Pion-nucleon scattering to order p 4 in SU(3) heavy baryon chiral perturbation theory
aa r X i v : . [ nu c l - t h ] J u l Pion-nucleon scattering to order p in SU(3) heavy baryonchiral perturbation theory Bo-Lin Huang ∗ Department of Physics, Jishou University, Jishou 416000, China
July 22, 2020
Abstract
We calculate the T -matrices of elastic pion-nucleon ( πN ) scattering up to fourth order inSU(3) heavy baryon chiral perturbation theory. The baryon mass in the chiral limit M andthe pertinent low-energy constants are determined by fitting to empirical πN phase shiftsbelow 200 MeV pion laboratory momentum in the physical region. The scattering lengthsand scattering volumes are also extracted from the chiral amplitudes, and turn out to be ingood agreement with those of other approaches and the available experimental values. Onthe basis of the various phase shifts and the threshold parameters, the convergence of thechiral expansion is analyzed in detail. A good convergence for πN scattering observables isobtained when working up to fourth order. In addition, we extend the fit to 250 MeV pionlaboratory momentum, and obtain also a good description of the empirical S - and P -wavephase shifts. PACS numbers:
Keywords:
Chiral perturbation theory, pion-nucleon scattering, chiral convergence
Chiral perturbation theory (ChPT) allows one to analyze hadronic processes at low energies, thatare not accessible by a perturbative expansion in the strong coupling constant α s of quantumchromodynamics (QCD)[1, 2, 3, 4]. ChPT is an efficient framework to calculate in a model-independent way e.g. the amplitudes of pion-nucleon ( πN ) scattering below the chiral symmetrybreaking scale Λ χ ≃ M in the chiral limit. Over the years, several approacheshave been proposed to solve this problem. Heavy baryon ChPT (HB χ PT) [5, 6], the infraredregularization of covariant baryon ChPT [7], and the extended-on-mass-shell scheme for baryonChPT [8, 9] are some popular approaches. The last two approaches are fully relativistic and havelead to substantial progress in many aspects as documented in ref. [10, 11, 12, 13, 14, 15, 16].However, the expressions from the loop diagrams become rather complicated in these fully rel-ativistic approaches [17, 18, 19]. On the other hand, HB χ PT is a well-established and versatiletool for the study of low-energy hadronic processes. The amplitudes in HB χ PT proceeds si-multaneously in terms of p/ Λ χ (non-relativistic contributions from tree- and loop-diagrams) and p/M (relativistic corrections), where p denotes the meson momentum (or mass) or the smallresidual momentum of a baryon in a low-energy process.In recent years, there has been renewed interest in theoretical studies of elastic meson-baryonscattering at low energies. These are not only concerned with the description of the strongmesonic interaction, but also with the chiral properties of the baryons. The low-energy processesbetween pions and nucleons have been investigated extensively in SU(2) HB χ PT in refs. [20,21, 22, 23]. Furthermore, the subthreshold parameters of pion-nucleon scattering have also beendeeply studied by combined with Roy-Steiner equations and SU(2) ChPT [24, 25]. For processesinvolving kaons or hyperons, the situation becomes more involved, since one has to work outthe consequences of three-flavor chiral dynamics. In a previous paper [26] we have investigated KN and ¯ KN elastic scattering up to one-loop order in SU(3) HB χ PT by fitting low-energy ∗ [email protected] KN -scattering and obtained quite reasonable results.This approach was then extended by predicting the amplitudes of pseudoscalar-meson octet-baryon scattering in all channels, with the pertinent low-energy constants fitted to partial-wavephase shifts of elastic πN and KN scattering [27]. At this point one should note that the detailedpredictions of SU(3) HB χ PT for the meson-baryon scattering lengths have been given previouslyin refs. [28, 29, 30, 31, 32]. Moreover, these studies in SU(3) HB χ PT has been extended topartial-wave phase shifts, the pion-nucleon sigma term, and other quantities. In a recent paper[33], we have calculated the complete T -matrices of pion-nucleon scattering up to third order inSU(3) HB χ PT and obtained as a byproduct nucleon properties, like the pion-nucleon sigma term σ πN . A good description of the phase shifts of πN scattering below 200 MeV has been obtained,and then it can serve as a consistency check to consider also the other meson-baryon scatteringchannels. However, no good convergence of the chiral expansion is observed in calculationsthat terminate at third order. Therefore, we will compute in this paper the T -matrices for πN scattering up to fourth order in SU(3) HB χ PT. The parameter M and the pertinent low-energy constants (LECs) will be determined by fitting to empirial S - and P –wave phase shiftsof πN scattering. In particular, the LECs related to contact terms of chiral dimension will beobtained separately. The convergence of the chiral expansions will be analyzed and discussed indetail. Since chiral expansions should become more accurate by increasing the chiral order, thedescription of the empirical πN phase shifts at somewhat higher energies will also be presented.The present paper is organized as follows. In Sec. 2, we summarize the Lagrangians involvedin the evaluation of the fourth-order contributions. In Sec. 3, we present some explicit expressionsfor the T -matrices of elastic πN scattering at order p . In Sec. 4, we outline how to derive phaseshifts, scattering lengths, and scattering volumes from the T -matrices. Section 5 contains thepresentation and discussion of our results and it also includes a brief summary. In order to calculate the pion-nucleon scattering amplitude up to order p in SU(3) heavy baryonchiral perturbation theory, one has to evaluate tree- and loop diagrams with vertices from theeffective chiral Lagrangian: L eff = L (2) φφ + L (1) φB + L (2) φB + L (3) φB + L (4) φB . (1)These Lagrangians are written in terms of the traceless hermitian 3 × φ and B thatinclude the pseudoscalar Goldstone boson fields ( π , K , ¯ K , η ) and the octet-baryon fields ( N ,Λ, Σ, Ξ), respectively. The explicit form of the L (2) φφ , L (1) φB , L (2) φB and L (3) φB can be found inref. [33]. The complete fourth-order heavy baryon Lagrangian L (4) φB naturally splits up into twoparts: relativistic corrections with fixed coefficients, and counterterms proportional to new low-energy constants. The relativistic terms can be obtained from the original leading-order, next-to-leading-order and next-to-next-to-leading-order lorentz-invariant Lagrangians through pathintegral manipulations [6]. For three-flavors and at chiral order four, the lorentz-invariant meson-baryon Lagrangian has been constructed in ref. [34] and we can obtain from it the countertermsrelevant for our purpose. Since the dimension-four chiral meson-baryon Lagrangian L (4) φB is verylengthy, the explicit expressions will not be given in this paper, where we consider only elasticpion-nucleon scattering. Of course, the expansions in SU(3) and SU(2) HB χ PT are consistentwith each other. For this reason the notation of low-energy constants ¯ e i ( i = 14 , ..., , , ..., χ PT introduced in ref. [21] will also be used inthis paper. T -matrices for pion-nucleon scattering We are considering the elastic pion-nucleon scattering process π ( q )+ N ( p ) → π ( q ′ )+ N ( p ′ ) in thecenter-of-mass system (cms). In states with total isospin I = 1 / I = 3 / T -matrix takes the following form in spin-space: T ( I ) πN = V ( I ) πN ( w, t ) + i σ · ( q ′ × q ) W ( I ) πN ( w, t ) (2)where w = q = q ′ = ( m π + q ) / is the pion cms-energy, and t = ( q ′ − q ) = 2 q ( z −
1) is theinvariant momentum-transfer squared with z = cos θ the cosine of the angle θ between q and q ′ .2ased on relativistic kinematics, one gets the relation: q ′ = q = M N p m π + M N + 2 M N p m π + p , (3)where p lab denotes the momentum of the incident meson in the laboratory system. Furthermore, V ( I ) πN ( w, t ) refers to the non-spin-flip pion-nucleon scattering amplitude and W ( I ) πN ( w, t ) is calledthe spin-flip pion-nucleon scattering amplitude.From leading order O ( p ) up to third order O ( p ), the expressions for the amplitudes V ( I ) πN ( w, t )and W ( I ) πN ( w, t ) can be found in ref. [33]. At fourth order O ( p ), the SU(3) results for the πN amplitudes from tree diagrams are essentially the same as those calculated in SU(2) HB χ PT inref. [21]. After an appropriate renaming of the low-energy constants, these contributions read: V (3 / , N3LO) πN = − ( D + F ) M w f π [( t + 7 t w + 11 t w − tw + 4 w ) − (11 t + 49 t w + 32 tw + 4 w ) m π + (45 t + 110 tw + 26 w ) m π − t + 26 w ) m π + 54 m π ]+ m π − w M f π (4 m π − t − w ) + 116 M f π [4 C ( t + 2 w − m π ) m π + C (2 t + 4 tw − tm π − w m π + 8 m π ) + 4 C (3 tw + 14 w − tm π − w m π + 8 m π )+ 2 C ( − t − tw + 4 tm π )] − M f π C C w m π + 12 M f π [ H (4 m π − t − w ) m π + H (4 w + t − m π ) t + 3 H (4 m π − t − w ) w + 2 H w t ]+ 1 f π [4¯ e (4 m π − m π t − t ) + 8¯ e (2 w m π − tw ) + 16¯ e w ] , (4) W (3 / , N3LO) πN = ( D + F ) M w f π [( t + 5 t w + 3 tw − w ) − (9 t + 25 tw + 4 w ) m π + 3(9 t + 10 w ) m π − m π ] + w − m π M f π + 18 M f π [8 C m π + C (2 t − m π ) − C w − C ( t + 2 w − m π )] + 12 M f π H (8 w + t − m π )+ 1 f π [¯ e ( − m π + 4 t ) − e w ] , (5) V (1 / , N3LO) πN = ( D + F ) M w f π [( t + 7 t w + 11 t w + 16 w ) − (11 t + 49 t w + 32 tw + 16 w ) m π + 5(9 t + 22 tw + 4 w ) m π − t + w ) m π + 60 m π ]+ w − m π M f π (4 m π − t − w ) + 18 M f π [2 C ( t + 2 w − m π ) m π + C ( t + 2 tw − tm π − w m π + 4 m π ) + 2 C (3 tw + 14 w − tm π − w m π + 8 m π )+ 2 C ( t + 4 tw − tm π )] − M f π C C w m π + 1 M f π [ H ( − m π + t + 4 w ) m π − H (4 w + t − m π ) t − H (4 m π − t − w ) w + H w t ]+ 1 f π [4¯ e (4 m π − m π t − t ) + 8¯ e (2 w m π − tw ) + 16¯ e w ] , (6) W (1 / , N3LO) πN = ( D + F ) M w f π [( − t − t w − tw + 4 w ) + (9 t + 25 tw + 4 w ) m π − t + 10 w ) m π + 24 m π ] + m π − w M f π + 14 M f π [ − C m π + C ( t − m π ) − C w + 2 C ( t + 2 w − m π )] + H M f π (8 w + t − m π ) + 8 f π [¯ e (2 m π − t ) + 2¯ e w ] , (7)3here C i ( i = 0 , , ,
3) and H i ( i = 1 , , ,
4) are linear combinations of low-energy constantsdefinded in eq.(22) and eq.(27) of ref. [33], respectively. For the dimension-four LECs, we followthe same strategy as in ref. [21]. The subset of dimension-four low-energy constants ¯ e i ( i =19 , , , , , , ,
38) can be absorbed onto the dimension-two LECs c i ( i = 1 , , , c i ( i = 1 , , ,
4) introduced in SU(2)HB χ PT can be replaced equivalently by the combinations of LECs C i ( i = 0 , , ,
3) of SU(3)HB χ PT. Thus, ¯ e i ( i = 14 , , , ,
18) are the five remaining dimension-four LECs that arerelevant in our calculation. At fourth order, additional πN amplitudes from one-loop diagramsmust be taken into account. The pertinent one-loop diagrams generated by the interactionvertices from L (2) φφ , L (1) φB and L (2) φB are shown in Figure 1. The expressions for these one-loop πN amplitudes are rather tedious and will therefore not be reproduced here. The explicit analyticalexpressions for the loop contributions of order O ( p ) to elastic πN scattering can be obtainedfrom the authors upon request. For orientation of the readers, we remark that these πN loop-amplitudes are composed of the following basic loop-functions: J ( w, m ) = 1 i Z d D l (2 π ) D v · l − w )( m − l ) = w π (cid:16) − mλ (cid:17) + π √ w − m ln − w + √ w − m m ( w < − m ) , − π √ m − w arccos − wm ( − m < w < m ) , π √ w − m iπ − ln w + √ w − m m ! ( w > m ) , (8)1 i Z d D l (2 π ) D { , l µ , l µ l ν } ( m − l )[ m − ( l − k ) ] = { I ( t, m ) , k µ I ( t, m ) , g µν I ( t, m ) + k µ k ν I ( t, m ) } , (9) I ( t, m ) = 18 π ( − ln mλ − r − m t ln √ m − t + √− t m ) , (10) I ( t, m ) = 148 π ( m − t
12 + (cid:16) t − m (cid:17) ln mλ − (4 m − t ) / √− t ln √ m − t + √− t m ) , (11) I ( t, m ) = 124 π ( − m t − ln mλ − (cid:16) − m t (cid:17)r − m t ln √ m − t + √− t m ) . (12)Note that terms proportional to the divergent constant λ D − [ D − + ( γ E − − ln 4 π )] have beendropped. Furthermore, one observes that the one-loop amplitudes of order O ( p ) involve thefull set of dimension-two LECs b i ( i = D, F, , ...,
11) and not just the few combinations of LECs C i ( i = 0 , , ,
3) arising from tree-diagrams.
The partial-wave amplitudes f ( I ) j ( q ), where j = l ± / l to orbital angular momentum, are obtained from the non-spin-flip and spin-flip amplitudesby the following projection formula: f ( I ) l ± / ( q ) = M N π √ s Z +1 − dz n V ( I ) πN ( w, t ) P l ( z ) + q W ( I ) πN ( w, t )[ P l ± ( z ) − zP l ( z )] o , (13)where P l ( z ) denotes the conventional Legendre polynomial, and √ s = p m π + q + p M N + q is the total center-of-mass energy. For the energy range considered in this paper, the phase shifts δ ( I ) l ± / are calculated as (see also refs. [20, 35]) δ ( I ) l ± / ( q ) = arctan h | q | Re f ( I ) l ± / ( q ) i . (14)4igure 1: Nonvanishing one-loop diagrams contributing at chiral order four. The heavy dotsrefer to vertices from L (2) φB . Crossed graphs are not shown.The scattering lengths for S -waves and the scattering volumes for P -waves are obtained bydividing out the threshold behavior of the respective partial-wave amplitude and approachingthe threshold [36] a ( I ) l ± / = lim | q |→ | q | − l f ( I ) l ± / ( q ) . (15) Before presenting results, we have to determine the M and the LECs. Unfortunately, the 14dimension-two LECs can not be regrouped up to fourth order in SU(3) HB χ PT. Then, in total,there are 24 unknown constants that need to be determined. Throughout this paper, we also use m π = 139 .
57 MeV, m K = 493 .
68 MeV, m η = 547 .
86 MeV, f π = 92 .
07 MeV, f K = 110 .
03 MeV, f η = 1 . f π , M N = 938 . ± .
29 MeV, M Σ = 1191 . ± .
86 MeV, M Ξ = 1318 . ± .
30 MeV, M Λ = 1115 . ± .
58 MeV, λ = 4 πf π = 1 .
16 GeV, D = 0 .
80 and F = 0 .
47 [37, 38, 39].We have various fitting strategies to determine the pertinent constants, and the related dis-cussion is given in Ref. [33]. One of them is using the octet-baryon masses ( M N, Σ , Ξ , Λ ) and thephase shifts of πN scattering simultaneously. This can no longer be done up to fourth orderdue to the appearance of the more constants for octet-baryon masses. Thus, we determine the M and LECs by using the phase shifts of the WI08 solution [40, 41] for πN scattering directly.We choose a common uncertainty of ±
2% to all phase shifts before the fitting procedure. Thedata points of the S and P waves in the range of 50-150 MeV pion lab. momentum are used.Therefore, there are 66 data in total for this fitting. The resulting M and LECs can be found inFit 1 of Table 1. The uncertainty for the respective parameter is purely estimated (for a detaileddiscussion, see, e.g., Refs.[42, 43]). It is not surprising that most of LECs are natural size, i.e. theabsolute values of these LECs are between one and ten when one introduces the dimensionless5ECs (e.g., b ′ i = 2 M b i ), whereas some of LECs come out fairly large. The situation also canbe found in SU(2) HB χ PT [21]. We can also find that the baryon mass in the chiral limit M appears small. It is smaller than any physical value of the octet-baryon mass. However, the M value is reasonable because it is a non-phyiscal quantity. The corresponding S - and P -wavephase shifts are shown by the solid lines of Fig. 2. Obviously, we obtain an excellent descriptionof all waves. Especially, the description of the P P M and LECs have different values in the respective order. Thus, for the amplitudes up to theTable 1: Values of the various fits. The fit 1, 2, 3 refer to the best fit upto fourth, third and second order, respectively. For a detailed descriptionof these fits, see the main text. Note that the ( ∗ ) values are calculated by b i . Fit 1 Fit 2 Fit 3 M (MeV) 370 . ± .
11 1530 . ± .
87 373 . ± . b D (GeV − ) − . ± . b F (GeV − ) 0 . ± . b (GeV − ) − . ± . b (GeV − ) − . ± . b (GeV − ) 0 . ± . b (GeV − ) − . ± . b (GeV − ) 3 . ± . b (GeV − ) − . ± . b (GeV − ) 0 . ± . b (GeV − ) 4 . ± . b (GeV − ) 1 . ± . b (GeV − ) − . ± . b (GeV − ) − . ± . b (GeV − ) − . ± . C (GeV − ) − . ± . ( ∗ ) − . ± . − . ± . C (GeV − ) − . ± . ( ∗ ) − . ± . − . ± . C (GeV − ) − . ± . ( ∗ ) . ± .
22 2 . ± . C (GeV − ) − . ± . ( ∗ ) . ± .
04 1 . ± . H (GeV − ) 31 . ± .
36 8 . ± . H (GeV − ) 5 . ± .
53 5 . ± . H (GeV − ) − . ± . − . ± . H (GeV − ) − . ± . − . ± . e (GeV − ) 28 . ± . e (GeV − ) 25 . ± . e (GeV − ) 4 . ± . e (GeV − ) 5 . ± . e (GeV − ) 5 . ± . χ / d.o.f. 0 .
62 1 .
63 4 . æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - - p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - - p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - p lab H MeV L P h ase S h i ft s H d e g i L S31 P31 P33S11 P11 P13
Figure 2: Fits and predictions for the WI08 phase shifts versus the pion lab. momentum | p lab | inpion-nucleon ( πN ) scattering. The dot-dashed, dotted, dashed and solid lines refer to the bestfits up to first, second, third and fourth order, respectively. Fitting for all πN waves are the datain the range of 50-150 MeV. For higher and lower energies, the phase shifts are predicted.given order, the corresponding fit to determine their respective constants is necessary.In the following, let us apply the chiral fourth order amplitudes to estimate the thresholdparameters. Since we do not fit data below 50 MeV, the pion-nucleon scattering lengths andscattering volumes are now predictions. The threshold parameters are obtained by using anincident pion lab. momentum | p lab = 10MeV | and approximating their values at the threshold.The results are shown in Table 2 in comparison with the values of the various analyses. Obviously,our results for both scattering lengths and scattering volumes are consistent with the ones fromSU(2) HB χ PT and SP98 [21]. The values of SP98 are obtained by the use of dispersion relationswith the help of a fairly precise tree-level model. In addition, there are two experimental valuesfor scattering lengths in Table 2. The values of EXP2001 are from pionic hydrogen and deuteriumin ref. [44]. The latter, EXPnew, are obtained by combining with the analysis of the results fromrefs. [45, 46, 47, 48, 49, 50]. As expected, our results for the threshold parameters are consistentwith those values within errors.Next, we can discuss the convergence of the chiral expansion. In Fig. 2, we show the bestfits up to the respective order. In all partial-waves, the fourth-order corrections are smallerthan the third-order ones, indicating convergence. In most cases, the third-order corrections aresmaller than the second-order ones. The second-order corrections are smaller than the first-orderones in a few partial-waves. Therefore, the convergence of the chiral expansion becomes betteralong with the increase of the order. However, with the energy increasing the convergence ofTable 2: Values of the S - and P -wave scattering lengths and scattering volumes in comparisonwith the values of the various analyses.Our results SU(2) SP98 EXP2001 EXPnew a / (fm) − . ± .
003 -0.119 − . ± . − . ± . − . ± . a / (fm) 0 . ± .
002 0.249 0 . ± .
002 0 . +0 . − . . ± . a / (fm ) 0 . ± .
021 0.586 0 . ± .
005 ... ... a / (fm ) − . ± .
004 -0.054 − . ± .
008 ... ... a / − (fm ) − . ± .
012 -0.113 − . ± .
006 ... ... a / − (fm ) − . ± .
010 -0.181 − . ± .
007 ... ...7able 3: Values of the S - and P -wave scattering lengths and scattering volumesfrom the different order. O ( q ) O ( q ) O ( q ) O ( q ) a / (fm) − . − . ± . − . ± . − . ± . a / (fm) 0 .
225 0 . ± .
067 0 . ± .
066 0 . ± . a / (fm ) 0 .
241 0 . ± .
008 0 . ± .
014 0 . ± . a / (fm ) − . − . ± . − . ± . − . ± . a / − (fm ) − . − . ± . − . ± . − . ± . a / − (fm ) − . − . ± . − . ± . − . ± . p/ Λ χ . We can also study the convergence of the threshold parameters, seeTable 3. The two scattering lengths a / and a / from S waves are almost unchanged in differentorder, indicating convergence in each order. The other four scattering volumes from P waves areconsistent from order O ( q ). The calculation results indicate the convergence of the thresholdparameters is fast that is due to m π / Λ χ ∼ /
7. To sum up, we obtain a good convergence in πN scattering on the basis of the various phase shifts and the threshold parameters.Inspired by the excellent description of all waves and the good convergence at fourth order,now we extend the fit to 250 MeV pion lab. momentum (corresponding to an energy in CMSof √ s ∼ . ±
2% to all phase shifts. The datapoints of the S and P waves in the range of the 50-250 MeV pion lab. momentum are used. Thus,there are 126 data in total for this fitting. The resulting M and LECs are shown in Table 4.We obtain a large χ / d.o.f. ≃ .
15 because we choose a small common uncertainty to all phaseshifts. The value of the baryon mass in the chiral limit M is more close to the physical valueand seems more reasonable. The values of the LECs are also different from the values which areobtained by fitting in range of 50-150 MeV pion lab. momentum, see Table 1. The corresponding S - and P -wave phase shifts are shown in Fig. 3. This time, we obtain a good description of allwaves except for the P P P πN phase shifts at high energies, the other hadronic contributions can not be ignored.In summary, we calculated the T matrices for pion-nucleon scattering to the fourth order inTable 4: Values of the fit in the range of 50-250MeV at fourth order. M (MeV) 512 . ± . b (GeV − ) 0 . ± . b D (GeV − ) 1 . ± . b (GeV − ) − . ± . b F (GeV − ) − . ± . b (GeV − ) 0 . ± . b (GeV − ) 1 . ± . H (GeV − ) 46 . ± . b (GeV − ) − . ± . H (GeV − ) 10 . ± . b (GeV − ) − . ± . H (GeV − ) − . ± . b (GeV − ) 0 . ± . H (GeV − ) − . ± . b (GeV − ) 4 . ± .
29 ¯ e (GeV − ) 6 . ± . b (GeV − ) − . ± .
66 ¯ e (GeV − ) − . ± . b (GeV − ) 0 . ± .
01 ¯ e (GeV − ) 8 . ± . b (GeV − ) 8 . ± .
45 ¯ e (GeV − ) 5 . ± . b (GeV − ) 1 . ± .
28 ¯ e (GeV − ) 5 . ± . æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - - - p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - - - - p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - - p lab H MeV L P h ase S h i ft s H d e g i L æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - p lab H MeV L P h ase S h i ft s H d e g i L S31 P31 P33S11 P11 P13
Figure 3: Fits for the WI08 phase shifts versus the pion lab. momentum | p lab | in pion-nucleon( πN ) scattering at fourth order. Fitting for all πN waves are the data in the range of 50-250MeV.SU(3) HB χ PT. We fitted the WI08 phase shifts of πN scattering in range of 50-150 MeV pionlab. momentum to determine the M and the LECs. This led to an excellent description of S - and P -wave phase shifts below 200 MeV pion lab. momentum. The scattering lengths andscattering volumes were also calculated at this order, which turned out to be in good agreementwith those of other approaches and available experimental data. We discussed the convergenceof the chiral expansion based on the phase shifts and threshold parameters from the best fits upto the respective order in detail. To sum up, we obtained a good convergence in πN scatteringat this order. Finally, we extended the description for phase shifts of πN scattering to 250 MeVpion lab. momentum. The reasonable M and LECS are also obtained. They can be used forthe other physical processes as input. A good description of all waves except for the P πN scattering may be achieved throughincluding the resonance ∆(1232) and the other hadronic contributions. Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant No.11947036. I thank Norbert Kaiser (Technische Universit¨at M¨unchen), Yan-Rui Liu (ShandongUniversity), Li-Sheng Geng (Beihang University), Jun-Xu Lu (Beihang University) and JingOu-Yang (Jishou University) for very helpful discussions.
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