# Results On Performing Hadron Spectroscopy With LQCD Figure 1: Graph with exponential fitting of the average correlators extracted from testing 4 different quark masses

Welcome to my third and final blog post. At the end of the previous post, I briefly explained the goals of this project. I talked about making LQCD simulations to extract a pion mass and find an input quark mass that would give a pion mass value close to the experimental one. Of course, all of these are once again ambiguous so let me describe them step by step.

In QCD theory, almost every problem has to do with the calculation of various physical quantities using the path integral. However, those integrals are what perturbative techniques fail to calculate, which forces us to use the LQCD formalism. In our case, we will focus on a class of those problems called hadron spectroscopy. In these problems, we intend to find the mass of a hadron(e.g. a pion) based on specific input parameters.

The most essential input parameters are the gauge field configuration and the quark mass. To obtain a solution that converges with this of the path integral, we must carefully select a quark mass and conduct the same experiment with many different gauge configurations. Next, we can take the average of all outputs, which should lead to a relatively accurate solution depending on how well we chose those input parameters.

## Path Towards Pion Mass Extraction

Since we discussed the necessary theoretical background, we can move forward with the procedure I followed to complete my project. Initially, I selected a quark mass equal to -0.01 and did simulations with many slightly different gauge configurations. Before you ask, yes the mass can be negative here! That happens due to an additive mass shift introduced during discretization, but it shouldn’t bother us. Also, I should clarify that all masses during the experiments are expressed in lattice units, which makes them dimensionless. That is yet another mathematical trick to make the calculations easier for the computer.

Upon completing those simulations, I extracted their pion correlators and calculated their average and standard error. A correlator is a function describing the relation of microscopic variables(e.g. spin) at different positions. The statistical observables we calculated are the key to the final step of this process. That step is to perform exponential fitting and acquire the pion mass. To achieve this, I utilized the tool called gnuplot. However, its pion mass error estimate was slightly inaccurate as it doesn’t consider the correlation among configurations. To remedy this, I replaced it by calculating the jackknife error, which is much more precise in such cases. Figure 2: Graph with the line representing the relation of the pion mass square and quark mass. The horizontal blue line is at the desired pion mass square

Finally, despite the already decent estimate of the pion mass value, I tried to get closer to the experimental value. That amounts to 135 MeV and translates into a lattice mass of 0.129. To this end, I exploited the approximately linear relation between the square of the pion mass and the quark mass to find an ideal target quark mass. I started by repeating the simulation with another quark mass equal to -0.05. Then, I created the line defined by the pairs we found, plugged in the desired pion mass square, and ended up with a quark mass of -0.087.