Abstract
In this section we will consider quarks with masses m « Λ. Because the only intrinsic dimensional parameter in QCD is, we believe, Λ,1 we may expect that to some approximations we may neglect the masses of such quarks which will only yield contributions of order m 2/Λ2 or m 2/Q2.
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Alphonse X “The Wise”, (1221–1284), King of Castille and León, on haying the Ptolemaic system of epicycles explained to him.
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It is, of course, unclear whether the meaningful parameter is A or A, defined by ax(A) 1. Likewise, the meaning of the expression m « A is ambiguous. Clearly, A A0, so there is really no guide but heuristics to help us decide which quarks are light in borderline cases. There is little doubt that u and d should be classed as “light,” but the situation is less definite for s
This seems to be the case in nature. As we will see one expects rri d Irriu ,--t 2, ’kirk z: 20, riiu.--z 6 MeV.
We consider the case n = 3. For n = 2, replace the ;:t by the a of Pauli. 4 To verify (7.1.11), we may use the free-field commutation relations (Appendix F), which is justified because QCD is free at distance 0. 5 Not all the diagonal elements are in SU x
Chiral symmetry and chiral dynamics is a subject in itself. Here we only touch upon some of its aspects that are related to QCD. This omits many important applications. The interested reader may consult the review of Pagels (1975) and, particularly, the excellent text of Georgi (1984).
The particles with zero flavor quantum numbers present problems of their own [the so-called U(1) problem] that will be discussed later.
Partially conserved axial current. In fact, in the limit m,r 2 0, the right-hand side of (7.3.1) vanishes.
This is properly the PCAC limit for in this limit, the axial current is conserved: 0,Au = 0. 10 See Weinberg (1978a); Dominguez (1978), and Zepeda (1978). The method originates in the work of Glashow and Weinberg (1968), Gell—Mann, Oakes, and Renner (1968), and Leutwyler (1974). Estimates of the quark masses essentially agreeing with (7.3.6, 7) had been obtained even before QCD by e.g. Okubo (1969).
A third set may be found in Kataev, Krasnikov and Pivovarov (1983) and it agrees with the other evaluations. This is an interesting consistency check, as these authors include order a,c. contributions.
We leave it as an exercise for the reader to verify this as well as that in this case, one can replace .ft TAu(x1 )A(x2)49(z) T(02,4u(x1 )02A1x2))0(z), i.e., that potential terms where the derivatives act on the 0(x0 1 — z0)... implicit in the T product make no contribution.
Actually, to all orders in any vectorlike interactions. The proof is essentially contained in the original paper of Adler and Bardeen (1969). See also Bardeen (1974), Crewther (1972), and Wilson (1969).
These commutation relations are actually self-contradictory. For example, using only the relations of Appendix A for D 4, we have Tr ,5yyyvγpγiγ = (6 — D)Tr γ5γuγvyPy’, while if we allow commutation, we can obtain Tr y5γayuyvγp,aγ° = — Tr y5yuγyyPy,rγ = (D — 2) Tr γ5yuγ.v,Pγ’, which differs from the former by terms 0(4 — D) These problems, however, only arise with at least four γu’s.
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 three-axial triangle has an anomaly closely related to the axial-vector one.
More about 0-vacua and the topics of this section will be found in Sections 8.3, 8.4, where the reasons for some seemingly peculiar names will become apparent.
A more rigorous derivation may be found in Crewther (1979a); later, in Section 8.4, we will present an alternate discussion.
A more detailed analysis shows that it is enough that one quark be massless. This result was first obtained by Peccei and Quinn (1977).
Another possibility is to use suitable Higgs systems that imply 0 = 0 [Peccei and Quinn (1977)]; this can be shown to lead to the existence of a new pseudoscalar boson [the “axion,” cf., Wilczek (1978); Weinberg (1978b)]. There is not enough experimental evidence to decide whether or not it exists.
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© 1993 Springer-Verlag Berlin Heidelberg
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Ynduráin, F.J. (1993). Light Quark Masses; PCAC; Chiral Dynamics; the QCD Vacuum. In: The Theory of Quark and Gluon Interactions. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02940-4_7
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DOI: https://doi.org/10.1007/978-3-662-02940-4_7
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