aa r X i v : . [ h e p - l a t ] N ov Quark masses and strong CP violation
Michael Creutz Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Abstract.
Two flavor QCD involves three independent mass parameters for which non-perturbative effects are not universal.This precludes matching lattice and perturbative results for non-degenerate quarks and eliminates a vanishing up quark massas a viable solution to the strong CP problem.
Keywords:
QCD, quark masses, strong CP violation
PACS:
In massless two-flavor QCD, chiral symmetry break-ing gives rise to three massless Goldstone pions. Incontrast, the two flavor analog of the eta prime me-son acquires a mass from the anomaly. Thus, as shownschematically in Fig. 1, meson exchange will contributeto a hypothetical quark spin-flip scattering experiment.Now turn on a small d quark mass. This allows con-necting the ingoing and outgoing d quark lines in Fig. 1,and gives a mixing between the left and right handed u quark. The presence of a non-zero d quark mass cre-ates an effective mass for the u quark, even if the latterinitially vanishes. Non-perturbative effects renormalize m u / m d . If this ratio is zero at some scale, it cannot re-main so for all scales. This cross talk between the massesof different quark species has been noted several times inthe past [1] and contradicts the lore that mass renormal-ization is flavor blind. The practice of matching latticecalculations to MS is problematic when m u = m d .A general mass term is an electrically neutralquadratic form that transforms as a Lorentz singlet. Thisleaves four candidates m yy + m yt y + im yg y + im yg t y . The massless limit should have the fla-vored chiral symmetry under y −→ e i g t a f a y . Withthe masses present, this mixes m with m and m with m . The four mass terms are not independent and onecan select any one of the m i to vanish and a second tobe positive. The chiral anomaly is responsible for thesinglet rotation y −→ e i g f y not being a valid symmetry[2]. This rotation does, however, allow one to removeany topological term from the gauge part of the action.Assume this has been done. I thank the Alexander von Humboldt Foundation for supporting visitsto the University of Mainz. This manuscript has been authored undercontract number DE-AC02-98CH10886 with the U.S. Department ofEnergy. Accordingly, the U.S. Government retains a non-exclusive,royalty-free license to publish or reproduce the published form of thiscontribution, or allow others to do so, for U.S. Government purposes. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) uu ddRL RL p, h ’ FIGURE 1.
Both pion and eta prime exchange contributetowards spin flip scattering between up and down quarks. Be-cause these mesons are non-degenerate, this scattering is nothelicity suppressed.
Adopt the common choice m = m as the av-erage quark mass. Then m is the quark mass differenceand m is CP violating. The possible presence of m rep-resents the strong CP problem.Strong interactions preserve CP to high accuracy. Withthe above conventions, it is natural to ask why is m so small? One proposed solution is that the up quarkmass might vanish, allowing a flavored chiral rotation toremove any phases from the quark mass matrix.Why is this not a sensible approach? From the above,one can define the up quark mass as m u ≡ m + m + im . But the quantities { m , m , m } are independent param-eters with different symmetry properties. As discussedearlier, the combination m + m = m + m + im = m = , this would depend on scale and shouldbe regarded as “not even wrong.” REFERENCES
1. H. Georgi and I. N. McArthur, Harvard University preprintHUTP-81/A011 (1981, unpublished); T. Banks, Y. Nir andN. Seiberg, arXiv:hep-ph/9403203; M. Creutz, arXiv:hep-th/0303254 (2003, unpublished); M. Creutz, Phys. Rev.Lett. , 162003 (2004) [arXiv:hep-ph/0312225].2. K. Fujikawa, Phys. Rev. Lett. , 1195 (1979); K. Fujikawa,Phys. Rev. D , 2848 (1980) [Erratum-ibid. D22