Singularities and accumulation of singularities of πN scattering amplitudes

Qu-Zhi Li and Han-Qing Zheng

1 Department of Physics Peking University,Beijing 100871,China

2 College of Physics,Sichuan University,Chengdu,Sichuan 610065,China

3 Collaborative Innovation Center of Quantum Matter,Beijing,China

Abstract It is demonstrated that for the isospin I=1/2 πN scattering amplitude,TI=1/2(s,t),are two accumulation points of poles on the second sheet of complex s plane,and are hence accumulation of singularities of TI=1/2(s,t).Foris the accumulation point of poles on the second sheet of the complex s plane.The proof is valid up to all orders of chiral expansions.

Keywords:dispersion relations,partial wave,analyticity

where dJis the standard Wigner d-function.Substituting equation(6)into equation(7),then using the relation between the Wigner d-function and the Legendre polynomial

one gets

with

where F stands for A or B.

(1)In the region s ∈(cR,sR),the S-matrixSJs

BP神經(jīng)網(wǎng)絡(luò)實質(zhì)上是一種采用反向傳播學(xué)習(xí)算法的多層前饋網(wǎng)絡(luò)。從結(jié)構(gòu)上講，BP網(wǎng)絡(luò)是一種分層型網(wǎng)絡(luò)，由輸入層、隱層和輸出層組成。層與層之間采用全互連方式，同一層單元之間不存在相互連接。

The sign of the infinity is independent of the angular momentum J.Things will be different,however,when s tends to cL.It turns out thattends toandt ends to ±(- 1)J+12∞.Taking the definition of Smatrix equation(15),and the fact that 2iρ(s)is negative at s=cR,cL,conclusions can be drawn that:5The emergence of S matrix zeros due to kinematical reasons is known,and χPT amplitudes can be consistently embedded into a unitary S matrix element.See for example the discussion in[4].

The kinematic factor ρ(s)is defined as

The scalar function AIand BImay be calculated by chiral perturbation theory at low energies.4In the region[sL,sR]chiral expansions are expected to work well.Here,we present the

which satisfy the optical theorem for partial waves,

A modification of the dispersion representation for function f defined in[1]in each channel may be needed accordingly,similar to what has been discussed in S11channel[1].Conclusions are similar in all channels:the contribution from the virtual pole to the phase shift is always exactly canceled by the additional contribution from the dispersion integral in f,when vL=cL(vR=cR).Taking for example the case that there exists only one virtual pole vLon the left.In this situation,

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1 2,

The parity eigenstates can be constructed by the linear combinations of helicity amplitudes

±()must contain a zero6This argument has been actually used a long time ago in[5].We thank one referee who brings our attention to this paper.,whiledoes not need to have.

(2)In the region s ∈(sL,cL),bothandcontain a zero for J=1/2,5/2,9/2,…,while bothandcontain a zero for J=3/2,7/2,11/2,….Or in other words,for I=1/2(I=3/2),there exists a virtual pole vL∈(sL,cL)in each partial wave amplitude with even(odd)angular momentum.

Taking a few examples,in S31channel there is no virtual pole,while in p waves,

· P11channel contains a virtual pole vR∈(cR,sR),as well as the nucleon bound state;

· P33channel contains a virtual pole in vL∈(sL,cL).

· P13channel contains a virtual pole in vR∈(cR,sR);

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whereSvLrepresents the‘S matrix’of the newly found virtual pole vL,while Sidepicts all other known poles.7For more details about the production representation,one can refer to[6—8].Such an Scutis real and negative when s ∈(sL,cL),8Sphy(sL)=+1 according to equation(15).This is because sL is the branch point caused by the u channel continuous absorptive singularities,T(sL)itself is regular.For resonances Si=+1,for bound state or virtual state Si=-1,when s=sL.Particularly is negative when s ∈(sL,vL)and positive when s ∈(vL,cL).so the ‘spectral

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數(shù)據(jù)的整理及分析軟件為Microsoft Office Excel 2013及SPSS 25.0軟件。各組間盜血程度及病變鎖骨下動脈收縮期峰值、對側(cè)椎動脈收縮期峰值的比較用非參數(shù)檢驗。合并椎動脈狹窄性病變組與對照組束臂試驗結(jié)果的比較用卡方檢驗。以P＜0.05為差異有統(tǒng)計學(xué)意義。

Hence,according to equation(28),as J →∞,the leading contribution ofis10The contribution from chiral corrections will not diverge,since the contribution itself is regular when s ∈(sL,sR),and notice that |Pl(x)|＜1 when x ∈[-1,+1]for arbitrary l—which is proved using the Laplace formula of Legendre function.

One further proves that the accumulation of zeros at cLoccurs and that s=cLis an accumulation of singularities of T3/2(s,t)on the second sheet of the s plane.

Acknowledgments

The authors would like to thank Wei-Nian Zhang at Sichuan University,and Wei-Da Zheng for helpful discussions.This work is supported in part by National Nature Science Foundations of China under contract numbers 11975028 and 10925522.