Effective Field Theory and Singular Potentials

1. Effective Field Theory and Singular Potentials Kregg Philpott Grove City College Advisor: Silas Beane University of Washington INT August 20, 2001 University of Washington INT REU Program

2. Introduction - Effective Field Theory In effective field theory, the interest lies not in constructing a theory of everything, but rather in constructing a field theory or model that accurately portrays the important physics within a certain region of energy. Energy Gravity ~ 1019 GeV ElectroWeak ~ 100 GeV Strong ~ 1 GeV

3. Introduction - Effective Field Theory Potential (V) Energy Distance (r)

4. Introduction - Effective Field Theory + + Introduce a momentum cutoff L to distinguish between “low energy” physics and “high energy” physics L=

5. Introduction - Effective Field Theory u(r) Distance (r) y(r)=Y(q,f)u(r)/r Energy L R Potential (V) Distance (r)

6. Introduction - Effective Field Theory Energy L R Potential (V) Distance (r)

7. Introduction - Effective Field Theory Energy L R Potential (V) Distance (r)

8. Introduction - Effective Field Theory Potential (V) Energy Distance (r) In the limit as R™0 (L™¥) we match to a delta function in position space.

9. The r-1 Potential An example of the r-1 potential is a nucleon-nucleon interaction via pion exchange. For cases concerning energy << mp, effective field theory may be applied.

10. The r-1 Potential Matching Equation ™0 as R™0 V0 Formula V0 Empirical

11. The r-1 Potential R=.00052 (MeV-1) R=.00228 (MeV-1) R=.00405 (MeV-1) R=.00583 (MeV-1) R=.00760 (MeV-1) R=.00937 (MeV-1) Experiment Plot of Scattering angle d for 1S0 versus energy (k)

12. Singular Potentials In momentum space, kinetic energy goes like k2, so that in position space, the kinetic energy term of the Hamiltonian goes like r-2. When the r-n potential is attractive and of order n>2 with coefficient greater than a certain critical value, or n³3 the potential is unbounded from below. H~r-2+V(r) H~r-2-ar-n

13. The r-2 Potential For the r-2 potential, if the coupling constant a is less than 1/4b, the potential is normal and behaves well. However, if a is greater than 1/4b, the potential is becomes singular.

14. The r-2 Potential Matching Equation V0 Empirical V0 Approximate V0 Approximate V0 R

15. Other Singular Potentials

16. Other Singular Potentials In general, for r-n potentials such that n³3, the eigenfunctions of the Schrödinger equation have infinite oscillation at the origin, preventing matching to a d potential.

17. Pauli-Villars Since the r-1 potential will match to a d function, perhaps the small r behavior of the long distance potential may be modified so that r-n potentials behave like r-1 for small r. e-Cr=1-Cr+… for small r

18. Other Possibilites  Modify the small r behavior of the coupling constant in some other way that preserves the long distance behavior of the potential but allows matching to a d function.  Possibility of using wavelets, which have resolution varying with frequency or scale, to transform the matching equation and/or solutions.