Renormalization of loop functions in QCD
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enormalization of Loop Functions in QCD1 M. Berwein, N. Brambilla, and A. Vairo PhysikDepartment, Technische Universität München, JamesFranckStr. 1, 85748 Garching, Germany email: [email protected] Abstract—We give a short overview of the renormalization properties of rectangular Wilson loops, the Poly akov loop correlator and the cyclic Wilson loop. We then discuss how to renormalize loops with more than one intersection, using the simplest nontrivial case as an illustrative example. Our findings expand on previ ous treatments. The generalized exponentiation theorem is applied to the Polyakov loop correlator and used to renormalize linear divergences in the cyclic Wilson loop. DOI: 10.1134/S1063779614040029 1
1. INTRODUCTION
We will discuss some loop functions with regard to their renormalization properties. By loop functions we mean the vacuum or thermal expectation value of a number of Wilson lines in an SU(Nc) gauge theory, which are closed and traced and each trace is normal ized by the number of colours Nc. Rectangular Wilson loops are of special interest, because they are related to the quarkonium potential, and therefore they have been studied in detail [1–5]. They consist of four straight Wilson lines, two along the time direction at fixed positions in space and at rel ative distance r, and two along the direction of r at fixed times and at temporal separation t. It is known that rectangular Wilson loops are UV divergent even after charge renormalization [6, 7]. They need to be renormalized by a multiplicative constant, which is given in the MS scheme by Z4c = exp[–2CFαs/ πε + ᏻ ( α s )], where CF is the quadratic Casimir of the fundamental representation, the space time dimension D = 4 – 2ε and 1/ ε = 1/ε – γE + ln4π. 2
This additional divergence comes from the four corners of the Wilson loop, where the contour has cusps of angle π/2. At finite temperature the Polyakov loop correlator Pc(r, T) plays a role similar to that of the rectangular Wilson loop in the vacuum. It is related to the free energy of a static quarkantiquark pair [8, 9]. Polyakov loops are Wilson lines spanning the whole of the imag inary time direction from τ = 0 to τ = β = 1/T at fixed spatial position. They are closed loops because of the periodic boundary conditions of the imaginary time formalism. The Polyakov loop correlator consists of two traced Polyakov loops at spatial distance r. This 1 The article is published in the original.
quantity is free of UV divergences in dimensional reg ularization after charge renormalization. In our publication [10] we have studied the renormal ization properties of the cyclic Wilson loop Wc(r, T), which is closely related to the two loop functions described above. It is a rectangular Wilson loop at finite temperature where the temporal Wilson lines are given by Polyakov loops. However, its divergence structure does not match that of a vacuum rectangular Wilson loop [11], because renormalization with a mul tiplicative constant fails. The reason for this behaviour lies in
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