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Unformatted text preview: UFTP–89- 20 October 19, 2003 The Orientation of the QCD Vacuum ? Pierre Sikivie and Charles B. Thorn Department of Physics University of Florida, Gainesville, FL 32611 ABSTRACT It is emphasized that when P and CP symmetry is imposed on the order parameter describing chiral symmetry breaking in massless QCD , there are two physically distinct orientations of that order parameter. Using a theorem of Vafa and Witten, it is decided which is chosen by QCD . ? Work supported in part by the Department of Energy, contract DE–FG05–86ER–40272. Although QCD is generally believed to be the correct theory of the strong interactions, there is as yet no thorough understanding of its nonperturbative aspects—in spite of the fact that it is precisely these aspects which play the crucial role in most manifestations of the strong force. As a result any theorems about nonperturbative properties of QCD , e.g. the anomaly constraints of ‘t Hooft,  are especially valuable. A particularly strong result of this kind was obtained a few years ago by Vafa and Witten.  They prove, using the positivity of the Euclidean path integrand for Yang-Mills theory, that QCD does not spontaneously break parity if ¯ θ = 0. In addition, they show that the vacuum energy density has its minimum value at ¯ θ = 0. In this note we clarify what this theorem tells us about the vacuum of QCD . Let us assume that QCD , with N f (usually one takes N f = 2 or 3) mass- less quarks and with the electroweak interactions turned off, breaks chiral SU ( N f ) × SU ( N f ) to SU ( N f ), so that the quark-antiquark condensate ¯ L i R j is proportional to a unitary N f × N f matrix ¯ L i R j = μ 3 U i j , (1) with μ of order the QCD scale. An SU ( N f ) × SU ( N f ) transformation changes the orientation of U but cannot change its determinant, so it is convenient to write U = e iγ/N f U (2) with det U = 1. Here we define γ at θ = 0, where θ...
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This note was uploaded on 09/25/2011 for the course PHY 6346 taught by Professor Staff during the Spring '08 term at University of Florida.
- Spring '08