Two-dimensional SYM theory with fundamental mass and Chern-Simons terms *

1. Two-dimensional SYM theory with fundamental mass and Chern-Simons terms* Uwe Trittmann Otterbein College OSAPS Spring Meeting at ONU, Ada April 25, 2009 * arXiv:0904.3144v1 [hep-th]

2. Supersymmetric Discretized Light-Cone Quantization • Simply put:SDLCQ is a practical scheme to calculate masses of bound states • use special quantization to make discretization easy • discretize the theory (“put system in a box”)  discretization parameterK - work (preferably) in low dimensions (two, three..) • supersymmetry to get rid of renormalization issues • typically solve problems numerically

3. Light-Cone Quantization • Use light-cone coordinates • Hamiltonian approach:ψ(t) = H ψ(0) • Theory vacuum is physical vacuum - modulo zero modes (D. Robertson)

4. The Theory: N=1 SYM in 3D with SYM & Chern-Simons couplings g & κ

5. Particle Content of the Theory • Adjoint gauge boson: (Aμ)ab • Adjoint (real) fermion: Λab • Fundamental complex scalar: ξa • Fundamental Dirac fermion: Ψa • Chern-Simons term gives effective mass proportional to coupling κ to the adjoint particles

6. Adding a VEV generates mass for the fundamental particles • Add vacuum expectation value (VEV) to perpendicular component of the gauge field in 3D theory • Shift field by its VEV, express theory in terms of new field: • Dimensionally reduce to 2D by dropping derivatives w.r.t. transverse coordinates

7. Extra Terms induced by the VEV • The shift by the VEV generates extra terms in the supercharge which are fairly simple: • In SDLCQ mode decomposition it reads

8. Symmetries • The original theory is invariant under • Supersymmetry (obviously) • Parity: P • Reversal of the orientation of the chain of partons: O • Shifting by the VEV destroys P and O, but leaves PO intact • Adding a CS term destroys P • Together, they only leave SUSY intact

9. Analytical Results • We can solve the theory for K=3 analytically because each symmetry sector has only 4 basis states • A quartic equation for the mass eigenvalues arises • Massless bound-states exist for

10. Limits: v,κ∞ • As the parameters get large we expect a free theory (SYM coupling g becomes unimportant) • Lightest states in the limit are short (2 fundamental partons), few • Heavy states (large relative momentum) are long, many

11. Bound-State Masses vs. VEV • Masses (squared) grow quadratically • Some masses decline • Massless states appear at some VEVs

12. Close-up at larger K • Combination of parabolic M2(VEV) curves yields light/massless states • As K grows more lighter states and more points of masslessness appear

13. Continuum limit • As K  ∞ the lowest state becomes massless even atVEV=1

14. Average number of partons in bound state • Ten lightest states at K=7 become “shorter” as VEV grows

15. Bound-State Masses with VEV vs. CS coupling • Masses (squared) grow quadratically • Some masses decline • No massless states appear

16. Continuum limit with CS term • As K  ∞ the lowest state remains massive (atVEV=1 andκ =1)

17. Structure Functions • Normalization: Sum over argument yields average number of type A partons in the state • Expectation: • Large momenta of fundamentals since state is short • To lower mass, have to have two fundamental fermions with same momentum  Fundamentals split momentum evenly  peaked around x=0.5 • Adjoints have small momenta • Few adjoints

18. Lightest state gaB • K= 8, v = 1, κ = 1 • #aB=0.67 • #aF=0.11 • #fB=1.08 • #fF=0.92 gfF gfB gaF

19. Second-Lightest state gaB • K=8, v=1, κ =1 • #aB=0.72 • #aF=0.07 • #fB=0.89 • #fF=1.11 gfB gfF gaF

20. Conclusions • Supersymmetric Discretized Light-Cone Quantization (SDLCQ) is a practical tool to calculate bound state masses, structure functions and more • Generated mass term for fundamentals from VEV of perpendicular gauge boson in higher dimensional theory • Studied masses and bound-state properties as a function of v (“quark mass”) &κ (“gluon mass”) • Spectrum separates into (almost) massless and very heavy states

21. Extra Slides

22. Discretization • Work in momentum space • Discretization: continuous line  K points (K=1,2,3…∞discretization parameter) integration  sum over values at K points (trapezoidal rule) operators  matrices “Quantum Field theory” “Quantum Mechanics” • E.g. two state system Hamiltonian matrix: E0 -D H= -D E0 • Now: “quarter-million state system” More states, more precision !

23. What does the Computer do? • works at specific discretization parameter K • generates all states at this K  basis • constructs Hamiltonian matrix in this basis • diagonalizes the Hamiltonian matrix, i.e. solves the theory for us  eigenvalues are masses of bound states  gets also eigenfunctions (wavefunctions) Repeat for larger and larger K !

24. Extracting Results • All observables (masses, wavefunctions) are a function of the discretization parameter K • Run as large a K as you can possible do • Extrapolate results: K  ∞ ”The next step in K is always the most important”

25. Computers and Codes • Runs on Linux PC and parallel computers • Typical computing times: • Test runs: few minutes • production runs: few days • Production runs also on: OSC machines, Minnesota Supercomputing Center • Code compatibility insured by tests on different machines (even Macintosh! ) • Evolution of the code: Mathematica  C++  data structure improved code