Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F...

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Origin and Status of INSTANTONS Utrecht University Gerard ’t Hooft, Spinoza Institute. Erice 2013

Transcript of Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F...

Page 1: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Origin and Status of INSTANTONS

Utrecht University

Gerard ’t Hooft, Spinoza Institute.

Erice 2013

Page 2: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

The pre-QCD age (before 1971)

π+ π− πo η η’

Ko

K+

Ko

K−

J PC = 0−+

ρ+ ρ− ρo ω ϕ

K*o

K*+

K*o

K*−

J P = 1−−

d u

s

Page 3: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Quark composition: the eta problem

J P = 0− :π 0 ≈ 1

2 (u u − d d )

η ≈ 16 (u u + d d − 2s s )

η ' ≈ 13 (u u + d d + s s )

J P = 1+ :ρ0 ≈ 1

2 (u u − d d )

ω ≈ 12 (u u + d d )

φ ≈ s s

J /ψ ≈ c cϒ ≈ b b

Compare:

Page 4: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Spontaneous breakdown of chiral symmetry

L = −ψ γ ∂ψ − ψ imijψ j

The kinetic term has U ( N ) × U ( N ) symmetry; the mass terms break that. This symmetry appears to be spontaneously broken towards U ( N )diagonal . This could explain perfectly why mπ

2 << mN2

But, this would require mu,d ≈ 0 , and this would keep U ( 2 ) × U ( 2 ) unbroken. 4 parameters: 4th pseudoscalar, η , should also be massless!

Page 5: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

The eta problem: Explain the eta mass, and the eta mixing.

Then came QCD

It had even worse problems: Explain quark confinement !

Page 6: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Nielsen - Olesen: Magnetic Confinement

( ) †14, ( )A F F D D Vµν µν µ µϕ ϕ ϕ ϕ= − − −L

In case of spontaneous "breakdown" of (1)U I→

N

S

Page 7: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Color Magnetic Super Conductivity

N S

+ _

Electric Super Conductor

Magnetic Super Conductor

Page 8: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

A.Polyakov, G. ’t Hooft (1974)

The Magnetic Monopole

Page 9: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

In 2 dimensions, we have a vortex (Niesen – Olesen)

You need a two-component (or complex U(1)) field In 3 dimensions, we have a particle

(magnetic monopole) You need a 3-component ( or isospin 1) field

In 4 dimensions ??

A topological event ?? Try a 4-component field, such as a (complex) isospin ½ field !

Hedgehogs:

Page 10: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

(2)SU

The Instanton Belavin, Polyakov, Schwartz, Tyupkin

Group of Gauge Transformations

But it so happens that then you can discard the isospin ½ field !

Page 11: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

The gauge transformation that transforms an isospin ½ hedgehog at infinity towards the form

creates a gauge potential field

with, by construction, a vanishing

φi →F0

⎛⎝⎜

⎞⎠⎟

F µνa

Aµa → 2

gηµνa xνx2

Now demand an extreme of the action

without singularity at x = 0, gives the instanton solution:

S = − 14 d 4x∫ F µν

a F µνa

Page 12: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Aµa (x) = 2

gηµνa (x − z)ν

(x − z)2 + ρ2

F µνa = −4

gρ2 ηµν

a

(x − z)2 + ρ2( )2ηµνa =

ε aµν , µ,ν = 1,2, 3

−δ aν , µ = 4

δ aµ , ν = 4

0 , µ = ν = 4

⎪⎪

⎪⎪

BPST observed:

F µνa = F µν

a = 12 εµναβ F αβ

a ;14 d 4x∫ F µν

a F µνa = 1

4 d 4x∫ F µνa F µν

a = −S = 8π 2g2

Page 13: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Triangle diagram:

∂µ JµAa = g2

16π 2 F µνa F µν

a

∂µ JµAa d 4∫ x = ± 2

Apparently, two units of axial charge are created (or destroyed) by the instanton If we have just one flavor, this is a mass term:

L R

for every flavor !

In case of many flavors: R

R

R L

L

L

Page 14: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

!"" !

""

Inst+= or

#uL

uR

dR

dL

#dL

dR

uR

uL

#

uL

uR

dR

dL

sR

sL

ms

+ ( u $ d )

π 00 = 1

2 (uu + dd )

Page 15: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Instanton

Fermi level

time LEFT

RIGHT

The case with massless fermions

Page 16: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

This topological transition is a tunnelling phenomenon,

FµνFµν∫ d4x = 32π 2

g2

ei Ld4x∫ → e

i Ld4x+i θg2

32π 2 Fµν Fµν∫ d4x∫ , or

L→L+ θg2

32π 2 FµνFµν

If the original state and the gauge rotated state have a relative phase , then this can be represented in the functional integrals by means of the substitution

ie θ

to be computed semiclassically in Euclidean spacetime ( ) ; the gauge fields take large values. One finds: t→ i x4

Page 17: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

(In the absence of massless fermions)

∂⋅(E + θg2 B

8π 2) = ρ ;

D =E + θg2 B

8π 2; H = 1

2 (E2 +

B2 )

L A,ϕ( ) = − 14 Fµν Fµν +

132π 2 θg2Fµν

Fµν =

= 12 (E 2 −

B 2 )+ 1

8π 2 θg2E ⋅B =

= 12 (E + 1

8π 2 θg2B)2 − 1

2 (1+ 164π 4 θ 2g4 )

B 2

The electric and magnetic charge quanta are:

Qe = g ; Qm = 4π

g

−θ2π

gConclude: magnetic monopoles

carry electric charge: ∂⋅B = Qmδ ( x)

Page 18: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

• • • • • •• • • • • •• • • • • •

• • • • • •• • • •

O O O O O

• •• • • • • •

electricQ

1 Dirac Unit

2m eQ Q πΔ ⋅Δ =

magneticQCondensation of Electric Charges

0θ =

Page 19: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

magneticQCondensation of Magnetic Charges

• • • • •• • • • •• • • • •• • • • •• • • • •• • •

O

O

O

O

O

• •• • • • •O

1 Dirac Unit

electricQ

2m eQ Q πΔ ⋅Δ =

0θ =

Page 20: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

(No massless fermions present) magneticQ

electricQ

Confinement when

• • • • •• • • • •• • • • •• • • • •

• • • • •• • • • •• •

O

O

O

O

O

O

O • • •

0θ ≠

Page 21: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

magneticQ Oblique Confinement:

• • • • •• • • • • • •• • • • •• • • • • •

• • • • •• • • • • ••

O

O

O

• • •

• • • • • •

θ π;

electricQ

(No massless fermions present)

Page 22: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

String tension for oblique confinement

0 2 π π

ρ ↑

θ→

Page 23: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Here, the true story about the instantons just begins!

P. van Baal D. Diakonov

Page 24: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Put QCD in a box with periodic boundary conditions

+ !

a)

b)q q

c)q q

d)

q q

e)

q q q q

f )

q

qq

Page 25: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Twisted boundary conditions: instantons calorons

Instanton number now may become fractional

Page 26: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or
Page 27: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

Yet another scalar meson problem: tetraquarks

Take a baryon, and replace one quark (in a 3-repres-entation of SU(3)-color) by an antisymmetric pair of antiquarks (also a color 3 state: ). We then have a 2quark+2antiquark state. Same tetraquarks can also be seen by replacing the quarks in a meson by antiquark bound states in the color-3 representation. They form a nonet:

[3.3]a = 3

Page 28: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

σ [0] = [ud][ud ]κ = [su][ud ]; [sd][ud ] +

f0[0] = 1

2 [su][s u ]+ [sd][s d ]( )a0 = [su][sd ]; 1

2 [su][s u ]− [sd][s d ]( ); [sd][s u ]

These states can be recognised by an inverted mass spectrum: the isotriplet has more strange quarks in it than the doublets. Why do they exist fairly prominently?

Page 29: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or

There’s mixing between tetraquarks and diquarks, due to the instanton:

Instan- ton

Mixing is largest where the levels are closest together:

ss ↔ [ud][u d ]

Page 30: Origin and Status o INSTANTONS · This topological transition is a tunnelling phenomenon F µν F µν d 4x=32π 2 g2 e i∫Ld4x→e iLd 4x+iθg 2 32π2 F µν F µνd ∫ 4x, or