mu-Squared Dependent Deviation of the Non Perturbative ZA,MOM from the True Axial Renormalisation Constant, Implied by Ward Identity
Ph.Boucaud, J.-P. Leroy, A. Le Yaouanc, J. Micheli, O. Pène, J. Rodriguez-Quintero
aa r X i v : . [ h e p - l a t ] A p r µ dependent deviation of the nonperturbative Z M OMA from the trueaxial renormalisation constant,implied by Ward identity
Ph. Boucaud a , J.P. Leroy a , A. Le Yaouanc a , J. Micheli a ,O. P`ene a , J. Rodr´ıguez–Quintero b January 10, 2018
LPT-Orsay-15-26UHU a Laboratoire de Physique Th´eorique ; Univ. Paris-Sud ; CNRS; UMR8627 ;Bˆatiment 210, Facult´e des Sciences, 91405 Orsay Cedex. b Dpto. F´ısica Aplicada, Fac. Ciencias Experimentales,Universidad de Huelva, 21071 Huelva, Spain.
Abstract
It is recalled why, as already stated in a previous paper, there seemsto be an inconsistency in identifying the non perturbative Z MOMA asthe renormalisation of the axial current, or equivalently, in setting asnormalisation condition that the renormalised vertex = 1 at p = µ at some renormalisation scale µ , where p is the momentum in the legs.Indeed, unlike the vector case, the Ward-Takahashi (WT) identity forthe axial current is shown to imply both the renormalisation scaleindependence of Z A and a µ dependence of Z MOMA . This µ depen-dence is simply related to certain invariants in the pseudoscalar vertexand can persist in the chiral limit due to the spontaneous breaking ofchiral symmetry (pion pole). It is seen clearly in the µ dependenceof some lattice calculations of Z MOMA /Z MOMV near the chiral limit. Introduction
The non perturbative MOM renormalisation scheme for lattice as introducedin Martinelli et al. Nucl.Phys. B445 (1995) 81-108, [1], and inspired by thecorresponding continuum scheme of Georgi and Politzer [9], has representedan imprtant progress in lattice QCD. It is very intuitive and easy to handle bypedestrians because it relies on the vertices and propagator functions whichare familiar in pertrubative QCD. This is to be compared with later schemeslike the one of Alpha [2], which is rigourous, but requires much more effortto understand and many technicalities.Nonetheless, it has features coming from its direct hadronic meaning thatprecisely complicate the matching with the usual perturbative schemes whichis the final goal. It has a non logarithmic, power dependence on µ generatedby OPE power corrections [4] and particle poles (although the latter arelying outside the Euclidean range). It may have critical chiral behaviourunlike the Alpha prescription, as has been underlined some time ago for thepseudoscalar Z P [3], leading to the recipe of ”extracting the pion pole”.Moreover, it seems to have often escaped the attention that there is also inprinciple an inconsistency in introducing Z MOMA as the renormalisa-tion of the axial current , as stated in our paper PhysRevD.81.094504 [5].This manifests itself in a µ dependence of Z MOMA /Z MOMV which persists inthe chiral limit, even with an explicitly chiral invariant action, due to thespontaneous breaking of the symmetry, see our earlier paper [6], section 8.1..Since this statement was presented in a rather long paper devoted toseveral topics, we think useful to recall the arguments in a clearer and moreexplicit manner, and in the same course to correct some sloppy notations ofthe paper.It must also be said that having reread the basic paper [1], we haverediscovered that there was a discussion in it having connection with thepresent one, although the conclusion seems different : we extract a finiteeffect in the chiral limit, which does not appear in their approach. Thereason will appear after having presented our own discussion, in a separatesection 5. It requires an examination of the interplay of the chiral and q → One must warn from the beginning about the practical meaning of the prob-lem . The problem is found to disappear at large µ , so that one may claimthat it is not real since anyway, the non perturbative MOM scheme is meant2recisely to be applied at such large µ . More precisely, for Z V,A one seemsto be free to choose any µ , therefore, it would suffice to work at such a large µ with Z MOMA . In fact the effect seems anyway to be small already about µ = 2 GeV or beyond, see fig. 5 in the quoted paper [6], which is now thecommonly adopted value for non perturbative renormalisation. Therefore,the effect is perhaps not worrying practically, in contrast to the pion pole in Z MOMP .However it is worth in general being clear on theoretical principles. Butalso working at large µ presents well-known practical problems.1) one may always wonder how large µ must be.2) measuring Green functions at large momenta requires the extractionof large artefacts, and this was the initial motivation for non perturbativerenormalisation : to avoid working at too large momenta.On the other hand, if µ is not sufficiently large, one has to “extract”a physical µ dependence, while artefacts may still be non negligible (onecannot exclude non canonical artefacts at small µ ).For all these reasons, it is useful to know about the possible causes ofmomentum dependence, either “physical” as the present one (physical withmany quotation marks), or artefactic.One must add that at present little effort has been devoted to determinethe actual magnitude of the effect (see the end of the text). Let us first fix the notations that we will use. We will use all along theEuclidean metrics. The continuum quark propagator is a 12 ×
12 matrix S ( p µ ) for 3-color and 4-spinor indices. One can take into account Lorentz (infact O (4)) invariance and discrete symmetries, as well as color neutrality ofthe vacuum by expanding the inverse propagator according to : S − ( p ) = δ a,b Z ψ ( p ) (cid:0) i p/ + M ( p ) (cid:1) (1)where a, b are the color indices. “ Z ψ ( p )” is a standard lattice notation,alluding to the role it plays as a renormalisation constant for the quark fieldin the standard Georgi-Politzer MOM renormalisation, where Z MOM ( µ ) = Z ψ ( µ ) − . But let us stress that here it is not by itself a renormalisationconstant. On the other hand, M ( p ) is the mass function, which is onepossible concept of mass, introduced by Georgi and Politzer. M ( p ) is UVfinite , since it is the ratio of two quantities renormalised by the same factor Z . It is identical with the MOM renormalised mass at scale µ = p , seealso below. 3et us consider a colorless local two quark operator ¯ q O q . The correspond-ing three point Green function G is defined by G ( p, q ) = Z d xd y e ip · y + iq · x < q ( y )¯ q ( x ) O q ( x )¯ q (0) > (2)It is a 4 × p, q ) = S − ( p ) G ( p, q ) S − ( p + q ) (3)In the note, we will often restrict ourselves to the case where the operatorcarries a vanishing momentum transfer q µ = 0. We will then omit to write q µ = 0 and we will moreover understand Γ( p ) without R index as the bare vertex function (computed on the lattice).Now, Lorentz covariance and discrete symmetries allow to write for theaxial vertex Γ Aµ ( p, q ) at q = 0:Γ Aµ ( p ) = δ a,b [ g (1) A ( p ) γ µ γ + ig (2) A ( p ) p µ γ + g (3) A ( p ) p µ p/γ + ig (4) A ( p )[ γ µ , p/ ] γ ] (4)which should be obeyed approximately on the lattice, as we checked.On the other hand, we need the pseudoscalar vertex at q = 0 :Γ ( p, q ) = δ a,b h g (1)5 ( p, q ) γ + ig (2)5 ( p, q ) γ ( p µ γ µ )+ ig (3)5 ( p, q ) γ γ µ q µ + g (4)5 ( p, q ) γ [ γ µ q µ , p/ ] i (5)where the quark momenta are p, p + q and the g ( i )5 ( p, q )’s are invariant func-tions of the momenta alone. For brevity, the first two invariants are denotedby the same symbol at q = 0, i.e. : g (1 , ( p ) = g (1 , ( p, q = 0) (6)More explicitly, the dependence of g ( i )5 ( p, q ) in p, q is : g ( i )5 ( p, q ) = g ( i )5 ( p , q , p.q ) (7)In the following, the color factors δ a,b will be skipped.4 .1 Renormalisation in general Without requiring any specific renormalisation scheme, we have to refer tothe renormalisation, because the Ward-Takahashi(W-T) identities should beimposed on the renormalised theory, and not on the bare quantities (wedo not consider anomalies). The corresponding renormalised quantities aredenoted by a sub- or superindex R. We then draw the consequences for thespecific MOM scheme. Z denotes as usual the fermion field or propagator renormalisation ac-cording to : q = p Z q R S ( p ) = Z S R ( p ) (8)Let us recall that the corresponding renormalised vertex functions are definedthrough: Γ( p ) = Z − Z − O Γ R ( p ) , (9)where the necessary subindices are implicit for each type of vertex; Z O is therenormalisation of the composite operator, namely a current or density oper-ator : O = j V , j A , P ; the Z factor is to take into account the amputation .The lattice calculations, being done at a finite cut-off, generate, as otherregularisation schemes, finite O ( g ) effects, due to additional divergenciesmultiplying the a terms (which have higher dimension), which vanish slowlywith the inverse cutoff or lattice unit a , and are included in the factors Z , Z V , Z A . There are also terms with powers of a which we do not write. Z V , Z A are independent of the renormalisation scheme up to such terms.The fact that we do not include such terms means that our equations shouldhold only sufficiently close to the continuum. Note that the standard definition of renormalisation constants is to divide the barequantity by the renormalisation constant to obtain the renormalised quantity (except forphoton or gluon vertex renormalisation factors Z which we do not use). In principle,renormalisation of composite operators, for instance Z V , should be defined similarly. Wehave followed this convention in our works on gluon fields, for the renormalisation of A . But, in the case of quark composite operators, an opposite convention has becomestandard in lattice calculations : (¯ q O q ) bare = Z − O (¯ q O q ) R ; we feel compelled to maintainthis convention for the sake of comparison with parallel works on the lattice. This explainsour writing of the renormalised vertex function. .2 The axial Ward identity Let us develop the consequences of the axial W-T identity. We define a baremass ρ through the equation ∂ µ ( j A ) µ = 2 ρP (10)(the notation ρ is old and unsuggestive of a mass, but it avoids any ambiguityin a world where there are so many masses). In renormalised form : ∂ µ ( j A ) µ R = 2 m R ( P ) R (11) m R is the renormalised mass in the considered scheme, which then satifiesthe relation : m R = Z − P Z A ρ. (12)To exploit fully the identities, one has to return first to the general case q = p ′ − p = 0. Since they reflect the symmetries of the physical theory,the naive Ward identities should a priori hold for the renormalised Greenfunctions (except for anomalies) and at infinite cutoff, which means : q µ (Γ A ) µ R ( p, q ) = − i ( S − ( p + q ) γ + γ S − ( p )) + i m R (Γ ) R ( p, q ) (13)Returning then to bare quantities which are the ones actually measured onthe lattice, one gets, multiplying both sides by Z − and using m R = Z − P Z A ρ : Z A q µ (Γ A ) µ ( p, q ) = − i ( S − ( p + q ) γ + γ S − ( p )) + i Z A ρ Γ ( p, q ) , (14)which depends only on one renormalisation constant Z A , and bare, renor-malisation scheme independent, quantities.Since eqn. (14 ) has been established without any specification of therenormalisation scheme, it shows that Z A also is independent of the renor-malisation scheme . Therefore, one should expect Z MOMA to be equal to Z A . But this is not the case , as the same identity eqn. (14) shows, see thedemonstration below. Let us now introduce the MOM scheme. It must be first defined for thepropagator, through conditions at some normalisation momentum p = µ ,originally due to Georgi and Politzer [9]: S − R ( µ ) = δ a,b (cid:0) i p/ + m MOM R (cid:1) | p = µ , (15)6hich means Z MOM = Z ψ ( µ ) − (16)according to eqn. (1). Also, it means that the renormalised mass is then m MOM R = M ( µ ) , (17)i.e. it is the mass function at p = µ .As to quark current vertices, it is then commonly accepted that they canbe renormalised analogously by setting ( g (1) V,A, ) R ( p = µ ) = 1 . This leadsto the well-known ”renormalisation constants”, for example :“ Z MOMV,A ” = Z ψ ( µ ) /g (1) V,A ( p = µ ) (18)The quotation marks are provocative and aim at signalling that they maynot be the true renormalisation constants. Indeed, it must be stressed thathaving fixed the renormalisation of the propagator, the renormalisationconditions of vertices cannot be imposed freely : they must be con-strained by the renormalised WT identities (similarly to the Slavnov identi-ties for QCD vertices); one is then not allowed to set ( g (1) V,A, ) R ( p = µ ) = 1freely. “ Z M OMA ” /Z A by derivationof the WT identity near q µ = 0 Let us first recall the very simple argument concerning the vector case. Inthe vector case, the Ward identity is very simple : Z V q µ (Γ V ) µ ( p, q ) = − i ( S − (( p + q ) − S − ( p )) . (19)Then, as is well known, by derivation with respect to q µ , one gets amongother relations: Z V = Z ψ ( p ) /g (1) V ( p ) (20)where g (1) V ( p ) is the coefficient of the γ µ term in the Lorentz decompositionof the vector vertex. Z V has thus be determined independently of any choiceof renormalisation scheme : it is indeed independent of the scheme, beingexpressed in terms of bare quantities. We set standard conditions on one invariant. We are aware that others may be set bycombining several invariants in a trace. They lead to complications in the discussion.
7t must be noticed, of course, that the r.h.s. of eqn. (20) is nothingelse than the MOM renormalisation constant at µ = p defined by therenormalisation condition ( g (1) V ( µ )) R = 1 , i.e. : Z V = “ Z MOMV ( µ )” (21)Therefore, “ Z MOMV ( µ )” is indeed the expected renormalisation of the vectorcurrent and we may abandon the quotation marks .Therefore also, up to now, everything is well with W-T identities in theMOM non perturbative scheme.Now, for the axial case, comes the inconsistency . As in the vector currentcase, the axial W-T identity will give a constraint on the axial vertex at q = 0by taking the derivative of eqn. 14 with respect to q at q = 0. We get : Z A (Γ A ) µ = − i ∂∂p µ S − ( p ) γ + 2 i Z A ρ ∂∂q µ Γ ( p, q ) (22)It must be stressed that not only this relation (22) is more complex thanin the vector case (19) , due to the Γ ( p, q ) contribution, but also the latterdoes not vanish in general even in the chiral limit, because of the pion polein Γ ( p, q ) . This is one more manifestation of the spontaneous breaking ofchiral symmetry. Also, of course, it must be recalled that, on the lattice, theWard identity is not exact, but holds only up to artefacts, because we workat finite cutoff, and the deviation could be found very large in some cases.Let us comment more on the chiral limit. The demonstration has beendone away from the chiral limit, and the relation is at q µ = 0. Now, the chirallimit of the r.h.s. of eqn. (22) is regular since the coupling of a pseudoscalarpion to the axial current has a factor q µ : h π | ( j A ) µ | i ∝ f π q µ . (23)and vanishes at q µ = 0. Therefore, the limit of the r.h.s. must be also regular,although non zero. This will be shown explicitly below.From the equation (22), one deduces that Z A = “ Z MOMA ”, where “ Z MOMA ”is defined, in parallel with Z MOMV , as Z ψ ( p = µ ) /g (1) A ( p = µ ). This is dueto the derivative of the pseudoscalar term. In fact Z ψ ( p = µ ) /g (1) A ( p = µ )is not even independent of µ ; one can hope only that it reaches Z A at large µ ; then, it would be perhaps better to discard this MOM definition, sincethe word is misleading. It must be observed that we stick strictly to definition of MOM condition through oneinvariant i = 1. We do not consider sums over several invariants as done sometimes A somewhat different expression was given in the previous paper, due to a confusionwith an older definition of Z MOMA through traces. γ µ γ structure of the undressed vertex, to match the invariant g (1) A ( p = µ ) in the axial vertex. From now on, we keep to the general p instead of setting p = µ , which is not useful. In Γ ( p, q ), written in full inequation (5) the relevant term is obviouslyΓ ( p, q ) = ... + ig (3)5 γ γ µ q µ + ... (24)The derivative at q = 0 gives a contribution 2 iZ A ρ × − ig (3)5 ( p ) γ µ γ to eqn.(22) . Therefore : Z A g (1) A ( p ) = Z ψ ( p ) + 2 Z A ρg (3)5 ( p ) (25)or Z A /Z MOMA ( p ) = 1 + 2 Z A ρ g (3)5 ( p ) /Z ψ ( p ) (26)The derivation on the propagator coefficients Z ψ ( p ) and Z ψ ( p ) M ( p ), aswell as on the other invariants in Γ is seen to give contributions to the otherinvariants in the expansion of the axial vertex, eqn. (4).One can express the result (26) in a more striking form, returning to therenormalised axial vertex, and to µ as the MOM renormalisation scale forthe propagator. Since Z ( µ ) = 1 /Z ψ ( µ ), Z A /Z MOMA ( µ ) is nothing elsethan Z A Z g (1) A ( µ ) = ( g (1) A ) R ( p = µ ). Then the relation is nothing but :( g (1) A ) R ( p = µ ) = 1 + 2 Z A Z ( µ ) ρ g (3)5 ( µ ) (27)which exhibits clearly the statement that ( g (1) A ) R ( p = µ ) cannot be chosenarbitrarily once the propagator has been renormalised : its value is completelydetermined, and in particular it cannot be set to g (3)5 ( p ) should vanish. Rather, it is clear that itcontains a pion pole contribution, since γ µ q µ γ is a known structure in theBethe-Salpeter vertex function of the pion, see for instance the appendix ofNambu and Jona-Lasinio, Phys.Rev. 124 (1961) 246-254 [7]. It is reason-able to suppose that the effect vanishes by powers at large p , since this isthe general behaviour of Green functions. One must note however that thetransition to 0 as seen from Fig. 5 of [6] is rather abrupt. One must be aware that since there are other possible definitions of Z MOMA , involvingtraces, one would obtain for them a similar equation, but with a contribution of differentinvariants of the pseudoscalar vertex, for instance the derivative of g (2)5 ( p, q ) for a trace on γ µ γ . /m π of g (3)5 ( p ) is multiplied by the factor m q .We have not heard of such an inconsistency of the M OM definition ofthe axial vertex renormalisation in perturbative QCD. At least the deviationshould vanish in the chiral limit in perturbative QCD, since there would nolonger be a pion pole to compensate the m q factor. On the other hand, itremains to be known by explicit calculation what happens at m q = 0.Finally, let us note that if one were using definitions of Z MOMA with traces,as has been usual for some time, one would have to include other invariantsof Γ in the expression of Z A /Z MOMA . Z M OMA /Z M OMV in lattice calculations
The effect may be exhibited most clearly with chiral invariant actions at m q = 0 where Z A /Z V = 1 is expected to hold exactly. And indeed, itseems to have been seen in certain lattice simulations of Z MOMA /Z MOMV withchiral symmetry preserving actions, showing near the chiral limit a decreasefrom 1 with decreasing q (Dawson, with domain wall fermions [8] ; ourpaper Phys.Rev.D74:034505,2006 [6] on the quark propagator with Ginsparg-Wilson action, especially fig. 5), while it reaches 1 at large momentum. Butin neither of these works, was it possible to separate cleanly the effect fromartefacts, and the works should be redone. In connection, a lattice calculationof the new invariants in the pseudocalar vertex should be made to ascertainthe estimate obtained in eqn. (26). In fact, there is in paper [1] a discussion also concluding to a difference Z MOMA = Z A . Apparently it has not led to further discussion. It has aconnection with ours. Their argument also rests on the derivative of theW-T identity at q = 0, and they also conclude that the problem shoulddisappear at large momenta. However, our finite result does not appear intheir calculation. There has been statements that Z MOMA /Z MOMV = 1 in the chiral limit, but they rest onthe assumption that the vacuum is chiral symmetric, so they are valid only asymptoticallyin p m q = 0, without taking a limit from m q = 0.At first sight, this approach is quite opposite and could be expected to beincompatible with ours. Moreover, one could feel dangerous to work at m q =0 and indeed we have preferred to start from m = 0. Nevertheless, we showfinally that one could obtain the same result as in our approach for m q → m q = 0 approach differently.1) Let us first compare the starting point of their discussion.Their argument is based on their eqn. (12) where the W-T identity hasno pseudoscalar term in the r.h.s . On the contrary, our pseudoscalarterm in the r.h.s., is non zero even in the chiral limit (in the form / ) . This r.h.s. pseudoscalar term is crucial in our argument, and in factthe effect is present at any mass m q , not only in the chiral limit.Moreover, in their treatment at m q = 0, they find by derivation of theW-T identity their eqn. (13), with some non explicited contribution to thederivative of q µ (Γ A ) µ ( p, q ) at q µ = 0 due to the q = 0 Goldstone pole in(Γ A ) µ ( p, q ).We do not find such a contribution in our treatment with m q = 0, because(Γ A ) µ ( p, q ) is not singular at all at q = 0, the pole being shifted by m ps .We obtain rather a q µ / ( q − m ps ), therefore a contribution q / ( q − m ps ) to q µ (Γ A ) µ ( p, q ), whose derivative is 0 : ∂∂q µ q | q µ =0 = 0 (28)(We work here in Minkowski space-time for easiness)2) In our opinion, a m q = 0 approach, although perhaps virtually dan-gerous, is nevertheless possible with various precautions, as we show below .But there is another problem in [1]. One decomposes ∂∂q ν q µ (Γ A ) µ into a sumof two terms (l.h.s. of their eqn.(13): ∂∂q ν ( q µ (Γ A ) µ ) = δ µν (Γ A ) µ + q µ ∂∂q ν (Γ A ) µ . (29)This decomposition has the drawback that both terms of the sum aresingular at q = 0, while their sum was regular : the singularities are q ν /q in their first term, − q ν /q in the second one. The finite difference is thennot made explicit. Let us recall that the mechanism of Nambu-Jona-Lasinio was partly illustrated withinthis strict chiral symmetric situation, at m q = 0 [7]
11) Let us now show that one can retrieve our result, at least in the chi-ral limit, through a calculation strictly done at m q = 0 but avoiding thisdecomposition.Let us consider the Goldstone contribution. There is no pseudoscalarterm in the r.h.s. of W-T identity in the m q = 0 method, and the pole isnow only present in (Γ A ) µ , and it has the form : q µ q f π Γ π ( p, q ) (30)where Γ π ( p, q ) is the full pseudoscalar vertex of the pion i.e. a 4 × . .It gives a regular contribution to q µ (Γ A ) µ : q µ q µ q f π Γ π ( p, q ) = f π Γ π ( p, q ) (31)The derivative of q µ (Γ A ) µ at q µ = 0 then has a term not present at m q = 0 : f π ∂∂q µ Γ π ( p, q ) | q µ =0 (32)However, this is exactly equivalent to what we get in our m q = 0 , → Z A (Γ A ) µ = ... , see the r.h.s. of our eqn. (22), second term, inΓ . The sign should be naturally opposite, since the contribution is on theother side of the W-T identity in the m q → h π | ¯ ψγ ψ | i q − m π Γ π ( p, q ); (33)and the contribution to eqn. (22 is then) : lim m π → f π ∂∂q µ m π q − m π Γ π ( p, q ) | q µ =0 = − f π ∂∂q µ Γ π ( p, q ) | q µ =0 (34)using f π m π = 2 Z A ρ h π | ¯ ψγ ψ | i . It is the same as in eqn. (32), except thatthe sign is duely opposite.Now, the effect is present at any quark mass but going to the chiral limitis much easier by our main method. 12 eferences [1] A general method for nonperturbative renormalization of latticeoperators. G. Martinelli, C. Pittori, Christopher T. Sachrajda,M.Testa, A. Vladikas,Nucl.Phys. B445 :81-108,1995 e-Print Archive:hep-lat/9411010[2] For example : Nonperturbative determination of the axial cur-rent normalization constant in O(a) improved lattice QCD. MartinLuscher, Stefan Sint, Rainer Sommer, Hartmut Wittig. Published inNucl.Phys.B491:344-364,1997. e-Print: hep-lat/9611015[3] Pseudoscalar vertex, Goldstone boson and quark masses on the lat-tice. Jean-Rene Cudell, A. Le Yaouanc, Carlotta Pittori. Published inPhys.Lett.B454:105-114,1999. e-Print: hep-lat/9810058, Pseudoscalarvertex and quark masses. Jean Rene Cudell, Alain Le Yaouanc, Car-lotta Pittori. Talk given at 17th International Symposium on LatticeField Theory (LATTICE 99), Pisa, Italy, 29 Jun - 3 Jul 1999. Publishedin Nucl.Phys.Proc.Suppl.83:890-892,2000.[4] Ghost-gluon running coupling, power corrections and the determinationof Λ ¯MS