Transcript Document

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]
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
• Use light-cone coordinates
• Hamiltonian approach:

1 0 1
x 
x x
2

ψ(t) = H ψ(0)
• Theory vacuum is physical vacuum
- modulo zero modes (D. Robertson)

The Theory: N =1 SYM in 3D
S21  SSYM  S fund  SCS
with SYM & Chern-Simons couplings g & κ
S SYM
i
 1 

  d xTr  F F    D  
2
 4

3

S fund   d 3 x D   D  iD     g (     )
SCS

ˆ   
2i

  d x   A  A  A A A   2 
3
2

3



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:
( A2 )'ab  ( A2 )ab  ( A2 )ab  ( A2 )ab  vˆ ab
• Dimensionally reduce to 2D by dropping
derivatives w.r.t. transverse coordinates
Extra Terms induced by the VEV
• The shift by the VEV generates extra terms in the
supercharge which are fairly simple:
Q

XS

gvˆ




dx






2

• In SDLCQ mode decomposition it reads

XS
Q
gvˆ
  1/ 4
2
L


n

1 
~
~
~ ~

Ca (n) Da (n)  Ca (n) Da (n)  Da (n)Ca (n)  Da (n)Ca (n)
n

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
v
33  5
 0.172
8
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
BoundState
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
BoundState
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
2
 q
 q nl A
g a (n)       ni  K   n  Al  (n1 , n2 ,...,nq )
q  2 n1 ,...,nq 1  i 1
 l 1
K
K q
• 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
Lightest
state
gaB
• K= 8, v = 1,
κ=1
• #aB=0.67
• #aF=0.11
• #fB=1.08
• #fF=0.92
gfF
gfB
gaF
SecondLightest
state
gaB
gfB
• K=8, v=1,
κ =1
• #aB=0.72
• #aF=0.07
• #fB=0.89
• #fF=1.11
gfF
gaF
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:
H=
E0 -D
-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