Preliminary study of the scattering length from lattice QCD

Abstract

The s-wave kaon–antikaon () scattering length is studied by lattice QCD using pion masses m _π =330–466 MeV. Through wall sources without gauge fixing, we calculate four-point functions in the I=1 channel with the “Asqtad”-improved staggered fermion formulation, and observe an attractive signal, which is consistent with pioneering lattice studies on potential. Extrapolating the scattering length to the physical point, we obtain , where the first error is statistical and the second is systematic. These simulations are conducted with MILC gauge configurations at lattice spacing a≈0.15 fm.

Appendices

Appendix A: I=1 scattering amplitudes in terms of quark propagators

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

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(A.10)

(A.11)

(A.12)

Appendix B: The analytic expression of s-wave scattering length for the isospin I=1 channel

In the isospin limit, both isospin amplitudes in can be described by two independent amplitudes T _ch (the amplitude for the processes K ⁺ K ⁻→K ⁺ K ⁻) and T _neu (that for ) [1, 11, 16]. Z.H. Guo and J.A. Oller derived two isospin amplitudes only in terms of T _neu [34]. Here we follow the notation and conventions in [34], since the chiral scale μ dependence in amplitude is canceled by the loops and the low-energy constants (LECs) [34].

These amplitudes can be decomposed into partial waves t _l (s) according to

$$T(s,t,u)=16\pi \sum_l (2l+1) t_l(s) P_l(\cos\theta) , $$

where l is total angular momentum, θ denotes scattering angle in the center-of-mass system. In the elastic region, the partial wave amplitude t _l (s) can be parameterized by real phase shifts δ _l (s),

$$t_l^I(s) = \sqrt{\frac{s}{s-4m_K^2}} \frac{1}{2i} \bigl\{ e^{2i\delta_l^I(s)}-1 \bigr\} . $$

Its real part can be expanded at threshold (namely, $s=4m_{K}^{2}, t=0, u=0$) in terms of the scattering lengths ($a_{l}^{I=1}$) and effective ranges ($b_{l}^{I=1}$),

$${\mathit{Re}}\,t_l^I(s) = \frac{\sqrt{s}}{2} \, q^{2l} \bigl\{ a_l^{I=1} + b_l^{I=1} q^2 + \mathcal{O} \bigl(q^4 \bigr) \bigr\} , $$

for the center-of-mass three-momentum of the kaons q. The s-wave scattering length is linked to the real part of the amplitude at threshold by

The one-loop order analytic expressions of scattering amplitudes in the isospin limit can be found in Ref. [34], however, no analytic formulas for s-wave scattering lengths were explicitly provided. We, therefore, will present it for the isospin I=1 channel. To this end, we first expand the scattering amplitude at threshold, namely,³

(B.13)

where are low-energy constants defined in Ref. [44] at the chiral symmetry breaking scale μ. The loop function is given in Ref. [44], namely,

(B.14)

with

$$\nu = \sqrt{ \bigl[4m_K^2 - (m_\pi + m_\eta )^2 \bigr] \bigl[4m_K^2 - (m_\pi - m_\eta )^2 \bigr] }. $$

By plugging the leading order relation $m_{\eta}^{2} = (4m_{K}^{2}-m_{\pi}^{2})/3$ [34] we can simplify the above formula as

(B.15)

The scattering length can be written in compact form,

(B.16)

where is for the known functions at NLO, which clearly depend on the chiral scale μ with chiral logarithm terms,

(B.17)