## The Topological Structure of the QCD Vacuum

Ting-Wai Chiu (趙挺偉) Department of Physics, National Taiwan University

### December 16, 2008

1 T.W. Chiu, LHC to Universe, Dec 16, '08

Yoichiro Nambu Nobel prize (1/2) Physics 2008

### Nobel prize in physics (2008)

### “for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics”

Chiral symmetry is one of the most subtle features of massless Dirac fermion,

as a consequence of relativity + QM.

Chiral symmetry leads to the conservation of the chiral charge (handedness).

If the chiral symmetry is spontaneously broken, then there exists massless Goldstone bosons (the number of GBs is equal to the number of broken generators).

T.W. Chiu, LHC to Universe, Dec 16, '08 3

### A longstanding problem in particle physics is to show D

### t hat the chiral sym. in C Q is spontaneously br oken.

1 0

### lim lim 1 Tr[( ) ]

*m* *V*

*D* *m*

### *V*

^{}

###

2

2

### 4

*PS*

*PS*

*m* *m*

*f*

###

### Now, if the fermion has a small mass (e.g., from EW breaking), then the mass of the Goldstone boson behaves a s

*m*

0 chiral symmetry is spontaneously broken

Spontaneous chiral symmetry breaking in QCD was reproduced on supercomputer IBM BlueGene/L (57 Tflops peak performance) at KEK, Japan.

[JLQCD and TWQCD Collaborations, Phys. Rev. Lett.98(2007)172001;

Phys. Rev. D76 (2007) 054503, Phys. Rev. Lett. 101(2008) 202004]

T.W. Chiu, LHC to Universe, Dec 16, '08 5

**Eigenvalues of quarks: QCD in a small box has discrete **
eigenvalues of quarks. This plot shows their ratios. "2/1",

"3/1" etc. denote the ratio of second-to-first or third-to- first eigenvalues.

MS 3

### (2 GeV) = (251 7 11 MeV)

###

[JLQCD and TWQCD Collaborations, Phys. Rev. Lett.98(2007)172001;

Phys. Rev. D76 (2007) 054503, Phys. Rev. Lett. 101(2008) 202004]

### OUTLINES

### • Introduction

### • Topological susceptibility

### • Topological structure of QCD vacuum

### • Outlooks

### The Challenge of QCD

At the hadronic scale, , perturbation theory is

incapable to extract any quantities from QCD, nor to tackle
the most interesting physics, namely, the spontaneously
**chiral symmetry breaking** and the color confinement

^{1}

*g r *

To extract any physical quantities from the first principles of QCD, one has to solve QCD nonperturbatively.

A viable nonperturbative formulation of QCD was first
**proposed by K. G. Wilson in 1974.**

But, the problem of lattice fermion, and to formulate
**exact chiral symmetry on the lattice had not been **
**resolved until 1992-98.**

7 T.W. Chiu, LHC to Universe, Dec 16, '08

### Basic notions of Lattice QCD

1. Perform Wick rotation: , then , and the expectation value of any observable

*O*

*t* *ix*

4 ### exp( ) *iS* exp( *S*

_{E}### )

###

1 , , ^{S}^{E}

*O* *dA d* *d* *O A* *e*

*Z* ^{}

###

###

^{S}

^{E}*Z*

###

*dA d*

*d*

*e*

^{}

( Recall that the divergences in QFT, which requires reg. and ren. , stemming from d.o.f.

###

, and proximity of any field operator )2. **Discretize the space-time as a 4-d lattice** with lattice
spacing a. Then the path integral in QFT becomes a well-defined
multiple integral which can be evaluated via Monte Carlo

###

^{4}

*L*4 *Na*

###

### 1

_{i}

_{j}

_{k}### , ,

^{S}

^{E}*i* *j* *k*

*O* *dA* *d* *d* *O A* *e*

*Z*

^{}

###

### Gluon fields on the Lattice

Then the gluon action on the lattice can be written as

where

*x* *a*ˆ *x* *a*ˆ *a*ˆ
*x* *x a* ˆ

The color gluon field are defined on each link
connecting and , through the link variable^{SU}

_{ }

^{3}

^{A}_{}

###

^{x}*x* *x a* ˆ

### ^{exp} ^{ˆ}

### 2

*U*

_{}

*x* ^{} *iagA*

_{}

^{} *x* *a* ^{} ^{}

###

2###

^{4}

^{ } ^{ }

plaquette

6 1 1

1 Re 0

3 2

*g* *p*

*S U* *tr U* *a* *d x* *tr F* *x F* *x*

*g* ^{} ^{}

###

^{}

###

###

^{ˆ}

###

^{†}

^{ˆ}

###

^{†}

*U**p* *U*_{} *x U*_{} *x* *a*

###

*U*

_{}

*x*

*a*

###

*U*

_{}

*x*

T.W. Chiu, LHC to Universe, Dec 16, '08 9

### Lattice QCD

,

The QCD action

where is the action of the gluon fields

( ) ( )

( )

( ) ( )

, , , , ,

fl

avor

*G*

*G*

*f* *f*

*a x* *f* *a x b y* *b y*

*S* *S U* *D U*
*S U*

*D U* *D U*

*f* *u d s c b t*

sites

, 1, 2, 3 , =1,

index color index

2,3,4

, 1

Dirac

, , =

inde s

x

*x* *y* *z* *t*

*a b*

*x y* *N* *N N N N*

3

1

16 32 1,572,864 1,

ite index

For example, on the lattice, is a complex matrix

of size 572,864

d

( , , ) ( , )

( , et( )

det( )

, )

*G*

*G*

*S*
*S*

*S*
*S*

*dUd d* *U e* *dU* *D* *U* *e*

*U* *dUd d e* *dU*

*D*

*D*
*D* *e*

###

###

T.W. Chiu, LHC to Universe, Dec 16, '08 11

### The Challenge of Lattice QCD

### So far, the lightest u/d quark cannot be put on the lattice.

### To use ChPT to extrapolate lattice results to physical ones.

### To have lattice volume large enough such that *m L*

_{}

### 1.

###

### To have attice spacing small enough such that l *m a*

_{q}### 1.

###

3

### The lattice size should be at least .

### The computing power should be aroun d

.### To meet the above two conditions:

### 100 200

### Petaflops

###

###

### Current Status / The State of the Art

### • RBC and UKQCD Collaborations (20-25 members)

with ~ (10+10) Tflops (peak)### ~(2+2) Tflops (sustained)

lattice fermion: domain-wall fermion largest lattice:

### • JLQCD and TWQCD Collaborations (10-15 members)

with ~ (57+4) Tflops (peak)### (8+2) Tflops (sustained)

lattice fermion: overlap fermion largest lattice:### 32

3### 64

### 24

3### 48

T.W. Chiu, LHC to Universe, Dec 16, '08 13

### The QCD Vacuum

### • The vacuum (ground state) of any nontrivial QFT constitutes various quantum fluctuations.

### • These quantum fluctuations are the origin of many interesting and important nonperturbative physics.

### • In QCD, each gauge configuration possesses a

*well-defined topological charge Q with integer value.*

### • Thus it is important to determine the topological charge fluctuation in the QCD vacuum.

, 1,0, 1,

( ) ^{i Q}_{Q}

*Q*

*Z*

###

*e*

^{}

^{}

*Z*

###

###

^{S}

^{E}^{[ ]}

^{Q}^{,}

*Z**Q*

###

*dA d*

*d*

*e*

^{}

### Topological susceptibility is defined as

###

4

### 0

*t*

*d x* *x*

###

### ^{1}

_{2}

^{tr}

*x* 32

_{}

*F*

_{}

*x F*

_{}

*x*

###

### ^{} ^{}

###

### where

4

### ( ) intege r *Q*

*t*

### *d x* *x*

###

^{2}

4

### 0

^{t}*t*

*d x* *x* *Q*

### *V* Thus

### However, on a lattice, it is difficult to extract unambiguously from the link variables !

### ^{x}

^{x}

###

T.W. Chiu, LHC to Universe, Dec 16, '08 15

4

### We turn to the Atiyah-Singer index theor em

### where is the number of zero modes with chirality of the massle

### ( ) i

### ss D

### ndex

### irac operator

### ( )

### ( ).

*Q*

*t*

*d x* *x* *D* *n* *n* *n*

*D*

^{}

_{}

*igA*

_{}

###

###

###

###

###

###

Michael Atiyah, Abel prize(2004) Fields Medal(1966)

Isadore Singer, Abel Prize(2004)

###

###

4 1 4

5 ,

1

5 ,

### tr ( )

### ( ) tr Since

### we can use

### to measure the topological susceptibility

*x x*

*x x*

*m d x* *D* *m* *n* *n* *d x* *x*

*x* *m* *D* *m*

###

###

###

###

###

###

##

0 5 2

### In order to compute at any , we need to compute the eigenmodes of

### In practice, we only need to project about 50 100 low-lying eigenmode

### ( )

### .

### s of 1

*x*

*D*

*D* *m* *H*

*H*

### *x*

###

###

###

###

###

###

###

###

T.W. Chiu, LHC to Universe, Dec 16, '08 17

### Veneziano-Witten relation

2 2

### (quenched)

*t*

### 4

*f*

*f m* *N*

###

^{}

### Leutwyler-Smilga relation

### (

2### ), 2 1

### 1 1 1

^{u}

^{f}*t*

*u* *d* *s*

*O m* *N*

*m* *m* *m*

### ^{} ^{ }

###

###

###

###

###

### Banks-Casher relation

### (0)

###

###

###

(0) is the density of near-zero modes of the massless Dirac operator*D*

^{}(

_{}

*igA*

_{})

### Some important relations

### Topological susceptibility in 2+1 flavors QCD with domain-wall fermions

[TWQCD, arXiv:0810.3406, Phys. Lett. B (2008) in press]

• 2+1 flavors of DWF on the lattice (L ≃ 2 fm) with , and

163 32

*s* 16

*N* *a*^{}^{1} 1.62(4) GeV

0.04, 0.01, 0.02, 0.03, total 800+809+717 conf.

*s* *u*

*m* *m*

0

5 2

0

1 ( )

1 , effective 4-dim Dirac operator

2 ( )

*H* *m*

*D* *r* *H* *m*

*• Q of each conf. is evaluated using the spectral flow method*

5

2

index( ) = Tr[ (1 )]

1 1

Tr ( )

2 2

*D* *n* *n* *rD*

*H* *h* *h*

*H*

T.W. Chiu, LHC to Universe, Dec 16, '08 19

The spectral flow of 12 lowest-lying eigenvalues of H.

There are 10 net crossings from negative to positive, so the index = -10.

### Histogram of topological charge distribution

T.W. Chiu, LHC to Universe, Dec 16, '08 21

Topological susceptibility in 2+1 flavors QCD with domain-wall fermions

[TWQCD, arXiv:0810.3406, Phys. Lett. B (2008) in press]

MS 3

ChPT (2 GeV) = [259(6)(9) MeV]

1 1 1

*t*

*u* *d* *s*

*m* *m* *m*

^{}

### Topological structure of the QCD vacuum

[TWQCD Collaboration, in preparation]

*• 2+1 flavors of DWF on the lattice (L* ≃ 2 fm)
with , and

163 32

*s* 16

*N* *a*^{}^{1} 1.62(4) GeV

0.04, 0.01, 0.02, 0.03, total 800+809+717 conf.

*s* *u*

*m* *m*

• For each conf., project 50 pairs of low-lying eigenmodes

###

^{1}

5 ,

( ) tr

Compute topological charge density with low-lying eigenmodes.

*x* *m* *D* *m* *x x*

^{} ^{} ^{}

• Identify topological clusters with positive or negative charge, by grouping any two neighboring sites with the same sign of

###

( )*x*

T.W. Chiu, LHC to Universe, Dec 16, '08 23

T.W. Chiu, LHC to Universe, Dec 16, '08 25

Quantum fluctuations in the QCD vacuum

*t* 2
*Q*

(a *≃ 0.12 fm, L ≃ 2 fm) *

*t* 7
*Q*

Quantum fluctuations in the QCD vacuum

(a *≃ 0.12 fm, L ≃ 2 fm) *

T.W. Chiu, LHC to Universe, Dec 16, '08 27

### Outlook

• To clarify the nature of QCD vacuum, whether it is more instanon-like, or more complicated 2-dim or 3-dim

sheet-like structure. Namely, to test the (anti-)self-duality, 1

*F*_{} *F*_{} 2_{}*F*^{}

• To identify the QCD vacuum fluctuations which are the most relevant to the mechanism of color confinement.