There is a Fifth Force, Beyond That Which is Known to Man…For Now

     Hi, my name is David and I am working with Professor Aldaihan this summer! Our project revolves around potential models of a fifth fundamental force. As a little background, physics has confirmed that four fundamental forces exist: the gravitational force, the electromagnetic force, the strong nuclear force, and the weak nuclear force. The Standard Model of particle physics unifies the electromagnetic, strong nuclear, and weak nuclear forces, and has been successful in describing much of the dynamics in the current universe. However, experimental anomalies have been observed ranging from the subatomic scale to the astronomical scale which are not explained by any of the currently known forces. As a result, explanations for these anomalies have been sought in the form of a fifth fundamental force.

One of the main questions that motivates our research is the lack of CP violation in nuclear interactions. CP symmetry is short for charge-parity symmetry. Charge symmetry means that if we are analyzing a system composed of charged particles and we decide to flip the signs of all the charges then the physics describing the system should not change. Similarly, parity symmetry means that if we impose coordinates on a physical system and flip the signs of one of the coordinate components, say, flipping the signs of all x-components of a system’s particles, the physics should not change. The dynamics of a nuclear interaction are predicted in theory by a mathematical expression known as the Lagrangian. The Lagrangian contains a term which suggests that CP symmetry ought to be broken during certain nuclear interactions. However, experimental evidence suggests that CP symmetry is maintained. The possible existence of a fifth force could explain this discrepancy between theory and experiment.

FIgure 1: Schematics of two different versions of the experiment for which we are calculating the functional form of the fifth force. Data concerning the interaction between the sphere and the plate was collected as the vertical distance between the two macroscopic samples was varied.

In order to have escaped detection so far, this new force must be extremely weak. Hence, there is a need for high precision measurements to be able to see any deviations from known forces. In the absence of a signal of a new interaction, experimental results are commonly reported in terms of limits on the strength of the interaction. Previously, theorists have derived 16 different potential force models. Of these 16 different models, 15 depend on spin. “Spin” is the angular momentum of a particle at rest. Mass and spin are the most fundamental properties of particles in spacetime: protons, neutrons, and electrons all have spin. We already know of one spin-dependent force (magnetism) which becomes visible when the electron spins in special materials (magnets) are lined up. The experimental limits on spin-dependent fifth force models have not been restricted particularly well. This is due to the fact that experimentally probing spin-dependent interactions requires spin-aligned (polarized) matter, which is achieved by introducing a magnetic field. This process is extremely challenging since magnetic fields are much stronger than the fifth force, and one must find a way to prevent magnetic effects from overshadowing the spin-dependent interaction. A large body of laboratory experiments are invested in this type of work. The ultimate goal of our project is to instead utilize strong boundaries of spin-independent experiments to also constrain the spin-dependent fifth force by considering new theoretical calculations.

Figure 2: Data collection from the experimental setup shown above. Consdering the shape of the data and the values of the force being so close to zero, it can be concluded that the fifth force is very weak and will have a functional form similar to that of a decaying exponential.

The spin of ordinary matter (electrons, protons, and neutrons) has two directions: up or down. When two particles interact via a spin-dependent fifth force, their spins flip. However, when one allows the same force to act twice, the original direction of the spin is restored. This allows us to investigate any spin-dependent interaction by the more stringent spin-independent searches. Professor Aldaihan used quantum mechanics (the analog of Newtonian mechanics for subatomic particles) to calculate the mathematical formula for the “double-flip” interaction between a pair of particles. A depiction of this interaction is shown below.

Figure 3: an alternative way of thinking about force suggests that a force may be thought of as an interaction between two particles (fermions) mediated by a tertiary particle (a force carrier. The interacting particles are represented by the solid lines whereas the exchange particle is the dashed line. (a) Exchanging a single force carrier particle between a pair of fermions with spin-dependent couplings generates a spin-dependent interaction. (b) With two exchanges, the spin is flipped twice giving rise to a spin-independent effect.

Currently, my research is focused on extending this “double flip” force between macroscopic masses for a particular spin-independent experiment done which involves a sphere and a plate interacting at short distances. The model of the fifth force previously calculated is not a force, but a quantity known as potential. The particular potential I am analyzing will take the form: V(r)=(g²_S,1g²_P,2/2m₂ + g²_S,2g²_P,1/2m₁)e^-2μr/32(3.14)²r² . The quantities g²_P,1, g²_P,2, g²_S,1 and g²_S,2 are constants known as couplings, which characterize the strength of the force. m₁ and m₂ are the masses of the particles interacting, and r is the separation distances of the particles. μ is the mass of a particle known as the axion. Physicist Hideki Yukawa offered an alternative way of thinking about forces. Rather than the typical way of thinking about force as a push or pull, Yukawa suggested a force may be thought of as an interaction between two objects mediated by a tertiary particle. The fifth force model I am looking at is based upon this idea that the fifth force is mediated by a particle called an axion.

Figure 4: Part of my work on a whiteboard in rederiving the results from a published paper. As can be seen, the expressions for the fifth force contain many terms and can take a long time to calculate. Thus, we swithced to doing calculations with a computer program called Mathematica.

     In the form previously shown, the potential describes the interaction between two point-like particles rather than a sphere and a plate. Luckily, the process of calculating the force between the sphere and the plate is simplified by the fact that we will be using the Proximity Force Approximation (PFA). The PFA gives a simple way to approximate the force within 0.1% of the actual force calculation by using the energy per unit area of two hypothetical plates interacting. The PFA formula for the interaction between the sphere and plate is F = 2(3.14)RE(a). In this formula, R is the radius of the sphere in the experiment and E(a) is the energy per unit area of two planar plates interacting. To obtain E(a), we need only to integrate V(r) over the volume of the two plates. The PFA provides a significantly quicker way of obtaining an equation for the force while still being quite accurate to the real calculation!

Figure 5: An excerpt of the work being done on Mathematica. Mathematica has become a very important tool in my work in cutting down time to do calculations. Evaluating these integrals would be extremely difficult and time consuming otherwise.

My day-to-day work largely involves me doing calculations with a pencil and paper or at a white board. However, in the past week or so, the calculations have gotten too complicated to do by hand. So, we have started using a computational program called Mathematica, which can do the calculations in a much shorter time! When not in the office doing calculations, I enjoy speaking with members of nearby labs about their research and eating lunch with friends. Although the math can be intensive, and sometimes frustrating, I find the work I do very satisfying as I make a little progress everyday to achieving our ultimate goal of lowering the upper-boundary of the fifth force’s magnitude.