An extension to the Lüscher’s finite volume method above inelastic threashold (formalism)

# An extension to the Lüscher’s finite volume method above inelastic threashold (formalism)

Noriyoshi ISHII
Department of Physics, The University of Tokyo
E-mail: ishii@ribf.riken.jp
Speaker.
for HAL-QCD Collaboration
###### Abstract

An extension of the Lüscher’s finite volume method above inelastic thresholds is proposed. It is fulfilled by extendind the procedure recently proposed by HAL-QCD Collaboration for a single channel system. Focusing on the asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave functions (equal-time) near spatial infinity, a coupled channel extension of effective Schrödinger equation is constructed by introducing an energy-independent interaction kernel. Because the NBS wave functions contain the information of T-matrix at long distance, S-matrix can be obtained by solving the coupled channel effective Schrödinger equation in the infinite volume.

An extension to the Lüscher’s finite volume method above inelastic threashold (formalism)

\abstract@cs

The XXVIII International Symposium on Lattice Field Theory, Lattice2010 June 14-19, 2010 Villasimius, Italy

## 1 Introduction

The standard method to calculate the scattering phase shift on the lattice is the Lüscher’s finite volume method[1]. It utilizes the shift of energy spectra in the finite periodic box to calculate the scattering phase. The formalism is restricted to the elastic scattering region. One attempted to extend it above inelastic thresholds [2]. Although it can lead to a constraint imposed on the matrix elements of S-matrix, S-matrix elements cannot be obtained separately.

We propose a different method to extend it above the inelastic threshold. This is fulfilled by extending the procedure recently proposed by HAL-QCD Collaboration [3, 4]. We focus on the asymptotic behaviors of the equal-time Nambu-Bethe-Salpeter (NBS) wave functions near the spatial infinity, where the information of T-matrix is contained. We construct a coupled channel effective Schrödinger equation by introducing an energy-independent interaction kernel, so that it can generate all the NBS wave functions simultaneously. Once such an interaction kernel is constructed in the finite volume, S-matrix is obtained by solving the effective Schrodinger equation in the infinite volume. (The resulting wave functions are nothing but the NBS wave functions, which contain the information of T-matrix in their long distance part.) In this paper, we use - coupled system to formalize the procedure.

Contents are organized as follows. In Sect. 2, we consider the asymptotic long distance behavior of the NBS wave functions for - coupled system. We will see that they contain T-matrix in their long distance part. In Sect. 3, we extend an effective Schrödinger equation for the coupled channel system by introducing an energy-independent interaction kernel. The extension is performed so that it can generate all the NBS wave functions simultaneously as solutions. In Sect. 4, we derive the factorization formula, which plays a key role in constructing an effective Schrödinger equation in a coupled channel system in Sect. 3.

## 2 Asymptotic behaviors of equal-time Nambu-Bethe-Salpeter (NBS) wave functions for a coupled channel system

We consider - coupled system with MeV, MeV, and MeV. For notational simplicity, we treat them as bosons. We are interested in the energy region

 mN+mΛ≤E≤mN+mΛ+mπ,\specialhtml:\specialhtml: (2.0)

which is a combined region of (i) elastic region: and (ii) - coupling region above the threshold: .

We consider two incoming states at the same energy as and . and denote the asymptotic momenta for and systems, respectively. They are related to as

 E=√m2N+→p2+√m2Λ+→p2=√m2N+→q2+√m2Σ+→q2. (2.0)

We define the NBS wave functions for an incoming state as

 ψNΛ;NΛ(x1,x2;E) ≡ Z−1/2NZ−1/2Λ⟨0|T[N(x1)Λ(x2)]|N(→p)Λ(−→p),in⟩ (2.0) ψNΣ;NΛ(x1,x2;E) ≡ Z−1/2NZ−1/2Σ⟨0|T[N(x1)Σ(x2)]|N(→p)Λ(−→p),in⟩, (2.0)

and the NBS wave functions for an incoming state as

 ψNΛ;NΣ(x1,x2;E) ≡ Z−1/2NZ−1/2Λ⟨0|T[N(x1)Λ(x2)]|N(→q)Σ(−→q),in⟩ (2.0) ψNΣ;NΣ(x1,x2;E) ≡ Z−1/2NZ−1/2Σ⟨0|T[N(x1)Σ(x2)]|N(→q)Σ(−→q),in⟩, (2.0)

where , and denote local composite interpolating fields for nucleon, and baryons, respectively. , and denote the normalization factors involved in the limit , , and , respectively, as . By using the reduction formula, these NBS wave functions are related to S-matrix as

 ⟨N(p′1)Λ(p′2),out|N(→p)Λ(−→p),in⟩ = disc.+∫d4x1d4x2eip′1x1(□1+m2N)eip′2x2(□2+m2Λ)ψNΛ;NΛ(x1,x2;E) ⟨N(q′1)Σ(q′2),out|N(→p)Λ(−→p),in⟩ = ∫d4x1d4x2eiq′1x1(□1+m2N)eiq′2x2(□2+m2Σ)ψNΣ;NΛ(x1,x2;E),

and

 ⟨N(p′1)Λ(p′2),out|N(→q)Σ(−→q),in⟩ = ∫d4x1d4x2eip′1x1(□1+m2N)eip′2x2(□2+m2Λ)ψNΛ;NΣ(x1,x2;E) ⟨N(q′1)Σ(q′2),out|N(→q)Σ(−→q),in⟩ = disc.+∫d4x1d4x2eiq′1x1(□1+m2N)eiq′2x2(□2+m2Σ)ψNΣ;NΣ(x1,x2;E),

where “disc.” stands for disconnected terms, and denotes d’Alembert operator. These relations lead us to the following asymptotic behaviors of equal-time restrictions of NBS wave functions near spatial infinity () as

 ψNΛ;NΛ(→x−→y;E) ≡ limx0→+0ψNΛ;NΛ(→x,x0,→y,y0=0;E) ≃ ei→p⋅→r+2EN(p)+EΛ(p)TNΛ;NΛ(s)eiprpr+⋯ ψNΣ;NΛ(→x−→y;E) ≡ limx0→+0ψNΣ;NΛ(→x,x0,→y,y0=0;E) ≃ 2EN(q)+EΣ(q)TNΣ;NΛ(s)eiqrqr+⋯,

and

 ψNΛ;NΣ(→x−→y;E) ≡ limx0→+0ψNΛ;NΣ(→x,x0,→y,y0=0;E) ≃ 2EN(p)+EΛ(p)TNΛ;NΣ(s)eiprpr+⋯ ψNΣ;NΣ(→x−→y;E) ≡ limx0→+0ψNΣ;NΣ(→x,x0,→y,y0=0;E) ≃ ei→q⋅→r+2EN(q)+EΣ(q)TNΣ;NΣ(s)eiqrqr+⋯,

where denotes the on-shell T-matrix, i.e., , , , , and .

Derivations of these asymptotic behaviors of NBS wave functions are similar to single-channel case [4, 5, 6]. Here, we give a brief example of case.

 ψNΛ;NΛ(→r;E) = Z−1/2NZ−1/2Λ⟨0|N(→r)Λ(→0)|N(→p)Λ(−→p),in⟩ = Z−1/2NZ−1/2Λ∫d3k(2π)32EN(k)⟨0|N(→r)|N(→k)⟩⟨N(→k)|Λ(0)|N(→p)Λ(−→p),in⟩+I(→r),

where is referred to as the inelastic contribution, which corresponds to contributions not associated with single nucleon intermediate state. Since does not propagate long distance, we will neglect this term below. Because of Eq. (2), we have an equality

 ⟨N(k1)|Λ(0)|N(p1)Λ(p2),in⟩ = ⟨0|Λ(0)aN,in(k1)|N(p1)Λ(p2),in⟩ +iZ−1/2N∫d4x1eik1x1(□1+m2N)⟨0|T[N(x1)Λ(0)]|N(p1)Λ(p2),in⟩ = Z1/2Λ2EN(p1)(2π)3δ3(k1−p1)+Z1/2ΛT(N(k1)Λ(p1+p2−k1);N(p1)Λ(p2))−(k1−p1−p2)2+m2Λ−iϵ,

which is used to replace the second factor in the integrand of Eq. (2) as

 ψNΛ;NΛ(→r;E) = ei→p⋅→r+∫d3k(2π)32EN(k) ×1EΛ(k)−EN(k)+EN(p)+EΛ(p)⋅T(N(→k)Λ(−→k);N(→p)Λ(−→p))EΛ(k)+EN(k)−EΛ(p)−EN(p)−iϵei→k⋅→r.

Eq. (2) is arrived at by performing the integration. The integration can be carried out by noticing that the propagating degrees of freedom come from the pole contribution, i.e., , where the off-shell can be replaced by the on-shell . The derivations of Eq. (2), Eq. (2) and Eq. (2) are similar.

## 3 An extension of effective Schrödinger equation for a coupled channel system

To construct an effective Schrödinger equation, we define the following functions by multiplying Helmholtz operators on the NBS wave functions as

 KNΛ;NΛ(→x;E) ≡ (△+→p2)ψNΛ;NΛ(→x;E) (3.0) KNΣ;NΛ(→x;E) ≡ (△+→q2)ψNΣ;NΛ(→x;E),

and

 KNΛ;NΣ(→x;E) ≡ (△+→p2)ψNΛ;NΣ(→x;E) (3.0) KNΣ;NΣ(→x;E) ≡ (△+→q2)ψNΣ;NΣ(→x;E).

These Helmholtz operators filter out the propagating degrees of freedom in the asymptotic forms of NBS wave functions. As a result, these functions become localized objects, i.e., is nonzero only within the range of interaction. We factorize these functions as

 [KNΛ;NΛ(→x;E)KNΛ;NΣ(→x;E)KNΣ;NΛ(→x;E)KNΣ;NΣ(→x;E)] = ∫d3y[UNΛ;NΛ(→x,→y)UNΛ;NΣ(→x,→y)UNΣ;NΛ(→x,→y)UNΣ;NΣ(→x,→y)]⋅[ψNΛ;NΛ(→y;E)ψNΛ;NΣ(→y;E)ψNΣ;NΛ(→y;E)ψNΣ;NΣ(→y;E)],

with an -independent kernel . (The proof of the factorization is given in Sect. 4.) Since is localized, has to be localized as well.

If such a factorization is possible, we have

 ⎡⎢⎣(△+→p2)ψNΛ;NΛ(→x;E)(△+→p2)ψNΛ;NΣ(→x;E)(△+→q2)ψNΣ;NΛ(→x;E)(△+→q2)ψNΣ;NΣ(→x;E)⎤⎥⎦ = ∫d3y[UNΛ;NΛ(→x,→y)UNΛ;NΣ(→x,→y)UNΣ;NΛ(→x,→y)UNΣ;NΣ(→x,→y)]⋅[ψNΛ;NΛ(→y;E)ψNΛ;NΣ(→y;E)ψNΣ;NΛ(→y;E)ψNΣ;NΣ(→y;E)].

This implies that any linear combinations of NBS wave functions with arbitrary complex numbers and as

 {ψNΛ(→x;E)≡αψNΛ;NΛ(→x;E)+βψNΛ;NΣ(→x;E)ψNΣ(→x;E)≡αψNΣ;NΛ(→x;E)+βψNΣ;NΣ(→x;E) (3.0)

satisfy

 ⎡⎢⎣(△+→p2)ψNΛ(→x;E)(△+→q2)ψNΣ(→x;E)⎤⎥⎦=∫d3y[UNΛ;NΛ(→x,→y)UNΛ;NΣ(→x,→y)UNΣ;NΛ(→x,→y)UNΣ;NΣ(→x,→y)]⋅[ψNΛ(→y;E)ψNΣ(→y;E)],\specialhtml:\specialhtml: (3.0)

which serves as an coupled channel extension of the effective Schrödinger equation.

(i) Eq. (3.0) does not depend on a particular choice of boundary condition, since and can be arbitrarily chosen. This is a desirable property, which makes it possible to determine the interaction kernel in the finite volume. (The volume has to be sufficiently large compared to the range of the interaction.) Once is determined in the finite volume, S-matrix can be obtained by solving Eq. (3.0) in the infinite volume. (Eq. (3.0) generates NBS wave functions as solutions, which contain the information of T-matrix in their long distance part.)

(ii) The interaction kernel is most generally a non-local object. As is described in LABEL:saoki, a practical way to construct such a non-local interaction kernel is to rely on the derivative expansion, which makes it possible to construct order by order in the derivative expansion by using a finite number of NBS wave functions of varying energy and angular momentum . The most challenging step in fulfilling this procedure is the construction of NBS wave functions at excited energies with variational method. This turns out to be feasible. (See LABEL:sasaki, where this procedure is applied to -- coupled system.)

(iii) In the non-relativistic limit, and are approximately related as , where and denote the reduced masses for and systems, respectively. In this limit, Eq. (3.0) reduces to an eigenvalue problem used in LABEL:liu:

 ⎡⎢⎣−△2μNΛ+˜UNΛ;NΛ˜UNΛ;NΣ˜UNΣ;NΛmΣ−mΛ−△2μNΣ+˜UNΣ;NΣ⎤⎥⎦⋅[ψNΛ(→x;E)ψNΣ(→x;E)]=ENR[ψNΛ(→x;E)ψNΣ(→x;E)], (3.0)

where , , , , and .

(iv) In constructing Eq. (3.0), we assumed the orthogonality of NBS wave functions. If the violation of orthogonality becomes severe, NBS wave functions should be orthogonalized before all the above procedures begin. The orthogonalization should be performed without affecting the long distance behaviors, where information of T-matrix is contained.

## 4 Proof of the factorization of K function

We assume that the following NBS wave functions are linearly independent

 {[ψNΛ;NΛ(→x;E)ψNΣ;NΛ(→x;E)],[ψNΛ;NΣ(→x;E)ψNΣ;NΣ(→x;E)]}mN+mΛ≤E≤mN+mΛ+mπ. (4.0)

There exists a dual basis

 {[˜ψNΛ;NΛ(→x;E),˜ψNΛ;NΣ(→x;E)],[˜ψNΣ;NΛ(→x;E),˜ψNΣ;NΣ(→x;E)]}mN+mΛ≤E≤mN+mΛ+mπ, (4.0)

which serves as a “left inverse” as an integration operator in the following sense:

 ∫d3y[˜ψNΛ;NΛ(→y;E′)˜ψNΛ;NΣ(→y;E′)˜ψNΣ;NΛ(→y;E′)˜ψNΣ;NΣ(→y;E′)]⋅[ψNΛ;NΛ(→y;E)ψNΛ;NΣ(→y;E)ψNΣ;NΛ(→y;E)ψNΣ;NΣ(→y;E)]=(2π)δ(E−E′). (4.0)

Now, the factorization can be verified in the following way:

 [KNΛ;NΛ(→x;E)KNΛ;NΣ(→x;E)KNΣ;NΛ(→x;E)KNΣ;NΣ(→x;E)] = ∫dE′2π[KNΛ;NΛ(→x;E′)KNΛ;NΣ(→x;E′)KNΣ;NΛ(→x;E′)KNΣ;NΣ(→x;E′)] ×∫d3y[˜ψNΛ;NΛ(→y;E′)˜ψNΛ;NΣ(→y;E′)˜ψNΣ;NΛ(→y;E′)˜ψNΣ;NΣ(→y;E′)]⋅[ψNΛ;NΛ(→y;E)ψNΛ;NΣ(→y;E)ψNΣ;NΛ(→y;E)ψNΣ;NΣ(→y;E)] = ∫d3y{∫dE′2π[[]llKNΛ;NΛ(→x;E′)KNΛ;NΣ(→x;E′)KNΣ;NΛ(→x;E′)KNΣ;NΣ(→x;E′)]⋅[˜ψNΛ;NΛ(→y;E′)˜ψNΛ;NΣ(→y;E′)˜ψNΣ;NΛ(→y;E′)˜ψNΣ;NΣ(→y;E′)]} ×[ψNΛ;NΛ(→y;E)ψNΛ;NΣ(→y;E)ψNΣ;NΛ(→y;E)ψNΣ;NΣ(→y;E)].

We arrive at the factorization formula Eq. (3) by defining an interaction kernel as

 [UNΛ;NΛ(→x,→y)UNΛ;NΣ(→x,→y)UNΣ;NΛ(→x,→y)UNΣ;NΣ(→x,→y)] ≡ ∫dE′2π[KNΛ;NΛ(→x;E′)KNΛ;NΣ(→x;E′)KNΣ;NΛ(→x;E′)KNΣ;NΣ(→x;E′)]⋅[˜ψNΛ;NΛ(→y;E′)˜ψNΛ;NΣ(→y;E′)˜ψNΣ;NΛ(→y;E′)˜ψNΣ;NΣ(→y;E′)].

Note that, due to the integration of , the interaction kernel does not depend on the energy .

## 5 Summary

We have proposed an extension of the Lüscher’s finite volume method above the inelastic threshold. We have considered - coupled system as an example. We have seen that equal-time Nambu-Bethe-Salpeter (NBS) wave functions contain the information of T-matrix in their long distance part. We have constructed an effective Schrödinger equation for a coupled channel system by introducing an energy-independent interaction kernel, so that it can generate all the NBS wave functions in the energy region . Since this interaction kernel is localized, it can be constructed by using lattice QCD calculation in a finite volume. Once it is constructed in the finite volume, S-matrix can be obtained by solving the effective Schrödinger equation in the infinite volume.

It is interesting to apply this method to the hyperon systems, where only a limited amount of experimental information is available. The first attempt to apply this method to -- coupled system is reported in LABEL:sasaki, where the interaction kernel is constructed at the leading order of the derivative expansion by using NBS wave functions obtained by the variational method.

## Acknowledgments

This work was supported in part by Grant-in-Aid of the Japanese Ministry of Education, Science, Sport and Culture (No. 22540268) and in part by a Grand-in-Aid for Specially Promoted Research (13002001).

## References

• [1] M. Lüscher, Nucl. Phys. B354, 531 (1991).
• [2] C. Liu, X. Feng, S. He, Int. J. Mod. Phys. A21, 847 (2006).
• [3] N. Ishii, S. Aoki, T. Hatsuda, Phys. Rev. Lett. 99, 022001 (2007).
• [4] S. Aoki, T. Hatsuda, N. Ishii, Prog. Theor. Phys. 123, 89 (2010).
• [5] C.-D.J. Lin, G. Martinelli, C.T. Sachrajda, M. Testa, Nucl. Phys. B619, 467 (2001).
• [6] S. Aoki et al. (CP-PACS Collaboration), Phys. Rev. D71, 094504 (2005).
• [7] K. Sasaki for HAL-QCD Coll., PoS LAT2010, 157 (2010).
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