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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]
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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Bound-State
Masses vs. VEV
• Masses (squared) grow quadratically
• Some masses decline
• Massless states appear at some VEVs
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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
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Continuum limit
• As K ∞ the lowest state becomes massless even at VEV=1
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Average number of partons in
bound state
• Ten lightest states at K=7 become “shorter” as VEV grows
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Bound-State
Masses with VEV vs. CS
coupling• Masses
(squared) grow quadratically
• Some masses decline
• No massless states appear
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Continuum limit with CS term
• As K ∞ the lowest state remains massive (at VEV=1 and κ =1)
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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
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Lightest state
• K= 8, v = 1, κ = 1
• #aB=0.67• #aF=0.11
• #fB=1.08• #fF=0.92 ggaFaF
ggaBaB
ggfBfB
ggfFfF
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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
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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
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Extra Slides
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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 !
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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 !
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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”
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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