Meson electromagnetic form factors

The electromagnetic structure of the pseudoscalar meson nonet is completely described by the sophisticated Unitary&Analytic model, respecting all known theoretical properties of the corresponding form factors.


I. INTRODUCTION
All hadrons are compound of constituent quarks. As a consequence in EM interactions they manifest a non-point-like structure, completely described by scalar functions F i (t), called electromagnetic (EM) form factors (FFs), where t is squared momentum transferred by the virtual photon γ * . If Mγ * → M ⇒ F i (t) are called elastic FFs. If Mγ * → A ′ or γ ⇒ F i (t) are called transition FFs.
According to SU(3) classification there are scalar meson, pseudoscalar meson, vector meson and tensor meson [1] multiplets to be bound states of light quarks u, d, s. For a description of their EM structure we use Unitary&Analytic (U&A) model [2], which is a consistent unification of pole and continuum contributions, depends on effective t in thresholds and the coupling constant ratios (f M M V /f V ) as free parameters. In order to determine them numerically one needs a comparison of the U&A model with some experimental data.

II. FIRST GENERALLY
Since pseudoscalar mesons M have spin 0 − there is only one FF F i (t) completely describing their EM structure, which is defined by the parametrization of the matrix element of the EM current.
Making use of the transformation J µ (x) and also the one-particle state vectors < p 2 | and |p 1 > | with regard to all three discrete C, P, T transformations simultaneously then . From the latter it follows for true neutral pseudoscalar mesons π 0 , η, for all values from the interval −∞ < t < +∞.
There is a general belief that all EM FFs are analytic in t-plane, besides branch points i.e. cuts on the positive real axis.
The U&A model is a consistent unification of finite number of complex conjugate pairs of poles contributions and just continua contributions represented by cuts on the positive real axis.
Experimental fact of the creation of ρ, ω, φ, ρ ′ , ω ′ , φ ′ , etc. in e + e − → hadrons in the first approximation can be taken into account by the standard V MD model with stable vector which automatically respects the asymptotic behavior of pseudoscalar meson EM FFs as predicted by the constituent quark model of hadrons.
Afterwards the V MD model is unitarized by an incorporation of two-cut approximation of the analytic properties of EM FFs with the help of the non-linear transformation where t 0 is the square-root branch point corresponding to the lowest possible threshold, t in is an effective square-root branch point simulating contributions of all higher relevant thresholds given by the unitarity condition and is the conformal mapping of the four-sheeted Riemann surface into one W -plane, to be just inverse to the previous non-linear transformation.
As a result every term , for | t |→ ∞ turning out to real constant. The subindex "0" means that still stable vector-mesons are considered. Generally one can prove if m 2 ) and in the second case to the following expression .
Finally, introducing the non-zero widths of resonances by a formal substitution m 2 r → (m r − Γ r /2) 2 i.e. simply one has to rid of 0 in subindices, one gets, when the resonance is where no more equality can be used in these relations.
Consequently, the U&A model of meson EM structure takes the form which is analytic in the whole complex Fig. 1 besides two cuts on the positive real axis.
IV. NOW ONE BY ONE π ± : The analytic properties of F π (t) are in Fig. 2. In comparison with expression there is additional left-hand cut on the II.Riemann sheet.
The latter is explained by the following way. Starting from the elastic unitarity condition is the P -wave isovector ππ-scattering amplitude, the analytic properties of which consist of right-hand unitary cut 4m 2 π < t < ∞ and of left-hand dynamical cut −∞ < t < 0.. Taking into account the fact that the contribution of any cut in Pad'e approximation can be represented by alternating zeros and poles on the place of the cut then we do it in U&A model of From the same elastic unitarity condition and δ 1 1 (t) q→0 ∼ a 1 1 q 3 one gets the threshold behavior of ImF π (t) to be transformed into 3 threshold conditions ImF π (t) q=0 = dImFπ(t) dq q=0 = d 2 ImFπ(t) dq q=0 ≡ 0, which reduce a number of (f vππ /f v ) as free parameters.
Taking into account both these notes and also the normalization explicitly one gets the Due to the ρ − ω interference effect one has to carry out the fit of existing data by | K ± , K 0 : The K + and K 0 belong to the same isomultiplet with I = 1/2. Then one can introduce, generally, the EM current of K, which splits into sum of isotopic scalar and isotopic vector.
The corresponding FFs suitable for a construction of the U&A models are F s follow. The specific 6 resonance (ρ, ω, φ, ρ ′ , φ ′ , ρ ′′ ) U&A model of the kaon EM structure has the Both functions are analytic in the whole complex t-planes besides two cuts on the positive real axis, generated by t s 0 = 9m 2 π and t s in in F s K [V (t)] and by t v 0 = 4m 2 π and t in in F v K [W (t)]. They are real on the whole real negative axis up to positive values t s 0 = 9m 2 π and t v 0 = 4m 2 π , respectively, automatically normalized to 1/2 with ImF s K (t) = 0 and ImF v K (t) = 0, starting from 9m 2 π and 4m 2 π , respectively, as it is required by the unitarity conditions. They possess complex conjugate pairs of poles on unphysical sheets of the Riemann surface, corresponding to considered vector-mesons with quantum numbers of the photon.
However, one can define nonzero single FF for each γ * → γP transition by a parametrization of the matrix element of the EM current < P (p) | J EM µ | 0 >= ε µναβ p ν ǫ α k β F γP (q 2 ) with ǫ α to be the polarization vector of γ, and ε µναβ is antisymmetric tensor.
The transition FFs are related to corresponding cross sections A straightforward calculation of F P γ (t) in QCD is impossible. One has to construct sophisticated phenomenological models.
In a construction of the U&A model it is again suitable to split F P γ (t) into two terms depending on the isotopic character of the photon F P γ (t) = F I=0 P γ (t) + F I=1 P γ (t) where F I=0 P γ (t) is saturated by isoscalar vector-mesons ω, φ, ω ′ , φ ′ etc. and F I=1 P γ (t) is saturated by isovector vector-mesons ρ, ρ ′ , ρ ′′ etc. However, there is a question how many vector-meson resonances have to be taken into account. It is prescribed by the existing data interval on the corresponding FF in t > 0 region. The data on π 0 γ transition FF allow to consider all 3 ground state vector mesons, ρ(770), ω(782), φ(1020) and also ω ′ (1420) and ρ ′ (1450), in order to construct automatically normalized U&A models.
With the aim of obtaining comparable results, the same number of resonances is considered also for η and η ′ .
In the analysis the resonance parameters are fixed at the The F P γ (t) FFs are analytic in t -plane besides the cut from t = m 2 π 0 up to +∞. Then the U&A model of F P γ (t) takes the form [5] and the normalization points V (0) = V N , W (0) = W N .

V. CONCLUSIONS
We have investigated EM structure of pseudoscalar mesons to be described by the corresponding EM FFs. Since there is no possibility to describe the latter in the framework of QCD, the universal U&A models have been elaborated.
More or less successful description of all existing data on the whole complete nonet of pseudoscalar mesons π − , π 0 , π + , K − , K 0 ,K 0 , K + , η, η ′ has been achieved in space-like and time-like regions simultaneously.