the many faces of qft, leiden 2007
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Chiral Symmetry on the Lattice. Utrecht University. Some remarks on an old problem Gerard ‘t Hooft. The Many Faces of QFT, Leiden 2007. P. van Baal, Twisted Boundary Conditions: A Non-perturbative Probe for Pure Non-Abelian Gauge Theories - PowerPoint PPT PresentationTRANSCRIPT
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.
The Many Faces of QFT, Leiden 2007
Gauge theory on the lattice:
1 2
4 3
Plaquette1234
Site x=1
Link 23
2
1(12)
ig A dxU e
P
The Many Faces of QFT, Leiden 2007
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 Many Faces of QFT, Leiden 2007
The Fermionic Action (first without gauge fields) :
Dirac Action
2 11 1
links
( ) ( )( ) ( )( )x xL 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
The Many Faces of QFT, Leiden 2007
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
The Many Faces of QFT, Leiden 2007
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
The Many Faces of QFT, Leiden 2007
(2)SU
3
The BPST instanton(A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin)
The Many Faces of QFT, Leiden 2007
Instanton
Fermi level
timetimeLEFT
RIGHT
The massless fermions
The Many Faces of QFT, Leiden 2007
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 Many Faces of QFT, Leiden 2007
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
The Many Faces of QFT, Leiden 2007
Each quark species makes oneleft - right transition at the instanton.
Leftu
Leftd
Lefts
Rightu
Rightd
Rights
Leftc Rightc
charmm
The Many Faces of QFT, Leiden 2007
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
The Many Faces of QFT, Leiden 2007
How many “small” instantons or anti-instantons are there inside any 4-simplex between the lattice sites? These numbers are ill-defined !
The Many Faces of QFT, Leiden 2007
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 ?
The Many Faces of QFT, Leiden 2007
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
( ) iagAU 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
The Many Faces of QFT, Leiden 2007
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
A a g
The Many Faces of QFT, Leiden 2007
Step #3: on the 3-simplices
Step #4: on the 4-simplices
We have on the entireboundary. 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 !!
The Many Faces of QFT, Leiden 2007
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 LOur 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)
The Many Faces of QFT, Leiden 2007
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 Many Faces of QFT, Leiden 2007
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 Many Faces of QFT, Leiden 2007
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 Many Faces of QFT, Leiden 2007
→ 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
g F F d x
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
We claim that this procedure is important for resolvingconceptual difficulties in lattice theories.
The END