First order formalism for spin one fields
aa r X i v : . [ h e p - ph ] S e p First order formalism for spin one field
Karol Kampf , Jiˇr´ı Novotn´y and Jaroslav Trnka Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics,Charles University, V Holeˇsoviˇck´ach 2, CZ-180 00 Prague 8, Czech Republic
Abstract
We study two general approaches how to describe spin one particles, using vectorand antisymmetric tensor fields within R χ T. In this paper we focus on the question ofan equivalence of both ways. The appearing problems lead us to the introduction ofa new type of the description - the first order formalism which naturally connects bothtraditional formalisms. Moreover, it gives a more general result on the level of the effectivechiral Lagrangian that contain all terms from effective chiral Lagrangians in vector andantisymmetric tensor formulations. for emails use: surname at ipnp.troja.mff.cuni.czPresented by J. T. at Petrov School 2007, organized by Kazan State University. Introduction
Chiral perturbation theory ( χ PT) [1, 2, 3] is the effective theory for strong interactions, itdescribes the dynamics of the lightest hadrons and their interactions at low energies. Thefundamental theory for strong interactions - QCD is invariant under the chiral symmetry SU (3) ⊗ SU (3) (with massless quarks). The process of spontaneous symmetry breaking givesrise to the octet of the Goldstone bosons. In χ PT we identify these Goldstone bosons withthe octet of the lightest hadrons, i.e. with the octet of the pseudoscalar mesons. In the lowenergy region (under some scale Λ that is typically Λ ≈ χ PT is formulated as a perturbative expansion in the small quantity p/ Λ . This is actuallythe derivative expansion in the momentum representation. The chiral Lagrangian can be thenwritten in the form: L χ = L + L + . . . where L n = O ( p n ). Weinberg formula [1] providesus with the rule which operators should be used when calculating concrete tree level or loopdiagrams of a given order.The chiral Lagrangian contains set of the coupling constants (called LEC - low energyconstants) . They effectively include the contributions of the heavy degrees of freedom. Forenergies p ≈ Λ χ PT loses its convergence and it is necessary to introduce phenomenologicalLagrangians that describe the direct interaction of resonances. Of course, when integratingthe resonances out and coming to low energies we reestablish the original χ PT Lagrangian.This can help us to learn how the χ PT coupling constants are saturated by the resonances.Consequently, the study of the Resonance chiral theory (R χ T) [4, 5, 6, 7, 8, 9, 10, 11] andthe matching it with experiments can give us the predictions of values of LEC [12].R χ T has not been yet formulated as a closed theory, despite a considerable progress hasalready been done. As in χ PT, an external momentum p is used as an expansion parameter.Finding the complete basis of operators up to a given order allows one to calculate variousphysical observables and to do the comparisons. However, some important questions remainwithout answers. For example, the loops - some calculations have been already done [13, 14]but the more systematic work is still missing.In this paper we focus on various types of descriptions of spin one resonances in R χ T.Specifically, we will discuss two mostly used ways - vector fields and antisymmetric tensorformalisms. The problem is that they are not completely equivalent (more in [6, 4, 5, 15, 16,17, 9, 18]) and therefore, it is not possible to convert one to the other without adding somecontact terms. As a third possibility, we introduce the first order formalism that in somesense connects both previous.All this business can be used in the context of R χ T or by itself as an interesting theoreticalfeature of the effective field theories.
The two main ways how to describe spin one fields are the formalisms using vector fields V aµ and antisymmetric tensor fields R aµν where a is a group index (for R χ T it is U(3) in large N C ).We will use the convention introduced in [6] in order to simplify the following expressions. In the massive case we do the expansion also in the masses of quarks which are of order m q = Ø( p ) For Ø( p ) we have 2 constants, for Ø( p ) 10 constants and for Ø( p ) approximately 100 constants V · V ) ≡ V aµ V a,µ . (1)Multiple dots and double dots stand for analogous objects( V · K · V ) ≡ V aµ K ab,µν V bν , R : J = R aµν J a,µν . (2)Antisymmetric derivative of the field b V is defined as b V aµν ≡ D abµ V bν − D abν V bµ . (3)Here the covariant derivative D abµ is constructed in order to b V have the right transformationproperties with respect to the symmetry group SU (3) L × SU (3) R [4]. Vector field formalism
The general Lagrangian that contains only kinetic and mass terms together with the linearcoupling to the external sources has the form [5] L V = −
14 ( b V : b V ) + 12 m ( V · V ) + ( j · V ) + ( j : b V ) . (4)Within R χ PT the external sources have the orders j = O ( p ) , j = O ( p ) (5)and consist of usual chiral blocks built of the pseudogoldstone fields and external sources[2, 3].Equations of motion in the leading order yield V = − m ( j − D · j ) (6)where the indices are suppressed. Moreover, we learned that V = O ( p ). Low energy effectivechiral Lagrangian is then defined as Z V [ j i ] = exp (cid:18) i Z d x L eff V (cid:19) = Z D V exp (cid:18) i Z d x L V (cid:19) . (7)with the result L (6) , eff V = − m ( j · j ) + 2 m ( D · j · j ) + 2 m (cid:16) D · j · j · ←− D (cid:17) (8)where the upper index indicates the chiral order of the effective Lagrangian. It is possible to eliminate the source j by redefining j . However it is convenient to preserve it due tobetter comparison with antisymmetric tensor formalism. ntisymmetric tensor field formalism The analogous form of Lagrangian in the antisymmetric tensor formalism has the followingform L T = −
12 ( W · W ) + 14 m ( R : R ) + ( J · W ) + ( J : R ) (9)where W aµ ≡ D abα R b,αµ . The orders of the external sources are J = O ( p ) , J = J (2)2 + J (4)2 = O ( p ) + O ( p ) . (10)where we divide the source J into two parts according to the order. Equation of motion inthe leading order is R = − m J (2)2 (11)which leads to R = O ( p ). Low energy effective chiral Lagrangian is then defined as Z V [ J i ] = exp (cid:18) i Z d x L eff T (cid:19) = Z D R exp (cid:18) i Z d x L T (cid:19) . (12)with the result L (4) , eff T = − m (cid:16) J (2)2 : J (2)2 (cid:17) (13) L (6) , eff T = − m (cid:16) J (2)2 : J (4)2 (cid:17) + 2 m (cid:16) D · J (2)2 · J (2)2 · ←− D (cid:17) − m (cid:16) D · J (2)2 · J (cid:17) (14)where the upper index indicates again the leading order of the effective Lagrangian. From the last section it can be seen that vector and antisymmetric tensor formalisms arenot equivalent because they produce different effective Lagrangians. The key observation isthat the effective Lagrangian starts at the order O ( p ) in the antisymmetric tensor formalismwhereas at the order O ( p ) in the vector formalism. Consequently, no adjusting of the sources j i and J i can establish the equivalence of L eff V and L eff T .Let us now consider the generating functional for the vector field Lagrangian Z V [ j i ] andintroduce auxiliary antisymmetric tensor field RZ V [ J i ] = Z D V exp (cid:18) i Z d x L V (cid:19) = R D V D R exp (cid:0) i R d x (cid:0) ( R : R ) + L V (cid:1)(cid:1)R D R exp (cid:0) i R d x (cid:0) ( R : R ) (cid:1)(cid:1) (15)After shifting R → mR − b V and integrating out the vector fields we obtain Z V [ j i ] = R D R exp (cid:0) i R d x L ′ R (cid:1)R D R exp (cid:0) i R d x (cid:0) m ( R : R ) (cid:1)(cid:1) (16)with L ′ T = −
12 ( W · W ) + 14 m ( R : R ) + ( J ′ · W ) + ( J ′ : R ) + L contact T (17)where J ′ = − m j , J ′ = mj , L contact T = − m ( j · j ) + ( j : j ) (18)4nalogously starting with the generating functional Z R [ J i ], introducing auxiliary field V andintegrating out the antisymmetric tensor field we finally get L ′ V = −
14 ( b V : b V ) + 12 m ( V · V ) + ( j ′ · V ) + ( j ′ : b V ) + L contact V (19)where j ′ = mJ , j ′ = − m J , L contact V = 12 ( J · J ) − m ( J : J ) (20)Now, we see the origin of the problem. When transforming from one formalism to anotherone some additional contact terms appear . This also leads to the differences at the order ofeffective Lagrangians.So, if we want to preserve the equivalence of L V and L T (after expressing j i in termsof J i or visa versa) it is necessary to add some contact terms to one or both Lagrangians.Moreover, we have learned that both formalisms lead to different effective Lagrangians andeach of them has some extra terms which are not present in the second one [12, 19]. It isoften necessary to add these terms in Lagrangian by hand in order to satisfy high energyconstraints. Therefore, we try to find a way how to get all terms in the effective Lagrangianin order not to lose any information and not to add anything by hand. As a solution, weintroduce the concept of the first order formalism.Simply saying, it is based on the rewriting of the Lagrangian in one of the formalismswhen the derivatives of the fields are replaced by the fields of the second type. Furthermore,instead of the standard kinetic term we include the “mixed” form. The complete Lagrangianin the first order formalism is then L V T = 14 m ( R : R ) + 12 m ( V · V ) − m (cid:16) R : b V (cid:17) + ( J · V ) + ( J : R ) (21)where we explicitly denote O ( p ) and O ( p ) parts of the source, J = J (2)2 + J (4)2 . Now wecan demonstrate the advantages of this improvement. After integrating out both the fieldswe obtain the effective Lagrangian L (4) , eff V T = − m (cid:16) J (2)2 : J (2)2 (cid:17) (22) L (6) , eff V T = − m ( J · J ) − m (cid:16) J (2)2 : J (4)2 (cid:17) + 2 m (cid:16) D · J (2)2 · J (2)2 · ←− D (cid:17) − m (cid:16) D · J (2)2 · J (cid:17) (23)We see that all terms in L eff V and L eff T were reestablished. The question is what happens if weintegrate out just one of the fields. Writing Z V T [ J i ] = Z D R D V exp (cid:18) i Z d x L V T (cid:19) = Z D R exp (cid:18) i Z d x L ′ T (cid:19) = Z D V exp (cid:18) i Z d x L ′ V (cid:19) (24)we obtain L ′ T = −
12 ( W · W ) + 14 m ( R : R ) + (cid:0) J ′ · W (cid:1) + (cid:0) J ′ : R (cid:1) + L contact T (25) Moreover, including the terms with two vector (or antisymmetric tensor) fields we obtain in the correspon-dence the infinite series of terms in the antisymmetric tensor (or vector) formalism. J ′ = − m J , J ′ = J , L contact T = − m ( J · J ) (26)and L ′ V = − (cid:16) b V : b V (cid:17) + 12 m ( V · V ) + ( j ′ · V ) + ( j ′ · b V ) + L contact V (27)with j ′ = J , j ′ = 1 m J , L contact V = − m ( J : J ) (28)It can be easily seen that contact terms in both formalisms are naturally derived from thefirst order Lagrangian. This supports the idea that the first order formalism is more generalthan vector and antisymmetric tensor formalisms. They both can be naturally obtained fromit with appropriate contact terms. Complete derivation is done in [6]. In this paper we have discussed vector and antisymmetric tensor formalisms for R χ T re-stricting ourselves to the Lagrangians with interaction terms linear in resonance fields. Afterintegrating out these fields we have obtained effective chiral Lagrangians which can be ex-panded in powers of p/M . We have illustrated this point in three possible formalisms. Thefact that the lowest term in the vector formalism is of the order O ( p ), whereas the an-tisymmetric tensor formalism has O ( p ) contribution contradicts the idea that vector andantisymmetric tensor approaches are completely equivalent (without the addition of somecontact terms). The effective chiral Lagrangian derived in the first order formalism containsall terms which are present both in L eff V and L eff T .It is shown in [6], [7], [10] and [8] that some problems with satisfying short-distanceconstraints can appear when calculating Green functions. This is a common feature of bothtraditional formalisms. We have seen, by the construction, that the results calculated inthe first order formalism are not expected to be worse than the results in the vector or theantisymmetric tensor formalisms. This was explicitly verified in [6] for VVP correlator andin [7] for the pion formfactor. It could be interesting to investigate also other correlators andformfactors. Acknowledgement
This work was supported in part by the Center for Particle Physics (project no. LC 527) andby the GACR (grant no. 202/07/P249).
References [1] S. Weinberg, Physica A (1979) 327.[2] J. Gasser and H. Leutwyler, Annals Phys. (1984) 142.[3] J. Gasser and H. Leutwyler, Nucl. Phys. B (1985) 465.[4] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B (1989) 311.65] G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B (1989)425.[6] K. Kampf, J. Novotny and J. Trnka, Eur. Phys. J. C , 385 (2007)[7] K. Kampf, J. Novotn´y and J. Trnka, arXiv:hep-ph/0701041.[8] M. Knecht and A. Nyffeler, Eur. Phys. J. C (2001) 659 [arXiv:hep-ph/0106034][9] M. Tanabashi, Phys. Lett. B (1996) 218 [arXiv:hep-ph/9511367].[10] P. D. Ruiz-Femenia, A. Pich and J. Portoles, JHEP (2003) 003[arXiv:hep-ph/0306157].[11] V. Cirigliano, G. Ecker, M. Eidemuller, R. Kaiser, A. Pich and J. Portoles[arXiv:hep-ph/0603205][12] K. Kampf and B. Moussallam, Eur. Phys. J. C , 723 (20006) [arXiv:hep-ph/0604125].[13] I. Rosell, J. J. Sanz-Cillero and A. Pich, JHEP , 042 (2004) [arXiv:hep-ph/0407240].[14] J. J. Sanz-Cillero, Phys. Lett. B , 180 (2007) [arXiv:hep-ph/0702217].[15] A. Abada, D. Kalafatis and B. Moussallam, Phys. Lett. B (1993) 256[arXiv:hep-ph/9211213].[16] D. Kalafatis, Phys. Lett. B (1993) 115.[17] J. Bijnens and E. Pallante, Mod. Phys. Lett. A , 1069 (1996) [arXiv:hep-ph/9510338].[18] E. Pallante and R. Petronzio, Nucl. Phys. B (1993) 205.[19] J. Bijnens, G. Colangelo and G. Ecker, Annals Phys.280