Thesis

State of the art spectroscopy of (un)physical gauge degrees of freedom using numerical experiments

• Call:

  IDPASC Portugal - PHD Programme 2015

• Academic Year:

  2015 / 2016

• Domain:

  Theoretical Particle Physics

• Supervisor:

  David Dudal

• Co-Supervisor:

  Orlando Oliveira

• Institution:

  Universidade de Coimbra

• Host Institution:

  U.Coimbra + KU Leuven (campus Kortrijk); Belgium

• Abstract:

  A popular and powerful way to study quantum chromodynamics (QCD) physics from first principles is the use of computer simulations: one can put a discretized version of the QCD action on a lattice (discretized space time in a finite volume), and use ever‐increasing computer power to extract numerical estimates, using a minimal amount of input. The use of lattice gauge theory as put forward by Wilson [1] evades the problem of gauge fixing, at least when it is applied to the study of gauge invariant correlation functions. A stringent weakness at times is the inherent Euclidean nature of lattice gauge theory. Having a quantum expectation value in Euclidean space, it can be necessary to know this quantity also for non‐Euclidean values of its variables. Let us stick here for reasons of definiteness to an Euclidean two‐point function with momentum squared as variable. Next to lattice computations, there are also continuum analytical approaches to strongly interacting gauge theories, but again, these mostly happen in Euclidean space for reasons of computability, as well as to rely on lattice input to solve the actual quantum equations of motion or to fix certain nonperturbative parameters in effective descriptions [2, 3]. But e.g. having obtained an estimate for the two‐point function F(p^2) for p^2 > 0 (Euclidean), we must recall that physical poles (corresponding to observable particles) happen at p^2 < 0 (Minkowskian), that is in a region where we in general have neither data nor a direct analytical estimate. Beyond perturbation theory, the validity of the usual Wick rotation is not automatically guaranteed [4]. This asks thus for an analytical continuation of a function over the positive real semi‐axis to the negative real semi‐axis through the complex momentum plane, which is a very delicate job, and highly instable at the numerical level. The "Maximum Entropy Method" (MEM) is the standard tool to extract spectral functions. Basically, one searches for the most probable spectral function that can reproduce the data [5]. Though, genuine MEM is inapplicable when probing spectral properties of unphysical degrees of freedom such as confined gluons or quarks. The last decade, an extremely important insight emerged from the (applied) math world about the usefulness of "sparsity". Pioneering work was presented in [6]. The main idea is that excessively large packs of information can be "sensed" using relatively few data (effectively compressing the signal), at least when the data is sensed in a suitable basis w.r.t. which it is sparse, meaning there are many zeroes in the coefficients of the represented quantity. This property of sparseness was then, surprisingly, sufficient to allow a full reconstruction of the a priori large signal from a limited amount of measurements of it. Compressed sensing is thus quite evidently very handy in real‐life applications. The field of compressed sensing found massive use in all fields of signal processing sciences, see [7] for an non‐exhaustive list. Though, in the specific field of elementary particle spectroscopy, it seems still a bit absent, and we wish to explore this potential new application in (lattice) QCD. More generally speaking, we will construct new methods to study the spectral properties of lattice correlation functions. This is a "real-life" example of an ill-posed problem: tiny deviations in the input data result in massive variations in the output, that is the associated spectral density. The latter quantity does however contain the physically relevant information as particle masses, decay constants etc., or important properties as transport coefficients are linked to it. We aim at a more general inversion strategy than usual, we also wish to probe unphysical correlation functions, see [8] for a first trial. This is of paramount relevance as input for the Bethe-Salpeter equation (BSE = relativistic quantum equation for a bound state, important for QCD which spectrum is built solely from bound states due to color confinement) or to explore how confined gluons/quarks vs. deconfined glueballs/mesons behave when passing through the deconfinement transition. This can shed more light on the quasiparticle properties of color degrees of freedom at finite temperature. References: [1] K. G. Wilson, Phys. Rev. D 10 (1974) 2445. [2] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477; C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45 (2000) S1; P. Maris and C. D. Roberts, Int. J. Mod. Phys. E 12 (2003) 297; R. Alkofer and L. von Smekal, Phys. Rept. 353 (2001) 281. [3] P. Boucaud, D. Dudal, J. P. Leroy, O. Pene and J. Rodriguez‐Quintero, JHEP 1112 (2011) 018; D. Dudal, O. Oliveira and J. Rodriguez‐ Quintero, Phys. Rev. D 86 (2012) 105005; M. Q. Huber, A. Maas and L. von Smekal, JHEP 1211 (2012) 035; M. Q. Huber and L. von Smekal, JHEP 1304 (2013) 149; A. Blum, M. Q. Huber, M. Mitter and L. von Smekal, Phys.Rev. D89 (2014) 061703 . [4] K. Osterwalder and R. Schrader, Commun. Math. Phys. 42 (1975) 281. [5] M. Asakawa, T. Hatsuda and Y. Nakahara, Prog. Part. Nucl. Phys. 46 (2001) 459. [6] E. Candes and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory 51 (2005) 4203; E. Candes and T. Tao, Near‐optimal signal recovery from random projections: Universal encoding strategies?, IEEE Transactions on Information Theory 52 (2006) 5406; E. Candes, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Communications on Pure and Applied Mathematics 59 (2006) 1207; [7] M. A. Davenport, M. F. Duarte, Y. C. Eldar and G. Kutyniok, Introduction to Compressed Sensing, in Compressed Sensing: Theory and Applications, Cambridge University Press (2012). [8] D. Dudal, O. Oliveira and P. J. Silva, Phys. Rev. D 89 (2014) 014010.