Kaon-nucleon scattering to one-loop order in heavy baryon chiral perturbation theory

# Kaon-nucleon scattering to one-loop order in heavy baryon chiral perturbation theory

Bo-Lin Huang    Yun-De Li Department of Physics, Yunnan University, Kunming 650091, China
August 24, 2019
###### Abstract

We calculate the T-matrices of kaon-nucleon () and antikaon-nucleon () scattering to one-loop order in SU(3) heavy baryon chiral perturbation theory (HBPT). The low-energy constants (LECs) and their combinations are then determined by fitting the phase shifts of scattering and the corresponding data. This leads to a good description of the phase shifts below 200 MeV kaon laboratory momentum. We obtain the LEC uncertainties through statistical regression analysis. We also determine the LECs through the use of scattering lengths in order to check the consistency of the HBPT framework for different observables and obtain a consistent result. By using these LECs, we predict the elastic scattering phase shifts and obtain reasonable results. The scattering lengths are also predicted, which turn out to be in good agreement with the empirical values except for the isospin-0 scattering length that is strongly affected by the resonance. As most calculations in the chiral perturbation theory, the convergence issue is discussed in detail. Our calculations provide a possibility to investigate the baryon-baryon interaction in HBPT.

PACS numbers

13.75.Jz,12.39.Fe,12.38.Bx

###### pacs:
Valid PACS appear here
preprint: APS/123-QED

## I Introduction

Chiral perturbation theory (ChPT) is the effective field theory of quantum chromodynamics (QCD) at energies below the scale of chiral symmetry breaking GeVmach2011; sche2012. As we all know, the relativistic framework for baryons in ChPT does not naturally provide a simple power-counting scheme as for mesons because of the baryon mass, which does not vanish in the chiral limit. Relativistic (such as infrared regularizationbech1999 and the extended on-mass-shell schemegege1999; fuch2003) and heavy baryonjenk1991; bern1992 approaches have been proposed and developed to solve the power-counting problem. Recently, the relativistic approaches have made some progress. For some observables, the chiral series even show a better convergence than the heavy baryon approachren2012; alar2013. However, the heavy baryon chiral perturbation theory (HBPT) is still a reasonable and useful tool in the study of the meson-baryon scattering. The expansion in HBPT is expanded simultaneously in terms of and , where represents the meson momentum or its mass or the small residue momentum of baryon in the nonrelativistic limit and denotes the baryon mass in the chiral limit.

Over the years, the low-energy processes have been widely investigated in the SU(2) HBPT. Fettes et al. have investigated pion-nucleon scattering up to the fourth orderfett1998; fett2000. The low-energy constants (LECs) of the SU(2) chiral pion-nucleon Lagrangian were determined by fitting various empirical phase shifts. The threshold parameters were also predicted in Refs. fett1998; fett2000. Krebs, Gasparyan, and Epelbaum calculated the chiral three-nucleon force at fifth order by using the LECs from scattering at fourth orderkreb2012, and Entem et al. considered peripheral nucleon-nucleon scattering at fifth order through using these LECsente2015. These predictions are in good agreement with the data.

For processes involving kaons or hyperons, the situation is more complicated. One has to use the SU(3) HBPT in comparison to the SU(2) sector of scattering. These involve several new problems. First, there are more unknown LECs needed to be determined through experimental data which are insufficient at present. Second, the kaon mass is larger than the pion mass duo to broken SU(3) symmetry. In fact, the pertinent expansion parameter results in a low convergence rate. Third, the and scattering are inelastic and elastic at low energies, respectively. These involve inconsistent predictions duo to the dynamical differences between and scattering. However, Kaiser achieved some success when analyzing the and scattering lengths in SU(3) HBPTkais2001. Then Liu and Zhu generalized this method to the predictions of meson-baryon scattering lengthsliu20071; liu20072; liu2011; liu2012. They obtained reasonable results. But higher-order corrections are needed to consider due to the complicated convergence. That leads to involving more LECs and needs more experimental meson-baryon scattering lengths which are unavailable for now. In this paper, we will determine the LECs by fitting the phase shifts of the elastic scattering and make predictions up to one-loop order, as the scattering in the framework of SU(2) HBPT.

In Sec. II, we summarize the Lagrangians involved in the evaluation up to one-loop order contributions. In Sec. III, we present the T-matrices of the elastic and scattering. In Sec. IV we explain how we calculate the phase shifts and the scattering lengths. Section V contains the results and discussions and also includes a brief summary. Appendix A contains the amplitudes from one-loop diagrams. Apppendix B contains the threshold T-matrices and the relation between the threshold T-matrices with the s-wave scattering lengths.

## Ii Lagrangian

Our calculation of the elastic and scattering is based on the effective SU(3) chiral Lagrangian in HBPT

 L=Lϕϕ+LϕB. (1)

Here, the SU(3) matrix and represent the pseudoscalar Goldstone fields () and the octet baryons fields, respectively. The lowest-order effective SU(3) chiral Lagrangians for meson-meson and meson-baryon interaction takes the formbora1997

 L(2)ϕϕ=f24tr(uμuμ+χ+), (2)
 L(1)ϕB = tr(i¯¯¯¯B[v⋅D,B])+Dtr(¯¯¯¯BSμ{uμ,B}) (3) +Ftr(¯¯¯¯BSμ[uμ,B]),

where denotes the covariant derivative

 [Dμ,B]=∂μB+[Γμ,B] (4)

and is the covariant spin operator à la Pauli-Lubanski

 Sμ=i2γ5σμνvν, (5)

with

 [Sμ,Sν]=iϵμνσρvσSρ,{Sμ,Sν}=12(vμvν−gμν), (6)

where is the completely antisymmetric tensor in four indices, . The chiral connection and the axial vector quantity contain even number meson fields and odd number meson fields, respectively. The SU(3) matrix collects the pseudoscalar Goldstone fields. The parameter is the pseudoscalar decay constant in the chiral limit. The axial vector coupling constants and can be determined by fitting the semileptonic decays ()bora1999. The combination with results in explicit chiral symmetry breaking. The complete heavy baryon Lagrangian at next-to-leading order can be written as

 L(2)ϕB=L(2,1/M0)ϕB+L(2,ct)ϕB, (7)

where denotes corrections of dimension two with fixed coefficients and stems from the expansion of the original relativistic leading-order Lagrangian bora1997. These read

 L(2,1/M0)ϕB = D2−3F224M0tr(¯¯¯¯B[v⋅u,[v⋅u,B]]) (8) −D212M0tr(¯¯¯¯BB)tr(v⋅uv⋅u) −DF4M0tr(¯¯¯¯B[v⋅u,{v⋅u,B}]) −12M0tr(¯¯¯¯B[Dμ,[Dμ,B]]) +12M0tr(¯¯¯¯B[v⋅D,[v⋅D,B]]) −iD2M0tr(¯¯¯¯BSμ[Dμ,{v⋅u,B}]) −iF2M0tr(¯¯¯¯BSμ[Dμ,[v⋅u,B]]) −iF2M0tr(¯¯¯¯BSμ[v⋅u,[Dμ,B]]) −iD2M0tr(¯¯¯¯BSμ{v⋅u,[Dμ,B]}),

where denotes the baryon mass in the chiral limit. The remaining heavy baryon Lagrangian proportional to the low-energy constants can be obtained from the relativistic effective meson-baryon chiral Lagrangianolle2006

 L(2,ct)ϕB = bDtr(¯¯¯¯B{χ+,B})+bFtr(¯¯¯¯B[χ+,B]) (9) +b0tr(¯¯¯¯BB)tr(χ+)+b1tr(¯¯¯¯B{uμuμ,B}) +b2tr(¯¯¯¯B[uμuμ,B])+b3tr(¯¯¯¯BB)tr(uμuμ) +b4tr(¯¯¯¯Buμ)tr(Buμ)+b5tr(¯¯¯¯B{v⋅uv⋅u,B}) +b6tr(¯¯¯¯B[v⋅uv⋅u,B])+b7% tr(¯¯¯¯BB)tr(v⋅uv⋅u) +b8tr(¯¯¯¯Bv⋅u)tr(Bv⋅u) +b9tr(¯¯¯¯B{[uμ,uν],[Sμ,Sν]B}) +b10tr(¯¯¯¯B[[uμ,uν],[Sμ,Sν]B]) +b11tr(¯¯¯¯Buμ)tr(uν[Sμ,Sν]B).

The first three terms proportional to the LECs result in explicit symmetry breaking. Notice that the LECs have dimension .

## Iii T-Matrices

We are considering only elastic kaon-nucleon and antikaon-nucleon scattering in the center-of-momentum system (CMS) with . The T-matrix takes the following form:

 T(I)KN,¯¯¯¯KN = (EN+MN2MN){V(I)KN,¯¯¯¯KN(q) (10) +iσ⋅(q′×q)W(I)KN,¯¯¯¯KN(q)},

with the nucleon mass, the nucleon energy, and the total isospin of the kaon-nucleon system. Furthermore, refers to the non-spin-flip kaon-nucleon or antikaon-nucleon amplitude, and refers to the spin-flip kaon-nucleon or antikaon-nucleon amplitude.

Now, we calculate the T-matrices order by order. Note that we choose for the sake of convenience throughout this paper. The leading-order amplitudes corresponding to diagrams (1a) and (1b) in Fig. 1 (including also the crossed diagram) read

 V(1)KN(q)=13f2K[−3w+(D2+3F2)q2zw], (11)
 W(1)KN(q)=−D2+3F23wf2K, (12)
 V(0)KN(q)=(2D2−6DF)q2z3wf2K, (13)
 W(0)KN(q)=−2D2−6DF3wf2K, (14)
 V(1)¯¯¯¯KN(q)=12f2K[w−(D−F)2q2zw], (15)
 W(1)¯¯¯¯KN(q)=−(D−F)22wf2K, (16)
 V(0)¯¯¯¯KN(q)=16f2K[9w−(D+3F)2q2zw], (17)
 W(0)¯¯¯¯KN(q)=−(D+3F)26wf2K, (18)

where denotes the kaon CMS energy and the angular variable between and . We also take the renormalized kaon decay constant instead of (the chiral limit value).

At next-to-leading order , one has the contribution from the second row diagrams of Fig. 1 (including also the crossed diagrams) involving the vertices from the Lagrangian and . First, for the vertices from the , we have

 V(1)KN(q) = 16M0f2K(D2+3F2)[−w2+2(z+2)q2 (19) −3(1+z)D2+3F2q2−2z(1+z)q4w2],
 W(1)KN(q) = −13M0f2K(D2+3F2)[1−(1+z)q2w2], (20)
 V(0)KN = 13M0f2K(D2−3DF)[−w2+2(z+2)q2 (21) −2z(1+z)q4w2],
 W(0)KN=−13M0f2K(2D2−6DF)[1−(1+z)q2w2], (22)
 V(1)¯¯¯¯KN = −14M0f2K[(D−F)2w2+2z(D−F)2q2 (23) −(1+z)q2],
 W(1)¯¯¯¯KN(q)=−(D−F)22M0f2K, (24)
 V(0)¯¯¯¯KN(q) = −112M0f2K[(D+3F)2w2 (25) +2z(D+3F)2q2−9(1+z)q2],
 W(0)¯¯¯¯KN(q)=−(D+3F)26M0f2K. (26)

Second, for the vertices from the , we introduce

 αη=4bDm2η+3b0(m2π+m2η), απ=4bDm2π+3b0(m2π+m2η) (27)

to make the following expressions more compact. The amplitudes read

 V(1)KN = −1f2K[4(bD+b0)m2K+(C1+C2)w2−C1zq2] (28) +zq212w2f2K[(D+3F)2αη+3(D−F)2απ],
 W(1)KN = −1f2KC3−112w2f2K[(D+3F)2αη (29) +3(D−F)2απ],
 V(0)KN = 1f2K[4(bF−b0)m2K+(C4+C5)w2−C4zq2] (30) +zq212w2f2K[9(D−F)2απ−(D+3F)2αη],
 W(0)KN = −1f2KC6−112w2f2K[9(D−F)2απ (31) −(D+3F)2αη],
 V(1)¯¯¯¯KN = 1f2K[(2bF−2bD−4b0)m2K−12(C1+C2 (32) −C4−C5)w2+12(C1−C4)zq2] +zq22w2f2K(D−F)2απ,
 W(1)¯¯¯¯KN=12f2K(C3+C6)+12w2f2K(D−F)2απ, (33)
 V(0)¯¯¯¯KN = −1f2K[2(3bD+bF+2b0)m2K+12(3C1+3C2 (34) +C4+C5)w2−12(3C1+C4)zq2] +zq26w2f2K(D+3F)2αη,
 W(0)¯¯¯¯KN=12f2K(3C3−C6)+16w2f2K(D+3F)2αη, (35)

where

 C1=−4b1−4b3−2b4, C2=−4b5−4b7−2b8, C3=4b10+b11, C4=−4b2+4b3−2b4, C5=−4b6+4b7−2b8, C6=−4b9−b11. (36)

The six combinations of LECs are introduced in order to reduce the number of LECs.

At the third order , we have the one-loop diagram contributions and the counterterm contributions. The nonvanishing one-loop diagrams generated by the vertices of and are shown in Fig. 2. The counterterm contribution estimated from resonance exchange was found to be much smaller than the chiral loop contribution in the case of threshold scatteringbern1993; bern1995. Kaiser assumed that similar features hold for threshold and scattering and also achieved some successkais2001. Liu and Zhu also ignored the counterterm contributions when they calculated meson-baryon scattering lengthsliu20071. Later, Liu and Zhu claimed that the counterterm contributions are larger than the one-loop diagrams contributions in some T-matrices in Ref. liu20072. But, Liu and Zhu did not consider the resonance contribution when determining the LECs and their combinations in Ref. liu20072. However, we are not considering the counterterm contributions when calculating T-matrices at in this paper. The nonvanishing one-loop amplitudes corresponding to loop diagrams are too tedious; thus, we present these amplitudes separately in Appendix A. In loop calculations, we use dimensional regularization and the minimal subtraction scheme to evaluate divergent loop integralshoof1979; bern19951; mojz1998; bouz2000; bouz2002. We use in all loops instead of corresponding decay constants in respective loops. The difference appears at higher order.

## Iv Calculating phase shifts and Scattering lengths

The partial wave amplitudes , where refers to the orbital angular momentum and to the spin, are given in terms of the invariant amplitudes via

 f(I)l±s(q) = EN+MN16π(w+EN)∫+1−1dz[V(I)KN,¯¯¯¯KN(q)Pl(z) (37) +q2W(I)KN,¯¯¯¯KN(q)(Pl±1(z)−zPl(z))],

where are conventional Legendre polynomials. For the energy range considered in this paper, the phase shifts are evaluated from (for discussions about the phase shifts, see Refs. gass1991; fett1998)

 δ(I)l±s(q)=arctan(qRef(I)l±s(q)). (38)

Based upon relativistic kinematics, there is a relation between the CMS on-shell momentum and the momentum of the incident kaon in the laboratory system ,

 q2=M2Nq2Km2K+M2N+2MN√m2K+q2K. (39)

Near threshold the scattering length for s waves and the scattering volume for p waves is given byeric1988

 a(I)l±s=limq→0q−2l−1% tanδ(I)l±s(q). (40)

## V Results and Discussion

Before calculating the phase shifts and the threshold parameters, we have to determine the LECs. There are 14 unknown LECs in and also need to be determined. Fortunately, after the regrouping, we determine only and the six LEC combinations which were defined by Eq. (III). Throughout this paper, we use MeV, MeV, MeV, MeV, MeV, MeV, and pdg2014, and for the axial vector coupling constants we use and . We also take as the chiral symmetry breaking scale.

We first determine , , and through the formulas of the octet-baryon masses and given in Ref.  bern19951. We take in the loops in these formulas, respectively. The baryon masses MeV, MeV, MeV, and MeV and the pion-nucleon () term hofe2015 are used to fit these four parameters. We obtain

 M0=646.30±47.72MeV, bD=0.043±0.008GeV−1, bF=−0.498±0.003GeV−1, b0=−1.003±0.047GeV−1 (41)

with . In our fitting, the new from Ref. hofe2015 is taken; thus, we obtain different values than those in Ref. liu20071. Note that the uncertainty of the th LEC (here, refers to one of the , , and ) is purely the statistical uncertainty that is a measure of how much this particular parameter can change while maintaining a good description of the fitted data, as detailed in Refs. doba2014; carl2015.

We now determine the six LEC combinations by using the phase shifts of the SP92 solution, GW Institute for Nuclear Studies, for kaon-nucleon () scattering analysisSAID; hysl1992. Since the SP92 give no uncertainties for the phase shifts, we set a common uncertainty of to all values before the fitting procedure. For the parameters , we use the data of the S11, P11 and P13 waves between 50 and 90 MeV (15 data points in total) to fit. As to the , we fit the data of the S01, P01 and P03 waves at MeV. The resulting LECs are given by

 C1=1.99±0.11GeV−1, C2=−0.45±0.11GeV−1, C3=6.36±0.09GeV−1, (42)

with and

 C4=3.01±0.21GeV−1, C5=−5.10±0.21GeV−1, C6=−5.13±0.12GeV−1, (43)

with . For the uncertainties, see the above description. The corresponding S- and P-wave phase shifts are shown in Fig. 3. For the P01 wave, the description of the phase shifts is surprisingly good even at higher and lower energies. The remaining waves are also in good agreement with the empirical phase shifts below 150 MeV and purely overestimated at large kaon momentum. However, to sum up, we obtain a good description for these six lowest partial waves in this one-loop order calculation of the scattering up to surprisingly large kaon momenta.

In order to check the consistency of the ChPT framework for different observables, we now determine the low-energy constants by the scattering lengths. However, there are six LEC combinations , but only four scattering lengths can be used. At this time, we take the threshold T-matrices to calculate the scattering lengths; see Appendix B. For comparison, we use the two scattering lengths and from the SP92hysl1992 to determine the two LEC combinations and . The resulting LECs are given by

 C12=C1+C2=1.59GeV−1, C45=C4+C5=−1.99GeV−1. (44)

The LEC combination determined by the phase shifts from Eq. (V) is , while the from Eq. (V) is . The results are consistent with the LEC combinations determined by the scattering lengths from Eq. (V) within the limit of error.

In the following, we make predictions for the scattering through the above LECs determined by the phase shifts and the corresponding data. At present, the existing empirical phase shifts of the scatteringarme1969; hemi1975; gopa1977; ks20131; ks20132 are all above the kaon laboratory momentum of 200 MeV (corresponding to the CM energy of around 1460 MeV); thus, the resulting S- and P-wave phase shifts are shown in Fig. 4 without the empirical phase shifts. From the plot of Fig. 4, none of the phase shifts shows the resonant behavior. The recent multichannel partial-wave analysis for scattering KS2013ks20131; ks20132 includes a variety of resonances, such as the S01, S11, P01, P03, P11, and P13 wave including the , , , , and resonances, respectively. But all the resonances considered by the KS2013 do not contribute to the phase shifts below the CM energy of 1460 MeV , because they are so far away. Thus, the predictions for the phase shifts of the partial waves in the scattering are reasonable. However, as we all know, there exists the resonance as a quasibound state below the threshold energy in a S01 wave. To solve this problem, the solution is given by the nonperturbative resummation approach with a phenomenologically successful description of the scattering amplitudekais1995; oset1998; olle2001; lutz2002 (for a review on this issue, see Ref. hyod2012).

Now let us apply the above LECs to estimate the kaon-nucleon and antikaon-nucleon scattering lengths. We have two approaches to predict the scattering lengths. One is through the use of the Eq. (40) and the LECs from Eqs. (V) and (V). As before, we do not fit data below MeV (for , ); hence, the scattering lengths are predictions. The scattering lengths are obtained by using an incident kaon momentum MeV and approximating its value at the threshold. As a result, no errors are provided. We present the values of the scattering lengths as “Prediction A” in Table 1. The other is through the use of the formalism in Appendix B and the LECs from Eq. (V). We show the values of the scattering lengths as “Prediction B” in Table 1. The values purely have slightly difference than Ref. liu20071 because different data are taken. In addition, for comparison, the various empirical values are also shown in Table 1. We successfully predict the isospin-1 scattering lengths. For the isospin-0 scattering length, we obtain very small negative values differing from the empirical values. However, the error will cover the difference. As expected, we fail to predict the isospin-0 scattering length that is dominated by the resonance. The situation is the same as the prediction for the phase shifts of the scattering. From this, it would be more reliable to predict the et al. scattering, although no empirical data are available. These works will be presented in our next publication.

Finally, we discuss the convergence. This issue is addressed for scattering in Fig. 5. For S01, the leading order is zero. The second-order contribution is much bigger than the third order and describes well the partial wave. For S11, we find that there is sizeable cancellation between the second and the third orders. This feature has also occurred in the chiral expansion of a threshold T-matrixkais2001. For P-waves, the second order is much more important than the others in all partial waves and nearly describes well the empirical phase shifts. The situation is simpler than the scatteringfett1998. According to these results, a higher-order calculation is needed.

In summary, we have calculated the T-matrices for and scattering to one-loop order in SU(3) HBPT. We then fit the , the SP92 phase shifts of scattering, and the corresponding data to determine the LECs. This leads to a good description of the phase shifts below 200 MeV kaon momentum in the laboratory frame. We also discuss the LEC uncertainties through statistical regression analysis. In order to check the consistency of the ChPT framework for different observables, we determine the LECs by the scattering lengths, make a comparison with the LECs determined by the phase shifts, and obtain a consistent result. By using these LECs, we predict the scattering phase shifts and obtain a reasonable result. The s-wave scattering lengths are predicted with the energy-dependent solution (Prediction A) and in the case of the threshold T matrices (Prediction B). As expected, we fail to predict the isospin-0 scattering length which is dominated by the resonance. This issue can be successfully solved by the nonperturbative resummation approach, and that is not the focus of this paper. Finally, we check the convergence of the scattering and find that the large cancellations occurred between the second and third orders in the S11 wave. In order to determine accurately the LECs and make better predictions, higher-order calculations are needed in SU(3) HBPT. In addition, the prediction for the octet meson and octet baryon interaction ( such as and scattering) will be calculated in the next publication. We also expect our calculations to provide a possibility to investigate the baryon-baryon interaction in HBPT.

###### Acknowledgements.
This work is supported by the National Natural Science Foundation of China under Grants No. 11465021 and No. 11065010. B. L. H. thanks Norbert Kaiser (Technische Universität München), Yan-Rui Liu (Shandong University) and Jia-Qing Zhu (Yunnan University) for very helpful discussions.

## Appendix A One-loop amplitudes

In this Appendix, we present the nonvanishing amplitudes from nonvanishing one-loop diagrams. The amplitudes are shown one diagram by one diagram (but similar diagrams are grouped together) due to the expressions being too tedious. For giving the expressions as many details as possible, we use several functions in the following expressions. The normal unit step function

 θ(x)={1x>0,0x<0 (45)

is used. We also define

 Q2=2q2(z−1), (46)
 r(m)=√|1−4m2Q2|. (47)

Figures 2(a)-2(d):

 V(1)KN = zq2144π2w2f4K{αDFπ[w3−wm2π+πm3π+(3wm2π−2w3)lnmπλ−2(w2−m2π)3/2lnw+√w2−m2πmπ] (48) +αDFK[w3−wm2K+πm3K+(3wm2K−2w3)lnmKλ−2(w2−m2K)3/2lnw+√w2−m2KmK] +αDFη[w3−wm2η+πm3η+(3wm2η−2w3)lnmηλ−2(m2η−w2)3/2(arccoswmη)θ(m2η−w2) −2(w2−m2η)3/2(lnw+√w2−m2ηmη)θ(w2−m2η)]},
 W(1)KN=−V(1)KNzq2, (49)
 V(0)KN = zq2144π2w2f4K{βDFπ[w3−wm2π+πm3π+(3wm2π−2w3)lnmπλ−2(w2−m2π)3/2lnw+√w2−m2πmπ] (50) +βDFK[w3−wm2K+πm3K+(3wm2K−2w3)lnmKλ−2(w2−m2K)3/2lnw+√w2−m2KmK] +βDFη[w3−wm2η+πm3η+(3wm2η−2w3)lnmηλ−2(m2η−w2)3/2(arccoswmη)θ(m2η−w2) −2(w2−m2η)3/2(lnw+√w2−m2ηmη)θ(w2−m2η)]},
 W(0)KN=−V(0)KNzq2, (51)
 V(1)¯¯¯¯KN = zq296π2w2