W. Hollik
Max-Planck-Institut für Physik
D-80805 Munich, GermanyA. Accomando, F. del Aguila, M. Awramik, A. Ballestrero
J. van der Bij, W. Beenakker, R. Bonciani, M. Czakon
G. Degrassi, A. Denner, K. Diener, S. Dittmaier, A. Ferroglia
J. Fleischer, A. Freitas, N. Glover, J. Gluza, T. Hahn
S. Heinemeyer, S. Jadach, F. Jegerlehner, W. Kilian
M. Krämer, J. Kühn, E. Maina, S. Moretti, T. Ohl
C.G. Papadopoulos, G. Passarino, R. Pittau, S. Pozzorini
M. Roth, T. Riemann, J.B. Tausk, S. Uccirati
A. Werthenbach, and G. Weiglein
NSCR Democritos Athens, CERN, Univ. Granada, Univ. Durham, DESY
Univ. Edinburgh, Univ. Freiburg, Univ. Karlsruhe, Univ. Katowice
HNINP Krakow, Max Planck Institut für Physik München, Univ. Nijmegen
Univ. Roma 3, Univ. Southampton, Univ. Torino, PSI Villigen, Univ. Würzburg
(Received October 18, 2004)
Work on electroweak precision calculations and event generators for
electroweak physics studies at current and future colliders is summarized.
PACS numbers: 12.15.Lk, 13.66.Jn, 14.70.Fm, 14.70.Hp
1. Introduction
Apart from the still missing Higgs boson, the Standard Model (SM)
has been impressively confirmed by successful collider experiments at the
particle accelerators LEP, SLC, and Tevatron during the last decade with
high precision, at the level of radiative corrections. Future colliders like
the upcoming LHC or an e
+
e
−
International Linear Collider (ILC), offer an
even greater physics potential, and in turn represent a great challenge for
∗
Presented at the final meeting of the European Network “Physics at Colliders”,
Montpellier, France, September 26–27, 2004.
(2533)
Page 2
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W. Hollik et al.
theory to provide even more precise calculations. Accurate predictions are
necessary not only to increase the level of precision of SM tests, but also to
study the indirect effects of possible new particles.
With increasing energies many new processes with heavy particle pro-
duction will be accessible. Such heavy particles are unstable, so that their
production leads to many-particle final states, with e.g. four or six fermions.
Predictions for such reactions should be based on full transition matrix el-
ements, improved by radiative corrections as far as possible, and call for
proper event generators.
Joint work done within the network is reviewed here; more details can
be found also in previous studies for future colliders [1–3].
2. Precision observables and multi-loop calculations
2.1. Muon lifetime and W–Z mass correlation
The precise measurement of the muon lifetime, or equivalently of the
Fermi constant G
µ
, sets an important constraint on the SM parameters,
G
µ
=
πα(0)
√2M
W
2
s
w
2
(1 + ∆r),
(1)
with s
w
2
= 1 − c
w
2
= 1 − M
W
2
/M
Z
2
, where the quantity ∆r comprises
the radiative corrections to muon decay (apart from the photonic correc-
tions in the Fermi model). Solving this relation for the W-boson mass M
W
yields a precise prediction for M
W
that can be compared with the directly
measured value. Recently the full electroweak two-loop calculation has been
completed, with the contributions from closed fermion loops [4, 5], from
bosonic loops involving virtual Higgs-bosons [4], and the complete bosonic
corrections [5, 6]. The two-loop fermionic contributions influence the M
W
prediction at the level of ∼ 50MeV, the two-loop bosonic corrections by
1−2MeV.
The predictions at the two-loop level have been further improved by
universal higher-order contributions to the ρ-parameter. The terms of the
order O(G
2
µ
m
4
t
α
s
) and O(G
3
µ
m
6
t
) have been obtained in [7] and were found
to change M
W
at the level of 5MeV and 0.5MeV, respectively. The leading
three-loop bosonic contribution to the ρ-parameter in the large Higgs mass
limit [8], yielding the power M
4
H
/M
4
W
, is opposite in sign to the leading
two-loop correction ∼ M
2
H
/M
2
W
. The two terms cancel each other for a
Higgs boson mass around 480 GeV. This interesting new result stabilizes the
perturbative expansion and makes a strongly interacting Higgs sector very
unlikely in view of the electroweak precision data.
The prediction for M
W
, including the above-mentioned two-loop and
leading three-loop effects, carries, besides the parametric uncertainty, an
Page 3
Electroweak Physics
2535
intrinsic theoretical uncertainty, which is estimated to be of 3−4MeV [9,10],
which has to be compared with the aimed precision of 7MeV in the M
W
determination at a future ILC [3].
2.2. Precision observables at the Z resonance
In order to describe the Z-boson resonance at LEP1 within satisfac-
tory precision it was possible to parametrize the cross section near the
resonance in such a way [11, 12] that a Born-like form with generalized
“effective” couplings is convoluted with QED structure functions model-
ing initial-state radiation (ISR). From these effective Z-boson–fermion cou-
plings so-called “pseudo-observables” were derived, such as various asymme-
tries, the hadronic Z-peak cross section, partial Z-decay widths, etc. The
precisely calculated pseudo-observables are implemented in the programs
Zfitter and Topaz0 [13]. A critical overview on high-precision physics at
the Z pole, in particular focusing on the theoretical uncertainties, can be
found in [14]. The status of precision pseudo-observables in the MSSM is
summarized in [15].
Following the formal tree-level like parametrization of the couplings, an
“effective weak mixing angle”, sin
2
θ
eff
f
, was derived for each fermion species.
Among these parameters the leptonic variable sin
2
θ
eff
l
plays a particularly
important role, since it is measured with the high accuracy of 1.7 × 10
−4
and is very sensitive to the Higgs-boson mass. Quite recently, a substantial
subclass of the electroweak two-loop contributions, the fermionic contribu-
tions to sin
2
θ
eff
l
, were calculated [16]; they reduce the intrinsic theoretical
uncertainty to ∼ 5 × 10
−5
.
Whether the pseudo-observable approach will also be sufficient for
Z-boson physics at the high-luminosity GigaZ option remains to be investi-
gated carefully. In any case, substantial theoretical progress will be needed
to match the aimed GigaZ precision on the theoretical side, e.g. for the ex-
pected experimental accuracy in sin
2
θ
eff
l
of about 1.3 × 10
−5
. A full control
of observables at the two-loop level, improved by leading higher-order effects,
seems indispensable.
An important entry for the precision observables with a large paramet-
ric uncertainty is the photonic vacuum polarization at the Z scale. The
hadronic part is determined via a dispersion relation from the cross sec-
tion for e
+
e
−
→ hadrons with experimental data as input in the low-energy
regime. Possible scans with the radiative-return method require a careful
theoretical treatment to reach the required precision [17].
Page 4
2536
W. Hollik et al.
2.3. Deep-inelastic neutrino scattering
An independent precise determination of the electroweak mixing angle
in terms of the M
W
/M
Z
ratio has been done in deep-inelastic neutrino scat-
tering off an isoscalar target by the NuTeV Collaboration [18], yielding a
deviation of about 3 standard deviations from the value derived from the
global analysis of the other precision observables. A new calculation of the
electroweak one-loop corrections was performed [19] to investigate the stabil-
ity and size of the quantum effects, showing that the theoretical error of the
analysis in [18] was obviously underestimated. The new theoretical result is
now being used for a re-analysis of the experimental data (see also [20] for
another recent recalculation of the electroweak radiative corrections).
2.4. At the 2-loop frontier
Although there are no complete next-to-next-to-leading (NNLO) pre-
dictions for 2 → 2 scattering reactions and 1 → 3 decays (with one truly
massive leg) available yet, significant progress was reached in this direction
in recent years.
Complete virtual two-loop amplitudes for (massless) Bhabha scattering
[21] and light-by-light scattering [23] have been worked out. Also first steps
have been made towards massive Bhabha scattering
In Ref. [22] the coefficient of the O(α
2
log(s/m
2
e
)) fixed-order contri-
bution to elastic large-angle Bhabha scattering is derived by adapting the
classification of infrared divergences that was recently developed within di-
mensional regularization, and applying it to the regularization scheme with
a massive photon and electron.
The subset of factorizable corrections, resulting from one-loop subrenor-
malization, is considered in [24]. This requires the evaluation of one-loop
diagrams in arbitrary dimension d = 4 − ε. The ε-expansion covering the
orders 1/ε and ε, in particular for the box graphs needed in Bhabha scat-
tering, was performed in [25], based on the work of [26]
1
. For the genuine
two-loop QED corrections to Bhabha scattering, the master integrals for the
box with the fermionic loop were calculated [28], and the cross section with
the corrections at two loop resulting from these diagrams [29].
A complete set of master integrals for massive two-loop Bhabha scatter-
ing has been derived in the meantime [30], and for form factors with mass-
less [31] and massive propagators [32]. Moreover, two-loop QCD corrections
for the electroweak forward-backward asymmetries were obtained [33]. Also
two-loop master integrals and form factors from virtual light fermions were
derived and applied to Higgs boson production and decays [34].
1
Techniques applied in the latter work have also been used in [27] for the analytic
calculation of Higgs boson decay into two photons.
Page 5
Electroweak Physics
2537
2.5. Electroweak radiative corrections at high energies
Electroweak corrections far above the electroweak scale, e.g. in the TeV
range, are dominated by soft and collinear gauge-boson exchange, leading to
corrections of the form α
N
log
M
(s/M
W
2
) with M ≤ 2N. The leading terms
(M = 2N) are called Sudakov logarithms. At the one-loop (N = 1) and
two-loop (N = 2) level the leading and subleading corrections to a 2 → 2
process at √s ∼ 1TeV typically amount to [35]
δ
1−loop
LL
∼ −
α
πs
w
2
log
2
( s
M
W
2
) ≃ −26%,
δ
1−loop
NLL
∼ +
3α
πs
w
2
log (
s
M
W
2
) ≃ 16%,
δ
2−loop
LL
∼ +
α
2
2π
2
s
w
4
log
4
( s
M
W
2
) ≃ 3.5%,
δ
2−loop
NLL
∼ −
3α
2
π
2
s
w
4
log
3
( s
M
W
2
) ≃ −4.2%,
(2)
revealing that these corrections become significant in the high-energy phase
of a future ILC. In contrast to QED and QCD, where the Sudakov logarithms
cancel in the sum of virtual and real corrections, these terms need not com-
pensate in the electroweak SM for two reasons. The weak charges of quarks,
leptons, and electroweak gauge bosons are open, not confined, i.e. there is
(in contrast to QCD) no need to average or to sum over gauge multiplets in
the initial or final states of processes. Even for final states that are inclusive
with respect to the weak charges Sudakov logarithms do not completely can-
cel owing to the definite weak charges in the initial state [36]. Moreover, the
large W- and Z-boson masses make an experimental discrimination of real
W- or Z-boson production possible, in contrast to unobservable soft-photon
or gluon emission.
In recent years several calculations of these high-energy logarithms have
been carried out in the Sudakov regime, where all kinematical invariants
(p
i
p
j
) of different particle momenta p
i
, p
j
are much larger than all particle
masses. A complete analysis of all leading and subleading logarithms at the
one-loop level can be found in [37]. Diagrammatic calculations of the leading
two-loop Sudakov logarithms have been carried out in [35,38]. Diagrammatic
results on the so-called “angular-dependent” subleading logarithms have been
presented in [35]. All these explicit results are compatible with proposed
resummations [39,40] that are based on a symmetric SU(2)×U(1) theory at
high energies matched with QED at the electroweak scale. In this ansatz,
improved matrix elements M result from lowest-order matrix elements M
0
Page 6
2538
W. Hollik et al.
upon dressing them with (operator-valued) exponentials,
M ∼ M
0
⊗ exp(δ
ew
) ⊗ exp(δ
em
) .
(3)
Explicit expressions for the electroweak and electromagnetic corrections δ
ew
and δ
em
, which do not commute with each other, can, for instance, be found
in [35]. For 2 → 2 neutral-current processes of four massless fermions,
also subsubleading logarithmic corrections have been derived and resummed
[40] using an infrared evolution equation that follows the pattern of QCD.
Applications to vector-boson pair production in proton–proton collisions can
be found in [41].
In supersymmetric models the form of radiative corrections at high en-
ergies has also been worked out for a broad class of processes [42]. Based on
one-loop results their exponentiation has been proposed.
2.6. Higher-order initial-state radiation
Photon radiation off initial-state electrons and positrons leads to large
radiative corrections of the form α
N
log
N
(m
2
e
/s). These logarithmic correc-
tions are universal and governed by the DGLAP evolution equations. The
solution of these equations for the electron-photon system yields the struc-
ture functions, generically denoted by Γ(x) below, which can be used via
convolution to improve hard scattering cross sectionsσ(p
e
+
,p
e
−
) by photon
emission effects,
σ(p
e
+
,p
e
−
) =
1
∫
0
dx
+
Γ(x
+
)
1
∫
0
dx
−
Γ(x
−
)
׈σ(x
+
p
e
+
,x
−
p
e
−
).
(4)
While the soft-photon part of the structure functions (x → 1) can be re-
summed, resulting in an exponential form, the contributions of hard pho-
tons have to be calculated order by order in perturbation theory. In [43]
the structure functions are summarized up to O(α
3
). Ref. [44] describes a
calculation of the (non-singlet) contributions up to O(α
5
) and of the small-x
terms [α log
2
(x)]
N
to all orders (for previous calculations see papers cited in
Ref. [44]).
Page 7
Electroweak Physics
2539
3. Radiative corrections to 2
→
3
,
4
, . . . processes
3.1. Four-fermion final states and W-pair production
Four-fermion final states in e
+
e
−
collisions, which involve electroweak
boson pair production, are of special interest since they allow the mechanism
of spontaneous symmetry breaking and the non-Abelian structure of the
Standard Model to be directly tested by the experiments. Moreover they
provide a very important background to most searches for new physics.
LEP2 has provided in this respect an ideal testing ground for the SM. The
W profile has been measured with great accuracy, new bounds on anomalous
trilinear gauge-boson couplings have been set, and single W, single Z, ZZ
and Zγ cross sections have been determined for the first time. These studies
will be continued with much higher statistics and energy at a future e
+
e
−
Linear Collider.
In this context, the Monte Carlo four fermion generator WPHACT 2.0 has
been completed [45], adapted to experimental requests and used for simu-
lation of the LEP2 data [46]. WPHACT 2.0 computes all Standard Model
processes with four fermions in the final state at e
+
e
−
colliders, it makes
use of complete, fully massive helicity amplitude calculations and includes
the Imaginary Fermion Loop gauge restoring scheme
2
. Thanks to these
features and new phase space mappings, WPHACT has been extended to all
regions of phase space, including kinematical configurations dominated by
small momentum transfer and small invariant masses like single W, single Z,
Zγ
∗
and γ
∗
γ
∗
processes. Special attention has been devoted to QED effects,
which have a large numerical impact, with new options for the description of
Initial State Radiation (ISR) and of the scale dependence of the electromag-
netic coupling. Moreover, there is the possibility of including CKM mixing,
and to account for resonances in q¯q channels.
The theoretical treatment and the presently gained level in accuracy in
the description of W-pair-mediated 4f production were triggered by LEP2,
as it is reviewed in Refs. [43,48]. The W bosons are treated as resonances
in the full 4f processes, e
+
e
−
→ 4f (+γ). Radiative corrections are split
into universal and non-universal corrections. The former comprise leading-
logarithmic corrections from ISR, higher-order corrections included by using
appropriate effective couplings, and the Coulomb singularity. These cor-
rections can be combined with the full lowest-order matrix elements easily.
The remaining corrections are called non-universal, since they depend on
the process under investigation. For LEP2 accuracy, it was sufficient to in-
clude these corrections by the leading term of an expansion about the two
W poles, defining the so-called double-pole approximation (DPA). Different
2
In [47] the incorporation of the fermion-loop corrections into e
+
e
−
→ n
fermions is
discussed in terms of an effective Lagrangian approach.
Page 8
2540
W. Hollik et al.
versions of such a DPA have been used in the literature [49–51]. Although
several Monte Carlo programs exist that include universal corrections, only
two event generators, YFSWW [50] and RacoonWW [51,52], include non-
universal corrections, as well as the option of anomalous gauge couplings [53].
In the DPA approach, the W-pair cross section can be predicted within
a relative accuracy of ∼ 0.5%(0.7%) in the energy range between 180GeV
(170GeV) and 500GeV, which was sufficient for the LEP2 accuracy of ∼ 1%
for energies 170−209GeV. At threshold (
√s <
∼
170GeV), the present state-
of-the-art prediction results from an improved Born approximation based
on leading universal corrections only, because the DPA is not reliable there,
and thus possesses an intrinsic uncertainty of about 2%, which demonstrates
the necessary theoretical improvements for the threshold region. At energies
beyond 500GeV, effects beyond O(α), such as the above-mentioned Sudakov
logarithms at higher orders, become important and have to be included in
predictions at per-cent accuracy.
At LEP2, the W-boson mass has been determined by the reconstruc-
tion of W bosons from their decay products with a final accuracy of about
30MeV. In [54] the theoretical uncertainty is estimated to be of the order
of ∼ 5MeV. Theoretical improvements are, thus, desirable.
The above discussion illustrates the necessity of a full one-loop calcu-
lation for the e
+
e
−
→ 4f process and of further improvements by leading
higher-order corrections.
3.2. Single-W production
The single-W production process e
+
e
−
→ eν
e
W → eν
e
+2f plays a par-
ticularly important role among the 4f production processes at high scatter-
ing energies. The process is predominantly initiated by eγ
∗
collision, where
the photon is radiated off the electron (or positron) by the Weizsäcker–
Williams mechanism, i.e. with a very small off-shellness q
2
γ
.
Consequently the cross section rises logarithmically with the scattering
energy and is of the same size as the W-pair production cross section around
√s = 500GeV; for higher energies single-W dominates over W-pair produc-
tion.
Theoretically the dominance of photon exchange at low q
2
γ
poses several
complications. Technically, q
2
γ
→ 0 means that the electrons (or positrons)
are produced in the forward direction and that the electron mass has to be
taken into account in order to describe the cross section there. Moreover,
the mere application of s-dependent leading-logarithmic structure functions
does not describe the leading photon-radiation effects properly, since ISR
and final-state radiation (FSR) show sizable interferences for forward scat-
tering. Thus, the improvement of lowest-order calculations by leading radi-
Page 9
Electroweak Physics
2541
ation effects is more complicated than for s-channel-like processes. Finally,
the running of the electromagnetic coupling α(q
2
γ
) has to be evaluated in the
region of small momentum transfer (q
2
γ
< 0) where the fit for the hadronic
part of the photon vacuum polarisation [55] should be used.
The Monte Carlo generator KoralW [56] has recently been updated to
include the ISR-FSR interference effects as well as the proper running of
α(q
2
γ
). Therefore, this program now has reached a level of accuracy similar
to the other state-of-the-art programs for single-W production: Grc4f [57],
Nextcalibur [58], Swap [59], Wphact [45,60], and Wto [61]. It should
be kept in mind that none of these calculations includes non-universal elec-
troweak corrections, leading to a theoretical uncertainty of about ∼ 5% in
cross-section predictions. Although the final solution for a high-energy Lin-
ear Collider certainly requires a full O(α) calculation of the 4f-production
process, a first step of improvement could be done by a careful expansion
about the propagator poles of the photon and W boson. The electroweak
O(α) corrections to the process eγ → ν
e
W, which are known [62], represent
a basic building block in this calculation.
3.3. Progress for multi-particle production processes
One-loop integrals become more and more cumbersome if the number N
of external legs in diagrams increases. For N > 4, however, not all exter-
nal momenta are linearly independent because of the four-dimensionality of
space-time. As known for a long time [63], this fact opens the possibility to
relate integrals with N > 4 to integrals with N ≤ 4. In recent years, various
techniques for actual evaluations of one-loop integrals with N = 5,6 have
been worked out [64,65] (see also references therein for older methods and
results). The major complication in the treatment of 2 → 3 processes at one
loop concerns the numerical evaluation of tensor 5-point integrals; in partic-
ular, the occurrence of inverse Gram determinants in the usual Passarino–
Veltman reduction to scalar integrals leads to numerical instabilities at the
phase-space boundary. A possible solution to this problem was worked out
in Ref. [65] where the known direct reduction [63] of scalar 5-point to 4-point
integrals was generalized to tensor integrals, thereby avoiding the occurrence
of leading Gram determinants completely. More work on one-loop N-point
integrals can be found in [66].
In the evaluation of real corrections, such as bremsstrahlung, a proper
and numerically stable separation of infrared (soft and collinear) divergences
represents one of the main problems. In the phase-space slicing approach
(see [67] and references therein) the singular regions are excluded from the
“regular” phase-space integration by small cuts on energies, angles, or invari-
ant masses. Using factorization properties, the integration over the singular
Page 10
2542
W. Hollik et al.
regions can be done in the limit of infinitesimally small cut parameters. The
necessary fine-tuning of cut parameters is avoided in so-called subtraction
methods (see [68–70] and references therein), where a specially tuned aux-
iliary function is subtracted from the singular integrand in such a way that
the resulting integral is regular. The auxiliary function has to be chosen
simple enough, so that the singular regions can be integrated over analyt-
ically. In [68] the so-called “dipole subtraction approach” has been worked
out for massless QCD. and subsequently extended for photon emission off
massive fermions [69] and for QCD with massive quarks [70].
Applications were preformed for complete one-loop calculations of elec-
troweak radiative corrections for specific 2 → 3 processes of special in-
terest for a future ILC, e
+
e
−
→ ν¯νH [71, 72] and e
+
e
−
→ t
¯
tH [73–75].
In [72, 73, 75] the technique [65] for treating tensor 5-point integrals was
employed. While [71, 73, 74] make use of the slicing approach for treat-
ing soft-photon emission, the results of Refs. [72,75] have been obtained by
dipole subtraction and checked by phase-space slicing for soft and collinear
bremsstrahlung. Analytic results for the one-loop corrections are provided
in [76].
The Yukawa coupling of the top quark could be measured at a future ILC
with high energy and luminosity at the level of ∼ 5% [2] by analyzing the
process e
+
e
−
→ t
¯
tH. A thorough prediction for this process, thus, has to
control QCD and electroweak corrections. Results on the electroweak O(α)
corrections of Refs. [74,75] show agreement within ∼ 0.1%. The results of
the previous calculation [73] roughly agree with the ones of Refs. [74,75] at
intermediate values of √s and M
H
, but are at variance at high energies (TeV
range) and close to threshold (large M
H
).
4. Event generators for multi-particle final states
4.1. Multi-purpose generators at parton level
The large variety of different final states for multi-particle production
renders multi-purpose Monte Carlo event generators rather important, i.e.
generators that deliver an event generator for a user-specified (as much as
possible) general final state based on full lowest-order amplitudes. As results,
these tools yield lowest-order predictions for observables, or more generally
Monte Carlos samples of events, that are improved by universal radiative
corrections, such as initial-state radiation at the leading-logarithmic level
or beamstrahlung effects. Most of the multi-purpose generators are also
interfaced to parton-shower and hadronization programs. The generality
renders these programs, however, rather complex devices and, at present,
they are far from representing tools for high-precision physics, because non-
universal radiative corrections are not taken into account in predictions.
Page 11
Electroweak Physics
2543
The following multi-purpose generators for multi-parton production, in-
cluding program packages for the matrix-element evaluation, are available:
• Amegic [77]: Helicity amplitudes are automatically generated by the
program for the SM, the MSSM, and some new-physics models. Vari-
ous interfaces (ISR, PDFs, beam spectra, Isajet, etc.) are supported.
The phase-space generation was successfully tested for up to six par-
ticles in the final state.
• Grace [78]: The amplitudes are delivered by a built-in package, which
can also handle SUSY processes. The phase-space integration is done
by BASES [79]. Tree-level calculations have been performed for up
to (selected) six-fermion final states. The extension of the system to
include one-loop corrections is the Grace-Loop [80] program.
• Madevent [81] + Madgraph [82]: The Madgraph algorithm can
generate tree-level matrix elements for any SM process (fully support-
ing particle masses), but a practical limitation is 9,999 diagrams. In
addition, Madgraph creates Madevent, an event generator for the
requested process.
• Phegas [83] + Helac [84]: The Helac program delivers amplitudes
for all SM processes (including all masses). The phase-space integra-
tion done by Phegas has been tested for selected final states with
up to seven particles. Recent applications concern channels with six-
fermion final states [85].
• Whizard [86] + Comphep [87] / Madgraph [82] / O’mega [88]:
Matrix elements are generated by an automatic interface to (older
versions of) Comphep, Madgraph, and (the up-to-date version of)
O’mega. Phase-space generation has been tested for most 2 → 6
and some 2 → 8 processes; unweighted events are supported, and a
large variety of interfaces (ISR, beamstrahlung, Pythia, PDFs, etc.)
exists. The inclusion of MSSM amplitudes (O’mega) and improved
phase-space generation (2 → 6) are work in progress.
• Alpgen [89] is a specific code for computing the perturbative part of
observables in high energy hadron-hadron collisions, which require a
convolution of the perturbative quantities with structure or fragmen-
tation functions that account for non perturbative effects.
Tuned comparisons of different generators, both at parton and detector level,
are extremely important, but become more and more laborious owing to the
large variety of multi-particle final states. Some progress to a facilitation
and automization of comparisons are made by MC-tester project [90] and
Java interfaces [91].
Page 12
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W. Hollik et al.
4.2. Event generators and results for e
+
e
−
→ 6f
Particular progress was reached in recent years in the description of
six-fermion production processes. Apart from the multi-purpose genera-
tors listed in the previous section, also dedicated Monte Carlo programs and
generators have been developed for this class of processes:
• Sixfap [92]: Matrix elements are provided for all 6f final states (with
finite fermion masses), including all electroweak diagrams. The gen-
eralization to QCD diagrams and the extension of the phase-space
integration for all final states is in progress.
• eett6f [93]: Only processes relevant for t
¯
t production are supported
(a new version includes e
±
in the final state and QCD diagrams); finite
fermion masses are possible.
• Lusifer [94]: All 6f final states are possible, including QCD diagrams
with up to four quarks; representative results for all these final states
have been presented. External fermions are massless. An unweighting
algorithm and an interface to Pythia are available.
• Phase [95]: All Standard Model process with six fermions in the final
state at the LHC. It employs the full set of tree-level Feynman dia-
grams, taking into account a finite mass for the b quark. An interface
to hadronization is provided.
A comparison of results [96] for some processes e
+
e
−
→ 6f relevant for t
¯
t
production for massless external fermions reveals good agreement between
the various programs, where minor differences are presumably due to the dif-
ferent treatments of the bottom-quark Yukawa coupling, which is neglected
in some cases.
A tuned comparison of results obtained with Lusifer and Whizard for
a large survey of 6f final states has been presented in Ref. [94].
5. Other developments
5.1. Automatization of loop calculations
Once the necessary techniques and theoretical subtleties of a perturbative
calculation are settled, to carry out the actual calculation is an algorithmic
matter. Thus, an automization of such calculations is highly desirable, in
order to facilitate related calculations. Various program packages have been
presented in the literature for automatized tree-level, one-loop, and multi-
loop calculations. A comprehensive overview can, e.g., be found in [97]; in
the following we have to restrict ourselves to a selection of topics, where
emphasis is put on electroweak aspects.
Page 13
Electroweak Physics
2545
The generation of Feynman graphs and amplitudes is a combinatorial
problem that can be attacked with computer algebra. The program pack-
ages FeynArts [98] (which has been extended in [99] for the MSSM) and
Diana [100] (based on Qgraf [101]) are specifically designed for this task;
also the Grace-Loop [80] system automatically generates Feynman dia-
grams and loop amplitudes. Moreover, the task of calculating virtual one-
loop and the corresponding real-emission corrections to 2 → 2 scattering
reactions is by now well understood. Such calculations are widely auto-
mated in the packages FormCalc combined with LoopTools [102], and
Grace-Loop [80].
An illustrating example was provided for the differential cross section
for e
+
e
−
→ t
¯
t in lowest order as well as including electroweak O(α) cor-
rections. A program FA+FC [103] was obtained from the output of the
FeynArts and FormCalc packages and makes use of the LoopTools
library for the numerical evaluation. Another program, Topfit [103,104],
was developed from an algebraic reduction of Feynman graphs (delivered
from Diana) within Form; for the numerics LoopTools is partially em-
ployed. A completely independent approach has been made by the Sanc
project [105]. The Sanc program contains another independent calculation
of the O(α) corrections to e
+
e
−
→ t
¯
t, the results of which are also included
in [103]. More details on comparisons, including also other fermion flavors,
can be found in [103,106]. The agreement between the numerical results at
the level of 10 digits reflects the enormous progress achieved in recent years
in the automatization of one-loop calculations.
The one-loop calculation for the process e
+
e
−
→ t
¯
t(γ) including hard
bremsstrahlung was originally performed in [104] without full automatiza-
tion; it was repeated in later course (apart from the hard bremsstrahlung
part) as an example for automatization in [107] . The extension of Di-
ana towards full automatization in terms of the package aıTalc is a new
development [107,108]; automatization of the full calculation is performed
including renormalization and the creation and running of a FORTRAN
code. Applications to the calculation of the processes e
+
e
−
→ f
¯
f(γ) for
various final fermions: t,b,µ,e and also s
¯
b,c
¯
t,µ¯τ are performed. For further
work in automatization see [109,110].
5.2. Numerical approaches to loop calculations
Most of the various techniques of performing loop calculations share the
common feature that the integration over the loop momenta is performed
analytically. This procedure leads to complications at one loop if five or more
external legs are involved, since both speed and stability of programs become
more and more jeopardized. At the two-loop level, already the evaluation of
Page 14
2546
W. Hollik et al.
self-energy and vertex corrections can lead to extremely complicated higher
transcendental functions that are hard to evaluate numerically.
An idea to avoid these complications is provided by a more or less purely
numerical evaluation of loop corrections. There are two main difficulties in
this approach. Firstly, the appearing ultraviolet and infrared divergences
have to be treated and canceled carefully. Secondly, even finite loop integrals
require a singularity handling of the integrand near particle poles, where
Feynman’s iϵ prescription is used as regularization.
In [111] a method for a purely numerical evaluation of loop integrals
is proposed. Each integral is parametrized with Feynman parameters and
subsequently rewritten with partial integrations. The final expression con-
sists of a quite simple part containing the singular terms and another more
complicated looking part that can be integrated numerically. The actual
application of the method to a physical process is still work in progress.
There are five papers in a series devoted to the numerical evaluation of
multi-loop, multi-leg Feynman diagrams. In [111] the general strategy is
outlined and in [112] a complete list of results is given for two-loop functions
with two external legs, including their infrared divergent on-shell derivatives.
Results for one-loop multi-leg diagrams are shown in [113] and additional
material can be found in [114]. Two-loop three-point functions for infrared
convergent configurations are considered in [115], where numerical results
can be found.
Ref. [113] presents a detailed investigation of the algorithms, based on
the Bernstein–Tkachov (BT) theorem [116], which form the basis for a fast
and reliable numerical integration of one-loop multi-leg (up to six in this
paper) diagrams. The rationale for this work is represented by the need
of encompassing a number of problems that one encounters in assembling
a calculation of some complicated process, e.g. full one-loop corrections to
e
+
e
−
→ 4fermions. Furthermore, in any attempt to compute physical ob-
servables at the two-loop level, we will have to include the one-loop part, and
it is rather obvious that the two pieces should be treated on equal footing.
All algorithms that aim to compute Feynman diagrams numerically are
based on some manipulation of the original integrands that brings the final
answer into something smooth. This has the consequence of bringing the
original (Landau) singularity of the diagram into some overall denominator
and, usually, the method overestimates the singular behavior around some
threshold. In these regions an alternative derivation is needed. Instead of
using the method of asymptotic expansions, a novel algorithm is introduced
based on a Mellin–Barnes decomposition of the singular integrand, followed
by a sector decomposition that allows one to write the Laurent expansion
around threshold.
Page 15
Electroweak Physics
2547
Particular care has been devoted to analyze those situations where a
sub-leading singularity may occur, and to properly account for those cases
where the algorithm cannot be applied because the corresponding BT factor
is zero although the singular point in parametric space does not belong to
the integration domain.
Finally, a description of infrared divergent one-loop virtual configura-
tions is given in the framework of dimensional regularization: here both the
residue of the infrared pole and the infrared finite remainder are cast into
a form that can be safely computed numerically. The collection of formu-
las that cover all corners of phase space have been translated into a set of
FORM codes and the output has been used to create a FORTRAN code
whose technical description will be given elsewhere.
Ref. [117] addresses the problem of deriving a judicious and efficient way
to deal with tensor Feynman integrals, namely those integrals that occur in
any field theory with spin and non trivial structures for the numerators of
Feynman propagators. This paper forms a basis for a realistic calculation of
physical observables at the two-loop level.
The complexity of handling two-loop tensor integrals is reflected in the
following simple consideration: the complete treatment of one-loop tensor in-
tegrals was confined to the appendices of [118], while the reduction of general
two-loop self-energies to standard scalar integrals already required a consid-
erable fraction of [119]; the inclusion of two-loop vertices requires the whole
content of this paper. The past experience in the field has shown the encom-
passed convenience of gathering in one single place the complete collection
of results needed for a broad spectrum of applications. In recent years, the
most popular and quite successful tool in dealing with multi-loop Feynman
diagrams in QED/QCD (or in selected problems in different models, charac-
terized by a very small number of scales), has been the Integration-By-Parts
Identities method [120]. However, reduction to a set of master integrals is
poorly known in the enlarged scenario of multi-scale electroweak physics.
In [121] another new method is presented in which almost all the work
can be performed numerically: the tensor integrals are numerically reduced
to the standard set of one-loop scalar functions and any amplitude is calcu-
lated simply contracting the numerically computed tensor integrals with the
external tensors. To this aim, a recursion relation is introduced that links
high-rank tensor integrals to lower-rank ones. Singular kinematical configu-
rations give a smoother behavior than in other approaches because, at each
level of iteration, only inverse square roots of Gram determinants appear.
Page 16
2548
W. Hollik et al.
6. Electroweak effects in hadronic processes
In Refs. [122–126] (see also [127]) it was proved the importance of elec-
troweak one-loop corrections to hadronic observables, such as b
¯
b, ‘prompt
photon + jet’ and ‘Z + jet’ production at Tevatron and LHC and jet and b
¯
b
production in linear colliders, which can compete in size with QCD correc-
tions. Their inclusion in experimental analyses is thus important, especially
in view of searches for new physics. In case of ‘Z + jet’ production they can
rise to O(15–20%) at large transverse momentum at the LHC, while being
typically half the size in case of ‘photon + jet’ production. As these two
channels are the contributors to the Drell-Yan process, and since the latter
is envisaged to be used as one of the means to measure the LHC luminosity,
it is clear that neglecting them in experimental analyses would spoil the
luminosity measurements.
Ref. [128] emphasised the importance of NLO electroweak effects in
three-jet production in e
+
e
−
scattering at the Z-pole (SLC, LEP and GigaZ),
showing typical corrections of O(2–4%) (e.g. in jet-rates and thrust), compa-
rable to the SLC and LEP experimental accuracy and certainly larger than
the one expected at GigaZ. They also introduce sizable parity-violating ef-
fects into the fully differential structure of three-jet events in presence of
beam polarisation, which are of relevance as background to new physics
searches in SLC and GigaZ data.
The complete set of electroweak O(α) corrections to the Drell–Yan-like
production of Z and W bosons have been studied in [129–131]. These cor-
rections are phenomenologically relevant both at the Tevatron and the LHC.
It is shown that the pole expansion yields a good description of resonance
observables, but it is not sufficient for the high-energy tail of transverse-
momentum distributions, relevant for new-physics searches. Horace and
Winhac are Monte Carlo event generators [132], developed for single W
production and decay, which in their current versions include higher-order
QED corrections in leptonic W decays, a crucial entry for precision deter-
mination of the W mass and width at hadron colliders.
Production of vector-boson pairs is an important probe for potential non-
standard anomalous gauge couplings. In order to identify possible deviations
from the SM predictions, an accurate knowledge of the electroweak higher-
order contributions is mandatory as well, in particular for large transverse
momenta. A complete electroweak one-loop calculation was performed for
γZ production [133]; for other processes like γW,... the large logarithms in
the Sudakov regime were derived [41].
A further aspect that should be recalled is that weak corrections nat-
urally introduce parity-violating effects in observables, detectable through
asymmetries in the cross-section. These effects are further enhanced if polar-
Page 17
Electroweak Physics
2549
isation of the incoming beams is exploited, such as at RHIC-Spin [134,135]
and will be used to measure polarised structure functions.
7. Conclusions
During the recent years there has been continuous progress in the devel-
opment of new techniques and in making precise predictions for electroweak
physics at future colliders. However, to be prepared for the LHC and a future
e
+
e
−
linear collider with high energy and luminosity, an enormous amount of
work is still ahead of us. Not only technical challenges, also field-theoretical
issues such as renormalization, the treatment of unstable particles, etc., de-
mand a higher level of understanding. Both loop calculations as well as the
descriptions of multi-particle production processes with the help of Monte
Carlo techniques require and will profit from further improving computing
devices. It is certainly out of question that the list of challenges and in-
teresting issues could be continued at will. Electroweak physics at future
colliders will be a highly exciting issue.
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