Glueballs and Tight Knots

Glueballs and
Tight Knots
Tom Kephart
Vanderbilt University
Seminar at Academia Sinica
14 May 2010
5/2/10
1
Knots, Links, and Physics
Examples of Knots and Links in Physics
Tight Knots Theory
Tight Knots/Links and Glueballs
Other Possible Tight Knot Applications
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Introduction: Tight
Knots and Physics
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Many Potential Physical Systems
can be Tightly Knotted or Linked
Examples from Several Areas of
Physics
3
Some Knots
All possible prime knots
ranging from 31 to 949.
Diagrams by Ali Roth/Cabinet,
from Knots and Links by
Dale Rolfsen
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4
List of Examples
CLASSICAL PHYSICS
 Plasma Physics
 DNA
 MacroBiology
QUANTUM PHYSICS
 QCD Flux Tubes
 Superconductors
 Superfluids and Superfluid Turbulence
 Atomic Condensates
 Cosmic Strings
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UNIVERSALITY for Tight Quantum Case 5
Flux Tubes

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Tubes under Tension Contract
Minimum Length Determined by
Topology
Taylor States in Plasma
Plasma Physics
L. Woltjer, PNAS, 44, 489 (1958)
H. K. Moffatt, J. Fluid Mech. 35, 117 (1969)
J. B. Taylor, PRL, 33, 1139 (1974)
Biology
A. Stasiak, Nature 384, 122 (1996)
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6
Plasma Physics
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Magnetic Fields at Sun Spots
SOHO image
7
Magnetic Helicity
Energy
Helicity

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E=
s B dV
H=
2
s B dA
Minimize Energy E Holding H Fixed
Find J = λB
Force Free Configurations
P. M. Bellan, ``Spheromaks,’’ (2000)
8
Helicity Calculation
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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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Sea Creatures in Knots
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Hagfish -- sightless eel
11
Feeding Hagfish
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Knotted DNA
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Tightly Knotted DNA
Tight knots first discussed and lengths
estimated in:
Katritch, V., Bednar, J., Michoud, D.,
Scharein, R.G., Dubochet, J. & Stasiak,
A. (1996) Nature
Katritch, V., Olson, W.K., Pieranski, P.,
Dubochet, J. & Stasiak, A. (1997) Nature.
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14
Proteins can also be Knotted
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Computer model of knotted protein in
methanobacterium thermoautotrophicum
Argonne National Laboratory
15
Knots and Particle Physics
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Lord Kelvin: Modeled “elementrary” atoms as
knotted fluid vortices in the aether.
16
Flux Tubes in
Quantum Chromodynamics
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Tight Knots and Links as Glueballs
Knot “Energy” (EK = L/r) Proportional
to Particle Mass
Quantized Flux
Semi-classical model at the level of
liquid drop model of nucleus or QCD
bag model
17
Tight Knots and Links in QCD
Roman V. Buniy and TWK, “A model of glueballs,’’
Phys. Lett. B576, 127 (2003)
Roman V. Buniy and TWK,“Glueballs and the
universal energy spectrum of tight knots and links,’’
Int. J. Mod. Phys. A20, 1252 (2005)
Martha J. Holmes, et al., “Rotational
energies of tight links’’ (to appear)
Jason Cantarella, Eric Rawdon, et
al., “On the lengths of tight knots and
links,’’ (to appear)
5/2/10
18
Tight Trefoil (31 knot)
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From E. Rawdon webpage
rope length 16.435
19
Tight Figure Eight Knot
(41 knot)
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rope length 21.2313
20
Tight 51 Knot
rope length 23.8431
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21
Tight 52 Knot
rope length 25.5724
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22
Tight 81 Knot
rope length 35.9874
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Growth of knot “mass”
Assume length L of tight knows grows
as number of crossings n to a power p ~ 1.
L = αnp
α is a constant.
Then the number of knots per unit length
grows very rapidly as seen in tables below.
Assume similar behavior for links.
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Counting knots
n = number of crossings
Hoste et al. 1998
N. Sloane, The On-Line Encyclopedia
of Integer Sequences!
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Counting knots
c
+
i
a
chiral noninvertible
amphichiral noninvertible
amphichiral invertible
chiral invertible
fully amphichiral and
invertible
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Glueballs
Hadrons - Strong Interactions
 No Valence Quarks, f states (JPC = 0++)
J
 Do not Decay Directly to Photons
 Decay via
1. String (Tube) Breaking
2. Reconnection
3. Tunneling
in Knotted Flux Tube Model of Glueballs
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fJ states as glueballs

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Lattice calculations
QCD sum rules
electric flux tube models
constituent glue models
consensus: states with no valence
quark are glueballs
quantum numbers J++ = 0 ++, ...
28
Identify:
Lightest glueball candidate --Shortest tight Knot/Link
f0(600)
<--->
Hopf link
Identify:
Next Lightest -- Next Shortest
f0(980)
Identify:
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<--->
etc.
trefoil knot
etc.
29
Knot energy vs glueball mass, 2003 results
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31
Angular momentum
Rotational excitations:
E.g., identify f0(1500), with the 52 knot and the f1(1510),
and f’2(1525) as the l = 1,2 rotational excitations of the 52.
Intrinsic angular momentum of solitons?
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32
Refit with New Data

New Glueball Data JPC = 0++ States
Particle Data Group 2008

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
New Tight Link/Knot Data
(still need composite knots and links!)
Continum of glueballs by ~3 GeV from
knot counts
Ted Ashton, Jason Cantarella, Michael Piatek, Eric Rawdon,
5/2/10
Math.DG/0508248
33
Rope lengths from Ashton et al.
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Rotational Energy Levels
Inertia tensor are known for some knots
Can be calculated exactly for some links
Examples:
Hopf link
Chains
Key chain links
Martha J. Holmes, Ph.D. thesis
Knots
J. Cantarella, E. Rawdon et al.
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Hopf link
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Inertia Tensor
I
I
Hopf
( )
Ia
=
=
Ib
Ic
( )
42 0 0
0 75 0
0 0 75
Hopf is Prolate:
πρa5/2
Ib = Ic > Ia
E(J,K) = BJ(J + 1) + (A - B)K2
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-1
I
B~ b
and
A ~ Ia-1
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Internal chain link
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Chain of three
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Chain of four
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N odd chains
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N even chains
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Key chain link of four
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Key chain link of five
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Hopf link with two units of flux in one
tube and one on the other
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Results (as of 2005)
Knot Length Errors 0.1%
ν
χ2 = 3.7 for n = 10 f0 States

Slope Parameter
S = 60.6 +/- 0.91 MeV
Intercept -9.0 +/- 26.1
S ~ ΛQCD/π

New States at E = EKxS

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E.g.,
412 at E = 1203 MeV,
772 at E = 1673 MeV,
etc.
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Lots of missing knots and links
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List of first 39 lengths (L/r) with knot
22 1
31
42 1
22 1 # 22 1
41
51
52
3 1 # 22 1
22 1
63 3
62 1
72 7
22 1 # 22 1 # * 22 1
62 2
61
42 1 # 22 1
62
72 8
4 1 # 22 1
63 1
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25.1334
32.7436
40.0122
41.8847
42.0887
47.2016
49.4701
49.6017
49.7716
50.5539
54.3768
55.5095
56.2655
56.7000
56.7058
56.9478
57.0235
57.7631
57.7971
57.8141
2010 Results
Ted Ashton, Jason Cantarella,
Michael Piatek, Eric Rawdon,
arXiv:1002.1723
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List of first 39 lengths (L/r) with knot-cont.
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63
632
623
22 1 # 22 1 # * 22 1
83 7
8 19
71
8 20
72
73
5 1m # 22 1
5 1 # 22 1
72 1
74
22 1 # 3 1 # 22 1a
82 15
42 1 # 31
52 m # 22 1
57.8392
58.0070
58.1013
56.2655
60.5754
60.9858
61.4067
63.0929
63.8556
63.9285
64.1491
64.1770
64.2345
64.2687
64.2798
64.2996
64.7711
64.9165
83 8
65.0042
48
MeV
Preliminary
L/d
Fit of all fJ states below 1950 MeV, but not all knots/links
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49
MeV
Preliminary
L/d
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Fit with all fJ states below 1950 MeV, and all knots/links
New glueball predicted at each circle
50
MeV
Preliminary
L/d
Just the predictions:
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Only 2 with M < 1675 MeV
But 23 with M < 1945 MeV51
New Masses
1219.66 MeV
1506.83
1675.66
1697.90
1710.69
1710.86
1717.98
1741.97
1742.97
1743.47
1744.21
1749.15
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Preliminary
1751.92
1836.79
1849.18
1921.24
1923.38
1929.88
1930.69
1932.39
1933.39
1933.72
1934.30
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Density of knot lengths
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9 knots/links, both prime and composite
with L/r < 50
27 knots/links, both prime and composite
with 50 < L/r < 65
~135 prime knots and links with 65 < L/r
< 80. (Including composites expect
~200.)
Implies approx one fJ state every 2 MeV
by L/r ~ 70
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A Few Other Systems
that may Support Tight
Knots with Quantized
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Superconductors

Type II Superconductors have Quantized
Magnetic Flux Tubes
Magnetic Field at Surface
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Braids, Weaves
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Superfluids
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Helium at Low Temperature
All Atoms are in the Same State
Ground State-Condensate
56
Superfluid Turbulence
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Vortex Flow Linkage
Path to Turbulence
(C. Ernst, TWK, and E. Rawdon, in progress)
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Vortex ring generator for superfluid 3He-B
From D. I. Bradley, et al., Phys. Rev. Lett. 95, 035302(2005)
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Atomic Condensates
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Some Atomic Gases form
Condensates at Low Temperature
and Density
78Ru
Quantized Angular Momentum
Vortex Lines
Knots and Links?
60
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Cosmic Strings
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Flux Tubes in Yang-Mills-Higgs
Gauge Theories
Can be Superconducting
Stable Knots and Links?
62
Super Strings
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3+1 Dimensions: Stable Knots and
Links ?
N+1 Higher Dimensions: Knotted and
Linked Branes in Codimension 2 ?
63
CONCLUSIONS
CLASSICAL SYSTEMS
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Knots and Links
Possibly Tight
Arbitrary Tube Radius
64
CONCLUSIONS
QUANTUM SYSTEMS

Knots and Links
Fixed Tube Radius (Quantized Flux)
Tight implies “Quantized Lengths” for tight knots
Quantized Energy

Universal Spectra
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One Parameter per System - the Slope
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END
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