The πN sigma term — an evaluation using staggered fermions

Abstract

A lattice calculation of the πN sigma term is described using dynamical staggered fermions. Preliminary results give a sea term comparable in magnitude to the valence term. Theoretical DiscussionPreprint HLRZ 93-70 November 1993 The πN sigma term, σ πN , is defined as that part of the mass of the nucleon coming from the expectation value of the up (u) and down (d) quark mass terms in the QCD Hamiltonian,where we have taken these quarks to have equal current mass (= m). Other contributions to the nucleon mass come from the chromo-electric and chromo-magnetic gluon pieces and the sea terms due to the s quarks. Experimentally this matrix element has been measured from low energy π-N scattering, [1]. A delicate extrapolation to the chiral limit [2] gives a result for the isospin even amplitude of Σ/f 2 π with Σ = σ πN , from which the πN sigma term may be found. The precise value obtained this way has been under discussion for many years. For orientation we shall just quote a range of results from later analyses of σ πN ≈ 56MeV, [3], down to 45MeV, [4].To estimate valence and sea contributions to σ πN , classical current algebra analyses assume octet dominance and make first order perturbation theory about the SU F (3) flavour symmetric Hamiltonian. This gives where we have first assumed that the nucleon wavefunction does not change much around the symmetric point. We then subtract and add a strange component. At the symmetric point the u and d quarks each have equal valence and sea part, while the s quark matrix element only has a sea component. Thus in the first term the sea contribution cancels, justifying the definitions given in eq. (2). Using first order perturbation theory for the baryon mass splittings σ val πN may be calculated to give σ val πN ≈ 25MeV and so σ sea πN ≈ 31 ∼ 20MeV. This in turn means that m s N |ss|N ≈ 400 ∼ 250MeV, which would indicate a sizeable portion of the nucleon mass (938MeV) comes from the strange quark contribution. Measuring σ πNWe now turn to our lattice calculation. We have generated configurations using dynamical staggered fermions on a 16 3 × 24 lattice at β = 5.35, m = 0.01 (plus some larger masses), [5]. Practically there are several possibilities open to us for the evaluation of the matrix element. The easiest is simply to differentiate the shift operator S 4 giving(the Feynman-Hellmann theorem). Thus we need to measure M N for different masses m and numerically estimate the gradient. This leads to σ πN ≈ 11.9(8)m| m=0.01 ≈ 0.12(1),

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