## Thesis etd-09302009-113353 |

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Thesis type

Tesi di dottorato di ricerca

Author

GIORDANO, MATTEO

URN

etd-09302009-113353

Thesis title

On a Euclidean approach to the problem of soft high-energy scattering in nonperturbative quantum chromodynamics

Academic discipline

FIS/02

Course of study

FISICA

Supervisors

**tutor**Prof. Meggiolaro, Enrico

**commissario**Prof. Nachtmann, Otto

**commissario**Prof. Papa, Alessandro

**commissario**Prof. Di Giacomo, Adriano

**commissario**Prof. Konishi, Kenichi

Keywords

- high-energy scattering
- lattice gauge theory
- nonperturbative QCD

Graduation session start date

20/10/2009

Availability

Withheld

Release date

20/10/2049

Summary

In this thesis we discuss the Euclidean nonperturbative approach to soft high-energy hadron-hadron scattering, in the framework of Quantum Chromodynamics (QCD). The main motivation for the study of soft high-energy scattering is the possibility to shed light on one of the oldest unsolved problems of strong interactions, namely the high-energy behaviour of total cross sections: a study of soft scattering at high energies can lead to an explanation of one of the most striking features of hadronic scattering processes, i.e., the fact that their total cross sections rise as the energy increases.

The presence of two different and widely separated energy scales, namely the total center-of-mass energy (hard, i.e., large scale) and the transferred momentum (soft, i.e., small scale), requires a fully nonperturbative approach to the problem, which has been formulated by Nachtmann in 1991, and further developed in the following years. Based on a description of hadrons in terms of the constituent partons, this approach leads to approximate but nonperturbative formulas which relate the partonic scattering amplitudes to properties of the QCD vacuum, namely the correlation functions (in the sense of the functional integral) of certain non-local operators, the so-called Wilson lines and Wegner-Wilson loops. Hadronic scattering amplitudes are recovered from the partonic amplitudes after folding the latter with appropriate wave functions, which describe the interacting hadrons. In Chapter 1 we concisely review the

theoretical framework, giving a short description of QCD, and introducing Nachtmann's approach.

Since most of the nonperturbative techniques in field theory are available only in the Euclidean formulation, it is necessary to relate the relevant correlation functions to corresponding Euclidean quantities via analytic continuation. This has been achieved by Meggiolaro, who has shown how to reconstruct the Minkowskian correlation functions starting from their counterpart in Euclidean space: a general introduction to this issue is given in Chapter 1. The analytic continuation relation has paved the way to the application of nonperturbative techniques, such as the so-called Stochastic Vacuum Model (SVM), or the AdS/CFT correspondence, or the Instanton Liquid Model (ILM), and, recently, also numerical simulations in Lattice Gauge Theory (LGT). The insights obtained in the various cases clearly rely on the validity of the analytic-continuation relations, which until recently had been only explicitly verified in the case of quenched QED to all perturbative orders, and in QCD up to the next-to-leading order of perturbation theory, but up to now not really justified on nonperturbative grounds.

Moreover, the various nonperturbative techniques, used in the calculation of the correlation functions relevant to soft high-energy scattering, involve a number of approximations which are often out of control. One then does not know how much information one is losing, and thus how much the approximate results depart from the ``true'' answer of QCD: to test the validity of the various approaches, one needs to compare their results with an ``exact'' calculation from the first principles of QCD. To our knowledge, the only technique which can be used for this purpose is LGT, which in principle allows for an ``exact'' numerical evaluation of the relevant functional integrals.

The main purpose of this thesis is to assess the validity of the Euclidean approach to soft high-energy scattering, focussing on the two above-mentioned issues. In particular, in Chapter 2 we discuss a fully nonperturbative foundation of the analytic continuation relation relating Minkowskian and Euclidean Wilson-loop correlation functions, directly at the level of the functional integral, and making use of an anisotropic lattice regularisation.

In Chapter 3 we discuss how the Euclidean loop-loop correlation functions can be evaluated in lattice QCD, and we compare numerical results, obtained with Monte Carlo simulations in the pure-gauge (i.e., quenched) theory, with some nonperturbative analytical estimates that appeared in the literature.

In Chapter 4 we critically repeat the calculation of instanton effects on the loop-loop correlation function in the ILM, and compare the results with the numerical data obtained on the lattice. We also derive an expression for the instanton-induced dipole-dipole potential, which is compared to some preliminary numerical results on the lattice.

Finally, in the Conclusions we review the results obtained, and we show some prospects for the future.

The presence of two different and widely separated energy scales, namely the total center-of-mass energy (hard, i.e., large scale) and the transferred momentum (soft, i.e., small scale), requires a fully nonperturbative approach to the problem, which has been formulated by Nachtmann in 1991, and further developed in the following years. Based on a description of hadrons in terms of the constituent partons, this approach leads to approximate but nonperturbative formulas which relate the partonic scattering amplitudes to properties of the QCD vacuum, namely the correlation functions (in the sense of the functional integral) of certain non-local operators, the so-called Wilson lines and Wegner-Wilson loops. Hadronic scattering amplitudes are recovered from the partonic amplitudes after folding the latter with appropriate wave functions, which describe the interacting hadrons. In Chapter 1 we concisely review the

theoretical framework, giving a short description of QCD, and introducing Nachtmann's approach.

Since most of the nonperturbative techniques in field theory are available only in the Euclidean formulation, it is necessary to relate the relevant correlation functions to corresponding Euclidean quantities via analytic continuation. This has been achieved by Meggiolaro, who has shown how to reconstruct the Minkowskian correlation functions starting from their counterpart in Euclidean space: a general introduction to this issue is given in Chapter 1. The analytic continuation relation has paved the way to the application of nonperturbative techniques, such as the so-called Stochastic Vacuum Model (SVM), or the AdS/CFT correspondence, or the Instanton Liquid Model (ILM), and, recently, also numerical simulations in Lattice Gauge Theory (LGT). The insights obtained in the various cases clearly rely on the validity of the analytic-continuation relations, which until recently had been only explicitly verified in the case of quenched QED to all perturbative orders, and in QCD up to the next-to-leading order of perturbation theory, but up to now not really justified on nonperturbative grounds.

Moreover, the various nonperturbative techniques, used in the calculation of the correlation functions relevant to soft high-energy scattering, involve a number of approximations which are often out of control. One then does not know how much information one is losing, and thus how much the approximate results depart from the ``true'' answer of QCD: to test the validity of the various approaches, one needs to compare their results with an ``exact'' calculation from the first principles of QCD. To our knowledge, the only technique which can be used for this purpose is LGT, which in principle allows for an ``exact'' numerical evaluation of the relevant functional integrals.

The main purpose of this thesis is to assess the validity of the Euclidean approach to soft high-energy scattering, focussing on the two above-mentioned issues. In particular, in Chapter 2 we discuss a fully nonperturbative foundation of the analytic continuation relation relating Minkowskian and Euclidean Wilson-loop correlation functions, directly at the level of the functional integral, and making use of an anisotropic lattice regularisation.

In Chapter 3 we discuss how the Euclidean loop-loop correlation functions can be evaluated in lattice QCD, and we compare numerical results, obtained with Monte Carlo simulations in the pure-gauge (i.e., quenched) theory, with some nonperturbative analytical estimates that appeared in the literature.

In Chapter 4 we critically repeat the calculation of instanton effects on the loop-loop correlation function in the ILM, and compare the results with the numerical data obtained on the lattice. We also derive an expression for the instanton-induced dipole-dipole potential, which is compared to some preliminary numerical results on the lattice.

Finally, in the Conclusions we review the results obtained, and we show some prospects for the future.

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