Tuesday, July 31, 2007

The problem of staggered fermions

In QCD blog Life on lattice there is a summary about Lattice-2007 conference. The talks of the second day touched also the problem of "staggered fermions". The talk Why rooting fails by Michael Creutz gives an enjoyable summary about the problem.

In QCD not only spinor fields in space-time are dynamical: also spinor structure itself becomes dynamical if space-time topology is non-trivial. In lattice QCD one effectively replaces space-time with 4-D torus. Non-trivial topology implies that there are 24 non-equivalent spinor structures, which means that one obtains 16-fold degeneracy of fermions. In the case of circle there would be 2-fold degeneracy: gamma matrix can be identified as γ or -γ: both choices anticommute to metric but they cannot be transformed to each other by a local Lorentz transformation. In lattice QCD this problem manifests as 16 different lattice Dirac operators γk sin(apk)/a, where a defines the lattice spacing and can have both signs.

The challenge is to get rid of this unphysical degeneracy ("tastes"). One could argue that the problem is an artifact of the periodic boundary conditions effectively replacing space-time with four-torus but the formalism of lattice gauge theory does not seem to agree with his view. From the lecture of Creutz one learns that in the perturbative approach the problem can be resolved by replacing the fermionic determinant by its 16:th root but that the situation changes in non-perturbative approach. This seems understandable since small perturbations mean only dynamics of spinor fields, not the discrete dynamics of spinor structure.

Why I decided to wrote this little comment is that this problem might not be a mere nasty technical detail but a clear signature for the fact that the notion of completely free ordinary spinor structure is not physical. In TGD framework induced spinor fields are dynamical but not the spinor structure, which is induced from that of H=M4×CP2. Whatever the TGD counterpart of lattice QCD might be, there is only single spinor structure to be induced to the space-time sheet so that the problem must disappear.

Note however that standard spinor structure in CP2 does not exist: a generalized spinor structure is obtained by coupling the two spinor chiralities to an odd multiple of Kähler gauge potential: this is absolutely essential for reproducing correct electroweak quantum numbers for the two H-chiralities of H-spinors identified as quark and lepton spinors (color is not spinlike quantum number but corresponds to CP2 partial waves). Baryon and lepton numbers are separately conserved since in TGD H-chirality remains exact symmetry whereas M4 chirality is broken from beginning: in gauge field theories in M4 one must break down exact M4-chiral invariance to make particles massive. Also electro-weak symmetry breaking is coded into the geometry of CP2 at the level of classical electroweak fields defined as projections of CP2 spinor connection.

P. S. Tommaso Dorigo has a posting New Higgs limits with taus from D0, which contains a summary of various production mechanisms for Higgs. Direct production by gluon-gluon fusion to Higgs via quark loop dominates in standard model Universe. In TGD Universe it could give only a small contribution since Higgs couplings to fermions could (but need not) be small (Higgs coupling does not explain fermion mass). Thus associate production as WH and ZH pairs for which rates are smaller by a factor of order ten could dominate.


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