the UV properties of the dimensional schemes
also IR divergences and their scheme dependence have been
investigated up to the multi-loop regime. The separate
treatment of -scalars has been used in Ref. [
19
] to clarify a
seeming non-factorization of QCD amplitudes observed
earlier in Refs. [
20–22
]. In Refs. [
6,23
], it is shown how dred
and fdh can be applied in the computation of NLO cross
sections in massless QCD. The scheme independence of a
cross section at NLO has also been studied in Ref. [24].
Regarding virtual contributions, these considerations have
been extended to NNLO in Refs. [
12–14,25,26
]. Moreover,
the latter references provide NNLO transition rules for
translating UV-renormalized virtual amplitudes from one
dimensional scheme to another. The IR factorization properties of
QCD including massive partons have been investigated at
NLO in Ref. [
27
] and recently up to NNLO in Ref. [
28
]. For
the real corrections, a formulation of the sector-improved
residue subtraction scheme in the hv scheme is presented in
Ref. [
29
].
Regarding supersymmetry, dred and fdh have
significant advantages as in many cases supersymmetry is
manifestly preserved although an all-order proof does not exist.
For reviews regarding applications of these schemes to
supersymmetry, we refer to Refs. [
30,31
].
3 FDF, SDF: four- and six-dimensional formalism
In the following we discuss some new (re-)formulations
of ds. In Sects. 3.1–3.3, we describe fdf, a strictly
fourdimensional formulation of the fdh scheme. The remaining
two subsections are dedicated to topics that are not directly
fdf but that are closely related to it, namely automated NLO
calculations using GoSam and the six-dimensional
formalism.
3.1 FDF: four-dimensional formulation of FDH
The four-dimensional formulation of the fdh scheme (fdf)
is a novel implementation of fdh. Its aim is to achieve the
ddimensional regularization of one-loop scattering amplitudes
in a purely four-dimensional framework [
32
]. The starting
point for the formulation of the scheme is the structure of the
quasi-ds -dimensional fdh space, Eq. (2.3), which we write
as
Accordingly, the underlying space of the fdh scheme is
written as an orthogonal sum of a strictly four-dimensional space
S[
4
] and a quasi-(n −2 )-dimensional space QS[n −2 ].
Similar to Eq. (2.5), metric tensors and γ matrices can then be
decomposed as
g[μdνs] = g[μ4ν] + g[μnν −2 ],
μ μ μ
γ[ds ] = γ[
4
] + γ[n −2 ], (3.2)
with
(g[
4
])μμ = 4 (g[
4
]g[n −2 ])μν = 0,
(g[n −2 ])μμ = (n − 2 ) ds→→4 0.
The algebraic properties of the matrices γ[μn −2 ] can be
obtained from Eq. (3.3) and read
{γ[n −2 ], γ[νn −2 ]} = 2g[μnν −2 ],
μ
γ[
4
], γ[νn −2 ] = 0, [γ[54], γ[μn −2 ]] = 0,
μ
Loop momenta, on the other hand, are treated in d dimensions
like in any dimensional scheme,
(3.1)
(3.3a)
(3.3b)
(3.4a)
(3.4b)
(3.5)
in Eq. (3.6) then suggests that any integral of the form
where i1 . . . ik are indices labeling the loop propagators. With
the decomposition of the integral measure in Eq. (3.8), any
one-loop integral in d dimensions has a four-dimensional
integrand, depending on an additional length μ2. The (radial)
integration over μ2 can be carried out algebraically by
redefining the number of dimensions [
33
],
r−1
j=0
Iid1···ik [(μ2)r ] = (2π )r Iid1+···2ikr [
1
]
(d − 4 + 2 j ),
(3.9)
so that powers of μ2 in the numerator of the integrand
generate integrals in shifted dimensions which are responsible for
the rational terms of one-loop amplitudes.
We remark that an (n − 2 )-dimensional metric tensor
cannot have a four-dimensional representation. This is due
to the fact that according to Eq. (3.3b), its square vanishes.
Additionally, in four dimensions the only non-null matrices
compatible with conditions (3.4) are proportional to γ 54 ,
[ ]
(3.7)
(3.8)
(3.10)
(3.11)
μ μ μ
k[d] = k[
4
] + k[−2 ],
with
k[2d] = (k[
4
] + k[−2 ])2 = k[
24
] + k[
2−2
] ≡ k[
24
] − μ2. (3.6)
Here and in the following, the square of the (−2
)dimensional component of a loop momentum is identified
with −μ2. The decomposition of the space-time dimension
γ[n −2 ] ∼ γ[54].
μ
γ[n −2 ](γ[n −2 ])μ
However, the matrices γ[n −2 ] fulfill the Clifford
algebra (3.4a), and thus
= (n − 2 ) ds→→4, while (γ[54])2 = I[
4
].
Equations (3.10) and (3.11) are therefore not compatible
with each other. Finally, the component k[μ−2 ] of the loop
momentum vanishes when contracted with a strictly
fourdimensional metric tensor, i.e. k[μ−2 ](g[
4
])μν = 0. In four
dimensions, the only four vector fulfilling this relation is the
null one.
The above arguments exclude any four-dimensional
representation of the (n −2 )- and (−2 )-dimensional subspaces.
It is possible, however, to find a representation by introducing
additional rules, in the following called (−2 ) selection rules,
(−2 )-SRs. Indeed, the Clifford algebra (3.4a) is equivalent
to
· · · (γ[n −2 ])μ · · · (γ[n −2 ])μ · · · ds→→4,
Therefore, any regularization scheme which is equivalent
of fdh has to fulfill conditions (3.3)–(3.6), and (3.12).
The orthogonality conditions (3.3) and (3.6) are fulfilled by
splitting a ds -dimensional vector field into a strictly
fourdimensional one and a scalar field, while the other conditions
are fulfilled by performing the substitutions
g[αnβ −2 ] → G AB , γ[αn −2 ] → γ[54] A, k[α−2 ] → i μQ A.
The (n − 2 )-dimensional and (−2 )-dimensional indices
are thus traded for (−2 )-SRs such that
G AB G BC = G AC , G A A = 0, G AB = G B A,
G AB A
The exclusion of terms containing odd powers of μ
completely defines the fdf scheme. It allows one to build
integrands which, upon integration, yield the same results as in
the fdh scheme. As mentioned before, the fdf scheme is
closely connected to the introduction of an additional scalar
field. The role of this field and its relation to the -scalar
present in the fdh scheme will be discussed in Sect. 3.3.
The rules in Eq. (3.14) constitute an abstract algebra which
is similar to an algebra related to internal symmetries. For
instance, in a Feynman diagrammatic approach, the (−2
)SRs can be handled as the colour algebra and performed for
each diagram once and for all. In each diagram, the indices of
the (−2 )-SRs are fully contracted and the outcome of their
manipulation is either 0 or ±1. It is worth to remark that
the replacement of γ[αn −2 ] with γ[54] takes care of the ds
dimensional Clifford algebra automatically. Thus, we do not
need to introduce any additional scalar field for each fermion
flavour.
Depending on the gauge we use, further simplifications
can arise. In Feynman gauge, for example, there are no
contributions coming from scalar loops, which is due to the (−2
)SRs,
G A1 A2 G A2 A3 . . . G Ak A1 = G A1 A1 = 0.
Similarly, for diagrams with internal scalars and fermions we
get the same cancellation,
A1 G A1 A2 . . . G Ak−1 Ak Ak =
A1 A1 = 0.
(3.15)
(3.16)
uλ,[ds ](k[d])u¯λ,[ds ](k[d]) = k/[d] + m,
vλ,[ds ](k[d])v¯λ,[ds ](k[d]) = k/[d] − m.
(3.20a)
(3.20b)
With the use of axial gauge, we obtain the opposite behaviour
since contributions from internal scalars have to be taken in
account,
G A1 A2 Gˆ A2 A3 . . . G Ak−1 Ak Gˆ Ak A1 = G A1 A2 Gˆ A2 A1
= −Q A1 Q A1 = −1,
(3.17)
where Gˆ AB ≡ G AB − Q A Q B . Diagrams that contain
interactions between generalized gluons and scalars are dropped
according to the (−2 )-SRs,
Q A1 Gˆ A1 A2 . . . Q Am . . . Gˆ Ak A1 = Gˆ A1 A2 Q A2 = 0.
(3.18)
Generalized-unitarity methods in dimensional regularization
require an explicit representation of the polarization
vectors and the spinors of ds -dimensional particles. The latter
ones are essential ingredients for the construction of the
tree-level amplitudes that are sewn along the generalized
cuts. In this respect, the fdf scheme is suitable for the
fourdimensional formulation of d-dimensional generalized
unitarity. The main advantage of fdf is that the four-dimensional
expression of the propagators in the loop admits an explicit
representation in terms of generalized spinors and
polarization expressions which is collected below.
In the following discussion, the d-dimensional
momentum k[d] will be put on-shell and decomposed according
to Eq. (3.5). Its four-dimensional component, k[
4
], will be
expressed as
m2 + μ2
k[
4
] = k[
4
] + qˆ[
4
], with qˆ[
4
] ≡ 2k[
4
] · q[
4
] q[
4
],
(3.19)
in terms of the two massless momenta k 4 and q[
4
].
[ ]
The spinors of a ds -dimensional fermion have to fulfill a
completeness relation which reconstructs the numerator of
the cut propagator,
(3.13)
3.2 Wave functions in FDF
Spinors
2(ds −2)/2
λ=1
2(ds −2)/2
λ=1
=
=
λ=±
λ=±
k/[d] − m = k/[
4
] + k/[−2 ] − m = k/[
4
] + i μγ[54] − m
in terms of generalized four-dimensional massive spinors
defined as
The spinors in Eq. (3.22a) are solutions of the tachyonic Dirac
equations [
34–37
]
(k/[4] + i μγ[54] + m)uλ k[
4
] = 0,
(k/[
4
] + i μγ[54] − m)vλ k[
4
] = 0.
It is worth to notice that the spinors in Eq. (3.22) fulfill the
Gordon identities
u¯λ(k[
4
]) γ[ν4] uλ(k[
4
])
2
=
v¯λ(k[
4
]) γ[ν4] vλ(k[
4
])
2
= k[ν4].
The substitutions (3.13) allow one to express the r.h.s. of
Eq. (3.20) as,
Polarization vectors
k/[d] + m = k/[
4
] + k/[−2 ] + m = k/[
4
] + i μγ[54] + m
μ kμd ην +k[νd]ημ
εi,[ds ](k[d], η)εi∗,ν[ds ](k[d], η) = −g[μdνs ] + [ ]
,
where η is an arbitrary d-dimensional massless momentum
such that k ·η = 0. Gauge invariance in d dimensions
guarantees that the cut is independent of η. In particular the choice
ημ
allows one to disentangle the four-dimensional contribution
from the (−2 )-dimensional one:
The second term of the r.h.s. of Eq. (3.27) is related to
the numerator of cut propagator of the scalar and can be
expressed in terms of the (−2 )-SRs as:
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
(3.21a)
(3.21b)
(3.22b)
(3.23)
(3.22a)
The first term is related to the cut propagator of a massive
gluon and can be expressed as
(3.24)
g[μnν −2 ] +
μ ν
k[−2 ]k[−2 ] → Gˆ AB ≡ G AB − Q A Q B . (3.31)
μ2
Therefore, we can define the cut propagators as
a, A
b, B
= Gˆ AB δab.
(3.32)
The generalized four-dimensional spinors and polarization
vectors defined above can be used for constructing tree-level
amplitudes with full μ-dependence.
3.3 Established properties and future developments of FDF
At one-loop, fdf has been successfully applied to compute
the scattering amplitudes for multi-gluon scattering gg →
n gluons with n = 2, 3, 4, and for gg → H + n gluons
with n = 2, 3 [
39,40
]. The use of dimensionally regularized
tree-amplitudes within fdf has been employed to study the
colour-kinematics duality [41] for one-loop dimensionally
regularized amplitudes [
42
].
The extension of fdf beyond the one-loop level is
currently under investigation. In particular at two loops, fdf
should be able to capture the dependence of the integrand
on the extra dimensional terms of the loop momenta, namely
on two mass-like variables, say μ21 and μ22, as well as on the
scalar product μ1 · μ2.
Equivalence of FDF and FDH at NLO: virtual contributions
to e+e− → γ ∗ → qq¯
To show that the strictly four-dimensional Feynman rules
of fdf together with the (−2 )-SRs indeed reproduce the
corresponding results in the fdh scheme for αe = αs ,
we consider virtual one-loop contributions to the process
e+e− → γ ∗ → qq¯ .
According to the discussion in Sect. 3.1, in fdf each
vector field is split into a strictly four-dimensional field and a
corresponding scalar field. The vertex correction subgraph
γ ∗ → qq¯ therefore receives two contributions in fdf; see
Fig. 6. The diagram including an internal fdf-scalar vanishes
according to the (−2 )-SRs since in Feynman gauge it is
proportional to A B G AB = A A = 0. Using only strictly
four-dimensional quantities, the amplitude is then given
by
(A(f1d)f)μ = −eQq gs2CF
dd k[d] u¯( pq )γ ν k/[
4
] + /pq,[
4
] +iμγ5 γμ k/[
4
] − /pq¯,[
4
] +iμγ5 γν u( pq¯ ) ,
(2π)d k[
24
] −μ2 (k[
4
] + pq,[
4
])2 − μ2 (k[
4
] − pq¯,[
4
])2 − μ2
(3.33)
where pq and pq denote the four-momenta of the massless
¯
quarks. Evaluating the strictly four-dimensional algebra and
Fig. 6 Virtual diagrams contributing to γ ∗ → qq¯ at NLO including
a strictly four-dimensional photon γ (wavy line) and an fdf scalar γ
(dashed line), respectively. Using Feynman gauge, the right diagram
vanishes according to the (−2 )-SRs
performing a tensor integral decomposition in d dimensions,
the amplitude can be written as
(A(f1d)f)μ = −i (A(f0d)f)μgs CF
2
d −d 4 I2d [
1
] − 2I3d [μ2] ,
(3.34)
,
(3.35a)
In the following we determine the β function related to the
coupling of the fdf-scalar to fermions in QED with NF
5 Since Mf(d0)f ≡ Mf(d0)h, this result coincides with the one obtained in
fdh for n = 2 and ge = gs at least up to O( 0), compare with
Eq. (2.19c).
(3.37)
Fig. 8 Diagrams contributing to the interaction of the fdf scalar with
fermions at the one-loop level. The right diagram vanishes according
to the (−2 )-SRs
fermion flavours, and compare it to the known
renormalization of the gauge and the evanescent coupling in the fdh
scheme given in Eq. (2.8).
To start, we consider the fermion self-energy, where two
diagrams contribute at the one-loop level; see Fig. 7. Using
the Feynman rules of Ref. [
32
] together with the (−2 )-SRs,
we obtain for the case of massless fermions
−i f(1d)f = μ4d−sd
ρ (g[
4
])μρ
× γ[
4
] (k[
4
] + p[
4
])2 − μ2
dd k[d] (−i )4e2γ μ4 k/[
4
] + i μγ5
(2π )d [ ] k[
24
] − μ2
dd k[d] e γ 4 γ ν4 γ ρ4 (g[
4
])μρ
2 μ
(2π )d [ ] [ ] [ ]
(k[
4
])ν
[k[
24
] − μ2][(k[
4
] + p[
4
])2 − μ2]
.
In particular, we applied relation (3.6) and made use of the
fact that terms containing odd powers of μ are set to zero. The
diagram including an internal fdf-scalar vanishes according
to the (−2 )-SRs since it is proportional to A B G AB =
A A = 0. Evaluating the d-dimensional integral in Eq.
(3.38), we then obtain6
(1) α
fdf = i /p 4
[ ] 4π
1
−
Using minimal subtraction, the renormalization of the fermion
field is therefore given by
α
4π
α
4π
α
4π
Z2 = 1 +
Z3 = 1 +
−
−
1
2
Finally, we consider the vertex correction. Again, in fdf two
diagrams contribute at the one-loop level; see Fig. 8.
According to the (−2 )-SRs, the diagram with an internal fdf-scalar
is proportional to B A B = − B B A + 2 B G AB =
2 A. Evaluating the strictly four-dimensional Lorentz
algebra and performing the d-dimensional loop integration, the
renormalization of the vertex is given by
−
4
In a similar way, the renormalization constants can be
obtained for the case of massive fermions. In the on-shell
scheme (os) they read,7
Z2|os = 1 +
α
4π
−
3
+ ln
6 This result can be compared to the one obtained in fdh; see Eq. (2.11).
After subtraction of the UV divergence, the limit d → 4 can be taken and
both results coincide. However, due to the vanishing scalar contribution
it is clear that the additional scalar field in fdf is different from the
-scalar of fdh.
7 The result of Z2 in the on-shell scheme has already been obtained in
Ref. [
28
] for the case of fdh. It coincides with Eq. (3.41a) for n = 2
and the case of equal couplings.
(3.40a)
(3.40b)
(3.40c)
(3.41a)
Z3|os = 1 +
Z1|os = 1 +
α
4π
α
4π
Combining the results in Eqs. (3.40) or (3.41), the β function
of the fdf-scalar coupling to fermions is finally given by
β = −
α
4π
2
[2 − 2NF ] + O(α3),
and therefore identical to the renormalization of the
evanescent coupling in fdh for ee = e, compare with Eq. (2.8b).
According to the discussion in Sect. 2.2, the different
renormalization of the couplings in the fdh scheme (and therefore
in fdf) does not play any role at the one-loop level. At higher
perturbative orders, however, it can lead to a breaking of
unitarity [
25
]. The way, how the different renormalization of the
scalar coupling can be consistently implemented beyond one
loop in the fdf framework is currently under investigation.
3.4 Automated numerical computation
To build a fully consistent procedure that is valid for every
Lagrangian is an issue for the complete automation of higher
order computations via numerical recipes. In the GoSam [
43
]
actual architecture we adopted a scheme that naturally
produces results in fdh.8 In this scheme, GoSam can
generate the full one-loop amplitude for every process originating
from every Lagrangian with the only condition that the power
of the loop momentum in the numerator of a diagram
cannot exceed the number of loop denominators plus one. On
the other hand, we still do not have a completely general
procedure for the renormalization. Technically, the algebraic
implementation of our procedure is extremely simple and can
be summarized in the following three points:
1. Assume that all Lorentz indices are four-dimensional,
even if in a following step the loop momentum k will be
treated as d-dimensional.
2. In all fermion chains, also in fermion loops, bring all
chiral projectors to the left and all loop momenta to the
right.
3. Apply the rule k/k/ = k[d] · k[d] = k[
4
] · k[
4
] − μ2.
This is a simplified version of what is effectively coded,
which has the same algebraic content and produces the same
8 The scheme is actually called dimensional reduction in GoSam and
in Ref. [
23
], but corresponds to what we call fdh in this article.
result. The μ2 parameter represents the length of the loop
momentum into the -dependent dimensions.
In GoSam, the generation of amplitudes starts from
diagram generation with QGRAF [
44
] that searches for
topologies and fills them with fields in all possible ways. This
construction paired with the few rules given above guarantees
that no spurious anomalies are generated and, most
important, it provides the correct result for all the computations
that are anomaly free. In full generality, for every diagram
we are then left with two ingredients: a number of
nonvanishing integrals with μ2, and a polynomial of the
fourdimensional part of the loop momentum sitting on every
number of denominators. Loop integrals with μ2 in the
numerator have been computed analytically since long, so that their
implementation is trivial. Furthermore, reduction programs
like Golem95 [
45,46
], Ninja [
47–49
] or Samurai [50] reduce
them easily. The polynomial in the four-dimensional
component of the loop momentum is the optimal representation of
the loop integral for the numerical reduction with programs
like CutTools [
51
], Golem95, Ninja or Samurai.
When we are computing higher-order differential cross
sections using some subtraction scheme [
52,53
] to
regularize IR divergences, the choice of the dimensional scheme
adopted is restricted to the virtual integration, and one can
exploit unitarity to derive the transition rules among
renormalized amplitudes computed in different (unitary) schemes;
see Refs. [
23,24
] for more details. For this reason it is
trivial to derive transition rules from fdh to cdr for example
deducing them from the different finite part of the integrated
dipoles computed in the two schemes. We refer to the
dipolessubtraction technique, but the reasoning is completely
general and provides the same conversion factors irrespective
of the subtraction scheme. To be definite, to convert a
oneloop amplitude in the Standard Model, one can start from
the massless gauge-boson emissions from QCD radiation to
determine the shift as nlq CF /2 + ngC A/6 times the
underlying tree-level interference, where nlq (ng) is the number
of the external light quarks (gluons) being part of the hard
scattering amplitude. This agrees with the shift found in
Ref. [
23
]. Similarly, for QED radiation the shift is again the
underlying tree-level interference times the sum of factors
δRS = −qi σi qk σk /2 for each pair of emitter (i) with electric
charge qi and spectator (k) with electric charge qk and σ being
1(−1) for an incoming fermion and outgoing anti-fermions
(vice versa).
Now we come to the renormalization. In GoSam, this is
still not fully automated. For the QCD part of the Lagrangian
that is renormalized with the ms prescription, subtracting
only the poles, with fdh or dred one is left with a different
definition for the renormalized coupling constants w.r.t. cdr.
A finite renormalization is needed to restore the customary
definition (cdr). There is of course no such problem with the
on-shell renormalization that is often used for electroweak
corrections. In GoSam we computed and implemented all the
renormalization constants of the Standard Model Lagrangian
and derived the conversion factors from fdh to cdr. They
can be found in Ref. [54].
To conclude, the fdh scheme appears optimal for
numerical computations and the conversion rules to other schemes
can easily be worked out once and for all exploiting unitarity.
Finally, we stress that on the path towards fully automated
computations for every Lagrangian, the automated
computation of the renormalization constants is mandatory.
3.5 SDF: six-dimensional formalism
In this section we discuss the possibility of implementing
dimensional regularization schemes via an embedding of the
loop degrees of freedom in a de-dimensional space, where de
(e stands for embedding) is an integer greater than 4 which
depends on the loop order. This is possible in dimensional
schemes such as fdh and hv, where the degrees of freedom
of the external particles live in the genuine four-dimensional
space S[
4
]. In particular, we focus on the case de = 6, which
is sufficient up to two loops [55].
Having a finite integer-dimensional embedding of the
loop degrees of freedom is especially useful in the
context of integrand reduction via generalized unitarity [56–
64], which provides an efficient way of generating loop
integrands from products of tree-level amplitudes summed over
the internal helicity states. In particular, the possibility of
using a de-dimensional spinor-helicity formalism provides a
finite-dimensional (six-dimensional in our case)
representation of both external and internal states. The six-dimensional
spinor-helicity formalism has been extensively developed in
Ref. [65], and used in the context of multi-loop
generalized unitarity for producing analytic results for five- and
sixpoint two-loop all-plus amplitudes in (non-supersymmetric)
Yang–Mills theory [64,66,67].
A useful property of this approach is that it gives both
internal and external states an explicit finite-dimensional
representation. This means that one can perform both analytic
and numerical calculations by working directly with the
components of momenta and spinors. Numerical calculations can
in turn be used to infer properties of the result before a full
analytic calculation, or in order to employ functional
reconstruction techniques (see e.g. Ref. [68]) which allow one to
reconstruct full analytic results from numerical calculations
over finite fields.
As mentioned, in this section we focus on a dimensional
regularization scheme where the external states live in the
physical four-dimensional space S[
4
], while we keep the
dimension ds of the space QS[ds ] undetermined. The
special cases of fdh and hv can be obtained by setting ds = 4
and ds = d, respectively, at the end of the calculation.
Internal degrees of freedom
We consider a generic contribution to an -loop amplitude
∞
−∞
i=1
dd ki
N (ki )
j D j (ki )
,
(3.43)
(3.44)
(3.45)
(3.46)
where N and D are polynomials in the components of the
loop momenta ki (a rational dependence on the external
kinematic variables is always understood). In particular, the
denominators Di correspond to loop propagators and have
the generic quadratic form
Di = i2 − mi2, liμ =
αi j k μj +
βi j p μj,
j=1
n
j=1
αi j , βi j ∈ {0, ±1},
with p j being the external momenta. It is often useful to split
the loop momenta kiμ into a four-dimensional part ki,[
4
] and
μ
μ
a (d − 4)-dimensional part ki,[d−4] as
kiμ = kiμ,[
4
] + kiμ,[d−4].
In a regularization scheme where the external states are
fourdimensional, a loop integrand can only depend on the (d − 4)
extra-dimensional components of each loop through scalar
products μi j defined as
μi j = −(ki,[d−4] · k j,[d−4]).
The scalar products μi j can in turn be reproduced by
embedding the loop momenta in an integer-dimensional space with
dimension de ≥ 4 + . In particular, as stated, the choice
de = 6 is sufficient up to two loops. Although we will focus
on the case de = 6 and scattering amplitudes at one loop or
two loops, unless stated otherwise our statements are valid
for any multi-loop amplitude, provided that the integer de is
sufficiently large.
In order to correctly reconstruct the dependence of the
integrand on the dimension ds of the space QS[ds ] where
internal gluon polarizations live, we add (ds − de) flavours of
scalar particles to the theory, which represent gluon
polarizations orthogonal to both the external and the loop momenta.
The Feynman rules for these scalars can easily be derived
from the ones of gluons (see e.g. Ref. [64]).
Internal states: six-dimensional spinor-helicity formalism
External states of helicity amplitudes can be efficiently
described using the well-known four-dimensional
spinorhelicity formalism [69,70]. After a higher-dimensional
embedding of internal states, one can similarly describe these by
means of a higher-dimensional spinor-helicity formalism.
In particular, the spinor-helicity formalism in six
dimensions has been developed in Refs. [65,71,72]. While a
comprehensive treatment of the subject is beyond the
purpose of this report (we refer the reader to Ref. [65] for
more details), it is worth pointing out a few properties of
six-dimensional spinors which are useful for providing an
integer-dimensional embedding of the loop internal states, in
particular for applications in the context of integrand
reduction via generalized unitarity, as we shall see in the next
section.
Six-dimensional Weyl spinors | pa and | pa˙ ] (with a, a˙ ∈
{0, 1} ≡ {+, −}) are defined as independent solutions of the
six-dimensional Dirac equation
pμσ (6) pa
μ |
= pμσ˜μ(6)| pa˙ ] = 0,
where σμ(6) and their dual σ˜μ(6) are six-dimensional
generalizations of the Pauli matrices (see Ref. [65] for an explicit
representation). Six-dimensional momenta can be built from
spinors,
pμ
Similarly, given a six-dimensional momentum pμ, a
representation for the spinors | pa and | pa˙ ] satisfying the previous
equations, while not unique, is not hard to find. Note that,
when building loop integrands, the internal spinors always
combine as on the r.h.s. of Eq. (3.48), hence the physical
results are always unambiguous and independent of the
chosen representation. Moreover, a subset of the six-dimensional
spinor components can be identified with the components of
four-dimensional Weyl spinors | p and | p], which ensures a
smooth four-dimensional limit.
Internal gluon states are described by six-dimensional
polarization vectors, which can be built out of these
spinors
aμa˙ ( p, η) = √2(1p · η) pa |σ μ|ηb ηc| pa˙ ] bc
with
(aa˙ ) ∈ {(00), (11), (01), (10)}
≡ {(++), (−−), (+−), (−+)}.
While (++) and (−−) correspond to positive and
negative helicity in the four-dimensional limit, respectively,
the polarizations (+−), (−+) only exist in six
dimensions. One can show [65] that these polarization vectors
satisfy all the expected properties, including the completeness
relation
aμa˙ ( p, η) ν aa˙ ( p, η) = gμν
When building an integrand via generalized unitarity, internal
polarization states always combine as on the l.h.s. of the
previous equation.
Applications to integrand reduction via generalized unitarity
Integrand reduction methods rewrite loop integrands as a sum
of irreducible contributions,
(3.47)
(3.48)
(3.49)
(3.50)
(3.51)
(3.52)
N (ki )
j D j (ki ) =
T
T (ki )
j∈T D j (ki )
,
where the sum on the r.h.s. runs over the non-vanishing
sub-topologies of the parent topology identified by a set
of denominators {D j }. The on-shell numerators or residues
T can be written as a linear combination of polynomials
qT = {qT,1, qT,2, . . .} which can be combined to form an
integrand basis up to terms proportional to the denominators
of the corresponding sub-topology T ,
T (ki ) =
cT,α(qT (ki ))α, qαT ≡
α
j
qTα,j j ,
(3.53)
where α = (α1, α2, . . .) runs over an appropriate set of
multiindices. Techniques for choosing an appropriate integrand
basis have been proposed e.g. in Refs. [61–63,67].
The coefficients cT,α only depend on the external
kinematics (they also have a polynomial dependence on ds ) and
they can be determined by evaluating the integrand on values
of the loop momenta such that the propagators of the
corresponding loop sub-topology are put on-shell {D j = 0} j∈T .
These constraints are also known as multiple cuts. On these
values of the loop momenta, the integrand factorizes as a
product of tree-level amplitudes summed over the internal
helicities corresponding to the cut on-shell loop momenta.
Hence, an efficient way of computing the integrands on the
cut conditions is by sewing together tree-level amplitudes.
This is known as generalized unitarity. As explained, by
means of a higher-dimensional spinor-helicity formalism,
one can build products of trees which contain the full
dependence of the integrand on the loop degrees of freedom.
More explicitly, the solutions of the cut conditions in de
dimensions can be expressed as a linear combination of terms
de ,
of a de-dimensional vector basis {ei j } j=1
kiμ =
μ
yi j ei j ,
de
j=1
(3.54)
where, in turn, the coefficient of this linear combination
can be expressed as yi j = yi j ({τk }), where {τk } is a set
of free variables which are not constrained by the cut
conditions. From these de-dimensional on-shell momenta, we
thus build the corresponding de-dimensional spinors, which
in turn are used to evaluate the tree-level helicity
amplitudes which define the integrand on the considered multiple
cut.
As we mentioned, the correct dependence of the integrand
on ds is obtained by adding to the theory (ds − de) flavours
of scalars representing additional polarizations of the
internal gluons. At two loops, an integrand can have at most a
quadratic dependence on scalar flavours
T =
(Tde,0) + (ds − de) (Tde,1) + (ds − de)2 (Tde,2).
(3.55)
More in general, each scalar loop can add at most one power
of (ds − de). We stress that the result for T does not depend
on the dimension de of the chosen embedding, unlike each
of the terms on the r.h.s. of the previous equation.
This setup has been used for the calculation of
planar five- and six-point two-loop amplitudes in Yang–Mills
theory presented in Refs. [64,66,67], as well as for the
first application of multivariate reconstruction techniques
to generalized unitarity presented in Ref. [68]. The latter
includes the calculation of the on-shell integrands of the
maximal cuts of the two-loop planar pentabox and the
nonplanar double pentagon topology, for a complete set of
independent helicity configurations. This shows that this
strategy is suitable for performing complex multi-leg
calculations at two loops, which is currently a very active field of
research.
4 IREG: implicit regularization
4.1 Introduction to IREG and electron self-energy at NLO
Implicit regularization (ireg) is a regularization framework
proposed by the end of the 1990s [
73–75
] as an alternative
to well-known dimensional schemes. A main characteristic
of the method is that it stays in the physical dimension of
the underlying quantum field theory, avoiding, in principle,
some of the drawbacks of ds such as the mismatch between
fermionic and bosonic degrees of freedom which leads to
the breaking of supersymmetry. ireg is proposed to work in
momentum space and relies on the following observation:
the UV divergent piece of any Feynman integral should not
(4.1)
(4.2)
(4.3)
(4.5)
depend on physical parameters such as external momenta or
particles masses.9 This simple fact leads to profound
consequences as we are going to see.
For ease of the reader, we will develop the basic concepts
of ireg by considering a familiar example of massless QED,
the one-loop corrections to the fermion propagator. We write
the initial (unregularized) expression as
−i (1)( p) = −e2
d4k 1
(2π )4 γ μ k1/ γμ (k − p)2 ,
where p is an external momentum. The first step is to perform
simplifications using Dirac algebra in strictly four
dimensions. In this example, the result is particularly simple
−i (1)( p) = 2e2γμ
d4k
kμ
The next step is just to introduce a fictitious mass in the
propagators which will allow us to control spurious IR divergences
introduced in the course of the evaluation. Thus, the integral
can be rewritten as
− i (1)( p)
= lim 2e2γμ
μ2→0
≡ lim [−i i(r1e)g( p, μ)].
μ2→0
d4k
kμ
(2π )4 (k2 − μ2)[(k − p)2 − μ2]
At this point one uses the main observation of iregthat the
intrinsic divergent integral should not depend on physical
parameters, the external momentum in this case. To achieve
that, one just uses the following identity as many times as
necessary to isolate the physical parameters in the finite
part:
1 1 (−1)( p2 − 2 p · k)
(k − p)2 − μ2 = (k2 − μ2) + (k2 − μ2)[(k − p)2 − μ2] .
(4.4)
In our example, one ends up with the following divergent
expression:
−i i(r1e)g( p, μ)|div = 2e2γμ
+ 2 pν
d4k
kμ
(2π )4 (k2 − μ2)2
d4k
kμkν
(2π )4 (k2 − μ2)3 ,
9 This point of view is shared by other methods as well, for instance by
fdr which is described in Sect. 5. In the latter scheme, these intrinsic
divergent pieces are called ‘vacua’.
in which all dependence on the external momenta is only
in the numerator. The latter can be therefore pulled outside
the integration. Focusing on the divergences, one notices the
existence of linear and logarithmic terms. The first piece is
automatically null (as in cdr) and we are left with only the
logarithmic term, whose integral is a particular example of
the general expression
Ilνo1g···ν2N (μ2) ≡
d4k kν1 · · · kν2N
(2π )4 (k2 − μ2)N +2
.
This is a characteristic of ireg that the UV divergence can be
always expressed in terms of a precise set of Basic Divergent
Integrals (BDI), composed of scalar and tensorial ones.
However, it can be shown that all tensorial integrals can be further
expressed in terms of the scalar ones plus surface terms. In
our particular example one has
ϒ0μν
=
d4k ∂ kν
(2π )4 ∂ kμ (k2 − μ2)2 = gμν Ilog(μ2)
− 4Ilμogν (μ2) ≡ gμν υ0,2,
where ϒ0μν is a surface term, arbitrary in principle. More
comments regarding the surface terms and their relation to
momentum routing invariance will be given at the end of this
section.
After all UV divergences are taken care of, one needs to
evaluate the finite part, for which we obtain
− i i(r1e)g( p, μ)
fin
= 2e2γμ
− p2
d4k kμ
(2π )4 (k2 − μ2)3
i
with b = (4π )2 .
It should be noticed that the limit μ2 → 0 has still to be
taken in the final result. However, it can easily be seen that
both Ilog(μ2) and the logarithm term then develop an IR
singularity which is spurious since our starting integral was
IR safe. To avoid this issue, one still needs to introduce a
scale λ2 = 0, which plays the role of a renormalization scale
in renormalization-group equations,
Ilog(μ2) = Ilog(λ2) − b ln
μ2/λ2 .
(4.6)
(4.7)
(4.8)
(4.9)
Combining the divergent and finite part and writing the
dimension of the external momentum explicitly, one finally
gets10
b−1 Ilog(λ2)
In summary, the treatment of UV divergent amplitudes in
ireg can be described as follows:
1. Introduce a fictitious mass μ2 in propagators to avoid
spurious IR divergences in the course of the evaluation.
2. Use Eq. (4.4) as many times as necessary to free the
divergent part from physical parameters like external momenta
and masses. In the case of massive theories, a similar
identity can be applied; see Ref. [
76
] for details.
3. Express the divergent part in terms of scalar and tensorial
basic divergent integrals.
4. Reduce tensorial BDIs to the scalar ones plus surface
terms.
5. Remove the μ2 dependence by introducing a scale λ2
which plays the role of a renormalization scale on
renormalization-group equations.
At this point, we would like to emphasize the role played
by the surface terms which, as defined, are just differences
between integrals with the same degree of divergence. As
shown in Ref. [
77
], these objects are at the root of
momentum routing invariance (the freedom one has in the
assignment of internal momenta inside a given Feynman diagram).
This can only be respected when the surface terms are set to
zero. It can also be shown that the same conclusion holds
for Abelian gauge invariance, allowing one to conjecture
that surface terms are at the root of symmetry breaking in
general. In Ref. [
77
], it is shown that this conjecture may
hold for supersymmetric theories as well. Similar analyses,
in many different theories and contexts, have been carried
out in Refs. [
78–91
].
4.2 Application example: e+e− → γ ∗ → qq¯ at NLO
In this section we perform the computation of the total cross
section of the process e+e− → γ ∗ → qq¯ , showing an
example on how ireg deals with different kinds of divergences.
We divide the presentation in two parts, as usual.
10 This result can be compared with the corresponding one obtained
in fdh; see Eq. (2.11). Setting the surface term υ0,2 to zero which is
necessary to preserve gauge invariance; see also Sect. 4.3, the finite
terms of the electron self-energy in ireg and ds are the same for
dby =b−14Ilaong(dλλ2) =↔ μ1 ds. The relation for the UV divergence is given
.
Page 20 of 39
The (unregularized) amplitude for the one-loop vertex
correction subgraph γ ∗ → qq¯ reads
A(μ1) = −e Qq gs2C F
d4k u¯ ( pq )γ ν (k/ + /pq )γμ(k/ − /pq )γν u( pq¯ ) ,
× (2π )4 k2(k + pq )2(k − pq¯¯)2
where pq and pq denote the four-momenta of the massless
¯
quarks. Using the Dirac equation for massless quarks, the
integral can be decomposed as
A(μ1) = −4e Qq gs2C F {u¯ ( pq )γμu( pq¯ )
× [( pq · pq¯ )I − ( pq,α − pq¯,α )I α − I2/2]
One notices the prescription of ireg to cancel denominators
as in I2 before introducing a regulating mass in the
propagators.11
The integrals in Eq. (4.13) are IR divergent for pq2 = pq2¯ =
0. In addition, the integral in Eq. (4.13a) carrying two Dirac
indices and the integral in Eq. (4.13b) are logarithmically
UV divergent. To deal with the latter, a regulating mass μ is
introduced in all propagators,
{Iireg, Iiαreg, Iiαrβeg}
d4k
=
{1, kα , kα kβ }
(2π )4 [k2 −μ2][(k + pq )2 −μ2][(k − pq )2 −μ2] ,
¯
(4.14)
and, after cancellation of one of the denominators, also in
d4k 1
(2π )4 [(k + pq )2 − μ2][(k − pq )2 − μ2] .
¯
(4.15)
11 This is a crucial difference compared to fdr, where μ2-terms remain
in the numerators. In a second step, they are then removed by so-called
‘extra integrals’. Further discussions can be found in Sect. 4.3
I2,ireg =
The limit μ2 → 0 in the divergent contributions is only to be
taken after the cross section of the whole process has been
evaluated. Endowed with the regulating mass, all integrals are
IR finite. Using μ0 ≡ μ2/s and s ≡ ( pq + pq¯ )2 = 2 pq · pq¯ ,
one obtains12
i 1
Iireg| pq2= pq2¯ =0 = (4π )2 s
×
In the UV divergent integrals, the BDI Ilog(μ2) has been
isolated, according to the rules of ireg. Inserting the integrals
from Eq. (4.16) into Eq. (4.12) and performing the remaining
contractions, one obtains for the one-loop vertex correction
(Ai(r1e)g)μ = (Ai(r0e)g)μ
αs
π C F −
7 − π 2 + 3i π
4
ln2(μ0)
4
−
where the UV divergent contributions ∼ Ilog(μ2) are
dropped. Taking twice the real part of the one-loop
correction, the virtual contribution to the total cross section is then
given by13
12 The results of the ireg integrals can be compared with the
corre2
sponding ones in ds. Setting μds = s, the integrals in Eqs. (4.16a) and
(4.16b), for example, are given by Ids pq2=pq2¯ =0 = c ( ) (4 πi)2 1s 12 +
iπ − π22 +O( ) and Idαs pq2=pq2¯ =0 = c ( ) (4 πi)2 (pq −spq¯ )α 1 +i π +2 +
O( ) . Using c ( = 0) = 1, the one-to-one correspondence between
double and single IR poles in ds and ireg then reads 12 ↔ 21 ln2(μ0)
and 1 ↔ ln(μ0); see also Sect. 5.4.
13 The virtual cross section in ireg can be compared with the ones
obtained in ds; see Eq. (2.21). Using the aforementioned translation
rules for IR divergences in ireg and ds, and 2( = 0) = c ( =
0) = 1, it follows that Eq. (4.18) coincides with the results obtained in
fdh and dred, Eq. (2.21c). In Sect. 5.3, it will be shown that the result
of the virtual cross section in ireg also coincides with the one obtained
in fdr.
with σ (0) given in Eq. (2.15). The divergences occurring in
the limit of a vanishing regulator mass μ0 will be exactly
canceled by the cross section related to the bremsstrahlung
diagrams, as shown in the next section.
Real contributions
In the following we obtain the bremsstrahlung contribution
to the total cross section, using the same regulator mass μ for
the gluon and the quarks, as in the previous section. At least at
NLO, apart from minor technical differences, the treatment
of IR singularities in ireg is equivalent to the fdr solution
proposed in Ref. [
92
] (see also Sect. 5.3).
The total cross section pertaining to the real emission
process e+( p )e−( p) → γ ∗(q) → q(k1)q¯ (k2)g(k3) is
obtained:
(r) 1
σireg = 2s
d3k1
(2π )32ω1
d3k3
(2π )32ω3
×
× Mi(r0e)g(qq¯ g),
d3k2
(2π )32ω2
(2π )4δ(4)(q − k1 − k2 − k3)
in terms of ki0 = ωi = ki2 + μ2.
Let us first analyze how the regulating mass enters the
phase-space integration boundaries. Using the CM frame of
the virtual photon, δ(4)(q − k1 − k2 − k3) = δ(q0 − ω1 −
ω2 − ω3)× δ(3)(k1 + k2 + k3), and after integrating out the
three-momentum of the gluon, the phase-space integration
P reduces to
P =
×
=
d3k1
(2π )32ω1
d3k3
(2π )32ω3
d3k1
(2π )32ω1
d3k2
(2π )32ω2
d3k2
(2π )32ω2
π
ω3
× δ(q0 − ω1 − ω2 − ω3),
with ω3 = (k1 + k2)2 + μ2. The integration over the angle
θ between k1 and k2 is performed, noting that ω3dω3 =
|k1||k2|dcos(θ ). In addition, with |ki |d|ki | = ωi dωi we
get
(2π )4δ(4)(q − k1 − k2 − k3), (4.20a)
(4.20b)
ω1M
ω1m
dω1
dω2
ω2M
ω2m
(4.18)
dω3 δ(0)(q0 − ω1 − ω2 − ω3).
(4.21)
×
ω3M
ω3m
The boundary values for the ω3 integration can be traced
back from the range of allowed θ angle values. At fixed k1
and k2 one thus obtains ω3m =
μ2 + (|k1| − |k2|)2
corresponding to θ = π and ω3M = μ2 + (|k1| + |k2|)2 for
θ = 0. In the first case, the quark and antiquark have
opposite momenta and thus a soft gluon momentum k3 can be
emitted together with hard fermion momenta. In the second
case, the fermions move parallel and soft gluon emission is
accompanied with soft fermion momenta. Introducing now
dimensionless variables
χi =
keeping in mind the interval allowed for non-vanishing
contributions of the δ-integration. The latter restrict the
boundaries of the χ2 integration to
χ2± =
with the notation χ2+ = χ2M , χ2− = χ2m . Finally, the
χ1 integration boundaries are obtained as follows. From
χ1 = 1 − 2 ωq01 , the upper limit is easily extracted, given when
The lower boundary is obtained for maximal ω1, i.e. for
ω1 M = μ2 + |k1M |2 = μ2 + (|k2| + |k3|)2, achieved when
the angle θ23 between the fermion and the gluon is zero.
Using further that energy conservation is expressed in the χ
variables as 1 = χ1 + χ2 + χ3 and rewriting Eq. (4.22) as
o|kqfi02|t2he=va(r1i−a4bχlie)2s −χ1,μqχ022 2,,oμn0e. cTahneemxpinreimssuωm1vMalounelyofinχ1tetrhmens
occurs for χ2 = (1−23μ0) , leading to
Page 22 of 39
Using Eqs. (4.25) and (4.24) together with q02 = q2 = s, we
obtain for the phase-space integral
.
(4.26)
(4.27)
(4.28a)
(4.28b)
(4.29)
(4.30)
(4.31a)
where we use the leptonic tensor of Eq. (2.13) and ω(r ) =
2 Qq2 C F /s2.
The result can be simplified by considering gauge
invariance, which implies that Gμν , after phase-space integration,
must be transverse to the photon momentum q. Thus, the
total cross section due to real contribution can be expressed
as
(r )
σireg = σ (0)
αs
π
C F
1
− 2 gμν Gμν .
After a tedious, yet straightforward computation, one obtains
χ1χ2
1
1
− 2
gμν Gμν = −
1
μ0 + χ1 + μ0 + χ2
χ2
1 χ1
+ 2 μ0 + χ1 + μ0 + χ2
1
+ (μ0 + χ1)(μ0 + χ2) + O(μ0),
where we use the definition of χi in Eq. (4.22) and ki2 = μ2.
Finally, the integrals can be evaluated with14
1
μ0 + χ1 =
χ1χ2
χ1χ2
1
μ0 + χ2
= − ln(μ0) − 3 + O(μ0),
14 These integrals are the same in ireg and fdr; see e.g. Eqs. (34)
and (35) of Ref. [
92
]. They can be compared with the corresponding
ones obtained in ds; see Eqs. (2.26). Again, the transition rules for the
IR divergences between ds and ireg/fdr read 12 ↔ 21 ln2(μ0) and
1 ↔ ln(μ0).
χ1χ2
χ2
μ0 + χ1 =
χ1
μ0 + χ2 = −
Finally, the total cross section due to the real contribution is
given by15
(r )
σireg = σ (0)
×
The procedure of obtaining the real corrections in ireg can be
summarized as follows: compute the matrix element squared
for massless external and internal particles as in Eq. (4.28b).
However, the on-shell limit ki2 = 0 should not be applied.
Instead, wherever a squared momentum appears it should
be replaced by ki2 = μ2. The phase-space integration is to
be carried out for massive external particles. IR divergences
appear as ln(μ0) terms.16
Finally, adding the virtual contribution, Eq. (4.18), one
obtains the well-known UV and IR finite result
σ (1) = σ (0) + σi(rve)g + σi(rre)g|μ0→0 =
× 1 +
3C F .
αs
4π
4.3 Established properties of IREG
Gauge invariance
In gauge theories, the initial structure of a given Feynman
diagram contains Dirac matrices, Lorentz contractions, etc.
These operations may generate terms with squared momenta
in the numerator which must be canceled against
propagators before applying the rules of ireg. This point was first
emphasized in differential regularization whose rules have a
one-to-one correspondence with the ireg prescription [
78
].
15 This result can be compared with the ones obtained in ds; see
Eq. (2.27). Using the rules for translating IR divergences between ireg
and ds together with 3( = 0) = 1, it follows that Eq. (4.32)
coincides with the results in fdh and dred, Eq. (2.27c). In Sect. 5.3, it will
be shown that Eq. (4.32) also coincides with the corresponding result
in fdr.
16 The only technical difference from the evaluation of real corrections
in fdr is that in fdr the matrix elements are computed in the strict
massless limit, i.e. using ki2 = 0. Thus, at least at NLO the two schemes
differ at most by terms O(μ0).
As an example, consider the (unregularized) off-shell
vacuum polarization tensor in massless QED at one loop
μν = −e2i 4Tr
d4k
,
which, after evaluating the Dirac algebra, can be expressed
as
μν = −4e2
d4k 2kμkν −gμν k2 −kμ pν −kν pμ +gμν (k · p)
(2π )4 k2(k − p)2
,
≡ −4e2 2I μν − gμν J − I μ pν − I ν pμ + gμν (Iα pα) .
The integrals, after applying the rules of ireg, are given as
Jireg =
=
μ
Iireg =
Iiμreνg =
d4k k2
(2π )4 k2(k − p)2
d4k 1
(2π )4 (k − p)2 = − p2υ0,2,
d4k
kμ
where we have suppressed quadratic divergences (in the
example they cancel exactly), and νi, j are surface terms
defined as
g{ν1···ν j }υi, j ≡ ϒiν1···ν j ≡
d4k
kν2 · · · kν j
where we use g{ν1···ν j } ≡ gν1ν2 · · · gν j−1ν j + symmetric
combinations. Inserting all results in μν , one obtains
ireg = − 43 e2(gμν p2 − pμ pν )
μν
× Ilog(λ2) − b ln
p2
− λ2
(4.38)
(4.39)
(4.40)
(4.34)
(4.35a)
(4.35b)
(4.36a)
(4.36b)
(4.36c)
(4.37)
1
− 3 (gμν p2 + 2 pμ pν )υ0,4
+ pμ pν υ0,2 − gμν υ2,2 .
As can be seen, to enforce gauge invariance (expressed in
the transversality of iμrνeg), surface terms should be null as
previously discussed [
77
].
We remark the appearance of a k2 term in Eq. (4.35),
defined as the divergent J integral, and the importance
of applying ireg rules only after cancelling such term
against propagators. Proceeding otherwise, by rewriting
k2 = gμν kμkν for instance, one would obtain
d4k k2
(2π )4 k2(k − p)2 = gμν
d4k kμkν
(2π )4 k2(k − p)2
which is different from the J integral, Eq. (4.36a), not only
by arbitrary terms encoded in the υi, j but also by a finite
term. In this way, gauge invariance would be broken even if
the surface terms are systematically set to zero. It should be
emphasized that the discussion above is restricted to
divergent integrals.
UV renormalization
We would also like to briefly show how
renormalizationgroup functions can be computed in the framework of ireg.
For simplicity, we adopt the background field method [
93
]
which relates the wave function renormalization of the
background field, B0 = Z B B, with the coupling renormalization,
e0 = Zee, through the equation Ze = Z −1/2. Therefore, by
B
applying this method to QED, the β function can be obtained
only with the knowledge of the vacuum polarization tensor.
Performing a minimal subtraction, which in ireg amounts
to subtract only basic divergent integrals as Ilog(λ2), and
remembering that λ plays the role of a renormalization-group
scale, one obtains17
β = λ2 ∂∂λ2 4eπ
4
= −
e
4π
2
e4 4 NF i λ2 ∂
= (4π )2 3 ∂λ2 Ilog(λ2) + O(e6)
17 This result coincides with the well-known value of the QED β
function of the gauge coupling obtained in ds; see Eq. (2.8a).
(iii) The global prescription in Eq. (5.2) should be made
compatible with a key property of multi-loop calculus:
In an -loop diagram, one should be able to
calculate a subdiagram,
insert the integrated form into the full diagram
and get the same answer.
(5.4)
We dub this subintegration consistency.
Finally, after limμ→0 is taken, ln μ → ln μR is understood
on the r.h.s. of Eq. (5.1), where μR is an arbitrary
renormalization scale. Note that inserting Eq. (5.3) into Eq. (5.1) gives
an alternative definition
= μli→m0 r d4q1 · · · d4q { J (q1, . . . , q , μ2)
− [ JINF(q1, . . . , q , μ2)]},
5 FDR: four-dimensional
regularization/renormalization
fdr [
96
] is a fully four-dimensional framework to compute
radiative corrections in QFT. The calculation of the loop
corrections is conceptually simplified with respect to more
traditional approaches in that there is no need to include UV
counterterms in the Lagrangian L. In fact, the outcome of an fdr
calculation at any loop order is directly a UV-renormalized
quantity. Moreover, this particular way of looking at the UV
problem may open new perspectives in the present
understanding of fundamental and effective QFTs [
97
]. In the
following, we review the fdr treatment of UV and IR
divergences, also using the e+e− → γ ∗ → qq¯ (g) process as an
explicit example.
5.1 FDR and UV infinities
Let J (q1, . . . , q ) be an integrand depending on integration
momenta q1, . . . , q . The fdr integral over J is defined as
follows:
[d4q1] · · · [d4q ] J (q1, . . . , q , μ2)
≡ μli→m0
d4q1 · · · d4q JF(q1, . . . , q , μ2),
(5.1)
where JF(q1, . . . , q , μ2) is the UV-finite part of J (q1, . . . ,
q , μ2) (specified below), μ is an infinitesimal mass needed
to extract JF from J , and [d4qi ] denotes the fdr integration.
The integrands J (q1, . . . , q , μ2) and JF(q1, . . . , q , μ2) are
obtained from J (q1, . . . , q ) with the help of the following
rules:
(i) Squares of integration momenta appearing both in the
denominators of J (q1, . . . , q ) and in contractions
generated in the numerator by Feynman rules are shifted by
μ2,
qi2 → qi2 − μ
2 ≡ q¯i2.
This replacement is called global prescription.
(ii) A splitting
J (q1, . . . , q , μ2) = [ JINF(q1, . . . , q , μ2)]
+ JF(q1, . . . , q , μ2)
(5.2)
(5.3)
is performed in such a way that UV divergences are
entirely parametrized in terms of divergent integrands
contained in [ JINF], which solely depend on μ2. By
convention, we write divergent integrands in square
brackets and call them fdr vacua, or simply vacua.
(5.5)
(5.6)
(5.7)
where r denotes an arbitrary UV regulator. Equation (5.5)
tells us that the UV subtraction is directly encoded in the
definition of fdr loop integration: no divergent integrand is
considered separately from its subtraction term.
fdr integration preserves shift invariance which is easy to
prove when using Eq. (5.5) with r = ds,
[d4q1] . . . [d4q ] J (q1, . . . , q , μ2)
=
[d4q1] . . . [d4q ] J (q1 + p1, . . . , q + p , μ2),
and the possibility of cancelling numerators and
denominators
4 4 q¯i2 − mi2
[d q1] · · · [d q ] (q¯i2 − m2)m · · ·
i
=
[d4q1] · · · [d4q ] (q¯i2 − m12)m−1 · · · ,
i
which are properties needed to retain the symmetries of L
[
98
]. From Eqs. (5.6) and (5.7) it follows that algebraic
manipulations in fdr integrands are allowed as if they where
convergent ones. This authorizes one to reduce complicated
multi-loop integrals to a limited set of Master integrals
(MI) by using four-dimensional tensor decomposition [
99
]
or integration-by-parts identities [
100
]. In other words, the
definition in Eq. (5.1) [or Eq. (5.5)] can be applied just at the
end of the calculation, when the actual value of the MIs is
needed.
An important subtlety implied by Eq. (5.7) is that the
needed cancellation works only if integrands involving
explicit powers of μ2 in the numerator are also subtracted
as if μ2 = qi2, where qi2 is the momentum squared which
generates μ2. For instance, one computes
[d4q] (q2 − M 2)3
μ2
= μli→m0 μ
2
d4q
in accordance with Eq. (5.5). In this case both integrals on
the r.h.s. are UV convergent and the only contribution which
survives the μ → 0 limit is generated by the subtraction term.
As a consequence, although only one kind of μ2 exists, one
has to keep track of its origin when it appears in the numerator
of J (q1, . . . , q , μ2). For this purpose we use the notation
μ2|i , which understands the same subtraction required for
the case μ2 = qi2. fdr integrals with powers of μ2|i in the
numerator are called ‘extra integrals’.18 Their computation
is elementary, as illustrated by Eq. (5.8). Additional one- and
two-loop examples can be found in Refs. [
96, 99
]. fdr extra
integrals play an important role in maintaining the theory
unitary without the need of introducing counterterms in L,
as will be discussed in Sect. 5.4.
As a simple example of an fdr integration, we consider the
scalar one-loop integrand
J (q) = (q2 − M 2)2
,
1
(5.9)
which diverges logarithmically for q →
define its fdr integral are as follows:
∞. The steps to
• Shift squares of the integration momentum,
J (q) → J (q, μ2)
1
≡ (q¯ 2 − M 2)2
,
with q¯ 2 ≡ q2 − μ2.
(5.10)
• Subtract the divergent part of the integrand [ JINF(q, μ2)]
= q¯14 in the μ → 0 limit, setting μ → μR in the
logarithms
18 This is different compared to ireg where no extra integrals are
introduced. While extra integrals are not strictly needed in fdr, they are
introduced for convenience to allow the decomposition of fdr tensor
integrals into MIs and to avoid introducing counterterms in L.
(5.11)
(5.12)
μ→μR
(5.13a)
(5.13b)
[d4q
1
(q¯ 2 − M 2)2 −
1
q4
¯
μ→μR
.
• The dependence on r is eliminated by using the partial
fraction identity
1
] (q¯ 2 − M 2)2 ≡ μli→m0
d4q
M 2
M 2
q¯ 4(q¯ 2 − M 2) + q¯ 2(q¯ 2 − M 2)2
= −i π 2 ln
M 2
μ2R .
In practice, one can directly start from the integrand in
Eq. (5.10) and expand it by means of Eq. (5.12). This
procedure allows one to naturally separate [ JINF(q1, . . . , q , μ2)]
from any integrand J (q1, . . . , q , μ2) and write down
definitions analogous to Eq. (5.13) at any loop order. Explicit
examples for the extraction of fdr vacua from two-loop
integrands are presented in Ref. [
99
].
Given the fact that the definition of fdr loop
integration is compatible with a graphical proof of the Slavnov–
Taylor identities through Eqs. (5.6) and (5.7) and can be made
congruent with the subintegration consistency of Eq. (5.4)
without the need of introducing UV counterterms in L (see
Sect. 5.4 and Ref. [
101
] for more details on this point),
fdr quantities are directly interpretable as UV-renormalized
ones. As an example, the correspondence between off-shell
two-loop QCD correlators computed in fdr and ds has been
worked out in Ref. [
101
].
5.2 FDR and IR infinities
The modification of the propagators induced by Eq. (5.2)
also regularizes soft and collinear divergences in the virtual
19 The alternative definition in Eq. (5.11) with, for example, r = ds
gives the same result, [d4q] (q¯2−1M2)2 = μ4d−sd dd q (q2−1M2)2 −
limμ→0 μ4d−sd dd q q¯14 μ→μds
can be directly set to zero since it is IR convergent.
= −i π 2 ln μM2d2s . In the first integral, μ
Fig. 10 Splitting regularized by μ-massive (thick) unobserved
particles. The one-particle cut contributes to the virtual part, the two-particle
cut to the real radiation
typical of the real radiation, and the last term generates IR
finite contributions. It is then clear that the fdr modification
q2 +ii0 → q¯2 +ii0 in the virtual contribution is matched by the
μ-massive version of Eq. (5.15), namely
,
which in turn is responsible for the correspondence q¯2 +ii0 →
(2π )δ(q¯ 2)θ (q0) depicted in Fig. 10; see also Ref. [
92
]. For
example, Eq. (5.16) can be used to rewrite the real part of
Eq. (5.14a) as an integral over an eikonal factor
(5.14b)
π 2
4
Re
1
i π 2
[d4q
1
,
Fig. 9 Massless scalar one-loop three-point function. Thick internal
lines denote the insertion of the infinitesimal mass μ, which generates
μ-massive propagators
integrals [
92
]. As an example, the massless one-loop
threepoint function corresponding to the Feynman diagram shown
in Fig. 9 is interpreted in fdr as
Ifdr =
d4q
1
q¯ 2 D¯ 1 D¯ 2
,
(5.14a)
with q¯ 2 = q2 − μ2 and D¯ i = (q + pi )2 − μ2. It is worth
noticing that this is the same definition as given in Eq. (5.5).
In fact, there is no [ JINF] term to subtract in this case since
the integrand is UV finite. It is easy to compute
Ifdr =
i π 2 ln2(μ0)
s
2
π 2
+ i π ln(μ0) − 2
with s = ( p2 − p1)2 and μ0 = μ2/s. Thus, IR
divergences take the form of logarithms of μ0. In the case at
hand, the squared logarithm is generated by an overlap of
soft and collinear divergences when q → 0 and q is collinear
to pi .
This prescription certainly allows one to assign a
precise meaning to virtual integrals also in the presence of
IR singularities. Nevertheless, the correct final result is
obtained only if the real part of the radiative corrections
is treated likewise. This is obtained by carefully analyzing
the Cutkowsky rules [
102
] relating real and virtual
contributions with different cuts of diagrams at a higher
perturbative level, where cutting a propagator means going on-shell,
q2 +ii0 → (2π )δ(q2)θ (q0). This correspondence is linked to
the identity20
i
q2 + i 0 = (2π )δ(q2)θ (q0) + q2
i
− i 0q0
.
(5.15)
In fact, IR singularities on the l.h.s. of Eq. (5.15) manifest
themselves as pinches of the integration path by two (or
more) singularities in the q0 complex plane, which occur
in the virtual part of the radiative corrections. On the other
hand, the first term on the r.h.s. gives end-point singularities,
20 This relation is also one of the starting points of the fdu scheme
described in Sect. 6.
where s¯i j = ( p¯i + p¯ j )2, p¯i2, j = μ2 → 0 and ¯ 3 denotes the
μ-massive 3-particle phase space.
In summary, the IR divergent 1 → 2 massless splitting
gets regularized by the introduction of an infinitesimal mass
μ for all unobserved particles. In the case of external
particles, this is equivalent to trade a massless phase space for a
μ-massive one. Furthermore, IR infinities cancel when
summing real and virtual contributions, for instance
σ =
m
dσ (v) + μli→m0
¯ m+1
dσ (r )({s¯i j }) = σ (v) + σ (r ),
(5.16)
(5.17)
(5.18)
(5.19)
as illustrated in Fig. 11. Finally, when dσ (r )({si j }) is
analytically known in terms of massless invariants si j = ( pi + p j )2
2
with pi, j = 0, Eq. (5.18) prescribes the replacement si j →
s¯i j . If, instead, dσ (r ) is known only numerically, one can
construct a mapping from a massive to a massless phase space,
mapping
¯ m+1 → m+1, use m+1 to compute massless
invariants and rewrite the real contribution as
σ (r )
= μli→m0
dσ (r )({si j })
¯ m+1
si j
Fig. 11 IR divergences drop
out when summing the
m-particle virtual piece σ (v) and
its real (m + 1)-particle
counterpart σ (r). Adding the
contributions gives the fully
inclusive NLO cross section
si j → 0.
In this way, dσ (r )({si j }) is gauge invariant, since it is
computed with massless kinematics and the fudge factor i< j ss¯ii jj
ecfofnefictgiuveralytiornesp.laTchesis siisj b→ecasu¯isje indσa(lrl)(r{esleijv}a)nt∼IRs1isjinwguhleanr
5.3 Application example: e+e− → γ ∗ → qq¯ at NLO
Virtual contributions
In this section we perform the computation of the total cross
section of the process e+e− → γ ∗ → qq¯ in QCD to
illustrate a typical fdr calculation. As for the virtual part of the
corrections, scaleless integrals vanish. More precisely, in fdr
they are proportional to ln μR/μ (where μ is the IR
regulator), which gives zero when choosing μR = μ [
99
]. Thus,
only the vertex diagram where a virtual gluon connects the
quark with the antiquark has to be considered. The only
subtle point of the calculation is the replacement
q/γ α q/ = −q2γ α + 2γβ qα qβ
→ −q¯ 2γ α + 2γβ qα qβ
in the fermion string, dictated by the global prescription. Note
that this is fully equivalent to the ireg recipe of performing
simplifications before introducing μ2 in the denominators. In
fact, the replacement in Eq. (5.20) produces a contribution
proportional to
+ 2γβ
−q¯ 2γ α + 2γβ qα qβ
[d4q
q¯ 2 D¯ 1 D¯ 2
qα qβ
,
= −γ α
[d4q]
1
D¯ 1 D¯ 2
which is the same result one would obtain by simplifying
before introducing μ-massive propagators. In both cases,
the gauge-preserving simplification between the
numerator and the denominator of the first integral on the r.h.s.
of Eq. (5.21) is achieved. Differences between fdr and
ireg start when evaluating the second integral. A
customary Passarino–Veltman tensor decomposition is possible in
(5.20)
(5.21)
fdr before using the definition of the fdr integral given in
Eq. (5.1):21
C αβ ≡
[d4q
qα qβ
= C00(gαβ ) + C11( p1α p1β )
+ C22( p2α p2β ) + C12( p1α p2β + p2α p1β ).
To obtain the coefficients Ci j , one needs to contract C αβ with
gαβ , resulting in
C αα =
[d4q
q2
.
Since q2 in the numerator is not generated by Feynman rules,
now it would be incorrect to simplify it with the q¯ 2
denominator, in the sense that one would not obtain the correct value
of C αβ . Here is the place where the fdr ’extra integrals’ play
an active role. In fact, by adding and subtracting μ2, one
rewrites
(5.22)
(5.23)
(5.24)
C αα =
[d4q
] q¯ 2 D¯ 1 D¯ 2
,
[d4q]
1
D¯ 1 D¯ 2
which produces the correct answer in terms of a minimum
set of scalar MIs. In other words, thanks to the introduction
of extra integrals, Eq. (5.1) can be considered as a convenient
way to define a loop integration for divergent integrals that
survives algebraic four-dimensional manipulations. This is a
peculiar property of fdr.
In the computation at hand, only C00 and C12 are needed.
The reduction gives
C00 =
I2,fdr
4
E I
+ 2
E I
s
,
21 In ireg, Cαβ is directly computed by subtracting its UV divergent
part.
(5.25)
obtaining
C1 = C2 =
qα
see Eqs. (5.13) and (5.8). Analogously, one reduces the
rankone tensor
with22
I2,fdr =
E I =
= −π
1
In summary, the virtual amplitude can be expressed as a linear
combination of the scalar integrals in Eqs. (5.14b) and (5.26).
Multiplying with the Born amplitude and taking the real part,
one obtains23
+ O(μ0) ,
where σ (0) is the Born total cross section given in Eq. (2.15)
and ln μ0 is the IR logarithm. The process at hand is UV
finite, so that the dependence on the logarithms has to drop
in the final result. As a consequence, the effect of all
scaleless integrals (nullified by our particular choice μR = μ) is
nothing but ln s/μ2R → ln s/μ2 in Eq. (5.26a), as can easily
be checked with an explicit calculation.
Real contributions
As for the bremsstrahlung contribution e+e− → γ ∗ →
q( p1)q¯ ( p2)g( p3), a tensor decomposition of the
three22 The value of the ‘extra integral’ E I is the same as the one of
(2π )4 I3d [μ2] obtained in fdf; see Eq. (3.36).
23 This result is identical to the one obtained in ireg, compare with
Eq. (4.18).
z¯ = s¯s12 − μ0, with μ0 = μs2 ,
1
ds¯13ds¯23 s¯13 =
R¯3
R¯3
= s
R¯3
1
y¯ + μ0
and they are listed in Ref. [
92
]. We report them here for
completeness25
= s
− ln(μ0) − 3 + O(μ0) ,
particle phase-space integrals produces the matrix element
squared24
Mf(d0)r(s12, s13, s23) =
16π αs
s
CF Mf(d0)r(s)
,
where Mf(d0)r(s) is the fully inclusive Born matrix element
squared of e+e− → γ ∗ → q(k1)q¯ (k2),
2
Mf(d0)r(s) = π
2
Mf(d0)r(k1, k2).
In accordance with Eq. (5.18), we now replace all the
invariants by their massive counterparts, si j → s¯i j , and integrate
over a μ-massive three-body phase space,
¯ 3
×
R¯3
Mf(d0)r(s¯12, s¯13, s¯23) =
4π 3αs
s2
CF Mf(d0)r(s)
ds¯13ds¯23
s s s¯13 s¯23 s2
− s¯13 − s¯23 + 2s¯23 + 2s¯13 + s¯13s¯23
(5.28)
(5.29)
(5.30)
(5.31)
(5.32)
(5.33)
(5.34a)
The quantity R¯3 represents the physical region of the
Dalitz plot for the μ-massive three-particle phase-space
parametrized in terms of s¯13 and s¯23. The limit μ → 0
is understood from now on. The needed integrals can be
expressed in terms of the scaled invariants
24 This corresponds to the usual matrix element squared for massless
particles computed in four dimensions, as given in Eq. (2.23).
25 Similar integrals have to be evaluated when using the ireg
framework to determine the real contributions; see Eqs. (4.31). Their
counterparts in ds are given in Eqs. (2.26).
The final result of the bremsstrahlung contribution reads26
CF
Adding the virtual contribution given in Eq. (5.29) produces
the total NLO correction,
σ (1) = σ (0) + σf(dvr) + σf(dr)r
Qq2 Nc
3s
e4
4π
(5.34b)
(5.34c)
(5.35)
(5.36)
(5.37)
.
(5.38)
26 This result is identical to the one obtained in ireg, compare with
Eq. (4.32).
Finally, we remark that it is possible to set up the entire
calculation in a fully local fashion. To achieve this, one has to
rewrite the double and single logarithms in Eq. (5.29) as local
counterterms to be added to the real integrand. For instance,
Eq. (5.34c) gives
ln2(μ0) − π 2 = 2
R¯3
The full counterterm needed for the case at hand can be
inferred uniquely from the factorization properties of the
matrix element squared,
Mfcdtr( p1, p2, p3)
16π αs
s
CF Mf(d0)r( pˆ1, pˆ2)
s s s¯13 s¯23 s2 17
− s¯13 − s¯23 + 2s¯23 + 2s¯13 + s¯13s¯23 − 2
R¯3
R¯3
s¯13
ds¯13ds¯23 s¯23 =
−
This equation is in agreement with Eq. (5.30) when
integrating over pˆ1 and pˆ2. The constant 127 is chosen in
such a way that only the logarithms and the π 2 term in
Eq. (5.35) are reproduced upon integration over R¯3. The
quantity Mf(d0)r( pˆ1, pˆ2) is computed with mapped quark and
antiquark momenta defined as
pα s23
ˆ1 = κ αβ p1β 1 + s12
, pˆ2α = κ αβ p2β 1 + ss1123 ,
κ =
ss12
(s12 + s13)(s12 + s23)
,
(5.39)
where αβ is the boost that brings the sum of the momenta
back to the original center of mass frame, pˆ1 + pˆ2 =
(√s, 0, 0, 0). After subtracting Mfcdtr( p1, p2, p3) from the
exact matrix element squared, μ can be set to zero before
integration. In this case, an analytic knowledge of Mf(d0)r(s12, s13,
s23) is not necessary. A simple flat Monte Carlo with 105
phase-space points reproduces the result in Eq. (5.36) at the
1 per mil level in a quarter of second.
5.4 Established properties and future developments of FDR
Correspondence between integrals in FDR and DS
At one loop, a one-to-one correspondence exists between
integrals regularized in fdr and ds. More precisely,
according to the definition of fdr, any result of a loop integration is
UV finite, whereas IR divergences are expressed in powers of
(logarithms of) μ0 = μ2/s. In ds, on the other hand, results
of an integration in d dimensions can be expanded in powers
of ; UV and IR divergences are then parametrized as poles
1/ n.
To provide an example for the relation between IR
divergences of integrals in fdr and ds, we consider the integral in
Eq. (5.14). Using d-dimensional integration, its result reads
Ids = c ( ) s
i π 2
1
2 +
i π
π 2
− 2 + O( ) .
The factor c ( ) is directly related to integration in d
dimensions. It is given in Eq. (2.20b). Comparing the result in
Eq. (5.40) with Eq. (5.14b), the relation between the
(regularized) IR divergences is given by
1 1 ln2(μ0),
2 ↔ 2
1
↔ ln(μ0).
Extending this to the ‘finite’ terms, the following generalized
relation for a (potentially UV and IR divergent) integral over
a generic integrand F holds:
(5.40)
(5.41)
1
(2π )4
[d4q]F (q¯ 2, q)
c ( )−1μ4−d
μ0
dd q
(2π )d
F (q2, q)
0
.
(5.42)
R¯3
μ2
s
R3
Analogously, for the real contribution one finds
d x d yd z F (x , y, z) δ(1 − x − y − z)
(x yz)
,
(5.43)
(5.44)
where R3, x , y, and z are the massless counterparts of
R¯3, x¯, y¯, and z¯, respectively; see also Eq. (5.33).
Finally, there exists a connection between the fdr ‘extra
integrals’ and fdf integrals containing powers of the (−2
)dimensional part of the loop momentum, q[−2 ] ≡ q˜ ,
[d4q]F (q¯ 2, q, −μ2) = μ4−d
dd q F (q2, q, q˜2).
For more comments on the interplay −μ2 ↔ q˜ 2; see also
the discussion around Eq. (3.6) and Ref. [
96
].
Global prescriptions, such as the one described at one loop
in Eq. (5.20), can be defined at any loop order. Their role is
maintaining the needed gauge cancellations. However, this
is not enough to guarantee that results are compatible with
unitarity. In fact, in a unitary QFT, all perturbative orders are
linked by unitarity relations, and any renormalization
procedure compatible with unitarity has to fulfill the following
two requirements:
(a) The UV divergences generated at any perturbative level
should have no influence on the next perturbative orders.
(b) The subintegration consistency in Eq. (5.4) should hold
true.
Schemes based on ds automatically respect subintegration
consistency when all objects (including γ matrices) are
treated in d dimensions, while requirement (a) is fulfilled
only if 1/ poles are subtracted order by order by introducing
counterterms in L. This forbids one to define ds loop
integrals beyond one loop by simply dropping 1/ poles. See the
discussion is Section 2.5 of Ref. [
99
] for more details.
On the other hand, fdr automatically respects requirement
(a) since the UV subtraction is embedded in the definition
of the fdr integral, so that there is no room for any UV
divergence to have any influence at higher perturbative levels.
For instance, products of two one-loop fdr integrals give the
same result at any perturbative order, which is not the case
in ds.
On the contrary, subintegration consistency is not
automatically obeyed in fdr. The reason for this can be traced
back to the fact that the global prescription needed at the level
of divergent subdiagrams (sub-prescription) clashes with the
global prescription required at the level of the full diagram
(full-prescription), so that one has to correct for this
mismatch. However, this can be done directly at a diagrammatic
level. This is possible thanks to the fdr extra integrals. They
can be used to parametrize, in an algebraic way, the
difference between the result one gets when cancellations do
or do not take place between numerators and denominators,
as illustrated, for example, in Eq. (5.24). In practice, one
looks at all possible UV divergent subdiagrams, adds the
piece needed to restore the sub-prescription and subtracts
the wrong behaviour induced in the subdiagram by the
fullprescription. The net result of this process is the addition
of fdr extra–extra integrals to the amplitude that enforce
requirement (b) without the need of an order-by-order
renormalization [
101
]. For example, a two-loop extra–extra
integral can be defined as the insertion of a one-loop extra
integral into a two-loop fdr integral. Thus, an fdr calculation
directly produces renormalized quantities, which is a unique
property of the fdr formalism.
Work is in progress to find the connection between fdr
extra–extra integrals and evanescent fdh couplings.
Preliminary results indicate that the introduction of fdr extra–extra
integrals is equivalent to a restoration of the correct behaviour
under renormalization in an fdh calculation in which one sets
equal gauge and evanescent couplings from the beginning.
6 FDU: four-dimensional unsubtraction
The four-dimensional unsubtraction (fdu) [
103–107
]
approach constitutes an alternative to the traditional subtraction
method. It is based on the loop–tree duality (LTD) theorem
[
108–111
], which establishes a connection among loop and
dual integrals, the latter being similar to standard phase-space
integrals. In this way, the method provides a natural way to
implement an integrand-level combination of real and virtual
contributions, thus leading to a fully local cancellation of IR
singularities. Moreover, the addition of local UV
counterterms allows one to reproduce the proper results in standard
renormalization schemes.
In the following, we describe briefly the general facts
about the method, using the computation of the NLO QCD
corrections to γ ∗ → qq¯ (g) as a practical guideline.
6.1 Introduction to LTD
The LTD theorem is based on Cauchy’s residue theorem.
Let us consider a generic one-loop scalar integral for an N
particle process, where the external momenta are labeled
as pi with i ∈ {1, 2, . . . N }, whilst the loop momentum is
denoted by . With these conventions, the internal virtual
momenta become qi = + ki where ki = p1 + · · · + pi and
kN = 0 because of momentum conservation. If the mass of
the internal particles is mi , a scalar integral can be expressed
as
L(1)( p1, . . . , pN ) =
G F (qi ),
N
i=1
(6.1)
(6.2)
(6.3)
with the Feynman propagators G F (qi ) = (qi2 − m2
i + i 0)−1.
As usual, qi represents a four momentum which can be
decomposed as qi,μ = (qi,0, qi ), independently of the
specific space-time dimension.27 The energy component is qi,0,
whilst qi denotes the spatial components.
At one-loop level, the dual representation of the loop
integral is obtained by cutting one by one all the available internal
lines and applying the residue theorem accordingly. The cut
condition is implemented by restricting the integration
measure through the introduction of
δ˜ (qi ) ≡ 2π i θ (qi,0)δ(qi2 − mi2),
which transforms the loop integration domain into the
positive energy section (i.e. qi,0 > 0) of the corresponding
onshell hyperboloid (i.e. qi2 = mi2). When the scattering
amplitude under consideration is composed of single powers of
the propagators, the computation of the residue simplifies to
removing the cut propagator and replacing the uncut ones
with their duals, i.e.
G D(qi ; q j ) = q 2j − m2j − i 0η · k ji
,
1
where i, j ∈ {1, 2, . . . N }, k ji = q j − qi and η is an arbitrary
future-like or light-like vector, η2 ≥ 0, with positive definite
energy η0 > 0. It is worth noticing that the dual
prescription takes care of the multiple-cut correlations introduced in
the traditional Feynman-tree theorem (FTT) [
112,113
], thus
allowing one to prove their formal equivalence.
In this way, the dual integrand looks like a tree-level
amplitude whose building blocks are the same as in the
stan27 In other words, we could be working in any of the ds schemes
mentioned in this article, with the only requirement that the associated
manifold is Lorentzian, i.e. that it only contains time component and an
arbitrary number of spatial ones.
dard theory with a modified i 0 prescription. Thus, the
oneloop scalar integral in Eq. (6.1) reads
N
i=1
j =i
L(1)( p1, . . . , pN ) = −
δ˜ (qi )
G D(qi ; q j ).
(6.4)
The existence of a dual representation for loop integrals
straightforwardly leads to a dual representation for loop
scattering amplitudes. As explained in Ref. [
108
], any loop
contribution to scattering amplitudes in any relativistic, local,
and unitary quantum field theory can be computed through
the decomposition into dual contributions. Of course, this
idea generalizes to multi-loop amplitudes, where dual
contributions involve iterated single-cuts [
108,110
].
For amplitudes containing higher powers of the
propagators, the previous result can be extended, as studied in
Ref. [
111
]. It is worth appreciating that higher powers of
the propagators explicitly manifest when dealing with
selfenergy corrections at one loop, self-energy insertions at
higher orders, and when computing the local version of the
UV counterterms [
104,105
].
6.2 Momentum mapping and IR singularities
The application of the LTD theorem to a virtual amplitude
leads to a set of dual contributions. From them, we can extract
useful information as regards the location of the
singularities in the corresponding integration domain, as well as the
components (or cuts) that originate them. As explained in
Refs. [
114–116
], the intersection of forward and backward
hyperboloids defined by the on-shell conditions allows one
to identify the IR (and threshold) singularities. Moreover,
this study is crucial to prove the compactness of the region
developing IR divergences [
103–105
], which constitutes a
very important result by itself. This is because the
realradiation contributions are computed on a physical phase
space, which is also compact.28 In consequence, since the
Kinoshita–Lee–Nauenberg (KLN) theorem states that there
is a cross-cancellation of IR singularities between real and
virtual terms, the compactness of the IR region inside the
dual integration domain allows one to implement a local
realvirtual cancellation of singularities by applying a suitable
momentum mapping. In this way, the singularities in the real
phase space (PS) are mapped to the dual integration domain
where the corresponding virtual singularities are generated;
then an integrand-level cancellation takes place and there is
no need of introducing any external regulator to render the
combination integrable.
In order to connect the Born kinematics (m-particle PS)
with the real- emission one ((m + 1)-particle PS), we rely on
28 This assumption is true whenever the incoming particles have fixed
momentum, thus leading to a global constraint on the energy available
for generating final-state radiation.
Page 32 of 39
techniques similar to those applied for the dipole method [
53,
117
]. To be more concrete, let us start thinking about the
virtual contribution. After obtaining the dual amplitudes, we
have a set of m external momenta and a free on-shell loop
momentum. In this way, the dual amplitudes introduce an
extra on-shell momentum. Since there are (m + 1) on-shell
momenta available, the kinematics of the dual components
exactly matches the kinematics of the real contribution.
Then it is necessary to isolate the real-emission IR
singularities by properly splitting the complete real PS. If piμ
are the momenta of the real-emission partons, we start by
defining the partition
m
i=1
Ri = {y ir < min y jk },
Ri = 1,
(6.5)
where y i j = 2 pi · p j /Q2, r is the radiated parton from parton
i , and Q is the typical hard scale of the scattering process.
It is important to notice that, inside Ri , the only allowed
collinear/soft configurations are i r or prμ → 0. Thus,
collinear singularities manifest in non-overlapping regions
of the real-emission PS which allows one to introduce an
optimized transformation to describe the collinear
configuration.
On the other hand, there are m dual contributions, each one
associated with a single cut of an internal line. So, we can
establish an identification among partitions and dual
amplitudes, based on the picture shown in Fig. 12. Concretely, the
cut-line in the dual amplitude must be interpreted as the
extraradiated particle in the real contribution; i.e. qi ↔ pr . Then
we settle in one of the partitions, for instance Ri . Because
the only collinear singularity allowed is originated by i r ,
we distinguish particle i and call it the emitter. After that,
we single out all the squared amplitude-level diagrams in the
real contribution that become singular when i r and cut
the line i . These have to be topologically compared with the
dual-Born interference diagrams whose internal momenta qi
are on-shell (i.e. the line i is cut), as suggested in Fig. 12.
In conclusion, the dual contribution i is to be combined with
the real contribution coming from region Ri .
The required momentum mapping is motivated by general
factorization properties in QCD [
114, 118
] and the
topological identification in Fig. 12. Explicitly, let us take the (m
+1)particle real-emission kinematics, with i as the emitter and r
as the radiated particle, and we introduce a reference
momentum, associated to the spectator j . For the massless case, the
generic multi-leg momentum mapping with qi on-shell is
given by
(6.6)
piμ = piμ − qiμ + αi p μj,
pkμ = pkμ k = i, j,
prμ = qiμ,
,
p μ
j = (1 − αi ) p μj,
with the primed momenta associated to the particles involved
in the real-emission process. In this case, note that pi2 =
p j2 = pr2 = 0 because we restrict ourselves to massless
particles. On the other hand, the initial-state momenta ( pa
and pb) are not altered by the transformation, neither is pk
with k = i, j . Besides that, since
pi + p j +
pk = pi + pr + p j +
pk ,
(6.7)
k=i, j
k=i, j,r
the transformation shows momentum conservation. It is
worth appreciating that this momentum mapping can be
extended to the massive case, even if the involved particles
have different masses, as we explained in Ref. [
105
].
6.3 Integrand-level renormalization and self-energies
Besides dealing with IR singularities, any attempt to provide
a complete framework for higher-order computations must be
able to treat UV divergences. In this case, a suitable local
version of the UV counterterms is required. This topic is deeply
discussed in Ref. [
104
] for the massless case, whilst the
massive one is studied in Ref. [
105
]. In the last case, the
selfenergy and vertex corrections become non-trivial and some
technical subtleties arise: there are noticeable changes in the
IR singular structure compared to the massless case. On one
hand, the mass acts as an IR regulator, preventing collinear
singularities to emerge. But, on the other hand, soft
singularities arising from gluon emissions become non-vanishing
because they are proportional to the mass of the emitting
with the renormalization scale μuv and ki,uv = qi − quv.
A similar expansion is carried out in the numerator, which
leads to the UV counterterm for the wave-function
renormalization,
Z2uv( p1)
= (2 − d)gs2CF
quv · p2
1 + p1 · p2
G F (quv)
2
[1 − G F (quv)(2quv · p1 + μ2uv)]
(6.8)
= −(4π ) (1 + ) αs CF
4π
2
μuv
μ2
− 1 −
2
.
(6.11)
leg. Since we are looking for a complete local cancellation
of singularities and a smooth massless transition, it is
necessary that the expressions for the massive case reduce to
those already available for massless processes, even at the
integrand level.
Let us start with the well-known expression for the
wavefunction renormalization. Working in Feynman gauge with
on-shell renormalization conditions, its integrated form is
given by29
αs
4π
Z2 =
CF −
1
uv
2
ir
+ 3 ln
M 2
where we kept track of the IR and UV origin of the poles
within ds. The unintegrated expression [
105
] is given by
Z2( p1)
which includes higher-order powers of the propagators, and
where we define q1 = + p1, q2 = + p1 + p2, and
q3 = . It is worth appreciating that there are many
equivalent integrand-level expressions to describe Z2( p1), but the
one presented in Eq. (6.9) develops the proper IR behaviour
to cancel singularities coming from real-emission processes.
Besides this, notice that the corresponding formula for the
massless case [
104
] is simply recovered by considering
M → 0 at the integrand level. The term proportional to
M 2 is responsible for soft divergences that appear when q1
is set on-shell, and it vanishes as M → 0 since soft
singularities are absent in the massless self-energy
computation. On the contrary, the collinear singularities that appear in
Z2(M = 0) manifest themselves as quasi-collinear
divergences, i.e. terms that behave like ln(M 2/μ2), as shown in
Eq. (6.8). Once we combine the self-energy contributions
with the virtual matrix elements, there are still UV
singularities present. These have to be removed by
performing an expansion around the UV propagator G F (quv) =
(qu2v − μuv + i 0)−1,
2
G F (qi ) = G F (quv)
1 −
2quv · ki,uv + ki,uv + μuv − m2
2 2
2 2 i + . . . ,
quv − μuv + i 0
(6.10)
29 The result of the field renormalization coincides with the one of cdr
and not with the one of fdh. In the latter scheme, the constant ‘−4’
would be replaced by ‘−5’; see e.g. Eq. (2.24) of Ref. [
28
].
The integrated form exactly reproduces the UV pole present
2
in Eq. (6.8). The subleading terms proportional to μuv are
chosen to subtract only the pole part from Eq. (6.8) and, in
this way, settle in the ms scheme. Finally, we define the
UVfree wave-function renormalization
Z ir
2 =
Z2 −
Z2uv,
(6.12)
that only contains IR singularities. To conclude this
discussion, it is important to emphasize that this construction is
completely general and that the subleading terms can be
adjusted to reproduce the desired scheme-dependent
contributions.
Besides the wave-function renormalization, it is also
necessary to remove the UV singularities associated to the vertex
corrections. The corresponding renormalization counterterm
in its unintegrated form is given by
(1) 2
A,uv = gs CF
G F (quv)
3
× γ ν q/uv (A0)q/uvγν − dA,uvμ2uv (A0) ,
(6.13)
where (A0) represents the tree-level vertex. Again, the term
proportional to μ2uv in the numerator is subleading in the UV
limit and its coefficient, dA,uv, must be adjusted in order to
implement the desired renormalization scheme [
105
].
6.4 Application example: e+e− → γ ∗ → qq¯ at NLO
In order to compute the NLO QCD corrections to e+e− →
γ ∗ → qq¯ , we start from the complete set of O(αs2) real and
virtual diagrams, including the self-energy ones. The total
unrenormalized virtual cross section is
1
σf(dvu) = 2s12
(0) (1)
d 1→2{2 Re Afdu|Afdu
Z2( p1) +
Z2( p2) Mf(d0)u},
(6.14)
where we distinguish contributions originated in the
triangle diagram from those related to self-energies. After that,
we must introduce the local UV counterterms which
implements the desired renormalization scheme and replace the
self-energy contributions by the wave-function
renormalization constants, Z 2ir. In this case, we apply LTD to Eq. (6.14)
and obtain a set of three dual contributions, σi(,vf)du.
Once the dual contributions are computed, we turn
attention to the real-emission terms. As explained in Sect. 6.2, we
need to isolate the different collinear singularities by
introducing a partition of the real phase space. This leads to
d 1→3 Mf(d0)u(qq¯ g)θ (y jr − yir )
(6.15)
which fulfills σ1(,rf)du +σ2(,rf)du = σf(dr)u. After that, we apply the
real-virtual mapping in each partition. This converts the real
terms into fully local IR counterterms for the dual
contributions; this guarantees a complete cancellation of IR
singularities at the integrand level, thus rendering the full expression
integrable in four dimensions. This is a really important fact,
because it allows one to put aside ds safely by directly
considering the limit → 0 at the integrand level [
103
]. Finally,
the master formula for computing the finite cross-section
correction is
− σ uv,
(6.16)
σi(,vf)du +
2
where σ uv is the dual representation of the local UV
counterterms and T is an operator that implements the unification
of dual coordinates at the integrand level (with the
corresponding Jacobians). If we add all the contributions at the
integrand level and deal with a single master integration, the
expression in Eq. (6.16) is directly implementable in four
space-time dimensions and leads to the correct result after
numerical computation. It is worth mentioning that, in order
to improve the numerical stability, it helps to compactify the
integration domain, applying a transformation as suggested
in Ref. [
105
].
6.5 Further considerations and comparison with other
schemes
As we depicted in the previous paragraphs, the fdu approach
is based on a fully local cancellation of IR and UV
singularities in strictly four dimensions. In this way, we avoid many of
the practical/conceptual problems related to the extension of
physical properties to d space-time dimensions. In particular,
the γ 5 issue is naturally absent here. Moreover, the idea of
using the mapped real contributions as local IR counterterms
for the dual part simplifies the treatment of IR divergences,
as well as it provides a better understanding of their origin.
On the other hand, the application of the traditional
renormalization procedure within this framework implies to
recompute the renormalization constants in an unintegrated
form (i.e. for the integrand-level implementation). In any
case, by fixing subleading terms in the UV expansion it is
possible to specify the finite part of the counterterms, thus
reproducing the results in any scheme (for instance, in MS).
Moreover, this algorithm is completely process-independent and,
in consequence, fully compatible with higher-order
computations. In this sense, the treatment of UV divergences is similar
to the procedure proposed within fdr. The main difference
is that we transform the local counterterms to the dual space,
in order to combine it with virtual amplitudes.
Besides this, it is worth mentioning that LTD can handle
loop amplitudes, as any other method described in this report,
but fdu is designed to work directly with physical
observables. For instance in Ref. [
119
], we applied our framework
to deal with the Higgs boson decay to massless gauge bosons,
which although known to be finite still requires a proper
regularization due to the fact that the amplitudes are UV singular
locally.
Finally, we would like to emphasize that fdu is compatible
with the desired requirements mentioned in the introduction.
In fact, since it is a four-dimensional approach which relies on
proper physically motivated changes of variables, fdu does
not alter the four-dimensional properties of the underlying
theory (i.e. unitarity, causality, and associated symmetries).
Moreover, it fulfills the crucial requirement of mathematical
consistency because singularities are completely removed by
a local mapping. In this way, all the singularities are canceled
before they manifest themselves in the integration.
7 Summary and outlook
The vast majority of higher-order calculations are done using
cdr. While there is no doubt that this made possible
impressive progress in perturbative calculations, there is a certain
danger that this success stifles the progress of other methods.
Whether such alternative methods will ever result in a viable
way to perform actual computations can only be established
by actually using them. In order to facilitate this, this article
provides an overview of recent (and not so recent)
developments of regularization schemes other than cdr. Some are
very close to cdr, for others the differences are much larger.
Using simple examples, we have illustrated the differences
and similarities of these methods and their relation to cdr.
Let us summarize the key points by means of the following
list.
FDH and DRED are perfectly consistent regularizations
schemes, at least up to NNLO. However, they require the
introduction of additional (evanescent) couplings with (in
general) different counterterms. In non-supersymmetric
theories, for dred this is already mandatory at NLO, for
fdh this is unavoidable only at NNLO and beyond.
Supersymmetry might protect the equivalence of the couplings
even beyond these approximations. Statements in the
literature that fdh is inconsistent always refer to ‘naive
fdh’, i.e. fdh without distinguishing the couplings.
Conversions between results in cdr, hv, fdh, and dred
can be made for individual parts contributing to a cross
section. For the virtual contributions this is known to
NNLO and can be elegantly described solely through
the scheme dependence of β functions and anomalous
dimensions. For real corrections and initial-state
factorization terms the explicit scheme dependence is only
known to NLO. These results have been used to explicitly
demonstrate the scheme independence of a cross section
at NLO.
FDF is an adaption of the (naive) fdh scheme that can be
used in strictly four dimensions. This enables the use of
unitarity methods, writing loop integrands as products of
tree-level amplitudes and performing numerical
calculations with the components of spinors and momenta. At
NLO, fdf gives results that are equivalent to fdh. How to
extend this beyond NLO is currently under investigation.
The scalars of fdf are not identical to the -scalars of
fdh.
GoSam makes use of fdf and other four-dimensional
techniques. The one-loop virtual amplitudes that are
called ‘dred’ and ‘cdr’ in GoSam correspond to what
we call ‘naive fdh’ and ’hv’, respectively, in this
article. Virtual one-loop amplitudes in other schemes are
obtained indirectly through conversion formulae.
SDF is based on the same idea as fdf. However, having
two-loop amplitudes in mind, the integer dimension is
set to de = 6. Hence, the spinor formalism has to be
extended to 6 dimensions.
UV singularities in IREG and FDR The basic idea of
ireg and fdr is similar and based on the observation that
UV singularities are independent of the kinematics. This
is used to isolate the UV singular part of loop integrals.
In ireg, the UV singular part is expressed in terms of
(implicit) integrals Ilog and boundary terms (which have
to be set to zero to respect gauge invariance), whereas in
fdr they are set to zero. The resulting UV-finite integrals
are evaluated in (strictly) four dimensions.
IR singularities in IREG and FDR are also treated in
strictly four dimensions. The matrix elements squared
are computed for massless particles (in four dimensions)
and the phase-space integration is also carried out in four
dimensions. IR singularities are regularized by
modifying the phase-space boundaries through a shift q → q +μ
and result in logarithms ln(μ0) = ln(μ2/s). In this sense
the method is similar to the introduction of a photon or
gluon mass. However, the procedures used by ireg and
fdr are superior as they preserve gauge invariance.
Differences between IREG and FDR In ireg, gauge
invariance is achieved by performing first the Dirac
algebra in the numerator and then cancel terms in the
numerator and denominator before the shift q → q +μ. In fdr,
the shift is done universally in the numerator and
denominator. Then additional terms with μ2 in the
numerator (called ‘extra integrals’) are included. ireg produces
expressions where the UV singularities are still present
in the form of implicit integrals Ilog. They have to be
removed by a suitable renormalization procedure, as in
ds. Applying fdr, on the other hand, results directly in
UV renormalized quantities.
Relation between IREG/FDR and dimensional schemes:
In ireg and fdr, ‘singularities’ related to real
contributions are encoded in powers of ln(μ0). At NLO, there is a
direct mapping between these terms and the 1/ n
singularities in the fdh scheme, namely 1/ 2 ↔ 1/2 ln2(μ0)
and 1/ ↔ ln(μ0). The extension to NNLO of such a
correspondence between the four-dimensional schemes
and the traditional dimensional schemes is under active
investigation. This also includes on how to compensate
for the absence of evanescent couplings in ireg and fdr.
FDU is an even more radical method in that it does
not split a cross section into (potentially IR divergent)
virtual and real parts. Rather, the combination of the
two parts (and thus the cancellation of IR singularities)
is done at the integrand level. Local counterterms are
used to perform ms renormalization. The extension to
initial-state singularities is also possible; the application
of a slightly modified momentum mapping allows one to
cancel the soft singularities. The remaining initial-state
collinear singularities can be canceled by adding
unintegrated initial-state counterterms. This is currently under
investigation.
Evanescent couplings are a fact of life! Even though they
can be avoided at NLO in some four-dimensional
formulations (like fdh) or do not show up in some particular
processes even at NNLO (like gg → gg in fdh), they
are present in all (partly) four-dimensional
regularizations of QED and QCD. In particular, they have an effect
at NNLO in fdh (like e.g. for gg → qq¯ ). The
connection of these effects to the ‘extra–extra integrals’ in fdr
is under investigation.
The list above illustrates that there are promising alternatives
available that at least at NLO are well understood. They can
and have been used for NLO calculations and in some cases
have proved to be more efficient.
Page 36 of 39
Currently, a huge effort in perturbative calculations is
made going beyond NLO towards automated computations
at NNLO. Many of the schemes above have been revisited
in the hope they provide a smoother road towards this goal.
We are convinced that this deserves to be investigated more
thoroughly. In any case, for an alternative scheme to be
consistent, there must – at least in principle – exist a well-defined
relation to cdr. At NNLO, these relations are fairly well
established for other traditional dimensional schemes like
hv, fdh, and dred. Regarding new formulations of
dimensional schemes like fdf or non-dimensional schemes such as
ireg and fdr, first steps towards establishing such relations
have been made. fdu has the advantage that a separate
regularization of the final-state IR singularities is not required, but
only the UV singularities have to be treated in a well-defined
way, such as ms.
Comparing to the impressive list of NNLO calculations
for physical cross sections that have been made using cdr, it
is fair to say, that none of the other methods has had a similar
impact so far. Since cdr is the best established scheme, it is
tempting to keep using it. However, it is not clear at all, if cdr
is really the most efficient scheme. Hence, the investigation
of other regularization schemes is an important aspect of
making further progress in perturbative computations. Are
there more efficient dimensional schemes? Or is it ultimately
advantageous to work completely in four dimensions?
That is the question.
Acknowledgements This article is a result of a Workstop/Thinkstart
that took place on 13–16 September 2016 at the Physik-Institut of
the University of Zürich (UZH). We gratefully acknowledge support
of UZH in general and its Physics Department in particular. A
special thank you to M. Röllin and C. Genovese for their help in
organizing the workshop. We thank S. C. Borowka for carefully reading
the manuscript and helpful comments. G. M. Pruna, Y. Ulrich and
A. Visconti are supported by the Swiss National Science Foundation
(SNF) under contracts 200021_160156 and 200021_163466,
respectively. W. J. Torres Bobadilla is supported by Fondazione Cassa di
Risparmio di Padova e Rovigo (CARIPARO). A. Cherchiglia is
supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior) - Brazil. B. Hiller acknowledges partial support from
the FCT (Portugal) project UID/FIS/04564/2016. M. Sampaio
acknowledges financial support from the Brazilian institutions CNPq (Conselho
Nacional de Desenvolvimento Cientfico e Tecnológico) and FAPEMIG
(Fundação de Amparo à Pesquisa do Estado de Minas Gerais). R.
Pittau was supported by the Research Executive Agency (REA) of the
European Union under the Grant Agreements PITN-GA2012-316704
(HiggsTools) and ERC-2011-AdG No 291377 (LHCtheory), and by
the MECD Proyects FPA2013-47836-C3-1-P and
FPA2016-78220-C33-P. F. Driencourt-Mangin, G. Rodrigo, G. Sborlini, and W. J.
Torres Bobadilla have been supported by CONICET Argentina, by the
Spanish Government, by ERDF funds from the European
Commission (Grants Nos. FPA2014-53631-C2-1-P and SEV-2014-0398) and
by Generalitat Valenciana (Grants Nos. PROMETEOII/2013/007 and
GRISOLIA/2015/035). G. Sborlini was supported in part by Fondazione
Cariplo under the Grant Number 2015-0761.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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