Hölder Inequalities and QCD Sum-Rule Bounds on the Masses of Light Quarks

QCD Laplace Sum-Rules must satisfy a fundamental Hölder inequality if they are to consistently represent an integrated hadronic spectral function. The Laplace sum-rules of pion currents is shown to violate this inequality unless the u and d quark masses are sufficiently large, placing a lower bound on mu + m d , the SU (2)-invariant combination of the light-quark masses. In this paper we briefly review the development of Hölder inequalities for QCD sum-rules 1 and their application to obtain light-quark (u, d) mass bounds. 2 Laplace sum-rules for pseudoscalar currents with quantum numbers of the pion relate a QCD prediction R 5 M 2 to the integral of the associated hadronic spectral function ρ 5 (t) R 5 M 2 = 1 π ∞ t0 ρ(t) exp − t M 2 dt , (1) where t 0 is the physical threshold for the spectral function. Since ρ 5 (t) ≥ 0, the right-hand (phenomenological side) side of (1) must satisfy integral inequalities over a measure dµ = ρ 5 (t) dt. Hölder's inequality over a measure dµ is t2 t1 f (t)g(t)dµ ≤ t2 t1 f (t) p dµ 1 p t2 t1 g(t) q dµ 1 q , 1 p + 1 q = 1 ; p, q ≥ 1 , (2) which for p = q = 2 reduces to the familiar Schwarz inequality, implying that the Hölder inequality is a more general constraint. The Hölder inequality can be applied to Laplace sum-rules by identifying dµ = ρ(t) dt, τ = 1/M 2 and defining S 5 (τ) = 1 π ∞ µ th ρ 5 (t)e −tτ dt (3) where µ th will later be identified as lying above m 2 π. Suitable choices of f (t) and g(t) in the Hölder inequality (2) yield the following inequality for S 5 (t):