1 Introduction
###### Abstract

In this article, we calculate the electromagnetic form-factor of the meson with the light-cone QCD sum rules. The numerical value is in excellent agreement with the experimental data (extrapolated to the limit of zero momentum transfer, or the normalization condition ). For large momentum transfers, the values from the two sum rules are all comparable with the experimental data and theoretical estimations.

Electromagnetic form-factor of the meson with light-cone QCD sum rules

[2mm] Zhi-Gang Wang 111 E-mail,wangzgyiti@yahoo.com.cn. , Zhi-Bin Wang

Department of Physics, North China Electric Power University, Baoding 071003, P. R. China

College of Electrical Engineering, Yanshan University, Qinhuangdao 066004, P. R. China

PACS numbers: 12.38.Lg; 13.40.Gp

Key Words: Electromagnetic form-factor, Light-cone QCD sum rules

## 1 Introduction

The meson, as both Nambu-Goldstone boson and quark-antiquark bound state, plays an important role in testing the quark models and exploring the low energy QCD. Its electromagnetic form-factor and electromagnetic radius are important parameters, and have been extensively studied both experimentally [1, 2, 3, 4, 5, 6, 7] and theoretically, for examples, the QCD sum rules [8, 9, 10, 11], the light-cone QCD sum rules [12, 13, 14, 15], perturbative QCD [16, 17, 18, 19, 20, 21], Schwinger-Dyson equation [22, 23, 24], etc222In Ref.[16], Radyushkin introduces the distribution amplitude of the meson for the first time, expresses the form-factor of the meson in terms of the distribution amplitudes asymptotically, and formulates the perturbative QCD parton picture for hard exclusive processes. .

In Refs.[12, 13, 14, 15], the axial-current is used to interpolate the meson, in Refs.[13, 14], the radiative corrections and higher-twist effects are taken into account. In this article, we choose the pseudoscalar current to interpolate the meson and calculate the electromagnetic form-factor of the meson with the light-cone QCD sum rules. In our previous works, we have studied the vector form-factors and scalar form-factors of the and mesons, the form-factors of the nucleons, and obtain satisfactory results [25, 26, 27, 28, 29]. The light-cone QCD sum rules carry out the operator product expansion near the light-cone instead of short distance , while the non-perturbative matrix elements are parameterized by the light-cone distribution amplitudes (which classified according to their twists) instead of the vacuum condensates [30, 31, 32, 33, 34, 35, 36]. The non-perturbative parameters in the light-cone distribution amplitudes are calculated with the conventional QCD sum rules and the values are universal [37, 38].

The article is arranged as: in Section 2, we derive the electromagnetic form-factor with the light-cone QCD sum rules; in Section 3, the numerical result and discussion; and in Section 4 is reserved for conclusion.

## 2 Electromagnetic form-factor of the π meson with light-cone QCD sum rules

In the following, we write down the definition for the electromagnetic form-factor ,

 ⟨π(p2)|Jμ(0)|π(p1)⟩=Fπ(q2)(p1+p2)μ, (1)

where the is the electromagnetic current and . We study the electromagnetic form-factor with the two-point correlation function ,

 Πμ(p,q) = i∫d4xe−iq⋅x⟨0|T{Jπ(0)Jμ(x)}|π(p)⟩, Jμ(x) = eu¯u(x)γμu(x)+ed¯d(x)γμd(x), Jπ(0) = ¯d(0)iγ5u(0), (2)

where we choose the pseudoscalar current to interpolate the meson. The correlation function can be decomposed as

 Πμ(p,q) = Πp(p,q)pμ+Πq(p,q)qμ (3)

due to Lorentz covariance. In this article, we derive the sum rules with the tensor structures and , respectively.

According to the basic assumption of the current-hadron duality in the QCD sum rules approach [37, 38], we can insert a complete series of intermediate states with the same quantum numbers as the current operator into the correlation function to obtain the hadronic representation. After isolating the ground state contribution from the pole term of the meson, the correlation function can be expressed in the following form,

 Πμ(p,q) = 2fπm2πFpπ(q2)(mu+md)[m2π−(q+p)2]pμ+ (4) fπm2πFqπ(q2)(mu+md)[m2π−(q+p)2]qμ+⋯,

where we introduce the indexes and to denote the electromagnetic form-factor from the tensor structures and respectively, and we use the standard definition for the decay constant ,

 ⟨0|Jπ(0)|π(p)⟩ = fπm2πmu+md.

In the following, we briefly outline the operator product expansion for the correlation function in perturbative QCD theory. The calculations are performed at the large space-like momentum regions and , which correspond to the small light-cone distance required by validity of the operator product expansion approach333In the frame where the meson has a finite 3-vector , , the and can be approximated as and , where , we obtain the relation and . , we take the values and to avoid strong oscillation, . For more details, one can consult Ref.[36] . We write down the propagator of a massive quark in the external gluon field in the Fock-Schwinger gauge firstly [39],

 ⟨0|T{qi(x1)¯qj(x2)}|0⟩=i∫d4k(2π)4e−ik(x1−x2) ⎧⎪⎨⎪⎩⧸k+mk2−m2δij−1∫0dvgsλaij2Gaμν(vx1+(1−v)x2) [12⧸k+m(k2−m2)2σμν−1k2−m2v(x1−x2)μγν]}, (5)

where the is the gluonic field strength. Substituting the above and quark propagators and the corresponding meson light-cone distribution amplitudes into the correlation function , and completing the integrals over the variables and , finally we obtain the representation at the level of quark-gluon degrees of freedom,

 Πi(p,q) = euΠui(p,q)+edΠdi(p,q), (6)

the explicit expressions of the and are given in the appendix. In calculation, we have used the two-particle and three-particle light-cone distribution amplitudes of the meson [30, 31, 32, 33, 34, 39, 40, 41, 42, 43], the explicit expressions of the light-cone distribution amplitudes are also presented in the appendix. The parameters in the light-cone distribution amplitudes are scale dependent and estimated with the QCD sum rules [30, 31, 32, 33, 34, 39, 40, 41, 42, 43]. In this article, the energy scale is chosen to be .

We take the Borel transformation with respect to the variable for the correlation functions and . After matching with the hadronic representation below the threshold, we obtain the following two sum rules for the electromagnetic form-factors and respectively,

 2fπm2πmu+mdFpπ(q2)e−m2πM2 = fπm2πmu+md∫1Δduφp(u)e−Ξ−(eumu−edmd)fπm2πM2∫1Δdu∫u0dtB(t)ue−Ξ +16fπm2πmu+md∫1Δduφσ(u){[1−uddu]1u+2(eum2u−edm2d)u2M2}e−Ξ +(eumu−edmd)fπ∫1Δduφπ(u)ue−Ξ−(eum3u−edm3d)fπm2π4M4∫1ΔduA(u)u3e−Ξ −euf3π∫10dv∫10dαg∫1−αg0dαuφ3π(αd,αg,αu)Θ(u−Δ) {(1+2v)m2πuM2−2(1−v)ddu1u}e−Ξ∣u=(1−v)αg+αu +edf3π∫10dv∫10dαg∫1−αg0dαdφ3π(αd,αg,αu)Θ(u−Δ) {(1+2v)m2πuM2−2(1−v)ddu1u}e−Ξ∣u=(1−v)αg+αd +2fπm4πM4∫10dvv∫10dαg∫αg0dβ∫1−β0dα eumuΦ(1−α−β,β,α)+edmd˜Φ(α,β,1−α−β)u2Θ(u−Δ)e−Ξ∣u=1−vαg −2eumufπm4πM4∫10dv∫10dαg∫1−αg0dαu∫αu0dα Φ(1−α−αg,αg,α)Θ(u−Δ)u2e−Ξ∣u=(1−v)αg+αu −2edmdfπm4πM4∫10dv∫10dαg∫1−αg0dαd∫αd0dα ˜Φ(α,αg,1−α−αg)Θ(u−Δ)u2e−Ξ∣u=(1−v)αg+αd +eumufπm2πM2∫10dv∫10dαg∫1−αg0dαuΨ(αd,αg,αu)Θ(u−Δ)u2e−Ξ∣u=(1−v)αg+αu +edmdfπm2πM2∫10dv∫10dαg∫1−αg0dαd˜Ψ(αd,αg,αu)Θ(u−Δ)u2e−Ξ∣u=(1−v)αg+αd,
 fπm2πmu+mdFqπ(q2)e−m2πM2 (8) = fπm2πmu+md∫1Δduφp(u)ue−Ξ−(eumu−edmd)fπm2πM2∫1Δdu∫u0dtB(t)u2e−Ξ −16fπm2πmu+md∫1Δduφσ(u)ddu1ue−Ξ −euf3πm2πM2∫10dv∫10dαg∫1−αg0dαuφ3π(αd,αg,αu) Θ(u−Δ)1+2vu2e−Ξ∣u=(1−v)αg+αu +edf3πm2πM2∫10dv∫10dαg∫1−αg0dαdφ3π(αd,αg,αu) Θ(u−Δ)1+2vu2e−Ξ∣u=(1−v)αg+αd +2fπm4πM4∫10dvv∫10dαg∫αg0dβ∫1−β0dα eumuΦ(1−α−β,β,α)+edmd˜Φ(α,β,1−α−β)u3Θ(u−Δ)e−Ξ∣1−vαg −2eumufπm4πM4∫10dv∫10dαg∫1−αg0dαu∫αu0dα Φ(1−α−αg,αg,α)u3Θ(u−Δ)e−Ξ∣u=(1−v)αg+αu −2edmdfπm4πM4∫10dv∫10dαg∫1−αg0dαd∫αd0dα Φ(α,αg,1−α−αg)u3Θ(u−Δ)e−Ξ∣u=(1−v)αg+αd,

where

 Ξ = m2q+u(1−u)m2π−(1−u)q2uM2, Δ = m2q−q2s0−q2, Θ(x) = 1forx≥0, (9)

and the is threshold parameter.

## 3 Numerical result and discussion

The input parameters of the light-cone distribution amplitudes are taken as , , , , , , , [30, 31, 32, 33, 34, 39, 40, 41, 42, 43]; and , , . The threshold parameter is chosen to be , which can reproduce the value of the decay constant in the QCD sum rules.

In this article, we take the values of the coefficients of the twist-2 light-cone distribution amplitude from the conventional QCD sum rules [40, 43]. The has been analyzed with the light-cone QCD sum rules and (non-local condensates) QCD sum rules confronting with the high precision CLEO data on the transition form-factor [44, 45, 46, 47, 48, 49]. We also study the electromagnetic form-factors and with the values and at GeV, which are obtained via one-loop renormalization group equation for the central values and at from the (non-local condensates) QCD sum rules with improved model [49].

The Borel parameters in the two sum rules are taken as , in this region, the values of the electromagnetic form-factors and are rather stable. In this article, we take the special value in numerical calculations, although such a definite Borel parameter cannot take into account some uncertainties, the predictive power cannot be impaired qualitatively.

In the two sum rules in Eqs.(7-8), the dominant contributions come from the two-particle twist-3 light-cone distribution amplitudes and due to the pseudoscalar current . The different values of the coefficients of the obtained in Ref.[43] and Ref.[49] respectively can lead to results of minor difference. If we choose the axial-vector current to interpolate the meson, the main contributions come from the twist-2 light-cone distribution amplitude [17, 18, 19, 20, 21]. The uncertainties concerning the denominator are canceled out with each other, see Eqs.(7-8), which result in small net uncertainties.

Taking into account all the uncertainties, finally we obtain the numerical values of the electromagnetic form-factors and , which are shown in the Figs.1-2, at zero momentum transfer,

 Fpπ(0) = 0.999±0.001, Fqπ(0) = 16.05±1.82, (10)

the parameters of the twist-2 light-cone distribution amplitude obtained in Ref.[49] can reduce the values of the form-factors and slightly, about .

Comparing the experimental data (extrapolated to the limit , or the normalization condition ) [1, 2, 3, 4, 5, 6, 7] and theoretical estimation with the vector meson dominance theory [50], our numerical value is excellent. The value is too large to make any reliable prediction, however, it is not un-expected. From the two sum rules, we can see that the terms of the are companied with an extra factor , for example,

 Fqπ(Q2) ∝ ∫1Δduφp(u)ue−m2q+u(1−u)m2π−(1−u)q2uM2=∫1Δduue−m2q+u(1−u)m2π−(1−u)q2uM2, Fpπ(Q2) ∝ ∫1Δduφp(u)e−m2q+u(1−u)m2π−(1−u)q2uM2=∫1Δdue−m2q+u(1−u)m2π−(1−u)q2uM2,

where we have taken the asymptotic distribution amplitude . The value of the is greatly enhanced in the region of small- due to the extra , in the limit , , the dominant contributions come from the end-point of the light-cone distribution amplitudes. We should introduce extra phenomenological form-factors (for example, the Sudakov factor [18, 19]) to suppress the contribution from the end-point. The value of the is more reliable at small momentum transfers.

In the light-cone QCD sum rules, we carry out the operator product expansion near the light-cone , which corresponds to and , the two sum rules for the form-factors and are valid at large momentum transfers. We take the analytical expressions of the and in Eqs.(7-8) as some functions which model the electromagnetic form-factor at large momentum transfers, then extrapolate the and to zero momentum transfer (or beyond zero momentum transfer) with analytical continuation in hope of obtaining some interesting results 444We can borrow some ideas from the electromagnetic -photon form-factor . The value of is fixed by partial conservation of the axial current and the effective anomaly lagrangian, . In the limit of large-, perturbative QCD predicts that . The Brodsky-Lepage interpolation formula [51] can reproduce both the value at and the behavior at large-. The energy scale () is numerically close to the squared mass of the meson, . The Brodsky-Lepage interpolation formula is similar to the result of the vector meson dominance approach, . In the latter case, the calculation is performed at the timelike energy scale and the electromagnetic current is saturated by the vector meson , where the mass serves as a parameter determining the pion charge radius. With a slight modification of the mass parameter, , the experimental data can be well described by the single-pole formula at the interval [52]. In Ref.[27], the four form-factors of have satisfactory behaviors at large , which are expected by naive power counting rules, and they have finite values at . The analytical expressions of the four form-factors , , and are taken as Brodsky-Lepage type of interpolation formulae, although they are calculated at rather large , the extrapolation to lower energy transfer has no solid theoretical foundation. The numerical values of , , and are compatible with the experimental data and theoretical calculations (in magnitude). In Ref.[28], the vector form-factors and are also taken as Brodsky-Lepage type of interpolation formulae, the behaviors of low momentum transfer are rather good in some channels.. It is obvious that the model functions and may have good or bad low- behaviors, although they have solid theoretical foundation at large momentum transfers. We extrapolate the model functions tentatively to zero momentum transfer, systematic errors maybe very large and the results maybe unreliable. The predictions merely indicate the possible values of the light-cone QCD sum rules approach, they should be confronted with the experimental data or other theoretical approaches. The numerical results indicate that the small- behavior of the is better than that of the , so we take the value of the at .

The electromagnetic form-factors and are complex functions of the input parameters, in principle, they can be expanded in terms of Taylor series of for large-. At large momentum transfer, for example, , the central values of the two form-factors and can be fitted numerically as

 Fpπ(Q2) = 0.285Q2, Fqπ(Q2) = 0.554Q2, (11)

which are comparable with the experimental data [1, 2, 3, 4, 5, 6, 7] and theoretical estimations, for examples, the light-cone QCD sum rules [12, 13, 14, 15], perturbative QCD [17, 18, 19, 20, 21]. In Fig.2, we plot the electromagnetic form-factor comparing with the experimental data in Refs. [3, 6, 7] and prediction of the light-cone QCD sum rules with the axial-vector current in Ref.[14]. For more literatures, one can consult Ref.[53].

The large- behavior is expected from the naive power counting rules [54, 55, 56]. At large-, the -th term in the form-factors and respectively can be expanded as , the terms proportional to with are canceled out approximately with each other, i.e. , , . Finally we obtain and for the and respectively. Due to partial conservation of the axial-vector current, the axial-vector current has no-vanishing coupling with the meson, we can choose either the axial-vector current or the pseudoscalar current to interpolate the meson. They can lead to different sum rules, in the case of the axial-vector current, the soft contributions proportional to manifest themselves at large- [13, 14], see Fig.2, more experimental data are needed to select the pertinent sum rules.

In the limit of large-, , which is consistent with the prediction of perturbative QCD theory, i.e. hard-gluon exchange between the and quarks dominates over Feynman mechanism.

## 4 Conclusion

In this article, we calculate the electromagnetic form-factor of the meson with the light-cone QCD sum rules. Our numerical value is in excellent agreement with the experimental data (extrapolated to the limit or the normalization condition ). For large momentum transfers, the values from the two sum rules are all comparable with the experimental data and theoretical estimations.

## Appendix

The explicit expressions of the correlation functions,

 Πup = fπm2πmu+md∫10duuφp(u)m2u−(q+up)2−mufπm2π∫10du∫u0dtuB(t)[m2u−(q+up)2]2 (12) +16fπm2πmu+md∫10duφσ(u)⎧⎨⎩[1−uddu]1m2u−(q+up)2+2m2u[m2u−(q+up)2]2⎫⎬⎭ +mufπ∫10du⎧⎨⎩φπ(u)m2u−(q+up)2−m2πm2u2A(u)[m2u−(q+up)2]3⎫⎬⎭ −f3π∫10dv∫10dαg∫1−αg0dαuφ3π(αd,αg,αu) +4mufπm4π∫10dvv∫10dαg∫αg0dβ∫1−β0dαuΦ(1−α−β,β,α)[m2u−(q+up)2]3∣1−vαg −4mufπm4π∫10dv∫10dαg∫1−αg0dαu∫αu0dαuΦ(1−α−αg,αg,α)[m2u−(q+up)2]3∣u=(1−v)αg+αu +mufπm2π∫10dv∫10dαg∫1−αg0dαuΨ(αd,αg,αu)[m2u−(q+up)2]2∣u=(1−v)αg+αu,
 Πdp = −fπm2πmu+md∫10duuφp(u)m2d−(q+up)2+mdfπm2π∫10du∫u0dtuB(t)[m2d−(q+up)2]2 (13) −16fπm2πmu+md∫10duφσ(u)⎧⎨⎩[1−uddu]1m2d−(q+up)2