Two-dimensional SYM theory with fundamental mass and

Chern-Simons terms*

Uwe TrittmannOtterbein College

OSAPS Spring Meeting at ONU, Ada

April 25, 2009

* arXiv:0904.3144v1 [hep-th]

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 parameter K- work (preferably) in low dimensions (two, three..) - supersymmetry to get rid of renormalization issues- typically solve problems numerically

Light-Cone Quantization 10

21 xxx • Use light-cone coordinates

• Hamiltonian approach: ψ(t) = H ψ(0)• Theory vacuum is physical vacuum - modulo zero modes (D. Robertson)

The Theory: N =1 SYM in 3D

CSfundSYM SSSS 12

with SYM & Chern-Simons couplings g & κ

232

2ˆ

)(

241

3

3

3

AAAiAAxdS

gDiDDxdS

DiFFxTrdS

CS

fund

SYM

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

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

ababababab vAAAA ˆ)()()()'( 2222

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

dxvgQXS 2ˆ

)(~)(~)()()(~)(~)()(12

ˆ4/1 nCnDnCnDnDnCnDnC

nLvgQ aaaaaaaa

nXS

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

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

172.08

533

v

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

Bound-State

Masses vs. VEV

• Masses (squared) grow quadratically

• Some masses decline

• Massless states appear at some VEVs

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

Continuum limit

• As K ∞ the lowest state becomes massless even at VEV=1

Average number of partons in

bound state

• Ten lightest states at K=7 become “shorter” as VEV grows

Bound-State

Masses with VEV vs. CS

coupling• Masses

(squared) grow quadratically

• Some masses decline

• No massless states appear

Continuum limit with CS term

• As K ∞ the lowest state remains massive (at VEV=1 and κ =1)

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

2

2112 11,...,

),...,,()(1

q

q

l

AA

nn

K

q

q

ii

qK

nna nnnKnng

l

l

q

Lightest state

• K= 8, v = 1, κ = 1

• #aB=0.67• #aF=0.11

• #fB=1.08• #fF=0.92 ggaFaF

ggaBaB

ggfBfB

ggfFfF

Second-Lightest

state

• K=8, v=1, K=8, v=1, κκ =1 =1

• #aB=0.72• #aF=0.07

• #fB=0.89• #fF=1.11

ggaFaF

ggaBaB

ggfBfB

ggfFfF

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

Extra Slides

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 !

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 !

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”

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