Mass of the Vector Meson in
Quantum Chromodynamics in the
limit of large number of colors
Rahul Patel
With
Carlos Prays and Ari Hietanen
Mentor: Dr. Rajamani Narayanan
Department of Physics
Florida International University
OUTLINE
• Quantum Chromodynamics
• Feynman Path Integral (Quantum)
• Parallel Transport (Gauge Theory)
- Abelian (EM) and Non-Abelian (Color)
• Particle Propagation
- Meson and Quark propagation
- Vector Meson Propagation
• Computer Simulation
Quarks and Gluons
Quarks
• Elementary particles constituting most matter in
universe
• Six different quarks
• Combinations of these quarks form composite
particles:
1) Baryons (3 quark combination):
ex: Protons, Neutrons
2) Mesons (quark, anti-quark combo):
ex: pion, rho
• Combinations must produce colorless
particles (due to Gauge symmetry)
Gluons
• Messenger particles between quarks
Quantum Chromodynamics (QCD)
• Study of interaction between quarks and gluons
• Interaction causes Strong Force
• Analogy:
Electromagnetic force: photon field interacting with electrons and
protons – only 1 field.
Strong force: Gluon field interacts with quarks – 8 fields defined by
color names (red, blue, green, etc.) – Confinement!!!
•N: refers to number of fields
•N2- 1 particles of that field
• This project studies QCD in more than 3 colors (N)  Final goal to set N
 infinity.
Summary thus Far…
Goal: To calculate the mass of a ρ particle at zero quark mass at Large N
(color fields) and 4 dim.
Large N :
• Large N : Results simplify for SU(N) where N  ∞
• Do not expect any qualitative difference between N = 3 and N = infinite
• QCD results obtained by setting N = 3
What is rho (ρ)?
• Subnuclear particle made up of two quarks
• Excited state of meson but a lot heavier
• pi meson mass goes to zero when quark mass goes to zero..
WHAT ABOUT ρ?
METHOD Summary: ρ Mass Calculation
1 ) Propagate composite meson through
space lattice.
2) Same time - Calculate propagation of
quarks in gluon field.
3) Look at energy-momentum
dependence of the composite particle
4) E2 = (mc2)2 + (pc)2
Note:
i) Particle being studied is ρ
meson
ii) 1 up and 1 anti – down
quark
Fig. 1 Overview of calculation of meson in lattice and gluon field
Lattice Gauge Theory
• Smooth, infinite continuous space is divided into finitely chopped up pieces  cells
Why?
• Purpose: Prevent infinities in calculation
• Quantum Field theory:
- Continuous space-time  ∞ particles (explain field theory)
- Get rid of ∞ particles by eliminating ∞ space
- Make it finite but large
• Vary lattice size to
and obtain thermodynamic limit
Fig. 1 Visualization of propagation of composite
particle through space lattice
Feynman Path Integral
Take path of particle through one path
• Operator propagation:
• Splice up time into equal segments
Fig. 1. Time spliced path of particle through
functional spaces
• Insert between each time spliced operator
• Feynman evaluated it
• After skipping tons of steps and introducing the Lagrangian :
-- time evolution can be written in integral form:
• Introduce time transformation:
• Went from operator to integral form – all that trouble  why?
• Kept Feynman busy – and is easier to compute
• All possible paths (ALL OF THEM) must be taken into account
• Path amplitudes nullified if dissproportionate to
1/h factor
Parallel Transport – Energy and Potentials
• Things look different in the world of sub
nuclear particles
• Presence of potential field (Aμ) curves the
space and changes particle’s orientation
• Total Change in particle’s wavefunction:
 Translation and phase change
Parallel Transport – Case 1
Case 1: EM Potential  Aμ
Propagating particle from point 1 
2 - akin to rotation like transform
• Full rotation around lattice cell
Counter-Clockwise
Fig. 1. Infinitestimal loop with gauge potential
Clockwise
• Full change in wavefunction:
• Where:
is the Field Strength Tensor
• Taylor expanding gives us:
• Replacing μ, ν = 0,1,2,3 in Gauge Field (1 time, 3 space)  sum over all
combinations
• Potentials take on form of electromagnetic potentials and by increasing
number of lattice cells, energy change derived:
Parallel Transport – Case 2
Case 2: Non-Abelian Gauge Potential  Aμ as matrix
• Repeat propagation of particle through small loop
• Counter-Clockwise and Clockwise propagation
• New Strength Tensor  Extra commutator indicative of non-Abelian
nature
Parallel Transport – N dim Potentials
• Aμν : N x N Hermitian matrix
•Aaν (x) where a:[1, N2-1]  N determines type of field and number of fields
- N = 2: Weak Field, N = 3: Strong Color Field
• At large N, commutator overpowers other terms – Field to Field
interaction
• New Action given by:
• Path integral for non – Abelian gauge potential:
•Need to be able to propagate this through the lattice
• Phase change through Gauge field replaced by unitary matrices  Traceless:
• Propagation in clockwise and counter-clockwise direction of original finite cell
replaced by unitary matrices:
• Partition function of Lattice QCD:
METHOD Summary: ρ Mass Calculation
1 ) Propagate composite meson through
space lattice.
2) Same time - Calculate propagation of
quarks in gluon field.
3) Look at energy-momentum
dependence of the composite particle
Fig. 1 Overview of calculation of meson in lattice and gluon field
Meson and Quark Propagation
Quark Propagation through Gluon Field
• Relativistic sister of Schrödinger equation: Dirac operator:
where
• Propagator for general quark: Propagation through lattice (dU) given by
inverse Dirac operator:
• Need to specify type of particle  ρ meson is vector particle with bilinear transformation
• Insert γμ for vector meson 
Vector Meson Propagation
• Propagator for particle in momentum space through Fourier transform  to
obtain energy 
• Inversely proportional to Dirac Operator
• Trace of propagator gives propagational amplitude of quarks in gluon field
Fig.2. Propagation of quarks and recombination to produce composite particle.
Calculation
• Code developed to obtain mass calculation
• Unitary matrix array (7 –dim) , [3 space, 1 time, motion, matrix coordinates]
• Lattice thermalized
• Initial momentum
; n = 2,3,4,5,6
• Calculate Propagational amplitude:
• Smear operator (Inverse of gauge laplacian)  spread given wavefunction for maximum
overlap with physical state
• 2 separate runs for different values of L (lattice size), N (color
fields), b (cell size) – in all 4 dimensions
• 1 run: Many weeks calculated on 31 nodes simultaneously
Fig. 1 Table of values used in calculating mass of the particle
• Inverse of propagation equation graphed to extract ρ mass
Fig. 2 Propagation equation
Fig. 1 Final graph of fitted data to final propagation points obtained.
• Calculations done with various momentum
•Mass of interest at y-intercept : Extrapolated to zero ρ momentum
RESULTS
•Final results for the mass of the rho particle while taking the mass of the quarks to be
zero is:
1082 +/- 70 MeV for N  ∞
•The experimental value is:
775.5 +/- 0.4 MeV
•Final results still correct by taking above factors into account
• Results - Published in Physics Letters B 674(2009): 80-82.
“The vector meson mass in the large N limit of QCD.”
arXiv: arXiv:0901.3752 [hep-lat]
REFERENCES
•
Cheng, Ta-Pei and Li, Ling-Fong. Gauge Theory of Elementary Particle Physics. New York:
Oxford UP, USA, 1988
•
M. Creutz, Quarks, Gluons, and Lattices, Cambridge University Press, 1985
•
M. Leon, Particle Physics: An Introduction, Academic Press, 1973
•
R. Narayanan, H. Neuberger, Phys.Lett. B616 (2005) 76-84
•
J. Marion, S. Thornton, Classical Dynamics of Particles and Systems, Thomson
Brooks/Cole, 2004
•
G. Arfken, H, Weber, Mathematical Methods for Physicists 6e., Elevier Academic Press,
2005
•
A. V. Manohar, in Les Houches 1997, Probing the standard model of particle interactions,
Pt. 2, arXiv:hep-ph/9802419
Thank You
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Dr. Rajamani Narayanan
Carlos Prays
Dr. Ari Hietenan
Richard Galvez
Dr. Rodriguez, Ricardo Leante
All the wonderful people in this room.