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PERTURBATIVE CURRENT QUARK MASSES in QCD V 421 V) PERTURBATIVE CURRENT QUARK MASSES IN QCD V"") — ; • - s • • by Michael D. Scadron • Physics Department University of Arizona Tucson, Arizona 85721, OSA ..Abstract Neutral PCAC current quark masses follow from the covarlant light plane or QCD requirement that OlCuu^lTD « m(M), which is not Inconsistent with the spontaneous breakdown of chiral symmetry. The resulting current quark nas3 ratio (ms/m)curr = 5 and scale ®ourr = ®2 at M = 2 GeV are compatible with the observed itîJ o-term, the Goldberger-Treiman discrepancy, the low-lying 0~, 1/2+, 1"». 3/2+ hadron mass spectrum, the flavor independence of the dynamically generated .quark mass and the perturbative weak binding I. Introduction Even though it appears that quarks are condemned to be bound in hadrons, it is now well understood that there are two distinctly different types of quark masses: a nonperturbative,; flavor-independent, dynamically generated nass mdyn and perturbative, flavor-dependent current quark masses ' m^^. In chiral field theories such as QCD, It is also clear that these (running) masses are momentum dependent and that mcurr = ®curr^M - 2 GeV) vanishes in the (^Ural limit while may. n does not, but the current quark mass ratio -422 remains constant for any value•of M. Despite this good start it has always amazed us that so many physicists faithfully believe in and do ao nuoh hard work employing the "strong PCAC" [1] current quark mass paradigm (SPCAC) •£-'2fii"2 ficurr <OlqqlO> (la) (Vfi)curî°= » 1 = 25 ' âeuSÎC=5MeV' (1b> based on so few (no) pieces of independent corroborating experimental evidence. In fact, we do not understand the dynamical origin of the current quark masses, so it is not at all clear what the mass scales (lb) — right or wrong — really mean. In our opinion, of far greater significance and utility for the theory of chiral symmetry and low energy QCD is the nonperturbative, dynamically generated quark mass scale [2-41 m^yn = 315 MeV - m^/3 obtained in at least eight independent ways (51 Nevertheless, there now exists two pieces of experimental evidence based on the pioneering work of . G. Höhler and collaborators [6] which should have a bearing on the current quark masses— the Goldberger-Treiman discrepancy A^ and the nN O-term: 5 1 06 1 01 (2a) \NN " (WV™) = °' °- a^ r 65 ± 5 MeV , o^/mjj = 0.07 ± 0.01 . (2b) Qualitatively these "large" chiral symmetry-breaking effects both are significantly greater than the very snail SPCAC nass scale of (1b), in terms of which one might expect ASNCAC - <<WVSPCAC - 0(Wfieon> - °'015 ' (3) where mcon s 1/2(mu+md)con = 340 MeV is the accepted average nonstrange constituent quark mass scale, only slightly larger than m^yn- For the past nine years I (along with R. F. Jones [7,81, J. F.^Gunion and P. C. McNamee [9], and N. H. Fuchs [1,10,11]) have struggled with the incompatibility between (1)-(3) ami have concluded that there is an alternative chiral-breaking -423 scheme to SPCAC which is reasonably consistent with the data (2) and the fundamental perturbation theory — QCD formulae for the perturbative chiral- breaking hamiltonian density 3C ' = q ""^eurr^ a2 = ^ <nl3C'lit> = £ (H) <n|(ûu)Jn> (4a) ir 2 curr M •J = \ <K| 3C'lK>" = J mcurr(M) <Kl(ïïu)HlK> + BSfCUrr(M) <Kl?s)H1K>] . (4b) This alternative, which we now call "neutral PC AC" [3-5,11,121, is not consistent with the additional SPCAC assumption which, e.g., converts (4a) to (la) while neglecting PCAC corrections. The QCD renormalization M-dependence on the HHS of (4) suggests that such PCAC corrections are of order unity..-—"*"vhence the association of "strong PCAC" with (1). On the other hand, if one does not invoke the additional PCAC operation to convert (4a) to (la) (neutral PCAC), the natural quark model mass dependence of the (mass dimension one) flavor- dependent matrix elements in (4) are the complete flavor-dependent and M- dependent QCD quark masses themselves: <u|(ïïu)JnK > « m(M) (5a) <Kl(üu)„IK> « m(M) , <Kl(ss)jK> ^ n (M) . (5b) M M S Then in the high M perturbative region vhere the quark masses In (5) are primarily of the current type, (4) reduces to [3,4,«>-121 m? « • , ml « (mf + m2) (6a) ir curr K s curr <v*Sc. = - J1'2 =5 » =62MeV- (6b) In a global sense we suggest that the quadratic NPCAC mass formulas (6) (for mesons and . for quarks) point to a consistent theoretical picture whereby the massless chiral limit, Bug 0, fficurr, ms eurr ••• 0 is achieved in the relatlvistic infinite momentum frame with E^ = p„[1 + mjj/2pjj1, etc. While the hybrid SPCAC mass formulas (1) (quadratic in meson mass, linear in quark mass) do not provide such a picture, an approximate linearized version of (6) (linear -424 in meson and In quark masses F133) leads to essentially the same (NPCAC) current quark oas3 ratio as (6b). Besides the relativistic quadratic mass structure of the NPCAC current quark formulas (6), a second natural feature of NPCAC is the weak binding current quark mass scale in (6b), i.e., mcurr - • m-n/2. Since the current quark mass scale is set for p2 above the confinement region of 300-500 MeV (i.e., at 2 a 2 pH = (2 GeV) ), one can apply perturbative QCD for which s(p ) is small and weak binding is therefore an obvious consequence. Given then mcurr - m^/2, we make the qualitative chiral symmetry breaking estimates .NPCAC ,„ . .NPCAC ...2 ..2 . - ft„, ,„ \NN - 'WV - curr eon " > (7} which are between two and three times the SPCAC estimates (3) and more in line with the phenomenologieal values (2). In this work we shall attempt to clarify and quantify the above differences between neutral and strong PCAC in the context of QCD. More specifically in See. H we review the argument [3,1,121 that in spite of the perturbative Quadratic relation mj} « fi§urr ^ the ohiral-broken world, QCD allows NPCAC to be consistent with nonperturbative spontaneous symmetry breakdown in the chiral limit. Then in Sec. HC we review - the quantitative theoretical and phenomenological determinations of the current quark mass ratio and suggest that only NPCAC provides a consistent picture. Next in See. IV we examine the SPCAC and NPCAC derivations of the current quark mass scale as set by mu and independently by the low-lying 0~, l/2+, 1-, 3/2+ hadron mass = spectrum. Again only the NPCAC masses mcurr 62 MeV, m3jCurr = 310 HeV are compatible with phenomenology. In Sec. V we independently investigate the momentum dependence of the constituent, current and dynamically generated quark masses and show that the flavor independence of the latter leads uniquely to the NPCAC current quark masses. Finally in Sec. VI we formulate an SU(3) x SU(3) alternative to chiral perturbation theory based upon (NPCAC) perturbative weak binding In QCD rather than on very small (SPCAC) current quark masses. H. Compatibility of Neutral PCAC with Spontaneous Symmetry Breakdown The greatest drawback to the NPCAC quadratic mass. dependence mÇ œ m§urr is the SPCAC assumption (la), the latter having its rigorous origins in the vacuum sum rule [1] i 25 -fi <Olïïu + ddlO> = f2 œ2 + imf/ d\ eiq** <0fT(v'n(*),vlr(0»|0> , (8) curr ir ir curp * vere where = q X^ yg q. Then If meurr "small enough" (we believe this is not the case in the real world), the background integral could be neglected, leading to (la) along with the "NPCAC fiasco" <0lqq!0> = mcurr which appears to violate the spontaneous symmetry breakdown condition <qq>0 £ 0 in the chiral limit. However it is QCD which saves the day and voids the above argument. What is important to stress in the fundamental chiral-breaking relations (4) is the renormalization M-dependenoe of the RHS. Once the corresponding M-dependence of (5) is established and m(M) is understood as the running QCD quark mass which is the sum of perturbative and nonperturbative parts [it! m(M) = ifi (M) + m . (M) , (9) curr dyn then two distinct limits may be achieved: mdyn(M) in the chiral limit (10a) ! ficurP(M) for M = 2 GeV . (10b) Again we repeat that M = 2 GeV is chosen; high enough to suppress the 2 nonperturbative dynamically generated quark mass n(jyn(p ) in (9) which in QCD 2 2 2 -1 2 0-1 falls off rapidly for large p — like (111 mdyn(p ) « (p ) (Inp ) with d = 1/2. On the other hand the perturbative current quark masses fall off much 2 2 d slower as [151 mCUI,r(p ) « (tnp )~ so that (10b) holds in the high M perturbative region as expected. Then in the chiral-broken world where (10b) is operative, (la) recovers the NPCAC mass dependence 2 Œ «C âcurr(M) <TT!(ÏÏU)MITT> <= Sgurr(M) for M = 2 GeV , (11a) while near the ehiral limit (ta) combined with (10a) leads to' Œ <itl(ûu) lit> « m „ (M) m. (M) . \ (Hb) ir curr M curMI r dyn 426' Thus the QCD M-dependence and (9) allows us to achieve the quadratic current quark mass dependence of (11a) while also formally obeying the linear current quark mass dependence (11b). The latter result is precisely what i3 needed so that the vacuum sun rule (8) is satisfied in the chiral limit with <qq>0 i 0. The final step in the argument is to repark that the NPCAC quadratic current quark mass dependence in (8) is recovered not by artificially demanding <0luu|0> « ®curr on the LHS of (8)? but instead follows naturally from the °lurr dependence of the HHS background integral.
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