ETD

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Tesi etd-06282018-105541


Tipo di tesi
Tesi di laurea magistrale
Autore
TODARO, ANTONINO
URN
etd-06282018-105541
Titolo
Numerical challenges in the study of topology in high T QCD: a Multicanonical approach
Dipartimento
FISICA
Corso di studi
FISICA
Relatori
relatore Prof. D'Elia, Massimo
Parole chiave
  • topology
  • topological susceptibility
  • QCD
  • axions
Data inizio appello
19/07/2018
Consultabilità
Completa
Riassunto
Quantum Chromodynamics (QCD) is a quantum gauge theory that describes the strong interactions between quarks and gluons, and is part of the Standard Model of elementary particles. The main difference with QED is that its gauge group, SU (3), is non-abelian. This feature gives rise to some of the less trivial properties of the theory, such as confinement, asymptotic freedom and non-trivial topology of gauge fields. This last property, in particular, allows to arrange gauge fields into homotopy classes, characterized by an integer Q, the so-called topological charge. The existence of a topological classification is related to the strong CP problem, namely the puzzling question of why strong interactions in nature preserve CP symmetries, though in principle a CP violating term in the Lagrangian of QCD is allowed by dimensional and symmetrical considerations (more specifically it has the form ∝ θ Q, where |θ| < 10e−9 from experimental measures of the electric dipole moment of the neutron).

Several attempts of explaining a possible θ = 0 value have been accomplished during the years; one of them (the Peccei-Quinn mechanism) predicts the presence of a new particle, the axion, not belonging to the SM, that represents also a promising candidate for explaining the observed dark matter abundance.
In order to further investigate the axion cosmological role, a precise estimation of its mass is required, in a range of temperatures T ∼ 100 MeV ÷ 1 GeV. Thanks to the proportional relation between the axion mass and the second derivative of the QCD free energy respect to θ, the so-called topological susceptibility χ, such a study can be carried out by means of lattice QCD simulations, a tool that, starting only from first principles, allows us to approach aspects, like topology, that have a deep non perturbative nature.
The key relation used in simulations for extracting the χ value is
χ(T)=<Q^2>_T/V_4
where V_4 is the four volume, therefore χ(T) is proportional to the width of the probability distribution P(Q). From the point of view of simulations, a correct measure of χ(T ) represents quite a challenging task, especially in simulating the realistic case where fermions are dynamical; two are the main difficulties:
• at high T , χ(T ) assumes very small values. This means that configurations at Q = 0 becomes rare and very long simulations are required to estimate a reliable value of Q^2 , clashing with the finite
computational capability. In a certain sense, this problem is related to the physics of the object of study.
• at small lattice spacings a, required for a correct continuum extrapolation, standard local updating algorithms fail in changing the topological sector of a gauge configuration. Thus simulations stay
frozen in a given sector and the algorithm loses ergodicity. This issue appears also at T = 0 and is strictly related to the algorithm employed. It is an algorithmic problem.
These difficulties can explain the discrepancy between results currently present in literature: works [1] and [2] show a power law behavior for χ(T) above T_c (the crossover temperature ∼ 180 MeV) of the type χ(T) ≈ T^(−2) , in contrast with analytic predictions from the Dilute Instanton Gas Approximation (which is valid at asymptotically high temperatures) and with results found in Ref. [3] and Ref. [4], that provides a χ(T) ≈ T^(−8) trend. In particular, this last work push its investigation to a wider range of temperatures and lattice spacings, overcoming the above issues by means of new algorithms which rest their bases on some assumptions and approximations. For this reason it would be desirable to provide an independent check with a method based only on first principles. This is the main aim of this work.

In particular, we check the feasibility of an already existing algorithm, the Multicanonical algorithm (Ref. [5]), usually employed in other contexts (statistical simulation in presence of strong first order transitions), in the QCD case, in order to solve, at least, the first of the two issues listed above, and mitigate the computational effort required for extracting reliable estimations of χ. Simulations are carried out by extracting configurations with a given statistical weight, proportional to e^(−S); the main idea of this new approach, is to modify this statistical weights (that for the topological susceptibility corresponds to a much tiny and peaked distribution) by the introduction of a potential that enhances Q > 0 topological sectors in order to obtain a flatter distribution, that can be sampled more easily. The resulting data are then re-weighted in order to recover the original distribution.

It is worth stressing that the shape of the potential highly affects the efficiency of the algorithm but not its exactness, because there are no approximations or further assumptions.
In the first part of our work, we test different ansatz for V (Q); it turns out from our preliminary simulations that a choice V (Q) ∝ |Q| is preferable to a quadratic choice and furnishes a good improvement with respect to the standard approach. Then we perform the continuum limit for χ(T) at two different temperatures, T = 430 MeV and T = 574 MeV. In doing that we also checked possible finite volume effects, obtaining that they are negligible. Data show a strong dependence on a, however continuum extrapolations are in agreement with results in Ref. [4], even if within large errors.

A natural continuation of our work consists in exploring higher temperature regions, which usually require a too high computational effort with standard algorithms; we expect that this algorithm could significantly mitigate the cost. Also the range of lattice spacings could be enlarged, in particular a better continuum extrapolation can be carried out having smaller values of a at disposal; at this point the topological freezing becomes the major issue and one may wonder if a good choice of V (Q) exists that can solve also this problem. More generally, another interesting development is to find an automatic procedure for determining the best choice of V ; during our simulations we have chosen the potential following rules of thumbs and reasonable guesses. An optimization of this procedure could in principle provide an improvement for the final uncertainty on χ(T).

References
[1] A. Trunin, F. Burger, E.-M. Ilgenfritz, et al. J. Phys. Conf. Ser., vol. 668, no. 1, p. 012123, 2016.
[2] C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, et al. JHEP, vol. 03, p. 155, 2016.
[3] P. Petreczky, H.-P. Schadler, and S. Sharma Phys. Lett., vol. B762, pp. 498–505, 2016.
[4] S. Borsanyi et al. Nature, vol. 539, no. 7627, pp. 69–71, 2016.
[5] B. A. Berg and T. Neuhaus Phys. Rev. Lett., vol. 68, pp. 9–12, 1992.
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