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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
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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 thesis: 4 July, 1984. SPIRES: 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
Gauge theory on the lattice: Site x=1 2 1 Plaquette 1234 Link 23 3 4
However, in the limit , the equation has several solutions besides the vacuum solution : since The Fermionic Action (first without gauge fields): Dirac Action Species doubling (and same for 2, 3 )
This forces us to treat the two eigenvalues of separately, and species doubling is then found to disappear. Wilson Action Effectively, one has added a “mass renormalization term” However, now chiral symmetry has been lost ! Nielsen-Ninomiya theorem
It could not have been otherwise: even in the continuum limit invariance is broken by the Adler-Bell-Jackiw anomaly. However, in the chiral limit, , the symmetry pattern is Can one modify lattice theory in such a way that symmetry is kept?
The BPST instanton (A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin)
Instanton The massless fermions LEFT time Fermi level RIGHT
The fermionic zero-mode: In Euclidean time: In Minkowski time: positive energy negative energy Thus, the number zero modes determines how many fermions are lifted from the Dirac sea into real space. Left– right: a left-handed fermion transmutes into a right-handed one, breaking chirality conservation / chiral symmetry
Each quark species makes oneleft - right transition at the instanton.
The interior is mapped onto 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 !
Therefore, the number of fermionic modes cannot depend smoothly on the gauge-field variables on the links! The number of instantons is ill-defined on the lattice! If one does keep this number fixed, one will never avoid the species-doubling problem. Domain-wall fermions are an example of a solution to the problem: there is an extra dimension, allowing for an unspecified number of fermions in the Kaluza-Klein tower! Is there a more direct way ?
Construct the gauge vector potential at all , starting from the lattice link variables(defined only on the links) We must specify # ( instantons) inside every 4-simplex. This can be done easily ! Step #1: on the 1-simplices Note: this merely fixes a gauge choice in between neighboring lattice sites, and does not yet have any physical meaning. Next: Step #2: on the 2-simplices
2 1 First choose local gauge : Here, we may now choose the minimal flux F , which means that all eigenvalues must obey: 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 ! Then subsequently, if so desired, gauge-transform back This procedure is local, as well as gauge- and rotation-invariant ( The subset of gauge- transformations needed to rotate is Abelian )
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 !) Step #3: on the 3-simplices This prescription is gauge-invariant and it is local !! Step #4: on the 4-simplices Exactly as in step #3, but then for the 4-simplices. Taking the absolute minimum of the action here fixes the instanton winding number !
Thus, there is a unique, gauge-independent and local way to define as a smooth function of starting from the link variables In principle, we can now leave the fermionic part of the action continuous: Our theory then is a mix of a discrete lattice sum (describing gauge fields and scalars) and a continuous fermionic functional integral. The fermionic integral needs no discretization because it is merely a determinant (corresponding to a single-loop diagram that can be computed very precisely)
1 + + + + + ··· • 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 ! The first four diagrams can be regularized in the standard way – giving only the standard U(1) anomaly The sum over the higher order diagrams can be bounded rigorously in terms of bounds on the A fields. (Ball and Osborn, 1985, and others)
The procedure proposed here is claimed to be non local in the literature. This is not true. The extended gauge field inside a d -dimensional simplex is uniquely determined by its (d – 1) -dimensional boundary What's the CATCH ?
The prescription is: solve the classical equations, and of all solutions, take the one that minimizes the total action. However, imagine squeezing an instanton in a 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 most essential part of the gauge field extrapolation procedure consists of determining the flux quanta on the 2-simplices, and the instanton winding numbers of the 4-simplices. We demand them to be minimal, which usually means that the Atiyah-Singer index on one simplex → 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 END 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 resolving conceptual difficulties in lattice theories.