Volume 212, number 2 PHYSICS LETTERS B 22 September 1988 THE TOPOLOGICAL SUSCEPTIBILITY ON THE LATTICE ~" M. CAMPOSTRINI, A. DI GIACOMO and H. PANAGOPOULOS INF]V;Sezionedi Pisa, and Dipartimento di Fisica, UniversRd di Pisa,2-56 ! O0 Pisa, Italy Received 5 July 1988 w~ s*.udy the multiNicative renormaIization of the topologicalcharge operator on the iat.zicewkh respect tc the continuum. Apgarent discrepanciesbetween different methods of determination &the topologica!charge on ~he lattice are exNained by our result. I. Introduction The existence of a nonzero topological susceptibil- ity of the QCD vacuum was proposed some years ago [I~3] to solve the so-ca'led U(I ) problem, i.e. the exp!icit breaking of the U ( 1 ) axial symmet~. The topological susceptibility is defined by the equation A= ( d"x(OiT(Q(x)Q(O))!O) , (d) j where Q(x) =: ~ Eu~'%FG(x~ F ~ ( x ) (2) 0~+%2 U ~ pa, is the topological charge density. O is the divergence of the topoiogical current K v [ 11 O(x) = GK,, K~,= .g2,e ~/jAg(~A~-}ga~a~A~A~) (3) the prescription defining the product of the operators ineq. (!) is [41 (Oi T(Q(x) Q(O) )tO) =- a. ( Oi T(KU(x) O(O ) ) iO ). (4) Chira! Ward identities lead to the prediction [ 3 ] Partly sappor.'.ed by M.P,I. (kalian Minis'..ry for PuNic Education ). 206 2Ny. . ~ A=m~ + m~, -- 2m~. (5) or, with three Iight flavours, A= (180MeV)4 ~ (6) The lattice is a unique too! to determine quant;.ties like A, which have nontrivial dimension in mass, and therefore cannot be computed by pegurbation theory. Three different methods have been used to extract .4 from iattice QCD. (i) The "naive" method [5-7] consists in the measurement (by Monte Carlo simulation) of the vacuun-_.- two-point correlation at zero momentum of an operator QL (x) which coincides with the topolog- ical charge Q(x) in the limit of zero lattice spacing (a-.0). A possibie choice of QL is (7) where//_"~ (x) is the parallel transport matrix aiong a ! × i Wilson loop; in the usual notation ~""(x) = u,(x) G,(x+¢) U;(x+~) g;(x) o (8) By this method either is measured [5,71, or the value Q,~=Z~Q:.(x) is 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V, ( North-Holland Physics Publishing Division )