Decay rates f 1 (1285)→ ρ 0 π + π − and a 1 (1260)→ω π + π − in the Nambu -- Jona-Lasinio model
DDecay rates f (1285) → ρ π + π − and a (1260) → ωπ + π − in the Nambu – Jona-Lasiniomodel A. A. Osipov, ∗ A. A. Pivovarov, † and M. K. Volkov ‡ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980, Russia
The anomalous decays f (1285) → ρ π + π − and a (1260) → ωπ + π − violating natural parity forvectors and axial-vectors are studied in the framework of the Nambu – Jona-Lasinio model. Weconsider the Lagrangian with U (2) L × U (2) R chiral symmetric four quark interactions. The theory isbosonized and corresponding effective meson vertices are obtained in the leading order of 1 /N c andderivative expansions. The uncertainties related with the surface terms of anomalous quark trianglediagrams are fixed by the corresponding symmetry requirements. We make a numerical estimateof the decay widths Γ( f (1285) → ρ π + π − ) = 2 .
78 MeV and Γ( a (1260) → ωπ + π − ) = 87 keV.Our result on the f (1285) → ρ π + π − decay rate is in a good agreement with experiment. It isshown that a strong suppression of the a (1260) → ωππ decay is a direct consequence of destructiveinterference between box and triangle anomalies. I. INTRODUCTION
The QCD perturbation theory is not applicable to thelow-energy physics of hadrons (
E <
E < m ρ . Toextend the calculational scheme up to order O ( p ), it in-corporates the lowest resonance spin-1 states implement-ing the appropriate QCD short-distance constraints [6–8]. Another well-known approach is the famous Nambu– Jona-Lasinio (NJL) model [9, 10] which incorporatesthe dynamical mechanism of spontaneous chiral symme-try breaking in hadron matter. Later on, this originalidea of Nambu has been reinterpreted in terms of quarksand successfully implemented to the construction of thelocal effective meson Lagrangians including not only spin-0 states, but also the vector and axial-vector resonances[11–22].Our study here is based on the NJL model approach.The most successful results in this model are obtainedfor pseudoscalar and vector mesons. The description ofscalars and axial-vectors is more problematic and still re-quires additional theoretical efforts. The recent progresshere is related with the study of the anomalous radiativedecays of the axial-vector f (1285) and a (1260) mesons[23]. These vertices belong to the AVV-type and haveseveral restrictions from the QCD low-energy theorems:the Adler-Bardeen theorem [24], the Landau-Yang the-orem [25, 26], and so on. Presently, there is a growinginterest in their theoretical and experimental investiga-tions. This includes a recent measurement of the branch-ing fraction of the τ → f πν τ decay [27] and its theo-retical description given in the different approaches [28– ∗ [email protected]; † tex [email protected]; ‡ [email protected]; f ργ and f ωγ have been considered [33–35]. The vertex f γγ is important in the study of the hyperfine struc-ture of muonic hydrogen [36]. There are also predictionsfor the f γ and a γ production from the e + e − primarybeams [37]. All these studies should clarify the nature of f (1285) and a (1260) mesons.In this work, we calculate the f (1285) → ρ π + π − and a (1260) → ωπ + π − decay widths assuming the q ¯ q natureof f (1285) and a (1260). The first process has been con-sidered in [38] in the massive Yang-Mills approach, andin [39] in the generalized hidden symmetry framework.In both cases the decay channel f → ρ ρ → ρ π + π − has been neglected. We take into account this mode here.One of the purposes is to test the structure of an effective AV V -vertex, obtained recently in [23], in the case whenone of the particles is off mass shell. The other goal isto study the structure of the box anomalous diagrams.Their contribution interferes with the triangle anomalies.We show that the result of this interference is controlledby the QED Ward identities through a mechanism of thevector meson dominance (VMD). The latter issue hasbeen also addressed in [40]. We are not aware of workswhere the decay width of a (1260) → ωπ + π − has beenobtained. So, we hope that our study of this mode ishelpful as a benchmark for future measurements.In our calculations we use the effective meson La-grangian derived by bosonization of the NJL quark model[15, 16]. The local vertices of this Lagrangian arise inthe long wavelength regime through a gradient expansionof the one-loop quark diagrams [41]. The coefficients ofthe gradient expansion [masses and coupling constants]are expressed in terms of the model parameters, i.e. areknown. That is essentially different from [39] where am-plitudes of f (1285) → ρ + π − π and a (1260) → ωπ + π − decays have been expressed in terms of unknown couplingconstants. To make a progress here one should calculatethese couplings. The NJL model gives such a possibility.We restrict our consideration to the tree-level Feynmandiagrams. It makes conclusions to be valid to lowest or-der in 1 /N c , where N c is the number of colors in QCD. a r X i v : . [ h e p - ph ] M a y Let us remind that resonances are narrow for large N c ,with widths of order 1 /N c [42–44]. This implies that oneshould neglect widths in the tree-level amplitudes unlessresonances reach their on-shell peaks in the physical re-gion. Since the zero-width propagators do not lead to theone-particle poles when one integrates over phase-spaceof the f (1285) → ρ π + π − and a (1260) → ωπ + π − de-cays, we may stay at leading 1 /N c order, and work in thezero-width approximation. This approximation can beimproved by considering the next to leading order correc-tions in 1 /N c . Such a step certainly would allow to takeinto account the finite widths of resonances. However, itwould also require to take into account the one-loop me-son diagrams. This seems too complicated for an initialstudy of these processes. That is why this issue will notbe addressed here.Can f (1285) → ρ π + π − decay be successfully de-scribed in leading order of 1 /N c expansion? It is quiteplausible that this is possible. The reasoning is that atree-level approximation has been already used in [38]and [39]. The derivation in [38] led to a rather low valuefor the decay width Γ( f (1285) → ρ π + π − ) (cid:39) m a = 1275 MeV, and m a = 1200 MeV corre-spondingly] compared to the experimental value quotedby the Particle Data Group (PDG) [45]Γ exp ( f (1285) → ρ π + π − ) = 2 . +0 . − . MeV . (1)However, this value is not actually a leading order 1 /N c result [the authors took into account a finite width ofthe a (1260) meson, which is a non-leading contribution].The zero-width calculations made in [39] showed that1 /N c expansion can be relevant to the question.Despite the obvious similarity of f (1285) → π + π − γ and f (1285) → ρ π + π − decays, the role of the interme-diate vector ρ (770) and axial-vector a (1260) states hereis different. In the radiative decay, the contribution ofthe ρ (770) exchange is dominated by the real pole in thephysical region [40]. On the contrary, a kinematic regionfor the process f (1285) → ρ π + π − is such that a tree-level amplitude has no one-particle pole. It makes the ρ -exchange contribution to be rather small. On the con-trary, a nearby on-shell singularity of the a propagatorenhances the a -exchange channel.The paper is organized as follows. In Sec. II we reviewbriefly the NJL model and establish our notations. InSec. III we derive the decay width of f (1285) → ρ π + π − .The different channels are analyzed in detail. Sec. IV isdevoted to the a (1260) → ωπ + π − mode. We follow herethe same strategy as for the f (1285) decay. We end witha short summary and conclusions in Sec. V. The moralseems to be that to describe decay widths of the processesconsidered one can use the leading order of the large N c expansion. However, if one wants to obtain the detailedinformation about other characteristics of a (1260) and f (1285) mesons one should go beyond the leading orderresult. II. THE LAGRANGIAN OF THE NJL MODEL
Let us consider the extended NJL model with the U (2) L × U (2) R chiral symmetric four quark interactions[13]. The Lagrangian density L = ¯ q ( iγ µ ∂ µ − M ) q + L S + L V , (2) L S = G S (cid:2) (¯ qq ) + (¯ qiγ (cid:126)τ q ) (cid:3) , (3) L V = − G V (cid:2) (¯ qγ µ (cid:126)τ q ) + (¯ qγ µ γ (cid:126)τ q ) (cid:3) (4)includes spin-0, G S , and spin-1, G V , four-quark cou-plings; M = ˆ mτ , ˆ m = ˆ m u = ˆ m d are the current quarkmasses (the isospin symmetry is assumed); τ is a unit2 × (cid:126)τ are the SU (2) Pauli matrices; γ µ are thestandard Dirac matrices in four dimensional Minkowskispace; in the notation of the quark field q the color,isospin and 4-spinor indices are suppressed.After introducing bosonic variables in the correspond-ing generating functional one obtains the equivalentbi-linearised form of multi-quark interactions, i.e., theYukawa-type vertices describing the couplings of the col-lective meson fields with the quark-antiquark pairs. Forour purpose here we need only the following part of theLagrangian density∆ L int = g ρ qγ µ (cid:2) γ (cid:0) f µ + (cid:126)τ(cid:126)a (cid:48) µ (cid:1) + ( ω µ + (cid:126)τ (cid:126)ρ µ ) (cid:3) q + ig π ¯ qγ (cid:126)τ(cid:126)πq. (5)Here q is the constituent quark field with up and downflavours; the (cid:126)π , (cid:126)ρ µ and ω µ are the field operators asso-ciated with the iso-triplet of pions π (140), vector ρ (770)and ω (782)-mesons; f µ describes the iso-singlet axial-vector f (1285)-meson (for simplicity we take f (1285)to be the ideally mixed combination, corresponding toits status as an axial ω ), and (cid:126)a (cid:48) µ stands for the unphys-ical axial-vector fields that should be redefined to avoidthe (cid:126)π − (cid:126)a (cid:48) µ mixing.Since the free part of the meson Lagrangian follow-ing from evaluation of the one-quark-loop self-energy dia-grams must preserve its canonical form, one should renor-malize the bare meson fields by introducing the Yukawacoupling constants g π and g ρ in Eq. (5). To absorb infini-ties of self-energy graphs, these couplings depend on thedivergent integral I which is regularized in a standardway [15] g ρ = (cid:114) I , g π = (cid:114) Z I , (6)where I = − i N c (2 π ) (cid:90) d k θ (Λ + k )( m − k ) = N c (4 π ) (cid:20) ln (cid:18) m (cid:19) − Λ Λ + m (cid:21) . (7)As usual, we assume that the quantum corrections arevalid only when the relevant momenta are less than thecut-off momentum Λ, which also has the meaning ofthe characteristic scale of spontaneous chiral symmetrybreaking, defining through the gap equation m − ˆ m = mG S I , (8)where I = N c π (cid:20) Λ − m ln (cid:18) m (cid:19)(cid:21) , (9)the masses m of constituent quarks q . It is assumed thatthe strength of the quark interactions is large enough, G S > π / ( N c Λ ), to generate a non-trivial, m (cid:54) = 0,solution of Eq. (8) [even if the current quarks would bemassless]. The non-zero value of m is held to signal thecondensation of quark-antiquark pairs in the vacuum, i.e.dynamical chiral symmetry breaking.The parameter Z in (6) appears as a result of elim-ination of the (cid:126)π − (cid:126)a (cid:48) transitions. For that one shouldredefine the axial-vector field (cid:126)a (cid:48) µ = (cid:126)a µ + (cid:114) Z κm∂ µ (cid:126)π, (10)where (cid:126)a µ represents a physical state a (1260). A dimen-sional parameter κ , related with Z by 1 − κm = Z − ,should be fixed by requiring that the meson Lagrangiandoes not contain the (cid:126)π − (cid:126)a µ mixing. It gives12 κ = m + 116 G V I = m + m ρ m a , (11)where the last two equalities are a consequence of themass formulas of the model.The model has four parameters: G S , G V , ˆ m , and Λ.To fix them we use the following empirical data. Fromthe ρ → ππ decay width we find that α ρ = g ρ / (4 π ) = 3.It gives I = 1 / (8 π ), and Λ /m = 4 .
48. Using the massof the ρ meson as a second input value, m ρ = 775 MeV,we find G V from the mass formula of the ρ meson G V = 38 m ρ I = 3 πm ρ = 1 . × − MeV − . (12)The coupling constant g π fulfilles at the quark levelthe celebrated Goldberger-Treiman relation g π = m/f π ,where f π = 93 MeV is a coupling of the π − → µ − ¯ ν µ weakdecay which we use as a third input. Then Eq. (6) gives6 m = Zg ρ f π . (13)Using (11) this equation can be transformed to the for-mula m ρ = (cid:18) ZZ − (cid:19) g ρ f π (14)that gives Z = 2 . κm = 0 . m = 344 . . m = 1544 . m π = 138 MeV, we are left with the system of two equa-tions, Eq. (8) and a pion mass formula m π = ˆ mg π mG S = ˆ mmG S f π , (15)to find the values of the current quark mass ˆ m , and thecoupling G S . Solving this system, we obtain G S = m m π f π + m I = 3 . × − MeV , (16)ˆ m = m (1 − G S I ) = 1 .
55 MeV . (17)It follows then that the mass of the a meson is given by m a = √ Zm ρ = 1146 MeV. This result agrees well withthe Weinberg’s prediction m a = √ m ρ = 1096 MeV [46]made on the basis of spectral-function sum rules, whichare valid in QCD for m π = 0, and KSRF formula [47, 48]for the ρ coupling to the isospin current [in our case thereis a similar relation (14)]. It also agrees with a theoreticalanalysis of [49], where the excellent agreement with ourpresent experimental knowledge of τ → πππν τ spectrumand branching ratio [50] has been obtained and the char-acteristics of a meson have been carefully extracted, giv-ing m a = 1120 MeV, and Γ a = 483 MeV. On the otherhand, our result is a little low compared to the value m a = 1230 ±
40 MeV quoted by the Particle Data Group[45]. About the larger value of the a (1260) mass hasbeen recently reported by the COMPASS collaboration: m a = 1298 +13 − MeV/c with Γ a = 400 +0 − MeV/c [51].Notice that their data are accumulated from the studyof the channel p + π − → π − π − π + + p recoil , for whichCOMPASS has acquired the so far world’s largest datasetof roughly 50M exclusive events using an 190 GeV/c π − beam.In the following, the necessary effective meson vertices[together with the corresponding coupling constants] willbe obtained from (5) by calculating the one-quark-loopdiagrams and taking out from them only the leadingterms in the derivative expansion which dominate in thelong-wavelength approximation. The decay amplitudesare given by a sum of tree-level diagrams involving theexchange of physical mesons. This approach is consis-tent with a picture arising in the large N c limit of QCD[42–44]. III. THE PROCESS f (1285) → ρ π + π − The partial width for the observed decay mode of theaxial-vector meson f (1285) → ρ π + π − can be estimatedin the NJL model by considering the following tree-levelcontributions: (a) the vector ρ -meson exchange channel f → ρ ρ → ρ π + π − ; (b) the axial-vector a ± -mesonexchange channel f → π ± a ∓ → π ± π ∓ ρ ; (c) the directdecay which is described by the quark box diagram. A. Kinematic invariants, the physical region and astructure of the amplitude
In the discussion of the decay f ( l ) → ρ ( p )+ π + ( p + )+ π − ( p − ) we will use the standard invariant quantitieswhich can be constructed from 4-momenta of particles l, p, p + and p − , namely s = ( l − p ) = ( p + + p − ) ,t = ( l − p + ) = ( p + p − ) ,u = ( l − p − ) = ( p + p + ) . (18)Only two of them are independent variables, because ofthe relation s + t + u = h , where h = m f + m ρ + 2 m π .From the law of conservation of 4-momentum one findsthe intervals for physical values of these variables4 m π ≤ s ≤ ( m f − m ρ ) , ( m ρ + m π ) ≤ t, u ≤ ( m f − m π ) . (19)The equation t − t ( h − s ) + 14 (cid:2) ( h − s ) − D ( s ) (cid:3) = 0 , (20)where D ( s ) = 1 s ( s − m π ) λ ( s, m f , m ρ ) , (21) λ ( x, y, z ) = ( x − y − z ) − yz = [ x − ( √ y + √ z ) ][ x − ( √ y − √ z ) ] , (22)defines a curve which is the boundary of the physical re-gion for the decay channel. There are two positive valuesof t for each value of s . These two roots of the quadraticEq. (20) are the endpoints of the closed interval for phys-ically permissible values of t − ≤ t ≤ t + t ± = 12 (cid:16) h − s ± (cid:112) D ( s ) (cid:17) . (23)Notice, that D (4 m π ) = D (( m f − m ρ ) ) = 0.One can see that the ρ -resonance exchange contri-bution has no pole at physical values of meson masses: m π = 138 MeV, m ρ = 775 MeV, m f = 1282 MeV. In-deed, the one-particle pole in the amplitude, if it is, comesout of the factor ( m ρ − s ) − . However, the physical val-ues of s belong to the interval 2 m π ≤ √ s ≤ m f − m ρ ,or numerically 276 MeV ≤ √ s ≤
507 MeV, which is quitedistant from the ρ -meson mass.The a meson exchange amplitudes include one of thefactors ( m a − t ) − , or ( m a − u ) − . The physical re-gion of the kinematic variables t and u is such that913 MeV ≤ √ t, √ u ≤ a meson. In particular,this channel will dominate if the mass of the a is aboutthe model estimate m a = 1146 MeV. On the contrary,at large values of m a = 1230 − a exchangemay lead approximately to the same order contribution as the ρ exchange. This reasoning show that the decaymode f → ρ π + π − may supply us with interesting in-formation on the a -meson characteristics.The amplitude of the process [as it follows from theNJL model calculations below] may be parametrized as T = ie µναβ (cid:15) β ( l ) (cid:15) ∗ γ ( p ) (cid:2) g αγ (cid:0) F l µ p ν + + F l µ p ν − + F p µ + p ν − (cid:1) + F p α l γ p µ + p ν − (cid:3) , (24)where (cid:15) β ( l ), (cid:15) γ ( p ) are the polarization vectors of the f and ρ mesons. In the following we will obtain the explicitexpressions for the form factors F a , a = 1 , , , /N c andderivative expansions. The different channels contributeto the sum independently F a = F ( ρ ) a + F ( a ) a + F ( d ) a . (25)Here, F ( ρ ) a is a contribution of the ρ -exchange channel(a), F ( a ) a describes the axial-vector a ± exchange mode(b), and the direct interaction (c) is presented by theform factor F ( d ) a . B. The ρ (770) exchange channel The resonance exchange mode f → ρ ρ → ρ π + π − in the NJL model can be described by the following La-grangian densities.The anomalous f ρ ρ vertex can be easily obtainedfrom the f ρ γ vertex [23]. For that one should replacethe electromagnetic field by the ρ field, electric charge e by the coupling g ρ , and introduce the factor 1 / ρ -meson states in the Lagrangian.As a result we obtain L f ρ ρ = g ρ N c πm ) e µναβ ρ µν (cid:18) ρ σα ↔ ∂ σ f β (cid:19) , (26)where ( a ↔ ∂ µ b ) = a∂ µ b − ( ∂ µ a ) b , and ρ µν stands for thefield strength ρ µν = ∂ µ ρ ν − ∂ ν ρ µ .Notice, that the effective vertex L f ρ ρ is given by thenext to the leading order term in the derivative expansionof the anomalous quark triangle diagram f ρ ρ shown inFig. 1. Actually, one would expect here the contributionlinear in momenta. Bose symmetry requires that it wouldhave a form L (cid:48) f ρ ρ ∝ e αβµν f α ρ µ ∂ β ρ ν . (27)This form, however, is not compatible with the ideaof vector dominance. In the real world with electro-magnetic interactions included, this vertex would gen-erate the gauge symmetry breaking contributions to the f → ρ γ and f → γγ amplitudes. So, in fact, (27) isnot consistent with the QED Ward identities. Let us alsonotice, that a superficial linear divergence appears in thecourse of evaluation of the overall finite f ρ ρ triangle f
00 +_
FIG. 1. The Feynman diagrams describing the ρ exchangemode for the f (1285) → ρ π + π − decay. It is assumed [for allfigures in the text], that each pion line represents the sum oftwo types of couplings of the pion with the quark-antiquarkpair: the pseudoscalar one ¯ qγ πq and the axial-vector one¯ qγ µ γ ∂ µ πq . integral. Shifts in the internal momentum variable of theclosed quark loop integrals induce an arbitrary finite sur-face term contribution of the type (27). Thus, one canalways choose the free coupling of the surface term tovanish (27). This avoids contradiction with Ward identi-ties.The nonanomalous ρππ vertex in Fig. 1 is describedby the Lagrangian density L ρ π + π − = − ig ρ ρ µ ( π + ↔ ∂ µ π − )+ ig ρ Z − m a ρ µν ∂ µ π + ∂ ν π − , (28)where, in the following, we neglect the second term in(28). The reasoning for this is that it has a small factor s ( Z − / (2 m a ) = sκm /m ρ [compared with the factor1 of the first term], which varies from 0.03 to 0.1 in thekinematic region of s .With the use of these Lagrangian densities we find the ρ exchange contribution to the amplitude of the processshown in Fig. 1. The result is F ( ρ )1 = (cid:32) α ρ m (cid:33) m f + m ρ − m f ( ε − ε − ) m ρ − s ,F ( ρ )2 = (cid:32) − α ρ m (cid:33) m f + m ρ − m f ( ε − ε + ) m ρ − s ,F ( ρ )3 = (cid:32) α ρ m (cid:33) m f + m ρ − m f εm ρ − s ,F ( ρ )4 = (cid:32) − α ρ m (cid:33) m ρ − s , (29)where ε, ε ± are the energies of the rho meson and chargedpions in the rest frame of the f (1285)-meson.Notice that this channel [through the ρ → γ tran-sition] gives the determining contribution to the de-cay width of f (1285) → π + π − γ [40]. Conversely, thediagram shown in Fig. 1 is not so important for the f +_+_ + _ a FIG. 2. Two Feynman diagrams describing the a +1 and a − exchange modes for the f (1285) → ρ π + π − decay. f (1285) → ρ π + π − decay. Indeed, its contribution tothe decay width is Γ( f (1285) → ρ π + π − ) = 37 keV. Weconclude that this channel is strongly suppressed in com-parison with a exchange channel [as it will be shown inSec. III C], but it is still worth to be taken into accountdue to their constructive interference. C. The a (1260) exchange channel To describe the a exchange modes f → π ± a ∓ → π + π − ρ , shown in Fig. 2, we use the nonanomalous La-grangian density [52] L a πρ = if π g ρ Z (cid:2) ρ µ a − µ π + + 1 m a (cid:0) a − µν ρ µ − a − µ ρ µν (cid:1) ∂ ν π + (cid:21) + h.c., (30)and the vertex which describes the anomalous f a π in-teraction L f a π = g a e αβµν f α ∂ µ (cid:126)a β ∂ ν (cid:126)π, (31)where g a = α ρ πf π (cid:2) − a ) κm (cid:3) . (32)The second term in the square brackets is due to thereplacement (10). The derivative coupling ¯ qγ µ γ ∂ µ (cid:126)π(cid:126)τ q makes the corresponding triangle quark diagram linearlydivergent, although the result of its evaluation is finite.As a consequence of this superficial divergence, an ar-bitrary finite surface term contribution proportional to(1 − a ) appears. Here a is a dimensionless constant, con-trolling the magnitude of an arbitrary local part [53, 54].A corresponding contribution to the amplitude (24) isgiven by T ( a ) = − ig a (cid:18) κm f π (cid:19) e µναβ (cid:15) β ( l ) (cid:15) ∗ α ( p ) l µ p ν + (cid:20) p m a − t (cid:21) − ( p + ↔ p − ) . (33)One can see that the contact part of this amplitude [thefirst term in the square brackets] would violate the gaugeinvariance, if one, following the idea of vector-mesondominance, switches to the related electromagnetic pro-cess [notice, that the second term in the square bracketsdoes not contribute to the radiative decay f (1285) → π + π − γ , because p = 0 for a real photon]. Indeed, in-troducing the 4-vector q ν = ( p + − p − ) ν , and using thefour-momentum conservation law l = p + p + + p − , oneobtains e µναβ l µ q ν = e µναβ (cid:0) p µ q ν − p µ + p ν − (cid:1) . If one replaces (cid:15) ∗ α ( p ) → p α in (33), one finds that the term ∝ p µ + p ν − survives. This violates Ward identities. Thepoint can be settled after considering the direct (box)part of the amplitude shown in Fig. 3 [see Sect. III D].To summarize, two diagrams with the a exchangeyield F ( a − )1 = − g a f π (cid:0) κm (cid:1) (cid:34) m ρ m a − t (cid:35) ,F ( a +1 )2 = g a f π (cid:0) κm (cid:1) (cid:34) m ρ m a − u (cid:35) . (34) D. The direct channel
Let us consider now the contribution to the decay am-plitude f (1285) → ρ π + π − due to the quark box dia-grams shown in Fig. 3. As usual, we will extract onlythe terms which are dominant at large distances, i.e. thelocal effective vertices with the smallest number of deriva-tives. This contribution contains information on the boxAAAV anomaly. The calculations performed in a wayexplained above lead us to the amplitude T ( d ) = i α ρ πf π e µναβ (cid:15) β ( l ) (cid:15) ∗ α ( p ) (cid:2) (1 − κm ) p µ q ν − κm (4 − κm ) p µ + p ν − (cid:3) . (35)It can be easily seen that if we again resort to the ra-diative decay f → π + π − γ amplitude the term ∝ p µ + p ν − will break the gauge symmetry. The most efficient way ofdealing with the issue is to sum all contact contributionsand fix the free parameter a by requiring the vanishingof the p µ + p ν − term. Combining contact terms of Eqs. (33)and (35), we find T ( a ) cont + T ( d ) = T ( c ) = iα ρ πf π e µναβ × (cid:15) β ( l ) (cid:15) ∗ α ( p ) (cid:0) A p µ q ν + A p µ + p ν − (cid:1) , (36)where A = 1 − κm − κm (cid:2) − a ) κm (cid:3) ,A = ( κm ) (5 − a ) . (37)At a = 5 /
12 one finds that A = 0. This solves theproblem. This gives for A A = 1 − κm + 12 ( κm ) = 2 − ZZ + ( Z − Z . (38) f +_ FIG. 3. The box Feynman diagrams for the f (1285) → ρ π + π − decay. We do not show the diagrams which can beobtained by permuting the final states. Thus, the contact terms contribute to the amplitudeas F ( c )1 = − F ( c )2 = 12 F ( c )3 = α ρ πf π A . (39)Correspondingly, the diagrams plotted in Figs. 2, 3 givethe following contributions to the pertinent form factors F ( a )1 + F ( d )1 = F ( c )1 − (cid:32) α ρ (cid:33) (4 − κm ) m a − t ,F ( a )2 + F ( d )2 = F ( c )2 + (cid:32) α ρ (cid:33) (4 − κm ) m a − u ,F ( a )3 + F ( d )3 = F ( c )3 , (40)where the relation g a = κm f π = α ρ m ρ (cid:0) − κm (cid:1) (41)has been used.Before we will present the result of our calculations infull details, it is instructive to show here the dominantrole of the a -exchange contribution. In fact, as it fol-lows from Eq. (39), the sum of contact contributions isnegligible: Γ ( c ) ( f (1285) → ρ π + π − ) = 1 . a = 5 /
12. The sizeof this effect is quite large. To understand how this works,let us compare the a -exchange (33), calculated with a = 5 /
12, Γ ( a ) ( f (1285) → ρ π + π − ) = 3 .
87 MeV withEq. (40), which gives lower value Γ ( d + a ) ( f (1285) → ρ π + π − ) = 2 .
22 MeV. The difference between these twonumbers is an effect of the box diagram, which is takeninto account in the latter case.
E. The f (1285) → ρ π + π − decay width The rate of the three-body decay f (1285) → ρ π + π − can be obtained from the standard formula d Γ = | T | m f (2 π ) dεdε + (42)where | T | = (cid:88) i ≤ j Re (cid:0) F i F ∗ j (cid:1) T ij , (43) F i = F ( ρ ) i + F ( a ) i + F ( d ) i , (44)and T = m f (cid:0) (cid:126)p + ∆ (cid:1) ,T = m f (cid:0) (cid:126)p − + ∆ (cid:1) ,T = 2[( p + p − ) − m π ] + ( m f + m ρ )∆ ,T = m f (cid:126)p ∆ ,T = 2 m f (2 (cid:126)p + (cid:126)p − − ∆) ,T = 4 m f [ m π ε − − ( p + p − ) ε + ] − m f ∆ ,T = − m f [ m π ε + − ( p + p − ) ε − ] + 2 m f ∆ ,T = − T = − m f ε ∆ ,T = 2 m f ( εm f − m ρ )∆ . (45)Notice that m ρ ∆ = ( (cid:126)p + × (cid:126)p ) = ( (cid:126)p − × (cid:126)p ) = ( (cid:126)p + × (cid:126)p − ) = (cid:126)p (cid:126)p − ( (cid:126)p + (cid:126)p ) . (46)Here all kinematic variables are given in the rest frameof the f meson. In this reference system the invariantvariables are s = m f + m ρ − m f ε,t = m f + m π − m f ε + ,u = m f + m π − m f ( m f − ε − ε + ) . (47)Thus, the physical region for independent variables ε and ε + is given by the inequalities m ρ ≤ ε ≤ m f (cid:0) m f + m ρ − m π (cid:1) ,m f − ε − (cid:112) Ω( ε )2 ≤ ε + ≤ m f − ε + (cid:112) Ω( ε )2 (48)whereΩ( ε ) = ( ε − m ρ ) (cid:32) − m π m f + m ρ − m f ε (cid:33) . (49)Integrating in (42) over energies taken in the givenintervals (48) we find that the decay width of the process f (1285) → ρ π + π − isΓ( f (1285) → ρ π + π − ) = 2 .
78 MeV . (50)Thus, the picture can be summarized as follows. The a -exchange gives the major contribution because it isenhanced by a nearby singularity of the a propagator.The box diagram almost cancels the contact part of (33)reducing decay width on 46%. The ρ -exchange (29) issmall but its interference with other channels increasesthe result from Γ ( d + a ) = 2 .
22 MeV to the final value(50). This value is obtained in the leading order of 1 /N c expansion and agrees well with empirical data (1). + _ a FIG. 4. A typical Feynman diagram describing the ρ -exchange mode for the a (1260) → ωπ + π − decay. IV. THE PROCESS a (1260) → ωπ + π − The calculation of the decay amplitude a ( l ) → ω ( p ) + π + ( p + ) + π − ( p − ), where l, p, p + , p − are the 4-momentaof corresponding particles, can be carried out in a sim-ilar way as was being done for the f (1285) → ρ π + π − decay in Sec. III. The amplitude accumulates contribu-tions from three different processes: (a) the ρ exchangechannel a → ωρ → ωπ + π − ; (b) the ρ ± exchange a → π ± ρ ∓ → π + π − ω ; and (c) the direct decay mode a → ωπ + π − . The kinematic variables and the phys-ical region can be easily obtained from the expressionspresented in Sec. III A and Sec. III E. A. The ρ exchange mode On the theoretical side, the only difference between f (1285) → ρ ρ → ρ π + π − and a (1260) → ωρ → ωπ + π − decay amplitudes is the replacement of f ρ ρ quark triangle by the a ωρ one [compare Fig. 1 andFig. 4]. These vertices are originated by the same quark-loop diagram, including an overall factor which comes outfrom the isospin trace calculations. In the case of a ωρ vertex we have tr[( a τ )( ωτ )( ρ τ )]=2 a ωρ . Thatshould be compared with tr[( f τ )( ρ τ )( ρ τ )]=2 f ρ ρ .Thus, for the channel (a) one can write immediately F ( ρ )1 = (cid:32) α ρ m (cid:33) m a + m ω − m a ( ε ω − ε − ) m ρ − s , (51) F ( ρ )2 = − (cid:32) α ρ m (cid:33) m a + m ω − m a ( ε ω − ε + ) m ρ − s , (52) F ( ρ )3 = (cid:32) α ρ m (cid:33) m a + m ω − m a ε ω m ρ − s , (53) F ( ρ )4 = (cid:32) − α ρ m (cid:33) m ρ − s , (54)where ε ω is the energy of the ω (782) meson in the restframe of a (1260) meson. In this reference frame, we have s = m ω − m a (2 (cid:15) ω − m a ) [In the following, for simplicity,we put m ω = m ρ .] This channel gives rather low valueΓ( a → ωρ → ωπ + π − ) = 12 keV. B. The ρ ± exchange modes The amplitude which describes the process shown inFig. 5 is the analog of the a ± exchange modes (b) forthe f → ρ π + π − decay. Here, there is a common vertex a ρπ , where the a (1260)-meson is on-shell now L a − massa πρ = i (cid:18) κm f π (cid:19) a µ (cid:0) ∂ ν ρ + µν π − − ∂ ν ρ − µν π + (cid:1) . (55)Another vertex, ρωπ , which is responsible for theunnatural-parity decay process, is similar to the vertex a f π [see Eq. (31)]. L ρωπ = 3 g a e αβµν ω ν ∂ β (cid:126)ρ µ ∂ α (cid:126)π, (56)where a coupling constant g a is given by Eq. (32).From these Lagrangian densities we find the amplitude T ( ρ ± ) = T ( ρ + ) + T ( ρ − ) corresponding to the diagramsshown in Fig. 5 T ( ρ ± ) = ig a (cid:18) κm f π (cid:19) e ·· αβµν (cid:15) β ( l ) (cid:15) ∗ α ( p ) p µ p ν + × um ρ − u − ( p + ↔ p − ) . (57)This result differs from the one we had previously, consid-ering the a ± exchange contributions to the f → ρ π + π − amplitude. In particular, this amplitude vanishes if onemakes a replacement (cid:15) ∗ α ( p ) → p α . Therefore the ampli-tude is a gauge invariant expression, and it is not possibleto fix the ambiguity in g a by insisting that this symmetryis preserved [the transition to the radiative decay am-plitude a → γπ + π − does not lead to any restrictionson the parameter a ]. However, one can fix a from the f → ρ π + π − decay, as we did in Sec. III D. There wegot a = 5 /
12. In doing this, we also improve the de-scription of ρ ± → π ± γ decay in the NJL model. Let usremind that the decay width of this process is given byΓ( ρ ± → π ± γ ) = αg a πα ρ (cid:32) m ρ − m π m ρ (cid:33) . (58)So, at a = 5 /
12 we find that Γ( ρ ± → π ± γ ) = 78 keV.This is a little high compared to the experimental valueΓ( ρ ± → π ± γ ) = 67 . ± . ρ ± → π ± γ ) = 87 keV obtained in [15].Finally, using Eq. (57) and Eq. (41), we come to thefollowing form factors F ( ρ ± )1 = 3 α ρ m ρ (cid:0) − κm (cid:1) um ρ − uF ( ρ ± )2 = − α ρ m ρ (cid:0) − κm (cid:1) tm ρ − tF ( ρ ± )3 = F ( ρ ± )1 − F ( ρ ± )2 . (59)It gives Γ( a (1260) → π ± ρ ∓ → ωπ + π − ) = 517 keV. +_ +_+ _ a FIG. 5. Two Feynman diagrams describing the ρ + and ρ − exchange modes for the a (1260) → ωπ + π − decay. +_ a FIG. 6. The box Feynman diagram describing the direct modefor the a (1260) → ωπ + π − decay. We do not show the dia-grams which can be obtained by permuting the final states. C. The box diagrams
In Fig. 6 there is drawn a typical diagram that de-scribes the direct decay mode. It depicts the pro-cess where pions interact with quarks without deriva-tive ¯ qγ (cid:126)τ(cid:126)πq . There are also diagrams which include thederivative coupling of pions with quarks ¯ qγ µ γ ∂ µ (cid:126)π(cid:126)τ q . Inthe corresponding amplitude (60), the contribution ofeach coupling with a derivative is proportional to κm .We also do not show the diagrams which can be obtainedby permuting the final states, although we take them intoaccount. The result of calculations of all box diagramsin the leading order of derivative expansion is T ( d ) = i α ρ N c πf π e µναβ (cid:15) β ( l ) (cid:15) ∗ α ( p ) (cid:2) (1 − κm ) p µ q ν ++ ( κm ) p µ + p ν − (cid:3) . (60)The corresponding form factors are F ( d )1 = − F ( d )2 = α ρ N c πf π (cid:0) − κm (cid:1) ,F ( d )3 = α ρ N c πf π (cid:18) − κm + 12 ( κm ) (cid:19) . (61)It follows then that Γ( a (1260) → ωπ + π − ) box = 52 keV.As we already know from Sec. III D, the last term in(60) can be a source of the gauge symmetry breaking[through the VMD mechanism]. We have checked gaugeinvariance for the a → γπ + π − decay amplitude. Thissymmetry is protected by contributions, which are notgenerated by the VMD mechanism. The details will begiven in the separate paper. D. The a (1260) → ωπ + π − decay width We have already shown that diagrams in Fig. 5 yieldthe dominant contribution to the a (1260) → ωπ + π − decay width. Our aim now is to clarify the interferenceeffects.Let us consider first the sum of diagrams plotted inFigs. 5 and 6. The corresponding form factors can becombined in the following structures F ( ρ ± )1 + F ( d )1 = (cid:32) α ρ (cid:33) − κm m ρ − u + 3 F ( c )1 ,F ( ρ ± )2 + F ( d )2 = − (cid:32) α ρ (cid:33) − κm m ρ − t − F ( c )1 ,F ( ρ ± )3 + F ( d )3 = 3 α ρ πf π (cid:2) − κm + ( κm ) (cid:3) + (cid:32) α ρ (cid:33) (4 − κm ) (cid:18) m ρ − u + 1 m ρ − t (cid:19) , (62)where F ( c )1 is given by (39). From that we deduce thatthere is destructive interference between the amplitudesarising from these two channels. As a result, their con-tribution to the decay width turns out to be essentiallysuppressed Γ ( ρ ± )+( d ) = (517 + 52 − ρ -exchange amplitude ofFig. 4 and the sum of diagrams shown in Figs. 5 and6. This leads to a rather low value Γ( a → ωπ + π − ) =(12 + 243 − V. CONCLUSIONS
The purpose of this paper has been to use our knowl-edge of the structure of the triangle quark f ργ anomalyfor studying f (1285) → ρππ and a (1260) → ωππ anomalous decays, where similar vertices f ρρ and a ρω arise as a part of more sophisticated chiral dynamics.As a result, it has been found that theoretical estima-tion for the f → ρππ decay width [Γ( f → ρ π + π − ) =2 .
78 MeV] agrees well with the experimental value (1).It has been also obtained [for the first time] a theo-retical prediction for the rate of the a → ωππ decay,Γ( a → ωπ + π − ) = 87 keV.Both processes receive contributions from the boxAAAV anomaly which is carefully calculated here. Thisanomaly is less studied experimentally and can be aninteresting subject for future investigations. Our calcu-lations indicate clearly that there is a large interferencebetween box and triangle anomalies. The strong suppres-sion of the a → ωππ decay found in our work is a directconsequence of such destructive interference. It wouldbe informative to measure the rate of this decay. Fromthis one could learn about the structure of the a (1260)state. If the experimental result will support the valuefound in this paper, one can conclude that q ¯ q compo-nent is dominated in a (1260). If not, it will reinforcethe idea of the dynamical, or molecular, nature of the a (1260) meson [55–57]. In fairness it has to be said thatthe internal structure of the f (1285) meson is also notwell understood. Thus, the obtained agreement with theexperimental result for its decay ratio is a significant andnon-trivial argument in favour of q ¯ q content of f (1285).Our estimates are based on the local vertices of the ef-fective meson Lagrangian of the NJL model where mesonstates are treated as the nearly stable quark-antiquarkparticles. Following the idea of 1 /N c expansion we as-sumed that in the long-wavelength regime only local con-tributions with minimal number of derivatives are im-portant. We suppose that the qualitative and quantita-tive features that emerge in our simplified considerationwould persist in a more accurate calculation. This can bedone in the future as soon as new empirical data will beavailable. Nonetheless, the undoubted merit of the de-scribed results, as compared to the already known ones inthe literature, is that they are relied on the more detaileddynamical picture. [1] S. Weinberg, Phys. Rev. Lett. 18, 188 (1967).[2] S. Weinberg, Physica A 96, 327 (1979).[3] J. Gasser, H. Leutwyler, Phys. Lett. B 125, 321 (1983).[4] J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984).[5] J. Gasser, H. Leutwyler, Nucl. Phys. B 250, 465 (1985).[6] G. Ecker, J. Gasser, H. Leutwyler, A. Pich, E. de Rafael,Phys. Lett. B 223, 425 (1989).[7] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl.Phys. B 321, 311 (1989).[8] M. Knecht, A. Nyffeler, Eur. Phys. J. C 21, 659678(2001).[9] Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).[10] Y. Nambu, G. Jona-Lasinio, Phys. Rev. 124, 246 (1961).[11] T. Eguchi, Phys. Rev. D 14, 2755 (1976).[12] D. Ebert and M. K. Volkov, Yad. Fiz. 36, 1265 (1982).[13] D. Ebert and M. K. Volkov, Z. Phys. C 16, 205 (1983). [14] M. K. Volkov, Ann. Phys. 157, 282 (1984).[15] M. K. Volkov, Sov. J. Part. Nucl. 17, 186 (1986).[16] D. Ebert and H. Reinhardt, Nucl. Phys. B 271, 188(1986).[17] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195(1991).[18] S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992).[19] M. K. Volkov, Phys. Part. Nucl. 24, 35 (1993).[20] D. Ebert, H. Reinhardt and M. K. Volkov, Prog. Part.Nucl. Phys. 33, 1 (1994).[21] T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221(1994).[22] M. K. Volkov, A. E. Radzhabov, Phys. Usp. 49, 551(2006).[23] A. A. Osipov, A. A. Pivovarov and M. K. Volkov, Phys.Rev. D 96, 054012 (2017). [24] S. L. Adler, and W. A. Bardeen, Phys. Rev. 182, 1517(1969).[25] L. D. Landau, Dokl. Akad. Nauk., USSR 60, 207 (1948).[26] C. N. Yang, Phys. Rev 77, 242 (1950).[27] J. P. Lees et al.et al.