hep-

lat/9

5060

08

3 Ju

n 95

IFUP-TH 4/94Lattice energy-momentum tensorwith Symanzik improved actions.Alessandra Buonanno1Dipartimento di Fisica dell'Universit�a, Piazza Torricelli 2, I-56126 Pisa, ItalyGiancarlo Cella2Dipartimento di Fisica dell'Universit�a, Piazza Torricelli 2, I-56126 Pisa, Italy, and IstitutoNazionale di Fisica Nucleare, Via Livornese 582/a I-56010 S. Piero a Grado (Pisa), ItalyGiuseppe Curci3Dipartimento di Fisica dell'Universit�a, Piazza Torricelli 2, I-56126 Pisa, Italy, and IstitutoNazionale di Fisica Nucleare, Via Livornese 582/a I-56010 S. Piero a Grado (Pisa), Italy(September 6, 1994)AbstractWe de�ne the energy-momentum tensor on lattice for the ��4 and for thenonlinear �-model Symanzik tree-improved actions, using Ward identities oran explicit matching procedure.The resulting operators give the correct one loop scale anomaly, and inthe case of the sigma model they can have applications in Monte Carlo simu-lations.PACS numbers: 11.10.Gh, 11.30.CpTypeset using REVTEX1

I. INTRODUCTION.The lattice regularization is the preferred framework for the non perturbative study of a�eld theory by means of numerical simulations. It gives the bonus of explicit preservation ofgauge invariance, which is believed to be the key to understand very important phenomena,as for example con�nement. The lattice however breaks other symmetries, as the translation(Poincar�e) invariance, and it is important to verify the correct restoration of these propertiesin the continuum limit.A classical symmetry is expressed at quantum level by a series of Ward identities involvingthe associated current operator. These identities can't be satis�ed in presence of a noninvariant regulator �. If the invariance must be preserved, we can expect that, after asuitable O(�h) operator rede�nition, these relations will be broken only by \irrelevant" (thatis to say, which vanish removing the regularization) terms.In general this program cannot be accomplished, as a classical invariance may not becompatible with the quantization. A well known example is the dilatation invariance, thatindependently of the regularization chosen cannot be restored in the renormalized theory.However, if an invariant regulator can be found, we must be able to restore the Wardidentities in another scheme, whatever it would be.This is the case of the translation invariance, for which an invariant scheme exists (forexample the dimensional regularization). The relative Ward identities are relations betweeninsertions of the energy-momentum tensor (EMT) that can be written as@x��(n)T�� (x;x1; � � � ; xn) = � nXi=1 @x� �(4)(x� xi)�(n)(x1; � � � ; xn) : (1)On the lattice, we can perturbatively impose the validity of these equations up to termsO(a)(or O(a2) in absence of fermionic �elds): explicit one loop calculations have been done forthe ��4 model [4], for the QED [5], and for the QCD [6,7].In a concrete lattice simulation the cuto� � = 1=a is obviously �nite. This gives rise tosystematic errors in the estimation of observable quantities that can be reduced using verylarge lattices. An alternative possibility, proposed by Symanzik [8] is to rede�ne the latticeaction: we can exploit the ambiguity inherent to the lattice transcription and perturbativelyeliminate the cuto� e�ects to a given power of a.We have explicitly calculated the �nite corrections necessary for the one loop de�nitionof the EMT with a Symanzik tree improved action. First, as an academic exercise, wehave analysed the simple ��4 model [9]. Next we have considered the nonlinear O(N)bidimensional � model [10]. This is of much interest as it shares some important propertieswith lattice gauge theories [11], �rst of all the asymptotic freedom which is an argument tojustify a perturbative approach.Since the EMT is related with the dilatation current, which as we said is broken atquantum level, we have also veri�ed that the so de�ned operator gives the correct one loopanomaly. 2

II. SYMANZIK IMPROVEMENT.We use the ��4 model to explain in detail the procedure. Starting from the classicalcontinuum action S we can write a lattice regularized version of it Slatt in in�nite ways,requiring only that lima!0Slatt[�] = S[�]: (2)The most simple option is the substitution of the integral with a sum over lattice sites n,and of the derivatives with the �nite di�erences �+��(n) = [�(n+ a�̂)� �(n)]=a,S(0;0)latt [�] = a4Xn "12X� �+� �(n)�+��(n) + 12m2�2(n) + 14!g�4(n)# = S[�] +O(a2): (3)If we limit ourselves to a tree level (classical) improvement we can easily do better. Di�e-rences proportional to powers of a between the two actions arise from the derivative term,and can be compensated adding to the lattice lagrangian suitable irrelevant operators, orequivalently using an improved �nite di�erence formally de�ned by@� = 1a log �1 + a�+� � (4)with the second term expanded at the desired order. The classical action with the O(a2)artifacts removed is given byS(1;0)latt [�] = S(0;0)latt [�] + a4Xn;� 14!a2�+��+��(n)�+��+��(n) = S[�] +O(a4); (5)as can be easily veri�ed expanding the �nite di�erences.In the quantized theory it is necessary to parametrize the divergences of the continuumtheory, for instance using dimensional regularization, and to de�ne an improvement criterion.We require that the lattice one particle irreducible functions � generated by S(n;l)latt coincideat small external momenta with the continuum ones, apart from o(�hl) + o(a2n) corrections.Symanzik has shown that it is possible to writeS(n;l)latt [�] = S(n;0)latt [�] + nXk=0 lXp=1 �hp NkXj fjp(g) aDXn a2kO(2k+D)j (n)! ; (6)where the O(k) are k-dimensional lattice operators, and the coe�cients fjp are calculableimposing the improvement criterion with a p loops calculation.A. Energy momentum tensor.From now on we work in the tree improved theory de�ned by S(1;0)latt , which is equivalentto an e�ective continuum (dimensionally regularized) action that can be written asSeff = Z dDx �12Z�@��@��+ 12m2Zm�2 + 14!gZg�4� : (7)3

The Z constant can be determined imposing the equivalence of super�cially divergent �functions. For the two points 1PI function we have on lattice (see app. A for the notations)�(2)latt(0; 0) = m2 + g2 Z d4l(2�)4Dlatt(l) = g2 "Z0a2 + m2(4�)2 �log a2m2 � F0000 � 1 + E � T0�#(8)and in the continuum�(2)cont(0; 0) = m2 + g�4�D2 Z dDl(2�)DDcont(l) = m2 + g2 " m2(4�)2 �1� � 1 + E + log m24��2!# :(9)Now we must subtract the pole part in (9) (we use minimal subtraction in the continuum),and dispose of the quadratically divergent piece in (8) by imposing the vanishing of selfenergy on lattice for p = m = 0 with a suitable counterterm�S(1;0)latt = �a4Xn g Z04a2�2(n): (10)Comparing the two results we obtain (we set the mass scale as � = 1=a for simplicity)Zm = 1 + g2(4�)2 [log 4� � T0 � F0000] : (11)The procedure for the determination of Zg is analogous. For the four points functions weobtain on lattice�(4)latt(0; 0; 0; 0) = g � 32g2 Z d4l(2�)4D2latt(l) = g � 32 g2(4�)2 h� E + F0000+ T0 � log a2m2i ;(12)and in the continuum�(4)cont(0; 0; 0; 0) = g � 32g2 Z dDl(2�)DD2cont(l) = g � 32 g2(4�)2 "1� � E � log m24��2# (13)so that Zg = 1 + 32 g2(4�)2 [log 4� � T0 � F0000] : (14)It is now possible to de�ne the EMT in a very simple way . In fact it can be shown [12,13]that, given the e�ective lagrangian Leff de�ned by (7), it is su�cient to write a latticeoperator which reduces in the leading order in a toT eff�� = Z�N [@��@��]� ���N [Leff ]; (15)where N [O] is an operator corrected accordingly to the prescription of the e�ective lagran-gian. We note that as we consider only the integrated T��, there are no problems of ambiguity4

in the de�nition of the classical operator, so that we can neglect the so-called \improvedColeman term" [17,16].We can also write at one loop orderT�� = Z1 ���� ����� a26 ����� ��3��+ ��3�� �����+� ��� "Z22 X� ���� ����� a26 X� ���� ��3��+ Z32 ���� ����+ Z42 m2�2 + Z54! g�4# ; (16)where we have used the symmetric �nite di�erence ����(n) = (�(n + a�̂)� �(n � a�̂))=2a,and we have imposed the Ward identities to determine the Z �nite corrections. We haveexplicitly checked that the two methods agree each other.The expression (16) reduces to the classical improved quantity if Z1 = Z2 = Z4 =Z5 = 1; Z3 = 0, and as we are considering only tree improvement we can neglect O(a2g)corrections. The resulting Feynman rules give two and four lines vertices, T (2)�� and T (4)�� . Therelevant identities de�ned by (1) involve the insertion of these vertices in the two and fourpoints irreducible functions (see �gure (1)). We start from the two points identity, writtenin momentum space for the integrated T��:�(2)T�� (0; p;�p) = p�2 @@p� + p�2 @@p� � ���!�(2)(p) : (17)We have already calculated the one loop correction to �(2) (see eq. (8)), while for the insertionwe obtain�(2)T�� (0; p;�p) = 2T (2)�� (p;�p) � g2 Z d4l(2�)4Dlatt(l) h��� + 2T (2)�� (l;�l)Dlatt(l)i ; (18)and substituing in the (17) we �nd the conditions(Z1 � 1) = (Z2 � 1) = Z3 = O(g2) (19)Z4 = 1 + g4 Z d4l(2�)4D2latt(l) "2 + �l2m2 + a23 �l4m2# == 1 + g2(4�)2 �T0 � T2 + 2�2Z0000 � 1 � 13T4�+ ga2m2 �T14 + T312� : (20)The determination of the Z5 correction can be done considering the four points identity�(4)T�� (0; 0; 0; 0; 0) = �����(4)(0; 0; 0; 0): (21)We calculate the relevant insertion which, taking into account the relations (20), is of theform �(4)T�� (0; 0; 0; 0; 0) = ����gZ5 + 3g2��� Z d4l(2�)4D2latt(l) ++3g2 Z d4l(2�)4D3latt(l) "2�l��l� + a23 ��l��l3� + �l��l3��� ��� �l2 + a23 �l4 +m2!# : (22)5

Using this result and the eq. (12) we obtainZ5 = 1 + 32g Z d4l(2�)4D2latt(l) "1�Dlatt(l) �l2 + 2m2 + a23 l4!# == 1 + 3 g2(4�)2 �T0 � T2 + 2�2Z0000 � 1 � 13T4� : (23)B. The dilatation current anomaly.From the Ward identity for the No�ether current connected to scale invarianceJ (D)� = x�T�� (24)one can show that the scaling equation is related to the EMT trace as follows a @@a �m @@m �N! �(N)(p1; � � � ; pN) ' �(N)T��(0; p1; � � � ; pN ) (25)Now, the mass derivative is equivalent to the insertion in � of a renormalized mass operator,and analogously the �eld number N can be generated inserting the renormalized operator� �S�� . It follows that scale anomaly is equivalent to the insertion of the evanescent operatora4Xn T�� +m2�2 + ��Slatt�� ! : (26)Using the de�ned EMT we obtain the same results one �nds with the not improved action [4].This is correct, as the anomaly is proportional to the � functions of the theory that, as wellknown, does not depend to this order from the regularization scheme.III. RESULTS FOR THE IMPROVED NONLINEAR SIGMA MODEL.The nonlinear O(N) sigma model is de�ned by the path integralZ = Z D~�(x) �(~�2(x)� 1) exp�� 12t Z d2y @�~�(y) � @�~�(y)� ; (27)where ~� = (�1; � � � ; �N) are N scalar �elds. In the continuumwe use as usual the dimensionalregularization scheme, with minimal subtraction at the scale � = 1=a, and in order toeliminate infrared divergences, which are an artefact of perturbative approximation, we addto the action a mass term which breaks the O(N) symmetry. With the chosen regularizationthere are no contributions from the path integral measure, and we can write the action asS = Z dDx�D�2 " 12t@�~� � @�~� + 12t@��@�� � m2t �# ; (28)with the parametrization 6

~� = (�1; � � � ; �N�1)� = �N = p1 � ~� � ~�: (29)This lagrangian generates an in�nite number of vertices, but at a given perturbative orderonly a restricted set of them gives a contribution. At one loop we need the propagator of �i�elds Dij(p) and the four points vertex Vijkl(p1; p2; p3; p4).The O(N) invariance gives strong constraints to the renormalized action. Solving expli-citly the relative Ward identities [14] one can prove that this can be written in terms of tworenormalization constant Z;Zt,S(R) = Z dDx�D�2 " Z2Ztt (@�~� � @�~� + @��@��)� m2t �# ;� = pZ�1 � ~�2 (30)On lattice the tree improved action can be written asS(1;0)latt = S0 + S1 + Smeas (31)with S0 = a2Xn " 12t�+�~�(n) ��+�~�(n) + 12t�+��(n)�+��(n)� m2t �(n)# (32)S1 = a2Xn " a224t�+��+�~�(n) ��+��+�~�(n) + a224t�+��+��(n)�+��+��(n)# (33)Smeas = a2Xn 1a2 log �(n); (34)the last term coming from the exponentiated path integral measure. This action is equivalentto an e�ective continuum one, that must be of the form (30). To determine the �niterenormalizations Z;Zt it is su�cient to compare the two point irreducible function on thecontinuum�(2)ij;cont(p;�p) = �ij "p2 +m2t � 14� �p2 + N � 12 m2� log a2m24� + E!# (35)and on the lattice�(2)ij;latt(p;�p) = �ijt "m2 +X� bp2� + a212 bp4�!� ta2 + tN + 12 m2 Z d2l(2�)2Dlatt(l)++ t2 Z d2l(2�)2Dlatt(l)X� d(p+ l)2� + d(p� l)2� + a212 d(p+ l)4� + a212 d(p� l)4�!# : (36)Using the lattice integrals properties and suppressing irrelevant terms we can write�(2)ij;latt(p;�p) ' �ijt hp2 +m2++ Z d2l(2�)2Dlatt(l) N � 12 m2 + (1 + 4 cos al1 � 2 cos2 al1)p23 !# ; (37)7

where the quadratic divergence in the integral has been canceled by the measure term. The�nal result isZt = 1 + t4� (N � 2)� E + E0 + log 324��+ t4� �83 � 43E5 + 23E0 + 23E3� (38)Z = 1 + t4� (N � 1)� E + E0 + log 324�� : (39)A. Energy momentum tensor.At �rst we note that we expect to obtain a O(N) invariant result, except for contributionsproportional to the symmetry breaking mass or to the motion equation�Slatt�~� = �1t "�+����~� � ~���+���� � �m2~��# : (40)So at the order we are working the operator can be written asT�� = T (0)�� + T (1)�� + T (m:e:)�� + T (l)�� (41)with T (0)�� = Z1t ���~� � ���~�� ��� "Z22t X� ���~� � ���~�� Zmt m2�# ; (42)T (1)�� = �a26t � ���~� � ��3�~�+ ��3�~� � ���~��+ ��� a26t X� ���~� � ��3�~�; (43)T (m:e:)�� = Z4~� � �Slatt�~� ; (44)T (l)�� = ����Z32t ���~� � ���~�: (45)When Z1 = Z2 = 1 the term T (0)�� reduces to the naive lattice transcription of the classicaloperator. T (1)�� cancels the O(a2) lattice artifacts at tree level, and we have not consideredO(t) corrections for it as they would be irrelevant to the order we are working. T (m:e:)�� is apossible contribution proportional to the motion equation, and as we said it doesn't need tobe O(N) invariant. It is not present at tree level, so Z4 = O(t). T (l)�� is an operator with onlythe hypercubic lattice symmetry that can be required to ripristinate the O(a2) rotationalinvariance.As in the ��4 model it is possible to calculate the Z corrections by using two equivalentmethods. We start evaluating the insertion with external momenta p = k and q = 0 (seethe graphs I0,I1,I2 in �g. (1)). In this regime the Ward identity reduces tok��(2)ij;T�� (k; k; 0) = �k��(2)ij (0; 0): (46)The contribution of the various insertions can be written as8

I0(k; 0) = �ij��� 1t hZ4 �k̂2 + 2m2�� Zmm2i (47)I1(k; 0) = ��ij2 Z d2l(2�)2 "2Dlatt(l) + (N � 1)Dlatt(l)Dlatt(l + k)Dlatt(k) #� [A��(k + l; l)+� ��� �12A��(k + l; k) +m2�� (48)I2(k; 0) = �ij Z d2l(2�)2Dlatt(l) �A��(k + l; l)� ��� �12A��(k + l; k) + N + 12 m2�� (49)A��(p; q) = p�q� + a26 �p3�q� + p�q3��+ [�$ �] (50)Summing the three contributions and using (46) we obtainZ4 = t8� (N � 1) (51)Zm = 1� t8� (N � 1): (52)The other constants can be calculated from the insertions of the integrated operator. Withthe same notation we haveI0(p;�p) = �ijt "2Z1p�p� + a23 �p�p3� + p3�p��� ��� Z2p2 + a23 p4 + Z3p2� + Zmm2++ 2Z4(p̂2 +m2)�i (53)I1(p;�p) = �ij���m22 Z d2l(2�)2D2latt(l) h2Dlatt(p+ l) + (N � 1)m2i (54)I2(p;�p) = �ij Z d2l(2�)2Dlatt(l) "2A��(p + l; p+ l)� ���2 �A��(l+ p; l + p) + (N + 1)m2�# (55)Taking all together, and using the Ward identity (17) we obtainZ1 = 1 + t12� (4E5 + E0 � 3E1 � 2E3 � 3E2 � 20) (56)Z2 = 1 + t12� (4E5 � 2E0 � 2E3 + 8E4 � 3(N � 1)� 5) (57)Z3 = t6� (3E0 � 3E1 � 3E2 � 8E4 � 12): (58)This result can be checked by an explicit determination of a lattice transcription of thee�ective EMT T eff�� = 1t ZZtN [@�~� � @�~�]� ���N [Leff ]; (59)where Leff is de�ned by (30). It is su�cient to match lattice and continuum insertion ofoperators in the two points function.For reference we list the results for the matching coe�cients9

b1 = 1 + t4� ��4� E2 � E1 � log 324� + E� ; (60)b2 = t8� �(N � 2)(� log 324� � E + E6 + 1) � E724 � ; (61)b3 = t12� [12 � 3E0 + 3E1 + 3E2 + 8E4] ; (62)b4 = b5 = t8� (N � 1) ��1� E6 + log 324� + E� ; (63)c1 = 1 + t4� �(N � 2)E6 � (N � 1)�log 324� + E�� 124E7 � E0 + 83E4� (64)c2 = c3 = t4� (N � 1) �log 324� + E � E6� ; (65)(66)de�ned byN [@�~� � @�~�] = b1 ���~� � ���~�� a26 � ���~� � ��3�~�+ ��3�~� � ���~��+ b2��� ���~� � ���~�++ b3��� ���~� � ���~�+ b4���m2� + b5���~��Slatt�~� ; (67)N [@�~� � @�~�] = c1 ���~� � ���~�� a23 ���~� � ��3�~�+ c2m2� + c3~��Slatt�~� : (68)B. The dilatation current anomaly.In analogy with the ��4 model, we �nd that the scale anomaly is proportional to theirrelevant operator a2Xn �T�� � 2tm2�� (69)whose insertions are equivalent to those ofa2Xn �(t)t @@t � �(t)2 ~� � @@~� � m(t)m2 @@m2! S: (70)Independently of the action (improved or standard) we �nd� = a2Xn (�(N � 2)2� ��12 ���~� � ���~�+m2��+ (N � 3)4� m2� � t(N � 1)4� ~� � �Slatt�~� ) ; (71)and we obtain the correct one loop (scheme independent) results:�(t) = �t(N � 2)2� ; (72)�(t) = t(N � 1)2� ; (73) m(t) = �t(N � 3)4� : (74)10

C. Results for the standard theory.We report the results obtained for the no improved theory. With the same notationsused previously we have for the renormalized lagrangianZt = 1 + t(N � 2)4� �log 324� + E�+ t4 ; (75)Z = 1 + t(N � 1)4� �log 324� + E� : (76)The relevant insertion of composite operators appearing in T�� givesb1 = 1 � t4� �4 + log 324� + E� ; (77)b2 = t8� �2 � � + (N � 2)�32 � log 324� � E�� ; (78)b3 = � t� [1 � �] ; (79)b4 = b5 = t(N � 1)8� ��32 + log 324� + E� ; (80)c1 = 1 + t4� �12(N � 2)� (N � 1)�log 324� + E�+ 3� � 6� ; (81)c2 = c3 = t(N � 1)4� �log 324� + E � 12� : (82)The renormalization constants for the energy-momentum tensor areZ1 = 1 � t4� [4 + �] ; (83)Z2 = 1 + t4� [3� � 8� (N � 2)] ; (84)Z3 = 2�t [1� �] ; (85)Z4 = t8� [N � 1] ; (86)Zm = 1 � t8� [N � 1] : (87)IV. CONCLUSIONS.We have de�ned on the lattice, to one loop, the energy-momentum tensor for two Syman-zik tree improved actions, the ��4 scalar theory and the bidimensional nonlinear � model.In the � model case, due to the asymptotic freedom of the theory, the calculation is reliablein the asymptotic scaling regime.We have found the correct scale anomaly, which as expected is independent of the regu-larization scheme. 11

For the � model the corrected EMT we have found can be a starting point for a successivenon-perturbative determination with numerical method [18]. As the T00 component of EMTis the energy density, it can be used on lattice for the determination of the mass spectrum.We think another interesting possibility is the test of EMT in variational computations, inorder to clarify what are in this framework the in uences of lattice artefacts.APPENDIX A: NOTATIONS AND INTEGRALS.The lattice propagator is de�ned asDlatt(l) = 1l̂2 +m2 ; with l̂2 = DX�=1 4a2 sin2 �a2 l�� : (A1)We use also l� = 1a sin al�: (A2)The lattice integral needed for calculations with the standard (not improved) action for ��4can be evaluated in terms of integrals of modi�ed Bessel functions [15]. All the relevantcases can be expressed in term of two parametersI1 = Z d4l(2�)4 1q̂2 +m2 = Z0000a2 � m2(4�)2 (1 + F0000 � E � log a2m2) +O(a2m4) (A3)I2 = Z d4l(2�)4 1(q̂2 +m2)2 = 1(4�)2 (F0000� E � log a2m2) +O(a2m2) (A4)I3 = Z d4l(2�)4 1(q̂2 +m2)3 = 1(4�)2 12m2 +O(a2) (A5)I4 = Z d4l(2�)4 q�q�(q̂2 +m2)2 = ���2a2 "�Z0000 � 18�+ a2m2(4�)2 (2�2Z0000 + E � 1� F0000+ (A6)+ log a2m2)i+O(a2m4) (A7)I5 = Z d4l(2�)4 q�q�(q̂2 +m2)3 = ���4(4�)2 (F0000� 2�2Z0000 � E � log a2m2) +O(a2m2) (A8)where F0000 = Z 10 ze�8zI40(2z)dz + Z 11 ze�8z "I40(2z)� e8z(4�z)2# ' 4:369 (A9)Z0000 = a2 Z ��� d4q(2�)4 1̂q2 ' 0:155 (A10)For the standard sigma model case all integrals are analitically calculable in term of �rstand second type elliptic functions. 12

S1 = Z d2l(2�)2 1l̂2 +m2 = � 14� log a2m232 +O(a2m2) (A11)S2 = Z d2l(2�)2 cos al1l̂2 +m2 = � 14� log a2m232 � 14 +O(a2m2) (A12)S3 = Z d2l(2�)2 cos al1 cos al2l̂2 +m2 = � 14� log a2m232 + 1� +O(a2m2) (A13)S4 = Z d2l(2�)2 cos2 al1l̂2 +m2 = � 14� log a2m232 � 12 + 1� +O(a2m2) (A14)The integrals which appear in calculations with the improved actions are not analiticallycalculable. For the ��4 model we need to extract the asymptotic a! 0 values ofM1 = Z d4l(2�)4Dlatt(l) = I1 � m2(4�)2T0 + Z0 � Z0000a2 +O(a2m4) (A15)M2 = Z d4l(2�)4D2latt(l) = I2 + 1(4�)2T0 +O(a2m2) (A16)M3 = Z d4l(2�)4D3latt(l) = I3 +O(a2m2) (A17)M4 = Z d4l(2�)4D2latt(l)q2 = T1a2 + 2m2(4�)2 (2�2Z0000 + E � 1 � T2 � F0000+� log a2m2) +O(a2m4) (A18)M5 = Z d4l(2�)4D3latt(l)q2 = T2(4�)2 + 1(4�)2 (F0000� 2�2Z0000 � E � log a2m2) +O(a2m2) (A19)M6 = a2 Z d4l(2�)4D2latt(l)q4 = T3a2 � 2m2(4�)2T4 +O(a2m4) (A20)M7 = a2 Z d4l(2�)4D3latt(l)q4 = 1(4�)2T4 +O(a2m2): (A21)The general strategy is to add and subtract integral resulting from the standard latticeformulation from the improved one. In this way we obtain a divergent piece that canbe calculated explicitly (as is a standard Ii integral), and a �nite expression that can beevaluated numerically. We give the values of the relevant constants in Tab. I. For the sigmamodel we need~S1 = Z d2l(2�)2Dlatt(l) = 14� E0 � log a2m232 !+O(a2) (A22)~S2 = Z d2l(2�)2Dlatt(l) cos al1 = 14� E5 � � � log a2m232 !+O(a2) (A23)~S3 = Z d2l(2�)2Dlatt(l) cos al1 cos al2 = 14� E1 + 4� log a2m232 !+O(a2) (A24)~S4 = Z d2l(2�)2Dlatt(l) cos2 al1 = 14� E3 + 4� 2� � log a2m232 !+O(a2) (A25)~S5 = Z d2l(2�)2Dlatt(l) sin4 al1 = 14�E4 +O(a2) (A26)13

~S6 = Z d2l(2�)2Dlatt(l) sin4 al1 cos al1 cos al2 = 14�E2 +O(a2) (A27)~S7 = Z d2l(2�)2D2latt(l) = 14�m2 +O(a2) (A28)which have been evaluated with the same technique, in terms of the constants reported inTab. II

14

TABLESZ0 0.129 1(4�)2T0 -0.008T1 0.045 1(4�)2T2 -0.005T3 0.029 T4(4�)2 0.009TABLE I. The constants Z0 and TiE0 -0.5928 E1 0.024514�E2 0.0443 E3 -0.312714�E4 0.1056 E5 -0.0577E6 0.8706 E7 7.041TABLE II. The constants Ei

15

REFERENCES[1] Electronic address: buonanno@sunthpi1.di�.unipi.it.[2] Electronic address: cella@sun10.di�.unipi.it.[3] Electronic address: curci@mvxpi1.di�.unipi.it.[4] S. Caracciolo, G. Curci, P. Menotti, and A. Pelissetto, Nucl. Phys. B 369, 612 (1988).[5] S. Caracciolo, G. Curci, P. Menotti, and A. Pelissetto, Phys. Lett. B 228, 375 (1989).[6] S. Caracciolo, G. Curci, P. Menotti, and A. Pelissetto, Ann. pf Phys. 197, 119 (1990).[7] S. Caracciolo, P. Menotti, and A. Pelissetto, Phys. Lett. B 260, 401 (1991).[8] K. Symanzik, in Mathematical problems in theoretical physics, edited by R. S. et al.(Springer, Berlin, 1982).[9] K. Symanzik, Nucl. Phys. B 226, 187 (1983).[10] K. Symanzik, Nucl. Phys. B 226, 205 (1983).[11] V. Novikov, M. Shifman, A. Vainshtein, and V. Zakharov, Phys. Rep. 116, 103 (1984).[12] G. Bonneau, Nucl. Phys. B 167, 261 (1980).[13] G. Bonneau, Nucl. Phys. B 171, 477 (1980).[14] E. Brezin and J. Zinn-Justin, Phys. Rev. 14, 2615 (1976).[15] A. Gonzales-Arroyo and C. P. K. Altes, Nucl. Phys. B 205 [FS5], 46 (1982).[16] J. C. Collins, Phys. Rev. D 14, 1965-1976 (1976).[17] C. G. Callan, S. Coleman and R. Jackiw, Ann. Phys. 59, 42-73 (1970).[18] A. Buonanno, G. Cella and G. Curci, In preparation.

16

FIGURESFIG. 1. The relevant insertions of T�� .I0

µ,ν

I1

µ,ν

I2µ,ν

J0

µ,ν

J1

µ,ν

J2

µ,ν

17