删除或更新信息,请邮件至freekaoyan#163.com(#换成@)

A discussion on vacuum polarization correction to the cross-section of e+e-→γ*/ψ→μ+μ-

本站小编 Free考研考试/2022-01-01

闂傚倸鍊搁崐鎼佸磹妞嬪海鐭嗗ù锝堟缁€濠傗攽閻樻彃鈧绱撳杈ㄥ枑闁哄啫鐗勯埀顑跨窔瀵粙顢橀悙鑼垛偓鍨攽閿涘嫬浠х紒顕呭灦瀵偊鎮╃紒妯锋嫼闂備緡鍋嗛崑娑㈡嚐椤栨稒娅犻柟缁㈠枟閻撶喖鏌熼崹顔兼殭濞存粍澹嗛埀顒冾潐濞叉牗鏅舵惔銊ョ闁告洦鍓氭慨婊堟煛婢跺顕滈柣搴㈠▕濮婂宕掑▎鎴犵崲闂侀€炲苯澧伴柛瀣洴閹崇喖顢涘☉娆愮彿婵炲鍘ч悺銊╂偂閺囥垺鐓熸俊顖濆吹閸欌偓闂佸憡鐟ョ€氼噣鍩€椤掑喚娼愭繛鎻掔箻瀹曞綊鎳為妷銈囩畾闂佸壊鍋呭ú鏍倷婵犲洦鐓忓┑鐐茬仢閸旀潙霉閸忓吋绀嬫慨濠冩そ閹筹繝濡堕崨顔锯偓楣冩⒑閼姐倕鏋傞柛搴㈠▕閸┾偓妞ゆ帊绀侀崵顒勬煕濞嗗繐鏆欐い鏇秮楠炲酣鎸婃径鎰暪闂備線娼ч¨鈧┑鈥虫喘瀹曘垽鏌嗗鍡忔嫼閻熸粎澧楃敮鎺撶娴煎瓨鐓曢柟鎯ь嚟閹冲洭鏌曢崱妤€鏆欓柍璇查叄楠炲鎮╃喊澶屽簥闂傚倷绀侀幉锟犳偡閿曞倹鏅柣搴ゎ潐閹哥ǹ螞濞戙垹鐒垫い鎺戝枤濞兼劖绻涢幓鎺旂鐎规洘绻堥獮瀣晝閳ь剟寮告笟鈧弻鐔煎礈瑜忕敮娑㈡煛閸涱喗鍊愰柡灞诲姂閹倝宕掑☉姗嗕紦闂傚倸鍊搁崐鎼佸磹閻戣姤鍊块柨鏃堟暜閸嬫挾绮☉妯诲櫧闁活厽鐟╅弻鐔告綇妤e啯顎嶉梺绋垮閺屻劑鍩為幋锕€纾兼慨姗嗗幖閺嗗牓姊虹粙娆惧剳闁哥姵鍔楅幑銏犫槈閵忕姷顓哄┑鐐叉缁绘帗绂掗懖鈺冪<缂備降鍨归獮鎰版煕鐎n偅宕屾慨濠呮閹风娀寮婚妷顔瑰亾濡や胶绡€闁逞屽墯濞煎繘濡搁敃鈧鍧楁煟鎼淬劍娑ч柟鑺ョ矋缁嬪顓奸崱鎰盎闂佸搫绋侀崑鍕閿曞倹鐓熼柟鎯х摠缁€鍐磼缂佹ḿ娲寸€规洖宕灒闁告繂瀚峰ḿ鏇炩攽閻橆偅濯伴悘鐐舵椤亞绱撴担铏瑰笡缂佽鐗撳畷娲焵椤掍降浜滈柟鐑樺灥椤忊晝绱掗悩顔煎姕闁靛洤瀚板顕€鍩€椤掑嫬纾块柛鎰皺閺嗭箓鏌曟径鍫濆姉闁衡偓娴犲鐓熼柟閭﹀墮缁狙勩亜閵壯冧槐闁诡喕绮欓、娑樷堪閸愌勵潟濠电姷顣介埀顒€纾崺锝嗐亜閵忊剝绀嬮柡浣稿€块幃鍓т沪閽樺顔囬梻鍌氬€烽懗鑸电仚闂佸搫鐗滈崜娑氬垝濞嗘挸绠婚悹鍥皺閻ゅ洭姊虹化鏇炲⒉闁荤噦绠撳畷鎴﹀冀閵娧咁啎闂佺硶鍓濊摫閻忓繋鍗抽弻锝夊箻鐎涙ḿ顦ㄦ繛锝呮搐閿曨亪銆佸☉妯锋瀻闁圭儤绻傛俊鎶芥⒒娴e懙鍦偓娑掓櫊瀹曞綊宕烽鐕佹綗闂佽宕橀褏澹曢崗鍏煎弿婵☆垰鎼幃鎴澪旈弮鍫熲拻濞撴埃鍋撻柍褜鍓涢崑娑㈡嚐椤栨稒娅犻悗娑欋缚缁犳儳霉閿濆懎鏆遍柛姘埥澶娢熼柨瀣垫綌闂備線娼х换鍡涘焵椤掆偓閸樻牠宕欓懞銉х瘈闁汇垽娼у瓭闂佹寧娲忛崐婵嬪箖瑜庣换婵嬪炊瑜忛弻褍顪冮妶鍡楃瑨閻庢凹鍙冮幃锟犲Ψ閳哄倻鍘介梺鍝勬川閸嬫盯鍩€椤掆偓濠€閬嶅焵椤掍胶鍟查柟鍑ゆ嫹
Hong-Dou Jin 1,2,
, Li-Peng Zhou 1,2,
, Bing-Xin Zhang 2,
, Hai-Ming Hu 2,1,
, 1.University of Chinese Academy of Science, Beijing 100049, China
2.Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
Received Date:2018-05-15
Accepted Date:2018-09-19
Available Online:2019-01-01
Fund Project: National Natural Science Foundation of China11275211National Natural Science Foundation of China11335008
Abstract:Vacuum polarization is a part of the initial-state radiative correction for the cross-section of e+e- annihilation processes. In the energy region in the vicinity of narrow resonances J/ψ and ψ(3686), the vacuum polarization contribution from the resonant component has a significant effect on the line-shape of the lepton pair production cross-section. This paper discusses some basic concepts and describes an analytical calculation of the cross-section of e+e-γ*/ψμ+μ- considering the single and double vacuum polarization effect of the virtual photon propagator. Moreover, it presents some numerical comparisons with the traditional treatments.

HTML

--> --> -->
1.Introduction
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 α2 is called Born cross-section. For accurate calculation, the contribution of higher level Feynman diagrams needs to be considered.
Among all the reactions in e+e? annihilation, e+e?e+e? and μ+μ? are the two simplest quantum electrodynamics (QED) processes. Calculations of the unpolarized e+e?e+e? and μ+μ? cross-sections to order α3 (${\mathcal{O}}(\alpha )\sim 1 \% $) correction were studied decades ago[1-4]. Typically, radiative correction includes vertex correction, electron self-energy, vacuum polarization (virtual photon self-energy), and bremsstrahlung [5].
For perturbative calculations up to order α3, 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, α, compared to that in the tree-level terms. Such approximations for the QED correction and non-resonant quantum chromodynamics (QCD) hadronic correction are reasonable. However, for the energy regions in the vicinity of narrow resonances, such as charmonium J/ψ and ψ(3686), the contribution of the resonant component of the vacuum polarization (VP) correction is neither a small quantity nor a smooth function of energy. This implies that the energy dependence of the VP correction factor has a significant influence on the line shape of the total cross-section. Therefore, the VP correction in the vicinity of narrow resonances has to be treated appropriately.
The radiative correction of process e+e?μ+μ? 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, me ? mμ[6]. The FSR correction can be neglected if one dose not require very high accuracy. In addition, the contributions of the two-photon-exchange diagrams and asymmetry of e± and μ± are less important. In this work, only the ISR correction of the process, e+e?μ+μ?, 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[7, 8].
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[9, 10]. Therefore, another motivation of this work is attempt to provide a scheme for extracting the bare electron widths of resonances J/ψ and ψ(3686) by fitting the measured cross-section of e+e?μ+μ? and then obtain the value of the Born-level Breit-Wigner cross-section.
The basic properties of a resonance with JPC = 1?? is characterized by its three bare parameters: nominal mass M, electron width Γe, and total width Γ. The values of the resonant parameters can be predicted by the potential model[11], but the theoretical uncertainties are difficult to estimate. A reliable method for obtaining accurate values of the resonant parameters is to fit the measured leptonic cross section[12, 13] or hadronic cross section[14] in the vicinity of these resonances. Extracting the bare values from experimental data can provide useful information to decide the theories and models.
The bare values of the resonant parameters are the input quantities for the calculation of ISR factor 1+δ(s) in the measurement of the R value, which is defined as the lowest level hadronic cross-section normalized by the theoretical μ+μ? production cross-section in e+e? annihilation[15, 16]. In fact, the total hadronic cross-section is measured with the experimental data:
$\begin{eqnarray}{\sigma }_{ex}^{{\rm{tot}}}(s)=\frac{{N}_{{\rm{had}}}}{L\epsilon }, \end{eqnarray}$
(1)
where Nhad is the number of hadronic events, L is the integrated luminosity of the data samples, ? is the detection efficiency for e+e?→ hadrons determined by the Monte Carlo method, and s is the square of the center-of-mass energy of initial state e+e?. However, the quantity of interest in physics is Born cross-section ${\sigma }_{ex}^{0}(s)$, which is related to ${\sigma }_{ex}^{{\rm{tot}}}(s)$ by ISR factor 1+δ(s) as follows:
$\begin{eqnarray}{\sigma }_{ex}^{0}(s)=\frac{{\sigma }_{ex}^{{\rm{tot}}}(s)}{1+\delta (s)}, \end{eqnarray}$
(2)
and R value is measured:
$\begin{eqnarray}R=\frac{{\sigma }_{ex}^{0}(s)}{{\sigma }_{\mu \mu }^{0}(s)}=\frac{{N}_{{\rm{had}}}}{{\sigma }_{\mu \mu }^{0}L\epsilon [1+\delta (s)]}, \, \, \, \, {\sigma }_{\mu \mu }^{0}(s)=\frac{4\pi {\alpha }^{2}}{3s}.\end{eqnarray}$
(3)
ISR factor 1+δ(s) indicates the fraction of all the high-order Feynman diagram contributions to the Born cross-section, which is a theoretical quantity by definition:
$\begin{eqnarray}1+\delta (s)\equiv \frac{{\sigma }^{{\rm{tot}}}(s)}{{\sigma }^{0}(s)}, \end{eqnarray}$
(4)
where σ0(s) and σtot(s) are the theoretical Born cross-section and total cross-section, respectively. The accurate calculation of 1+δ(s) is a key factor for obtaining the R value from the measured ${\sigma }_{ex}^{{\rm{tot}}}(s)$. The calculation of σtot(s) needs the values of σ0(s′) from ${s}^{\prime}=4{m}_{\pi }^{2}$ to s as inputs. If the correlation between the continuum and resonant states can be neglected, the hadronic Born cross-section can be written as:
$\begin{eqnarray}{\sigma }^{0}(s)={\sigma }_{{\rm{con}}}^{0}(s)+{\sigma }_{{\rm{res}}}^{0}(s), \end{eqnarray}$
(5)
where ${\sigma }_{{\rm{con}}}^{0}(s)={\sigma }_{\mu \mu }^{0}(s)\mathop{R}\limits^{\sim }(s)$, $\mathop{R}\limits^{\sim }(s)$ is the R value from which the resonant contribution has been subtracted. Generally, the Born-level resonant cross-section is expressed in the Breit-Wigner form:
$\begin{eqnarray}{\sigma }_{{\rm{res}}}^{0}(s)=\frac{12\pi {\varGamma }_{e}\varGamma }{{(s-{M}^{2})}^{2}+{M}^{2}{\varGamma }^{2}}, \end{eqnarray}$
(6)
where the resonant parameters (M, Γe, Γ) must be bare quantities. The value of the electron width cited in the PDG is, in fact, the experimental value of ${\varGamma }_{e}^{ex}$, which absorbs the VP effect, but uses the same notation, Γe, as the bare one. If the users directly use the dressed value of ${\varGamma }_{e}^{ex}$ as the bare one, Γe, in Eq. (6), then the value of 1+δ(s) calculated by Eq. (4) is incorrect. In this regard,
$\begin{eqnarray}{\mathop{\sigma }\limits^{\sim }}_{{\rm{res}}}(s)=\frac{12\pi {\varGamma }_{e}^{ex}\varGamma }{{(s-{M}^{2})}^{2}+{M}^{2}{\varGamma }^{2}}\ne {\sigma }_{{\rm{res}}}^{0}(s), \end{eqnarray}$
(7)
and
$\begin{eqnarray}\frac{{\sigma }^{{\rm{tot}}}(s)}{{\sigma }_{{\rm{con}}}^{0}(s)+{\mathop{\sigma }\limits^{\sim }}_{{\rm{res}}}(s)}\ne 1+\delta (s).\end{eqnarray}$
(8)
Obviously, the obtained value from the left-hand-side of Eq. (8) is VP double deducted. Even if a user notices that the ${\varGamma }_{e}^{ex}$ cited in the PDG is a dressed value, he does not know how to extract the bare value, Γe, from ${\varGamma }_{e}^{ex}$. If a user uses the value of Γe predicted by the theoretical model, then it becomes difficult to control the uncertainty of Γe. Some models, for example, the potential model introduced in reference[11], do not provide the theoretical uncertainty of Γe. Therefore, extracting Γe from the data is necessary for the R value measurement.
The discussion in the following sections will be concentrated on the VP correction of σtot(s) for the process, e+e?μ+μ?. The outline of this paper is as follows: In section 2, the related Born cross-sections are presented. In section 3, the VP correction to the virtual photon propagator described in text books and references is reviewed. In section 4, the experimental lepton width with different conventions is reviewed. In section 5, the properties of the VP-modified Born cross-section are discussed and the line-shapes are shown graphically. In section 6, the analytical expressions of the total cross-section of e+e?μ+μ? with single and double VP corrections are deduced, and the numerical results are presented. Section 7 presents some discussions and comments.
2.Born cross-section
In the energy region containing resonance ψ, final state μ+μ? can be produced in the e+e? annihilation via two channels:
$\begin{eqnarray*}{e}^{+}{e}^{-}\Rightarrow \left\{\begin{array}{c}{\gamma }^{\ast }\\ \psi \end{array}\right\}\Rightarrow \, {\mu }^{+}{\mu }^{-}.\end{eqnarray*}$
The mode via virtual photon γ* is the direct electromagnetic production, and another mode is the electromagnetic decay of intermediate on-shell resonance ψ. The tree-level Feynman diagram for this process is the coherent summation of the two diagrams in Fig. 1:
Figure1. Tree-level Feynman diagrams for processes e+e?μ+μ? via modes γ* (left) and ψ (right). Charge e at the vertex expresses the coupling strength between a lepton and photon.

Virtual photon propagator γ* is unobservable in the experiment, and its role is transferring the electromagnetic interaction between e+e? and μ+μ?. Intermediate resonance ψ is a real particle, which is a $c\bar{c}$-bound state with well-defined mass, life-time, spin, and parity JPC = 1??. Resonances J/ψ and ψ(3686) are identified with the 1S and 1P levels of the charmonium family predicted by the potential model[11]. Unstable J/ψ and ψ will decay into different final states via five modes[17]; here, only electromagnetic decay ψμ+μ? is discussed.
2
2.1.Cross-section of e+e?γ*μ+μ?
-->

2.1.Cross-section of e+e?γ*μ+μ?

Channel e+e?γ*μ+μ? is a pure QED process, which corresponds to the left diagram in Fig. 1, and the expression of the Born cross-section can be found in any QED text book[5]:
$\begin{eqnarray}{\sigma }_{{\gamma }^{\ast }}^{0}(s)=\frac{4\pi {\alpha }^{2}}{3s}.\end{eqnarray}$
(9)

2
2.2.Cross-section of e+e?ψμ+μ?
-->

2.2.Cross-section of e+e?ψμ+μ?

The channel via intermediate resonance ψ corresponds to the right diagram in Fig. 1, which concerns the production and decay of ψ. This section will provide some description about this mode.
In general, the wavefunction of time for an unstable particle is expressed as a plane wave with a damping amplitude:
$\begin{eqnarray}\begin{array}{ll}\Psi (t)&=\theta (t)\Psi (0)\cdot {{\rm{e}}}^{{\rm{i}}\omega t}\cdot {{\rm{e}}}^{-t/2\tau }\\&=\theta (t)|\Psi (0)|\cdot {{\rm{e}}}^{{\rm{i}}\delta }\cdot {{\rm{e}}}^{-{\rm{i}}t(M-{\rm{i}}\varGamma /2)}, \end{array}\end{eqnarray}$
(10)
where θ(t) is a step-function of time, Ψ(0) is the wave function at origin t = 0, ω is the circular frequency, τ is the life-time, and δ is the intrinsic phase angle of Ψ(0). Here, the relations of mass M=ω and total decay width Γ = 1/τ in natural unit ? = c = 1 are used. For a free particle, its parameters are bare quantities.
Performing the Fourier transformation on t for Ψ(t), the amplitude of an unstable particle is transformed to nonrelativistic wavefunction of energy W:
$\begin{eqnarray}{\mathcal{T}}(W)=\displaystyle {\int }_{-\infty }^{+\infty }\Psi (t)\cdot {{\rm{e}}}^{{\rm{i}}Wt}{\rm{d}}t=\frac{i|\psi (0)|{{\rm{e}}}^{{\rm{i}}\delta }}{(W-M)+i\varGamma /2}, \end{eqnarray}$
(11)
where the following formula is used:
$\begin{eqnarray}\displaystyle {\int }_{0}^{\infty }{{\rm{e}}}^{-pt}{\rm{d}}t=\frac{1}{p}, \, \, \, \, \, \, ({\rm{Re}}\, p\gt 0).\end{eqnarray}$
(12)
Origin wavefunction Ψ(0) can be determined from the normalization condition and production cross-section[5]. Considering a distinct production and decay process with initial state e+e? and final state f, the corresponding nonrelativistic amplitude is[18]:
$\begin{eqnarray}{{\mathcal{T}}}_{f}(W)=\frac{i\sqrt{{\varGamma }_{e}\cdot {\varGamma }_{f}}{{\rm{e}}}^{{\rm{i}}\delta }}{(W-M)+i\varGamma /2}, \end{eqnarray}$
(13)
where Γe and Γf are the bare electronic and final state widths. For final state μ+μ?, Γf = Γμ. The lepton universality implies Γe = Γμ under limit ${m}_{l}^{2}/s\to 0$.
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 M but with negative energy state ?W in the Dirac sea. Therefore, the relativistic amplitude, which includes particle-antiparticle, is:
$\begin{eqnarray}\begin{array}{ll}{{\mathcal{T}}}_{f}(W)&=\frac{i\sqrt{{\varGamma }_{e}\cdot {\varGamma }_{f}}{{\rm{e}}}^{{\rm{i}}\delta }}{(W-M)+i\varGamma /2}+\frac{i\sqrt{{\varGamma }_{e}\cdot {\varGamma }_{f}}{{\rm{e}}}^{{\rm{i}}\delta }}{(-W-M)+i\varGamma /2}\\&=\frac{i\sqrt{{\varGamma }_{e}\cdot {\varGamma }_{f}}(2M-i\varGamma ){{\rm{e}}}^{{\rm{i}}\delta }}{{W}^{2}-{M}^{2}+{\varGamma }^{2}/4+i\varGamma M}\\&\approx \frac{i2M\sqrt{{\varGamma }_{e}{\varGamma }_{f}}{{\rm{e}}}^{{\rm{i}}\delta }}{{W}^{2}-{M}^{2}+i\varGamma M}.\end{array}\end{eqnarray}$
(14)
For narrow resonances J/ψ and ψ(3686), the value of Γ is assumed much smaller than M and the energy dependence of the total width can be neglected, i.e., Γ is treated as a constant.
The Born cross-section for the resonant mode corresponding to the right diagram in Fig. 1 is generally written in the Breit-Wigner form:
$\begin{eqnarray}{\sigma }_{\psi }^{0}(s)=\frac{4\pi {\alpha }^{2}}{3s}|{{\mathcal{A}}}_{{\rm{BW}}}{|}^{2}, \, \, \, {{\mathcal{A}}}_{{\rm{BW}}}=\frac{Fr{{\rm{e}}}^{{\rm{i}}\delta }}{\Delta +ir}, \end{eqnarray}$
(15)
where the following notations are used:
$\begin{eqnarray}\Delta =\frac{s-{M}^{2}}{{M}^{2}}=t-1, \, \, \, \, t=\frac{s}{{M}^{2}}, \end{eqnarray}$
(16)
$\begin{eqnarray}r=\frac{\varGamma }{M}, \end{eqnarray}$
(17)
$\begin{eqnarray}F=\frac{3\sqrt{s{\varGamma }_{e}{\varGamma }_{f}}}{\alpha \varGamma M}=\frac{3}{\alpha }\sqrt{t{B}_{e}{B}_{f}}.\end{eqnarray}$
(18)
Combination parameter F ensures Eq. (15) provides the accurate Breit-Wigner cross-section.
Starting with the Van Royen-Weisskopf formula, Γe can be expressed by the following formula[17, 19, 20]:
$\begin{eqnarray}{\varGamma }_{e}=\frac{16}{3}\pi {\alpha }^{2}{e}_{c}^{2}{N}_{c}\frac{|R(0){|}^{2}}{{M}^{2}}\left(1-\frac{16{\alpha }_{s}}{3\pi }\right), \end{eqnarray}$
(19)
where ec = 2/3 is the charge of the charm quark in units of electron charge e, Nc = 3 is the number of colors, αs is the strong coupling constant evaluated at s = M2, and R(0) is the radial wavefunction of R(t) at origin t = 0. Some phenomenological models can provide a rough estimation for the value of R(0), but its accurate value has to be extracted based on the measurements of Γe and Γf.
2
2.3.Total Born cross-section
-->

2.3.Total Born cross-section

The total production amplitude of μ+μ? should be a coherent summation of the two channels:
$\begin{eqnarray}{{\mathcal{A}}}_{{\rm{eff}}}=1+\frac{Fr{{\rm{e}}}^{{\rm{i}}\delta }}{\Delta +ir}.\end{eqnarray}$
(20)
The total Born cross-section can be written as:
$\begin{eqnarray}{\sigma }^{0}(s)=\frac{4\pi {\alpha }^{2}}{3s}|{{\mathcal{A}}}_{{\rm{eff}}}{|}^{2}.\end{eqnarray}$
(21)
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[10, 18]. However, the interesting values in physics are the bare ones. The following sections will deduce the total cross-section formula for e+e?γ*/ψμ+μ?, 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.
3.Vacuum polarization correction
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 [5]:
$\begin{eqnarray}{\gamma }^{\ast }:\, \, \frac{-i{g}_{\mu \nu }}{{q}^{2}}\, \, \to \, \, {\mathop{\gamma }\limits^{\sim }}^{\ast }:\, \, \frac{-i{g}_{\mu \nu }}{{q}^{2}[1-\Pi ({q}^{2})]}, \end{eqnarray}$
(22)
where gμν is the metric tensor and Π(q2) is the VP function. For the e+e? annihilation process, q2 = s. Eq. (22) can be expressed graphically as the bare propagator, γ*, is modified to be the full propagator, ${\mathop{\gamma }\limits^{\sim }}^{\ast }$:
The original algorithm of Π(s) is an infinite integral of fermion-loops (leptons and quarks) in the four-momentum space. The integral for the QED lepton-loops (e+e?, μ+μ?, τ+τ?) can be calculated perturbatively according to the Feynman rules[5, 21]. The divergence of the infinite integral is canceled by electric charge renormalization ${e}_{0}\to \sqrt{{Z}_{3}}{e}_{0}=e$, where e0 is the bare electric charge in the original Lagrangian, e is the physical charge, and the renormalization constant is
$\begin{eqnarray}{Z}_{3}\equiv \frac{1}{1-\Pi (0)}, \, \, \, (\Pi (0)\to \infty ).\end{eqnarray}$
(23)
The remaining finite part of Π(s) is $\hat{\Pi }(s)=\Pi (s)-\Pi (0)$, which is used to define running coupling constant α(s) to the leading order:
$\begin{eqnarray}\alpha (s)=\frac{{e}_{0}^{2}/4\pi }{1-\Pi (s)}=\frac{\alpha }{1-[\Pi (s)-\Pi (0)]}\equiv \frac{\alpha }{1-\hat{\Pi }(s)}.\end{eqnarray}$
(24)
This formula expresses an important physics characteristic: finite part $\hat{\Pi }(s)$ in Eq. (24) is not the entire VP function; infinite part Π(0) is absorbed into the definition of physical charge e.
After the charge renormalization, the effect of the VP correction can be explained as bare charge e0 is redefined as physical charge e and simultaneously fine-structure constant α is replaced by effective energy-dependent running coupling factor α(s). Therefore, finite part $1-\hat{\Pi }(s)$ of the VP factor should be combined with α to yield effective running constant α(s). Thus, α and $1-\hat{\Pi }(s)$ should not be separated in the physical explanations and practical calculations.
In one-photon exchange and chain approximation, the finite part of VP function $\hat{\Pi }(s)$ can be expressed as the summation of all of fermion-loop contributions[7, 8, 10]:
$\begin{eqnarray}\hat{\Pi }(s)=\displaystyle \sum [{\Pi }_{l\bar{l}}(s)+{\Pi }_{q\bar{q}}(s)], \end{eqnarray}$
(25)
where $l\bar{l}={e}^{-}{e}^{+}, {\mu }^{-}{\mu }^{+}, {\tau }^{-}{\tau }^{+}$, and $q\bar{q}=u\bar{u}, d\bar{d}, s\bar{s}, c\bar{c}, b\bar{b}, t\bar{t}$. The QED terms of the lepton-loops can be calculated analytically[5, 21]. However, for the QCD quark-loops, analytic calculations cannot be used owing to the strong nonperturbative interaction. The solution for this issue is to use the optical theorem and dispersion relation[22, 23].
The optical theorem relates the imaginary part of the QCD component of the photon self-energy to the inclusive hadronic Born cross-section[23]:
$\begin{eqnarray}{\rm{Im}}{\Pi }_{q\bar{q}}(s)=\frac{s}{4\pi \alpha }{\sigma }_{{\rm{had}}}^{0}(s).\end{eqnarray}$
(26)
The dispersion relation relates the QCD contribution of the VP function to the integral of the imaginary part of the VP function about the quark-loops:
$\begin{eqnarray}{\Pi }_{q\bar{q}}(s)=\frac{s}{\pi }\displaystyle {\int }_{0}^{\infty }\frac{{\rm{Im}}{\Pi }_{q\bar{q}}({s}^{\prime})}{{s}^{\prime}({s}^{\prime}-s-i\epsilon )}{\rm{d}}{s}^{\prime}.\end{eqnarray}$
(27)
Inserting Eq. (26) in Eq. (27), the nonperturbative QCD VP term can be calculated using the hadronic cross-section,
$\begin{eqnarray}{\Pi }_{q\bar{q}}(s)=\frac{s}{4{\pi }^{2}\alpha }\displaystyle {\int }_{0}^{\infty }\frac{{\sigma }_{{\rm{had}}}^{0}({s}^{\prime})}{{s}^{\prime}-s-i\epsilon }{\rm{d}}{s}^{\prime}.\end{eqnarray}$
(28)
If the interference between the inclusive continuum and resonant hadronic states can be neglected, the contribution of the quark-loops can be written as:
$\begin{eqnarray}{\Pi }_{q\bar{q}}(s)={\Pi }_{{\rm{con}}}(s)+{\Pi }_{{\rm{res}}}(s).\end{eqnarray}$
(29)
Πcon(s) can be calculated by the numerical integral:
$\begin{eqnarray}{\Pi }_{{\rm{con}}}(s)=\frac{\alpha }{3\pi }\displaystyle {\int }_{0}^{\infty }\frac{\mathop{R}\limits^{\sim }({s}^{\prime})}{{s}^{\prime}-s-i\epsilon }{\rm{d}}{s}^{\prime}.\end{eqnarray}$
(30)
Generally, $\mathop{R}\limits^{\sim }(s)$ uses experimental values below 5 GeV[15, 24, 25], whereas $\mathop{R}\limits^{\sim }(s)$ adopts the perturbative QCD (pQCD) prediction above 5 GeV.
Πres(s) includes all the contributions of the resonances with JPC = 1??. If the interference between different resonances having the same decay final states are neglected for simplicity, resonant cross-section ${\sigma }_{{\rm{res}}}^{0}(s)$ can be written as the summation of the Breit-Wigner cross-sections:
$\begin{eqnarray}{\sigma }_{{\rm{res}}}^{0}(s)=\displaystyle \sum _{j}\frac{12\pi {\varGamma }_{ej}{\varGamma }_{j}}{{(s-{M}_{j}^{2})}^{2}+{M}_{j}^{2}{\varGamma }_{j}^{2}}, \, \, \, (j=\rho, \omega \ldots \psi \ldots ), \end{eqnarray}$
(31)
and the final analytical result is:
$\begin{eqnarray}\begin{array}{ll}{\Pi }_{{\rm{res}}}(s)&=\frac{s}{4{\pi }^{2}\alpha }\displaystyle {\int }_{0}^{\infty }\frac{{\sigma }_{{\rm{res}}}^{0}({s}^{\prime})}{{s}^{\prime}-s-i\epsilon }{\rm{d}}{s}^{\prime}\\&=\displaystyle \sum _{j}\frac{3s}{\alpha }\frac{{\varGamma }_{ej}}{{M}_{j}}\frac{1}{s-{M}_{j}^{2}+i{M}_{j}{\varGamma }_{j}}.\end{array}\end{eqnarray}$
(32)
In the vicinity of J/ψ and ψ(3686), their overlap can be neglected and only one resonance needs to be considered. However, in higher charmonia regions, wide ψ(4040), ψ(4160) and ψ(4415) overlap significantly, and all their contributions and interference effects should be included[14].
Figure 3 exhibits the energy dependence of running coupling constant α(s) expressed by Eq. (24) around resonances J/ψ and ψ(3686). The resonant shape of α(s) is due to the virtual VP effect, instead of the real resonance produced.
Figure2. Bare propagator γ* is replaced by full propagator ${\mathop{\gamma }\limits^{\sim }}^{\ast }$ with the VP correction.

Figure3. Energy dependence of α(s) around J/ψ (left) and ψ(3686) (right).

It should be noticed that in experiment measurements, there is no strict partition between the continuum and resonant states, as expressed in Eq. (5). For example, observed final state π+π? may be direct production e+e?π+π? or via intermediate mode e+e?ρ0π+π?. Therefore, Eqs. (5) and (29) are only roughly divided for simplicity.
It should be stressed that the dispersion relation and optical theorem merely provide a practical algorithm for calculating QCD nonperturbative VP function ${\Pi }_{q\bar{q}}(s)$, which does not provide extra physics explanation. However, the procedures for calculating ${\Pi }_{q\bar{q}}(s)$ from the dispersion relation and optical theorem may be misleading. Some users considered that cross-sections ${\sigma }_{{\rm{con}}}^{0}(s)$ and ${\sigma }_{{\rm{res}}}^{0}(s)$ in the expressions of ${\Pi }_{q\bar{q}}(s)$ imply that the VP effect also produces real continuum and resonant hadronic states in the virtual photon propagator. In fact, the fermion-loop integral of the VP function is the virtual quantum fluctuation by its definition, and it does not have characteristic quantum numbers (such as, mass, spin, parities), which are necessary for any real particle. A real physics state must be able to be measured in detectors, but the fermion-loops with infinite four-momentum fluctuations in the VP cannot be observed.
In general, the Born cross-sections of the γ* mode and intermediate ψ mode are proportional to α2. Considering the VP effect, running coupling constant α(s) leads to an additional energy-dependence of the cross-section. Moreover, for the energy region around J/ψ and ψ(3686), the value of Πres(s) is very sensitive to s, Γe, and Γ, which implies that the bare values of Γe and Γ will influence the line-shape of e+e?γ*/ψμ+μ? significantly.
4.Effective leptonic width
In most references, the value of the electron width in the Breit-Wigner cross section adopts experimental partial width ${\varGamma }_{e}^{ex}$ (which is represented as Γe in the PDG without declaring), with the VP effect being absorbed into the electron width. There are two different conventions for ${\varGamma }_{e}^{ex}$.
In reference [9], the experimental electron width is defined as:
$\begin{eqnarray}{\varGamma }_{e}^{ex}=\frac{{\varGamma }_{e}}{|1-\hat{\Pi }({M}^{2}){|}^{2}}, \end{eqnarray}$
(33)
where the entire VP function is absorbed in ${\varGamma }_{e}^{ex}$. In reference [10], the following definition is adopted:
$\begin{eqnarray}{\varGamma }_{e}^{ex}=\frac{{\varGamma }_{e}}{|1-{\hat{\Pi }}_{0}({M}^{2}){|}^{2}}, \, \, {\hat{\Pi }}_{0}(s)={\hat{\Pi }}_{{\rm{QED}}}(s)+{\hat{\Pi }}_{{\rm{QCD}}}(s).\end{eqnarray}$
(34)
This convention implies that electron width ${\varGamma }_{e}^{ex}$ absorbs the contribution of the non-resonant components only, whereas the resonant component of the VP correction is absorbed in parameters M and Γ, introducing dressed values $\mathop{M}\limits^{\sim }$ and $\mathop{\varGamma }\limits^{\sim }$. Thus, $\mathop{M}\limits^{\sim }$ and $\mathop{\varGamma }\limits^{\sim }$ deviate the original physical relevance of the mass and total width (life-time). The conventions in Eqs. (33) and (34) are not equivalent for ${\varGamma }_{e}^{ex}$. It is important to clarify which convention is adopted for the appropriate application of ${\varGamma }_{e}^{ex}$ cited in the PDG.
It is seen from the discussion in the above section, it is not necessary to introduce quantity ${\varGamma }_{e}^{ex}$ in the expression of the cross-section if α is replaced by α(s). The following sections will discuss this in more detail. Using α(s) to replace α can keep the bare value Γe in the analysis, which is more natural for understanding the VP effect than introducing ${\varGamma }_{e}^{ex}$. However, if bare value Γe is measured using the scheme proposed in this paper, one may obtain ${\varGamma }_{e}^{ex}$ by the definition in Eq. (33) or Eq. (34) and extract radial wave function R(0) according to Eq. (19).
5.VP-modified Born cross-section
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 α to running coupling constant α(s). In this section, the single and double VP effects will be discussed and their differences will be compared numerically.
The VP-corrected total Born cross-section is:
$\begin{eqnarray}{\mathop{\sigma }\limits^{\sim }}^{0}(s)=\frac{{\sigma }^{0}(s)}{|1-\hat{\Pi }(s){|}^{2}}=\frac{{\sigma }_{{\gamma }^{\ast }}^{0}(s)+{\sigma }_{\psi }^{0}(s)}{|1-\hat{\Pi }(s){|}^{2}}.\end{eqnarray}$
(35)
The next two sections will discuss the effect of VP on ${\sigma }_{{\gamma }^{\ast }}^{0}(s)$ and ${\sigma }_{\psi }^{0}(s)$, respectively.
2
5.1.VP-modified cross-section of γ* channel
-->

5.1.VP-modified cross-section of γ* channel

Born cross-section ${\sigma }_{\gamma \ast }^{0}(s)$ of γ* channel expressed in Eq. (9) is a smooth function of the energy. When the VP correction is applied to it,
$\begin{eqnarray}{\sigma }_{{\gamma }^{\ast }}^{0}(s)\, \to \, {\mathop{\sigma }\limits^{\sim }}_{{\gamma }^{\ast }}^{0}(s)=\frac{{\sigma }_{\gamma \ast }^{0}(s)}{|1-\hat{\Pi }(s){|}^{2}}=\frac{4\pi {\alpha }^{2}(s)}{3s}.\end{eqnarray}$
(36)
Figure 4 shows the line-shapes of ${\sigma }_{\gamma \ast }^{0}(s)$ given in Eq. (9) and of ${\mathop{\sigma }\limits^{\sim }}_{{\gamma }^{\ast }}^{0}(s)$ in Eq. (36). The line-shape of ${\sigma }_{\gamma \ast }^{0}(s)$ is smooth for s, and ${\mathop{\sigma }\limits^{\sim }}_{{\gamma }^{\ast }}^{0}(s)$ gives the obvious resonant structure. Clearly, the resonant structure of ${\mathop{\sigma }\limits^{\sim }}_{{\gamma }^{\ast }}^{0}(s)$ is owing to the VP effect or the sensitive energy-dependence of α(s) in the vicinity of ψ, and ${\mathop{\sigma }\limits^{\sim }}_{{\gamma }^{\ast }}^{0}(s)\lt {\sigma }_{\gamma \ast }^{0}(s)$ for s < M2, ${\mathop{\sigma }\limits^{\sim }}_{{\gamma }^{\ast }}^{0}(s)\gt {\sigma }_{\gamma \ast }^{0}(s)$ for s > M2. Thus, the resonant shape of the γ* channel cross-section does not imply that real resonant state J/ψ or ψ(3686) is produced but that resonant component Πres(s) affects the VP function. In the vicinities of narrow resonances, both Born cross-section σ0(s) expressed in Eq. (21) and VP function $\hat{\Pi }(s)$ are sensitive to energy. Therefore, the energy dependence of effective cross-section ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ is not only determined by σ0(s) but also by Πres(s) or α(s).
Figure4. (color online) Line-shape of ${\sigma }_{\gamma \ast }^{0}(s)$ (dashed line) and ${\mathop{\sigma }\limits^{\sim }}_{{\gamma }^{\ast }}^{0}(s)$ (solid line) in the vicinity of J/ψ (left) and ψ(3686) (right).

2
5.2.VP-modified cross-section of ψ channel
-->

5.2.VP-modified cross-section of ψ channel

Generally, the cross-section of a resonance is expressed in the Breit-Wigner form. If the value of the electron width adopts bare value Γe, the effective Breit-Wigner cross-section is modified by the VP correction. The reference [9] adopted the convention defined by Eq. (33), which corresponds to the VP effect-modified Breit-Wigner cross-section:
$\begin{eqnarray}{\mathop{\sigma }\limits^{\sim }}_{\psi }^{0}(s)=\frac{{\sigma }_{\psi }^{0}(s)}{|1-\hat{\Pi }({M}^{2}){|}^{2}}.\end{eqnarray}$
(37)
The numerator and denominator in Eq. (37) are evaluated at different energy scales; the numerator is evaluated at s, and the denominator is evaluated at peak M2. It is inappropriate to make line-shape scan measurements in the vicinity of J/ψ and ψ(3686) because most energy points si deviate from peak value M2. In fact, a more natural VP correction for Breit-Wigner cross-section ${\sigma }_{\psi }^{0}(s)$ should be
$\begin{eqnarray}{\mathop{\sigma }\limits^{\sim }}_{\psi }^{0}(s)=\frac{{\sigma }_{\psi }^{0}(s)}{|1-\hat{\Pi }(s){|}^{2}}, \end{eqnarray}$
(38)
which corresponds to the convention:
$\begin{eqnarray}{\varGamma }_{e}^{ex}(s)=\frac{{\varGamma }_{e}}{|1-\hat{\Pi }(s){|}^{2}}, \end{eqnarray}$
(39)
which according to Eq. (19) and Eq. (24 requires VP effect-modified Γe to be energy-dependent:
$\begin{eqnarray}{\varGamma }_{e}\to {\mathop{\varGamma }\limits^{\sim }}_{e}(s)=\frac{16}{3}\pi {[\alpha (s)]}^{2}{e}_{c}^{2}{N}_{c}\frac{|R(0){|}^{2}}{{M}^{2}}\left(1-\frac{16{\alpha }_{s}}{3\pi }\right).\end{eqnarray}$
(40)
Figure 5 shows the line-shape comparison of ${\sigma }_{\psi }^{0}(s)$ defined in Eq. (15) and ${\mathop{\sigma }\limits^{\sim }}_{\psi }^{0}(s)$ defined in Eq. (37) and Eq. (38) for J/ψ and ψ(3686), respectively. In the calculations for Fig. 5, M and Γ adopt the PDG values, whereas Γe uses theoretical values Γe = 4.8 keV for J/ψ and Γe = 2.1 keV for ψ(3686)[26]. The difference in the line-shapes based on Eqs. (15) and (37) is small. The peak positions of ${\sigma }_{\psi }^{0}(s)$ and ${\mathop{\sigma }\limits^{\sim }}_{\psi }^{0}(s)$ defined by Eq. (37) are the same, and the relative difference in their cross-sections at the peak is approximately 6% for both J/ψ and ψ(3686). The shift in the peak positions between ${\sigma }_{\psi }^{0}(s)$ and ${\mathop{\sigma }\limits^{\sim }}_{\psi }^{0}(s)$ defined by Eq. (38) is approximately 1.0 Mev and 0.4 MeV, and the relative difference in their cross-section at the peak is approximately 31% and 3% for J/ψ and ψ(3686), respectively. J/ψ is narrower than ψ(3686), and thus, the shift in the vicinity of J/ψ is much larger than that near ψ(3686). The line-shape of the VP-modified Breit-Wigner cross-section adopting Eq. (37) and Eq. (38) is different. It is clear that adopting Eq. (38) is reasonable, and it is consistent with the VP correction to the γ* channel, see Eq. (36).
Figure5. (color online) Line-shape comparison of resonant channels e+e?J/ψμ+μ? (left) and e+e?ψ(3686)→μ+μ? (right) between Born-level Breit-Wigner cross-section ${\sigma }_{\psi }^{0}(s)$ (dashed line) and VP-modified cross-section ${\mathop{\sigma }\limits^{\sim }}_{\psi }^{0}(s)$ (solid line).

2
5.3.Single VP correction case
-->

5.3.Single VP correction case

The Feynman diagram with a single VP correction is shown in Fig. 6, where e at the vertex is the electron charge, which represents the coupling strength between the leptons (e± or μ±) and photon (γ*). The grey bubble represents the VP correction in the 1PI approximation, and the hollow oval represents resonance ψ. For the ψ channel in the Feynman diagram in Fig. 6, only the virtual photon propagator between the initial e+e? and intermediary ψ is corrected by the VP. There is no VP correction for the virtual photon between ψ and final state μ+μ?, which is same as the traditional treatment, i.e., only a single VP correction is considered for the ψ channel.
Figure6. Feynman diagram with a single VP correction.

A coherent amplitude is given by sum of two diagrams:
$\begin{eqnarray}{\mathop{{\mathcal{A}}}\limits^{\sim }}_{{\rm{eff}}}\sim \frac{1}{1-\hat{\Pi }(s)}(1+\frac{Fr{{\rm{e}}}^{{\rm{i}}\delta }}{\Delta +ir}).\end{eqnarray}$
(41)
Considering the VP effect and that the electromagnetic coupling strength still expresses as α, the Born cross-section is modified as the following expression:
$\begin{eqnarray}{\sigma }^{0}(s)\to {\mathop{\sigma }\limits^{\sim }}^{0}(s)=\frac{4\pi {\alpha }^{2}}{3s}|{\mathop{A}\limits^{\sim }}_{{\rm{eff}}}{|}^{2}=\frac{{\sigma }^{0}(s)}{|1-\hat{\Pi }(s){|}^{2}}, \end{eqnarray}$
(42)
where σ0(s) is given by Eq. (21). The energy dependence of σ0(s) and ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ in the vicinity of J/ψ and ψ(3686) is displayed in Fig. 7. It is clear that the VP correction or equivalent α(s) distorts the line-shape of the original resonant structure of σ0(s).
Figure7. Line-shape of σ0(s) by Eq. (21) (dashed line) and ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ expressed by Eq. (42) (solid line) in the vicinity of J/ψ (left) and ψ(3686) (right).

The Feynman diagram with a single VP correction in Fig. 6 can also be replotted as Fig. 8 equivalently, which has the same topological structure as the tree level in Fig. 1. The black-dot at the vertex is effective running electron charge:
$\begin{eqnarray}{e}^{2}(s)=\frac{{e}^{2}}{|1-\hat{\Pi }(s)|}.\end{eqnarray}$
(43)
Figure8. Equivalent Feynman diagram of Fig. 6. The single VP correction is absorbed into effective electric charge e(s) defined in Eq. (43).

For the right Feynman diagram of channel e+e?ψμ+μ? in Fig. 6 or Fig. 8, coupling strength of three-line vertex e+e?γ* is e(s) corresponding to α(s), and for μ+μ?γ*, it is e corresponding to α:
$\begin{eqnarray}\alpha =\frac{{e}^{2}}{4\pi }, \, \, \, \, {\rm{and}}\, \, \, \, \alpha (s)=\frac{{e}^{2}(s)}{4\pi }.\end{eqnarray}$
(44)

2
5.4.Double VP correction case
-->

5.4.Double VP correction case

In the quantum field theory, processes e+e?μ+μ? and μ+μ?e+e? should be invariant under time reversal T??T, and both processes have the same cross-section if masses me and μμ can be neglected compared to energy $\sqrt{s}$. However, the right Feynman diagrams in Fig. 6 and Fig. 8 violate this basic requirement. This issue can be simply solved by the double VP correction.
Resonant channel e+e?ψμ+μ? has two independent virtual photons, one is between e+e? and ψ, and another is between ψ and μ+μ?. 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. 9.
Figure9. Feynman diagram with double VP correction.

The coherent amplitude for the Feynman diagram, as shown in Fig. 9, after the contraction of the Lorentz indices of the virtual photons γ* and intermediary vector meson ψ, can be written as:
$\begin{eqnarray}{\mathop{{\mathcal{A}}}\limits^{\sim }}_{{\rm{eff}}}\sim \frac{1}{1-\hat{\Pi }(s)}+\frac{1}{1-\hat{\Pi }(s)}\frac{Fr{{\rm{e}}}^{{\rm{i}}\delta }}{\Delta +ir}\frac{1}{1-\hat{\Pi }(s)}, \end{eqnarray}$
(45)
and the corresponding cross-section is:
$\begin{eqnarray}{\mathop{\sigma }\limits^{\sim }}^{0}(s)=\frac{4\pi {\alpha }^{2}}{3s}|{\mathop{{\mathcal{A}}}\limits^{\sim }}_{{\rm{eff}}}{|}^{2}.\end{eqnarray}$
(46)
Figure 10 presents the line-shape comparison of σ0(s) expressed in Eq. (21) and ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ in Eq. (46).
Figure10. (color online) Line-shape of σ0(s) (dashed line) and ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ expressed in Eq. (45) and Eq. (46) (solid line) in the vicinity of J/ψ (left) and ψ(3686) (right).

Comparing Figs. 7 and 10, the single and double VP correction lead to different line-shapes for the cross-section. This issue will yield different results when extracting the resonant parameters from experimental data.
The Feynman diagram in Fig. 9 with double VP correction can be replotted equivalently as Fig. 11, which is symmetrical for the two time-reversal leptonic processes:
$\begin{eqnarray}{e}^{+}{e}^{-}\, \, \rightleftarrows \, \, {\gamma }^{\ast }/\psi \, \, \rightleftarrows \, \, {\mu }^{+}{\mu }^{-}.\end{eqnarray}$
(47)
Figure11. Equivalent Feynman diagram with double VP correction; the black spots represent effective charge e(s) defined by Eq. (43).

The tree-level Feynman diagrams in Fig. 1 and double VP-corrected equivalent diagram in Fig. 11 have the same topology, but the coupling vertexes possess different coupling strengths e and e(s), respectively.
6.Total cross-section
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 e+e?μ+μ? 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.
2
6.1.General form
-->

6.1.General form

In the Feynman diagram scheme, the total cross-section up to order ${\mathcal{O}}({\alpha }^{3})$ can be written as[7, 8]:
$\begin{eqnarray}{\sigma }^{{\rm{tot}}}(s)=(1-{x}_{m}^{\beta }+{\delta }_{{\rm{vert}}}){\mathop{\sigma }\limits^{\sim }}^{0}(s)+\displaystyle {\int }_{0}^{{x}_{m}}{\rm{d}}xH(x;s){\mathop{\sigma }\limits^{\sim }}^{0}({s}^{\prime}), \end{eqnarray}$
(48)
where $x\equiv {E}_{\gamma }/\sqrt{s}$ is the energy fraction carried by a Bremsstrahlung photon, ${x}_{m}=1-4{m}_{\mu }^{2}/s$ is the maximum energy fraction of the radiative photon, s′ = (1 ? x)s is the effective square of the center-of-mass energy of the final μ+μ? pair after radiation, δvert is the vertex correction factor, and the radiative function is:
$\begin{eqnarray}H(x;s)=\beta \frac{{x}^{\beta }}{x}\left(1-x+\frac{{x}^{2}}{2}\right), \, \, \, \, \beta =\frac{2\alpha }{\pi }\left(\text{ln}\frac{s}{{m}_{e}^{2}}-1\right).\end{eqnarray}$
(49)
In principle, the integral in Eq. (48) can be calculated using a numerical method. However, in the application for narrow resonances J/ψ and ψ(3686) scan experiment, the e± beam energy spread effect must be considered. The effect total cross-section that matches the experiment data is:
$\begin{eqnarray}{\sigma }_{th}^{{\rm{tot}}}({s}_{0})=\displaystyle \int {\rm{d}}sG(s;{s}_{0}){\sigma }^{{\rm{tot}}}(s), \end{eqnarray}$
(50)
where G(s; s0) is the Gaussian function representing the energy spread distribution of the initial e± beams and $\sqrt{{s}_{0}}$ is the nominal center-of-energy of e±. Eq. (50) is a two-dimensional integral in variables x and s. Integral Eq. (50) contains Eq. (48) and the outer integral in s about energy spread has to be calculated numerically. However, the inner integral in Eq. (48) of x can be evaluated analytically. The analytical calculation in Eq. (48) can save much CPU time and achieve high numerical accuracy.
In the following sections, the analytical expression of integral Eq. (48) is deduced for the two cases of single and double VP corrections, and total cross-section σtot(s) is evaluated using the analytical results.
2
6.2.Analytical calculation for single VP
-->

6.2.Analytical calculation for single VP

If the initial e± radiates a photon with energy fraction x, the notations in Eqs. (16) and (32) are changed:
$\begin{eqnarray}\Delta \Rightarrow \Delta (x)=(1-x)t-1, \end{eqnarray}$
(51)
$\begin{eqnarray}{\Pi }_{{\rm{res}}}(s)\Rightarrow {\Pi }_{{\rm{res}}}(x;s)=h\frac{1-x}{\Delta (x)+ir}.\end{eqnarray}$
(52)
The Born cross-section with VP correction is:
$\begin{eqnarray}{\sigma }^{0}(s)\Rightarrow {\mathop{\sigma }\limits^{\sim }}^{0}(x;s)=\frac{4\pi {\alpha }^{2}}{3s}\frac{1}{1-x}\cdot \frac{U(x)}{V(x)}, \end{eqnarray}$
(53)
where the quadratic polynomials have the forms:
$\begin{eqnarray}U(x)={u}_{2}{x}^{2}+{u}_{1}x+{u}_{0}, \end{eqnarray}$
(54)
$\begin{eqnarray}V(x)={v}_{2}{x}^{2}+{v}_{1}x+{v}_{0}.\end{eqnarray}$
(55)
The integrand in Eq. (48) has the following polynomial form:
$\begin{eqnarray}H(x){\mathop{\sigma }\limits^{\sim }}^{0}(x;s)=\frac{4\pi {\alpha }^{2}}{3s}\beta \frac{{x}^{\beta }}{x}\left[\frac{1}{1-x}\displaystyle \sum _{n=0}^{4}{w}_{n}{x}^{n}+\frac{1}{V(x)}\displaystyle \sum _{n=0}^{5}{d}_{n}{x}^{n}\right], \end{eqnarray}$
(56)
where coefficients ui, vi, wn, and dn are the combinations of known constants and resonant parameters. The integral of Eq. (48) can be performed analytically. The results of the analytical integrals of σtot(s) are shown in Fig. 12, and the line-shape of σ0(s) is plotted to exhibit the effect of the ISR correction.
Figure12. (color online) Line-shapes of σ0(s) (dashed line) and σtot(s) for single VP correction (solid line) in the vicinity of J/ψ (left) and ψ(3686) (right).

2
6.3.Analytical calculation for double VP
-->

6.3.Analytical calculation for double VP

The integrand of Eq. (48) for the double VP correction can be expressed as the following elementary function:
$\begin{eqnarray}\begin{array}{ll}H(x){\mathop{\sigma }\limits^{\sim }}^{0}(x;s)=&\frac{4\pi {\alpha }^{2}}{3s}\beta \frac{{x}^{\beta }}{x}\left[\frac{1}{1-x}\displaystyle \sum _{n=0}^{4}{p}_{n}{x}^{n}\right.\\&\left.+\frac{1}{V(x)}\displaystyle \sum _{n=0}^{5}{q}_{n}{x}^{n}+\frac{1}{{V}^{2}(x)}\displaystyle \sum _{n=0}^{5}{r}_{n}{x}^{n}\right], \end{array}\end{eqnarray}$
(57)
where coefficients pn, qn, and rn are the combinations of known constants and resonant parameters. The integral of Eq. (48) can be performed analytically, and the analytical results are displayed in Fig. 13.
Figure13. (color online) Line-shapes of σ0(s) (dashed line) and σtot(s) for double VP correction (solid line) in the vicinity of J/ψ (left) and ψ(3686) (right).

7.Discussions
This work discusses two issues: (1) treating the VP correction of the γ* channel and ψ channel by a natural and consistent scheme; (2) comparing the cross-sections of e+e?γ*/ψμ+μ? evaluated by the single and double VP corrections schemes.
The tree-level Feynman diagram in Fig. 1 for e+e?γ*/ψμ+μ? is the coherent summation of the γ* channel and ψ channel. The VP-modified Born cross-section is given in Eq. (46), the γ* channel is modified by a single VP factor, and the ψ channel is modified by double VP factors.
Figure 14 exhibits the comparison of original Born cross-section σ0(s) and single and double VP-modified Born cross-sections ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ in the vicinity of J/ψ and ψ(3686). The line-shapes of ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ for the single and double VP corrections are significantly different.
Figure14. (color online) Line-shapes of σ0(s) (dashed line), single (dot-dashed line), and double (solid line) VP-modified ${\mathop{\sigma }\limits^{\sim }}^{0}(s)$ in the vicinity of J/ψ (left) and ψ(3686) (right).

Reference [10] discusses the VP-modified Born cross-section of process e+e?μ+μ?, where the tree-level Feynman diagram is only a continuum γ* channel and there is no resonant ψ channel. In fact, this is the case discussed in section 5.1 in this paper. The VP-modified Born cross-section in reference [10] is same as expressed in Eq. (36) in our paper. Eq. (36) is a very concise and natural expression, and it is easy to understand in physics. Reference [10] made a skillful mathematic identical transformation to VP correction, where the full factor of $1/(1-\hat{\Pi })$ was divided to two terms: the term with 1/(1?Π0) explained as the continuum amplitude, and term ${\mathop{\Pi }\limits^{\sim }}_{{\rm{res}}}/{(1-{\Pi }_{0})}^{2}$ as the resonant amplitude. In this explanation, only non-resonant component Π0 is viewed as the VP correction factor, whereas resonant component ${\mathop{\Pi }\limits^{\sim }}_{{\rm{res}}}$ is viewed as the resonant amplitude. Thus, the original one-continuum channel is transformed into two channels, which implies that a pure identical transformation in mathematics leads to a new physics picture. Resonant amplitude ${\mathop{\Pi }\limits^{\sim }}_{{\rm{res}}}$ contains non-resonant components Π0 of $\hat{\Pi }$ in the following form:
$\begin{eqnarray}{\mathop{\Pi }\limits^{\sim }}_{{\rm{res}}}(s)=\frac{3{\varGamma }_{e}}{\alpha }\frac{s}{M}\frac{1}{s-{\mathop{M}\limits^{\sim }}^{2}+i\mathop{M}\limits^{\sim }\mathop{\varGamma }\limits^{\sim }}, \end{eqnarray}$
(58)
where mass $\mathop{M}\limits^{\sim }$ and width $\mathop{\varGamma }\limits^{\sim }$ are called dressed values:
$\begin{eqnarray}{\mathop{M}\limits^{\sim }}^{2}={M}^{2}+\frac{3{\varGamma }_{e}}{\alpha }\frac{s}{M}{\rm{Re}}\frac{1}{1-{\Pi }_{0}}, \end{eqnarray}$
(59)
$\begin{eqnarray}\mathop{M}\limits^{\sim }\mathop{\varGamma }\limits^{\sim }=M\varGamma -\frac{3{\varGamma }_{e}}{\alpha }\frac{s}{M}{\rm{Im}}\frac{1}{1-{\Pi }_{0}}.\end{eqnarray}$
(60)
Therefore, the value of ${\varGamma }_{e}^{ex}$ defined with convention Eq. (34) cannot be adopted all alone because Π0 is only a partial VP correction and not the full one, $\hat{\Pi }$. In this case, ${\varGamma }_{e}^{ex}$ must be used together with $\mathop{M}\limits^{\sim }$ and $\mathop{\varGamma }\limits^{\sim }$ for completeness and consistency. It is noticed that only Γe is present in initial state e+e? in the numerator of Eq. (58) and that there is no Γf for the appointed final state, μ+μ?. If ${\mathop{\Pi }\limits^{\sim }}_{{\rm{res}}}$ can be interpreted as the resonant amplitude of e+e?ψμ+μ?, why it cannot be for the other final states, such as e+e?, τ+τ? or hadrons? In fact, the true resonant amplitude is written in the Breit-Wigner form in Eq. (14). The VP effect is the quantum fluctuation of vacuum, and it does not refer to any final state. Convention Eq. (34) and the explanation in [10] convert a simple and clear problem as a complex and an obscure one. However, the convention in Eq. (33) is clear and natural.
The bare resonant parameters (M, Γ, Γe, δ) are the basic quantities in the Breit-Wigner formula, and they characterize the main properties of a resonance. The values of these parameters can be estimated from phenomenological potential models [26, 27]. However, their accurate values have to be measured by fitting the experimental data.
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, ${\sigma }_{th}^{{\rm{tot}}}(s)$, depends on the VP correction scheme, and it is also the function of the resonant parameters. ISR correction factor 1+δ is a theoretical quantity defined in Eq. (4), and it affects the Born cross-section according to Eq. (2).
The values of the resonant parameters of J/ψ and ψ(3686) can be extracted by fitting the measured cross-section in the line-shape scan experiment based on the least square method:
$\begin{eqnarray}{\chi }^{2}=\displaystyle \sum _{i=1}^{n}\frac{{[{\sigma }_{ex}^{{\rm{tot}}}({s}_{i})-{\sigma }_{th}^{{\rm{tot}}}({s}_{i})]}^{2}}{{\Delta }_{i}^{2}}, \end{eqnarray}$
(61)
where ${\sigma }_{ex}^{{\rm{tot}}}$ can be measured using Eq. (1) and Δi is the uncertainty of ${\sigma }_{ex}^{{\rm{tot}}}({s}_{i})$ at energy point si. The optimized values of (M, Γ, Γe, δ) correspond to the optimized minimum of χ2.
When the value of Γe is extracted, one may obtain ${\varGamma }_{e}^{ex}$ by any convention, but it is not necessary in physics and nor in experiments. Γe connects to original radial wave function R(0) of $c\bar{c}$ bound state ψ according to Eq. (19). The value of Γe can deduce the value of R(0) and can test potential models. Γe can be used to calculate the correct ISR factor in the R measurement.
It is expected that if the values of the resonant parameters (M, Γ, Γe, δ) are extracted using the scheme proposed in this paper, the results will not be the same as in previous measurements. Therefore, which scheme is reasonable should be determined by experiments and further studies.
相关话题/discussion vacuum polarization

濠电姷鏁告慨鐑藉极閹间礁纾婚柣鎰惈閸ㄥ倿鏌涢锝嗙缂佺姳鍗抽弻鐔虹磼閵忕姵鐏堢紒鐐劤椤兘寮婚悢鐓庣鐟滃繒鏁☉銏$厽闁规儳顕埥澶嬨亜椤撶偞鍋ラ柟铏矊椤曘儱螖婵犱線鍋楅梺璇″枟閿曘垽骞冮埡鍐<婵☆垳鍘х敮楣冩⒒娴gǹ顥忛柛瀣噽閹广垽宕熼姘К闂佹寧绻傞ˇ浼存偂濞嗘垟鍋撶憴鍕婵炲眰鍊濋崺銏ゅ醇閳垛晛浜鹃悷娆忓缁€鍐磼椤旇偐效妤犵偛绻樺畷銊╁级閹寸偛绁舵俊鐐€栭幐楣冨窗閹伴偊鏁婇煫鍥ㄧ⊕閳锋帡鏌涚仦鎹愬闁逞屽墮閸㈡煡婀侀梺鎼炲労閻忔稑鈽夐姀鐘殿槹濡炪倖鍔戦崐鏍р枔閹屾富闁靛牆妫楅崸濠囨煕鐎n偅灏伴柕鍥у椤㈡洟鏁愰崶鈺冩澖濠电姷顣介崜婵嬪箖閸岀偛鏄ラ柍鈺佸暞婵挳鏌ц箛鏇熷殌妤犵偐鍋撳┑鐘殿暜缁辨洟宕戦幋锕€纾归柡宥庡幗閸嬪淇婇妶鍛櫤闁稿绻濋弻鏇㈠醇濠靛洨鈹涙繝娈垮枟婵炲﹪寮婚埄鍐ㄧ窞濠电姴瀚惃鎴濃攽閳╁啫绲婚柣妤佹崌瀵鏁撻悩鑼槰闂佹寧绻傞幊宥嗙珶閺囩喓绡€闁汇垽娼цⅷ闂佹悶鍔嶅浠嬪极閸愵喖顫呴柣妯虹仛濞堥箖姊洪崨濠勭畵閻庢凹鍣e鎶藉幢濞戞瑧鍘遍梺鍝勬储閸斿本鏅堕鐣岀闁割偅绻勯悞鍛婃叏婵犲啯銇濈€规洏鍔嶇换婵嬪磼濮f寧娲樼换娑氣偓娑欋缚閻矂鏌涚€c劌鈧洟鎮惧畡鎳婃椽顢旈崟顓濈礈闂備礁鎼崐鍫曞磿閺屻儻缍栫€广儱顦伴埛鎴︽偡濞嗗繐顏╅柛鏂诲€濋弻锝嗗箠闁告柨瀛╃粋宥夊箹娓氬洦鏅濋梺闈涚墕濞层劑鏁嶅⿰鍐f斀閹烘娊宕愰弴銏犵柈妞ゆ劧绠戦崙鐘绘煛閸愩劎澧涢柣鎾寸懃椤啰鈧綆浜妤呮煃鐠囪尙澧涙い銊e劦閹瑩寮堕幋鐘辩礉婵°倗濮烽崑娑樏洪鈧偓浣糕枎閹惧厖绱堕梺鍛婃处娴滐綁宕洪崨瀛樷拻闁稿本鑹鹃埀顒勵棑缁牊绗熼埀顒勭嵁閺嶎収鏁冮柨鏇楀亾缁炬儳婀遍幉鎼佹偋閸繄鐟查梺绋款儜缁绘繂顕i崼鏇為唶婵﹩鍘介悵鏇烆渻閵堝骸浜濇繛鑼枛瀵濡搁埡鍌氫簽闂佺ǹ鏈粙鎴︻敂閿燂拷
2濠电姷鏁告慨鐑藉极閹间礁纾婚柣鎰惈閸ㄥ倿鏌涢锝嗙缂佺姵澹嗙槐鎺斺偓锝庡亾缁扁晜绻涘顔荤盎閹喖姊洪崘鍙夋儓妞ゆ垵娲ㄧ划娆掔疀濞戞瑢鎷洪梺闈╁瘜閸樺ジ宕濈€n偁浜滈柕濞垮劜椤ャ垻鈧娲滈弫濠氬春閳ь剚銇勯幒鎴濐仾闁抽攱鍨块弻娑樷槈濮楀牆浼愭繝娈垮櫙缁犳垿婀佸┑鐘诧工閹冲孩绂掓潏鈹惧亾鐟欏嫭绀冩俊鐐扮矙瀵偊骞樼紒妯轰汗閻庤娲栧ú銈夌嵁濡ゅ懏鈷掑〒姘e亾婵炰匠鍛床闁割偁鍎辩壕褰掓煛瀹擃喒鍋撴俊鎻掔墢閹叉悂寮崼婵婃憰闂佹寧绻傞ˇ顖炴倿濞差亝鐓曢柟鏉垮悁缁ㄥジ鏌涢敐搴″箻缂佽鲸鎸婚幏鍛村礈閹绘帒澹堥梻浣瑰濞诧附绂嶉鍕靛殨妞ゆ劧绠戠壕濂告煟閹邦厽缍戞繛鍫熷姍濮婃椽宕橀崣澶嬪創闂佸摜濮甸懝鎯у祫闂佸憡顨堥崑鎰板绩娴犲鐓冮柦妯侯槹椤ユ粌霉濠婂懎浠滄い顓″劵椤﹁櫕銇勯妸銉含鐎殿噮鍋嗛埀顒婄秵閸撴稓澹曢挊澹濆綊鏁愭径瀣敪婵犳鍠栭崐鎼佹箒濠电姴锕ゅΛ妤呮偂閹邦儮搴ㄥ炊瑜濋崝鐔兼煃瑜滈崜姘辩矙閹烘洘鎳屽┑鐘愁問閸ㄤ即顢氶鐘愁潟闁圭儤鍨熷Σ鍫熸叏濡も偓濡宕滄潏鈺冪=闁稿本姘ㄥ瓭闂佹寧娲忛崕鑼矚鏉堛劎绡€闁搞儴鍩栭弲婵嬫⒑闂堟稓澧曢柟宄邦儔瀵娊顢橀姀鈾€鎷洪梺鍛婃崄鐏忔瑩宕㈠☉銏$厱闁靛ǹ鍎抽崺锝団偓瑙勬礃濡炰粙宕洪埀顒併亜閹哄秹妾峰ù婊勭矒閺岀喐娼忛崜褏蓱缂佺虎鍙€閸╂牠濡甸崟顖涙櫆闁兼祴鏅濋弳銈夋⒑閸濆嫭婀扮紒瀣灴閸┿垺鎯旈妶鍥╂澑闂佸搫娲ㄦ刊顓㈠船閸︻厾纾介柛灞剧懅缁愭梻绱撻崒娑滃閾荤偤鏌涢弴銊ユ灓濞存粍鐟╁缁樻媴閸涘﹤鏆堝┑鐐额嚋缁犳挸鐣烽姀锝冧汗闁圭儤鍨归敍娑㈡⒑閸︻厼鍔嬫い銊ユ閸╂盯骞嬮敂鐣屽幈濠电娀娼уΛ妤咁敂閳哄懏鐓冪憸婊堝礈濞嗘垹绀婂┑鐘叉搐缁犳牠姊洪崹顕呭剱缂傚秴娲弻宥夊传閸曨偂绨藉┑鐐跺亹閸犲酣鍩為幋锔绘晩閻熸瑦甯為幊鎾诲煝閺傚簱妲堥柕蹇娾偓鍐插婵犲痉鏉库偓鎰板磻閹剧粯鐓冮悷娆忓閻忔挳鏌熼瑙勬珚妤犵偞鎹囬獮鎺楀幢濡炴儳顥氶梻浣哥秺濡法绮堟笟鈧弻銊╁Χ閸涱亝鏂€闂佺粯蓱瑜板啴寮搁妶鍡欑闁割偅绮庨惌娆撴煛瀹€瀣М妤犵偛娲、妤佹媴閸欏浜為梻鍌欑劍閹爼宕愬Δ鍛獥闁归偊鍠楀畷鍙夌節闂堟侗鍎忛柣鎺戠仛閵囧嫰骞掗幋婵愪患闂佺粯甯楀浠嬪蓟濞戙垹绠涙い鏍ㄧ〒閵嗗﹪姊哄ú璇插箺妞ゃ劌鎳橀崺鐐哄箣閿旂粯鏅╃紓浣圭☉椤戝棝鎮鹃崼鏇熲拺缂備焦锕╁▓鏃傜磼缂佹ê绗ч柛鎺撳浮瀹曞ジ鎮㈡搴g嵁闂佽鍑界紞鍡涘礈濞戙埄鏁婇柡鍥ュ灪閳锋垿鏌i悢鐓庝喊闁搞倗鍠庨埞鎴︻敊閻愵剚姣堥悗娈垮枟婵炲﹪宕洪敓鐘茬<婵犲﹤鎷嬮崯搴ㄦ⒑閼姐倕孝婵炲/鍥х妞ゆ劦鍋傜槐锟�547闂傚倸鍊搁崐鎼佸磹妞嬪海鐭嗗ù锝夋交閼板潡姊洪鈧粔鏌ュ焵椤掆偓閸婂湱绮嬮幒鏂哄亾閿濆簼绨介柨娑欑洴濮婃椽鎮烽弶搴撴寖缂備緡鍣崹鍫曞春濞戙垹绠虫俊銈勮兌閸橀亶姊洪崫鍕妞ゃ劌妫楅埢宥夊川鐎涙ḿ鍘介棅顐㈡祫缁插ジ鏌囬鐐寸厸鐎光偓鐎n剙鍩岄柧缁樼墵閹鏁愭惔鈥茬盎濡炪倕楠忛幏锟�4濠电姷鏁告慨鐑藉极閹间礁纾婚柣鎰惈閸ㄥ倿鏌涢锝嗙缂佺姵澹嗙槐鎺斺偓锝庡亾缁扁晜绻涘顔荤凹闁哄懏鐓¢弻娑㈠Ψ閵忊剝鐝栧銈忓瘜閸ㄨ泛顫忓ú顏呭仭闂侇叏绠戝▓鍫曟⒑缁嬫鍎戦柛鐘崇墵瀹曟椽濮€閵堝懐鐫勯梺閫炲苯澧村┑锛勬暬瀹曠喖顢欓崜褎婢戦梻浣筋潐閸庢娊顢氶鈶哄洭鏌嗗鍡忔嫼缂備礁顑嗛娆撳磿閹扮増鐓欓柣鐔哄閹兼劙鏌i敐鍛Щ妞ゎ偅绮撻崺鈧い鎺戝閳ь兛绶氬顕€宕煎┑鍡氣偓鍨攽鎺抽崐鏇㈠疮椤愶妇宓侀柟鎵閳锋帡鏌涚仦鍓ф噮妞わ讣绠撻弻娑橆潩椤掑鍓板銈庡幖閻忔繈锝炲⿰鍫濈劦妞ゆ巻鍋撻柣锝囧厴椤㈡盯鎮滈崱妯绘珖闂備線娼х换鍫ュ垂閸濆嫧鏋斿Δ锝呭暞閳锋垿姊婚崼鐔剁繁婵$嫏鍐f斀闁炽儴娅曢崰姗€鏌涢埞鍨伈鐎殿噮鍣e畷濂告偄閸濆嫬绠ラ梻鍌欒兌椤㈠﹪锝炴径鎰闁哄洢鍨洪崕宥嗙箾瀹割喕绨奸柣鎾跺枛閺岋綁寮崼鐔告殸闁荤姵鍔х槐鏇犳閹烘挻缍囬柕濞垮劤閻熸煡鎮楅崹顐g凡閻庢凹鍣i崺鈧い鎺戯功缁夐潧霉濠婂懎浠︾紒鍌涘浮閹剝鎯斿Ο缁樻澑闂備胶绮崝妯衡枖濞戞碍顫曢柨鏇炲€归悡鏇熶繆閵堝懎顏柣婵愪簻鑿愰柛銉戝秴濮涢梺閫炲苯澧紒瀣笩閹筋偅绻濆▓鍨仭闁瑰憡濞婇獮鍐ㄧ暋閹佃櫕鐎诲┑鐐叉閸ㄧ敻宕虹仦鍓х閻庢稒岣块惌鎺旂磼閻樺磭澧电€殿喛顕ч埥澶愬閻樼數鏉搁梻浣呵圭换鎰板箺濠婂牆鏋侀柡宥庡幗閳锋垹绱掗娑欑婵炲懏姊荤槐鎺旂磼濡偐鐤勯悗娈垮枦椤曆囧煡婢跺ň鍫柛娑卞灡濠㈡垿姊绘担鐟邦嚋缂佽鍊块獮濠冩償椤帞绋忛梺鍐叉惈閹冲繘鍩涢幋锔界厱婵炴垶锕崝鐔兼煙閾忣偅绀堢紒杈ㄥ笚濞煎繘濡搁敂缁㈡Ч婵°倗濮烽崑娑氭崲濮椻偓楠炲啴鍩¢崘鈺佺彴闂佽偐鈷堥崜锕傚疮鐎n喗鈷掑ù锝呮啞閸熺偛銆掑顓ф疁鐎规洖缍婇獮搴ㄥ礈閸喗鍠橀柛鈺嬬節瀹曘劑顢欑憴鍕伖闂備浇宕甸崑鐐电矙閸儱鐒垫い鎺嗗亾闁告ɑ鐗楃粩鐔煎即閵忊檧鎷绘繛杈剧到閹诧紕鎷归敓鐘插嚑妞ゅ繐妫涚壕濂告煏婵炲灝濡煎ù婊冩贡缁辨帡顢氶崨顓炵閻庡灚婢樼€氫即鐛崶顒夋晣闁绘ɑ褰冪粻濠氭⒒閸屾瑧顦﹂柟纰卞亞閳ь剚鍑归崜娑㈠箲閵忋倕绠抽柡鍐ㄦ搐灏忛梻浣告贡鏋紒銊у劋缁傚秴饪伴崼鐔哄幐闂佹悶鍎洪悡渚€顢旈崼鐔封偓鍫曟煠绾板崬鍘撮柛瀣尭閳绘捇宕归鐣屽蒋闂備胶枪椤戝懘鏁冮妶澶樻晪闁挎繂娲﹀畷澶愭偠濞戞帒澧查柣搴☆煼濮婅櫣鎷犻垾宕団偓濠氭煕韫囧骸瀚庨柛濠冪箓椤繒绱掑Ο璇差€撻梺鑽ゅ枛閸嬪﹪宕电€n剛纾藉ù锝呭閸庢劙鏌涢妸銊ュ姷婵☆偆鍠庨—鍐Χ閸℃ê钄奸梺鎼炲妼缂嶅﹪骞冮悙鍝勫瀭妞ゆ劗濮崇花濠氭⒑閸︻厼鍔嬮柛鈺侊躬瀵劍绻濆顓炩偓鍨叏濡厧浜鹃悗姘炬嫹40缂傚倸鍊搁崐鎼佸磹閹间礁纾瑰瀣捣閻棗銆掑锝呬壕闁芥ɑ绻冮妵鍕冀閵娧呯厒闂佹椿鍘介幐楣冨焵椤掑喚娼愭繛鍙夌墪鐓ら柕濞у懍绗夐梺鍝勫暙閻楀﹪鎮″▎鎾寸厵妞ゆ牕妫楅懟顖氣枔閸洘鈷戠€规洖娲ㄧ敮娑欐叏婵犲倻绉烘鐐茬墦婵℃悂濡烽钘夌紦闂備線鈧偛鑻晶鐗堢箾閹寸姵鏆鐐寸墬閹峰懘宕ㄦ繝鍕ㄥ亾椤掑嫭鐓熼幖鎼灣缁夐潧霉濠婂啰鍩i柟顔哄灲瀹曞崬鈽夊▎蹇庡寲闂備焦鎮堕崕鑽ゅ緤濞差亜纾婚柟鎹愵嚙缁€鍌炴煕濞戝崬寮炬俊顐g矌缁辨捇宕掑顑藉亾瀹勬噴褰掑炊閵婏絼绮撻梺褰掓?閻掞箓宕戦敓鐘崇厓闁告繂瀚崳褰掓煢閸愵亜鏋旈柍褜鍓欓崢婊堝磻閹剧粯鐓曢柡鍥ュ妼娴滅偞銇勯幘瀛樸仢婵﹥妞介獮鎰償閿濆洨鏆ゆ繝鐢靛仩鐏忔瑦绻涢埀顒傗偓瑙勬礃閸ㄥ潡鐛Ο鑲╃<婵☆垵顕ч崝鎺楁⒑閼姐倕鏋戦柣鐔村劤閳ь剚鍑归崜鐔风暦閵忥絻浜归柟鐑樻尨閹锋椽姊洪崨濠勭畵閻庢凹鍘奸蹇撯攽鐎n偆鍘遍柟鍏肩暘閸ㄥ綊鎮橀埡鍌欑箚闁告瑥顦慨鍥殰椤忓啫宓嗙€规洖銈搁幃銏ゅ传閸曨偄顩梻鍌氬€烽懗鍓佹兜閸洖绀堟繝闈涙灩濞差亜鍐€妞ゆ劑鍎卞皬缂傚倷绶¢崑鍕偓娈垮墴濮婂宕掑顑藉亾妞嬪孩顐芥慨姗嗗厳缂傛氨鎲稿鍫罕闂備礁鎼崯顐﹀磹婵犳碍鍎楅柛鈩冾樅瑜版帗鏅查柛顐亜濞堟瑩姊洪懡銈呮瀾閻庢艾鐗撳顕€宕煎┑鍡欑崺婵$偑鍊栧Λ渚€锝炴径灞稿亾閸偆澧垫慨濠勭帛閹峰懘宕ㄦ繝鍌涙畼濠电偞鎸荤喊宥夈€冩繝鍌滄殾闁靛繈鍊栫€电姴顭跨捄鐑橆棡闁诲孩妞介幃妤呭礂婢跺﹣澹曢梻浣告啞濞诧箓宕滃☉銏犲偍闂侇剙绉甸埛鎴︽煕濠靛棗顏╅柡鍡欏仱閺岀喓绮欓崹顔规寖婵犮垼顫夊ú鐔肩嵁閹邦厽鍎熸繛鎴烆殘閻╁酣姊绘笟鈧ḿ褎顨ヨ箛鏇燁潟闁哄洠鍋撻埀顒€鍊块幊鐘活敆閸屾粣绱查梻浣告惈閸燁偊宕愰幖浣稿嚑婵炴垶鐟f禍婊堟煏韫囧﹤澧茬紒鈧€n喗鐓欐い鏃囶潐濞呭﹥銇勯姀鈩冪闁挎繄鍋ら、姗€鎮滈崱姗嗘%婵犵數濮烽弫鎼佸磻閻樿绠垫い蹇撴缁€濠囨煃瑜滈崜姘跺Φ閸曨垼鏁冮柕蹇婃櫆閳诲牓姊虹拠鈥虫珯缂佺粯绻堝畷娲焵椤掍降浜滈柟鐑樺灥椤忣亪鏌嶉柨瀣诞闁哄本绋撴禒锕傚箲閹邦剦妫熼梻渚€鈧偛鑻崢鍝ョ磼椤旂晫鎳囬柕鍡曠閳诲酣骞囬鍓ф闂備礁鎲″ú锕傚礈閿曗偓宀e潡鎮㈤崗灏栨嫼闂佸憡鎸昏ぐ鍐╃濠靛洨绠鹃柛娆忣槺婢ц京绱掗鍨惞缂佽鲸甯掕灒闂傗偓閹邦喚娉块梻鍌欐祰椤鐣峰Ο琛℃灃婵炴垯鍩勯弫浣衡偓鍏夊亾闁告洦鍓涢崢鍛婄箾鏉堝墽鍒板鐟帮躬瀹曟洝绠涢弬璁崇盎濡炪倖鎸撮崜婵堟兜閸洘鐓欏瀣閳诲牓鏌涢妸鈺冪暫鐎规洘顨婂畷銊╊敍濞戞ḿ妯嗛梻鍌氬€搁崐椋庢濮樿泛鐒垫い鎺戝€告禒婊堟煠濞茶鐏︾€规洏鍨介獮鏍ㄦ媴閸︻厼骞橀梻浣告啞閸旀ḿ浜稿▎鎾虫槬闁挎繂鎳夐弨浠嬫煥濞戞ê顏柡鍡╁墴閺岀喖顢欓悾灞惧櫚閻庢鍠栭悥濂哥嵁鐎n噮鏁囬柣鎰儗閸熷本绻濋悽闈浶fい鏃€鐗犲畷鏉课旈崨顔芥珖闂佸啿鎼幊搴g矆閸屾稓绠鹃柟瀵稿仧椤e弶銇勯锝嗙闁哄被鍔岄埞鎴﹀幢濡桨鐥柣鐔哥矌婢ф鏁Δ鍛柧闁哄被鍎查悡鏇㈡煃閳轰礁鏆熼柟鍐叉嚇閺岋綁骞橀崘娴嬪亾閹间讲鈧棃宕橀鍢壯囨煕閹扳晛濡垮ù鐘插⒔缁辨帡鎮欓浣哄嚒缂備礁顦晶搴ㄥ礆閹烘鐓涢柛娑卞枛娴滄粎绱掗悙顒€顎滃瀛樻倐瀵彃鈹戠€n偀鎷洪梻鍌氱墛缁嬫挻鏅堕弴鐔虹閻犲泧鍛殼濡ょ姷鍋涘Λ婵嬪极閹邦厼绶為悗锛卞嫬顏归梻鍌欑濠€杈ㄧ仚濠电偛顕崗姗€宕洪妷锕€绶為悗锝冨妺缁ㄥ姊洪幐搴㈩梿妞ゆ泦鍐惧殨妞ゆ洍鍋撻柡灞剧洴閸╃偤骞嗚婢规洖鈹戦敍鍕杭闁稿﹥鐗滈弫顕€骞掑Δ浣规珖闂侀潧锛忛埀顒勫磻閹炬剚娼╅柣鎰靛墮椤忥拷28缂傚倸鍊搁崐鎼佸磹閹间礁纾瑰瀣椤愪粙鏌ㄩ悢鍝勑㈢痪鎹愵嚙椤潡鎳滈棃娑樞曢梺杞扮椤戝洭骞夐幖浣哥睄闁割偁鍨圭粊锕傛⒑閸涘﹤濮﹂柛鐘崇墱缁粯绻濆顓犲幈闂佽宕樼亸娆戠玻閺冨牊鐓冮柣鐔稿缁犺尙绱掔紒妯肩疄濠殿喒鍋撻梺鎸庣箓濡盯濡撮幇顑╂柨螖婵犱胶鍑归梺鍦归崯鍧楁偩瀹勬壋鏀介悗锝庝簻缁愭盯鏌f惔銏⑩姇瀹€锝呮健瀹曘垽鏌嗗鍡忔嫼闂佸憡绻傜€氼剟寮虫繝鍥ㄧ厱閻庯綆鍋呯亸鐢电磼鏉堛劌绗ч柍褜鍓ㄧ紞鍡涘磻閸涱厾鏆︾€光偓閳ь剟鍩€椤掍緡鍟忛柛锝庡櫍瀹曟垶绻濋崶褏鐣烘繛瀵稿Т椤戝懘宕归崒娑栦簻闁规壋鏅涢悘鈺傤殽閻愭潙鐏存慨濠勭帛閹峰懘宕ㄦ繝鍐ㄥ壍婵犵數鍋犻婊呯不閹达讣缍栨繝闈涱儏鎯熼梺鍐叉惈閸婂憡绂掗銏♀拺閻庡湱濮甸妴鍐偣娴g懓绲婚崡閬嶆煕椤愮姴鍔滈柣鎾寸懇閺岋綁骞囬棃娑橆潽缂傚倸绉甸崹鍧楀蓟閻旂厧绀傞柛蹇曞帶閳ь剚鍔欓弻锛勪沪閻e睗銉︺亜瑜岀欢姘跺蓟濞戙垹绠婚柛妤冨仜椤洤螖閻橀潧浠滅紒缁橈耿瀵偊骞樼紒妯绘闂佽法鍣﹂幏锟�1130缂傚倸鍊搁崐鎼佸磹閹间礁纾瑰瀣捣閻棗銆掑锝呬壕闁芥ɑ绻冮妵鍕冀閵娧呯厒闂佹椿鍘介幑鍥蓟濞戙垹绠婚柤纰卞墻濡差噣姊洪幖鐐插缂佽鐗撳濠氬Ω閳哄倸浜滈梺鍛婄箓鐎氬懘濮€閵忋垻锛滈梺閫炲苯澧寸€规洘甯¢幃娆戔偓鐢登归獮鍫熺節閻㈤潧浠﹂柛銊ョ埣閺佸啴顢曢敃鈧紒鈺冪磽娴h疮缂氱紒鐘荤畺閺屾盯顢曢敐鍥╃暭闂佺粯甯楅幃鍌炲蓟閿涘嫪娌紒瀣仢閳峰鎮楅崹顐g凡閻庢凹鍣i崺鈧い鎺戯功缁夐潧霉濠婂嫮鐭掗柨婵堝仱瀹曞爼顢楁担鍙夊闂傚倷绶¢崑鍡涘磻濞戙垺鍤愭い鏍ㄧ⊕濞呯姴螖閿濆懎鏆為柣鎾寸懇閺屾盯骞嬪▎蹇婂亾閺嶎偀鍋撳鐐