### Project Outcomes Report for the General Public:

Exploring the Origin of Mass with High-Performance Computing

(Also available at Research.gov)

The research I carried out in Taiwan during the summer of 2011, through the NSF EAPSI program, concerns the origin of elementary particles’ masses. This is a longstanding mystery in particle physics, which I will briefly explain before discussing my own work.

More than 50 years ago, physicists realized that a fundamental symmetry of nature (the electroweak unification of electromagnetism and the weak nuclear force) appeared to be incompatible with the observed existence of massive elementary particles. This difficulty was overcome in the 1960s with the discovery that the electroweak symmetry can be hidden, a process popularly known as the Higgs mechanism. While the Higgs mechanism as a general process has been strongly supported by experiments, the physics underlying it remains unknown. Understanding what drives the Higgs mechanism is a central challenge in particle physics today, and is the main goal of CERN’s Large Hadron Collider (LHC).

My research focuses on theories in which the Higgs mechanism results from the existence of some new, strongly-interacting force. Any elementary particles feeling this new force must be bound together into extremely small composite particles. It would take roughly ten tons of force to separate two such elementary particles by as little as a billionth of a nanometer, which gives an idea of what we mean by “strong”. Because the composite particles predicted by these types of theories are so small, they have not yet been experimentally identified, or ruled out. Experiments at the LHC are actively searching for such particles, and may be able to determine whether they exist in nature.

On the theoretical side, strong interactions are extremely difficult to analyze. For technical reasons, large-scale supercomputing is at present the only way to obtain reliable, quantitative predictions from strongly-interacting theories. This approach, known as lattice field theory, is the focus of my research. The basic idea is to discretize space and time, replacing them by a regular, finite lattice of discrete sites connected by links. The theory under consideration is likewise discretized, and defined on the spacetime lattice in such a way that we recover the original continuum theory when the lattice is taken to be infinitely large, with its sites infinitesimally close together. The discretized system involves a finite (but very large) number of degrees of freedom, which allows us to calculate predictions through standard numerical algorithms.

A particularly important quantity to predict this way is an abstract parameter known as *S*. Simple (but simplistic) arguments suggest that a new strongly-interacting force would imply a value for *S* at least four standard deviations different than its experimental value. For my PhD I performed the first lattice calculation of *S* for representative theories of a new strong interaction, which resulted in predictions significantly closer to experiment, illustrated in Figure 1.

*Figure 1: S is related to the origin of elementary particles’ masses. New, strongly-interacting forces often predict a value for S very different (red points) than the experimental value of -0.15 +/- 0.10. Recent lattice calculations found results (blue points) much closer to the experimental value.*

My work in Taiwan focused on improving this lattice calculation of *S*. One limitation of calculations on a lattice of finite extent *L* is that only certain momenta *Q* (nonzero multiples of *2π/L*) can be accessed directly, as shown in Figure 2. Because the *S* parameter is defined at zero momentum, we extract it from our data by performing an extrapolation based on a simple model, which produces the red band in Figure 2. This extrapolation is one of the main sources of uncertainty in our results.

*Figure 2: In the standard approach, only relatively few momenta Q can be accessed by lattice calculations, introducing significant uncertainty (red band) from a long extrapolation to zero momentum.*

A recently-developed numerical technique manipulates the lattice formulation to allow us to access arbitrary momentum, without significantly increasing computational costs. The goal of my summer in Taiwan was to apply this technique to my lattice calculation of the *S* parameter, by working with one of the world’s experts. We were able to complete this project successfully, and the results are displayed in Figure 3: all of the blue points have been added to the three red points that were previously accessible. As a result, the extrapolation to zero momentum is much better controlled, and can be treated in a more rigorous manner.

*Figure 3: From the same data used in Figure 2, a recently-developed numerical technique allows us to access many more momenta Q (blue) than the standard approach (red), improving the extrapolation to zero momentum.*

Of course, the reason for meeting face-to-face with other researchers is to benefit from interacting in person, and these interactions can often go in unexpected directions. While completing the project described above, I took advantage of working with my host to learn another technique he uses in his research, known as finite-size scaling. Without getting into technical details, finite-size scaling determines the character of physical systems by considering how they behave in different volumes. (In the context of lattice field theory, we compare calculations on lattices of different sizes.) Figure 4 is a typical result of this sort of analysis: the positions and depths of the minima in the curves reveal valuable information about the nature of the system under consideration. Finite-size scaling doesn’t tell us about the *S* parameter, but can help guide us to those systems for which the more computationally expensive calculation of *S* would be most interesting.

*Figure 4: Each colored line indicates the quality of finite-size scaling for a different observable (with “scaling exponent” y_m); good scaling produces D=1. The black line is the sum of the colored results. The positions and depths of the minima in the curves tell us about the nature of the system.*

Last modified 15 December 2011