Redalyc.A Local Non-Abelian Gauge Invariant Action Stemming from The

Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil

Dudal, D.; Capri, M. A. L.; Gracey, J. A.; Lemes, V. E. R.; Sobreiro, R. F.; Sorella, S. P.; Verschelde, H. A Local Non-Abelian Gauge Invariant Action Stemming from the Nonlocal Operator F(D2)-1F Brazilian Journal of Physics, vol. 37, núm. 1B, march, 2007, pp. 232-238 Sociedade Brasileira de Física Sâo Paulo, Brasil

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How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative 232 Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007

A Local Non-Abelian Gauge Invariant Action Stemming from the 2 −1 Nonlocal Operator Fµν(D ) Fµν

D. Dudala,∗ M. A. L. Caprib, J. A. Graceyc, V. E. R. Lemesb, R. F. Sobreirob, S. P. Sorellab,† and H. Verscheldea a Ghent University Department of Mathematical Physics and Astronomy, Krijgslaan 281-S9, B-9000 Gent, Belgium

b UERJ - Universidade do Estado do Rio de Janeiro, Rua Sao˜ Francisco Xavier 524, 20550-013 Maracana,˜ Rio de Janeiro, Brazil

c Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, United Kingdom (Received on 12 September, 2006) 2 −1 We report on the nonlocal gauge invariant operator of dimension two, Fµν(D ) Fµν. We are able to localize this operator by introducing a suitable set of (anti)commuting antisymmetric tensor ﬁelds. Starting from this, we succeed in constructing a local gauge invariant action containing a mass parameter, and we prove the renormalizability to all orders of perturbation theory of this action in the linear covariant gauges using the algebraic renormalization technique. We point out the existence of a nilpotent BRST symmetry. Despite the additional (anti)commuting tensor ﬁelds and coupling constants, we prove that our model in the limit of vanishing mass is equivalent with ordinary massless Yang-Mills theories by making use of an extra symmetry in the massless case. We also present explicit renormalization group functions at two loop order in the MS scheme. Keywords: Yang-Mills gauge theory; BRST symmetry; Renormalization; Mass

I. INTRODUCTION However, in spite of the progress in the last decades, we still lack a satisfactory understanding of the behaviour of Yang- We shall consider pure Euclidean SU(N) Yang-Mills theo- Mills theories in the low energy regime. Here the coupling ries with action constant of the theory is large and nonperturbative effects have Z to be taken into account. 1 S = d4x Fa Fa , (1) The introduction of condensates, i.e. the (integrated) vac- YM 4 µν µν uum expectation value of certain operators, allows one to parametrize certain nonperturbative effects arising from the a 2 where Aµ, a = 1,...,N − 1 is the gauge boson ﬁeld, with as- infrared sector of e.g. the theory described by (1). Via the sociated ﬁeld strength Operator Product Expansion (OPE) (viz. short distance ex-

a a a abc b c pansion), which is applicable to local operators, one can re- Fµν = ∂µAν − ∂νAµ + g f AµAν . (2) late these condensates to power corrections which give nonperturbative information in addition to the perturbatively cal- The theory (1) is invariant with respect to the local gauge culable results. If one wants to consider the possible effects transformations of condensates on physical quantities in a gauge theory, quite a ab b clearly only gauge invariant operators should be considered. δAµ = Dµ ω , (3) The most famous® example is the dimension 4 gluon condensate α F2 , giving rise to 1 power corrections. Via the with s µν Q4 SVZ (Shifman-Vainshtein-Zakharov) sum rules® [3], one can ab ab abc c 2 Dµ = ∂µδ − g f Aµ , (4) extract phenomenological estimates for αsFµν . In recent years, a great deal of interest arose in dimension 2 denoting the adjoint covariant derivative. condensates in gauge theories.® Most attention was paid to the 2 As it is well known, the theory (1) is asymptotically free gluon condensate Aµ in the Landau gauge, due to the work [1, 2], i.e. the coupling becomes smaller at higher energies of [4, 5], as the quantity and vice versa. At very high energies, the interaction is weak Z D E 2 ® 1 4 ¡ U ¢2 and the gluons can be considered as almost free particles. Amin ≡ min d x Aµ , (5) U∈SU(N) VT which is gauge invariant due to the minimization along the ∗ gauge orbits, could be physically relevant. In fact, as shown® Postdoctoral fellow of the Special Research Fund of Ghent University. 2 †Work supported by FAPERJ, Fundaçao˜ de Amparo a` Pesquisa do Es- in [4, 5] in the case of compact QED, the quantity Amin tado do Rio de Janeiro, under the program Cientista do Nosso Estado, E- seems to be useful in order to detect the presence of nontrivial 2 26/151.947/2004. field configurations like monopoles. One can show that Amin D. Dudal et al. 233 can be written as an infinite series of nonlocal terms, see [6, 7] and references therein, namely

Z · µ ¶ µ ¶µ ¶ ¸ ∂ ∂ ∂ 1 A2 = d4x Aa δ − µ ν Aa − g f abc ν ∂Aa ∂Ab Ac + O(A4) . (6) min µ µν ∂2 ν ∂2 ∂2 ν

2 Since the operator Amin is nonlocal, it falls beyond the ap- In other gauges, there can be found other renormalizable plicability of the OPE annex sum rules, which refer to local local operators, which condense and give rise to a dynamical operators. gluon mass. Next to the Landau gauge [14, 16–19, 37, 38] the However, in the Landau gauge, ∂µAµ = 0, all nonlocal terms maximal Abelian [39, 40], linear covariant [41–43] and Curci- 2 Ferrari gauges [44, 45] have been investigated in the past. of expression (6) drop out, so that Amin reduces to the local operator A2, hence the interest in the Landau gauge and µ ® 2 The relevant operators in these other gauges are however its dimension two gluon condensate Aµ . A complication is 2 gauge variant, and as a consequence also the effective gluon that the explicit determination of the absolute minimum of Aµ mass. From this perspective, it is worthwhile to find out along its gauge orbit, and moreover of its vacuum expectation whether a gauge invariant framework might be found for a value, is a very delicate issue intimately related to the problem dynamical mass, and related to it for 1 power corrections. of Gribov copies [8–13]. Q2 Nevertheless, some nontrivial results were proven concern- 2 In order to have a starting point, we need a dimension 2 ing the operator Aµ. In particular, we mention its multi- plicative renormalizability to all orders of perturbation the- operator that is gauge invariant. This necessarily implies a ory, in addition to an interesting and numerically verified re- nonlocal operator, since gauge invariant local operators of di- lation concerning its anomalous dimension [14, 15]. An ef- mension 2 do not exist. We would also need a consistent cal- fective potential approach consistent with renormalizability culational framework, which requires an action only contain- and renormalization group requirements for local compos- ing local terms. Therefore, we should find an operator that ite operators (LCO) has also been worked out for this oper- can be localized by means of a finite set of auxiliary fields, in ator, giving further evidence of a nonvanishing condensate such a way that the local gauge invariance is respected. As ® A2 looks a bit hopeless from this viewpoint as it is a infinite A2 6= 0, which lowers the nonperturbative vacuum energy min µ series of nonlocal terms (6), we moved our attention instead [16]. The ® LCO method yields an effective gluon mass squared 2 2 to the nonlocal gauge invariant operator mg ∼ Aµ of a few hundred MeV [16–19]. In [20], it was already argued that gauge (in)variant condensates could also influence gauge variant quantities such as the gluon propagator. An OPE argument based on lattice simula- Z h iab 1 4 a ¡ 2¢−1 b tions in the Landau ® gauge has indeed provided evidence that O ≡ d xFµν D Fµν . (7) 2 VT a condensate Aµ could account for quadratic power corrections of the form ∼ 1 , reported in the running of the coupling Q2 constant as well as in the gluon propagator, see e.g. [21–26]. This OPE approach® allows one to obtain an estimate for the This operator caught already some attention in 3 dimensional soft part A2 originating from the infrared sector. The OPE µ IR gauge theories in relation to a dynamical mass generation can also be employed to relate this condensate to an effective [46]. gluon mass [21]. The presence of mass parameters in the gluon propagator In the following sections, we shall show that the operator have also been advocated from the lattice perspective several (7) can be localized, giving rise to a local, classically gauge times [27–32], whilst effective gluon masses also found phe- invariant action. Afterwards, we discuss how to investigate nomenological use [33, 34]. the renormalizability of the action once quantized. Eventually, 2 A somewhat weak point about the operator Amin is that it is we need to introduce a slightly more general classical action in unclear how to deal with it in gauges other than the Landau order to obtain a quantum action that is renormalizable to all gauge. Till now, it seems hopeless to prove its renormaliz- orders of perturbation theory. In the case of vanishing mass, ability out of the Landau gauge. In fact, at the classical level, the equivalence of our action with usual Yang-Mills theories adding (5) to the Yang-Mills action is equivalent to add the can be shown. We shall point out the existence of a naturally so-called Stueckelberg action, which is known to be not renor- extended version of the usual BRST symmetry. Before turn- malizable [35, 36]. We refer to [7] for details and references. ing to conclusions, we explicitly give various renormalization As already mentioned, also the OPE becomes useless outside group functions, verifying the renormalizability at the practi- the Landau gauge for this particular operator. cal level. 234 Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007

a a II. CONSTRUCTION OF THE ACTION AND ITS using (11) with m coupled to the composite operators BµνFµν a a RENORMALIZABILITY AT THE QUANTUM LEVEL and BµνFµν, we introduce 2 suitable external sources Vρσµν and V ρσµν and replace Sm by A. The action at the classical level Z 1 ³ ´ d4x V Ba Fa −V Ba Fa . (14) We can add the operator (7) to the Yang-Mills action as a 4 σρµν σρ µν σρµν σρ µν mass term via

SYM + SO , (8) At the end, the sources Vσρµν(x), V σρµν(x) are required to at- tain their physical value, namely with ¯ ¯ im ¡ ¢ 2 Z h iab ¯ ¯ m 4 a ¡ 2¢−1 b V σρµν¯ = Vσρµν¯ = − δσµδρν − δσνδρµ , (15) SO = − d xFµν D Fµν . (9) phys phys 2 4 so that (14) reduces to Sm in the physical limit. As we have discussed in [7], the action (8) can be localized by From now on, we assume the linear covariant gauge fixing, introducing¡ a pair¢ of complex bosonic antisymmetric tensor implemented through fields, Ba ,Ba , and a pair of complex anticommuting anti- µν µν ¡ ¢ Z a a ³α ´ symmetric tensor fields, Gµν,Gµν , belonging to the adjoint S = d4x baba + ba∂ Aa + ca∂ Dabcb , (16) representation, according to which g f 2 µ µ µ µ Z · µ Z 1 e−SO = DBDBDGDGexp − d4xBa DabDbcBc In [7], we wrote down a list of symmetries enjoyed by the 4 µν σ σ µν Z Z ¶¸ action 1 a im ¡ ¢ − d4xG DabDbcGc + d4x B − B a Fa 4 µν σ σ µν 4 µν µν SYM + SBG + Sg f , (17) (10)

Therefore, we obtain a classical local action which reads i.e. in absence of the sources. Let us only mention here the BRST symmetry, generated by the nilpotent transformation s SYM + SBG + Sm , (11) given by where g sAa = −Dabcb , sca = f abccacb , Z ³ ´ µ µ 2 1 a a S = d4x B DabDbcBc − G DabDbcGc , a abc b c a a abc b c BG 4 µν σ σ µν µν σ σ µν sBµν = g f c Bµν + Gµν , sBµν = g f c Bµν , Z ¡ ¢ a abc b c a abc b c a im 4 a a sGµν = g f c Gµν , sGµν = g f c Gµν + Bµν , Sm = d x B − B µν Fµν , (12) 4 sca = ba , sba = 0 , s2 = 0 . (18) which is left invariant by the gauge transformations

a ab b It turns out that one can introduce all the necessary exter- δAµ = −Dµ ω , nal sources in a way consistent with the starting symmetries. a abc b c a abc b c δBµν = g f ω Bµν , δBµν = g f ω Bµν , This allows to write down several Ward identities by which a abc b c a abc b c the most general counterterm is restricted using the algebraic δGµν = g f ω Gµν , δGµν = g f ω Gµν . (13) renormalization formalism [7, 49]. After a very cumbersome analysis, it turns out that the action (11) must be modiﬁed to B. The action at the quantum level Sphys = Scl + Sg f , (19) In order to discuss the renormalizability of (11), we relied on a method introduced by Zwanziger in [47, 48]. Instead of with

Z · ³ ´ 1 im 1 a S = d4x Fa Fa + (B − B)a Fa + Ba DabDbcBc − G DabDbcGc cl 4 µν µν 4 µν µν 4 µν σ σ µν µν σ σ µν abcd ³ ´³ ´¸ 3 ¡ a ¢ λ ¡ ¢2 λ a c − m2λ Ba Ba − G Ga + m2 3 Ba − Ba + Ba Bb − G Gb Bc Bd − G Gd , (20) 8 1 µν µν µν µν 32 µν µν 16 µν µν µν µν ρσ ρσ ρσ ρσ D. Dudal et al. 235 in order to have renormalizability to all orders of perturbation physical operators shall not depend on the choice of the gauge theory. We notice that we had to introduce a new invariant parameter [49]. quartic tensor coupling λabcd, subject to the generalized Jacobi B. Existence of a “supersymmetry” when m ≡ 0 identity

man mbcd mbn amcd mcn abmd mdn abcm We define a nilpotent (anti-commuting) transformation δs f λ + f λ + f λ + f λ = 0, (21) as and the symmetry constraints a a a δsBµν = Gµν , δsGµν = 0 , abcd cdab a a a λ = λ , δsGµν = Bµν , δsBµν = 0 , abcd bacd λ = λ , (22) δs(rest) = 0 . (25) as well as two new mass couplings λ1 and λ3. Without the new Then one easily verifies that (25) generates a “supersymme- couplings, i.e. when λ ≡ 0, λ ≡ 0, λabcd ≡ 0, the previous m≡0 1 3 try” of the action Sphys since action would not be renormalizable. We refer to [7, 50] for a all the details. We also notice that the novel fields Ba , B , m≡0 µν µν δsS = 0 , (26) a a phys Gµν and Gµν are no longer appearing at most quadratically. As it should be expected, the classical action Scl is still gauge with invariant w.r.t. (13). 2 δs = 0 . (27)

III. FURTHER PROPERTIES OF THE ACTION Taking another look at the transformations s and se, respec- tively given by (18) and (24), one recognizes that A. Existence of a BRST symmetry with a nilpotent charge s = se+ δs , The BRST transformation (18) no longer generates a sym- {δs,se} = 0 . (28) metry of the action Sphys. However, we are able to define a natural generalization of the usual BRST symmetry that does Since δs is a nilpotent operator, it possesses its own cohomology, which is easily identified with polynomials in the original constitute an invariance of the gauge fixed action (19). Indeed, a a a a after inspection, one shall find that Yang-Mills fields {Aµ,b ,c ,c }. The auxiliary tensor fields, a a a a {Bµν,Bµν,Gµν,Gµν}, do not belong to the cohomology of δs, sSe phys = 0 , because they form pairs of doublets [49]. This fact can be se2 = 0 , (23) brought to use in the following subsection. with C. Equivalence with Yang-Mills gauge theory when m ≡ 0 g sAe a = −Dabcb , sce a = f abccacb , µ µ 2 a abc b c a abc b c If the mass m ≡ 0, we would expect that the action (19) sBe µν = g f c Bµν , seBµν = g f c Bµν , would be equivalent with the usual Yang-Mills gauge the- a abc b c a abc b c sGe µν = g f c Gµν , seGµν = g f c Gµν , ory, since in the nonlocal formulation (9), we would have in- seca = ba , sbe a = 0 . (24) troduced “nothing”. In the local renormalizable formulation (19), this would also be trivially true when λabcd ≡ 0 as then we would only have added a -although quite complicated- Hence, the action Sphys is invariant with respect to a nilpotent BRST transformation se. We obtained thus a gauge field unity to the Yang-Mills action. Unfortunately, since renor- abcd theory, described by the action S , (19), containing a mass malization forbids setting λ = 0, we must find another ar- phys m≡0 term, and which has the property of being renormalizable, gument to relate Sphys to the usual Yang-Mills gauge theory. while nevertheless a nilpotent BRST transformation express- As proven in [50], the “supersymmetry” δs of (25) can be used ing the gauge invariance after gauge fixing exists simultane- to show that ously. It is clear that sestands for the usual BRST transforma- hGn(x1,...,xn)i ≡ hGn(x1,...,xn)i m≡0 , (29) tion, well known from literature, on the original Yang-Mills SYM+Sg f Sphys fields, whereas the gauge fixing part Sg f given in (20) can be written as a se-variation, ensuring that the gauge invariant where

Gn(x1,...,xn) = A(x1)...A(xi)c(xi+1)...c(x j)c(x j+1)...c(xk)b(xk+1)...b(xn) , (30) 236 Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 is a generic Yang-Mills functional. The expectation value of [52, 53], is devised to extract the divergences from massless ©any Yang-Millsª Green function, constructed from the fields 2-point functions. The propagators of the massless fields in an a a a a Aµ,c ,c ,b and calculated with the original (gauge fixed) arbitrary linear covariant gauge are Yang-Mills action S + S , is thus identical to the one cal- YM g f · ¸ culated with the massless action Sm≡0, where it is of course δab p p phys hAa(p)Ab(−p)i = − δ − (1 − α) µ ν , assumed that the gauge freedom of both actions has been fixed µ ν p2 µν p2 by an identical gauge fixing. δab p/ The foregoing result also reflects on the renormalization a b hc (p)c (−p)i = 2 , hψ(p)ψ(−p)i = 2 , group functions. As usual, we employ a massless renormal- p p ab ization scheme known as the MS scheme. As a consequence, b δ £ ¤ hBa (p)B (−p)i = − δ δ − δ δ , we can set m = 0 in order to extract the ultraviolet behaviour. µν σρ 2p2 µσ νρ µρ νσ Using (29), we conclude that all the renormalization group ab a b δ £ ¤ functions of the original Yang-Mills quantities are not affected hGµν(p)Gσρ(−p)i = − 2 δµσδνρ − δµρδνσ , (31) by the presence of the extra fields or couplings. This fact shall 2p be explicitly verified in the next section. where p is the momentum. The necessary Feynman diagrams were generated automatically with QGRAF [54]. IV. EXPLICIT RENORMALIZATION AT TWO LOOP We first checked that the same two loop anomalous dimen- ORDER sions emerge for the gluon, Faddeev-Popov ghost and quarks in an arbitrary linear covariant gauge as when the extra local- Having proven the renormalizability of the action (19), we izing fields are absent. It was also explicitly verified that the shall now compute explicitly the two loop anomalous dimen- correct coupling constant renormalization constant is found. sion of the fields and the one loop β-function of the tensor These results are in agreement with the general argument of coupling λabcd. The corresponding details can be found in [7] the previous subsection. We have implemented the properties (21) and (22) of the for one loop results, while two loop results are discussed in abcd [50]. λ coupling in a FORM module, while it was assumed that We have regarded the mass operator as an insertion and split 1 the Lagrangian into a free piece involving massless fields with λacdeλbcde = δabλpqrsλpqrs , the remainder being transported to the interaction Lagrangian. NA 1 To renormalize the operator, we insert it into a massless Green λacdeλbdce = δabλpqrsλprqs , (32) function, after the fields and couplings have been renormal- NA ized in the massless Lagrangian. An attractive feature of the massless field approach is that we can use the MINCER al- which follows from the fact that there is only one rank 2 in- gorithm to perform the actual computations. This algorithm, variant tensor in a classical Lie group. [51], written in the symbolic manipulation language FORM, At two loops in the MS scheme, we find that

·µ 2 ¶ ¸ α 61 2 10 2 1 abcd acbd γB(a,λ) = γG(a,λ) = (α − 3)a + + 2α − CA + TF Nf a + λ λ , (33) 4 6 3 128NA

where NA is the dimension of the adjoint representation of the that to deduce its renormalization constant, we need to con- 2 colour group, a = g and we have also absorbed a factor of sider a 4-point function. However, in such a situation the 16π2 MINCER algorithm is not applicable since two external mo- 1 into λabcd here and in later anomalous dimensions. These 4π menta have to be nullified and this will lead to spurious in- anomalous dimensions are consistent with the general obser- frared infinities which could potentially corrupt the renormal- vation that these fields must have the same renormalization ization constant. Therefore, for this renormalization only, we constants, in agreement with the output of the Ward identities have resorted to using a temporary mass regularization intro- [7]. A check on (33) is that after the renormalization of the duced into the computation using the algorithm of [55] and 3-point gluon Ba vertex, the correct gauge parameter inde- µν implemented in FORM. Consequently, we find the gauge pa- pendent coupling constant renormalization constant emerges. rameter independent anomalous dimension We also determined the one loop β-function for the λabcd couplings. As this is present in a quartic interaction it means D. Dudal et al. 237

· 1 ³ ´ βabcd(a,λ) = λabpqλcpdq + λapbqλcdpq + λapcqλbpdq + λapdqλbpcq λ 4 i abcd abp cdp 2 adp bcp 2 abcd 2 − 12CAλ a + 8CA f f a + 16CA f f a + 96dA a , (34)

abcd a bµν c d σρ abcd abcd from both the λ BµνB BσρB and λ -independent terms. If we had not included the λ - abcd a bµν c d σρ abcd interaction term in the original action, then such a term would λ BµνB GσρG vertices where dA is the totally symmetric rank four tensor deﬁned by inevitably be generated at one loop through quantum correc- ³ ´ tions, meaning that in this case there would have been a break- dabcd = Tr T aT (bT cT d) , (35) down of renormalizability. A A A A A Finally, we turn to the two loop renormalization of the mass a m. The corresponding operator can be read off from (19) and with TA denoting the group generator in the adjoint representation, [56]. Producing the same expression for both these is given by 4-point functions, aside from the gauge independence, is a ¡ a a ¢ a strong check on their correctness as well as the correct im- M = Bµν − Bµν Fµν . (36) plementation of the group theory. Moreover, as it should be, abcd abcd a b β enjoys the same symmetry properties as the tensor λ , We insert M into a Aµ-Bνσ 2-point function and deduce summarized in (22). the appropriate renormalization constant, leading to the MS We notice that λabcd = 0 is not a ﬁxed point due to the extra anomalous dimension

µ ¶ µ ¶ 2 11 4 77 2 2 1 abe cde adbc 1 abcd abcd γO(a,λ) = − 2 TF Nf − CA a − TF Nf CA + 4TF Nf CF − CA a + f f λ a − λ λ (37) 3 6 3 12 8NA 128NA

as the two loop MS anomalous dimension. The gauge para- ply unitarity. The nilpotent BRST symmetry (24) might be meter independence is again a good check, as the operator M useful for this. is gauge invariant. The model (19) is also asymptotically free, implying that at low energies nonperturbative effects, such as confinement, could set in. Proving and understanding the possible confine- V. CONCLUSIONS ment mechanism in this model is probably as difficult as for usual Yang-Mills gauge theories. We added a nonlocal mass term (9) to the Yang-Mills ac- It would also be interesting to find out whether a dynami- tion (1), and starting from this, we succeeded in construct- cally generated term m(B − B)F might emerge, which in turn ing a renormalizable massive gauge model, which is gauge could influence the gluon Green functions. This might also invariant at the classical level and when quantized it enjoys be relevant in the context of gauge invariant 1 power correc- a nilpotent BRST symmetry. This BRST symmetry ensures Q2 tions, an issue that recently has also attracted attention from that the expectation value of gauge invariant operator is gauge the gauge/string duality side, the so-called AdS/QCD [57, 58]. parameter independent. We have also proven the equivalence of the massless version of our model with Yang-Mills gauge theories making use of a “supersymmetry” existing between the extra fields in that case. We presented explicit two loop renormalization functions, thereby verifying that the anom- Acknowledgments alous dimensions of the original Yang-Mills quantities remain unchanged. D. Dudal and H. Verschelde are grateful to the organizers Many things could be investigated in the future concerning of the stimulating IRQCD06 conference. The Conselho Na- the gauge model described by (19). cional de Desenvolvimento Cient´ıfico e Tecnologico´ (CNPq- At the perturbative level, it could be investigated which (as- Brazil), the Faperj, Fundaçao˜ de Amparo a` Pesquisa do Es- ymptotic) states belong to a physical subspace of the model, tado do Rio de Janeiro, the SR2-UERJ and the Coordenaçao˜ and in addition one should find out whether this physical sub- de Aperfeiçoamento de Pessoal de N´ıvel Superior (CAPES) space can be endowed with a positive norm, which would im- are gratefully acknowledged for financial support. 238 Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007

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