ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaGlobally Conformal Invariant Gauge Field Theorywith Rational Correlation Fun tionsN.M. NikolovYa.S. StanevI.T. Todorov

Vienna, Preprint ESI 1335 (2003) November 11, 2003Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

Globally onformal invariant gauge �eld theorywith rational orrelation fun tions�N.M. Nikolov1), Ya. S. Stanev1)2), I.T. Todorov1)3)1) Institute for Nu lear Resear h and Nu lear EnergyTsarigradsko Chaussee 72, BG-1784 So�a, Bulgaria2) I.N.F.N. { Sezione di Roma IIVia della Ri er a S ienti� a 1, I-00133 Roma, Italy3) Se tion de Math�ematiques, Universit�e de Gen�eve2-4 rue du Li�evre, p 240, CH-1211 Gen�eve, SuisseIHES/P/03/16CERN{TH/2003-056ROM2F/2003/12

Abstra tOperator produ t expansions (OPE) for the produ t of a s alar �eldwith its onjugate are presented as in�nite sums of bilo al �elds V�(x1; x2)of dimension (�; �). For a globally onformal invariant (GCI) theory wewrite down the OPE of V� into a series of twist (dimension minus rank)2� symmetri tra eless tensor �elds with oeÆ ients omputed from the(rational) 4-point fun tion of the s alar �eld.We argue that the theory of a GCI hermitian s alar �eld L(x) ofdimension 4 in D = 4 Minkowski spa e su h that the 3-point fun tions of apair of L's and a s alar �eld of dimension 2 or 4 vanish an be interpretedas the theory of lo al observables of a onformally invariant �xed point ina gauge theory with Lagrangian density L(x).Mathemati al Subje t Classi� ation. 81T40, 81R10, 81T10Key words. 4{dimensional onformal �eld theory, rational orrelation fun tions,in�nite{dimensional Lie algebras, non-abelian gauge theorye-mail addresses: mitov�inrne.bas.bg, stanev�roma2.infn.it, todorov�inrne.bas.bg�published in Nu l. Phys. B 670 [FS℄ (2003) 373{400; a brief preview of this paper is ontained in [21℄. 1

1 Introdu tionThe present paper o�ers a new step in the realization of the program set upin [19℄ and [20℄ of onstru ting a 4{dimensional onformal �eld theory (CFT)model with rational orrelation fun tions of observable �elds.Global onformal invarian e (GCI) allows to write down lose form expres-sions for orrelation fun tions involving (ea h) a �nite number of free parame-ters. Thus, the trun ated 4{point fun tionwt4 of a neutral s alar �eld of (integer)dimension d depends on hd23 i real parameters ([a℄ standing for the integer part ofthe positive number a). In addition to GCI it satis�es the onstraints of lo alityand energy positivity. Operator produ t expansion (OPE) provides a methodof taking the remaining ondition of Wightman (i. e. Hilbert spa e) positivityinto a ount. We organize systemati ally OPE into mutually orthogonal bilo al�elds V� (x1; x2) of dimension (�; �) de�ned by the ondition that V� an beexpanded into an (in�nite) sum (of integrals) of twist 2� symmetri tra elesstensor lo al �elds. An algorithm is given for omputing the full ontribution ofea h V� to the 4{point fun tion. The e�e tiveness of this approa h is enhan edby the fa t that V1 (x1; x2) , whi h involves an in�nite sum of onserved tensor�elds, satis�es the d'Alembert equation in ea h argument and the 4{point fun -tion h0 j V1 (x1; x2)V1 (x3; x4) j 0i is rational by itself. For the model of a d = 2neutral s alar �eld � (x) the OPE of two �'s an be summed up simply as [20℄� (x1) � (x2) = h0 j � (x1)� (x2) j 0i+ (12)V1 (x1; x2)+ :� (x1)� (x2) : (1.1)where (12) is the free 0{mass 2{point fun tion,(12) = 14�2�12 ; �12 = x 212 + i 0x012 ; x12 = x1 � x2 ; x2 = x2 � x20 ;(1.2)and the normal produ t : � (x1)� (x2) : de�ned by (1.1) is non{singular forx1 = x2. The simpli ity of V1 in this model has been exploited in [20℄ to provethat it an be written as a sum of normal produ ts of (mutually ommuting)free 0{mass �elds.We fo us in the present paper on the study of a model generated by a(neutral) s alar �eld whi h an be interpreted as a (gauge invariant) Lagrangiandensity L(x). Taking into a ount the above ited triviality result for d = 2 werequire that the OPE of L(x1)L(x2) � h0 j L(x1)L(x2) j 0i does not involveany �eld of dimension lower than 4 (whi h just amounts to ex luding a possibles alar �eld of dimension 2). This requirement de reases the number of freeparameters in wt4 from �ve to four. It already ex ludes most of the standardrenormalizable intera tion Lagrangians (like Yukawa and '4) but allows for theLagrangian L and for the pseudos alar topologi al term eL of a pure gauge theoryL(x) = �14 tr(F��(x)F ��(x)) ; (1.3)eL(x) = 14 "���� tr(F��(x)F��(x)) : (1.4)2

The invarian e of L under onformal res aling of the metri is made manifestby writing the a tion density in terms of the Yang-Mills urvature formF (x) := 12 F��(x) dx� ^ dx� (1.5)and its Hodge dual, �F :L(x)pjgjdx0 ^ dx1 ^ dx2 ^ dx3 = tr(�F (x) ^F (x)) (1.6)(where jgj is the absolute value of the determinant of the metri tensor, jgj = 1for the Minkowski metri ). (It does not seem super uous to reiterate that the onformal invarian e of the gauge �eld Lagrangian singles out D = 4 as thedimension of spa e time [6℄.)It seems appropriate to demand further invarian e of the theory under\ele tri -magneti duality"1, { i.e., under the hange F ! �F . Taking intoa ount the fa t that the Hodge star de�nes a omplex stru ture in Minkowskispa e, we dedu e that the Lagrangian (1.6) hanges sign under �:�(�F ) = �F ) L(�F ) = �L(F ) : (1.7)It follows that odd point orrelation fun tions of L(x) should vanish. Requiringspa e re e tion invarian e we also �nd that the va uum expe tation values ofan odd number of eL's (and any number of L's) should vanish. This leads us tothe assumption that no (pseudo) s alar �eld of dimension 4 appears in the OPEof two L's thus eliminating one more parameter in the expression for wt4.The ase of a non-abelian gauge �eld is distinguished by the existen e ofa non-trivial 3-point fun tion of F whi h agrees with lo al ommutativity ofBose �elds. Consisten y of the equations of motion satis�ed by the 2- and 3-point fun tions of F with OPE, however, requires the use of inde omposablerepresentations of the onformal group C whi h makes unpra ti al exploitingthe ompositeness of L (Se . 5.2).The paper is organized as follows.We outline basi ideas and review earlier work and notation in Se . 2. Inparti ular, we make pre ise the notation of an elementary positive energy rep-resentation of C, and indi ate an extension of the present approa h to any evenspa e-time dimension D.Se . 3.1 provides a general treatment of the OPE of a pair of onjugate s alar�elds of dimension d 2 N in terms of bilo al �elds V�(x1; x2). It also reprodu esthe formula for the expansion of V� into an in�nite series of symmetri tra elesstensor �elds of twist 2�. Se . 3.2 displays the ( rossing symmetrized) ontribu-tion of twist 2 ( onserved) tensors and their light one expansions for d = 2 andd = 4.In Se . 4.1 we study systemati ally the trun ated 4-point fun tion of thed = 4 �eld L(x) and dis uss the spe ial ase of the Lagrangian L0 (4.17) of the1The authors thank Dirk Kreimer for this suggestion.3

free Maxwell �eld. In Se . 4. 2 we analyse the onformally invariant (rational)fa tor f1 of the harmoni 4-point fun tionh0 j V1(x1; x2)V1(x3; x4) j 0i = (13)(24)f1(s; t) ; (1.8)displaying a basis of solutions j�(s; t) of the \ onformal Lapla e equation" (4.7)whi h admits a natural rossing symmetrization. Se 4.3 displays the operator ontent and the light one expansion in twist 4 and twist 6 tensor �elds in the ase of general L.The restri tions on the parameters of the trun ated 4-point fun tion omingfrom Wightman positivity of the singular part of the the s- hannel OPE areanalysed in Se . 5.1. Se . 5.3 is devoted to a summary of results and on ludingremarks.Appendix A summarizes the derivation of the main result of [20℄ referred toin the text dis ussing on the way the possibility for a similar treatment of thed = 4 ase of interest (and the diÆ ulties lying ahead).2 Consequen es of GCI (a synopsis)2.1 Elementary positive energy representations of the on-formal groupThe idea that quantum �elds should be subdivided into lo al observables and\ harged �elds" (that are relatively lo al to the observables and a t as inter-twiners among di�erent supersele tion se tors) emerged from several de ades ofwork of Haag and ollaborators reviewed in [16℄. It has been implemented in2-dimensional onformal �eld theory (CFT) models in whi h hiral urrents (in- luding the stress-energy tensor) play the part of lo al observables while primary(with respe t to a given hiral algebra) \ hiral vertex operators" orrespond to\ harged �elds" intertwining between di�erent supersele tion se tors. (Amongthe numerous reviews on 2D CFT we ite the textbook [7℄ and the earlier arti le[15℄ whi h refers to the axiomati quantum �eld theory framework.) Lo al ob-servable �elds are assumed to satisfy Wightman axioms [24℄. More spe i� ally,we demand that their set ontains both the stress energy tensor T��(x) ( f. [18℄)and the Lagrangian density L(x).To set the stage we �x the lass of lo al �eld representations of the quantumme hani al onformal group C(D) = Spin(D; 2) under onsideration ( f. [9℄,[17℄, [25℄). The forward tube,T+ = nx+ iy ; y0 > jyj := �y21 + : : :+ y2D�1� 12o ; (2.1)the primitive analyti ity domain of the ve tor valued fun tion (x+ iy) j 0i forany Wightman �eld , appears as a homogeneous spa e of C (D),T+ = C (D) =K (D) for K (D) = Spin (D) � U (1) ; (2.2)4

K (D) being the maximal ompa t subgroup of C (D) . An elementary pos-itive energy representation of C (D) is one indu ed by a (�nite dimensional)irredu ible representation of K (D). Any su h representation gives rise to atransformation law for lo al �elds de�ned (as operator valued distributions) onthe ( onformally) ompa ti�ed Minkowski spa e MD whi h appears as a (partof the) boundary of T+ . MD also appears as a homogeneous spa e of C(D)with respe t to a �D+12 � + 1 parameter \paraboli subgroup" H(D) � C(D)de�ned as a semidire t produ t H(D) = ND o (Spin(D � 1; 1) � SO(1; 1)) ofa D dimensional abelian subgroup ND of \spe ial onformal transformations"with the dire t produ t of the (quantum me hani al) Lorentz group and the1-parameter subgroup of C(D) of uniform dilations:MD ' C(D)=H(D) ' (S1�SD�1)=Z2 : (2.3)The resulting elementary (lo al �eld) representation [25℄ an be de�ned as a pos-itive energy representation of C(D) indu ed by a �nite dimensional irredu iblerepresentation of H(D) (energy positivity �xing the representation of the dis- rete entre {Z4 for even D { of C(D) { see [17℄). In the ase D = 4 of interest(studied in [17℄),C � C(4) = SU (2; 2)(�= Spin(4; 2)) ; M (�M4) = U (2)(= (S1�S3)=Z2) ; (2.4)the elementary representations of C are labeled by triples (d; j1; j2) of non-negative half integers, whi h admit two equivalent to ea h other interpreta-tions. Viewed as labels of a K{indu ed representation d stands for the minimaleigenvalue of the onformal Hamiltonian (de�ned below) while (j1; j2) labelthe irredu ible representations of SU (2) � SU (2). Alternatively, referring toa H{indu ed representation, d is the onformal dimension (whi h may take allpositive values if we substitute C by its universal overing) and (j1; j2) standsfor a (2j1+1)(2j2+1) dimensional representation of the Lorentz group (a spin-tensor of 2j1 undotted and 2j2 dotted indi es). The elementary positive energyrepresentation of C are, in general, neither unitary nor irredu ible. They areproven [17℄ to be unitary (or to admit an irredu ible unitary subrepresentation)i� (a) d � 1 + j1 + j2 for j1j2 = 0 ;(b) d � 2 + j1 + j2 for j1 j2 > 0 : (2.5)For a symmetri tra eless tensor with ` indi es (i.e. for 2j1 = 2j2 = `) the ounterpart of (2.5) an be readily written for any spa e-time dimension D:d � d0 := D � 22 for ` = 0; d � D � 2 + ` for ` = 1; 2; : : : ; (2.6)d0 being the dimension of a free 0{mass s alar �eld. It is quite remarkablethat onventional relativisti (quantum) �elds �t pre isely the above de�nition(rather than transforming under Wigner's unitary irredu ible representations of5

the Poin ar�e group { f. [5℄ or [24℄). Furthermore, a free s alar, Weyl spinorand Maxwell tensor orrespond to the lower limit of the above series (a), forweights (1; 0; 0), �32 ; 12 ; 0� and (2; 1; 0) + (2; 0; 1), respe tively, while onservedsymmetri tra eless tensors (starting with the ve tor urrent �3; 12 ; 12�) are twisttwo �elds (�tting the lower limit of the series (b) in (2.5) and of (2.6)).In all the latter ases it is the subspa e of solutions of the orresponding�eld equations or onservation law that arries a unitary representation of C.Note further that the onformal Hamiltonian, the generator of the entre,U (1), of the maximal ompa t subgroup, Spin(D)�U (1) =Z2 of C(D) (S(U (2)�U (2)) for D = 4) is positive de�nite whenever the Minkowski spa e energy is( f. [23℄) and has a dis rete spe trum belonging to the set fd+n ; n = 0; 1; 2; : : :gfor the positive energy representation of C of weight (d; j1; j2).2.2 Huygens prin iple and rationality of onformally in-variant orrelation fun tionsGlobal onformal invarian e (GCI) and lo al ommutativity in four dimensional(4D) Minkowski spa e M imply the Huygens prin iple whi h an be stated inthe following strong form [19℄. Let (x) be a GCI lo al (Bose or Fermi) �eld ofweight (d; j1; j2); then d+ j1 + j2 should be an integer and(x212)d+j1+j2 ( (x1) �(x2) � (�1)2j1+2j2 �(x2) (x1)) = 0 (2.7)for all x1; x2 inM . The Huygens prin iple and energy positivity (more pre isely,the relativisti spe tral onditions) imply rationality of orrelation fun tions([19℄, Theorem 3.1).Wightmanpositivity in 4D restri ts the degree of singularities of trun ated (� onne ted) n-point fun tions (whi h an only o ur for oin iding arguments):they must be stri tly lower than the degree of the pole of the orresponding2-point propagator. This allows to determine any orrelation fun tion up toa �nite number of ( onstant) parameters. In parti ular, the trun ated 4-pointfun tion of a hermitian s alar �eld � of (integer) dimension d 2 N an be writtenin the form ([20℄ Se . 1)wt4 � wt(x1; x2; x3; x4) := h1234i � h12ih34i � h13ih24i � h14ih23i= (12)(23)(34)(14)� �13�24�12�23�34�14�d�2 Pd(s; t) (2.8)where h1 : : :ni := h0 j �(x1) : : : �(xn) j 0i ; (2.9)(12) is the free massless propagator (1.2), s and t are the onformally invariant ross-ratios (so that the prefa tor in (2.8) is a rational fun tion of �ij):s = �12�34�13�24 ; t = �14�23�13�24 ; (2.10)6

�ij are de�ned in (1.2) and Pd(s; t) is a polynomial in s and t of total degree2d� 3 (P1 (s; t) � 0): Pd(s; t) = Xi;j�0i+j�2d�3 ij si tj : (2.11)Furthermore, lo al ommutativity implies the rossing symmetry onditionss12Pd(s; t) = Pd(s; t) = s23 Pd(s; t) (2.12)where sij is the substitution ex hanging the arguments xi and xj :s12 Pd(s; t) := t2d�3Pd�st ; 1t� ;s23 Pd(s; t) := s2d�3 Pd�1s ; ts� : (2.13)Invarian e under (2.13) implies the symmetry property s13Pd(s; t) := Pd(t; s) =Pd(s; t) where s13 = s12s23s12 = s23s12s23. (The Wightman fun tion w4 ofa neutral s alar �eld is in fa t symmetri { as a rational fun tion { underthe group S4 of permutations of the four position variables xa, a = 1; 2; 3; 4.The normal subgroup Z2�Z2 of S4, however, generated by s12s34 and s14s23leaves the onformal ross ratios s and t invariant, so that only the fa tor groupS4=Z2 �Z2 ' S3 a ts e�e tively on Pd(s; t).) The number of independent rossing symmetri polynomials is hd23 i (the integer part of d23 : 1 for d = 2, 3for d = 3, 5 for d = 4).3 OPE in terms of bilo al �elds3.1 Bilo al �elds as in�nite series of symmetri tensor�elds of a given twistIt appears onvenient, at least in analysing a (trun ated) 4-point fun tion, toorganize the in�nite series of integrals of lo al tensor �elds in an OPE into a�nite sum of bilo al �elds. In order to avoid purely, te hni al ompli ations weshall exhibit the basi idea in the simplest ase of the produ t of a d-dimensionals alar �eld (x) (in M = M4) with its onjugate. We look for an expansion ofthe form �(x1) (x2) = h12i+ d�1X�=1(12)d�� V�(x1; x2)+ : �(x1) (x2) : ; (3.1)where (12) is the free massless propagator de�ned in (1.2), h12i is the 2-pointfun tion h12i = h0 j �(x1) (x2) j 0i = N (12)d ; and V�(x1; x2) is a bilo al7

onformal �eld of dimension (�; �) whi h is assumed to have an expansion in aseries of lo al (hermitian) symmetri tra eless tensor �eldsO2�;`(x; �) = O2�;�1:::�` (x) ��1 : : : ��` ; �� O2�`(x; �) = 0 (3.2)(for �� := �2��2 � �2��20 ) of dimension 2�+ ` (i.e. of �xed twist 2�). We shall makeuse of the fa t that the harmoni polynomialO2�;`(x; �) in the auxiliary variable� is uniquely determined by its values on the light one �2 = 0 [2℄. Wheneversu h an expansion is valid, it an be written quite expli itly2:V�(x1; x2) = 1X̀=0 C�` Z 10 K�`(�; �12�2)O2�;`(x2 + �x12;x12) d� (3.3)where K�`(�; z) = 1Xn=0 [�(1� �)℄`+�+n�1B(` + �; `+ �) �� z4�nn!(`+ 2�� d0)n�(�)n = �(�+ n)�(�) � ; (3.4)B is the Euler beta fun tion, �(�+ �)B(�; �) = �(�) �(�), and the d'Alembertoperator �2 a ts on x2 for �xed x12. The normal produ t in (3.1) has a similarexpansion : � (x1) (x2) : = 1X�= d ���d12 V� (x1; x2) (3.5)thus involving all higher (even) twists.Remark 2.1. A �eld V (x1; x2) is alled bilo al if it ommutes with any lo al�eld �(x3) for spa elike x13 and x23. We note that GCI does not imply theHuygens prin iple for bilo al �elds; hen e, their orrelation fun tions need notbe rational. It is noteworthy that for any even spa e-time dimension D, Vd0does have rational orrelation fun tions as will be demonstrated in Se . 3.2below using onservation of twist 2d0 tensors. We shall also see (in Se . 4. 3)that this is not the ase for V� if � > d0.In order to verify (3.3), (3.4) we need the general onformally invariantexpression for the 2- and 3-point fun tions [27℄h0 j O2�;`(x1; �1)O2�;`0(x2; �2) j 0i = Æ``0N�`(12)2�(�1R(x12)�2)`for �21;2 = 0 ; (3.6)h0 j V�(x1; x2)O2�;`(x3; �) j 0i = A�`(13)�(23)�(X312�)` (3.7)2Covariant onformal OPE were proposed in [13℄, soon after the pioneer work of Wilson[29℄. Further developments are reviewed in [9℄, [27℄, [22℄, and [14℄; we found useful the re entpresentation [10℄. 8

where we have introdu ed the symmetri tensor�12�1R(x12)�2 = �1r(x12)�2 := �1�2 � 2(�1x12)(�2x12)�12 (r(x)2 = 1I) (3.8)and the ve tor X312 = x13�13 � x23�23 of square (X312)2 = �12�13�23 : (3.9)The integrodi�erential operator in the right hand side of (3.3) is hara terizedby the property to transform a 2-point fun tion of type (3.6) into a 3-point one,(3.7) (see [10℄):Z 10 d�K�`(�; �12�2)(x12R(y(�))�)`��2�y(�) = (X312�)`(�13�23)� for �2 = 0 ; (3.10)where y(�) = x23 + �x12 = �x13 + (1 � �)x23, �y = y2 + i0y0. The aboverelations are parti ularly easy to verify for �12 = 0 as the se ond argument ofK�` then vanishes and we obtain the light- one expansion of V�.Eqs. (3.3), (3.6) and (3.10) imply that the onstants A�` in (3.7) are propor-tional to the expansion oeÆ ients C�` in (3.3). Furthermore, a simple analysisshows that only their produ t is invariant under res aling of O2�;` (for �xed V�).We emphasize that the expansion (3.3) is universal: only the oeÆ ients C�`depend on the �eld in (3.1).It follows from (3.1) that ea h V� satis�es the symmetry ondition[V�(x1; x2)℄� = V�(x2; x1) ; (3.11)taking into a ount the reality of O2�;` we dedu eC��` = (�1)`C�` : (3.12)If is hermitian then so is V�, and C�2`+1 vanish:C�` = 0 for odd ` if V�(x1; x2) = V�(x2; x1) ( � = ) : (3.13)The tensor �elds O2�;` transform under elementary representations of C. Itis onvenient to work with su h onformal �elds, sin e they are mutually orthog-onal under va uum expe tation values (see, e.g., [25℄ and referen es therein). Itfollows that ea h term in the expansion (3.1) is orthogonal to all others:h0 j V� j 0i = 0 = h0 j V�(x1; x2)V �� (x3; x4) j 0i for � 6= � ;h0 j V�(x1; x2) : (x3) �(x4) :j 0i = 0 ; (3.14)as the normal produ t (3.5) is expanded in higher twist �elds.From now on we restri t attention to the study of the OPE algebra of ahermitian GCI s alar �eld of dimension d.9

For a given (rational) trun ated 4-point fun tion we an ompute the invari-ant under res aling produ ts B�` := A�2` C�2` (3.15)(for O2�2` ! �O2�2`; A�2` ! �A�2`; C�2` ! ��1C�2`; B�` ! B�` with A andC introdu ed in (3.7), (3.3)) using the light one expansion of V�(x3; x4) in the4-point fun tionh0 j V�(x1; x2)V�(x3; x4) j 0i = (13)�(24)�f�(s; t) : (3.16)As a onsequen e of (3.13) the onformally invariant amplitude f� is s12 sym-metri : s12f�(s; t) := t��f� �st ; 1t� = f�(s; t) : (3.17)Combining (3.3), (3.7), (3.15) and using the standard integral representationfor the hypergeometri fun tion we �ndf�(0; t) = 1X̀=0B�`(1� t)2`F (2`+ �; 2`+ �; 4`+ 2�; 1� t) : (3.18)The symmetry ondition (3.17) is re e ted in a known transformation formulafor the hypergeometri fun tion:F �a; a; 2a; zz � 1� = (1� z)aF (a; a; 2a; z) for a = 2`+ � ; z = 1� t : (3.19)3.2 Crossing symmetrized ontribution of onservedtensorsThe ase � = d0 (2.6) is of parti ular interest sin e the expansion then omprisesall onserved tensors O2d0;`(x; �) =: T`(x; �) (T2 being the stress-energy tensor).The 3-point fun tions (3.7) are all harmoni in x1 and x2; hen e, so is Vd0 :�1Vd0 (x1; x2) = 0 = �2Vd0 (x1; x2) : (3.20)Applying �1 to both sides of (3.16) for � = d0 we �nd�1 n(13)d0 (24)d0 fd0 (s; t)o = 4st�12 �14 �24 (13)d0 (24)d0 � fd0 (s; t) = 0(3.21)where � = s �2�s2 + t �2�t2 + (s + t� 1) �2�s�t + D2 � ��s + ��t� : (3.22)For D = 4 the operator � has been introdu ed in [11℄ (in the ontext of N = 4supersymmetri Yang-Mills theory3). The general solution of (3.21) an be3The authors thank Emery Sokat hev for this remark.10

written in terms of the hiral variables u and v exploited in [10℄ (a similarpro edure is used in Se . 7 of [11℄; see also Appendix A to [20℄). It is given byfd0 (s; t) = g(u) � g(v)(u� v)d0 for s = uv ; t = (1� u)(1� v) : (3.23)(For eu lidean xj the variables u and v are omplex onjugate to ea h other.)For a d dimensional �eld (d > d0) we are looking for a solution f = fd0 of �f = 0(3.21) su h that the produ t of td�1f is a polynomial in s and t of overall degreenot ex eeding 2d�3 (as a onsequen e of (2.11)). It an be obtained by viewingEq. (3.21) as a Cau hy problem with initial ondition (satisfying the symmetryproperty (3.17))f(0; t) = (1 + t�d0) d�d0�1X�=0 a� (1� t)2�t� �= t�d0f �0; 1t�� : (3.24)It thus depends on d � d0 parameters a0; a1; : : : ; ad�d0�1. Noting that for s =0 = v we have t = 1� u, we an write the solution in the form (3.23) withu�d0 g (u) = f (0; 1� u) = h1 + (1� u)�d0i d�d0�1X�=0 a� u2�(1� u)� : (3.25)Proposition 2.1. The equation�1 Z 10 Kd0; `(�; �12�y) O2d0; `(y; x12) d� = 0 for y = x2 + �x12 (3.26)where K�` is given by (3.4) is ne essary and suÆ ient for the onservation ofthe (tra eless) twist 2d0 tensor O2d0`:�2�y �x12 O2d0; `(y; x12) = 0 : (3.27)Sket h of proof. We shall verify that (3.4) and (3.27) imply (3.26). (The proofof the suÆ ien y of (3.26) for the validity of (3.27) uses the same omputation.)The statement follows from the di�erentiation formula�1 f�n12O2d0; `(y; x12)g = �12( 4n�n+ `+ d0 + � ���� ++��12� �2�y �x12 + ��y�) O2d0; `(y; x12) (3.28)by integrating by parts the term involving ��� .11

In the ase D = d = 4 (d0 = 1) of interest we �nd a 3-parameter family ofsolutions:f1 (s; t) = a0 i0 (s; t) + a1 i1 (s; t) + a2 i2 (s; t) ;i0 (s; t) := 1 + t�1 ; i1 (s; t) := �1� tt �2 (1 + t � s) � 2 st ;i2 (s; t) := (1� t)4t3 (1 + t� 2s)�6s (1 � t)2t2 + s2t2 (1 + t)�1 + (1� t)2t �: (3.29)In both ases, d = 2 and d = 4 (D = 4), we an ompute B1`(d) from (3.18).The result is B1`(2) = 2a0�4`2`� (3.30)(we have used for a0 in [20℄)B1`(4) = 1�4`2`� h2a0 + 2`(2` + 1)�2a1 + (2`+ 3)(` � 1)a2�i : (3.31)The ondition for the absen e of a d = 2 s alar �eld in the OPE is given bya0 = 0 implying the vanishing of V1 for x12 ! 0:a0 = 0) V1(x1; x2) = C x�12x�12T��(x1; x2) : (3.32)The proportionality oeÆ ient C an be hosen in su h a way that T��(x) :=T��(x; x) is the stress energy tensor of the theory, normalized by the standardWard-Takahashi identity ( f. [18℄).We note that the system (3.18) is overdetermined: ea h B1` has to satisfy two onditions to �t the oeÆ ients to (1�t)2` and (1�t)2`+1. Thus the existen e ofa solution provides a non-trivial onsisten y he k. The result (3.30) was provenanalyti ally using the integral representation for the hypergeometri fun tions(see Appendix A of [20℄); Eq. (3.31) was derived analyti ally for small values of` and veri�ed numeri ally for 2` � 300.Wightman (i.e. Hilbert spa e) positivity impliesB�` = A�2` C�2` = N�2` C2�2` � 0 (3.33)(sin e C�2` is real, a ording to (3.12), while N�2` is the normalization of thepositive de�nite 2-point fun tion (3.6) so it should be positive). The onditionB1` > 0 is indeed veri�ed for non-negative a� with a positive sum. This isonly a ne essary ondition for Wightman positivity. A ne essary and suÆ ientpositivity ondition was established in the d = 2 ase; it says that (= a0) shouldbe a positive integer (see [20℄ Theorem 5.1). As the ingredients in the derivationof this result have a more general signi� an e we review them in Appendix Awith an eye to a possible generalization to the theory of the Lagrangian �eldL(x) (of dimension d = 4). 12

Knowing the singular part of the OPE (3.1) allows, after rossing sym-metrization to re onstru t the omplete 4-point fun tion. The result for thed = 2 ase is:h1234i(= h0 j �(x1)�(x2)�(x3)�(x4) j 0i)= (1 + s23 + s13)�h12ih34i+ 12(12)(34)h0 j V1(x1; x2)V1(x3; x4) j 0i�= h12ih34i+ h13ih24i+ h23ih14i+ wt(x1; : : : ; x4) (3.34)where, in the onventions of [20℄ (for d = 2),hj`i = 2(j`)2 ; �(j`) = 14�2�j` ; j < `� ; (3.35)wt(x1; x2; x3; x4) = f(12)(23)(34)(14) + (13)(23)(24)(14)+ (12)(13)(24)(34)g (3.36)for h0 j V1(x1; x2)V1(x3; x4) j 0i = f(13)(24) + (14)(23)g : (3.37)Remark 2.2. The fa tor 12 in the se ond term in the bra es in Eq. (3.34)re e ts a spe ial ase of the following phenomenon. In general, we wish thatthe symmetrized ontribution of V�(x1; x2) to the trun ated 4-point fun tion,F�(x1; x2; x3; x4) = S �(12)d��(34)d��h0 j V�(x1; x2)V�(x3; x4) j 0i ; (3.38)where S stands for an yet unspe i�ed symmetrization, � = 1; : : : ; d� 1, is notjust symmetri under any permutation of its arguments but that the di�eren eF�(x1; x2; x3; x4) � (12)d��(34)d��h0 j V�(x1; x2)V�(x3; x4) j 0iis of order (12)d��(34)d��s (or smaller) for s! 0. This se ond ondition for esus to use the standard symmetrization h12ih34i 7! (1 + s23 + s13)h12ih34i =h12ih34i + h13ih24i + h14ih23i for the dis onne ted term in the middle partof (3.34) while applying 12 (1 + s23 + s13) to the se ond term. We shall seein Se .4.2 below that there exist a basis of rational harmoni fun tions whosesymmetrized ontribution satisfying the above ondition is proportional to thestandard symmetrization. We further note that the fun tion F1, whi h is ra-tional by itself, an be viewed as providing a minimal model for the trun ated4-point fun tion of �. 13

4 General form of the 4-point fun tion of L(x).Operator ontent4.1 s- hannel ontributions to the 4-point fun tion for ar-bitrary twistsWe shall now exploit a result of Dolan and Osborn [10℄ whi h permits to writedown losed form expressions for f�(s; t) (3.16) given the rational 4-point fun -tion for d = 4(= D). More pre isely, keeping in mind the analysis of s- hannelpositivity we shall study the di�eren ew (x1; x2; x3; x4) � h12ih34i = h13ih24i+ h14ih23i+ wt (x1; x2; x3; x4) == (2�)�8 (�12�23�34�14)�2 h 1stP (s; t) + N2s2 �t2 + t�2�i : (4.1)with P (s; t)(= P4(s; t)), a polynomial in s and t of overall degree 5 satisfyingthe rossing symmetry onditions (2.12)(2.13) (see (4.28) below). The rationalfun tion in the bra kets in the right hand side of (4.1) an be expressed as anin�nite sum of even twist ontributionsN2 s3 t �t2 + t�2� + P (s; t) = t3 1X�= 1 s��1 f� (s; t) ; (4.2)thus extrapolating (3.16) beyond the range � = 1; 2; 3. Albeit f� are, in general,not rational fun tions of s and t they are determined by the polynomial P andthe onstant N2. Indeed Eq. (3.10) of [10℄ allows to write down the followingextension of (3.23) (d0 = 1) to any � > 1:f� (s; t) = 1u� v nF (�� 1; �� 1; 2�� 2; v) g� (u) ��F (�� 1; �� 1; 2�� 2; u) g� (v)o : (4.3)We note that the normalization N ontributes to � � d (= 4) only. Thenit has to be taken into a ount when verifying Wightman positivity ondition(3.33). The hypergeometri fun tions F (a; a; 2a; v), a = 1; 2; : : :, are expressedas linear ombinations of log(1 � v) with rational in v oeÆ ients:F (1; 1; 2; v) = 1Xn=1 vn�1n = 1v ln 11� v ;F (2; 2; 4; v) = 1Xn=1 6n vn�1(n+ 1) (n+ 2) = 6v2 �2� vv ln 11� v � 2� ; et :The OPE of the bilo al �eld V�(x1; x2) in terms of twist 2� rank 2` symmetri tra eless tensors orresponds, a ording to (3.18), to an expansion of g�(u) interms of hypergeometri fun tions of the same type:g� (u) = uf� (0; 1� u) = 1X̀=0 B�` u2 `+1 F (2` + �; 2` + �; 4` + �; u) (4.4)14

(Eq. (4.4) is obtained from (4.3) just using g�(0) = 0, F (a; a; 2a; 0) = 1.) Thefun tions f�(0; t) an be determined re ursively from the relationsf� (0; t) = lims!0 �s1�� �t�3 P (s; t)� ��1X�=1 s��1 f� (s; t)�� ;f1 (0; t) = t�3 P (0; t) : (4.5)and (4.3) whi h imply that the operator bh,bh t�3 P (s; t) := 1u�v� u(1�u)3 P (0; 1�u)� v(1�v)3 P (0; 1�v)�=f1 (s; t)(4.6)de�nes a (rational) harmoni proje tion of the produ t t�3P (s; t). In fa t, thefun tion (13)(24)f1(s; t) is harmoni in both x12 and x34 for �xed x23 while itsdi�eren e with (13)(24)t�3P (s; t) vanishes on either light one (�12 = 0 and�34 = 0) sin e t�3P (0; t) = f1(0; t) a ording to (4.5). Thus Eq. (4.3) an beviewed as an extension of the notion of harmoni proje tion to arbitrary twists.4.2 A basis of rossing symmetrized onformal harmoni fun tionsWe shall now display a basis j�(s; t) of rational solutions of the onformal Lapla eequation (3.21),� j� (s; t) = 0 ; � = 0; 1; 2 (� = s �2�s2 +t �2�t2 +(s+t�1) �2�s�t +2 ��s +2 ��t ) :(4.7)that are symmetri under s12t�1 j� �st ; 1t� = j� (s; t) (4.8)and are eigenfun tions of the operatorbh (1 + s23 + s13) : (4.9)We shall verify that su h a basis is given byj� (s; t) = i� (s; t) ; � = 0; 1 ; j2 (s; t) = i1 (s; t) + i2 (s; t) ;t3 j2(s; t) = (1 + t3)[(1 + s � t)2 � s℄� 3 s (1 � t) ; (4.10)where i� are de�ned in (3.29). It will be ome lear on the way that i2 is not aneigenve tor of the operator (4.9) as j1 and j2 orrespond to di�erent eigenvalues.Set indeed I� = (1 + s23 + s13)�t3 i� (s; t) ; � = 0; 1; 2 (4.11)15

we �ndI0(s; t) = s2(1 + s) + t2(1 + t) + s2t2(s + t) ;I1(s; t) = s(1 + s)(1� s)2 + t(1 + t)(1 � t)2 ++ st[(s � t)(s2 � t2)� 2Q1℄ ; Q1 := 1 + s2 + t2I2(s; t) = (1 + s)(1 � s)2(2� 3s + 2s2) + (1 + t)(1� t)2(3� 3t+ 2t2)� 2 ++st �4Q1 � (s + t) �5s2 � 8st+ 5t2�� : (4.12)The simplest way to ompute the harmoni proje tion bh �t�3I�(s; t)� onsistsin looking at the \initial onditions" for s = 0. We see that while I� (0; t) =t3i� (0; t) for � = 0; 1 , we haveI2(0; t) = 2 (1�t)4(1+t)+t(1�t)2(1+t) = t3 (2 i2(0; t) + i1(0; t)) : (4.13)It follows that i2 is not an eigenfun tion of the operator (4.9) but j2 (4.10) isone (with eigenvalue 2). This suggests introdu ing a new basis of symmetrizedtwist two polynomial fa tors in the trun ated 4-point fun tion:J�(s; t) (= I� (s; t)) = (1 + s23 + s13) �t3j� (s; t)� ; � = 0; 1;J2(s; t) = 12 (1 + s23 + s13) �t3j2 (s; t)� = 12 (I1 (s; t) + I2 (s; t)) == (1 + t3) [(1 + s� t)2 � s℄ � 3 s (1� t) + s3 [(1� s + t)2 � t℄ ; (4.14)the J� are distinguished by the propertiesbh �t�3J�(s; t) = j� (s; t) ; � = 0; 1; 2;J0 (s; t) � t3 j0 (s; t) = s2 (1 + t3) + s3 (1 + t2)J1 (s; t) � t3 j1 (s; t) = s (1� t)(1� t3) � s2 (1 + t3) � s3 (1 + t)2 + s4 (1 + t)J2 (s; t) � t3 j2 (s; t) = s3 (1 + t+ t2)� 2s4 (1 + t) + s5 : (4.15)Comparing (4.15) with (4.5) we dedu e that the di�eren es J1 � t3j1, J0 �t3j0, and J2 � t3j2 involve twist 4 and higher, twist 6 and higher, and twist 8and higher, respe tively. The expressions (3.29) for f1 and the orresponding rossing symmetri amplitudeF1 an be rewritten in the basis (4.10) (4.14) withthe result f1 (s; t) = a0 j0 (s; t) + a12 j1 (s; t) + a2 j2 (s; t) ; a12 = a1 � a2 ;F1 (x1; x2; x3; x4) = (12)2 (23)2 (34)2 (14)2 1st ��fa0 J0 (s; t) + a12 J1 (s; t) + a2 J2 (s; t)g : (4.16)We note that the trun ated 4-point fun tion of the (va uum) Maxwell La-grangian, L0(x) = �14 : F��(x)F ��(x) : ; (4.17)16

(or a sum of mutually ommuting expressions of this type) is a spe ial ase of(4.16). Here F�� is a free ele tromagneti �eld with 2-point fun tionh0 j F�1�1(x1)F�2�2(x2) j 0i = 4D�1�1�2�2(x12)= f��1(��2��1�2 � ��2��1�2 )� ��1(��2��1�2 � ��2��1�2)g(12) ; (4.18)we have4�2D�1�1�2�2(x) = R�1�2 (x)R�1�2(x)� R�1�2(x)R�1�2 (x) ; (4.19)the symmetri tensor R(x) being de�ned in (3.8). The OPE of the produ t oftwo L0's has the formL0(x1)L0(x2) = h12i0 + 8�2�312 T (x1; x2;x12)+ : L0(x1)L0(x2) : (4.20)where h12i0 = 3(��12)�4, and T (x1; x2;x12) (a multiple of V1) is the harmoni bilo al �eldT (x1; x2; x12) = 14 : F �� (x1)F�� (x2) : x212 � x�12 : F ��(x1)F��(x2) : x�12 == 14 x 212 (r12)�� (r12)�� : F��(x1)F�� (x2) : : (4.21)In verifying �1T (x1; x2;x12) = 0 one should use both the tra elessness of T ,i. e. �yT (x1; x2; y) = 0 and the two Maxwell equations,dF (x) = 0 = d �F (x) ; (4.22)where F (x) is the 2-form (1.5). The se ond expression for T (x1; x2; x12) (4.21)makes obvious its onformal invarian e. The 4-point fun tion of L0 is omputedfrom (4.17){(4.19) using repeatedly the triple-produ t formula (of [22℄)r(x12)r(x23)r(x13) = r(X123) ; X123 = x13�13 � x12�12 (4.23)(see Appendix B of [20℄). The ontribution of T to the 4-point fun tion is givenby the solution (3.29) of (3.21):�4h0 j T (x1; x2;x12)T (x3; x4;x34) j 0i = (13)(24)f1(s; t)for a0 = 0 ; a1 = a2 ; (4.24)it follows that wt4 from (4.1) assumes the formwtL0(x1; x2; x3; x4) = (1 + s12 + s23) 0tr(D12D23D34D14)= 08�8 (�12�23�34�14)�2 (st)�1 J2 (s; t) (4.25)D12 := D�1�1�2�2(x12) ; D23 := D�2�2�3�3(x23) et : (4.26) orresponding to a1 = a2 = 25 0, a0 = a12 = 0. In fa t, a �nite sum ofexpressions of type (4.17) will give rise to a dis rete subset of su h values witha positive integer 0. 17

4.3 General form of wt4. Higher twist ontributionsThe most general GCI trun ated 4-point fun tion of a d = 4 s alar �eld ontains,in addition to F1 (4.16), two more terms orresponding to rossing symmetrizedtwist 4 ontributions. They an be written as s tQi(s; t), i = 1; 2, where Qi are rossing symmetri polynomials of degree 2:Q1 = 1 + s2 + t2 ; Q2 = s + t+ st ;t2Qj �st ; 1t� = Qj(s; t) = s2Qj �1s ; ts� ; j = 1; 2 : (4.27)Thus, the polynomial P (s; t) of Eq. (4.1) an be presented in the form:P (s; t) = a0J0(s; t) + a12J1(s; t) + a2J2(s; t) + s t [b1Q1(s; t) + b2Q2(s; t)℄ :(4.28)Remark 3.1. The above basis of rossing symmetri polynomials J� ; Qj andthe P�; Qj basis of [20℄, whereP0(s; t) = 1 + s5 + t5 ; P1(s; t) = s(1 + s3) + t(1 + t3) + st(s3 + t3) ;P2(s; t) = s2(1 + s) + t2(1 + t) + s2t2(s + t) (= I0(s; t)) ; (4.29)are related by:J0 = P2 ; J1 = P1 � P2 � 2 s tQ1 ; J2 = P0 � 2P1 + P2 + s tQ1 : (4.30)We shall now apply the pro edure outlined in Se .4.1 to ompute the twist 4and 6 ontributions in (4.28)). It follows from (4.4) (4.15) and (4.28) thatf2 (0; 1�u) = a12�1�u+ 1(1�u)3�+ (b1�a12)�1+ 1(1�u)2�+ b2 11�u �� g2 (u)u = 1P`=0B2` u2` F (2` + 2; 2` + 2; 4` + 4; u) (4.31)yieldingB2` = �4`+ 12` ��1 h` (`+ 1) (2`+ 1) (2` + 3) a12 + 2 (`+ 1) (2`+ 1) b1 + b2i :(4.32)We see, in parti ular, that the absen e of a s alar �eld of dimension 4 in theOPE of two L's impliesB20 = 2 b1 + b2 = 0 () h123i = 0) : (4.33)The di�eren e between P (s; t) (4.28) and the twist 2 (Eq. (4.16)) and twist 4part, omputed from (4.3), de�nes t3f3(0; t) as the oeÆ ient to the s2 term:P (s; t)� t3[f1(s; t) + sf2(s; t)℄ = s2t3f3(0; t) +O(s3) : (4.34)18

A omputer aided al ulation (using Maple) givesg3 (u)u = 1P`=0 B3` u2` F (2`+ 3; 2`+ 3; 4` + 6; u) == 12 a12 + a0 + 12 b2 � b11� u + 12 b1 + 2 b2(1� u)2 + 12 a12 + 2 a0(1� u)3 �� 12 (2 b1 + b2)�1u + 2u2� � (2 b1 + b2) ln (1� u)u3 ; (4.35)with B3` = 12�4`+32`+1��1 h (`+ 1) (`+ 2) (2`+ 1) (2`+ 3) (2a0 + a12) ++2 (`+ 1) (2`+ 3) (b1 + 2b2)� 2b1 � b2i : (4.36)Remarkably, g3 (u) is rational pre isely when the ondition (4.33), re e ting theele tri -magneti duality, is satis�ed.5 Impli ations for the gauge �eld Lagrangian5.1 Restri tions fromOPE, Hodge duality, and WightmanpositivityAs dis ussed in the Introdu tion the gauge �eld Lagrangian (1.3) is hara terizedby the absen e of s alar �elds of dimension 2 and 4 in the OPE of two L's.A ording to (3.32) and (4.33) this impliesa0 = 0 = 2b1 + b2: (5.1)This allows to write the polynomial P (s; t) in (4.1) asP (s; t) = a0J1(s; t) + aJ2(s; t) + b[Q1(s; t) � 2Q2(s; t)℄ (5.2)where we have set a2 = a, a12 = a0, b1 = b(= �b22 ); the di�eren eQ1(s; t) � 2Q2(s; t) = (1� s � t)2 � 4 s t (5.3)is hara terized by having a (se ond order) zero at t = 1 for s = 0 (and beingnegative for eu lidean xij).The restri tions on the three remaining onstants, a, a0, and b, oming fromthe positivity ondition (2.33) for Bkl given by (3.31) (4.32) and (4.36) area+ a0 > 0; a; a0 � 0; (5.4)(` + 1)(2` + 1) a0 + 2b � 0, for l > 0; (` + 2)(2` + 1) a0 � 6b for ` � 0. The lasttwo inequalities imply �3 a0 � b � 13 a0 : (5.5)19

In parti ular, if a0 = 0 it would follow that also b = 0 and we would end upwith the trun ated 4-point fun tion that is a multiple of the one of the freeele tromagneti Lagrangian (4.17).Remark 4.1. If we allow for a positive a0 in the 4-point fun tion (in luding,say, a s alar �eld ontribution in the Lagrangian) then, a ording to (4.36), therestri tion (5.5) would be repla ed by a more general one�3 a0 � b � 13 (2 a0 + a0) (5.6)whi h leaves room for a (non-zero) positive b even for a0 = 0. A numeri alanalysis indi ates that the most general GCI 4-point fun tion involves a non-trivial open domain in the 5-dimensional proje tive spa e of the parameters a� ,bj and the 2-point fun tion normalization,N2, de�ned up to a ommon positivefa tor, in whi h Wightman positivity is veri�ed for all twists.5.2 DiÆ ulties in exploiting the ompositeness of LIt is instru tive to understand the restri tions, oming from (5.1), within anaxiomati treatment of non-abelian gauge �eld theory. At the same time we shalltry to answer the question: an we extra t more information from the expression(1.3) for L in terms of F�� than just saying that the OPE of L(x1)L(x2) ontainsneither L nor a s alar �eld of dimension 2?In (perturbative) quantum ele trodynami s amplitudes with an odd numberof photon external legs (and no external harged parti les) vanish be ause of harge onjugation invarian e (Furry's theorem). The general onformally in-variant 3-point fun tion of a Maxwell �eld F��(x), on the other hand, violateslo al ommutativity of Bose �elds and should hen e be also set equal to zero(without having to assume any dis rete symmetry). This last argument fails fora non-abelian gauge �eldF (x; !) := 12!��F a��(x)ta ; [ta; tb℄ = ifab t ; (!�� = �!��) (5.7)where ta are orthonormal hermitianmatri es generating the de�ning representa-tion of a ( ompa t, semi-simple) non-abelian gauge group G, fab is the totallyantisymmetri tensor of (real) stru ture onstants of the Lie algebra G of G(fab = "ab , the Levi-Civita tensor for G = su(2), ta = 12 �a, a; b; = 1; 2; 3), !is a onstant skew-symmetri tensor (or di�erential form) introdu ed for nota-tional onvenien e. While the general (gauge and) onformally invariant 2-pointfun tion of F a(x; !) = 12 !��F a��(x) oin ides with the free Maxwell one, (4.18),h0 j F a(x1; !1)F b(x2; !2) j 0i = NF ÆabD(x12;!1; !2) (5.8)where we have introdu ed a ontra ted form of (4.19),4�2D(x;!1; !2) = R�1�2 (x)R�1�2(x)!�1�11 !�2�22 ; (5.9)20

there is a 2-parameter family of lo al gauge and onformally invariant 3-pointfun tions: W ab (x1; !1; x2; !2; x3; !3) = fab (N1W (1) +N2W (2)) ; (5.10)W (1)(x1; !1; x2; !2; x3; !3) = (X123�1R�1�3(x13)X231�2R�2�3(x23)� X123�1R�1�2 (x12)X312�3R�2�3(x23)+ X231�2R�1�2(x12)X312�3R�1�3(x13))!�1�11 !�2�22 !�3�33 ; (5.11)W (2)(x1; !1; x2; !2; x3; !3) = R�1�2 (x12)R�1�3 (x13)R�2�3(x23)!�1�11 !�2�22 !�3�33 ; (5.12)where R�� and X312 are de�ned in (3.8) and (3.9).The expression (5.10) an, sure, be used to produ e a non-zero 3-point fun -tion of L (1.3). Thus, the se ond ondition (5.1) is not, at �rst sight, an au-tomati onsequen e of (1.3), onformal invarian e and lo ality, Hodge dualityproviding an independent restri tion on orrelation fun tions. It turns out, how-ever, that the ombination of (5.8) and (5.10) implies the existen e of a �eld Iin the OPE of F (x1) F (x2) transforming under an inde omposable represen-tation of C (whi h would severely ompli ate the use of onformal invarian e).This statement follows from the observation that the 3-point fun tion (5.10)does not satisfy the free Maxwell equation d�F = 0 for any non-zero hoi e ofN1 and N2 while the 2{point fun tion does:h0 j d�F (x1) F (x2) j 0i = 0;h0 j d�F (x1) F (x2) F (x3) j 0i 6= 0 : (5.13)It follows that the OPE of F (x2) F (x3) should involve a lo al �eld I thatis not orthogonal to d�F . I annot be a derivative of F be ause of the �rstequation (5.13). On the other hand, I annot be an elementary onformal �eldtransforming under an inequivalent representation of C sin e then it should againbe orthogonal to F , and hen e to d�F thus violating the se ond equation (5.13).Thus the vanishing of odd point orrelation fun tions of L is a natural prop-erty of the Lagrangian (1.3) if the lo al observable algebra is spanned by ele-mentary onformal �elds (and their derivatives).Remark 4.2. Let Aa�(x), a = 1; 2; 3 be three ommuting purely longitudinalgauge potentials, { i.e., generalized free �elds su h thath0 j Aa�(x1)Ab�(x2) j 0i = 12 Æabr��(x12)(12) = Æab8�2 R��(x12) ;��Aa�(x) = ��Aa�(x) : (5.14)21

Then both the 2-point fun tion (5.8) and the 3-point fun tion (5.12) are repro-du ed by the orresponding su(2) Yang-Mills urvature tensorF a��(x) = ��Aa�(x)� ��Aa�(x)� g"ab : Ab�(x)A �(x) := �g"ab : Ab�(x)A �(x) : : (5.15)This example is already ex luded, however, by our requirement that no d = 2s alar �eld appears in the OPE of two L's (neither does F (5.15) satisfy theYang-Mills equation with onne tion A). Even for the more general d = 4 omposite �eldL��(x) = � : (Aa�(x)A�a(x))2 : �� : Aa�(x)A�b (x)Ab�(x)A�a(x) : (5.16)(whi h in ludes (1.3) with F given by (5.15) for � = � = g24 ) we �nd that theleading term in the OPE of two L's,L��(x1)L��(x2) � 4(14�2 � 16�� + 11�2)(12)3(Aa�(x1)r��(x12)Aa�(x2)); (5.17)involves a s alar �eld of dimension d = 2 (with the same oeÆ ient as the 2-point fun tion of L�� { whi h is non-zero for any not simultaneously zero real�; �).5.3 Con luding remarksWe have presented in the pre eding se tions two intertwined developments:(i) a systemati study of the theory of a GCI lo al s alar �eld of any (integer)dimension d in terms of bilo al �elds V�(x1; x2) of dimension (�; �) appearing inthe OPE �(x1) (x2) (3.1) (3.5); (ii) �rst steps in an attempt to onstru t ina non-perturbative, axiomati approa h a onformally invariant �xed point of agauge �eld theory, formulated entirely in terms of gauge invariant lo al observ-ables of dimension 4: the Lagrangian density L(x) and the stress-energy tensorT��(x) or, rather, its polarized bilo al ounterpart that determines V1(x1; x2)a ording to (3.32).(i) The use of bilo al �elds simpli�es substantially the analysis of the op-erator ontent of GCI orrelation fun tions. In parti ular, V1(x1; x2), de�nedas the (in�nite) sum of twist 2 onserved symmetri tensor �elds, satis�es as a onsequen e the d'Alembert equation (3.20). This allows to ompute the ontri-bution of the orrelator h0 j V1(x1; x2)V1(x3; x4) j 0i to the (trun ated) 4-pointfun tion wt4 of L. (Its expression for the d = 2 neutral s alar �eld � has been omputed in [20℄.) A minimal model for the onne ted part of the 4-point fun -tion of a general neutral s alar �eld of dimension 4 is given by F1 (4.16) whi h isdetermined by its harmoni proje tion. A general pro edure is outlined - basedon the Dolan-Osborn formula (4.3) - for omputing the expe tation values (3.16)of V�. The ase � = 1 is distinguished by the fa t that the onformal harmoni fun tion f1(s; t) (and hen e, the orresponding symmetrized ontribution F1)22

is rational. For � > 1 f�(s; t) are linear ombinations (with rational fun tion oeÆ ients) oflog(1� u) and log(1� v) for s = uv; s + 1� t = u+ v : (5.18)We have displayed these fun tions (taking into a ount also the additional termb1Q1 + b2Q2 in (4.28)) and the asso iated (in�nite) OPE expansions in termsof symmetri tra eless tensor �elds of twist 2� for � = 1; 2; 3 (see Eqs. (3.31),(4.31)(4.32), (4.35)(4.36), respe tively). The same pro edure applies to higher� as well, when f� also depend on the sum of produ ts of 2-point fun tionsh13ih24i+ h14ih23i (whi h introdu es one more paprameter, the normalizationN of the 2-point fun tion). The resulting expressions (together with other omputer aided results - on erning 6-point fun tions) will be presented in aforth oming publi ation, in ollaboration with K.-H. Rehren.(ii) Conditions (5.1) ex lude ontributions of s alar �elds of dimension 2 and4 in the OPE of L(x1)L(x2) leaving us with a 3-parameter family of trun ated4-point fun tions. We assert that L(x) then has the properties of the Lagrangiandensity of a gauge �eld urvature (without matter �elds). The study of Wight-man positivity for twists 4 and 6 ontributions leads to a rather strong onstraint(5.5) for the remaining parameters. Should, in parti ular, the analysis of the2n-point fun tion of L for n � 3 yield the onstraint a0 = 0 Eq.(5.5) wouldalso imply b = 0 and leave us with a multiple of the 4-point fun tion of theLagrangian of a free abelian gauge �eld. This would on�rm the general belief(see, e.g. [12℄ [30℄) that a (pure) non-abelian gauge theory ne essarily involvesa mass gap, thus violating onformal invarian e. By ontrast, if we do notimpose (5.1) { i.e., if we allow for the presen e of (at least) a s alar �eld inthe Lagrangian, a full a ount of Wightman positivity for the 4-point fun tionappears to allow for an open set of the 5-dimensional (proje tive) parameterspa e.The eviden e that we are displaying a gauge invariant 4-point fun tion ina (non-abelian) gauge theory is rather indire t. One veri�es on a ase by asebasis that any other renormalizable Lagrangian would involve �elds of d < 4 inthe OPE and that is ex luded by ondition (5.1). The diÆ ulty in identifyingthe theory in terms of basi (gauge dependent) �elds like F a�� (brie y reviewedin Se . 5.2) lies in the fa t that the model we are trying to onstru t is ne es-sarily non-perturbative (if it exists at all). From this point of view our modelis not in ompatible with the N = 4 supersymmetri Yang-Mills theory (see e.g.[1℄, [3℄, [4℄, [11℄ and referen es therein) for suÆ iently large value of the ou-pling onstant g, su h that the anomalous dimension of the Konishi �eld (whi happears also in the OPE of two �elds of the supermultiplet of the stress-energytensor) is a positive even integer.A knowledgments. The authors thank Dirk Kreimer for a stimulating dis- ussion and Karl-Henning Rehren for an enlightening orresponden e. N.N. andI.T. a knowledge the hospitality of the Erwin S hr�odinger International Insti-tute for Mathemati al Physi s (ESI) as well as partial support by the Bulgarian23

National Coun il for S ienti� Resear h under ontra t F-828. The resear hof Ya.S. was supported in part by I.N.F.N., by the EC ontra ts HPRN-CT-2000-00122 and -00148, by the INTAS ontra t 99-0-590 and by the MURST-COFIN ontra t 2001-025492. All three authors a knowledge partial supportby the Resear h Training Network within the Framework Programme 5 of theEuropean Commission under ontra t HPRN-CT-2002-00325 and by a NATOlinkage grant PST.CLG.978785. I.T. thanks l'Institut des Hautes Etudes S i-enti�ques (Bures-sur-Yvette), the Theory Division of CERN and Se tion deMath�ematiques, Universit�e de Gen�eve for hospitality during the �nal stage ofthis work.

24

Appendix A. Va uum representation of the alge-bra generated by the harmoni bilo al �eldV1. The ase d = 2There are two main ingredients in the proof of the entral result, Theorem 5.1, of[20℄ to be reviewed in the two se tions of this Appendix. The �rst is a ( omputeraided) study of the (5- and) 6-point fun tion of the basi �eld �(x) of dimension2, ombined with the expansion (3.3) (for � = 1) and the onservation law forthe twist two �elds T`(x; �) (` = 2; 4; : : :). Here we sum up a modi�ed version ofthe argument whi h uses Proposition 2.1. The (te hni al) diÆ ulty of su h ananalysis in reases drasti ally with in reasing the dimension of the underlyings alar �eld and it has not been ompleted for d = 4. The se ond ingredientis quite general: it uses the dis rete mode expansion of V1 with respe t to the onformal Hamiltonian whi h an be arried out for any d.A.1 Analysis of 5- and 6-point fun tions for d = 2The general GCI and rossing symmetri 5-point fun tion of �(x) (for d� = 2)involves two independent terms: the sum of twelve 1-loop graphsw(1) = 2 X�2Perm(2;:::;n)(1�2) p q�2�3 : : : p q�n�1�n (1�n) for n = 5 (A.1)where (ij) is de�ned in (2.9) andp q�i�j= (min(�i; �j);max(�i; �j)) ; (A.2)and a sum, w(2), of 10 produ ts of 7 fa tors ea h:w(2) = � X1�i<j�5�ij Y1�k�5i6=k6=j pqik pqkj : (A.3)A rather nasty ( omputer aided) al ulation shows that the 5-point fun tion ofthree �'s and a V1 satis�es the d'Alembert equation in the last two arguments�j h0 j �(x1)�(x2)�(x3)V1(x4; x5) j 0i = 0 for j = 4; 5 (A.4)i� � = 0. In view of Proposition 2.1 (A.4) is equivalent to demanding the in�niteset of onservation laws�2�x�4��� h0 j �(x1)�(x2)�(x3)T2`(x4; �) j 0i = 0 : (A.5)Similarly, only the rossing symmetri sum of �12 � 5! = 60� 1-loop graph ontributions to the trun ated 6-point fun tion is onsistent with T2` onserva-tion. The 1-loop expression for the 6-point fun tion allows to prove that the25

limit V (x1; x2) = lim�13!0�23!0f(2�)4�13�23(�(x1)�(x2)�(x3)� (13)�(x2) � (23)�(x1)� h123i)g (A.6)exists, does not depend on x3 and de�nes a harmoni in ea h argument bilo al�eld V1(x1; x2); moreover, the trun ated n-point fun tions of � will be given by(A.1) for all n (see [20℄, Proposition 2.3).Remark A.1. The above sket hed analysis for the 5-point fun tion has been arried out in the d = 4 ase, too, with the following results. There are 37independent GCI and rossing symmetri 5-point fun tions of L ( ompared to 2for d = 2(!)). After imposing onservation of the twist 2 tensors T2`(x; �) thereremain only 8, just 1 among them a tually ontributing to the twist 2 part ofthe OPE. The latter is non-zero only if the ondition (5.1) is violated and thenit yields a2 = 0.The analysis of the general 6-point fun tion has to deal with 31990 (insteadof 8 for d = 2) independent stru tures (most of them onsisting of 6! = 720terms ea h).A.2 Analyti ompa t pi ture �elds. Conformal Hamilto-nian. Mode expansionsCompa ti�ed Minkowski spa eM = (S1�S3)=Z2 (A.7)(see [8℄) has a onvenient realization in terms of (eu lidean) omplex 4-ve tors([26℄):M = fz� = e2�i�u�; � = 1; 2; 3; 4; � 2 R; u2S3; i.e. u2 := u2 + u24 = 1g : (A.8)Real Minkowski spa e M is mapped on an open dense subset of M :M 3 (x0;x)! z = !�1(x)x ; z4 = 1� x22!(x) ; !(x) = 1 + x22 � ix0 : (A.9)M an be obtained by adding to the image of M the 3- one at in�nity:K1 = fz 2M ; 1 + 2z4 + z2(= 2(u4 + os 2��)e2�i�) = 0g : (A.10)Note that Eq. (A.9) an be interpreted as the Cayley map from the Lie algebrau(2)(' R4) to the group U (2) { f. [28℄.We note that the map (A.9) extends to omplex arguments x 7! x+ iy andis regular in the forward tube (2.1) whi h is mapped (for any D) on the domainT+ =nz 2 CD : ��z 2��< 1; 2 jzj2 �= DX�=1 jz�j2� < 1 + ��z 2��2o: (A.11)26

where z2 = h�1� y0 + ix0�2 + (x + iy)2i h�1 + y0 � ix0�2 + (x + iy)2i�1 . Themaximal ompa t subgroup K (= K (4)) a ts by linear homogeneous transfor-mations on z�: SU (2) � SU (2) is represented by real SO (4) rotations of theve tor (z�) while the U (1) fa tor a ts by phase transformations: z� 7! ei�z�.Thus the point z = 0 2 T+ (the image of (x+ iy) = (i; 0) 2 T+) is left invariantby K.A s alarM -spa e �eld �M(x) of dimension d is related to its z-pi ture oun-terpart, �(z) (for z = z(x) given in (A.9)) by:�M(x) = [2�!(x)℄�d�(z(x)) : (A.12)The numeri al fa tor 2� in (A.12) is hosen for onvenien e so that the freemassless s alar propagator assumes a simple z-pi ture form(12) = 1z212 : (A.13)A z-pi ture s alar �eld �(z) of dimension d transforms in su h a way thatthe form �(z)(dz2)d=2 remains invariant. (The transformation properties of theLagrangian L(z), for d = 4, an be read o�, alternatively, from the invari-an e of the volume form L(z) dz1^ dz2^ dz3^ dz4 - f. (1.6).) This implies,in parti ular, that orrelation fun tions are invariant under omplex eu lideantransformations. The onformal Hamiltonian H (that generates translations inthe onformal time �) a ts on �(z) and on the bilo al �elds V�(z1; z2) a ordingto the law [H;�(z)℄ = �d+ z ��z��(z) ;[H;V�(z1; z2)℄ = �2�+ z1 ��z1 + z2 ��z2�V�(z1; z2) : (A.14)Here V� are again related to � by an expansion of type (3.1) with (12) substitutedby its z-pi ture ounterpart (A.13).The mode expansion of V1(z1; z2) is, in parti ular, an expansion in homoge-neous harmoni polynomials:V1(z1; z2) = Xn;m2ZVm�1n�1(z1; z2) ;Vm�1n�1(�1z1; �2z2) = Vm�1n�1(z1; z2)�m1 �n2 for �1; �2 2 C � (A.15)�1Vmn(z1; z2) = 0 = �2Vmn(z1; z2) for �a = 4X�=1� ��z�a �2 : (A.16)For a hermitian s alar �eld � the modes Vmn have the following simple onju-gation property. For m;n � 0V�m�1;�n�1(z; w) = V �1 :::�m�1:::�n�m�1;�n�1 z�1 : : : z�mw�1 : : :w�n (A.17)27

where V �1:::�m�1:::�n�m�1�n�1 are symmetri tra eless tensors in �1; : : : ; �m and in �1; : : :,�n (separately) whi h are mapped under hermitian onjugation into tensors withthe same properties, (V �1:::�m�1:::�n�m�1;�n�1 )� = V �1 :::�m�1:::�nm+1n+1 (A.18)so thatVm+1n+1(z; w) = 1z2w2V �1:::�m�1:::�nm+1n+1 z�1z2 : : : z�mz2 w�1w2 : : : w�nw2 : (A.19)We are interested in the va uum representation of the algebra of V1 modesfor whi h Vmn j 0i = 0 = h0 j V�m�n if either m � 0 or n � 0 : (A.20)For d = 2 the algebra generated by Vnm(z; w) oin ides with a entral exten-sion bsp(1;R) of the in�nite symple ti Lie algebra. This is parti ularly simpleto see when the (invariant under res aling) stru ture onstant is a natural num-ber: := 8 h12ih13ih23i(h123i)2 = N 2 N (A.21)and V1(z1; z2) = NXj=1 : 'j(z1)'j(z2) : (V1(z; z) = 2�(z)) ; (A.22)where 'j(z) are free ( ommuting for di�erent j) massless s alar �elds. Themodes 'n(z) (n 2Z) of ea h '(z) generate the in�nite Heisenberg algebra:'n(z) = e�2�i(n+1)�'n(u) ; ['n(u); 'm(v)℄ = Æn;�m "(n)C1jnj�1(uv) (A.23)where "(n) = signn�= njnj ; "(0) = 0� C1n are the Gegenbauer polynomials gen-erated by the 2-point fun tion of ':C1n�1( os 2��) = sin 2�n�sin 2�� ; 1z212 = 1z21 (1� 2xz + z2) = 1z21 1Xn=0 znC1n(x) ;z =sz22z21 ; x = uv : (A.24)It is well known that the quadrati ombination of the generators of a Heisenbergalgebra give rise to a symple ti Lie algebra. The entral extension omes, asusual, from the normal produ ts.The algebra bsp(1;R) of Vnm(u; v) has a simple diagonal subalgebra, gener-ated by vnm := Vnm(u; u), u 2S3,[vn1m1 ; vn2m2 ℄ = n1m1(Æn1 ;�n2Æm1 ;�m2 + Æn1;�m2Æm1 ;�n2)+ n1(Æn1 ;�n2vm1m2 + Æn1;�m2vm1n2)+ m1(Æm1 ;�n2vn1m2 + Æm1;�m2vn1n2) : (A.25)28

The proof of Theorem 5.1 of [20℄ is now based on the onstru tion of a sequen eh�n j, n = 1; 2; : : : of ve tors (Lemma 5.2)h�n j= 1n! h0 j ������� v11 v12 : : : v1nv21 v22 : : : v2n: : : : : : : : : : : :vn1 vn2 : : : vnn �������su h that h�n j �ni = (n + 1)! ( � 1) : : : ( � n+ 1) ; (A.26)implying 2 N for unitary va uum representations of bsp(1;R). Then onededu es that Wightman positivity implies that V1 has the form (A.22) for someN . The diÆ ulty in extending this analysis to the ase d = 4 again lies in thene essity of having the 6-point fun tion of L in order to be able to omputethe stru ture onstants of the in�nite Lie algebra generated by the modes ofV1(z1; z2).

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