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

2. 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 Gerard 't HooftUtrecht U. SPIRES: P. van Baal, Twisted Boundary Conditions: A Non-perturbative Probe for Pure Non-Abelian Gauge Theories thesis: 4 July, 1984.

3. The Many Faces of QFT, Leiden 2007 Gauge theory on the lattice: 1 2 4 3 Plaquette 1234 Site x=1 Link 23

4. The Many Faces of QFT, Leiden 2007 After symmetrization :

5. The Many Faces of QFT, Leiden 2007 The Fermionic Action (first without gauge fields) : Dirac Action Species doubling (and same for 2, 3 ) However, in the limit, the equation has several solutions besides the vacuum solution : since

6. The Many Faces of QFT, Leiden 2007 Wilson Action 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

7. The Many Faces of QFT, Leiden 2007 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?

8. The Many Faces of QFT, Leiden 2007  The BPST instanton (A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin)

9. The Many Faces of QFT, Leiden 2007 Instanton Fermi level time LEFT RIGHT The massless fermions

10. The Many Faces of QFT, Leiden 2007 The fermionic zero-mode: In Euclidean time: In Minkowski time: negative energy positive 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

11. The Many Faces of QFT, Leiden 2007 The instanton breaks chiral symmetry explicitly:

12. The Many Faces of QFT, Leiden 2007 Each quark species makes one left - right transition at the instanton.

13. The Many Faces of QFT, Leiden 2007 The interior is mapped onto  The number of left-minus-right zero modes of the fermions = the number of instantons there. Atiyah-Singer index

14. 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 !

15. 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 never avoid the species-doubling problem. Therefore, the number of fermionic modes cannot depend smoothly on the gauge-field variables on the links! 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 ?

16. 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) 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

17. 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 : 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 ) 1 2 Here, we may now choose the minimal flux F, which means that all eigenvalues must obey:

18. The Many Faces of QFT, Leiden 2007 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 ! ) Exactly as in step #3, but then for the 4-simplices. Taking the absolute minimum of the action here fixes the instanton winding number ! This prescription is gauge-invariant and it is local !!

19. The Many Faces of QFT, Leiden 2007 Thus, there is a unique, gauge-independent and local way to define as a smooth function ofstarting 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)

20. The Many Faces of QFT, Leiden 2007 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 bounded rigorously 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 !

21. The Many Faces of QFT, Leiden 2007 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

22. 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 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.

23. 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 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

24. 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/9411228Gerard 't HooftUtrecht U. We claim that this procedure is important for resolving conceptual difficulties in lattice theories. The END