A method to measure the antikaon-nucleon scattering length in lattice QCD

We propose a method to determine the isoscalar \bar K N scattering length on the lattice. Our method represents the generalization of L\"uscher's approach in the presence of inelastic channels (complex scattering length). In addition, the proposed approach allows one to find the position of the S-matrix pole corresponding the the Lambda(1405) resonance.

1. The antikaon-nucleon scattering amplitude is of fundamental importance in nuclear, particle and astrophysics, see e.g. Refs. [1][2][3][4]. In particular, theKN system at threshold provides an interesting testing ground of the chiral dynamics of QCD with strange quarks due to the Λ(1405) resonance just below the scattering threshold. In fact, experimental information on the K − p scattering length from scattering data and kaonic hydrogen level shifts is contradictory, as first stressed in [5] and further elaborated on in Refs. [6][7][8]. A clarification is expected from the upcomimg SIDDHARTA experiment at DAΦNE, that intends to remeasure kaonic hydrogen with unprecedented accuracy and is expected to give further constraints to the isoscalar and isovector kaon-nucleon scattering lengths from the first measurement of the energy spectrum of kaonic deuterium.
On the theoretical side, effective field theory methods in various disguises are employed to pin down thē KN scattering length. The scattering information is usually analyzed in terms of unitarized versions of chiral perturbation theory, that lead to a dynamic generation of various resonances, in particular the Λ(1405). Such schemes have been worked out over the years by various groups at various levels of sophistication (see, e.g. [6][7][8][9]). While an impressive amount of data (cross sections, threshold ratios, mass distributions, etc.) is described in such approaches with good precisison, unitarization of course introduces some unwanted model-dependence. On the other hand, the extraction of theKN scattering length from kaonic hydrogen is firmly rooted in non-relativistic bound-state effective field theory and thus is devoid of the above-mentioned model-dependence (for a recent review, see [10]). However, it entirely rests on the availablity of precise kaonic atom data. Unfortunately, the existing data from DEAR [11] and KEK [12] are conflicting. It would therefore be most welcome to have another tool at hand that would allow one to determine this fundamental quantity.
As we will argue in this letter, lattice QCD provides such a framework. As first shown by Lüscher, finite volume simualations of the energy levels of two-particle states can give access to scattering information [13,14]. The idea of Lüscher is very elegant and simple. For two-particle states that are well separated from bound states or resonances in the given channel, the 1/L expansion of the energy levels takes the generic form (here, L is the size of the box with volume L × L × L) where a is the desired scattering length and c 1 , c 2 are pure numbers (see below). The method has e.g. been used to extract the ππ, πK and KN S-wave scattering lengths from lattice data [15]. Note that an alternative proposal to extract the scattering length from the two-particle wave function is e.g. given in Ref. [16].
However, for the extraction of theKN scattering length, a generalization of this scheme is called for since there is a strong channel coupling betweenKN and Σπ, the latter channel having its threshold about 100 MeV below the opening of theKN one. In addition, the appearance of the Λ(1405) just between these two thresholds further complicates the picture. All these features can be captured by a two-channel Lippmann-Schwinger equation. As we will show in the following, a suitable formulation of this equation in the finite volume allows for an unambigous extraction of the complex-valued isoscalar K − N scattering length. Note also that our method is similar to the approach adopted in Ref. [17], where the problem was treated within the potential scattering theory. In this paper, we use instead the language of the non-relativistic effective field theory (NR EFT), which enables one to systematically address the effects of particle creation/annihillation and relativistic corrections.

2.
To set the stage, we consider a two-channel Lippmann-Schwinger (LS) equation in NR EFT in the infinite volume. Note that we are using a covariant version of the NR EFT, considered in Refs. [18]. The channel number 1 refers toKN and 2 to Σπ with total isospin I = 0. The resonance Λ(1405) is located between two thresholds, on the second Riemann sheet, close to the real axis (these thresholds are defined by The coupled-channel LS equation for the T -matrix elements T ij (s) in the dimensionally regularized NR EFT reads (we only consider S-waves here) with and λ(x, y, z) stands for the Källen function. Furthermore, the H ij (s) denote the driving potential in the corresponding channel. Continuation of the center-of-mass momentum Fig. 1 for the corresponding analytical structure) The resonance corresponds to a pole on the second Riemann sheet in the complex s-plane, its position can de determined from the secular equation The energy and width of the resonance are then given by Thus, to pin down its complex value, we need to determine the three real quantities H 11 , H 12 , H 22 at s = s t appearing in Eq. (5) 3. We now consider the same problem in a finite volume. The rotational symmetry is broken to a cubic symmetry so that the infinite volume version of the LS equation Eq. (2) takes the form (we consider only S-waves here, neglecting the small mixing to higher partial waves. The mixing can be easily included at latter stage, see e.g., Ref. [19].), #5 In the following, we work in the isospin limit and thus do not resolve the further splitting of these thresholds. with Here, we have neglected the terms that vanish exponentially at a large L. The secular equation that determines the spectrum can be brought into the form This is rewritten as δ(s, L) = −φ(k 2 ) + n π , n = 0, 1, 2, . . .
δ(s, L) is called the pseudophase. It is a function of the energy √ s and the level index n, δ n (s) = δ(s, L n (s)).
The dependence of the pseudophase on s and L (or, equivalently, on the level index n) is very different from that of the usual scattering phase. Namely, the elastic phase extracted from the lattice data by using Lüscher's formula is independent of the volume modulo terms that exponentially vanish at a large L. Further, the energies where the phase passes through π/2 lie close to the real resonance locations. In contrast with this, the pseudophase contains terms which are only power suppressed at a large L. Moreover, it contains the tower of resonances which are not related to the dynamics of the system in the infinite volume and merely reflect the existence of the discrete energy levels in the "shielded" channel.
Measuring the pseudophase on the lattice can be used to determine theKN scattering length. It can be directly seen from the expression of the pseudophase, which depends on real functions H 11 , H 12 , H 22 . Extracting these from the data, we then find the scattering length by using Eq. (5). Note that in the expression for the scattering length we need H ij (s) evaluated at threshold s = s t . We shall however demonstrate below that replacing H ij (s) by H ij (s t ) in certain observables, related to the pseudophase, introduces very small correction, since the effective range term proportional to (s − s t ) is suppressed by L −3 as compared to the leading order result. To be specific, we consider the following three observables: 1. For some chosen value of n, we measure the value of the pseudophase δ(s t ; L(s t )) .
= δ t at threshold s t = (m N + M K ) 2 and E t = √ s t (see Fig. 3). On the other hand, we may express δ t through H ij at s = s t in the following way. At threshold, Z 00 is singular, so that tan δ(s t ; L(s t )) = q 2 (s t ) H 22 (s t ) − H 2 12 (s t )/H 11 (s t ) Thus, measuring δ t , we may extract the combination H 22 − H 2 12 /H 11 .
2. Suppose that tan δ(s; L(s)) is infinite at s = s 3 = E 2 3 and L = L 3 = L(s 3 ) (see Fig. 2 for a specific representation of the pseudophase based on a two-channel K-matrix model described below). This occurs at the energy where the denominator of Eq. (8) vanishes We solve this equation by expanding both H 11 (s) and Z 00 (1; k 2 1 (s)) in Taylor series in the vicinity with Substituting Eqs. (15) and (16) into Eq. (14), we obtain This means that measuring the value of s, where the tan δ(s; L(s)) becomes infinite, we may extract H 11 at s = s t . Note that the effective-range term, which contains H ′ 11 (s t ), contributes first at O(L −6 ).
Using finally Eq. (5), we can express the scattering length in terms of the three quantities H 11 , I and G, all taken at s = s t This is the central result of this letter.  Figure 2: The pseudophase δ. The energy E 1 at which the pseudophase passes through π/2 corresponds to the Λ(1405).
Finally, note that in the analysis of the lattice data it may be more convenient to directly fit the explicit expression of the pseudophase given in Eq. (8) to the measured values on the lattice around s = s t , replacing H ij (s) by H ij (s t ) and considering H ij (s t ) (ij = 11, 12, 22) as three independent fitting parameters. From the above discussion one may expect that such a fit will lead to the precise determination of these parameters. The effective range terms can be neglected since their contribution is suppressed by three powers of L. In this case, one does not need to measure the pseudophase in the whole interval between s 2 and s 3 .

4.
Given the parameters H ij determined from fitting to the pseudophase, the position of the pole on the second Riemann sheet of the complex variable s, which corresponds to the Λ(1405)-resonance, can be determined from the secular equation (4). We expect that replacing H ij (s) by H ij (s t ) allows one to locate the pole position at a reasonable accuracy.

5.
In order to demonstrate the above-described proposal in practice, we have investigated a coupledchannel model with an explicit Λ(1405) resonance located at Re √ s R = 1406 MeV and −2 Im excited energy levels is caused by the opening of theKN threshold. At higher energies, the picture repeats -an avoided level crossing emerges, if theKN system has a discrete eigenvalue at this energy in a finite volume. If the volume changes, the avoided level crossing moves (in difference to the avoided level crossing corresponding to the "true" resonance). For L → ∞ the bifurcation lines accumulate at threshold s = s t . In this limit, the scattering amplitude is not analytic at s = s t (unitary cusp).
In Fig. 2 gives the pseudophase derived from the second (n = 2) energy level. It shows the expected behavior. First, it crosses π/2 at √ s = E 1 , very close to the mass of the Λ(1405). Then, it passes π at √ s = √ s 2 = E 2 , close to the threshold where its value is δ t > π. At E 2 , the tangent of the pseudophase vanishes at since q 2 1 (s 2 ) < 0, we can conclude that G(s t ) > 0, cf. Eq. (19). Finally, the value of 3π/2 is reached at √ s = √ s 3 = E 3 . Here, q 2 1 (s 3 ) > 0 and consequently H 11 (s t ) < 0. This can be deduced from Eq. (19) after the substitutions s 2 → s 3 and L 2 → L 3 .

6.
In this letter, we have generalized Lüscher's algorithm for the extraction of the scattering length from the finite-volume energy spectrum measured on the lattice. The modified algorithm applies to the case when the scattering length is complex due to the presence of the open channel(s) below threshold. In the case of theKN scattering with total isospin I = 0, the scattering length can be determined by measuring the volume dependence of the first excited level around the threshold energy.