Single-W Production and Fermion-Loop Scheme: Numerical Results

27 Pages·0.35 MB·English

by  G. Passarino

W
Single- Production and Fermion-Loop Scheme:
Numerical Results
Giampiero Passarino
0
Dipartimento di Fisica Teorica, Universit`a di Torino, Italy
0
0 INFN, Sezione di Torino, Italy
2

n
a
J

0
2 Abstract

1 The single-W production mechanism is synonymous to the e+e− annihilation into eν and
e
v
a W boson with the outgoing electron lost in a small cone around the beam direction. It
2
1 requires a Renormalization Scheme that preserves gauge invariance and fermion masses
2
cannot be neglected in the calculation. A recently proposed generalization of the so-called
1
0 Fermion-Loop scheme is applied to the evaluation of observables at LEP 2 energies. The
0 total contribution to single-W processes can be split, in a gauge invariant manner, into
0
a s-channel component and a t-channel one. The latter is dominated by a regime of low
/
h momentumtransferoftheoutgoingelectronandanyhigh-energyRenormalization Scheme,
p
as the G -one, fails to give the correct description of the scale. The Fermion-Loop scheme
- F
p automatically converts, among other things, all couplings of the theory into couplings that
e
h are running at the appropriate scale. Therefore, in addition to represent the only scheme
: fully justified on a field-theoretical basis, the Fermion-Loop is the best starting point to
v
i include radiative corrections into single-W production. Numerical results are presented,
X
showing a decrease in the predictions that can be sizeable. There is no naive and overall
r
a rescaling of α in any pragmatic scheme, as the Fixed-Width one, that can reproduce
QED
the Fermion-Loop results, at the requested accuracy, for all configurations and for all
kinematical cuts.
Pacs: 11.15.-q, 13.10.-q, 13.38.-b, 13.40.-f, 14.70.Fm
1
1 Introduction
An interesting process at LEP 2 is the so-called single-W production, e+e− Weν which can be
→
seen as a part of the CC20 process, e+e− qq(µν , τ ν )eν , or as a part of the Mix56 process,
µ τ e
→
e+e− e+e−ν ν . For a theoretical review we refer to [1] and to [2]. For the experimental aspects
e e
→
we refer to the work of Ref. [3].
TheCC20processisusuallyconsideredintworegimes, cosθ(e−) corLACC20and cosθ(e−)
| | ≥ | | ≤
c or SACC20. Strictly speaking the single W production is defined by those events that satisfy
cosθ(e−) 0.997 and, therefore is a SACC20.
| | ≥
The LACC20 cross-section has been computed by many authors and references can be found in
[4]. It represents a contribution to the e+e− W+W− total cross-section, in turn used to derive a
→
valuefor M , theW boson mass. Thispointdeserves acomment: by e+e− W+W− itis meant the
W →
ideal cross-section obtained with the three double-resonant CC03 diagrams, see Fig. 4, and therefore
the background, i.e. the full cross-section minus the CC03 one, is evaluated with the help of some
MonteCarlo, estimatingtheerroronthesubtractionbycomparingwithsomeotherMonteCarlo. Then
M is derived from a fit to σ(CC03) with the help of a third calculation. From a theoretical point of
W
view the evaluation of LACC20 is free of ambiguity, even in the approximation of massless fermions,
as long as a gauge-preserving scheme is applied and θ(e−) is not too small.
For SACC20 instead, one cannot employ the massless approximation anymore and this fact makes
the calculation much moredifficult. In other words, in addition to double-resonant W-pair production
with one W decaying into eν , there are t-channel diagrams that give a sizeable contribution for small
e
values of the polar scattering angle of the t-channel electron.
Single-W processes are sensitive to the breaking of U(1) gauge invariance in the collinear limit,
as described in Ref. [5] (see also [6]). The correct way of handling them is based on the so-called
Fermion-Loop (FL)scheme [7], the gauge-invariant treatment of the finite-width effects of W and Z
bosons in LEP2 processes. However, till very recently, the Fermion-Loop scheme was available only
for the LACC20 regime.
For e+e− e−ν f f , the U(1) gauge invariance becomes essential in the region of phase space
→ e 1 2
where the angle between the incoming and outgoing electrons is small, see the work of [5] and also an
alternative formulationsin[8]. Inthislimitthesuperficial1/Q4 divergenceofthepropagatorstructure
is reduced to 1/Q2 by U(1) gauge invariance. In the presence of light fermion masses this gives raise
to the familiar ln(m2/s) large logarithms.
e
Furthermore, keeping a finite electron mass through the calculation is not enough. One of the
main results of [1] was to show that there are remaining subtleties in CC20, associated with the zero
mass limit for the remaining fermions.
In Ref. [9] (hereafter I) a generalization of the Fermion-Loop scheme has been given (for previous
work we refer to [10]). It consists of the re-summation of the fermionic one-loop corrections to the
vector-boson propagators and of the inclusion of all remaining fermionic one-loop corrections, in
particular those to the Yang–Mills vertices. In the original formulation, the Fermion-Loop scheme
requires that vector bosons couple to conserved currents, i.e., that the masses of all external fermions
can be neglected. There are several examples where fermion masses must be kept to obtain a reliable
prediction. As already stated, this is the case for the single-W production mechanism, where the
outgoing electron is collinear, within a small cone, with the incoming electron. Therefore, m cannot
e
be neglected.
2
Furthermore, among the 20 Feynman diagrams that contribute (for eν ud final states, up to 56
e
for e+e−ν ν ) there are multi-peripheral ones that require a non-vanishing mass also for the other
e e
outgoing fermions. In I a generalization of the Fermion-Loop scheme is introduced to account for
external, non-conserved, currents. Dyson re-summed transitions are introduced without neglecting
the p p -terms and including the contributions from the Higgs-Kibble ghosts in the ’t Hooft-Feynman
µ ν
gauge. In I we have introduced running vector boson masses and studied their relation with the
corresponding complex poles. Always in I it is shown that any -matrix element takes a very simple
S
form when written in terms of running masses. Finally, the relevant Ward identity, the U(1) Ward
identity for single-W, is derived in the situation of interest, when all currents are non-conserved and
when the top quark mass is not neglected inside loops.
For all details concerning the formal construction of the fully massive Fermion-Loop scheme we,
therefore, refer to [9]. Here, instead, we concentrate on its implementation within the FORTRAN
programWTO[11],onthecorrespondingnumericalresultsandonthecomparison,forsingle-W,between
the Fermion-Loop (hereafter FL) scheme and the Fixed-Width (hereafter FW) one.
To be specific the name of Fixed-Width scheme is reserved for the following: the cross-section is
computed using the tree-level amplitude. The massive gauge-boson propagators are given by 1/(p2+
m2 iΓ). This gives an unphysical width for p2 > 0, but retains U(1) gauge invariance in the CC20
−
process.
Themostrecentnumericalresultsproducedforsingle-W productionarefromthefollowingcodes[12]:
CompHEP, GRC4F, NEXTCALIBUR,1 PVALPHA, WPHACT and WTO.
Among these codes, WPHACT is the only one to employ the Fermion-Loop scheme in its imaginary
version [14], where the full imaginary part of the Fermion-Loop corrections is used.
Inviewofarequested,inclusivecross-section, accuracy of2%wemustincluderadiativecorrections
to the best of our knowledge, at least the bulk of any large effect. As we know, the correct scale of
the couplings and their differentiation between s and t-channel is connected to the real part of the
corrections, so that the imaginary FL is not enough, we need a complete FL for single-W. Having
all the parts, the tree-level couplings are replaced by running couplings at the appropriate momenta
and the massive gauge-boson propagators are modified accordingly. The vertex coefficients, entering
through the Yang–Mills vertex, contain the lowest order couplings as well as the one-loop fermionic
vertex corrections.
Apartfromsomerecent development, each calculation aimed toprovidesomeestimate for e+e−
→
4f production is, at least nominally, a tree level calculation. Among other things it will require the
choice of some Input Parameter Set (IPS) and of certain relations among the parameters. In the
literature, althoughimproperly,thisisusuallyreferredtoasthechoiceoftheRenormalization Scheme.
Typically, we have at our disposal four experimental data points (plus α ), i.e. the measured
s
vector boson masses M ,M and the coupling constants, G and α. However, we only have three
Z W F
bare parameters at our disposal, the charged vector boson mass, the SU(2) coupling constant and the
sinus of the weak mixing angle. While the inclusion of one loop corrections would allow us to fix at
least the value of the top quark mass from a consistency relation, this cannot bedone at the tree level.
Thus, different choices of the basic relations among the input parameters can lead to different results
with deviations which, in some case, can be sizeable.
1 NEXTCALIBUR with masses, uses a massive matrix element program called HELAC, that is a recursive algorithm for
electroweakamplitudecalculationsbasedonDyson-SchwingerequationsdevelopedbyA.KanakiandC.G.Papadopou-
los [13].
3
For instance, a possible choice is to fix the coupling constant g as
4πα πα
g2 = , s2 = , (1)
s2 W √2G M2
W F W
where G is the Fermi coupling constant. Another possibility would be to use
F
g2 = 4√2G M2 , (2)
F W
but, in both cases, we miss the correct running of the coupling. Ad hoc solutions should be avoided,
and the running of the parameters must always follow from a fully consistent scheme. Therefore,
the only satisfactory solution can be found in the extension of the full Fermion-Loop to having non-
zero external masses, or non-conserved currents. Unless, of course, one can compute the full set of
corrections.
Note that the Fixed-Width scheme behaves properly in the collinear and high-energy regions of
phase space, to the contrary of the Running-Width scheme, but it completely misses the running of
the couplings, an effect that is expected to beabove the requested precision tag of 2%. By considering
the impact of the FL-scheme on the relevant observables we will be able to judge on the goodness of
naive rescaling procedures.
2 The ingredients in the Fermion-Loop scheme.
There are several building blocks that enter into the construction of the Fermion-Loop scheme, see
for instance [15]. In the ’t Hooft–Feynman gauge, the δ part of the vector–vector transitions can be
µν
cast in the following form [15], where s (c ) is the sine(cosine) of the weak mixing angle:
θ θ
g2s2 g2
S = θ Π (p2)p2, S = Σ (p2),
γγ 16π2 γγ ZZ 16π2c2 ZZ
θ
g2s g2
S = θ Σ (p2), S = Σ (p2). (3)
Zγ 16π2c Zγ WW 16π2 WW
θ
Next we have can transform to the (3,Q) basis [15], where one writes
Σ (p2) = Σ (p2) 2s2Σ (p2)+s4Π (p2)p2,
ZZ 33 − θ 3Q θ γγ
Σ (p2) = Σ (p2) s2Π (p2)p2. (4)
Zγ 3Q − θ γγ
We now consider three parameters, the e.m. coupling constant e, the SU(2) coupling constant g and
the sine of the weak mixing angle s . At the tree level they are not independent, but rather they
θ
satisfy the relation g2s2 = e2. The running of the e.m. coupling constant is easily derived and gives
θ
1 1 1
= Π (s). (5)
e2(s) g2s2 − 16π2 γγ
θ
However, we have a natural scale to use since at s = 0 we have the fine structure constant at our
disposal. Therefore, the running of e2(s) is completely specified in terms of α by
1 1 α
= 1 Π(s) , with Π(s)= Π (s) Π (0). (6)
e2(s) 4πα − 4π γγ − γγ
(cid:20) (cid:21)
4
Figure 1: Runningof coupling constants. The sign of Img2(s) is reversed and the corresponding curve
is magnified by a factor 35. Here q is the absolute value of the momentum.
For the running of g2 we derive a similar equation:
1 1 1
= Π (s). (7)
g2(s) g2 − 16π2 3Q
The running of the third parameter, s2(s), is now fixed by
θ
e2(s)
s2(s) = . (8)
θ g2(s)
In Fig. 1 we show the running of e2(q2) for q2 0 , compared with the fixed value of e2 that one
+
→
would use in the G -scheme. Furthermore, in the same figure, we show the evolution of g2(q2) for q2
F
time-like or space-like, again compared with g2 . The sizeable difference that one gets between e2
GF
runninginthet-channelande2 fixedintheG -schemeisexpectedtobeoneofthemajorimprovements
F
induced by the full Fermion-Loop scheme.
The re-summed propagators for the vector bosons are:
S (p2) 2 −1
G (p2) = p2 S (p2) Zγ ,
γ − γγ − p2+M2 S (p2)
(cid:26) (cid:2) 0 − Z(cid:3)Z (cid:27)
5
S (p2)
G (p2) = Zγ ,
Zγ [p2 S (p2)][p2+M2 S (p2)] [S (p2)]2
− γγ 0 − ZZ − Zγ
S (p2) 2 −1
G (p2) = p2+M2 S Zγ ,
Z 0 − ZZ − p2 S (p2)
(cid:26) (cid:2) − γγ (cid:3) (cid:27)
−1
G (p2) = p2+M2 S (p2) . (9)
W − WW
h i
ThequantityM = M/c isthebareZ mass. Anessentialingredientintheconstruction ofthescheme
0 θ
is represented by the location of the complex poles. Substituting the corresponding results into the
expressions for the propagators, Eq.(9), we see that all ultraviolet divergences not proportional to p2
cancel. We obtain
−1
G (s) = s+p Z(s)+Z(p ) ,
Z − Z − Z
h i −1
G (s) = s+p S (s)+S (p ) ,
W − W − WW WW W
h S (s) i
G (s) = Zγ G (s),
Zγ − s+S (s) Z
γγ
2
1 S (s)
G (s) = + Zγ G (s). (10)
γ − s+Sγγ(s) "s+Sγγ(s)# Z
The vector boson propagators are now expressed as
g2(s) ω (s) c2 g2(s) ω (s)
G (s) = W , G (s) = θ Z ,
W − g2 s Z −g2 c2(s) s
s s2(s)
θ
G (s) = 1 G (s),
Zγ cθ " − s2θ # Z
2
e2(s) s2 s2(s)
G (s) = + θ 1 G (s), (11)
γ e2 c2θ " − s2θ # Z
where the propagation functions are
g2(s) p 1
ω−1(s) = 1 W f (s) f (p ) ,
W − s g2(p ) − 16π2 W − W W
(cid:26) W h i(cid:27)
g2(s) c2(p ) 1
ω−1(s) = 1 Z p f (s) f (p ) . (12)
Z − c2(s)s (g2(p ) Z − 16π2 Z − Z W )
Z h i
The explicit form of the f -functions is given in [9]. The complete one-loop re-summation in the
W,Z
’t Hooft-Feynman gauge is equivalent to some effective unitary-gauge W-propagator. The whole
amplitude can be written in terms of a W-boson exchange diagram, if we make use of the following
effective propagator:
1 pµpν
∆µν = δµν + . (13)
eff p2+M2 S0 M2(p2)
− W h i
The explicit form of M2(p2) is, again given in [9]. We now define the following line-shape functions:
s2 1
L (s) = ,
FL (s Rep )2+(Imp )2 R(s)2
− W W | |
6
Figure 2: Comparison between the Fixed-Width and the Fermion-Loop line-shapes, Eq.(14).
s2
L (s) = . (14)
FW 2
s M2 +M2 Γ2
− W W W
(cid:16) (cid:17)
with a running ρ-factor
R(s)= 1+ g2(s) fW(s)−fW(pW)+pW Π3Q(pW)−Π3Q(s) . (15)
16π2 s ph i
− W
HereM ,Γ aretheon-shellW massandtotalwidth,respectively. Thetwoline-shapesarecompared
W W
in Fig. 2 for different values of the top quark mass. We are using complex-mass renormalization but
we only include fermionic corrections. Therefore, we can start with the Fermi coupling constant but
also with M as an input parameter. Equating the corresponding renormalization conditions yields a
W
relation between M , G , Re α(M2)−1 , M , and m , see [7]. This relation can be solved iteratively
Z F { Z } W t
for m . For the following input parameter set,
t
M = 80.350 GeV, M = 91.1867 GeV, G = 1.16639 10−5 GeV−2, (16)
W Z F ×
we obtain the following solution:
Im(p )
µ = Re(p ) = 80.324 GeV, γ = W = 2.0581 GeV, m = 148.62 GeV. (17)
W W W − µ t
q W
7
Figure 3: Running of α (q2) in the t-channel.
QED
The 26 MeV difference between M and µ is responsible for the sharp transition around 80GeV
W W
that can be seen in Fig. 2. Apart from that, the Fermion-Loop line-shape can be few percents above
the Fixed-Width line shape. The behavior of e2(q2) in the t-channel and, to some extent, the ratio
between Fermion-Loop and Fixed-Width cross-sections is crucially dependent on which regime we are
considering. A careful examination of Fig. 3 shows the following: if we choose
s
< q2 >=(1 < cosθ >) , (18)
e
− 2
and use 10◦ or 0.1◦ as representative for LACC20 or SACC20, we obtain a ratio
α(q2) 2
1 6% 4%, <θ >= 0.1◦,
e
α − → − ÷−
h GF i
α(q2) 2
1 +0.5% +5%, < θ >= 10◦ (19)
e
α − → ÷
h GF i
for √s ranging from 150 GeV to 1 TeV. Therefore, the effect of a runningα in the t-channel is to
QED
decrease/increase the cross-section, with respect to the Fixed-Width scheme, for SACC20/LACC20.
A final ingredient, existing in the Fermion-Loop scheme, is represented by the inclusion of vertices
that are m -dependent. The top-quark contributions are particularly important for delayed-unitarity
t
effects. Inthisrespectalso termsinvolving thetotally-antisymmetric ε-tensor (originating fromvertex
corrections) are relevant. While such terms are absent for complete generations of massless fermions
owing to the anomaly cancellations, they show up for finite fermion masses.
8
All details concerning vertices can be found in [9] where one can find the explicit expressions for
the vertex form-factors in terms of standard C-functions [16].
3 The diagrams.
As already mentioned, there are 20 Feynman diagrams that contribute for eν ud final states, and
e
56 for e+e−ν ν . They are well known to the expert working in the field, but we present them for
e e
convenience of the less specialized reader. For simplicity, we only consider the CC20 family. The most
familiar part of CC20 is the CC03 one, with three diagrams as depicted in Fig. 4, The next set of
e− e−
e+
e+
W W ν
γ,Z ν e
e
+ νe
u
u
W W
e− e−
d
d
Figure 4: The CC03 family of diagrams, annihilation conversion.
⊕
diagrams needed to complete the s-channel component shows up with the CC11 class where we add
to the annihilation and conversion CC03 diagrams all topologies corresponding to pair production of
fermions. TheCC11diagramsnotinCC03aresingle-resonantandtheyareshowninFig.5Finally, we
u
e− e−
e+ e+
ν
γ,Z γ,Z e W
+ d
u
e− W
e− e−
ν d ν
e e
u u e−
e+ e+
d
γ,Z γ,Z W
ν
+ e
e−
u W
e− e−
ν
e
d d
Figure 5: Diagrams belonging to the CC11 CC03 family.
−
have the 10 diagrams forming the t-channel part. In the latter part, one diagram has a W-exchange,
9
five a Z-exchange and four a γ-exchange. As we have shown in [9], this picture is not changed by the
inclusion of one-loop fermionic corrections.
e+ νe e+ νe
d W d
γ,Z
W
+
γ,Z
u u
e− e− e− e−
e+ νe e+ νe
W W
d d
u + d
u
γ,Z γ,Z u
e− e− e− e−
e+ νe e+ νe
d d
W
Z W
+
W
u u
e− e− e− e−
Figure 6: The t-channel component of the CC20 family of diagrams: fusion, bremsstrahlung and
multi-peripheral.
4 Implementing the Fermion-Loop scheme.
Here we follow closely the spirit of Sect. 6 of [9]: once gauge invariance is preserved then we are
allowed to investigate the numerical relevance of masses and to neglect some, if convenient. For the
photon t-channel diagrams, the amplitude squared, summed over spin and integrated over the phase
space of ν ud, can be written as follows:
e
1 m2 1
LγγWµν = 2 e W1 W1
4 µν γγ (Xys)2 γγ − Xys γγ
2 1 2 1 1 1
+ 1 W2 + + +1 W2
Xy2s − y γγ y2s (X +1)2 X +1 y γγ
(cid:18) (cid:19) h i (cid:18) (cid:19)
1 1
+ W2 (20)
ys X +1 γγ
10