Mixing of pseudoscalar-baryon and vector-baryon in the J P =1/ 2 − sector and the N ∗ (1535) and N ∗ (1650) resonances
aa r X i v : . [ h e p - ph ] N ov Mixing of pseudoscalar-baryon and vector-baryon in the J P = 1 / − sector and the N ∗ (1535) and N ∗ (1650) resonances. E. J. Garzon and E. Oset Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad deValencia-CSIC, Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia,SpainOctober 27, 2018
Abstract
We study the meson-baryon interaction with J P = 1 / − using the hidden-gauge Lagrangiansand mixing pseudoscalar meson-baryon with the vector meson-baryon states in a coupled channelsscheme with πN , ηN , K Λ, K Σ, ρN and π ∆ (d-wave). We fit the subtraction constants of eachchannel to the S partial wave amplitude of the πN scattering data extracted from experimentaldata. We find two poles that we associate to the N ∗ (1535) and the N ∗ (1650) resonances and showthat the subtraction constants are all negative and of natural size. We calculate the branching ratiosfor the different channels of each resonance and we find a good agreement with the experimentaldata. The cross section for the π − p → ηn scattering is also evaluated and compared with experiment. Partial wave analyses of πN data [1, 2] have provided us with much data on amplitudes, cross sections andresonance properties. It has also been the subject of intense theoretical investigations (see Refs. [3, 4] forrecent updates on the subject). The introduction of the chiral unitary techniques to study these reactionsin Ref. [5] resulted in surprising news that the N ∗ (1535) resonance was dynamically generated from theinteraction of meson baryon, with a price to pay: coupled channels had to be introduced. Some of thechannels were closed at certain energies, like the K Λ and K Σ in the region of the N ∗ (1535), but they wereshown to play a major role in the generation of this resonance, to the point of suggesting in Ref. [5] thatthe N ∗ (1535) could qualify as a quasibound state of K Λ and K Σ. Work on this issue followed in Ref. [6],corroborating the main findings of Ref. [5], and posteriorly in Refs. [7, 8, 9, 10]. In the chiral unitaryapproach the loops of the Bethe-Salpeter equation must be regularized, and this is done with cut offs orusing dimensional regularization. The cut off, or equivalently the subtraction constants in dimensionalregularization in the different channels should be of “natural size”, as discussed in Ref. [10, 11, 12], ifone wishes to claim that the resonances have been generated dynamically from the interaction. However,this is not the case of the N ∗ (1535), where different cut offs in Ref. [5], or different subtraction constantsin Ref. [6] for different channels must be used. This is unlike the case of the Λ(1405), where a uniquecut off in all channels leads to a good reproduction of the data [13, 11, 14, 15]. This fact was interpretedin Ref. [10] as a manifestation of the nature of the two resonances, where the Λ(1405) would be largelydynamically generated, while the N ∗ (1535) would contain a nonnegligible component of a genuine state,formed with dynamics different from the pseudoscalar meson interaction. One might think of remnantsof an original seed of three constituent quarks, but this is not necessarily the case. It could also be dueto the missing of important channels different than pseudoscalar-baryon. Actually this has been a sourceof investigation recently, where the mixing of pseudoscalar-baryon and vector-baryon channels has ledto interesting results and some surprises. In Ref. [16] the vector-baryon interaction was studied usingthe method developed in Ref. [17] but mixing also pseudoscalar-baryon components. It was found thatthe mixing produced a shift of some of the resonance positions of Ref. [17] and led to some increasein the width. Similar results have been obtained recently in Refs. [18, 19, 20]. One of the interesting1 B V V V V V B B ( a ) ( b ) Figure 1: Vector meson-baryon interaction: (a) with vector meson exchange (b) contact term. πN ηN K Λ K Σ πN − ηN − − K Λ 0 0 K Σ 2Table 1: Coefficients of
P B transition with I = 1 / πN (d-wave), ρN , π ∆(s and dwaves) in the sector of spin-parity 3 / − with chiral dynamics led to a good reproduction of the πN datain d-waves and to the generation of the N ∗ (1520) and N ∗ (1700) resonances [21]. In Ref. [25] N ∗ (1535)and N ∗ (1650) are obtained with only pseudoscalar - baryon states using an offshell approach, which isin principle equivalent to having different subtraction constants in different channels. By means of thatone can effectively take into account the effect of missing channels. In our work we explicitly introducefrom the beginning a larger space of channels. In the present work we want to extend the results ofRef. [21] to the sector of 1 / − , with the aim to see if the mixture of the pseudoscalar-baryon and vector-baryon channels can remove the pathology observed by the need of different subtraction constants indifferent channels. We will show that this is the case and then we shall be able to conclude that themissing components of the wave function in the N ∗ (1535) noted in Ref. [10] are due to vector-baryonand additional π ∆ states that we shall also mix in the present coupled channel approach. The most important coupled channels of N ∗ (1535) and N ∗ (1650) are πN , ηN , K Λ, K Σ, ρN and π ∆(d-wave). Some of the matrix elements of the interaction between these channels have been well studiedas the P B → P B transition mediated by a vector meson exchange addressed in Ref. [6]. The diagraminvolved in this transition is show in Fig. 1 and the potential of this transition is given by V ij = − C ij f (cid:0) k + k ′ (cid:1) (1)The P B transition coefficients are taken from Ref. [6]. However, since those coefficients are in chargebasis we need to convert them to isospin basis, as show in Table 1. Similarly, the ρN → ρN transitionhas been studied in Ref. [21] and the coefficients are given in Appendix A of Ref. [17]. The transitionto V B → P B is implemented following the formalism described in Ref. [21], where the interactionis mediated by a pseudoscalar meson as shown in Fig. 2(c). Furthermore we also include the Kroll-Ruderman term shown in Fig. 2(e). The evaluation of the diagrams shown in Fig. 2 leads us to the2 a) ρN ( s ) → ρN ( s ) (b) π ∆( s ) → π ∆( s ) (c) ρN ( s ) → πN ( d )(d) ρN ( s ) → π ∆( s, d ) (e) ρN ( s ) → π ∆( s ) Figure 2: Diagrams of the channels involved in the calculation for N ∗ (1520) and N ∗ (1700).Figure 3: Diagrams of s − and u − channels exchange with the nucleon propagator.following vertices for the transitions (See Appendix A of Ref. [21] for details) t ρN ( s ) → πN ( s ) = − √ g D + F f ( ~q πN ( P V + q πN ) − m π + 1 ) (2) t ρN ( s ) → ηN ( s ) = 0 (3) t ρN ( s ) → K Λ( s ) = − √ g D + 3 F f ( ~q K Λ ( P V + q K Λ ) − m K + 1 ) (4) t ρN ( s ) → K Σ( s ) = − √ g D − F f ( ~q K Σ ( P V + q K Σ ) − m K + 1 ) (5)(6)where we take F = 0 .
51 and D = 0 .
75 [22, 23], an q i is the momentum of the pseudoscalar meson in thecenter of mass. The factor 1 that appears inside the braces corresponds to the Kroll-Ruderman vertex. Incomparison with the results of Ref. [24], where the authors only take into account the Kroll-Rudermannterm, we obtain the same coefficients for the P B → V B transition.Moreover we find interesting to include the contribution to the s-wave from the s − and u − channelscontaining the nucleon propagator, shown in Fig. 3, given by t πN → N → πN = ( E π ) (cid:18) D + F f (cid:19) (cid:18) √ s + M N − √ s − E π + M N (cid:19) (7) t ηN → N → ηN = ( E η ) (cid:18) √ D − F f (cid:19) (cid:18) √ s + M N + 1 √ s − E η + M N (cid:19) (8)3his is easily obtained by separating the relativistic nucleon propagator into positive and negative energycomponents /p + mp − m = ME ( ~p ) X r (cid:26) u r ( ~p )¯ u r ( ~p ) p − E ( ~p ) + iǫ + v r ( − ~p )¯ v r ( − ~p ) p + E ( ~p ) − iǫ (cid:27) (9)Then the positive energy part contributes to p-wave and the negative energy part to s-wave. We shouldnote that these terms, as well as a possible isoscalar seagull contribution [26, 27] give a very smallcontribution.On the other hand we have the transition of ρN → π ∆( d ) that has been already studied in Ref. [21].The diagram of this transition is given in Fig. 2(d) and the evaluation of this diagram gives the transitiongiven by t ρN ( s ) → π ∆( d ) = g √ f πN ∆ m π ( ~q ( P V + q ) − m π ) (10)Here we do not have the 1 factor from the Kroll-Ruderman term since this transition only involves L = 2.As done in Ref. [21] the transitions involving L = 2 are introduced with a parameter γ . This isdone for the diagonal transition π ∆( d ) → π ∆( d ) and for the transition channel that we consider relevant π ∆( d ) → πN ( s ). t π ∆( d ) → π ∆( d ) = − γ m π q π ∆ (11) t π ∆( d ) → πN ( s ) = − γ m π q π ∆ (12)The parameters are normalized with the corresponding power of the pion mass to be dimensionless.Both are parametrized, but since the π ∆( d ) → π ∆( d ) transition are both d-wave channels, the potentialhas four momenta while the transition π ∆( d ) → πN ( s ) is a transition from s-wave to d-wave has onlytwo momenta. As done before, the divergence of the momenta is controlled with the Blatt-Weisskopfbarrier-penetration factors as done in Ref. [21]. We have some unknown parameters in our theory that we need to determine fitting the data. First we havethe subtraction constant for each channel which, are expected to be around − µ = 630 MeV. We have also two undetermined parameters in the potential γ and γ corresponding tothe transition of π ∆( d ) → π ∆( d ) and π ∆( d ) → πN . We perform a fit of the S partial wave amplitudeof the πN scattering data extracted from experimental data of Ref. [28]. We need to normalize theamplitude of the T-matrix using Eq. (7) of Ref. [29], which relates our amplitude with the experimentalone by ˜ T ij ( √ s ) = − s M i q i π √ s s M j q j π √ s T ij ( √ s ) (13)where M is the mass of the baryon for the specific channel and q is the on-shell momentum. In Fig. 4we show the fit of both the real and imaginary parts of ˜ T for the diagonal channel of πN .In Table 2 we show the results obtained with the fit. As we commented before, we have used aregularization scale different for each channel, corresponding to the mass of the baryon of that channel.We consider interesting to show also the subtraction constant with a regularization scale of µ = 630MeV in order to compare them with other results found in the literature. It is interesting to see thatthe introduction of the ρN and π ∆ channels has had an important qualitative effect in the subtractionconstants, which now are all negative and of the same order of magnitude, while in Ref. [6] some of thesubtraction constants were even positive. Using the parameters determined with the fit, we evaluate the T matrix using the Bethe-Salpeter equationand show in Fig. 5 the result of | T | for all the diagonal channels. Analysing the T matrix, using themethod explained in detail in Ref. [21], we found two poles that can be associated to the resonances4igure 4: Fit to the data extracted from Ref. [28]. We show the real part (circles) and imaginary part(crosses) of the data and the result of our fit of ˜ T πN for the real (solid) and imaginary (dashed) parts. µ [MeV] a Nπ a Nη a Λ K a Σ K a Nρ a ∆ π γ γ M B -1.203 -2.208 -1.985 -0.528 -0.493 -1.379 0.595 1.47630 -2.001 -3.006 -3.128 -1.799 -1.291 -2.720 0.595 1.47Table 2: Parameters obtained with the fit. The first row are the parameters with a regularization scale µ that corresponds to the mass of the baryon of each channel. Second row are the same results but withthe natural regularization scale µ = 630 MeV. The parameters γ i are not changed.5igure 5: Results of the | T | matrix for the diagonal channels.6 ∗ (1535) N ∗ (1650)Channel g i | g i | g i G i g i | g i | g i G i Nπ Nη K K Nρ π Table 3: Couplings of the different channels to each resonance N ∗ (1535) and N ∗ (1650). This is a remarkable novelty, since in Ref. [6] the N ∗ (1535) appears but notthe N ∗ (1650). The poles are used to calculate the couplings of all channels to each resonances. Theseresults are compiled in Table 3. With the couplings one can determine the decay width and branchingratio to each channel of both resonances. The results of the branching ratios for the resonances N ∗ (1535)and N ∗ (1650) are shown in Tables 4 and 5 respectively. In the Tables we show the position of the polesand the branching ratios for each channel found in this work. We compare them with the experimentalresults of the PDG [30], and as the PDG average has big uncertainties we also compare them with singleresults of the experiments and analysis [31, 32, 33, 34].Looking at Table 3 we see that the ρN channel has the strongest coupling to N ∗ (1535) but theresonance is below the threshold, however due to the width of the ρ we can generate enough phase spaceand obtain a small width, but in a very good agreement with the experimental results, as seen in Table 4.The K Σ channel has a coupling as big as the ρN but as the resonance is below the threshold it hasnot phase space to decay. Similarly the coupling to channel K Λ is big but again there is not phasespace for decay. On the other hand the π ∆ channel has a very small coupling but around 200 MeV ofphase space, so this gives it a small branching ratio which agrees with the experimental values. Theother channels πN and ηN have smaller couplings but since they have much momentum to decay theyhave big branching ratios in good agreement with the experimental results. Concerning the width of the N ∗ (1535) in Table 4 we should note that, although the theoretical width obtained from the pole in thecomplex plane is smaller than the experimental one, the apparent width from Im ˜ T πN in the real axis,seen in Fig. 4, is much closer to the experiment.For the case of N ∗ (1650) the K Σ channel has now the biggest coupling to this resonance, but as theresonance is below the threshold it has no phase space for decay. The same as before happens to the ρN channel, the small momentum generated with the mass convolution of the ρ gives a small width but inqualitative agreement with the experimental value. Although the channels πN , ηN and π ∆ have smallercouplings, due to the huge phase space that they have, the branching ratios are quite big, which is ina very good agreement with the experimental results of PDG, but with single experiments as well, asseen in Table 5. Now, the K Λ channel is open and the value found for the branching ratio is in goodagreement with the only experimental value available of Ref. [32].We consider interesting to include in Table 3 the value of the wave function in coordinate space atthe origin, defined in Ref. [35] as (2 π ) / ψ i ( ~
0) = g i G i ( z R ) (14)where the G function is evaluated in the pole. This magnitude represents the wave function at theorigin for s-wave channels. For d-wave channels the wave function goes as r at the origin and vanishes.The magnitude gG then represents the relative strength of the channel for coupling of the resonance toexternal sources [36]. The results show information about how relevant is each channel for the resonances.The first surprise is to see that, although the K Σ channel has the second biggest coupling, the value ofthe wave function at the origin reveals that this channel is not relevant in the N ∗ (1535). We also seethat the most important channels are the ηN and πN , as one can expect of the experimental results ofthe branching rations. Moreover the K Λ channel has an important contribution but since the resonance7 ∗ (1535) J P = 1 / − Theory PDG Cutkosky Anisovich Vrana Thoma[31] [32] [33] [34]Re(Pole) 1508.1 1490 – 1530 1510 ±
50 1501 ± + − ±
80 134 ±
11 102 165 ± i / Γ(%)]N π (1077) 58.6 35 – 55 50 ±
10 54 ± ± ± η (1487) 37.0 42 ±
10 33 ± ± ± ρ (1714) 1.0 2 ± ± π (1370) 3.3 0 – 4 2.5 ± ± ± Table 4: Results for the pole position and branching ratios for the different channels of N ∗ (1535) J P =1 / − and comparison with experimental results.is below the threshold, this fact is not noticeable experimentally. For the N ∗ (1650) case the πN channelis now the most important but the ηN channel is very important as well. However we can see that nowthe K Σ has a very important contribution since the pole is very close to the K Σ threshold. The K Λhas a moderate relevance and this is in agreement with the experimental results for the width, as seenin Table 5. The ρN and π ∆ channels have a small relative contribution and this is in good agreementwith experimental values.One reaction that filters the I = 1 / J P = 1 / − , that we study here, is the π − p → ηn . In Fig. 6we show the result for the cross section of the π − p → ηn , where we can see a good agreement withthe experimental data of Ref. [37]. In comparison with the results of Ref. [6] for the same reaction, wesee a good improvement of our work, since in Ref. [6] the experimental data of the cross section above1550 MeV is not well reproduced. In this work we generate dynamically the N ∗ (1650) resonance, whichfills up the region of the cross section above 1550 MeV that is not well obtained in Ref. [6], where the N ∗ (1650) was not generated. We have studied the meson-baryon interaction with J P = 1 / − including the coupled channels consideredin the experimental analysis, πN , ηN , K Λ, K Σ, ρN and π ∆ (d-wave). We have studied the interactionusing the hidden gauge formalism, where the interaction is mediated by the exchange of vector mesons.Other extensions of this formalism involving pseudoscalar and vector mesons are also used as explainedin the text. The loops are regularized using dimensional regularization with subtraction constants foreach channel. These constants are treated as free parameters and fitted to reproduce the experimentaldata of the S πN scattering data extracted from Ref. [28].Two poles are found and the couplings for each channel, as well as the wave function at the origin, arecalculated. These couplings are used to obtain the branching ratios to all channels of both resonances.The results are then compared with several experimental values and there is a good agreement for mostof them. It must be noted that the consideration of the ρN and π ∆ channels has had an importantqualitative change with respect to the work of Ref. [6] where only the pseudoscalar-baryon octet channelswere considered. The first one is that now we are able to generate both the N ∗ (1535) and the N ∗ (1650)resonances, while in Ref. [6] only the N ∗ (1535) appeared. The second one is that now the subtractionconstants are all negative and of natural size. From the perspective of Ref. [10] we can say that theconclusion in Ref. [10] that the N ∗ (1535) had an important component of a genuine state in the wavefunction, can be translated now by stating that the missing components can be filled up by the ρN and8 ∗ (1650) J P = 1 / − Theory PDG Cutkosky Anisovich Vrana Thoma[31] [32] [33] [34]Re(Pole) 1672.3 1640 – 1670 1640 ±
20 1647 ± ± ±
30 103 ± ± i / Γ(%)]N π ±
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