^{+}

^{−}→

^{*}/

^{+}

^{−}

Supported by National Natural Science Foundation of China (11275211) and (11335008)

^{+}

^{−}→

^{*}/

^{+}

^{−}

^{*}

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

Vacuum polarization is a part of the initial-state radiative correction for the cross-section of ^{+}^{−} annihilation processes. In the energy region in the vicinity of narrow resonances ^{+}^{−} →^{*}/^{+}^{−} considering the single and double vacuum polarization effect of the virtual photon propagator. Moreover, it presents some numerical comparisons with the traditional treatments.

Article funded by SCOAP^{3}

In quantum field theory, tree-level Feynman diagrams represent a basic process of elementary particles reaction from the initial state to the final state, and the corresponding lowest order cross-section with order ^{2} is called Born cross-section. For accurate calculation, the contribution of higher level Feynman diagrams needs to be considered.

Among all the reactions in ^{+}^{−} annihilation, ^{+}^{−}→^{+}^{−} and ^{+}^{−} are the two simplest quantum electrodynamics (QED) processes. Calculations of the unpolarized ^{+}^{−}→^{+}^{−} and ^{+}^{−} cross-sections to order ^{3} (

For perturbative calculations up to order ^{3}, the radiative correction terms are the interferences between the tree level and higher level (one-loop) Feynman diagrams. In the references mentioned above, all the radiative correction terms were treated as small quantities owing to the extra factor,

The radiative correction of process ^{+}^{−}→^{+}^{−} includes the initial-state and final-state corrections. The final-state radiative (FSR) correction is much smaller than the initial-state radiative (ISR) correction owing to the mass relation, _{e}_{μ}^{±} and ^{±} are less important. In this work, only the ISR correction of the process, ^{+}^{−} → ^{+}^{−}, is considered to keep the discussion succinct, and the discussions only concentrate on the VP correction. The calculations for other correction terms follow the expressions given in the related references[

The calculations of the resonant cross-section and VP correction need the bare value of the electron width of the resonance, but the value cited in the particle data group (PDG) is the experimental electron width, which absorbs the VP effect[^{+}^{−} → ^{+}^{−} and then obtain the value of the Born-level Breit–Wigner cross-section.

The basic properties of a resonance with ^{PC}^{−−} is characterized by its three bare parameters: nominal mass _{e}

The bare values of the resonant parameters are the input quantities for the calculation of ISR factor 1+^{+}^{−} production cross-section in ^{+}^{−} annihilation[_{had} is the number of hadronic events, ^{+}^{−}→ hadrons determined by the Monte Carlo method, and ^{+}^{−}. However, the quantity of interest in physics is Born cross-section ^{0}(^{tot}(^{tot}(^{0}(_{e}_{e}_{e}_{e}_{e}_{e}_{e}_{e}

The discussion in the following sections will be concentrated on the VP correction of ^{tot}(^{+}^{−}→^{+}^{−}. The outline of this paper is as follows: In section ^{+}^{−}→^{+}^{−} with single and double VP corrections are deduced, and the numerical results are presented. Section

In the energy region containing resonance ^{+}^{−} can be produced in the ^{+}^{−} annihilation via two channels:
^{*} is the direct electromagnetic production, and another mode is the electromagnetic decay of intermediate on-shell resonance

Tree-level Feynman diagrams for processes ^{+}^{−} →^{+}^{−} via modes ^{*} (left) and

Virtual photon propagator ^{*} is unobservable in the experiment, and its role is transferring the electromagnetic interaction between ^{+}^{−} and ^{+}^{−}. Intermediate resonance ^{PC} = 1^{−−}. Resonances ^{+}^{−} is discussed.

Channel ^{+}^{−}→^{*}→^{+}^{−} is a pure QED process, which corresponds to the left diagram in Fig.

The channel via intermediate resonance

In general, the wavefunction of time for an unstable particle is expressed as a plane wave with a damping amplitude:

Performing the Fourier transformation on ^{+}^{−} and final state _{e}_{f}^{+}^{−}, _{f}_{μ}_{e}_{μ}

The relativistic amplitude can be obtained easily by adopting the physics picture of the Dirac sea. Dirac considered that an antiparticle corresponded to a hole with same mass

The Born cross-section for the resonant mode corresponding to the right diagram in Fig.

Starting with the Van Royen–Weisskopf formula, _{e}_{c}_{c}_{s}^{2}, and _{e}_{f}

The total production amplitude of ^{+}^{−} should be a coherent summation of the two channels:

In practical evaluations, the parameter values in the Breit–Wigner cross-section typically adopt the experimental values published in the PDG, which contain the radiative effect[^{+}^{−}→^{*}/^{+}^{−}, in which all the parameters are bare quantities. Based on this formula, the bare parameter values can be extracted by fitting the measured cross-section.

From the viewpoint of quantum field theory, two charged particles interact by exchanging quanta of the electro-magnetic field, which corresponds to the virtual photon propagator between the two charges. The VP effect modifies the photon propagator, which is equivalent to a change in the coupling strength between two charges. In the one-particle-irreducible (1PI) chain approximation, an infinite series of 1PI diagrams is summed, and the photon propagator is modified by the VP correction in following manner [_{μν}^{2}) is the VP function. For the ^{+}^{−} annihilation process, ^{2} = ^{*}, is modified to be the full propagator,

Bare propagator ^{*} is replaced by full propagator

The original algorithm of Π(^{+}^{−}, ^{+}^{−}, ^{+}^{−}) can be calculated perturbatively according to the Feynman rules[_{0} is the bare electric charge in the original Lagrangian,

After the charge renormalization, the effect of the VP correction can be explained as bare charge _{0} is redefined as physical charge

In one-photon exchange and chain approximation, the finite part of VP function

Energy dependence of

The optical theorem relates the imaginary part of the QCD component of the photon self-energy to the inclusive hadronic Born cross-section[_{con}(

Π_{res}(^{PC}^{−−}. If the interference between different resonances having the same decay final states are neglected for simplicity, resonant cross-section

Figure

It should be noticed that in experiment measurements, there is no strict partition between the continuum and resonant states, as expressed in Eq. (^{+}^{−} may be direct production ^{+}^{−} → ^{+}^{−} or via intermediate mode ^{+}^{−} → ^{0} → ^{+}^{−}. Therefore, Eqs. (

It should be stressed that the dispersion relation and optical theorem merely provide a practical algorithm for calculating QCD nonperturbative VP function

In general, the Born cross-sections of the ^{*} mode and intermediate ^{2}. Considering the VP effect, running coupling constant _{res}(_{e}_{e}^{+}^{−}→^{*}/^{+}^{−} significantly.

In most references, the value of the electron width in the Breit–Wigner cross section adopts experimental partial width _{e}

In reference [

It is seen from the discussion in the above section, it is not necessary to introduce quantity _{e}_{e}

From the viewpoint of Feynman diagrams, the VP correction modifies the photon propagator, which can be understood from another perspective: the VP effect modifies fine structure constant

The VP-corrected total Born cross-section is:

Born cross-section ^{*} channel expressed in Eq. (

Figure ^{2}, ^{2}. Thus, the resonant shape of the ^{*} channel cross-section does not imply that real resonant state _{res}(^{0}(^{0}(_{res}(

(color online) Line-shape of

(color online) Line-shape comparison of resonant channels ^{+}^{−} →^{+}^{−} (left) and ^{+}^{−}→^{+}^{−} (right) between Born-level Breit–Wigner cross-section

Generally, the cross-section of a resonance is expressed in the Breit–Wigner form. If the value of the electron width adopts bare value _{e}^{2}. It is inappropriate to make line-shape scan measurements in the vicinity of _{i}^{2}. In fact, a more natural VP correction for Breit–Wigner cross-section _{e}

Figure _{e}_{e}_{e}^{*} channel, see Eq. (

The Feynman diagram with a single VP correction is shown in Fig. ^{±} or ^{±}) and photon (^{*}). The grey bubble represents the VP correction in the 1PI approximation, and the hollow oval represents resonance ^{+}^{−} and intermediary ^{+}^{−}, which is same as the traditional treatment, i.e., only a single VP correction is considered for the

Feynman diagram with a single VP correction.

Line-shape of ^{0}(

A coherent amplitude is given by sum of two diagrams:
^{0}(^{0}(^{0}(

The Feynman diagram with a single VP correction in Fig.

Equivalent Feynman diagram of Fig.

For the right Feynman diagram of channel ^{+}^{−} →^{+}^{−} in Fig. ^{+}^{−}^{*} is ^{+}^{−}^{*}, it is

In the quantum field theory, processes ^{+}^{−}→^{+}^{−} and ^{+}^{−}→^{+}^{−} should be invariant under time reversal _{e}_{μ}

Resonant channel ^{+}^{−} →^{+}^{−} has two independent virtual photons, one is between ^{+}^{−} and ^{+}^{−}. According to the Feynman rule and ISR correction principle, each independent virtual photon propagator will be modified by a single VP correction factor, and the two VP factors cannot be combined into one. A Feynman diagram with time reversal symmetry can be plotted as Fig.

Feynman diagram with double VP correction.

(color online) Line-shape of ^{0}(

The coherent amplitude for the Feynman diagram, as shown in Fig. ^{*} and intermediary vector meson ^{0}(

Comparing Figs.

The Feynman diagram in Fig.

Equivalent Feynman diagram with double VP correction; the black spots represent effective charge

(color online) Line-shapes of ^{0}(^{tot}(

(color online) Line-shapes of ^{0}(^{tot}(

(color online) Line-shapes of ^{0}(

The Born cross-section corresponding to the tree-level Feynman diagram reflects the basic property of an elementary particle reaction process, which is interesting in physics. However, in experiments, the measured property is the total cross-section. In this section, the general form of the total cross-section for ^{+}^{−}→^{+}^{−} is given first. Subsequently, the analytical expression of the total cross-section is deduced for the cases of single and double VP corrections, and they are compared numerically.

In the Feynman diagram scheme, the total cross-section up to order ^{+}^{−} pair after radiation, _{vert} is the vertex correction factor, and the radiative function is:

In principle, the integral in Eq. (^{±} beam energy spread effect must be considered. The effect total cross-section that matches the experiment data is:
_{0}) is the Gaussian function representing the energy spread distribution of the initial ^{±} beams and ^{±}. Eq. (

In the following sections, the analytical expression of integral Eq. (^{tot}(

If the initial ^{±} radiates a photon with energy fraction _{i}_{i}_{n}_{n}^{tot}(^{0}(

The integrand of Eq. (_{n}_{n}_{n}

This work discusses two issues: (1) treating the VP correction of the ^{*} channel and ^{+}^{−}→^{*}/^{+}^{−} evaluated by the single and double VP corrections schemes.

The tree-level Feynman diagram in Fig. ^{+}^{−}→^{*}/^{+}^{−} is the coherent summation of the ^{*} channel and ^{*} channel is modified by a single VP factor, and the

Figure ^{0}(

Reference [^{+}^{−} → ^{+}^{−}, where the tree-level Feynman diagram is only a continuum ^{*} channel and there is no resonant _{0}) explained as the continuum amplitude, and term _{0} is viewed as the VP correction factor, whereas resonant component _{0} of _{0} is only a partial VP correction and not the full one, _{e}^{+}^{−} in the numerator of Eq. (_{f}^{+}^{−}. If ^{+}^{−}→^{+}^{−}, why it cannot be for the other final states, such as ^{+}^{−}, ^{+}^{−} or hadrons? In fact, the true resonant amplitude is written in the Breit–Wigner form in Eq. (

The bare resonant parameters (_{e}

Generally, the cross-section directly measured in experiments is the total cross-section, which includes all the radiative effects. To extract the bare resonant parameters from the measured cross-section correctly, an appropriate treatment of the ISR correction is crucial.

As seen in the previous sections, the value of the total cross-section,

The values of the resonant parameters of _{i}_{i}_{e}^{2}.

When the value of _{e}_{e}_{e}_{e}

It is expected that if the values of the resonant parameters (_{e}