The Many Faces of QFT, Leiden 2007 Some remarks on an old problem Gerard ‘t Hooft Utrecht University

The Many Faces of QFT, Leiden 2007The Many Faces of QFT, Leiden 2007

Some remarks on an old problemGerard ‘t Hooft

Utrecht University

The Many Faces of QFT, Leiden 2007

Lattice regularization of gauge theories without loss of chiral symmetry.Gerard 't Hooft (Utrecht U.) . THU-94-18, Nov 1994. 11pp. Published in Phys.Lett.B349:491-498,1995. e-Print: hep-th/9411228

SPIRES:

P. van Baal, Twisted Boundary Conditions: A Non-perturbative Probe for Pure Non-Abelian Gauge Theories thesis: 4 July, 1984.


Gauge theory on the lattice:

1 2

4 3

Plaquette1234

Site x=1

Link 23

2

1(12)

ig A dxU e

P


1234(12) (23) (34) (41)

2 2 4 2112 122

Tr ( ) Tr

Tr ( )

( )ig A dx

U U U U e

C iga F g a F

P

1 14 2Tr ( ) Tr ( )F F UUUU

2

1(12)

ig A dxU e

P

After symmetrization :


The Fermionic Action (first without gauge fields) :

Dirac Action

2 11 1

links

( ) ( )( ) ( )( )x x

L x m xa

Species doubling

(and same for 2, 3 )

12 1( )U x12 1( )U x

However, in the limit , the equation

has several solutions besides the vacuum solution :

since

( ) ( )( ) 0x e x stC

11 4 1 1 4 1 5 1'( , , ) ( 1) ( , , ) ;xx x C x x C

1 1 1 1C C

0m


Wilson Action

1 1links

( ) ( )( ) ( ) (1 ) (1 )

x e x eL x m x

a a

This forces us to treat the two eigenvalues of separately,and species doubling is then found to disappear.

Effectively, one has added a “mass renormalization term”

However, now chiral symmetry has been lost !

Nielsen-Ninomiya theorem


ABJ anomaly

Lattice

( ) (1) ( ) (1)

( ) ( ) (1)

( ) (1)

F L L F R R

F L F R V

F V V

SU N U SU N U

SU N SU N U

SU N U

It could not have been otherwise: even in the continuum limit invariance is broken by the Adler-Bell-Jackiw anomaly.5

However, in the chiral limit, , the symmetry pattern is0M

Can one modify lattice theory in such a way thatsymmetry is kept?( ) ( ) (1)F L F R VSU N SU N U


(2)SU

3

The BPST instanton(A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin)


Instanton

Fermi level

timetime

LEFT

RIGHT

The massless fermions


The fermionic zero-mode:

In Euclidean time:

e e

In Minkowski time: i te i te

negativeenergy

positiveenergy

Thus, the number zero modes determines how many fermions are lifted from the Dirac sea intoreal space.Left – right: a left-handed fermion transmutes into a right-handedone, breaking chirality conservation / chiral symmetry


The instanton breaks chiral symmetry explicitly:

2 2 implies

(2) (2) (2) (2) (1)

m m

U U SU SU U

2 2' implies

(3) (3) (3) (3) (1)

Km m

U U SU SU U


Each quark species makes oneleft - right transition at the instanton.

Leftu

Leftd

Lefts

Rightu

Rightd

Rights

LeftcRightc

charmm


The interior is mapped

onto 4

(2)SU

3

The number of left-minus-right zero modes of the fermions = the number of instantons there.

Atiyah-Singer index


How many “small” instantons or anti-instantons are there inside any 4-simplex between the lattice sites? These numbers are ill-defined !


The number of instantons is ill-defined on the lattice!

If one does keep this number fixed, one will neveravoid the species-doubling problem.

Therefore, the number of fermionic modes cannot dependsmoothly on the gauge-field variables on the links!( )U x

Domain-wall fermions are an example of a solution to theproblem: there is an extra dimension, allowing for anunspecified number of fermions in the Kaluza-Klein tower!

Is there a more direct way ?


We must specify # ( instantons) inside every 4-simplex.

This can be done easily !

Construct the gauge vector potential at all , starting

from the lattice link variables (defined only on the links)

( )A x x

( )U x

Step #1: on the 1-simplicesdef

( )iagA

U x e

Note: this merely fixes a gauge choice in between neighboringlattice sites, and does not yet have any physical meaning.

Next: Step #2: on the 2-simplices


This is unambiguous only in the elementary, faithful representation,which means that we have to exclude invariant U(1) subgroups – the space of U variables must be simply connected

– we should not allow for a clash of the fluxes !

First choose local gauge :

12F a A

Then subsequently, if so desired, gauge-transform back

This procedure is local, as well asgauge- and rotation-invariant

( The subset of gauge- transformations needed

to rotate is Abelian )

U I iagAU e

U I

U I

1

2

Here, we may now choose the minimal flux F , which means that

all eigenvalues must obey:

2 1 2 1( , ) ( / )A x x A x a

Aa g


Step #3: on the 3-simplices

Step #4: on the 4-simplices

We have on the entire

boundary. Extend the field in the 3-d bulk by choosing it to obey sourceless 3-d field eqn’s

(extremize the 3-d action , and in

Euclidean space, take its absolute minimum ! )

( )A x

3 ( ) ( )( )ij ijd x F x F x

Exactly as in step #3, but then for the 4-simplices. Taking theabsolute minimum of the action here fixes the instantonwinding number !

This prescription is gauge-invariant and it is local !!


Thus, there is a unique, gauge-independent and local way todefine as a smooth function of starting from the

link variables ( )A x x

( )U x

In principle, we can now leave the fermionic part of theaction continuous:

fermion ( )( )( )xm igA L

Our theory then is a mix of a discrete lattice sum(describing gauge fields and scalars) and acontinuous fermionic functional integral.

The fermionic integral needs no discretization because it ismerely a determinant (corresponding to a single-loop diagramthat can be computed very precisely)


0 0 1log det ( ) log det log 1iiD ig A D C A

The first four diagrams can be regularized in the standard way – giving only the standard U(1) anomaly

1 + + + + + ···

The sum over the higher order diagrams can be boundedrigorously in terms of bounds on the A fields.

(Ball and Osborn, 1985, and others)

- one might choose to put the fermions on a very dense lattice: , to do practical lattice calculations, but this is not necessary for the theory to be finite !

fermion gaugea a


The procedure proposed here is claimed to be non localin the literature. This is not true.

The extended gauge field inside a d -dimensional simplexis uniquely determined by its (d – 1) -dimensional boundary


The prescription is: solve the classical equations, and of all solutions, take the one that minimizes the total action.

However, imagine squeezing an instanton ina 4-simplex, using a continuous process (such as gradually reducing its size).

As soon as a major fraction of the instanton fits inside the 4-simplex, a solution with different winding number will show up, whose action is smaller.


→ the gauge field extrapolation procedure itself is discontinuous ! Depending on the configuration of the link variables U, the number of instantons within given 4-simplices may vary discontinuously.

This is as it should be!

The most essential part of the gauge field extrapolation procedureconsists of determining the flux quanta on the 2-simplices, andthe instanton winding numbers of the 4-simplices. We demandthem to be minimal, which usually means that the Atiyah-Singerindex on one simplex 2

42

1232

gF F d x


Lattice regularization of gauge theories without loss of chiral symmetry. Gerard 't Hooft (Utrecht U.) . THU-94-18, Nov 1994. 11pp. Published in Phys.Lett.B349:491-498,1995. e-Print: hep-th/9411228

We claim that this procedure is important for resolvingconceptual difficulties in lattice theories.

The END

Documents

The Many Faces of QFT, Leiden 2007 Some remarks on an old problem Gerard ‘t Hooft Utrecht University