Physics Letters B 543 (2002) 183–188 www.elsevier.com/locate/npe Connecting the quenched and unquenched worlds via the large Nc world Jiunn-Wei Chen Department of Physics, University of Maryland, College Park, MD 20742, USA Received 16 July 2002; accepted 5 August 2002 Editor: H. Georgi Abstract In the large Nc (number of colors) limit, quenched QCD and QCD are identical. This implies that, in the effective field theory framework, some of the low energy constants in (Nc = 3) quenched QCD and QCD are the same up to higher-order corrections in the 1/Nc expansion. Thus the calculation of the non-leptonic kaon decays relevant for the I = 1/2 rule in the quenched approximation is expected to differ from the unquenched one by an O(1/Nc) correction. However, the calculation relevant to the CP-violation parameter / would have a relatively big higher-order correction due to the large cancellation in the leading order. Some important weak matrix elements are poorly known that even constraints with 100% errors are interesting. In those cases, quenched calculations will be very useful. 2002 Elsevier Science B.V. All rights reserved. Quantum chromodynamics (QCD) is the under- fermion determinant arising from the path integral to lying theory of strong interaction. At high energies be one. This approximation cuts down the comput- ( 1 GeV), the consequences of QCD can be studied ing time tremendously and by construction, the re- in systematic, perturbative expansions. Good agree- sults obtained are close to the unquenched ones when ment with experiments is found in the electron–hadron the quark masses are heavy ( ΛQCD ∼ 200 MeV) deep inelastic scattering and hadron–hadron inelastic so that the internal quark loops are suppressed. Thus scattering (Drell–Yan process) [1]. At low energies the quenched approximation is an efficient and reli- ( 1 GeV), however, QCD becomes non-perturbative. able tool to use as long as all the quark masses are Quantitative results not based on symmetry properties heavy. However, in the real world, the light quark along can only be calculated directly from QCD via (u, d and s) masses are less than ΛQCD so the jus- computer simulations on a Euclidean lattice. tification for quenching does not apply. In this range In lattice QCD simulations, it is common to per- of quark mass, quenching effects can be systemat- form the quenched approximation which is to drop ically studied by quenched chiral perturbation the- the internal quark loop contributions by setting the ory (QχPT) [2], an effective field theory of quenched QCD (QQCD). In QχPT, quenching could induce a double pole to the flavor singlet meson propagator and E-mail address: [email protected] (J.-W. Chen). could make quenched quantities more singular than 0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02461-9 184 J.-W. Chen / Physics Letters B 543 (2002) 183–188 Fig. 1. Feynman diagrams contributing to the pion two point function. The solid lines and curly lines denote the quark and gluon propagators, respectively. the unquenched ones [2] in the infrared region. In the poorly known weak matrix elements for which even ultraviolet region, QCD and QQCD are different as 100% error constraints are interesting. well because QQCD can be considered as having in- It is known that the quenched world has some finite quark masses for the internal loops so its ultra- similarity to the large Nc world. Take a pion two- violet behavior is different from that of QCD. Those point function 0|π(x)π(0)|0, for example: there are uncontrolled effects make it hard to see what we can contributions from diagrams shown in Fig. 1. For learn about QCD from QQCD. convenience, the quarks connected to the external However, there is a known connection between sources are called “valence quarks”; the quarks not QCD and QQCD in the limit that the number of connected to the external sources are called “sea colors (Nc) becomes infinite; the large Nc limit of quarks”. In the quenched approximation, the diagrams QQCD is indistinguishable to the large Nc limit of with sea quark loops such as the one in Fig. 1(b) QCD. In this paper, we study the finite Nc quenching are zero. (Then the gluon–quark coupling in surviving errors systematically using effective field theories with diagrams such as the ones in Fig. 1(a), (c) are usually 1/Nc expansions. Generally, in the real world where rescaled to mimic the situation that the sea quark Nc = 3, some of the contributions which are formally masses are set to infinity. We will not do this rescaling higher order in the 1/Nc expansion could become at this moment but leave it for later discussion.) In numerically important for certain observables near contrast, in the large Nc limit proposed by ’t Hooft the chiral limit (zero quark mass limit). Fortunately, [7], these big contributions are typically non-analytic in 2 quark masses and can be computed in effective field Nc →∞,g→ 0,Ncg → constant, (1) theories of QCD and QQCD. After these big non- analytic contributions in QQCD are replaced by those where g is the quark gluon coupling, both the Fig. 1(b) in QCD, it is plausible that the errors which are diagram with one sea quark loop and the “non-planar” formally O(1/Nc) in the quenched approximation diagram of Fig. 1(c) with one crossing in gluon lines could imply numerically ∼ 30% errors. Sometimes are suppressed by one power of 1/Nc compared to that even smaller errors can be achieved by forming ratios of Fig. 1(a). The diagrams with more sea quark loops of observables, such that the O(1/Nc) errors from or more gluon crossings are further suppressed by the leading terms in the chiral expansion cancel. The even higher powers of 1/Nc. So, in the large Nc limit ratio of B meson decay constants fBs /fB [3,4] is of QCD, only the planar diagrams such as Fig. 1(a) an explicit example. We then study the quenched survive, while in QQCD both Fig. 1(a) and (c) survive. calculations of the non-leptonic kaon decays K → Now if we further take the large Nc limit of QQCD, ππ [5,6]. We find that the quenching error in the the non-planar diagrams vanish as well. From this we part relevant to the I = 1/2 rule is consistent conclude that QQCD and QCD have the same large Nc with O(1/Nc) ∼ 30%. However, the CP violation limit—at least in the mesonic two-point function case. parameter / calculation could have a big quenching It is easy to generalize the above observation to an error due to the large cancellation in the leading terms arbitrary n-point function even involving baryons. To in the 1/Nc expansion. Finally, for illustration, we generalize the baryons to the large Nc case, we follow identify some cases that quenched calculations can be the prescription suggested by Dashen and Manohar [8] very useful. Those are calculations of important but such that there is no valence strange quark in a proton. J.-W. Chen / Physics Letters B 543 (2002) 183–188 185 Q Q Q The identity between the large Nc QCD and large where the low energy constants B , fπ and K3 Nc QQCD imposes some constraints on the low en- have different values than those in QCD and where 2 ergy effective field theories of QCD and QQCD, which M0 is the strength of the η η coupling. In QCD, are the chiral perturbation theory (χPT) [9–12] and the 2 M0 can be iterated to all orders to develop a shifted quenched chiral perturbation theory (QχPT) [2], re- η pole such that η becomes non-degenerate with spectively. χPT and QχPT have the same symmetries the Goldstone bosons in the chiral limit. However, and symmetry breaking patterns as QCD and QQCD. in QQCD, all the sea quark loops are dropped such In both cases, low energy observables are expanded 2 that the iteration stops at one insertion of M0 .This in a power series of the small expansion parameters gives the η propagator a double pole dependence and (p, MGB)/Λχ ,wherep is the characteristic momen- makes physical quantities in QQCD more singular tum transfer in the problem, MGB is the Goldstone than those in QCD in the infrared region [2]. The boson mass and Λχ is an induced chiral perturbation quenched pion mass in Eq. (4) is an explicit example. scale. In QCD, Λ ∼ 1 GeV and the Goldstone bo- 2 χ The quenched chiral logarithm (the M0 term) is more son masses are Mπ ∼ 140 MeV, MK ∼ 500 MeV and singular than the QCD chiral logarithm in Eq. (2). As M ∼ 550MeV.IntheQCDcase(N = 3), χPT gives 2 η c Nc becomes large, M0 scales as 1/Nc while the other [10] quenched low energy constants scale the same√ way as the corresponding unquenched ones, f Q = O( N ), 1 M2 π c M2 = 2m B 1 + M2 log π BQ and KQ = O(1). Thus as N →∞, π u 2 π 2 3 c (4πfπ ) µ 2 2 Q → Q + Q + 3 1 M Mπ 2muB 1 2muK3 O mq . (5) − M2 log η 3 η µ2 SinceQCDandQQCDhavethesamelargeN c 2mu + ms limits, Eqs. (3) and (5) imply + 2muK3 + K4 3 = Q = Q →∞ B B ,K3 K3 as Nc . (6) + 3 O mq , (2) Thus some low energy constants (to be more precise, those that are leading in the 1/N expansion) of χPT where mq is the mass of the flavor q(= u, d, s) c and QχPT are identical in the large N limit.