Supported in part by National Nature Science Foundations of China (NSFC)(10925522, 11021092), and by the Spanish Ministerio de Economía y Competitividad (MINECO) and the European Regional Development Fund (ERDF)(FIS201784038C21P, FIS201784038C22P, SEV20140398)
A production representation of partialwave
Article funded by SCOAP^{3}
Pionnucleon (
At low energies, baryon chiral perturbation theory (BChPT) is one of such modelindependent methods in the study of
A rigorous manner is to construct partialwave
The advantage of the use of the PKU representation is twofold. On the one hand, it is suitable for pole analysis. The PKU representation separates partial waves into various terms contributing either from poles or branch cuts. The corresponding phase shifts extracted from PKU representation are sensitive to subthreshold poles, enabling one to determine pole positions rather accurately. Furthermore, each phase shift contribution has a definite sign, which makes possible the disentanglement of hidden poles from a background. On the other hand, it respects causality honestly. In the PKU representation, the pole contributions are regarded either as hidden poles or as known poles fixed by experiments, while the cut contribution is estimated from perturbative BChPT amplitudes and uncertainties from such an estimation is known to be severely suppressed. Importantly, the philosophy of the PKU representation is not to directly unitarize the amplitude itself, instead, it unitarizes the lefthand (and inelastic) cuts of the perturbative amplitude and hence hazardous spurious poles, violating causality, can be avoided [
In Ref. [
In Sect.
The isospin structure of the
where
As for the Lorentz structure, for an isospin index
where
One can substitute the nucleon spinors
the first and second subscripts refer to the helicities of the initial and final nucleon respectively, and subscripts "
where
Throughout this work, the partialwave label
The branchcut structure of the partialwave pionnucleon scattering amplitudes is generally discussed in Refs. [
The branch cuts of the partial wave
The
where
Here
with
With the above Lagrangians, it is readily to derive the
The tree and loop amplitudes are collected in Appendices. A.1 and A.2 respectively. As pointed out in Ref. [
The analytical expressions for all the involved loop integrals are compiled in Appendix A.4. Those expressions are obtained by means of dispersion relations with the spectral functions calculated using Cutkosky rule. Therefore, in principle, the BChPT amplitudes shown in this paper are calculable on the whole complex Mandelstam plane.
The PKU representation of the partial wave
where
with
where
Specifically, the partialwave
Taking into account the realistic
where
The fact that
The production presentation of the
For easy explanation of the polehunting strategy, the
In above, the term
The knownpole contribution can be fixed with their corresponding experimental information. The known poles contain bound states and abovethreshold resonances, both of which are well determined experimentally. In our case, the
Known poles in the two
−  { 

−  {Δ(1620), Δ(1900)}  
{ 
{ 

−  {Δ(1910)}  
−  { 

−  {Δ(1232), Δ(1600), Δ(1920)} 
which contribute to
The cut contribution can be estimated from BChPT amplitudes as demonstrated in subsection
throughout this work:
In above we discussed the validity region of BChPT. However, in no way one should expect that the contribution beyond the validity region of BChPT should be negligible. The two parameters
From the discussions of the constant
Furthermore, to compute the cut contribution, the masses, the nucleon axial charge and the pion decay constant in the BChPT amplitudes take the following values:
while the relevant LECs,
Fit I is in absence of explicit
With the above preparations, we are now in the position to perform phase shift analyses, and explore possible hidden poles with the equipment of the PKU representation.
In Ref. [
In what follows, it will be shown that the
The knownpole and cut contributions in
The figure indicates that the convergence property of the current method is not so satisfactory, i.e.
(color online) PKUrepresentation analysis of the
To proceed, the position of the hidden pole can be determined by a fit to the RS phase shift data. The fit results in a crazy resonance below threshold, which is quite stable against the variation of the cutoff parameters. The hidden pole positions with various cutoff parameters are listed in
The chisquares here are calculated using the bootstrap method, and so as the other five channels.
. The fit results are plotted inThe
The
pole position /MeV 



−0.08  814(3)i141(8)  1.46 
−1.00  882(2)i190(4)  1.31 
−9.00  960(2)i192(2)  1.14 
−25.0  976(2)i187(1)  1.14 
(color online) Fit with an extra hidden pole in the
The pole position of this crazy resonance is given by
The numbers in the first brackets are systematical errors responsible for the variation of the cutoff parameter
In the fits for different channels the statistical errors are much more smaller than the systematical errors and hence are negligible. Such small errors may come form the way we use to handle the data and may not be so trustworthy. Anyhow, the systematical errors from the evaluation of lefthand cuts are of course more important physically.
. By adding the two uncertainties in quadrature, our final reported result is
It is compatible with the determination reported by the
Note that we have also tried to fit the data of a larger energy region, i.e. below
Likewise in the
(color online) PKUrepresentation analysis of the
Following Ref. [
Even if we put a resonance (usually appearing in pair according to analyticity), it automatically converts into two virtual states. This amazing fact supports crucially the stability of our whole program. As already discussed in Ref. [
The

pole position /MeV 


−1.00  983  1.63 
−9.00  962(1)  1.37 
−25.0  948(1)  2.08 
(color online) The fit with two extra virtual states in
It should be pointed out that no satisfied fit can be achieved with
Note that similar to what is discussed in Ref. [
In principle, one may also perform fit with the cutoff parameter
The physical mechanism of the
Case in
(color online) PKUrepresentation analysis of the
This observation may indicate that the evaluation of
However if one insists on getting a good fit to the data, the easiest way is to employ a pole term as an extra background, similar to what have been done in the
The

pole position /MeV 


−9.00  875  2.13 
−25.0  887  1.86 
(color online) The fit with one extra virtual state as background in
It is interesting to find that the location of the virtual state serving as background is not so far away from the threshold (although it lies on the left of the
Similar to the
(color online) PKUrepresentation analysis of the
The
The

pole position /MeV 


−1.00  832, 799  4.62 
−9.00  784(1)i 164(2)  1.80 
−25.0  807(1)i 219(2)  4.35 
(color online) The fit with extra poles as background in
Only when
The situation in
(color online) PKU analysis of
The fit results with an extra resonance as background are shown in
The

pole position /MeV 


−9.00  755(1)i, 116(1)  3.64 
−25.0  756(1)i, 167(2)  2.48 
(color online) The fit with one extra resonance as background in
The discrepancy between known contributions and data in
(color online) PKUrepresentation analysis of
In this channel the
(color online) The known contributions with
From the above discussions, it is found that, to obtain good description of the data in
●
●
●
The poles in
(color online) The relationship between the location of a resonance pole
All the
where
For the scattering lengths (volumes)
channel  

(−169.0~−165.4)  −169.9(19.4)  
(79.7~86.0)  86.3(10.4)  
(51.0~67.3)  70.7(4.3)  
(45.1~57.0)  41.0(3.1)  
(30.1~31.1)  29.4(3.9)  
−198.8  −211.5(2.8) 
channel  

(1.0~2.5)  
(15.2~22.3)  
(39.0~206.6)  
(−227.0~−40.3)  
(−31.2~−29.9)  
4.8 
Furthermore, the nonrelativistic limit of PKU representation (i.e., Ning Hu representation [
where
channel  

0.56  0.47  0.52  0.38  0.38  0.30 
Note that in
In summary, this paper, as a followup of Ref. [
The other four channels are also investigated quantitatively: a good description of the data in
Although the analyses of lowenergy pionnucleon scatterings based on PKU representation with lefthand cuts simply from perturbative calculation are completed, the present work could still be improved in future. The major weakness of the present work, though not vital in drawing the current conclusions as we believe, comes from the uncertainty when calculating distant lefthand cuts. Hence through our work, we only cautiously draw physical conclusions extracted from discrepancies at qualitative level, and avoid any results drawn only depending on minor discrepancies between data and the cuts. It is worth stressing that our major conclusions are actually established based on the negative definiteness of the background contribution, which is supported by quantum mechanical scattering theory. The
The tree diagrams of
The tree diagrams at
The tree diagrams at
The tree diagrams at
● At
● At
● At
● At
Note that Eq. (A3) can be absorbed in Eq. (A1) by redefinition of the
According to the power counting rule of BChPT, for a loop diagram with
Consequently, the diagrams, which are classified into three groups and displayed in
Oneloop diagrams at
Oneloop diagrams at
Oneloop diagrams at
To be specific, the first group (
In this paper we use PassarinoVeltman notations [
● One point function
● Two point functions
● Three point functions
where
● Four point functions
where
For brevity, the abbreviations of the loop functions in
Abbreviations of the loop functions.
full notations  abbreviations 



























●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
The UV divergent pieces in loop functions are dealt with in dimensional regularization and expressed in terms of
Such conditions can be fulfilled by choosing suitable values of the nucleon wave function renormalization constant
To be specific, the bare parameters can be split as follows
where the LECs with superscript "0" denote the bare quantities and the ones with "
and in addition,
where
See
where
According to the spirit of EOMS scheme, the PCB terms hidden in the loop function come from the infrared regular parts and behave as polynomials of the external momenta. The infrared regular parts of related loop functions can be found in Ref. [
where the quantities with no superscripts refer to the physical ones. To cancel the power counting breaking terms, the
The counterterms for the cancellations of the PCB terms can be obtained by the replacements:
In our numerical computations, the tensor integrals like
●
●
●
Now only the expressions of those scalar loop functions are needed. One point and two point functions are simple when the dimension
where
with
where
In Eqs. (A56) and (A58) we set
Those spectral functions can be obtained by Cutkosky rule in physical region^{⑨}
Generally speaking Cutkosky rule on physical cut may not give the complete spectral functions due to the existence of some anomalous singularities like the anomalous triangle singularity. Fortunately, in twobody scattering processes with stable particles such singularities do not appear on the physical sheet, so Cutkosky rule is valid.
and are listed below. For three point functions,
and for the four point function,
In this subsection, the influence of the variations of the cutoff parameters is discussed. The set
(color online) The
As already mentioned in the preceding paper [
In principle one can also figure out the locations of the spurious poles given by conventional unitarization approaches at
At
where
(color online) Phase shifts from
In the end we emphasize that, as claimed in the preceding paper [