Z. Phys. A 357, 325–331 (1997) ZEITSCHRIFT FUR¨ PHYSIK A c Springer-Verlag 1997
Rho meson properties in the Nambu-Jona-Lasinio model
A. Polleri1,?, R.A. Broglia1,2,3, P.M. Pizzochero1,2, N.N. Scoccola2,4,??
1 Dipartimento di Fisica, Universita` degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy 2 INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy 3 The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 4 Departamento de F´ısica, Comision´ Nac. de Energ´ıa Atomica,´ Av. Libertador 8250, 1429 Buenos Aires, Argentina
Received: 30 April 1996 / Revised version: 18 October 1996 Communicated by W.Weise
Abstract. Some properties of the rho vector meson are cal- approximate low-energy symmetries, such as chiral symme- culated within the Nambu-Jona-Lasinio model, including try) of QCD are respected. The basic difference with QCD processes that go beyond the random phase approximation. is that gluons are eliminated in favor of an effective local To classify the higher order contributions, we adopt 1/Nc as quark-quark interaction. For convenient values of the cou- expansion parameter. In particular, we evaluate the leading pling constants, chiral symmetry is spontaneously broken, order contributions to the ρ ππ decay width, obtaining the a dynamical, “constituent” quark mass is generated and the value Γ = 118 MeV, and to→ the shift of the rho mass which pion appears as a massless Goldstone boson. The pion mass turns out to be lowered by 64 MeV with respect to its RPA can be brought to its physical value by adding a small bare, value. A set of model parameters is determined accordingly. “current” quark mass in the effective lagrangian. Due to the local character of the interaction, the NJL PACS: 12.39.Fe; 13.25.-k model in 3 + 1 dimensions is not renormalizable. Therefore, some kind of regularization scheme has to be introduced. One of the methods widely used is to set a cut-off Λ in the quark 4-momentum. Typical values are Λ 1 GeV. In a ≈ 1 Introduction sense, the cut-off gives an idea of the range of applicability of the model. Since the pion mass is well below 1 GeV, Vector mesons are known to play an important role in differ- one might think that the predictions of the model should be ent aspects of low-energy hadronic physics. In fact, effective reasonable in that sector. On the other hand, the rho mass is lagrangians including nucleons, pions and vector mesons as not far from the typical cut-off values and one might worry fundamental degrees of freedom have been successful in de- about the validity of the model to describe the properties of scribing a large amount of empirical data [1]. On the other this meson. Within the NJL model, meson modes are viewed hand, with the introduction of QCD as the fundamental the- as collective solutions of a Bethe-Salpeter equation in the ory of strong interactions, it became clear that the structure random phase approximation (RPA). While the pion can be and properties of both baryons and mesons should be un- considered as a collective mode, the situation concerning the derstood in terms of quark and gluon degrees of freedom. vector mesons is less clear. Since at the mean field level the Although much effort has been made in trying to predict dynamical quark mass is momentum independent, the NJL low energy hadron observables directly from QCD, one is model does not confine. Therefore, depending on the model still far from reaching this goal. In such a situation it proves parameters, the rho mass can be above the unphysical qq¯ convenient to turn to the study of effective models. threshold and consequently its decay into a qq¯ configuration One of the models that has received considerable atten- (Landau damping) is possible within the model. In spite of tion in recent years is the Nambu and Jona-Lasinio (NJL) these potential inconveniences, at least at the RPA level, the model [2]. It was originally proposed as a way to explain vector mesons seem to emerge as poles in the corresponding the origin of the nucleon mass together with the existence -matrix channel with predicted weak decay constants in T of the (almost) massless pion in terms of the spontaneous reasonable agreement with their empirical values [7, 8]. breakdown of chiral symmetry. Later on, as it became clear To make more definite statements about the validity of that quarks are the fundamental components of the hadrons, the NJL model in the vector meson sector, one should study the model was reformulated by replacing the nucleon fields other properties. In particular the width of the rho meson by quark fields. In its modern version [3–6], the NJL model in its main decay channel, namely Γρ ππ. This requires, is constructed in such a way that all the symmetries (or however, to go beyond the RPA approximation→ and brings ? Present address: Niels Bohr Institute, Blegdamsvej 17, us into the problem of introducing a reasonable classifica- DK-2100 Copenhagen Ø, Denmark tion of higher order diagrams. Recently [9, 10], it has been ?? Fellow of the CONICET, Buenos Aires, Argentina suggested that a convenient way to do so is in terms of 326
an expansion in the inverse of the number of colors, Nc. The integral in Eq. (3) is logarithmically divergent. There- Within QCD, the idea of using 1/Nc as an expansion pa- fore a cut-off Λ has to be introduced in order to regularize it. rameter has a long history. In [11, 12] it was shown that, In the case of the covariant regularization method, that will in order to have a sensible theory for large values of Nc, be used throughout this paper, this integral has an explicit the quark-gluon coupling constant has to scale as 1/√Nc.It analytical form [8]. follows that, for example, the baryon masses turn out to be As well-known, when m0 = 0 there is a certain critical of (N ), the meson masses of (1), and the coupling of value of G1 above which Eq. (3) has a non-trivial solution O c O a meson to two-meson states or to qq¯ states of (1/√Nc). with mq =/ 0. This corresponds to the dynamical breakdown Independent of these developments, around theO same time of chiral symmetry. Correspondingly, a non-vanishing value very similar ideas were introduced in a rather different con- of the quark condensate < υ/¯υ>/ develops. For small val- text. Namely, it was shown that the traditional Hartree + ues of m0, the transition is no longer sharp but the same RPA treatment of many-body problems could be considered qualitative behaviour is obtained. as the leading order of an expansion in terms of the inverse From Eq. (3), it is seen that the dynamically generated of the degeneracy of the available Hilbert space, N [13]. contribution to mq in the Hartree approximation scales with Moreover, a method to go beyond the leading terms based G1Nc. Therefore, in order to have a sensible large Nc limit on the 1/N expansion was developed [14]. Since the meth- we should impose G1 (1/N ). This type of requirement ∼O c ods applied in the study the NJL model are basically the is completely equivalent to the one used in large-Nc QCD. same as those used in many-body physics, it is quite natural In fact, the scaling G1 (1/Nc) was to be expected since to expect that the equivalent of the 1/N expansion could be the effective quark-quark∼O interaction can be thought of as the performed. In fact, the results of [10] seem to confirm this heavy-gluon (i.e. strongly dressed) limit of a gluon mediated expectation. Some previous estimates of the ρ decay width interaction. Therefore, in the Hartree approximation mq into two pions within the NJL model already exist [15]. In (1). It is easy to verify [10] that the Fock contribution∼ to O general, however, these calculations show that this quantity the self-energy is of (1/Nc). is largely underestimated. As we will see, a consistent treat- Making use of theO vacuum in the mean field approxima- ment in terms of 1/Nc tends to improve this situation. tion, we shall now study the associated meson fluctuations. In the present paper we report on a calculation of 1/Nc For that purpose, we solve the Bethe-Salpeter equation for corrections to the RPA description of the ρ meson within the the -matrix T NJL model. The paper is organized as follows. In Sect. II we d4p briefly review the main features of the NJL model. Namely, i (q2)= i Tr i iS(p + 1 q) (4) − (2π)4 2 × we show how quark self-energy and meson masses, together T K Z K with their couplings, are obtained in the Hartree + RPA ap- i (q2) iS(p 1 q) , T − 2 proximation and we give the values of the model parameters where S(p)= 1 is the fermion Feynman propagator. to be used in our numerical calculations. In Sect. III we cal- p m The -matrix as6− well as the kernel , can be decomposed culate the leading contribution to the ρ ππ decay width. in termsT of scalar, pseudoscalar, vectorK and axial vector In Sect. IV we calculate the 1/N correction→ to the rho mass. c channels.1 Conclusions are given in Sect. V. Some details of the cal- Due to the presence of the explicit chiral symmetry culations are collected in Appendices A and B. breaking induced by the current mass m0, we have to deal with the mixing between pseudoscalar and longitudinal axial fields, called π A1 mixing. Therefore, in the pseudoscalar 2 Generalized NJL model − in the Hartree + RPA approximation channel one obtains a 2 2 matrix equation, while in the vector channel a 1 1 equation× is obtained. The solution of In this work we consider the two-flavor version (up and these equations can× be found by summing the corresponding down quarks) of the extended NJL model, defined by the geometrical series. One finally gets effective Lagrangian 2 a a 1 i π(q )=(iγ5τ iγ5τ ) 2 (5) NJL =υ/¯(x)(i ∂ m0)υ/(x)+ int . (1) T ⊗ D(q ) × L 6 − L In writing Eq. (1) we have assumed that the bare quark 2 2 G1 1 G2JAA(q ) G1G2JPA(q ) masses are degenerate and set mu = md = m0. The four- − 2 2 , × G1G2JAP (q ) G2 1 G1JPP(q ) fermion interaction term int is given by − L where G1 2 a 2 int = (υ/¯ (x)υ/ (x)) + (υ/¯ (x)iγ5τ υ/ (x)) + (2) 2 2 2 L 2 D(q )= 1 G1JPP(q ) 1 G2JAA(q ) h i − − − G2 a 2 a 2 2 2 + υ/¯ (x)γµτ υ/ (x) + υ/¯ (x)γµγ5τ υ/ (x) . G1G2JPA(q ) JAP (q ) , 2 − and Following theh usual treatment, we first apply the mean i field formalism in the Hartree approximation. This leads to the so- i (q2)= γ τa γ τa (6) ρ µ ⊗ ν × called “gap equation” for the quark self-energy (also called T G 2 µν µ ν 2 constituent quark mass) 2 g q q /q . 1 G2 JVV(q ) − Λ 4 − d p 1 1 In what follows we will concentrate only on the pseudoscalar and mq = m0 + mq G1 Nc 8i . (3) (2π)4 p2 m2 vector channels, since the others are not relevant for our purposes Z − q 327
p’ p’ q
ρ ρ π iΓπ q iΓρ q
p + k1 p - k2 ←→ -( k1 k2 ) p p Fig. 1. Effective meson-qq¯ vertices obtained within the RPA approximation. p a 5 a π π We have iΓρ = igρ qq¯γµτ and iΓπ = igπ qq¯(1 aπ qˆ )iγ τ − − − 6
Λ k - k In these equations, Jαβ are the polarization functions (i.e. 1 2 quark-antiquark loops) of the corresponding channels. They Fig. 2. The ρ ππ amplitude used to calculate the decay width. It corre- − µ are divergent functions, that require regularization. For con- sponds to the function T (q; k1,k2) as in Eq.(10) sistency, the same regularization method used in the mean field approximation is to be used here. Explicit expressions for these functions can be found in the literature [8]. gauge calculations. Our prediction for mq, although large The masses of the bound states and resonances are ob- with respect to some recent estimates [6, 16], is well within tained from the position of the poles of the -matrix. In the range of values usually used in the literature [3, 4, 17]. order to determine the meson-qq¯ coupling constants,T we ap- With such value of mq the RPA prediction for the ρ mass, (0) proximate the -matrix close to the pole position. We there- mρ , is below the qq¯ threshold and therefore this particle fore write T can be clearly identified as a pole in the corresponding - (0) T 2 a a channel. Note that mρ is larger than the experimental value i π(q ) (iγ5τ iγ5τ ) (7) Emp T ≈ ⊗ × mρ = 770 MeV. As shown in Sect. IV, the ρ mass will be ˆ i ˆ 2 igπ qq(1 + aπ q) igπ qq(1 aπ q) , lowered to this value by self-energy corrections . − 6 q2 m2 − − 6 − π Before concluding this section it is worth recalling the where qˆ =q/ q2, and Nc-behaviour of the meson properties obtained in the RPA 6 6 approximation. It can be shown that all the polarization func- 2 a a i (q ) γpτ γ τ (8) tions are proportional to Nc. As a consequence, the meson ρ ≈ µ ⊗ ν × T µν µ ν 2 masses are of (1), while the meson-qq¯ coupling constants i g q q /q O ig − − ig . are of (1/√Nc). This is in agreement with the Nc counting ρ qq 2 2 ρ qq O − q mρ − rules obtained in QCD [11, 12] and also with those found − from which the meson-qq¯ vertices can be obtained (see in many-body theories [13, 14]. Fig.1). For the vector channel, the rho-quark gρ qq coupling − 3 The ρ ππ decay constant can be defined as the corresponding residue at the → pole and expressed in terms of the function JVV. Due to the The main decay channel of the rho meson is into two pions. π A1 mixing, the situation for the pion is slightly more The leading contribution to this process corresponds to the complicated,− but one can still define the standard quantities diagram shown in Fig. 2, which is of (1/√Nc) in agree- gπ qq and aπ in terms of the functions JPA and JAA. O − ment with the large Nc QCD result. This behaviour results The four parameters of the model can be fixed by fit- from the factor N due to the quark loop and from the factors ting four physical quantities, namely the quark condensate c ¯ 1/√Nc associated with the meson-qq¯ vertices (cf. Fig. 1). < qq¯ > (=
ˆ 5 i√2gπ qq¯(1 + aπ k1)iγ − 6 × i (k1 k2) . p+ k1 m − ↔ 6 6 − q Evaluating the trace one gets µ 2 T (q; k1,k2)= 2iNc gρ qq gπ qq¯ (11) − − − × [kµ G(q2; k2,k2) kµ G(q2;k2,k2)], 1 1 2 − 2 2 1 2 2 2 where the function G(q ; k1,k2) can be written in terms of the standard regularized functions which appear in the NJL model. Its explicit expression is given in Appendix A. Making use of Fermi’s Golden Rule, the corresponding de- cay width is found to be
1 2 Γ = dφ(2) m2 ,m2 , (12) 2m ρ π ρ Z M 3 3 (2) d k1 d k2 4 4 2 2 where dφ = 3 3 (2π) δ (q k1 k2) is the Fig. 3. The 4-momentum dependence of the function G(q ,mπ), used in (2π) 2Ek1 (2π) 2Ek2 − − 2 standard two-body phase-space-measure. Eq. (13). Here the deviation from its value at q = 0 is shown R R The bar over the squared transition amplitude implies an average over the initial polarizations and a sum over the final We conclude this section with a short remark on the eval- ones. Moreover, all particles are put on their mass shells. The uation of using the naive low momentum expansion, calculation of expression (12) is straightforward and details Γ ρ ππ as sometimes→ done in the literature. Within this approxima- are given in Appendix A. tion, the form factor G2 q2; m2 in Eq. (13) is replaced by One obtains π its value at q2 = 0. This is based on the assumption that this 2 2 3/2 Nc 2 4 4mπ function is only weakly dependent on q and leads, of course, Γ ρ ππ = gρ qq gπ qq mρ 1 2 (13) → 12π − − − m × to an important simplification in the evaluation of the corre- ρ 2 2 2 sponding decay amplitudes. In order to investigate whether G mρ; mπ . this approximation could have been used in the present cal- In order to be fully consistent with the 1/Nc expansion, all culation we display in Fig. 3 the momentum dependence of 2 2 2 the quantities appearing in Eq. (13) have to be taken at the G q ; mπ as calculated using Eq. (A9). As we see, in the values obtained in Hartree + RPA approximation, namely range of momenta we are interested in ( q 800 MeV), 2 2 2 ≈ those given at the end of the previous section. In particular, G q ; mπ differs from its value at zero momentum for we have to take RPA value for the ρ mass. This is equiva- about 30 40%. This clearly indicates that, at least within − lent to use the Rayleigh-Schroringer¨ (RS) perturbation theory the present regularization scheme, the use of the low mo- [19]. The final result is mentum approximation would have been inappropiate. Th Γ ρ ππ = 118 MeV. (14) → As we see, already at leading order in 1/Nc we obtain a 4 Self-energy correction to the rho mass value which compares reasonably well with empirical value [20] The leading self-energy corrections (i.e. beyond RPA) to the Emp rho meson propagator are of order 1/Nc, as can be seen from Γ ρ ππ = 151.2 1.2 MeV. (15) → ± the corresponding diagrams displayed in Fig. 4a-c. The self- It is interesting to note that, instead of using the RPA energy, which we will call Πµν (q), can be decomposed into value, we could have evaluated the width using the cor- its real and imaginary parts which can be calculated using rected ρ mass (i.e. its value once 1/Nc corrections are in- cutting rules and dispersion relations. cluded; see next section). This corresponds to use the so- As already mentioned, the NJL model does not confine, called Brillouin-Wigner (BW) perturbation theory [19]. This i.e. it allows hadrons to decay into free quarks. In order to type of expansion implies the inclusion of diagrams of or- cure this unphysical behaviour we follow [22] and impose der higher than 1/√Nc, in which a ρ meson line appears confinement “by hand”, that is we neglect contributions from as an intermediate state. At least in the case of many-body cuts of the diagrams across quark lines thus taking into ac- systems, this leads to a poor convergence of the perturbative count only cuts across meson lines. This is consistent with series [21] since for a given order in 1/Nc, such diagrams the pole approximation for the meson lines already used in usually have opposite signs with respect to all the other (dis- the previous sections. Within such an approximation, the di- regarded) diagrams of the same order. Within this scheme, agrams (b) and (c) of Fig. 2 do not contribute, while only the resulting width is the cut across the pion lines in diagram (a) will give a con- Th tribution to the imaginary part of Πµν (q). Therefore Γ ρ ππ(BW) = 96 MeV, (16) → which is in somewhat worse agreement with the empirical d4p 1 Πµν (q)= iT µ(q; p + 1 q, p 1 q) (17) value. − i (2π)4 2 − 2 × Z 329
Cut m2 =(m(0))2 + eΣ (m(0))2 . (21) ρ ρ R ρ π Using Eq. (21) together with our set of model parameters, ρ ρ we find a shift of 64 MeV from the RPA value. The new, ( a ) “physical” rho mass− is therefore
mρ = 770 MeV, (22) π which justifies a posteriori the choice made at the end of section II. Here again we have used RS perturbation theory to evaluate the mass shift. Our number is surprisingly con- ρ ρ ρ ρ π sistent with the value obtained in [23], even though in that π work quark degrees of freedom are not included. The use of BW perturbation theory would correspond to solve the equation ( b ) ( c ) m2 =(m(0))2 + eΣ m2 . (23) ρ ρ R ρ Fig. 4a-c. Rho self-energy diagrams to order 1/Nc . As explained in the As we have already stressed this method is not completely text, within our approximations only diagram a, with a cut across the pion lines, does contribute consistent with the 1/Nc expansion. Numerically, however, the resulting shift 60 MeV is not so different from that obtained by using RS− perturbation theory. In a way this be- i iT µ(q; p 1 q, p + 1 q) haviour reminds of that found in the study of nuclear giant (p 1 q)2 m2 − 2 2 × resonances, where the predictions for the resonance width − 2 − π i are more sensitive than the shift of its centroid to the per- , (p + 1 q)2 m2 turbation method used. 2 − π The values of the rho mass shift reported above agree where iT µ is the ρ ππ amplitude of Eq. (10) and it repre- well with those obtained in [23] using an effective mesonic − sents the quark triangular loops. Making use of the cutting action. On the hand, arguments based on the assumption rules, one obtains that the ρ0 ω mass splitting ( 12 MeV empirically) is basically− due to the rho mass shift≈ (see e.g. [24]) seem µν 1 (2) µ ν mΠ (q)= dφ iT (q) (iT (q))∗ , (18) to indicate that our value is too large. It is clear that since I −2 Z the origin of such splitting is not yet fully understood this where the pion momenta k1 and k2 are set on shell. Using the type of argument has to be taken with some care. In any transversality property of the self-energy, that is Πµν (q)= case, if one relaxes the constraint imposed above on the rho gµν qµqν /q2 Σ(q2), it is possible to write (see Appendix momentum cut-off and reduces it by about 15%, a new self- − B) consistent calculation shows that the shift can be reduced ρ ω mΣ(q2)= qΓ (q), (19) down to a value consistent with the 0 mass splitting. I − ρ This variation has only a minor effect on− the predicted rho where Γ ρ(q) has the same expression as in Eq. (12) except decay width which is lowered by about 10%. that the rho momentum is not on shell. The real part of the self-energy can be calculated from the above expression making use of the Kramers-Kronig¨ re- 5 Conclusions lation, leading to3
Λ2 2 In this paper we have evaluated the rho decay width into two 2 1 2 Im Σ(µ ) pions and the corresponding mass shift within the Nambu eΣ(q)= dµ 2 2 . (20) R πP 4m2 µ q and Jona-Lasinio model. Since the calculation of these quan- Z π − tities requires to go beyond the usual Hartree + RPA approx- It should be noticed that a priori there is no reason to use in imation, we have introduced the inverse of the number of Eq. (20) the same cutoff as in, for example, Eq. (3). How- colors 1/N as a good expansion parameter. In terms of this ever, if one assumes that the cutoff represents the momen- c parameter, the Hartree + RPA approximation corresponds to tum scale up to which the low energy effective model can the inclusion of diagrams of (1), while leading contribu- be used, the fact that certain particles can be described up tions to the decay width andO mass shift are of (1/N ). to momenta higher than such scale while others not is quite c In order to be fully consistent within 1/N , we haveO used hard to justify. Consequently, we choose to define our model c the values obtained in the Hartree + RPA approximation to by using the same cutoff for all the possible types of loops. compute the 1/N quantities. This is equivalent to the use The pole position is now moved in the complex plane c of Rayleigh-Schrodinger¨ perturbation theory in many-body with respect to the RPA value, the imaginary part being physics. related to the decay width and the real part with the mass The calculated decay width turns out to be 118 MeV of the particle. Then, the expression of the rho mass which to be compared with the experimental value 151.2 1.2 includes corrections up to order 1/N reads c MeV. This corresponds to a 20% accuracy, which is± what 3 Strictly speaking Eq. (20), which becomes exact in the limit Λ2 , one could expect from a leading order calculation. For the defines an approximation to eΣ(q2) →∞ mass shift we have obtained 64 MeV, which is of the R − 330 order of 10% of the RPA value. This result, together with In the last two equations we have used those obtained in the quark sector [10], seems to indicate a Λ 4 rather good convergence of the perturbative series in 1/N . 2 dp 1 c I2(q )=4i , (A4) In a way, this fact also justifies a posteriori our choice of (2π)2 [p2 m2 ][(p q)2 m2 ] Z − q − − q model parameters, which was based on the assumption that next-to-leading corrections to the mass shift were negligible. Λ d4p 1 Of course, to make more definite statements about the con- 2 2 2 I3(q ; k1,k2)=4i 2 2 2 (A5) (2π) [(p k1) m ] × vergence of the 1/Nc expansion other properties as well as Z − − q higher orders in the expansion would have to be calculated. 1 , In the evaluation of the mass shift we have disregarded 2 2 2 2 [p mq][(p + k2) mq] the diagrams that contain cuts across quark lines. Such an − − approximation, which is consistent with the pole approxima- 2 2Nc 2 2 2 2 J (q )= (q +2m )I2(q ) 2m I2(0) , (A6) tion, is certainly a rather primitive way to implement con- VV 3 q − q finement. Quite recently [25], some variations of the NJL where q = k1 + k2. When pions are on shell, one obtains a model have been introduced in which quarks are confined much simpler expression through a scalar confinement mechanism. An obvious ex- µ 2 2 2 µ tension of our work would be the evaluation of the rho me- T (q)= 2iNc gρ qq gπ qq¯ G(q ;mπ)(k1 k2) , (A7) son properties in such type of models. In fact, shortly after − − − − the present paper was submitted for publication, some work where now we have along this line has been reported [24]. 1 h G(q2; m2 )=(1 h ) I(m2)+ π (A8) We conclude that the Nambu-Jona-Lasinio model pro- π π 2 π 2 − 2 − q 4mπ × vides an economic and at the same time reasonably accu- 2 2 2{ 2 − 4 2 2 (q 2m ) I2(q ) I2(m ) +2m I3(q ;m ) rate picture of the ρ meson, in particular of its decay width − π − π π π − into two pions. This is true, provided that a single cut-off h2 J (q2) π VV . energy of the order of 1 GeV is introduced to regularize 2 − 2(1 hπ) 4mq the different ultraviolet divergences and that cuts in the di- − } agrams are taken only across meson lines, to avoid decay This pion-on-shell expression agrees with the result obtained into free quarks. With these approximations, all the many- in [16]. Note, however, the different type of regularization body techniques can be used to systematically explore the used in that reference. consequences of the variety of couplings of meson among To calculate the decay width, one has to evaluate themselves, as well as with fermions. 2 dφ(2) (m2 ,m2) (A9) M ρ π We would like to thank P.F. Bortignon and W. Weise for enlightening Z (2) (λ) µ 2 discussions. = dφ µ (q) T (q) , Z λ=0,+, X− which can be easily done by using the relations Appendix A: The ρ ππ Amplitude and Decay − (λ)(q) (λ)(q)= gµν qµqν /q2 (A10) µ ν − − As seen in Sect. III, the ρ ππ decay amplitude can be λ=0,+, expressed in terms of → X− and µ 2 T (q; k1,k2)= 2iNc gρ qq gπ qq¯ (A1) − 2 − − × (2) 1 4mπ µ 2 2 2 µ 2 2 2 dφ = 1 , (A11) [k1 G(q ; k1,k2) k2 G(q ;k2,k1)]. 8π − m2 − Z s ρ Here obtaining as final result the expression given in Eq. (14). 2 2 2 (0) 2 2 2 2 G(q ; k ,k )=G (q ; k ,k )+hπ[I2(k) (A2) 1 2 1 2 1 − (0) 2 2 2 2 2 (0) 2 2 2 2 G (q ; k1,k2)]+ hπ[ I2(k1)+G (q ; k1,k2)+ Appendix B: The rho self-energy − 2 2 2 − (q k1+k2) 2 + − 2 2 JVV(q )], From Eq. (18) in the text we have, relabelling momenta, 8mqq 4 4 2mq µν dk1 d k2 4 4 with hπ = aπ and Π (q)= i (2π) δ (q k1 k2) (B1) mπ − (2π)4 (2π)4 − − × Z 1 (0) 2 2 2 2 µ i µ G (q ; k1,k2)=I2(k1)+ (A3) iT (q; k1,k2) iT (q; k2,k1) q2 2(k2 + k2) × k2 m2 × − 1 2 2 π 2 2 − 2 2 2 2 I2(k1)+I2(k2) i q (k + k ) [ I2(q ) ]+ . − 1 2 − 2 k2 m2 1 − π { 2 2 2 2 k2+k1 2 2 2 Cutting the diagram across pion lines, it is possible to ob- +(3k k) I3(q;k,k ) . 2− 1 2 1 2 tain the imaginary part of the self-energy. This consists in } 331 re-writing the amplitude with appropriate rules and putting 7. V. Bernard and U. G. Meissner, Nucl. Phys. A489, 647 (1988) pions on shell with the prescription 8. S. Klimt, M. Lutz, U. Vogl and W. Weise, Nucl. Phys. A516, 429 (1990) i 1 2πδ(k2 m2)θ(k0) . 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