Partially Quenched Chiral Perturbation Theory and the Replica Method

Transcription

1 Partially Quenched Chiral Perturbation Theory and the Relica Method P. H. Damgaard and K. Slittorff The Niels Bohr Institute Blegdamsvej 7 DK-200 Coenhagen Ø Denmark March 24, 2000 Abstract We describe a novel framework for artially quenched chiral erturbation theory based on the relica method. The comutational rules are exceedingly simle. We illustrate these rules by comuting the artially quenched chiral condensate to one-loo order. By considering arbitrary chiral k-oint functions we show exlicitly to one-loo order the equivalence between this method and the one based on suersymmetry. It is ossible to go smoothly from the conventional relica method to a suersymmetric variant by choosing the number of valence quarks to be negative. NBI-HE-00-3 he-lat/000307

2 Introduction The question of non-erturbative analytical redictions for quenched or artially quenched lattice gauge theory comutations has been thoroughly studied in the context of effective chiral Lagrangians [, 2, 3, 4]. So far the most systematic framework has been the suersymmetric formulation of Bernard and Golterman [, 2], which builds on an idea first introduced in the context of staggered lattice fermions [5]. Here one introduces k additional quark secies (of conventional statistics) on to of the hysical sea quarks, and k ghost quarks of oosite statistics to cancel the effects of the additional quarks. When is taken to vanish this gives the fully quenched theory, while for non-zero it gives the artially quenched theory. Both are accessible to a study by Monte Carlo techniques in lattice gauge theory. The chiral flavor symmetry grou is in that formulation extended to a suer Lie grou which in erturbation theory can be taken as SU( + k k). Based on the usual assumtion of sontaneous chiral symmetry breaking (here extended to the suer grou case) the effective low-energy theory of the lowest-lying hadronic excitations is that of a chiral Lagrangian, now with fields living on the coset of suer Lie grous. This effective Lagrangian can be studied by the conventional methods of chiral erturbation theory. In what follows we denote fully and artially quenched chiral erturbation theory by QChPT and PQChPT, resectively. The suersymmetric framework has also roven to be an efficient means of deriving analytical results for the soft art of the Dirac oerator sectrum in finite volume, by taking an aroriate discontinuity of the artially quenched chiral condensate [6, 7, 8]. This has brought earlier results derived entirely from universal Random Matrix Theory [9] (for a very recent comrehensive review, see ref. [0]) in direct contact with the effective Lagrangian of QCD. In articular, a series of very comact relations that described general k-oint sectral correlation functions of low-lying Dirac oerator eigenvalues in terms of effective artition functions with additional quark secies [] can now be understood as due to the cancelling airs of fermionic and bosonic valence quarks. When taking the same discontinuity near the origin in PQChPT it has also been shown that one recovers among other terms the analytical rediction for the sloe of the sectral density of the Dirac oerator at the origin [6, 7], a formula first derived in the QCD case by Smilga and Stern [2]. The same analysis has recently been extended to the two other major chiral symmetry breaking classes by Toublan and erbaarschot [8]. There are thus also lenty of hysical alications of PQChPT that have nothing to do with the artifacts of the quenched aroximation at all. While the suersymmetric aroach to QChPT and PQChPT has been well tested, and is by now quite well understood, it is still of interest to find alternative means of formulating the same roblem. In articular, the suersymmetry itself is not fundamental and not an inherent roerty of QChPT and PQChPT. Indeed, it has recently been shown in the context of the finite-volume effective chiral Lagrangian related to the soft art of the Dirac oerator sectrum [9] that the so-called relica method can rovide a useful alternative technique [3]. Here full or artial quenching is instead achieved by adding N v valence quarks (of usual statistics), and then taking the limit N v 0 at the end of the calculation. In ordinary QCD erturbation theory this rocedure trivially kills all valence quark loos. In the framework of the effective Lagrangian of Goldstone bosons it is far from obvious that such a rocedure can be carried out exlicitly. It entails an extension of the chiral symmetry grou U(N) to non-integer N, and integrals over such a grou are not known in closed form. Nevertheless, it turns out that in series exansions the required analytical continuation can be carried out exlicitly [3], and results agree with what was earlier established by the suersymmetric method [6, 7]. This suggests that also conventional QChPT and PQChPT can be erformed by simly taking the limit N v 0. 2

3 In this aer we shall show that this is indeed the case. We shall give the very simle Feynman rules, and exlain the intimate relationshi to QChPT and PQChPT in the suersymmetric formulation. As a simle illustration we show how to derive the artially quenched chiral condensate to one-loo order using this relica method. This fully or artially quenched chiral condensate is a articularly convenient observable on which to test the non-erturbative finite-volume scaling results discussed above [4, 5]. The way artially quenched chiral erturbation theory smoothly matches on to this regime has been exlained in ref. [6]. After roviding the Feynman rules, it becomes quite obvious how the relica method in erturbation theory is equivalent to the suersymmetric method. We illustrate a few of the counting rules by considering a chiral k-oint function below. Mainly out of curiosity, we also show how a variant of the relica method that is suersymmetric can be used to rovide identical results. This suersymmetric variant is however slightly more cumbersome than the conventional N v 0 relica method, and we do not roose to use that articular variant for ractical calculations. 2 The relica method As exlained above, with the relica method one adds N v valence quarks to the QCD Lagrangian, which here can be taken as any SU(N c 3) gauge theory with hysical (sea) quark flavors. Deending on the alications, it can be convenient to introduce k sets of such valence quarks with k different masses m vj, each set containing N v new quark flavors. The hysical quark masses are denoted by m f. The QCD artition function with these kn v additional quark secies reads Z ( +kn v) = [da] k det(i /D m vj ) Nv det(i /D m f ) e SYM[A]. () j= f= This artition function can be viewed as an unnormalized average of k sets of N v identical relicas of the following artition functions of quarks in a fixed gauge field background A µ : [ ] [d ψ j dψ j ] ex d 4 x ψ j (i /D m vj )ψ j (2) Z vj in the sense that k ] Z ( +kn Nv v) = [da] [Z vj det(i /D m f )e S YM. (3) j= f= Clearly, if we set N v = 0 this just reroduces the original QCD artition function. But the theory extended with kn v additional quark secies in this way is a generating functional for artially quenched averages of ψ j ψ j and mixed averages also involving hysical quark fields. One simly sets N v to zero after having erformed the functional differentiations χ(m v,...,m vk,m f,...,m fl, {m f }) lim N v 0 N k v N l f m v m vk ln Z ( +N v). (4) m f m fl Technically, it can be convenient to add local sources for both scalar and seudoscalar quark bilinears ψ j (x)ψ j (x) and ψ j (x)γ 5 ψ j (x) and similarly for the vector and axial vector currents (for simlicity taken flavor diagonal). If needed, one can of course introduce corresonding sources in the hysical quark sector. Because such terms have no bearing on our arguments resented below, we shall for simlicity omit them here. 3

4 2. Adating the relica method to the chiral Lagrangian For N v integer, and + kn v small enough, chiral symmetry is assumed to be sontaneously broken according to the standard attern of SU L ( + kn v ) SU R ( + kn v ) SU( + kn v ). The effective low-energy theory can therefore be described in the entirely conventional framework of a chiral Lagrangian based on SU( + kn v ), with no new assumtions about the attern of chiral symmetry breaking. The cases =0and = are obviously very secial here. For = there is not any sontaneous breaking of chiral symmetry in the theory after taking N v to zero, and the case = 0 (which would corresond to full quenching) is so unusual that we shall discuss it searately. Having in mind a ossibly non-trivial rôle layed by the flavor singlet meson, the lowest-order effective chiral Lagrangian is taken to be the usual O( 2 ) exression L = F 2 4 Tr( µu µ U ) Σ 2 TrM(U + U )+ µ2 Φ α µ Φ 0 µ Φ 0. (5) 2N c 2N c Here the field U ex[i 2Φ/F ]isanelementofsu( + kn v ),andwehavekettheflavor-singlet field Φ 0 TrΦ. As in the suersymmetric method [], it roves convenient to work in a quark basis where Φ ij corresonds to ψ i ψ j. With all external sources set to zero, this gives a simle roagator for the off-diagonal mesons corresonding to Φ ij ψ i ψ j, i j: D ij ( 2 ) = 2 + M 2 ij, (6) while for the diagonal mesons Φ ii ψ i ψ i the roagator can be written in the form [2] G ij ( 2 ) = δ ij ( 2 + M 2 ii ) (µ 2 + α 2 )/N c ( 2 + M 2 ii )(2 + M 2 jj )F(2 ). (7) Here M ij (m i + m j )Σ/F 2 and F( 2 ) + µ2 + α 2 k N v N c j= 2 + Mv 2 + j v j 2 + M 2. (8) f= ff Note that N v enters as a arameter due to the mass degeneracy of the valence quarks in each of the k sets. This is exactly what is required in order to aly the relica method. We remark that the unusual form of the roagator (7) just stems from using the quark basis and including the flavor singlet field Φ 0 = TrΦ, and not from any eculiarities of artial quenching. Although we borrow the result (7) from ref. [2], it is also unrelated to the suersymmetry of the method discussed there. By including the Φ 0 field in the Lagrangian we have ket oen the ossibility of studing various exansion schemes (see e.g. the second reference of []). The Φ 0 terms affect only G ij. For G ii the flavor-singlet Φ 0 can give rise to double oles, but the aearance of such double oles is not secial to the relica method. Indeed such double oles are also resent in the suersymmetric formulation The reader might worry about the assumtion that +kn v should be taken small enough for the theory to suort sontaneous chiral symmetry breaking. Actually, there will be no new constraint from this. We simly analyze the chiral Lagrangian for arbitrary + kn v even though this Lagrangian is only the low-energy theory of QCD for + kn v sufficiently small. However, we take the limit N v 0 in the end. Then we must meet only the usual constraint that the number of hysical light quarks should be small enough to lead to sontaneous chiral symmetry breaking. 4

5 Proagator relica PQChPT suersymmetric PQChPT D ij ( 2 ) 2 +M 2 ij ɛ i 2 +M 2 ij G ij ( 2 ) δ ij ( 2 +M 2 ii ) (µ 2 +α 2 )/N c ( 2 +M 2 ii )(2 +M 2 jj )F(2 ) ɛ i δ ij ( 2 +M 2 ii ) (µ 2 +α 2 )/N c ( 2 +M 2 ii )(2 +M 2 jj )F(2 ) F( 2 ) + µ2 +α 2 N c ( kj= N v 2 +M 2 v j v j + f= ) 2 +Mff 2 + µ2 +α 2 Nf +2k N c f= ɛ i 2 +M 2 ii Table : The roagators for relica PQChPT, SU( + kn v ), and suersymmetric PQChPT, SU( + k k). The sign ɛ i is defined as ɛ i for i =,..., + k and ɛ i for i = + k +,..., +2k. Note that F coincides in the artially quenched limit of the two aroaches. where a thorough study has been done [, 2, 3]. As we rove in the next section the two formulations have equivalent erturbative exansions. The aearance of the double ole in the relica method is therefore comletely analogous to the case of the suersymmetric formulation. In articular, we note that also in the relica formalism the case = 0 is quite secial since in that case F( 2 )simly becomes unity, and the double ole in G ii is unavoidable. Moreover, in just that case there is no decouling as the scale µ is sent to infinity. In Table we give the exlicit relation between the Feynman rules based on the relica method, and those based on the suersymmetric formulation. The suersymmetry Feynman rules are sulemented by the standard relative minus sign between boson and fermion loos. Desite the additional minus signs in the Feynmann rules of the suersymmetric formulation, the Green functions are identical in the two formulations. As we show below, the signs due to combinatorics in the relica method match those arising from statistics and the suertrace in the suersymmetric formulation. 3 The equivalence between relica and suersymmetric PQChPT In this section we formulate the equivalence between the generating functional of PQChPT in the relica and suersymmetric formulations. The equivalence roof is by default restricted to erturbation theory (exressed in terms of the Feynman rules), and we can in rincile not make any statements at the non-erturbative level. But this is as it should be, as our whole framework in any case is restricted to chiral erturbation theory. The Lagrangian itself contains an infinitely long string of interactions that become relevant with increasing loo order, and we shall only demonstrate the equivalence at the one-loo level. However, seeing how the equivalence roof roceeds, it is retty obvious how to generalize this to arbitrarily high order. 5

6 Our claim is: The generating functional of relica PQChPT for +kn v flavors with k sets of N v massdegenerate quarks is in erturbation theory equivalent to the generating functional of suersymmetric PQChPT for + k fermionic and k bosonic quarks. By equivalence between the SU( + kn v )andthesu( + k k) generating functionals is meant that the chiral exansions are equivalent order by order. Of course, the resective limits, N v 0and mass degeneracy between the k bosons and k of the fermions, are to be introduced at the end of the calculations. While we believe that this statement is true we will as mentioned above only address the equivalence at the one-loo level. At this one-loo level the contributions from the O( 4 )chiral Lagrangian act as counter terms and we can base the discussion on the Lagrangian of (5). Let us first consider the sea sector. (The term sea sector is used when only sea quark masses are involved in differentiations of the generating functional.) For this sector both methods are equivalent to SU( ) ChPT. In the relica formulation the contributions from the valence quarks at one-loo to any of the correlators χ(m f,...,m fl, {m f }) N k f m f m fk ln Z ( +N v), (9) are necessarily roortional to ositive owers of N v. Hence the deendence on the valence quarks vanishes as N v 0, leaving the sea sector equivalent to SU( ) ChPT. The analogous statement in suersymmetric PQChPT was roven in ref. [2]. This equivalence was formulated as three theorems in that reference. At the risk of making some oversimlifications we state them comactly as follows: I) The sea sector of SU( + k k) PQChPTisequivalenttoSU( )ChPT. II) The suer-η is equivalent to the conventional η of SU( )ChPT. III) ThedoubleoleofG ii arise at a given fermionic quark mass if and only if all fermionic quarks with this mass are aired u by bosonic quarks. In the suersymmetric formalism theorem I is established by noting that k of the fermionic quarks and the k bosonic quarks only aear as virtual loos in the sea sector. Since these 2k quarks are aired u in masses the virtual loos cancel exlicitly. This cancellation is also resonsible for establishing theorem II in the suersymmetric formalism, only now it takes lace in the quark loo corrections to the η -roagator. Finally theorem III follows directly from the structure of the last term in G ii. We emhasize here that the obvious analogs of both theorems I and II are comletely trivial in the resent relica formalism. Theorem III, when re-stated in the language of the relica formalism, stiulate under what circumstances the otential double ole of G ii is cancelled: By insection this occurs when M ii = M ff for at at least one hysical meson labelled by ff. The roof of theorem III is then almost identical in the relica and suersymmetric formulations. In the hrasing of refs. [, 2, 3] the double oles can only occur at mass scales that are comletely quenched. In the remaining quark sectors the equivalence is far less trivial. However, the suersymmetric bosonic Green functions equal the fermionic ones u to a well defined sign. So we can focus on the sectors involving fermionic valence quarks. The equivalence in these sectors is not just of academic interest. As mentioned in the introduction, differentiations with resect to valence quark masses may be related to hysical quantities. For 6

7 instance the artially quenched chiral condensate for the valence quarks, Σ(m v, {m f }) lim ln Z ( +N v), (0) N v 0 N v m v can be used to determine the Dirac sectral density. This density is given by the discontinuity of the artially quenched chiral condensate across a cut on the imaginary axis [6]: ρ(λ; {m f }) = 2π Disc m Σ(m v=iλ v, {m f }) = 2π lim[σ(iλ + ɛ, {m f }) Σ(iλ ɛ, {m f })]. () ɛ 0 (This identification holds when one considers Σ(m v, {m f }) as a function of a real mass m v, and then relaces m v iλ ± ɛ.) In the valence sector and the mixed sector the equivalence is established in two stes. First, noticethat the roagator (7) of relica PQChPT for N v = 0 is identical to the one for the fermionic sector of the corresonding suersymmetric PQChPT in the limit where each of the boson masses is aired u with a fermion mass, see Table. (This equivalence holds trivially for the off-diagonal quark anti-quark roagators.) Second, the signs arising from combinatorics in the relica method is exactly matched by the oosite signs of boson and fermion loos occurring in the suersymmetric formulation. In order to see exactly how the signs come to match in the two aroaches, we exlicitly give the derivation of the k-oint function in the valence sector. The generalization to the mixed sector follows in comlete analogy. 3. The one-oint function in the valence sector In this first examle we give the contributions to the valence quark mass deendent chiral condensate defined in (0). We show how the cancellations that occur exactly match those of the suersymmetric formulation. It turns out that this simle -oint function actually is ideally suited for illustrating the equivalence between the relica method and the suersymmetric method, as all essential roerties of the roagators and of the combinatorics come into lay. To evaluate the one-oint function we need to introduce just one set of relica fermions. Exlicitly erforming the differentiation of the generating functional, see Eq. (0), or alternatively counting the number of realizations of quark flow diagrams we have to one-loo Σ(m v, {m f }) = lim (N v N Σ N v 0 N v F 2 v (Mvf 2 )+N v(n v ) (Mvv 2 )+N v G vv ( 2 ) ) f= (2) where (Mij) Mij 2 D ij ( 2 ) (3) is a one-loo integral of the standard diagonal roagator for the off-diagonal mesons, Φ ij ψ i ψ j, i j. (We write everything in finite-volume notation, having also in mind alications of the kind discussed in refs. [6, 7, 8].) The first term in G vv( 2 )issimly (Mvv 2 ). For arbitrary N v this term is seen to cancel against the term just before G vv. In the N v 0 limit we also get rid of the term roortional to N v,leavingsimly Σ(m v, {m f }) = Σ F 2 (Mvf 2 ) (µ 2 + α 2 )/N c ( 2 + Mvv 2 )(2 + Mvv 2. (4) )F(2 ) f= 7

8 This is comletely analogous to the result obtained in the suersymmetric formulation. In that case a similar cancellation takes lace between the first term in G vv() and the loo of the meson built u by the fermionic and bosonic valance quark. It is also instructive to trace the cancellation of valence quark loos. In the suersymmetric formulation this cancellation occurs because of a matching boson loo, while in the resent formulation it is due to the lack of a relica fermion. Pictorially seaking, this lack of a relica fermion acts like a boson. 3.2 The k-oint function in the valence sector As for the condensate, the k-fold derivative, k 2, of ln Z ( +kn v) with resect to each of the valence quark masses is related to the sectral k-oint function. The evaluation of the k-fold derivative is quite simle but we need to treat the case k = 2 searately. The reason is simle: The roduct +2N v j,k= Φ ij (x )Φ ji (x )Φ lk (x 2 )Φ kl (x 2 ), i l (5) occurring in the two oint function includes two connected terms, namely and Φ v v (x )Φ v2 v 2 (x 2 )Φ v2 v 2 (x 2 )Φ v v (x ) Φ v v 2 (x )Φ v2 v (x 2 )Φ v v 2 (x 2 )Φ v2 v (x ). For k>2 there is no connected analogue of the latter crossed diagram, since k of the indices must be different (we differentiate with resect to different masses). The 2-oint function is thus different from higher k-oint functions because meson loos corresond to just quark-antiquark lines. In terms of the roagators the two-oint function is 2 χ(m v,m v2, {m f }) Σ 2 = lim N v 0 N 2 v ( F 4 Nv 2 D v v 2 ( 2 )D v2 v ( 2 )+Nv 2 Whereas for k>2 there is no crossed diagram, and we are left with χ(m v,...,m vk, {m f }) Σ k = lim N v 0 N k v ( ) k F 2k N v k ) G v v 2 ( 2 )G v2 v ( 2 ). (6) G v v 2 ( 2 ) G vk v ( 2 ). (7) We observe that in both cases the N v -deendence is such that the limit N v 0 becomes trivial. The corresonding exressions in the suersymmetric formalism are identical. Note that sea fermion and ghost -loos only aear in the one-oint function. 4 From relicas to suersymmetry Interestingly, in erturbation theory it is ossible to use a eculiar variant of the relica method that is suersymmetric. This is because all N v -deendence in the roagators and vertices is entirely arametric. We can thus make relicas of an arbitrary real number of valence quarks. Moreover, artial quenching can be achieved not only by taking N v 0, but also by taking N v to any fixed 2 This chiral 2-oint function has been analyzed in the suersymmetric formulation by Osborn, Toublan, and erbaarschot (rivate communication). 8

9 number of quarks N v, and re-interreting the remaining + N v as hysical quarks (of which it just haens that at least N v are degenerate in mass). Because the N v -deendence is arametric in erturbation theory, we can trivially go one ste further and consider a artially quenched theory of hysical fermions as the limit N v Ñv of a theory based on + Ñv + N v quarks, out of which the Ñv + N v quarks are degenerate in mass m v = m v. This corresonds to considering the effective theory of a fundamental artition function that is artially suersymmetric (for simlicity considering only one such set of relica quarks): Z ( +Ñv+Nv) +Ñ v Nv= Ñv = [da] det(i /D m v ) Nv det(i /D m f ) e S YM[A] = det(i /D m v)ñv [da] det(i /D m v )Ñv f= f= Nv= Ñv det(i /D m f ) e S YM[A]. (8) At this level the artition function is exactly as the starting oint of the suersymmetric method. However, when we consider the effective artition function in terms of the Goldstone bosons, the working rules are entirely different. We kee our Feynman rules of Table, and just remember to take the limit N v Ñv in the end. The fact that this rocedure works is of course a direct consequence of the fact that in erturbation theory we can get bosons from fermions by letting the number of (degenerate) secies go from ositive to negative (also the statistics sign of closed fermion loos relative to closed boson loos comes out right in this way). It is instructive to see how this suersymmetric variant of the relica method works in detail. Consider again our rototye of a Green function, that of the artially quenched chiral condensate. Using the notation of above, we find Σ(m v, {m f }) Σ lim Nv Ñv mv mv m v lnz ( +Ñv+Nv) = lim (N v N Nv Ñv N v F 2 v (Mvf)+Ñv (M 2 2 vṽ) + N v (N v ) (Mvv) 2 mv mv f= ) +N v G vv ( 2 ) ) (9) where Mvṽ 2 (m v + m v )Σ/F 2,andG vv ( 2 ) is as in Table, excet for the obvious change that now F( 2 ) + µ2 + α 2 N v Ñ v N c 2 + Mvv Mṽṽ 2 + f= 2 + Mff 2. (20) Taking the degenerate mass limit m v = m v and letting N v Ñv we note that terms cancel out exactly as in the revious N v 0 relica method. For instance, in F( 2 ) the terms linear in N v and Ñv just cancel each other. In eq. (9) the term roortional to Nv 2, which reviously droed out trivially in the N v 0 limit, is now recisely cancelled by a similar term roortional to N v Ñ v Ñ v 2.All unwanted terms thus exactly cancel as they should, and we are left with the correct one-loo result (4). As we mentioned earlier, this examle of the one-oint function is actually the most instructive for illustrating the cancellations. The other k-oint functions clearly roceed analogously. Although it is thus ossible to make a suersymmetric variant of the relica method, it is obviously rather ointless to do so. The simlest Feynman rules come from using just the conventional N v 0 9

10 limit. We also note that although the starting artition function (8) is identical to that forming the basis for the suersymmetric chiral Lagrangian [, 2], the effective theory one works with in the analogous suersymmetric relica scheme is of a very different nature, and has in fact here only been defined by means of the erturbative exansion. 5 Conclusions We have shown how the relica method can been adated to chiral erturbation theory. This rovides a new and systematic realization of quenched and artially quenched chiral erturbation theory. We have demonstrated how the relica method is equivalent to the suersymmetric formulation in erturbation theory. This equivalence is quite trivial in the sector of hysical quarks, and has allowed us to extend the three theorems of [2] to the resent relica formulation of PQChPT. The equivalence between the relica and the suersymmetric formalisms also extends outside the sea sector. The comlete agreement (at least to one-loo order) of the two aroaches offers a non-trivial consistency check. In articular, the assumed extension of the standard symmetry breaking attern to the suergrou case is avoided in the resent context. The fact that results agree can be taken as indeendent confirmation of the validity of both aroaches. As an equivalent but nevertheless indeendent formulation of PQChPT the relica method illustrates the fact that suersymmetry is a technical tool for quenching rather than of fundamental nature. For ractical uroses the usefulness of the relica method as comared to the suersymmetric formulation is erhas a matter of taste. The advantage of having fewer sign-rules using the relica method is to some extent traded for the marginally simler combinatorics in the suersymmetric formulation. Finally, the relica method resented here gives the background and the justification for the emirical rules observed by Colangelo and Pallante in [4]. Within the suersymmetric formulation they studied fully quenched chiral erturbation theory to one loo. Based on an exlicit calculation of the divergent arts of the generating functional for both SU(k k) (and the additional U() of the Φ 0 ) and standard SU( ) chiral erturbation theory (without the Φ 0 ), they roosed a set of rules for guessing large arts of the SU(k k) generating functional from that of SU(N v ). The equivalence between the SU(k k) and SU(N v 0) theories (when the Φ 0 is included in both), is a secial case of the general equivalence established here. This formally establishes the rules suggested in [4] and furthermore shows that the terms missing in SU( 0) chiral erturbation are just those roduced by including the Φ 0. The rocedure to comute in artially quenched chiral erturbation theory to any order is now extremely simle. One must take a usual chiral SU( + N v ) chiral Lagrangian and add the contributions from Φ 0. For examle, to order 6 the whole list of divergent contributions in the case of a degenerate SU(N v ) theory is rovided in ref. [6]. This can form the basis for a fully quenched calculation once the contributions from the flavor singlet have been included (for a discussion of the large-n c limit, see e.g. ref. [7]). Acknowledgement: This work was suorted in art by EU TMR grant no. ERBFMRXCT References [] C. Bernard and M.F.L. Golterman, Phys. Rev. D46 (992) 853; Phys. Rev. D53 (996) 476. M.F.L. Golterman and K.-C. Leung, Phys. Rev. D57 (998)

11 M.F.L. Golterman, Acta. Phys. Polon. B25 (994) 73. [2] C. Bernard and M.F.L. Golterman, Phys. Rev. D49 (994) 486. [3] S.R. Share, Phys. Rev. D46 (992) 346; Phys. Rev. D56 (997) S.R. Share and Y. Zhang, Phys. Rev. D53 (996) 525. [4] G. Colangelo and E. Pallante, Nucl. Phys. B520 (998) 433. E. Pallante, JHEP 990 (999) 02. [5] A. Morel, J. Physique 48 (987). [6] J.C. Osborn, D. Toublan and J.J.M. erbaarschot, Nucl. Phys. B540 (999) 37. [7] P.H. Damgaard, J.C. Osborn, D. Toublan and J.J.M. erbaarschot, Nucl. Phys. B547 (999) 305. [8] D. Toublan and J.J.M. erbaarschot, Nucl. Phys. B560 (999) 259. [9] E.. Shuryak and J.J.M. erbaarschot, Nucl. Phys. A560 (993) 306. J.J.M. erbaarschot and I. Zahed, Phys. Rev. Lett. 70 (993) J.J.M. erbaarschot, Phys. Rev. Lett. 72 (994) 253. G. Akemann, P.H. Damgaard, U. Magnea and S. Nishigaki, Nucl. Phys. B487 (997) 72. [0] J.J.M. erbaarschot and T. Wettig, he-h/ [] P.H. Damgaard, Phys. Lett. B424 (998) 322. G. Akemann and P.H. Damgaard, Nucl. Phys. B59 (998) 682; Phys. Lett. B432 (998) 390; he-th/ [2] A. Smilga and J. Stern, Phys. Lett. B38 (993) 53. [3] P.H. Damgaard and K. Slittorff, he-th/ P.H. Damgaard, he-lat/ [4] J.J.M. erbaarschot, Phys. Lett. B368 (996) 37. [5] P.H. Damgaard, R.G. Edwards, U.M. Heller and R. Narayanan, he-lat/ P. Hernandez, K. Jansen and L. Lellouch, he-lat/ [6] J. Bijnens, G. Colangelo and G. Ecker, he-h/ [7] P. Herrera-Siklody, J.I. Latorre, P. Pascual and J. Taron, Nucl. Phys. B497 (997) 345.