Glueballs and Tight Knots - Academia Sinica

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Glueballs and Tight Knots - Academia Sinica

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

5/2/10 2


Introduction: Tight

Knots and Physics


Many Potential Physical Systems

can be Tightly Knotted or Linked


Examples from Several Areas of

Physics

5/2/10 3


Some Knots

All possible prime knots

ranging from 3 1

to 9 49

.

Diagrams by Ali Roth/Cabinet,

from Knots and Links by

Dale Rolfsen

5/2/10 4


List of Examples

CLASSICAL PHYSICS

Plasma Physics

DNA

MacroBiology

QUANTUM PHYSICS

QCD Flux Tubes

Superconductors

Superfluids and Superfluid Turbulence

Atomic Condensates

Cosmic Strings

UNIVERSALITY for Tight Quantum Case

5/2/10 5


Flux Tubes




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)

5/2/10 6


Plasma Physics


Magnetic Fields at Sun Spots

SOHO image

5/2/10 7


Magnetic Helicity

Energy

E = s B 2 dV

Helicity

H = s B dA


Minimize Energy E Holding H Fixed

Find J = λB

Force Free Configurations

P. M. Bellan, ``Spheromaks,’’ (2000)

5/2/10 8


Helicity Calculation

5/2/10 9


Knot/Link stabilty

Conserved quantum numbers

Gaussian linking--Hopf link, trefoil knot

Genealized linking--Borromean rings, etc.

5/2/10 10


Sea Creatures in Knots

Hagfish -- sightless eel

5/2/10 11


Feeding Hagfish

5/2/10 12


Knotted DNA

5/2/10 13


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.

5/2/10 14


Proteins can also be Knotted

Computer model of knotted protein in

methanobacterium thermoautotrophicum

Argonne National Laboratory

5/2/10 15


Knots and Particle Physics

Lord Kelvin: Modeled “elementrary” atoms as

knotted fluid vortices in the aether.

5/2/10 16


Flux Tubes in

Quantum Chromodynamics





Tight Knots and Links as Glueballs

Knot “Energy” (E K = L/r) Proportional

to Particle Mass

Quantized Flux

Semi-classical model at the level of

liquid drop model of nucleus or QCD

bag model

5/2/10 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 (3 1 knot)

From E. Rawdon webpage

rope length 16.435

5/2/10 19


Tight Figure Eight Knot

(4 1 knot)

rope length 21.2313

5/2/10 20


Tight 5 1 Knot

rope length 23.8431

5/2/10 21


Tight 5 2 Knot

rope length 25.5724

5/2/10 22


Tight 8 1 Knot

rope length 35.9874

5/2/10 23


Growth of knot “mass”

Assume length L of tight knows grows

as number of crossings n to a power p ~ 1.

L = αn p

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.

5/2/10 24


Counting knots

n = number of crossings

Hoste et al. 1998

N. Sloane, The On-Line Encyclopedia

of Integer Sequences!

5/2/10 25


Counting knots

c chiral noninvertible

+ amphichiral noninvertible

- amphichiral invertible

i chiral invertible

a fully amphichiral and

invertible

5/2/10 26


Glueballs


Hadrons - Strong Interactions

No Valence Quarks, f J

states (J PC = 0 ++ )



Do not Decay Directly to Photons

Decay via

1. String (Tube) Breaking

2. Reconnection

3. Tunneling

in Knotted Flux Tube Model of Glueballs

5/2/10 27


f J states as glueballs





Lattice calculations

QCD sum rules

electric flux tube models

constituent glue models

consensus: states with no valence

quark are glueballs

quantum numbers J ++ = 0 ++ , ...

5/2/10 28


Identify:

Lightest glueball candidate --Shortest tight Knot/Link

f 0 (600) Hopf link

Identify:

Next Lightest -- Next Shortest

f 0 (980) trefoil knot

Identify:

etc.

etc.

5/2/10 29


Knot energy vs glueball mass, 2003 results

5/2/10 30


5/2/10 31


Angular momentum

Rotational excitations:

E.g., identify f 0 (1500), with the 5 2 knot and the f 1 (1510),

and f’ 2 (1525) as the l = 1,2 rotational excitations of the 5 2 .

Intrinsic angular momentum of solitons?

5/2/10 32


Refit with New Data





New Glueball Data J PC = 0 ++ States

Particle Data Group 2008

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,

Math.DG/0508248

5/2/10 33


Rope lengths from Ashton et al.

5/2/10 34


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.

5/2/10 35


Hopf link

5/2/10 36


Inertia Tensor

(

I a

)

I = Ib

Ic

I Hopf

=

( )

42 0 0

0 75 0

0 0 75

πρa 5 /2

Hopf is Prolate:

I b = I c > I a

E(J,K) = BJ(J + 1) + (A - B)K 2

B ~ I b -1 and A ~ I a

-1

5/2/10 37


Internal chain link

5/2/10 38


Chain of three

5/2/10 39


Chain of four

5/2/10 40


N odd chains

N even chains

5/2/10 41


Key chain link of four

5/2/10 42


Key chain link of five

5/2/10 43


Hopf link with two units of flux in one

tube and one on the other

5/2/10 44


Results (as of 2005)

Knot Length Errors 0.1%

ν χ 2 = 3.7 for n = 10 f 0 States

Slope Parameter

S = 60.6 +/- 0.91 MeV

Intercept -9.0 +/- 26.1

S ~ Λ QCD /π

New States at E = E K xS

E.g.,

4

2

1 at E = 1203 MeV,

7

2

7 at E = 1673 MeV,

etc.

5/2/10 45


Lots of missing knots and links

5/2/10 46


List of first 39 lengths (L/r) with knot

2 2 1

25.1334

3 1

32.7436

4 2 1

40.0122

2 2 1

# 2 2 1

41.8847

4 1

42.0887

5 1

47.2016

5 2

49.4701

3 1

# 2 2 1

49.6017

2 2 1

49.7716

6 3 3

50.5539

6 2 1

54.3768

7 2 7

55.5095

2 2 1

# 2 2 1

# * 2 2 1

56.2655

6 2 2

56.7000

6 1

56.7058

4 2 1

# 2 2 1

56.9478

6 2

57.0235

7 2 8

57.7631

4 1

# 2 2 1

57.7971

6 3 1

57.8141

2010 Results

Ted Ashton, Jason Cantarella,

Michael Piatek, Eric Rawdon,

arXiv:1002.1723

5/2/10 47


List of first 39 lengths (L/r) with knot-cont.

6 3

57.8392

6 3 2

58.0070

6 2 3

58.1013

2 2 1

# 2 2 1

# * 2 2 1

56.2655

8 3 7

60.5754

8 19

60.9858

7 1

61.4067

8 20

63.0929

7 2

63.8556

7 3

63.9285

5 1m

# 2 2 1

64.1491

5 1

# 2 2 1

64.1770

7 2 1

64.2345

7 4

64.2687

2 2 1

# 3 1

# 2 2 1a

64.2798

8 2 15

64.2996

4 2 1

# 3 1

64.7711

5 2 m

# 2 2 1

64.9165

8 3 8

65.0042

5/2/10 48


MeV

Preliminary

Fit of all f J states below 1950 MeV, but not all knots/links

5/2/10 49

L/d


MeV

Preliminary

L/d

Fit with all f J states below 1950 MeV, and all knots/links

New glueball predicted at each circle

5/2/10 50


MeV

Preliminary

L/d

Just the predictions:

Only 2 with M < 1675 MeV

5/2/10 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

Preliminary

1751.92

1836.79

1849.18

1921.24

1923.38

1929.88

1930.69

1932.39

1933.39

1933.72

1934.30

5/2/10 52


Density of knot lengths





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 f J state every 2 MeV

by L/r ~ 70

5/2/10 53


A Few Other Systems

that may Support Tight

Knots with Quantized

5/2/10 54


Superconductors



Type II Superconductors have Quantized

Magnetic Flux Tubes

Magnetic Field at Surface


Braids, Weaves

5/2/10 55


Superfluids




Helium at Low Temperature

All Atoms are in the Same State

Ground State-Condensate

5/2/10 56


Superfluid Turbulence



Vortex Flow Linkage

Path to Turbulence

(C. Ernst, TWK, and E. Rawdon, in progress)

5/2/10 57


Vortex ring generator for superfluid 3 He-B

From D. I. Bradley, et al., Phys. Rev. Lett. 95, 035302(2005)

5/2/10 58


5/2/10 59


Atomic Condensates






Some Atomic Gases form

Condensates at Low Temperature

and Density

78 Ru

Quantized Angular Momentum

Vortex Lines

Knots and Links?

5/2/10 60


5/2/10 61


Cosmic Strings




Flux Tubes in Yang-Mills-Higgs

Gauge Theories

Can be Superconducting

Stable Knots and Links?

5/2/10 62


Super Strings



3+1 Dimensions: Stable Knots and

Links ?

N+1 Higher Dimensions: Knotted and

Linked Branes in Codimension 2 ?

5/2/10 63


CONCLUSIONS

CLASSICAL SYSTEMS




Knots and Links

Possibly Tight

Arbitrary Tube Radius

5/2/10 64


CONCLUSIONS

QUANTUM SYSTEMS





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

5/2/10 65


END

5/2/10 66

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