Infrared finitness and analyticity properties of the loop-loop scattering amplitudes in gauge theories

We shall discuss about the infrared finitness and some analyticity properties of the loop-loop scattering amplitudes in gauge theories, when going from Minkowskian to Euclidean theory, and we shall see how they can be related to the still unsolved problem of the s-dependence of the hadron-hadron total cross-sections.Comment: Talk given at the ``International Workshop on Diffraction in High-Energy Physics'' (DIFFRACTION 2004), Cala Gonone (Sardinia, Italy), 18-23 September 2004; 3 pages, LaTeX fil

The analytic continuation of the high-energy parton-parton scattering amplitude with an IR cutoff

The high-energy parton-parton scattering amplitude can be described, in the c.m.s., by the expectation value of two infinite Wilson lines, running along the classical trajectories of the two colliding particles. The above description suffers from IR divergences (typical of 3 + 1 dimensional gauge theories), which can be regularized by considering finite Wilson lines, extending in proper time from -T to T (and eventually sending T to infinity). Generalizing the results of a previous paper, we give here the general proof that the expectation value of two IR-regularized Wilson lines, forming a certain hyperbolic angle in Minkowski space-time, and the expectation value of two IR-regularized Euclidean Wilson lines, forming a certain angle in Euclidean four-space, are connected by an analytic continuation in the angular variables and in the IR cutoff T. This result can be used to evaluate the IR-regularized high-energy scattering amplitude directly in the Euclidean theory.Comment: 18 pages, LaTeX fil

Remarks on the U(1) axial symmetry in QCD at zero and non-zero temperature

This paper is organized in two parts. The first part (Sections 2-5) is dedicated to the theory at T=0 and contains a pedagogical review of some fundamental aspects related with the chiral symmetries of QCD, the U(1) problem and its solution proposed by 'tHooft, Witten and Veneziano. In the second part (Sections 6-14) we discuss the role of the U(1) axial symmetry for the phase structure of QCD at finite temperature. One expects that, above a certain critical temperature, also the U(1) axial symmetry will be restored. We will try to see if this transition has (or has not) anything to do with the usual chiral transition: various possible scenarios are discussed. In particular, we analyse a scenario in which the U(1) axial symmetry is still broken above the chiral transition. We will show that this scenario can be consistently reproduced in the full respect of the relevant QCD Ward Identities and also using an effective Lagrangian model. A new order parameter is introduced for the U(1) axial symmetry.Comment: Expanded version of a talk given at the ``Workshop on Quark-Gluon Plasma and Relativistic Heavy Ions'', Frascati (Italy), January 14th-18th, 2002 (QGP2002); 61 pages, LaTeX file, + 6 PS figure

A remark on the high--energy quark--quark scattering and the eikonal approximation

In this paper we calculate the high--energy quark--quark scattering amplitude, first in the case of scalar QCD, using Fradkin's approach to derive the scalar quark propagator in an external gluon field and computing it in the eikonal approximation. (This approach was also recently used by Fabbrichesi, Pettorino, Veneziano and Vilkovisky to study the four--dimensional Planckian--energy scattering in gravity.) We then extend the results to the case of ``real'' ({\it i.e.} fermion) QCD, thus deriving again, in a rather direct way, the results previously found by Nachtmann. The abelian case (QED) is also discussed in the Appendix.Comment: 24 pages (no figures), LaTeX--file. New references have been added and commented and some other minor corrections have been performe

On the loop-loop scattering amplitudes in Abelian and non-Abelian gauge theories

The high-energy scattering amplitude of two colour-singlet quark-antiquark pairs is governed by the correlation function of two Wilson loops, which follow the classical straight lines for quark (antiquark) trajectories. This quantity is expected to be free of IR divergences, differently from what happens for the parton-parton scattering amplitude, described, in the high-energy limit, by the expectation value of two Wilson lines. We shall explicitly test this IR finitness by a direct non-perturbative computation of the loop-loop scattering amplitudes in the (pedagogic, but surely physically interesting) case of quenched QED. The results obtained for the Abelian case will be generalized to the case of a non-Abelian gauge theory with N_c colours, but stopping to the fourth order in perturbation theory. In connection with the above-mentioned IR finitness, we shall also discuss some analytic properties of the loop-loop scattering amplitudes in both Abelian and non-Abelian gauge theories, when going from Minkowskian to Euclidean theory, which can be relevant to the still unsolved problem of the s-dependence of hadron-hadron total cross-sections.Comment: 20 pages, LaTeX file, + 1 PS figure. Completely revised version, to be published in Nucl. Phys.

Deriving the pomeron from a Euclidean-Minkowskian duality

After a brief review, in the first part, of some relevant analyticity properties of the loop-loop scattering amplitudes in gauge theories, when going from Minkowskian to Euclidean theory, in the second part we shall see how they can be related to the still unsolved problem of the s-dependence of the hadron-hadron total cross-sections.Comment: 10 pages. Contribution to "Sense of Beauty in Physics - a volume in honour of Adriano Di Giacomo" (Pisa University Press, Pisa, 2006), on the occasion of his 70th birthda

Euclidean formulation of diffractive scattering

After a brief review (in the first part) of some relevant properties of the high-energy parton-parton scattering amplitudes, in the second part we shall discuss the infrared finiteness and some analyticity properties of the loop-loop scattering amplitudes in gauge theories, when going from Minkowskian to Euclidean theory, and we shall see how they can be related to the still unsolved problem of the s-dependence of the hadron-hadron total cross-sections.Comment: Talk given at the ``XIth International Conference on Elastic and Diffractive Scattering'' (EDS 2005), XVIIth Rencontre de Blois, Chateau de Blois (France), 15-20 May 2005; 6 page

Theta dependence of the vacuum energy density in chiral effective Lagrangian models at finite temperature, above TcT_c

In this work, extending a previous study at zero temperature ($T=0$), we perform a systematic study of the modifications to the QCD vacuum energy density $\epsilon_{vac}$ in the finite-temperature case, above the chiral transition at $T_c$, caused by a nonzero value of the parameter $\theta$, using two different effective Lagrangian models which implement the $U(1)$ axial anomaly of the fundamental theory and which are both well defined also above $T_c$. In particular, we derive (and critically compare) the expressions for the topological susceptibility $\chi$ and for the second cumulant $c_4$ starting from the $\theta$ dependence of $\epsilon_{vac}(\theta)$ in the two models.Comment: 23 pages; revised versio

Eikonal propagators for the high-energy parton-parton scattering in gauge theories

By a direct resummation of perturbation theory in the limit of very high energy and small transferred momentum (the so-called ``eikonal'' limit), we derive expressions for the truncated-connected quark, antiquark and gluon propagators in an external gluon field, both for scalar and fermion gauge theories. These are the basic ingredients to derive ``soft'' high-energy parton-parton scattering amplitudes, using the LSZ reduction formulae and a functional integral approach.Comment: Talk given at the ``Sixth Workshop on Non-Perturbative Quantum Chromodynamics'', Paris (France), June 5th-9th 2001 (NPQCD 01); 8 pages, LaTeX file, + 1 PS figur

High-energy scattering amplitudes in QCD: from Minkowskian to Euclidean space

We shall discuss about some analytic properties of the high-energy parton-parton (and hadron-hadron) scattering amplitudes in gauge theories, when going from Minkowskian to Euclidean theory, and we shall see how they can be related to the still unsolved problem of the s-dependence of the total cross-section.Comment: Talk given at the ``26th Johns Hopkins Workshop on current problems in particle theory: high energy reactions'', Heidelberg (Germany), 1-3 August 2002 (JHW2002); 7 pages, LaTeX file. Revised version with two errors in Eqs. (26) and (27) correcte