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Knotted flux tubes in QCD
Tom Kephart Vanderbilt University
MIAMI'12 Fort Lauderdale, FL
14 December 2012
Work with R. Buniy, J. Cantarella, and E. Rawdon
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Introduction: Tight Knots and Physics
● Potential for Many Physical Systems to be Knotted or Linked, some tightly
● Examples in Several Areas of Physics
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List of Knot/Link ExamplesCLASSICAL PHYSICS/BIOPHYSICS● Plasma Physics● DNA● MacroBiology● Fluid Vortices
QUANTUM PHYSICS● QCD Flux Tubes ● Superconductors ● Superfluids and Superfluid Turbulence● Atomic Condensates● Cosmic Strings
UNIVERSALITY for Tight Quantum Case
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Plasma Phys: Magnetic Flux Tubes● Tubes under Tension Contract● Minimum Length Determined by Topology● Taylor States in Plasma● Differs for QCD where radius is fixed
L. Woltjer, PNAS, 44, 489 (1958)Plasma Physics
H. K. Moffatt, J. Fluid Mech. 35, 117 (1969)J. B. Taylor, PRL, 33, 1139 (1974)
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Knot / Link stabilty
Conserved quantum numbers
Gaussian linking--Hopf link, trefoil knot
Genealized linking--Borromean rings, etc.
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Flux Tubes in Quantum Chromodynamics
● ``Glueballs'' as Tight Knots and Links ● Quantized Flux● Knot “Energy” (EK = L/r) Length
Proportional to Particle Mass● Semi-classical model at the level of
liquid drop model of nucleus or QCD bag model
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Confinement
● Weak coupling at high energy scalesαS is ~ 0.11 at 100 GeV
● Strong coupling at low energy scalesαS is ~ 1 at 1 GeV
● Quarks and gluons are confined ● bags and tubes
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Chromoelectric Flux
● Flux between q, qbar pair is chromoelectric● Open for normal mesons● Closed in our case● Tightly knotted and/or linked
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Counting knotsn = number of crossings
Hoste et al. 1998N. Sloane, The On-Line Encyclopedia of Integer Sequences!
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Types of knots and links
● Prime knots● Composite knots (connected sums of
prime knots)● Prime links ● Composite links● Physics should allow all types !
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Non-q-qbar bosonic hadrons
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Glueballs● Hadrons - Strong Interactions● No Valence Quarks, fJ states (JPC = J++)● Do not Decay Directly to Photons
● Decay in Knotted Flux Tube Model of Glueballsvia QM processes:
1. String (Tube) Breaking 2. Reconnection 3. Tunneling
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String tension and tube radius
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Tight Knots and Links in QCD
T. Ashton, J. Cantarella, M. Piatek, and E. J. Rawdon, “Knot Tightening by Constrained Gradient Descent,” Experimental Mathematics, 20, 57 (2011). arXiv:1002.1723
Roman V. Buniy and TWK, “A model of glueballs,’’ Phys. Lett. B576, 127 (2003)
R. Buniy and TWK,“Glueballs and the universal energy spectrum of tight knots and links,’’ Int. J. Mod. Phys. A20, 1252 (2005)
R. Buniy, J. Cantarella, TWK and E. Rawdon,“The tight knot spectrum in QCD,” arXiv:1212.1500
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2012 Refit with New Data● New Glueball Data JPC = J++ States
Particle Data Group 2011● New Tight Prime Knot/Link Data● New! Composite Knots and Links● Expect continuum of glueballs by
~3 GeV from knot counts
R. Buniy, J. Cantarella, TWK and E. Rawdon,“The tight knot spectrum in QCD,” arXiv:1212.1500
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Fit● We fit one set of apparently sporadic
numbers to another● I.e., f-state masses <---> knot lengths● Either one-parameter: slope● Or two parameters: slope and intercept
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fJ's vs knot lengths
One parameter fit with uncorrected knot lengths
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fJ's vs knot lengths
One parameter fit with uncorrected knot lengths
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Curvature Corrections● We have physical knots, not ideal knots● Flux is uniform across cross sections of
straight tubes● But, knotted and linked tubes are curved● Flux is not uniform across cross sections of
curved tubes● Correct energy for curvature
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Toroidal Solenoid-Cross Section
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Curvature Correction as a Function of R2
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Histogram of the magnitudes of the curvature corrections to the first 72 knot and link lengths.
31Histogram of the magnitudes of the curvature corrections to all 945 currently tabulated knot and link lengths.
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Now fit fJ's to knots with curvature corrections included
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Options● 1. Linear fit: 1-1 in order of mass/length● 2. Move f states with large error bars out of
order to improve fit to: a. Account for tension in data by leaving gaps b. Find best fit
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Uncorrected lengths
Fitting Summary
One parameter fit Two parameter fit
Low f(1370) fit1-1 for 1st 12
High f(1370) fit1190 MeV state
Curvature corrected lengths
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fJ's vs knot lengths
One parameter fit with curvature corrected knot lengths (high 1370)
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fJ's vs knot lengths
One parameter fit with curvature corrected knot lengths
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High f(1370) fit
● This previous figure is best fit allowing for resolution of the tension in the data at ~1220 MeV
● Free knot/link: i.e., 421 unassigned and
used to predict a state near 1190 MeV
381 parameter fit of f-states with high f(1370)
391 parameter fit predictions with high f(1370)
401-p fit of f-states plus predictions with high f(1370)
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Predicted state at ~1190 MeV
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One parameter curvature corrected vs uncorrected fits
● f(1270) and f(1285) poorly fit without curvature correction
● Fit of f(1270) and f(1285) considerably improved with curvature correction
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Return to uncorrected length
● Two parameters● f(1270) and f(1285) fit somewhat improved ● Large intercept
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fJ's vs knot lengths
Two parameter fit with uncorrected knot lengths
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fJ's vs knot lengths
Two parameter fit with uncorrected knot lengths
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fJ's vs knot lengths
Two parameter fit with uncorrected knot lengths
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Two parameter fit for curvature corrected lengths
● Fit virtually unchanged from one parameter fit—intercept goes through origin
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fJ's vs knot lengths
Two parameter fit with curvature corrected knot lengths
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fJ's vs knot lengths
Two parameter fit with uncorrected knot lengths
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Summary of two parameter fits
● Two parameter fit does not pass through origin for uncorrected lengths
● Curvature corrected two parameter fit is consistent with zero intercept
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Corrections to Knot Lengths● Curvaturecorrections ~ 1.5 % variation● Constriction corrections few %● Distortion corrections few %● QCD confinement effects ?
● Conservative assumption:
3% error on knot energy● Note: only the spread in corrections is
important
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one parameter fit—curvature corrected, low f(1370)
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Best Fit
● The previous figure is the best overall fit● Curvature corrected● One parameter● Low f(1370) ● First 12 are 1-1
54two parameter fit—curvature corrected, low f(1370)
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Predictions for masses (1-parameter, CC), low f(1370)
56Fit plus predictions (1-parameter, CC), low f(1370)
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Quality of the fit
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Fit and predicted masses for 1st 22 knots/links
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Predicted fJ's from knot energies
From one parameter fit with curvature corrected knot lengths
● No new states below ~1690 MeV, low f(1370) (But not inconsistent with one new state at ~1190 MeV upon reordering f(1370)-high)
● Five new states 1690—1705 MeV range● Many new states above ~1720 MeV
● Estimated masses to 3% accuracy
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Comments about Predictions
● There is tension in the data near f(1710), f(1810), and some other high mass f-states.
● 50 years of HEP data to reconcile ● Multiple states possible in these regions● Attempts should be made to resolve them
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CONCLUSIONS
Systems of Quantized Flux Tubes ● Knots and Links ● Fixed Tube Radius (Quantized Flux)● Tight implies “Quantized Lengths” for tight knots● Quantized Energy
● Universal Spectra
One Parameter per System - the Slope
Predictions for QCD!
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END
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fJ's vs knot lengths
One parameter fit with uncorrected knot lengths
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Partial list of f0(1710) experiments
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Partial list of f0(1710) experiments
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Fit of f(1710) mass
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Other Corrections
Constriction:
Balance tension with magnetic pressure
Estimate 5% correction for Hopf link
QCD may be more complicated-confinement effects, etc.
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Other Corrections
Toroidal flux tube (minor radius R1, major radius R2) constricting a cylindrical flux tube of radius R(z)
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Other Corrections
Distortion of tube cross section
E.g., wrap a rope tightly around a post and it'sCross section distorts
Estimate few % corrections
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Fit and predicted masses for 1st 22 knots/links
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Smallest knot on a cubic lattice
Minimal trefoil hits 24 vertices, Y. Diao (1993)
From knotplot, Rob Scharein
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Trefoil energy
● Minimal lattice trefoil energy = 24 (Y. Diao)● Minimal lattice trefoil energy ~ 16.44● Lattice could predict knot energies on the
high side
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Lattice
Conventional glueball on lattice--difficult problem
Lattice knotted and linked flux tubes--open problem
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CONCLUSIONS
CLASSICAL SYSTEMS ● Systems can Knot and Link ● Possibly Tight● Varying Tube Radius and Energy ● No discrete spectrum
