Light Quarks; PCAC; Chiral Dynamics; the QCD Vacuum

  • Francisco J. Ynduráin
Part of the Texts and Monographs in Physics book series (TMP)


In this section we will consider quarks with masses m « , to be referred to as light quarks.1. Because the only dimensional pa ameter intrinsic to QCD is, we believe, , we may expect that to some appr ximation we may neglect the masses of such quarks, which will yield only con ributions of order m 2/ 2 or m 2/Q 2


Quark Masse Chiral Symmetry Light Quark Topological Charge Axial Current 
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  1. 5.
    Chiral symmetry and chiral dynamics is a subject i itself. Here we only touch upon some of its aspects. This omits many important applications. The interested reader may consult the review of Pagels (1975), the e ellent text of Georgi (1984) and, more recently, the basic paper of Gasser and Leutwyler (1984) and the reviews of Pich (1995) and Ecker (1995) .Google Scholar
  2. 6.
    The particles with zero flavour quantum numbers pr sent problems of their own (the so-called U(1) problem) that will be discussed 1 ter.Google Scholar
  3. 7.
    Partially conserved axial current. In fact, in the limit m2 → 0, the right hand side of ( 7. 3. l a) vanishes.Google Scholar
  4. 8.
    The equation below should have been written with subtractions, to compensate for the growth of π(q 2 ) for large q 2; but these do not alter the conclusions.Google Scholar
  5. 9.
    Properly speaking, this is the PCAC limit, for in this limit the axial current is conserved.Google Scholar
  6. 10.
    The method originates in the work of Glashow and Weinberg (1968) and GellMann, Oakes and Renner (1968) . In QCD, see Weinberg (1978a), Domínguez (1978) and Zepeda (1978) . Estimates of the quark masses essentially agreeing with (7.3.6, 7) below had been obtained even before QCD by e.g. Okubo (1969), but nobody knew what to do with them. The first evaluation in the context of QCD is due to Leutwyler (1974) .Google Scholar
  7. 11.
    Broadhurst (1981) and Chetyrkin et al. (1995) for subleading mass corrections; Becchi, Narison, de Rafael and Ynduráin (1981), Generalis (1990), Sugurladze and Tkachov (1990), Chetyrkin, Groshny and Tkachov (1982), Groshny, Kataev, Larin and Sugurladze (1991) and Pascual and de Rafael (1982) for radiative corrections to various terms.Google Scholar
  8. 12.
    The proof is essentially contained in the original paper of Adler and Bardeen (1969) . See also Wilson (1969), Crewther (1972) and Bardeen (1974) .Google Scholar
  9. 14.
    For a detailed discussion, see the reviews of Adler (1971) and Ellis (1976). The triangle graph is the only one that has primitive anomalies; it does, however, induce secondary anomalies in square and pentagon graphs. The triangle with three axial currents has an anomaly closely related to the one we have discussed, cf. the text of Taylor (1976). An elegant discussion of currents with anomalies for arbitrary interaction may be found in Wess and Zumino (1971). The derivation of the anomaly in the context of the path integral formulation of field theory, where it is connected with the divergence of the measure, may be found in Fujikawa (1980, 1984, 1985). Google Scholar
  10. 15.
    More about the θ-vacua and the topics of this section will be found in Sect. 8.4 where, in particular, the reasons for some seemingly peculiar names will become apparent.Google Scholar
  11. 16.
    A more detailed analysis shows that it is enough that o n e quark is massless. This result was first obtained by Peccei and Quinn (1977) .Google Scholar
  12. 17.
    Another possibility to obtain θ = 0 is to use a system of Higgs fields which is nonminimal (Peccei and Quinn, 1977) . This can be shown to lead to the existence of a new pseudoscalar boson, the “axion” (Weinberg, 1978b; Wilczek, 1978) . There is not enough experimental evidence to rule out completely the existence of this particle.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Francisco J. Ynduráin
    • 1
  1. 1.Departamento de Física Teórica, C-XIUniversidad Autónoma de Madrid, Canto BlancoMadridSpain

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