τ→ ν τ ρ 0 π − decay in the Nambu - Jona-Lasinio model
ττ → ν τ ρ π − decay in the Nambu - Jona-Lasinio model A. A. Osipov ∗ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980, Russia
Within the context of an extended Nambu - Jona-Lasinio model, we analyze the role of the axial-vector a (1260) and a (1640) mesons in the decay τ → ν τ ρ π − . The contributions of pseudoscalar π and π (1300) states are also considered. The form factors for the decay amplitude are determinedin terms of the masses and widths of these states. To describe the radial excited states π (1300)and a (1640) we introduce two additional parameters which can be estimated theoretically, or fixedfrom experiment. The decay rate and ρπ mass spectrum are calculated. I. INTRODUCTION
Semihadronic decay modes of the tau lepton remain topresent date a topic of interest to theoreticians as wellas experimentalists [1]. One mode of particular interestis the decay τ → ν τ π + π − π − . This decay is governed bythe axial-vector hadronic current j Aµ and gives a uniquepossibility to scrutinize our understanding of chiral dy-namics in the energy range of 1 − J P C = 0 − + , ++ , ++ , ++ . The nature of some of thesestates is not yet well understood.The specific mode τ → ν τ ρ π − → ν τ π + π − π − , whichis the main subject of our present investigation, is mostsuitable to study the role of 0 − + and 1 ++ states inthe hadronization process. Besides the pion, these are π (1300), a (1260), and a (1640) resonances. In theNambu - Jona-Lasinio (NJL) model a (1260) is consid-ered to be a member of the basic axial-vector nonet, i.e. a (1260) is a pure q ¯ q state, with a (1640) being its firstradial excitation. The pseudoscalar π (1300) is the firstradial excitation of the pion. One of our goals here is toclarify the role of these resonances in the τ → ν τ ρ π − decay.In fact, the considered q ¯ q picture agrees with the lead-ing order of the 1 /N c expansion [at large N c , where N c isthe number of colors, mesons are pure q ¯ q states, ratherthan, for instance, qq ¯ q ¯ q [2, 3]]. Of course, a more detaileddescription of these states would require implementationof mixing scenarios, in which the q ¯ q components mixwith the four-quark components. This step requires totake into account the next to leading order 1 /N c correc-tions and will not be considered here. Let us also noticethat for the a (1260) axial-vector meson, there is no es-tablished understanding whether it is a quark-antiquarkstate or dynamically generated hadronic molecule [4–6].Thus, it is useful to study how far one can go with the q ¯ q picture of a (1260).Another goal of this work is to attract the attentionof experimentalists to the important information con-tained in the specific mode τ → ν τ ρ π − → ν τ π + π − π − , ∗ [email protected]; which is shown to be sensitive only to the a (1260) and a (1640) contributions. The experimental data on thespectral function [see Fig.3] would clarify the specific roleof the a (1640) state. It is quite difficult to study the a (1260) − a (1640) interference through the fit of the3 π invariant mass spectra of the τ → ν τ π + π − π − mode,because the corresponding amplitude has too many pa-rameters to fit [7]. The major subprocess of the channel τ → ν τ ρ π − → ν τ π + π − π − is the τ → ν τ ρ π − decay.The amplitude of this three-particle decay has much lessnumber of parameters.The relevant approximation to this question is the1 /N c expansion which provides the solid theoreticalgrounds for the description of the q ¯ q resonance states.In accord with this idea, all q ¯ q meson states [includ-ing q ¯ q -resonances] are stable, free, and non-interactingat N c = ∞ . It is from the point of view of the 1 /N c expansion the theoretical idea about an on-shell ρ (770)state makes the sense, and the τ → ν τ ρ π − decay am-plitude, at leading order, can be described by the treeFeynman diagrams.The τ → ν τ ρ π − → ν τ π + π − π − mode contains theall necessary information about the τ → ν τ ρ π − decay,that relates our study to the experiment. For instance,the sequential decay formula [8] which is correct in thenarrow width approximationΓ( τ → ν τ ρ π − → ν τ π + π − π − )= Γ( τ → ν τ ρ π − )Br( ρ → π + π − ) (1)and the fact that the ρ decays into π + π − to hundredpercent yield Γ( τ → ν τ ρ π − ) = Γ( τ → ν τ ρ π − → ν τ π + π − π − ). Since the latter value can be extractedfrom the data on τ → ν τ π + π − π − , the theoretical es-timate of Γ( τ → ν τ ρ π − ) has a definite sense [eg. re-cently [9], the theoretical result for Γ( η (cid:48) → ργ ) has beenused to quantify Γ( η (cid:48) → ργ → π + π − γ ) ]. It is nec-essary to notice that a similar situation occurs for the τ → ν τ ρ K − decay, where the sequential decay mode τ → ν τ ρ K − → ν τ π + π − K − has already been mea-sured [the PDG quoted value is Br(Γ( τ → ν τ ρ K − → ν τ π + π − K − )) = (1 . ± . × − [10]]. The measure-ment of the Br(Γ( τ → ν τ ρ π − → ν τ π + π − π − )) will notonly fill the gap in the existing data, but, as it is shown inthis work, will clarify the role of the a (1640) resonancein the underlying chiral dynamics.In the NJL model, there is a nonlocal extension which a r X i v : . [ h e p - ph ] F e b deals with the excited states of the 0 − + , ++ , −− and1 ++ ground state nonets [11, 12]. The nonlocal four-quark interactions lead to the nonlocal effective mesonLagrangian which describes the physics of these excitedstates. Nonetheless, here we will apply a more modestdescription, which should ideally arise from [11, 12] inthe large N c limit, namely we suppose that excited statesat leading 1 /N c order can be described by the local La-grangian, like, for instance, in the extended linear sigmamodel approach [13]. In this case the propagators of ex-cited states have the same form as the ground state prop-agators, but with different couplings to the weak axial-vector current. Such a simplified treatment of exitedstates is not new. Notably, it is exactly how the con-tribution to the τ → ν τ πππ amplitude from the vector ρ (1450) resonance exchange has been estimated in [14].One of the first theoretical studies of the role of the a (1260) axial-vector state in the τ → ν τ ρ π − decay ispresented in [15], where the current algebra sum ruleshave been used to clarify whether experimental data arecompatible with a contribution of the a (1260) resonanceto the τ → ν τ ρ π − decay mode. This decay has beenalso considered in [16], where the a (1260) dominancehas been revealed. There is no doubt, nowadays, aboutthe dominant role of a (1260) in this process. However,we still need to understand the nature and parameters ofthe a (1260) resonance. Besides that, the role of its radialexcited state a (1640) must be clarified. The measure-ments of the branching ratio and ρ π − mass spectrumcan provide insight into the issue.In this paper we calculate the τ → ν τ ρ π − decay am-plitude in the framework of NJL model with SU (2) × SU (2) chiral symmetry. The first attempt to use the NJLapproach to study this decay was made in [17], where thefinite terms in the derivative expansion of quark loopscorresponding to vertices a ρπ and ρππ have been takeninto account. Though the analysis in [17] allows to re-produce the experimental value of the τ → ν τ ρ π − decaywidth [mainly due to the contribution of finite terms]the procedure of extracting these finite terms, used in[17], is not compatible with the chiral symmetry restric-tions imposed on such contributions by the chiral invari-ant Schwinger-DeWitt expansion at large distances [theconsistent Schwinger-DeWitt approach requires also totake into account the finite terms of self-energy diagramswhich will redefine the coupling constants of the theory,what has not been done there]. As opposed to this, wedo not consider here the contributions of the problem-atic finite terms, but show instead that the decay canbe described by the standard effective meson Lagrangian[18–20] provided the first radial excitations π (1300) and a (1640) are taken into account.The material of the paper is presented in the followingway. In Sec. II we describe the relevant meson verticesof the effective Lagrangian, obtain the amplitude of the τ → ν τ ρ π − decay, and discuss the partial conservationof the axial-vector current (PCAC). This important re-lation should be fulfilled in the chiral approach. In Sec. III the radial excited states are considered. We showthat the inclusion of these states can be done withoutcontradiction with the PCAC condition. In Sec. IV weintroduce the momentum dependent widths of the reso-nances and calculate the differential decay width of theprocess. In Sec. V the results of our numerical calcula-tions are presented in Tables I and II and Fig. 3. Weconclude with Sec. VI. II. LAGRANGIAN AND AMPLITUDE
Our starting point is the effective meson Lagrangianobtained on the basis of the NJL model with the global U (2) R × U (2) L chiral symmetric four-quark interactionswhich also possesses the gauge SU (2) L × U (1) R symme-try of the electroweak interactions. For convenience, werefer to the paper [21] which contains all necessary detailsrelated with the obtention of such effective Lagrangian.The appropriate weak hadronic part of the Lagrangiandensity is L = − G F V ud l µ (cid:32) m ρ g ρ a +1 µ + f π ∂ µ π + (cid:33) + H.c. , (2)where G F is the Fermi constant, V ud (cid:39) cos θ c is an el-ement of the Cabibbo-Kobayashi-Maskawa matrix, θ c =13 ◦ is the Cabibbo angle, l µ = ¯ ν τ γ µ (1 − γ ) τ is the weakcharged lepton current, m ρ is the mass of the ρ (770) me-son, g ρ is the coupling which arises due to a redefinitionof the spin-1 fields, f π = 92 MeV is the pion decay con-stant.Notice that the Lagrangian density (2) has the stan-dard form of the axial-vector dominance, i.e. it does nothave a contact term ∼ ρ π + . In the covariant formulation[21] the corresponding part of the hadronic weak axial-vector current j Aµ is proportional to π + ( ρ µ − ∂ ν ρ µν /m ρ ).This factor is zero on the mass shell of the ρ -meson. Thisdoes not mean that the matrix element (cid:104) ρπ | j Aµ | (cid:105) has nocontact part. Hereafter, in Eq.(7), it is clear that it does[see the first term ∼ g µν ]. This term is effectively origi-nated by the a -exchange contribution (cid:104) ρπ | a (cid:105)(cid:104) a | j Aµ | (cid:105) .Thus the fact that the on-shell ρ (770) cancels the directterm ∝ ρπ in j Aµ is a part of the fundamental mechanismwhich is responsible for the PCAC relation in the model.The hadronic weak axial-vector current j Aµ in formula(2) can be also obtained in the standard noncovariant ap-proach [22] by using the variational method of Gell-Mannand L´evy [23, 24]. However, it requires some work, be-cause the direct application of this technique leads onlyto the pion exchange. To arrive at the axial-vector dom-inance one should use the Lagrangian equations for theaxial-vector field, and after that neglect the total deriva-tives of the antisymmetric tensors. Another way to ob-tain the Lagrangian density (2) is described in [25].Thus we need only two additional vertices to find theamplitude of the τ → ν τ ρ π − decay. This is the a ρπ FIG. 1. The Feynman diagram describing the contribution ofthe mode τ − → ν τ a − → ν τ ρ π − to the decay amplitude (6).FIG. 2. The Feynman diagram describing the contribution ofthe mode τ − → ν τ π − → ν τ ρ π − to the decay amplitude (6). vertex L a ρπ = i Zf π g ρ tr { a µ [ ρ µ , π ]+ 1 m a ( ρ µν [ a µ , ∂ ν π ] + a µν [ ρ µ , ∂ ν π ]) (cid:27) , (3)where ρ µν = ∂ µ ρ ν − ∂ ν ρ µ , Z = (1 − m /m a ) − , m is themass of the constituent quark; m a = √ Zm ρ is the massof the a (1260) meson; it is assumed that all fields arecontracted with Pauli matrices, for instance, π = π i τ i ,and trace is calculated over products of tau-matrices.Notice that the Lagrangian density (3) obtained in thecovariant approach [21] coincides with the result of thestandard noncovariant approach [22].The second vertex that we need, in the covariant ap-proach, is given by the Lagrangian density L ρππ = − ig ρ m ρ (cid:18) Z + 1 Z (cid:19) tr { ∂ µ ρ ν [ ∂ µ π, ∂ ν π ] } . (4)On the mass shell of the ρ -meson it yields L ρ − massρππ = ig ρ (cid:18) Z + 1 Z (cid:19) tr { ρ µ [ ∂ µ π, π ] } . (5)Now we have all necessary ingredients to find the am-plitude A of the τ ( Q ) → ν τ ( Q (cid:48) ) + ρ ( p ) + π − ( q ) decay,where Q, Q (cid:48) , p and q are the 4-momenta of the particles.The corresponding Feynman diagrams are shown in Figs.1 and 2.In this way we have A = − iG F V ud g ρ f π ¯ ν τ ( Q (cid:48) ) γ µ (1 − γ ) τ ( Q ) F µ , (6)where the pure hadronic part is given by the 4-vector F µ = g µν + m ρ g µν − k µ k ν m a m a − k + (cid:18) Z (cid:19) k µ k ν m π − k (cid:15) ∗ ν ( p ) . (7) Here (cid:15) ν ( p ) is a polarization vector of the ρ -meson, and k = q + p . The invariant subsidiary condition on thecomponents of the vector state is assumed p ν (cid:15) ν ( p ) = 0.Notice that k µ F µ = (cid:32) m ρ m a + Z + 1 Z k m π − k (cid:33) k ν (cid:15) ∗ ν ( p )= Z + 1 Z m π m π − k k ν (cid:15) ∗ ν ( p ) , (8)i.e. k µ F µ = (cid:104) πρ | ∂ µ j Aµ | (cid:105) is dominated by the pion polein accord with PCAC.We further stress the presence of a contact contributionin (7). The first term with g µν results from the diagramof Fig.1. Its appearance is partly due to the first term ofthe Lagrangian density (3).The most general Lorentz-covariant form of F µ is F µ = (cid:15) ∗ µ ( p ) F + ( p + q ) µ ( (cid:15) ∗ q ) F − + ( p − q ) µ ( (cid:15) ∗ q ) F + . (9)In particular, the NJL model yields F = 1 + m ρ m a − k , F + = 0 ,F − = 1 + g A m π − k − g A m a − k , (10)where g A = 1 /Z . Thus, the NJL approach is quite re-strictive: the form factor F + does not contribute at lead-ing order in 1 /N c and derivative expansions. All formfactors are the functions of only one variable k . III. RADIALLY EXCITED STATES AND PCAC
The region between 1 . − . ρ π − spectrumis still poorly described by the standard NJL model. Onecan improve our description of the τ → ν τ ρ π − decayamplitude by including the contributions of the radiallyexcited states of the pion and the a (1260) meson, i.e.the π (cid:48) = π (1300) and a (cid:48) = a (1640) resonances. Fol-lowing [14] we perform the substitutions in the pion and a (1260) propagators:1 m π − k →
11 + β π (cid:48) (cid:18) m π − k + β π (cid:48) m π (cid:48) − k (cid:19) , m a − k →
11 + β a (cid:48) (cid:32) m a − k + β a (cid:48) m a (cid:48) − k (cid:33) . (11)Notice that the limit β → m π (cid:48) → m π and m a (cid:48) → m a lead to the same result. The substitutions are written interms of physical states, therefore, the coupling β is theonly parameter which absorbs contributions arising dueto the redefinition of the primary meson fields [this in-cludes the diagonalization of π − π (cid:48) and a − a (cid:48) quadraticforms and the pseudoscalar - axial-vector mixing effects].As a result, the factor 1 / (1 + β ) rescales the contributionof the ground state.Doing these replacements, we must ensure that sub-stitutions (11) do not destroy the PCAC condition. Forthat, together with (11), one should modify the contactterm g µν → (1 + δ ) g µν , (12)where δ is the constant approximating the higher-mass1 ++ contribution to F in such a way that the PCACcondition is fulfilled.Indeed, in this case the modified hadronic part of theamplitude A → A (cid:48) , F µ → F (cid:48) µ has the form F (cid:48) µ = (cid:15) ∗ ν ( p ) [(1 + δ ) g µν + m ρ β a (cid:48) g µν − k µ k ν m a m a − k + β a (cid:48) g µν − k µ k ν m a (cid:48) m a (cid:48) − k + 1 + g A β π (cid:48) k µ k ν (cid:18) m π − k + β π (cid:48) m π (cid:48) − k (cid:19)(cid:21) . (13)The divergence of the hadronic current k µ F (cid:48) µ should van-ish in the chiral limit m π →
0. This requirement is ful-filled only if δ is uniquely fixed as δ = g A β a (cid:48) β a (cid:48) (cid:32) − m a m a (cid:48) (cid:33) . (14)As a result we obtain that k µ F (cid:48) µ = 1 + g A β π (cid:48) (cid:18) m π m π − k + β π (cid:48) m π (cid:48) m π (cid:48) − k (cid:19) ( (cid:15) ∗ k ) . (15)This is a modified PCAC relation, which, in particular,tells us that in the chiral limit β π (cid:48) = f π (cid:48) /f π →
0, where f π and f π (cid:48) are the weak decay constants of the π and π (cid:48) mesons.Thus, the consideration above shows that to lowest or-der in 1 /N c our procedure introduces only four additionalparameters: the two masses of π (1300) and a (1640) res-onances and two mixing parameters β π (cid:48) and β a (cid:48) whichshould be fixed theoretically or from experimental data. IV. DECAY WIDTH
Let us proceed now with calculations of the decay rate.For that we need the appropriate spin-averaged matrixelement squared | A (cid:48) | = (2 G F V ud g ρ f π ) [2( QF (cid:48) )( Q (cid:48) F (cid:48) ) − ( F (cid:48) F (cid:48) )( QQ (cid:48) )] . (16)Here we averaged over initial τ -lepton states. It is eas-iest to perform the calculation in invariant form beforespecializing to the rest frame of tau. The invariant Man-delstam variables are s = ( Q − Q (cid:48) ) = k , t = ( Q − q ) , and u = ( Q − p ) . It gives | A (cid:48) | = G F V ud (1 − g A ) (cid:8) | F (cid:48)− | m τ ( m τ − s ) λ ( s, m ρ , m π )+4 | F (cid:48) | (cid:2) t + t ( s − m π − m ρ ) + m ρ ( m τ − s + m π ) (cid:3) +2Re( F (cid:48) F (cid:48)∗− ) (cid:2) ( m τ − s ) (cid:0) ( s − m π ) − λ ( s, m ρ , m π ) (cid:1) + m τ (cid:0) t ( s + m ρ − m π ) − m ρ ( m ρ + 4 m τ ) (cid:1) + m ρ s (cid:0) s + m π ) − m ρ (cid:1)(cid:3)(cid:9) , (17)where the NJL model relation (1 − g A ) m ρ = g ρ f π hasbeen used, and the K¨all´en function λ ( x, y, z ) is definedas follows λ ( x, y, z ) = ( x − y − z ) − yz .In the physical range ( m ρ + m π ) ≤ √ s ≤ m τ the formfactors in Eq. (13) contain zero-width a , a (cid:48) and π (cid:48) prop-agator poles, which lead to divergent phase-space inte-grals in the calculation of the τ → ν τ ρ π − decay width.In order to regularize the integrals one should includethe finite widths of these resonances through the typicalBreit-Wigner form of propagators. This is a step beyondthe leading order in the 1 /N c expansion which we areforced to make in connection with the above-mentionedproblem. We consider the substitutions:1 m R − k → m R − k − im R Γ R ( k ) . (18)A k dependence for Γ R ( k ) is required by unitarity. Thedescription of a set of resonances with the same quantumnumbers as a sum of Breit-Wigner amplitudes may vio-late unitarity and is a good approximation only for well-separated resonances with little overlap. This conditionis fulfilled here.Following [15], we have chosen to use the formΓ R ( k ) = Γ R (cid:115) λ ( k , m ρ , m π ) λ ( m R , m ρ , m π ) m R + k R k + k R , (19)where Γ R = Γ R ( m R ). The function Γ R ( k ) has a thresh-old factor in the proper position, i.e. at k = ( m ρ + m π ) .The value of k R is determined by the following integralcondition1 π ∞ (cid:90) ( m ρ + m π ) dk Im (cid:2) m R − k − im R Γ R ( k ) (cid:3) − = 1 . (20)In the narrow-width approximation this equation is au-tomatically fulfilled. If the resonance is broad, Eqs. (19)and (20) make our results less sensitive to the details ofΓ R ( k ). A rigorous form can only be obtained if the totalwidth is completely understood, but this is not the caseat the moment.The differential decay rate can be written in the form d Γ ds = (cid:90) t + ( s ) t − ( s ) dt | A (cid:48) | m τ (2 π ) , (21)where t ± ( s ) = 12 (cid:20) m τ + m ρ + m π − s + m τ s ( m ρ − m π ) ± (cid:112) D ( s ) (cid:105) , (22) D ( s ) = (cid:18) m τ s − (cid:19) λ ( s, m ρ , m π ) . (23)The integral over t in (21) can be done explicitly: d Γ ds = ( G F V ud ) m τ (2 π ) (cid:112) D ( s ) (cid:18) m τ s − (cid:19) (1 − g A ) × (cid:26) m ρ | F (cid:48) | (cid:20) s + 2 m τ sm ρ λ ( s, m ρ , m π ) + m τ + 2 s (cid:21) + m τ λ ( s, m ρ , m π ) (cid:0) s | F (cid:48)− | + 2Re( F (cid:48) F (cid:48)∗− ) (cid:1)(cid:9) . (24)Integrating this expression over s one finally obtains the τ → ν τ ρ π − decay width. V. NUMERICAL RESULTS
Our consideration above shows that we are able to de-scribe the tree τ → ν τ ρ π − decay amplitude in terms ofthe known masses: m π , m ρ , m a , m π (cid:48) , m a (cid:48) , m τ , two mix-ing parameters β π (cid:48) , β a (cid:48) , and three widths Γ a , Γ a (cid:48) , Γ π (cid:48) .The value of g A is not free due to the mass formula g A m a = m ρ which is valid in the NJL model. This pa-rameter is also related with the value of the constituentquark mass m , which, in the case of exact isospin sym-metry m = m u = m d , is given by m = m ρ √ (cid:114) g A − . (25)In the following we will vary the value of m a in theinterval 1120 MeV ≤ m a ≤ g A will be changed correspondingly. The upperboundary is inspired by the recent measurements of theCOMPASS Collaboration m a = 1299 +12 − MeV, Γ a =380 ±
80 MeV [26, 27]. The lower boundary is a resultof a comparison between the theoretical m π -spectra ofthe τ → ν τ π + π − π − decay [14] with ALEPH data [28],that yields m a = 1120 MeV and Γ a = 483 ±
80 MeV.The PDG averaged values: m a = 1230 ±
40 MeV,Γ a = 250 −
600 MeV [10], and the parameters ex-tracted by the JPAC group m a = 1209 ± +12 − MeV,and Γ a = 576 ± +80 − MeV [29] are also considered.
A. Ground states contribution
Our numerical calculations we start from the simplestcase, when only ground states are considered. With thispurpose we use the form factors given by Eq. (10) modi-fied by the substitutions (18). As can be seen from TableI, the higher value of m a and the lower value of Γ a , TABLE I. The width of the τ → ν τ ρ π − decay obtained inthe NJL model with only the ground state contributions. Thefirst two columns contain the phenomenological input valuesof m a and Γ a taken from [10, 26–29].set m a (MeV) Γ a ( m a ) (MeV) Γ τ → ν τ ρ π − (MeV)a) 1299 300 0.67 × − b) 1299 380 0.58 × − c) 1230 250 0.72 × − d) 1230 400 0.52 × − e) 1209 576 0.40 × − f) 1120 483 0.35 × − the better agreement with experimental data. We makethis conclusion by confronting our results to the old valueof the branching ratio Br( τ → ν τ ρ π − ) = (5 . ± . τ → ν τ ρ π − mode]. This corresponds to the fol-lowing decay widthΓ expτ → ν τ ρ π − = (1 . ± . × − MeV . (26)It is worth pointing out that the latest measurements ofthe CLEO Collaboration of the τ → ν τ πππ decay [31, 32]and the results of the COMPASS experiment in diffrac-tive production [26, 27, 33] can be used [although thisis a model dependent procedure which is also influencedby the production mechanism] to extract the branchingratios of the specific decay channels. In particular, theJPAC [29] made a rough estimate for the dominant ρπS -wave channel. The found branching ratio is of 60% –80%. It corresponds to the decay widthΓ JP ACτ → ν τ ρ π − = (1 . − . × − MeV . (27)In the sets (a) and (b) of the Table I we use the inputdata of the COMPASS Collaboration [26, 27]. Two val-ues of Γ a are considered: the lowest one Γ a = 300 MeV,and the central one Γ a = 380 MeV. The PDG averagedvalues (c) and (d) [10] are selected in the same way. Theinput (e) is taken in accord with the results of the JPACgroup [29]. In the set (f) the output of the analyses [14]is used.To summarize the above, it should be noted that, theground states contribution, where the a (1260) exchangedominates, is too low to explain the experimental dataon Γ τ → ν τ ρ π − . The best estimate here is given by theset (c), but even this prediction of the NJL model isslightly below the lower boundary of the experimentalvalue (26). Therefore, one should take into account theexited states which also contribute to the τ → ν τ ρ π − decay amplitude in leading 1 /N c order. B. Exited states contribution
Let us turn now to the study of the contributions aris-ing from the exited π (1300) = π (cid:48) and a (1640) = a (cid:48) states to clarify their role in the decay τ → ν τ ρ π − .The characteristics of π (1300) quoted by the PDGare m π (cid:48) = 1300 ±
100 MeV, and Γ π (cid:48) = 200 −
600 MeV[10]. The impact of this state on the τ → ν τ ρ π − decay width is controlled by the parameter β π (cid:48) . Inaccord with the PCAC relation, one can expect that | β π (cid:48) | ∼ ( m π /m π (cid:48) ) = 0 .
01 [11]. This is too small tohave an appreciable impact on the τ → ν τ ρ π − decaywidth. Hence, the only contribution which may affectthe description presented in Table I is the contributionof the exited a (1640) state.The PDG lists the a (1640) as ”omitted from summarytable”, nonetheless they give the average world values m a (cid:48) = 1654 ±
19 MeV, and Γ a (cid:48) = 240 ±
27 MeV. Ontop of that, the COMPASS Collaboration has recentlyreported on the Breit-Wigner a (cid:48) -resonance parameters: m a (cid:48) = 1700 +35 − MeV, and Γ a (cid:48) = 510 +170 − MeV [27].It is worth mentioning that the value of the param-eter β a (cid:48) is not so strongly suppressed as β π (cid:48) . This fol-lows from the crude estimate | β a (cid:48) | ∼ ( m a /m a (cid:48) ) = 0 . β ρ (cid:48) considered in [14] in the context of an effec-tive description of the role of the ρ (cid:48) = ρ (1450) exited stateof ρ (770) gives | β ρ (cid:48) | ∼ ( m ρ /m ρ (cid:48) ) = 0 .
28, in harmonywith their result β ρ (cid:48) = − .
25, obtained by fitting experi-mental data]. In the following, the free parameter β a (cid:48) willbe fixed in accord with our estimate β a (cid:48) = − .
56 above.Notice the increase of the impact of the ground state a (1260) due to the factor 1 / (1 + β a (cid:48) ) in (11). Again,the similar effect took place for the ground ρ (770) statecontribution when the exited state ρ (cid:48) = ρ (1450) had beentaken into account [14].Our goal now is to show that the known experimentaldata allow for a meaningful evaluation of the impact ofthe a (cid:48) -resonance propagator on the τ → ν τ ρ π − decay.To this end, we consider the sets of experimentally knowncharacteristics of a and a (cid:48) resonances. The results ofsuch numerical calculations are collected in Table II.In the sets (a) and (b), the data of the COMPASS Col-laboration are considered [27]. Notice that COMPASShas performed the so far most advanced partial-waveanalysis of diffractively produced π + π − π − final states,using the isobar model. That has allowed them, in par-ticular, to determine mass and width of a and a (cid:48) reso-nances with high confidence. Their interpretation of a (cid:48) as a first radial excitation of a is in line with our theo-retical consideration.In Fig.3 we show the typical behaviour of the spectralfunction (24) for the case (b), which agrees well both withthe experimental value (26) and with the JPAC estimate(27). The a (1640) resonance contributes mostly throughits interference with a (1260). The distractive interfer- × - × - × - FIG. 3. The predicted spectral distribution (24) (in GeV − on the ordinate) as a function of s (in GeV on the abscissa).The parameters correspond to the set (b) in Table II.TABLE II. The effect of the exited axial-vector a (1640) stateon the τ → ν τ ρ π − decay width obtained in the NJL modelwith the use of Eq.(11) and Eq.(18), where β π (cid:48) = 0 and β a (cid:48) = − .
56. The ground state contribution Γ ( a ) τ → ν τ ρ π − is shown forcomparison. The masses and widths are given in MeV.set m a Γ a m g A m a (cid:48) Γ a (cid:48) Γ ( a ) τ → ν τ ρ π − Γ τ → ν τ ρ π − a) 1299 300 426 0.356 1700 510 1 . × − . × − b) 1299 380 426 0.356 1700 510 1 . × − . × − c) 1230 250 390 0.397 1654 240 2 . × − . × − d) 1230 400 390 0.397 1654 240 1 . × − . × − e) 1209 576 379 0.411 1654 240 1 . × − . × − f) 1120 483 330 0.479 1654 240 1 . × − . × − ence suppresses the a -exchange contribution Γ ( a ) τ → ν τ ρ π − on 16%.The sets (c) and (d) are based on the PDG averagedvalues: m a = 1230 ±
40 MeV, Γ a = 250 −
600 MeV, and m a (cid:48) = 1654 ±
19 MeV, Γ a (cid:48) = 240 ±
27 MeV [10]. Thewidth of a shows large uncertainties, but large valuesfor the width are known to be ruled out by COMPASSmeasurements. Thus, in our estimates, we use two valuesΓ a = 250 MeV and 400 MeV. The latter value is morepreferable. The distractive interference suppresses the a -exchange contribution Γ ( a ) τ → ν τ ρ π − on 20%.In the set (e) we use the a parameters extracted bythe JPAC group m a = 1209 ± +12 − MeV, and Γ a =576 ± +80 − MeV [29]. In the set (f) the data of thetheoretical fit for the ground state of a are considered[14]. In both cases the characteristics of a (cid:48) are takenfrom the PDG [10].Let us summarize the results presented in Table II andFig. 3.1) The a (1260) resonance dominates the τ → ν τ ρ π − process, while the a (1640) contributes less than the 20%.2) The contribution of the π (1300) resonance to the τ → ν τ ρ π − decay is negligible. This is a directconsequence of the PCAC relation. In particular, thevalue β π (cid:48) = 0 in Table II can be replaced by β π (cid:48) = − ( m π /m π (cid:48) ) = − .
01 without a noticeable effect. Tohave a noticeable effect the value of β π (cid:48) should be about β π (cid:48) = − .
4. At this stage, however we do not see anyvalid theoretical reason why | β π (cid:48) | would be so large.3) The comparison of Tables I and II shows that theinclusion of the excited axial-vector a (1640) state is anecessary element for the successful description of the τ → ν τ ρ π − decay width. A reason for the improvementis contained in the substitutions (11) and (18) which in-crease substantially the contribution of the a groundstate, although this growth is partly suppressed due toa destructive interference with the exited a (cid:48) state. Thisour conclusion agrees with the CLEO Collaboration re-sult [31]. Their studies show that adding of the a (cid:48) terminto the Breit-Wigner function improves significantly theagreement with the τ → ν τ π data.4) The set c) overestimates the τ → ν τ ρ π − decayrate. This is a consequence of a very low a -resonancewidth Γ a = 250 MeV. The other sets with larger valuesof Γ a are in agreement with the experimental value (26).Apparently, this points out that the value Γ a (cid:39)
400 MeVis preferable one. This observation is consistent with thedeterminations from COMPASS.5) The spectral distribution shown in Fig. 3 is a pre-diction of the NJL model. It is assumed that this resultwould be checked in the study of the tau decay into threepions and neutrino, where events with pion pairs over the ρ (770) mass would be selected. VI. CONCLUSIONS
The purpose of this paper has been to describe the τ → ν τ ρ π − decay by using an extended U (2) L × U (2) R chiralsymmetric NJL model with spin-0 and spin-1 four-quarkinteractions. The channel τ → ν τ ρ π − → ν τ π − π − π + dominates the tau decay into three pions and neutrino.That explains our interest to the problem.We have used the covariant approach [21] to describethe weak interactions of mesons in leading order in 1 /N c and derivative expansions. It has been shown that in thisapproximation the axial-vector current is dominated bythe a (1260) and the pion exchanges. However, we havefound that the τ → ν τ ρ π − decay width is too low if the physical value of Γ a is considered.The contributions of the first radial excitations of thepion and the a states have been taken into account toimprove the description. For that we supplemented theregular π and a propagators with new terms correspond-ing to the propagators of excited π (1300) and a (1640)states. Our treatment of these excitations is similar tothe successful description of the ground ρ (770) and ex-cited ρ (1450) vector resonances in [14]. The momentumdependent off shell widths of all resonances have beenapproximated by the functions introduced in paper [15].This procedure can be further elaborated as soon as thenew more precise experimental data will be reported onthe τ → ν τ ρ π − decay.As a result, we obtain that the contribution of the π (1300) resonance is negligible, and conclude that thechannel τ → ν τ ρ π − → ν τ π − π − π + is a source of suffi-ciently clear information on a (1260) and a (1640) states.The a (1260) resonance dominates the intermediate pro-cess, while the a (1640) contributes less than 20%. InTable II, we present our estimations for the decay widthΓ( τ → ν τ ρ π − ) which correspond to the different inputvalues of a and a (cid:48) characteristics. The spectral distri-bution shown in Fig. 3 can be used for comparison withthe data, as soon as those would be available.Our result indicates on the important role which the a (1640) state plays in the theoretical description of this τ → ν τ ρ π − decay. It means, in particular, that oneshould carefully estimate its contribution and role in the τ → ν τ π − π − π + decay. This will be done somewhereelse. The results obtained here could be useful for suchstudies. ACKNOWLEDGMENTS
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