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. Gravity ~ 10 19 GeV ElectroWeak ~ 100 GeV Strong ~ 1 GeV Energy

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

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

5. Potential (V) Introduction - Effective Field Theory Distance (r) Energy  R u(r) Distance (r)  ( r )=Y( ,  )u(r)/r

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

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

8. Potential (V) Introduction - Effective Field Theory Distance (r) Energy 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 << m , effective field theory may be applied.

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

11. The r -1 Potential R=.00228 (MeV -1 ) R=.00052 (MeV -1 ) R=.00405 (MeV -1 ) R=.00583 (MeV -1 ) R=.00760 (MeV -1 ) R=.00937 (MeV -1 ) Experiment Plot of Scattering angle  for 1 S 0 versus energy (k)

12. Singular Potentials In momentum space, kinetic energy goes like k 2, 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 n ³3 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  is less than 1/4 , the potential is normal and behaves well. However, if  is greater than 1/4 , the potential is becomes singular.

14. The r -2 Potential Matching Equation R V0V0 Empirical V 0 Approximate V 0

15. Other Singular Potentials

16. In general, for r -n potentials such that n ³ 3, the eigenfunctions of the Schrödinger equation have infinite oscillation at the origin, preventing matching to a  potential.

17. Pauli-Villars Since the r -1 potential will match to a  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  function. Possibility of using wavelets, which have resolution varying with frequency or scale, to transform the matching equation and/or solutions.