Electromagnetic pion form factor at finite temperature

The electromagnetic form factor of the pion in the space-like region, and at finite temperature, Fπ(Q 2, T ), is obtained from a QCD Finite Energy Sum Rule. The form factor decreases with increasing T, and vanishes at some critical temperature, where the pion radius diverges. This divergence may be interpreted as a signal for quark deconfinement. John Simon Guggenheim Fellow 1994-1995 The possibility that QCD exhibits chiral-symmetry restoration and quark-gluon deconfinement at finite temperature has triggered interest in the thermal behaviour of QCD in general, and of hadronic propagators in particular. In this respect, some time ago a proposal was made to consider the imaginary part of any hadronic propagator as a phenomenological order parameter for the deconfinement phase transition [1]. According to this proposal, with increasing T one should expect resonances to become broader, and stable-hadron propagators to develop a non-zero imaginary part. This behaviour has been confirmed later for pions, nucleons, and rho-mesons using a variety of theoretical approaches [2]. Independent phenomenological evidence for the deconfinement phase transition in QCD may be obtained e.g. by studying the thermal behaviour of the electromagnetic form factor of the pion, Fπ. In this case one would expect the size of the pion to increase with increasing temperature. At the critical temperature the pion radius should presumably diverge, indicating quark-gluon deconfinement. A recent calculation in the framework of the Nambu-Jona Lasinio [3] model supports this scenario. In this note we determine the temperature dependence of Fπ in the space-like region using a Finite Energy QCD Sum Rule (FESR). The pion form factor at T = 0 has been extensively studied in the past with FESR, as well as with Laplace transform QCD sum rules [4]. In order to establish some notation, as well as the T = 0 normalization, we briefly describe the method at T = 0 before introducing thermal corrections. The appropriate object to study is the three-point function Πμνλ(p, p ′, q) = i ∫ dx dy e x−qy) < 0|T (A†ν (x) Vλ(y) Aμ(0))|0 > , (1) where Aμ(x) = ū(x)γμγ5d(x) is the axial-vector current, Vλ is the electromagnetic current, and q = p − p the momentum transfer. On general analyticity grounds, the three-point function (1) satisfies the double dispersion relation Πμνλ(p , p, Q) = 1 π2 ∫ ∞