Preprint[pdf] - HU Berlin

e + e − → hadrons transition amplitude is proportional to e 2 from the e + e − γ ∗ and the γ ∗ q¯q (hadrons) vertices
which in the cross section appears in quadrature ∝ e 4 or α 2 . Due to the running of the electromagnetic charge
the physical (dressed) cross section is ∝α 2 (s). Undressing requires one to replace the running α(s) by the
classical α:
σ (0)
tot (e+ e − → hadrons) = σtot(e + e − → hadrons)
and, using Eq. (101) we obtain
Π ′ γ(k 2 ) − Π ′ γ(0) = k2
4π 2 α
�
0
∞
ds σ(0)
tot(e + e − → hadrons)
(s − k 2 − iε)
� �2 α
α(s)
(114)
. (115)
It should be stressed that using the physical cross–section in the DR gives a nonsensical result. In order to
get the photon propagator we have to subtract in any case the effective charge from the external e + e − γ
vertex at the correct scale. Thus if we would use
k 2
4π 2 α
�
0
∞
ds σtot(e + e − → hadrons)
(s − k 2 − iε)
we would be double counting the VP effects. In contrast with the linearly in α/α(s) rescaled cross–section
k 2
4π 2
�
0
∞
ds 1
α(s)
σtot(e + e − → hadrons)
(s − k 2 − iε)
we obtain the hadronic shift Eq. (70) for the full photon propagator.
, (116)
Since what we need is the hadronic blob, in processes as displayed in Fig. 26, it is evident that a precise
extraction of the desired object requires a subtraction of all radiative corrections not subsumed in the
blob. Thus besides the VP effects, in particular, the initial state radiation (ISR) has to be subtracted.
It is well known that photon radiation leads to infrared (IR) singularities if not virtual and real (soft)
radiation are included on the same footing (Bloch-Nordsieck prescription). Thereby soft photon radiation
has to be included at least up to energies 0 < E < Ecut where Ecut is the detection threshold of the
detector utilized. Any charged particle cross–section measurement requires some detector dependent cuts
in photon phase space and the detector dependent radiation effects must be subtracted in order to obtain
a detector independent meaningful physics cross–section. Besides the ISR there is final state radiation
(FSR) as well as initial–final state interference effects, where the latter to leading order drop out in total
cross–sections for C symmetric cuts. While ISR from the e + e − initial state is calculable in QED to any
desired order of precision, the calculation of the FSR for hadronic final states is not fully under control.
In addition, experimentally it is not possible to distinguish ISR from FSR photons. Fortunately the most
important channel contributing to a (4)
µ (vap, had) is the two–body π + π − one and FSR usually is calculated
in scalar QED (sQED) which however is appropriate only for relatively soft photons which see the pions as
point particles. In fact a generalized version of sQED is applied where the point form–factor F point
π = 1 is
replaced by the experimentally determined pion form–factor Fπ(s). The FSR contribution will be discussed
in Sect. 4.2. On the level of the quarks the leading order FSR contribution would be given by the diagrams
19) to 21) of Fig. 10 and corresponds to the inclusion of the final state radiation correction of R(s). The
Kinoshita-Lee-Nauenberg (KLN) theorem infers that these fully inclusive corrections are not enhanced by
any logarithms. This also infers that the model dependence of the FSR contribution is at worst moderate.
For a more detailed discussion see Refs. [198]–[202] and references therein.
4.1.2. Hadronic τ-Decays and Isospin Violations
In principle, the I = 1 iso–vector part of e + e − → hadrons can be obtained in an alternative way by using
the precise vector spectral functions from hadronic τ–decays τ → ντ + hadrons which are related by an
46