In this talk we report on the use of a chiral unitary approach for the
interaction of the octets of meson and baryon and the octet of mesons with the
decuplet of baryons. Two octets of $J^{\pi}=1/2^-$ baryon states and a singlet
are generated dynamically in the first case, resulting in the case of
strangeness $S=-1$ in two poles of the scattering matrix close to the nominal
$\Lambda(1405)$ resonance. In the second case many resonances are also
generated, among them an exotic baryon with S=1 corresponding to a $\Delta K$
resonance. We make suggestions of experiments which could show evidence for the
existence of these states.Comment: Talk at the MENU04 Symposium, Beijing, September 200

Jido, D.

Meissner, Ulf-G.

Oller Berber, José Antonio

Oset Báguena, Eulogio

Ramos Gómez, Àngels

Sarkar, S.

Vicente Vacas, Manuel José

English

arXiv

In this talk we report on the use of a chiral unitary approach for the interaction of the octets of meson and baryon and the octet of mesons with the decuplet of baryons. Two octets of J(pi) = 1/2(-) baryon states and a singlet are generated dynamically in the first case, resulting in the case of strangeness S = -1 in two poles of the scattering matrix close to the nominal A(1405) resonance. In the second case many resonances are also generated, among them an exotic baryon with S = I corresponding to a Delta K resonance. We make suggestions of experiments which could show evidence for the existence of these states

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/0411036v1  2 Nov 2004August 4, 2013 8:17 WSPC/INSTRUCTION FILE menuInternational Journal of Modern Physics Ac© World Scientific Publishing CompanyDYNAMICALLY GENERATED RESONANCES IN THE CHIRALUNITARY APPROACH TO MESON BARYON INTERACTIONE. OSET, S. SARKAR, M.J. VICENTE VACASDepartamento de F´ısica Teo´rica and IFIC, Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigacio´n de Paterna, Aptd. 22085, 46071 Valencia, SpainA. RAMOSDepartament d’Estructura i Constituents de la Mate`ria, Universitat de Barcelona,Diagonal 647, 08028 Barcelona, SpainD. JIDOPhysik-Department, Technische Universita¨t Mu¨nchen D-85747 Garching, GermanyJ.A. OLLERDepartamento de F´ısica, Universidad de Murcia, 30071 Murcia, SpainU.G. MEISSNERUniversita¨t Bonn, Helmholtz-Institut fu¨r Strahlen- und Kernphysik (Theorie)Nußalle 14-16, D-53115 Bonn, GermanyReceived (Day Month Year)Revised (Day Month Year)In this talk we report on the use of a chiral unitary approach for the interaction of theoctets of meson and baryon and the octet of mesons with the decuplet of baryons. Twooctets of Jpi = 1/2− baryon states and a singlet are generated dynamically in the firstcase, resulting in the case of strangeness S = −1 in two poles of the scattering matrixclose to the nominal Λ(1405) resonance. In the second case many resonances are alsogenerated, among them an exotic baryon with S = 1 corresponding to a ∆K resonance.We make suggestions of experiments which could show evidence for the existence of thesestates.Keywords: Keyword1; keyword2; keyword3.1. Description of the meson baryon interactionsThe introduction of unitarity constraints in coupled channels in chiral perturbationtheory has led to unitary extensions of the theory that starting from the sameeffective Lagrangians allow one to make predictions at much higher energies . Oneof the interesting consequences of these extensions is that they generate dynamicallylow lying resonances, both in the mesonic and baryonic sectors. By this we mean that1August 4, 2013 8:17 WSPC/INSTRUCTION FILE menu2 Authors’ Namesthey are generated by the multiple scattering of the meson or baryon components,much the same as the deuteron is generated by the interaction of the nucleonsthrough the action of a potential, and they are not preexistent states that remainin the large Nc limit where the multiple scattering is suppressed.Starting from the chiral Lagrangians for the interaction of the octets of mesonand baryon 1 and using the N/D method to obtain a scattering matrix fulfillingexactly unitarity in coupled channels 2, the full set of transition matrix elementswith the coupled channels is given in matrix form byT = [1− V G]−1 V . (1)Here, the matrix V , obtained from the lowest order meson–baryon chiral Lagrangian,contains the Weinberg-Tomozawa or seagull contribution, as employed e.g. in Ref. 3,Vij = −Cij 14f2(2√s−Mi −Mj)(Mi + E2Mi)1/2 (Mj + E′2Mj)1/2, (2)For strangeness S = −1 the Cij coefficients are given in Ref. 4, and an averagedmeson decay constant f = 1.123fpi is used3, with fpi = 92.4MeV the weak piondecay constant.The diagonal matrix G stands for the loop function of a meson and a baryonand is defined by a subtracted dispersion relation in terms of phase space with acut starting at the corresponding threshold 2. It corresponds to the loop functionof a meson and a baryon once the logarithmic divergent constant is removed. Theanalytical properties of G are properly kept when evaluating the previous loop func-tion in dimensional regularization and an explicit subtraction constant,ai, appearsin the expression.This meson baryon loop function was calculated in Ref. 4 with a cut-off regu-larization. The values of the ai constants are found to be around −2 to agree withthe results of the cut–off method for cut–off values of the order of the mass of theρ(770) 2, which we call of natural size.2. Poles of the T-matrixThe study of Ref. 3 showed the presence of poles in Eq. (1) around the Λ(1405)and the Λ(1670) for isospin I = 0 and around the Σ(1620) in I = 1. The sameapproach in S = −2 leads to the resonance Ξ(1620) 5 and in S = 0 to the N∗(1535)6,7. One is thus tempted to consider the appearance of a singlet and an octet ofmeson–baryon resonances. Nevertheless, the situation is more complicated becauseindeed in the SU(3) limit there are two octets and not just one, as we discuss below.The presence of these multiplets was already discussed in Ref. 2 after obtaining apole with S = −1 in the I = 1 channel, with mass around 1430 MeV, and two poleswith I = 0, of masses around that of the Λ(1405).The appearance of a multiplet of dynamically generated mesons and baryonsseems most natural once a state of the multiplet appears. Indeed, one must re-call that the chiral Lagrangians are obtained from the combination of the octet ofAugust 4, 2013 8:17 WSPC/INSTRUCTION FILE menuInstructions for Typing Manuscripts (Paper’s Title) 3pseudoscalar mesons (the pions and partners) and the octet of stable baryons (thenucleons and partners). The SU(3) decomposition of the combination of two octetstells us that8⊗ 8 = 1⊕ 8s ⊕ 8a ⊕ 10⊕ 10⊕ 27 . (3)Thus, on pure SU(3) grounds, should we have a SU(3) symmetric Lagrangian, onecan expect e.g. one singlet and two octets of resonances, the symmetric and anti-symmetric ones.The lowest order of the meson–baryon chiral Lagrangian is exactly SU(3) in-variant if all the masses of the mesons, or equivalently the quark masses, are setequal. As stated above [see Eq. (2)], in Ref. 3 the baryon masses take their physicalvalues, although strictly speaking at the leading order in the chiral expansion theyshould be equal to M0. For Eq. (2) being SU(3) symmetric, all the baryons massesMi must be set equal as well. When all the meson and baryon masses are equal, andthese common masses are employed in evaluating the Gl functions, together withequal subtraction constants al, the T –matrix obtained from Eq. (1) is also SU(3)symmetric.If we do such an SU(3) symmetry approximation and look for poles of the scat-tering matrix, we find poles corresponding to the octets and singlet. The surprisingresult is that the two octet poles are degenerate as a consequence of the dynamicscontained in the chiral Lagrangians. Indeed, if we evaluate the matrix elements ofthe transition potential V in a basis of SU(3) states, we obtain something propor-tional to Vαβ = diag(6, 3, 3, 0, 0,−2) taking the following order for the irreduciblerepresentations: 1, 8s, 8a, 10, 10 and 27, with positive sign meaning attraction.Hence we observe that the states belonging to different irreducible represen-tations do not mix and the two octets appear degenerate. The coefficients in Vαβclearly illustrate why there are no bound states in the 10, 10 and 27 representations.In practice, the same chiral Lagrangians allow for SU(3) breaking. In the caseof Refs. 4,3 the breaking appears because both in the Vij transition potentials asin the Gl loop functions one uses the physical masses of the particles as well asdifferent subtraction constants in Gl, corresponding to the use of a unique cut-off inall channels. In Ref. 2 the physical masses are also used in the Gl functions, althoughthese functions are evaluated with a unique subtraction constant as corresponds tothe SU(3) limit. In both approaches, physical masses are used to evaluate the Glloop functions so that unitarity is fulfilled exactly and the physical thresholds forall channels are respected.By following the approach of Ref. 3 and using the physical masses of the baryonsand the mesons, the position of the poles change and the two octets split apart infour branches, two for I = 0 and two for I = 1, as one can see in Fig. 1. In the figurewe show the trajectories of the poles as a function of a parameter x that breaksgradually the SU(3) symmetry up to the physical values. The dependence of massesand subtraction constants on the parameter x is given byMi(x) = M0 + x(Mi −M0),August 4, 2013 8:17 WSPC/INSTRUCTION FILE menu4 Authors’ Names250200150100500Im zR   [MeV]17001600150014001300Re zR   [MeV] 1390     (I=0)  1426     (I=0) 1680     (I=0) 1580     (I=1) disappear     (I=1)Singlet OctetFig. 1. Trajectories of the poles in the scattering amplitudes obtained by changing the SU(3)breaking parameter x gradually. At the SU(3) symmetric limit (x = 0), only two poles appear, oneis for the singlet and the other for the octet. The symbols correspond to the step size δx = 0.1.The results are from 8.m2i (x) = m20 + x(m2i −m20),ai(x) = a0 + x(ai − a0), (4)where 0 ≤ x ≤ 1. In the calculation of Fig. 1, the values M0 = 1151 MeV, m0 = 368MeV and a0 = −2.148 are used.The complex poles, zR, appear in unphysical sheets. In the present search wefollow the strategy of changing the sign of the momentum ql in the Gl(z) loopfunction for the channels which are open at an energy equal to Re(z).The splitting of the two I = 0 octet states is very interesting. One moves tohigher energies to merge with the Λ(1670) resonance and the other one moves tolower energies to create a pole, quite well identified below the K¯N threshold, witha narrow width. On the other hand, the singlet also evolves to produce a pole atlow energies with a quite large width.We note that the singlet and the I = 0 octet states appear nearby in energy andwhat experiments actually see is a combination of the effect of these two resonances.Similarly as for the I = 0 octet states, we can see that one branch of the I = 1states moves to higher energies while another moves to lower energies. The branchmoving to higher energies finishes at what would correspond to the Σ(1620) reso-nance when the physical masses are reached. The branch moving to lower energiesfades away after a while when getting close to the K¯N threshold.The model of Ref. 2 reproduces qualitatively the same results, but the I = 1pole also stays for x = 1. Nevertheless, in both approaches there is an I = 1amplitude with an enhanced strength around the K¯N threshold. This amplitudeAugust 4, 2013 8:17 WSPC/INSTRUCTION FILE menuInstructions for Typing Manuscripts (Paper’s Title) 5has non negligible consequences for reactions producing piΣ pairs in that region.This has been illustrated for instance in Ref. 9, where the photoproduction of theΛ(1405) via the reaction γp → K+Λ(1405) was studied. It was shown there thatthe different sign in the I = 1 component of the | pi+Σ−〉, | pi−Σ+〉 states leads,through interference between the I = 1 and the dominant I = 0 amplitudes, todifferent cross sections in the various charge channels, a fact that has been confirmedexperimentally very recently 10.Once the pole positions are found, one can also determine the couplings of theseresonances to the physical states by studying the amplitudes close to the pole andidentifying them withTij =gigjz − zR . (5)The couplings gi are in general complex valued numbers. In the Table we summarizethe pole positions and the complex couplings gi obtained from the model of Ref.3for isospin I = 0. The results with the model of 2 are qualitatively similar.zR 1390 + 66i 1426 + 16i 1680 + 20i(I = 0) gi |gi| gi |gi| gi |gi|piΣ −2.5− 1.5i 2.9 0.42− 1.4i 1.5 −0.003− 0.27i 0.27K¯N 1.2 + 1.7i 2.1 −2.5 + 0.94i 2.7 0.30 + 0.71i 0.77ηΛ 0.010 + 0.77i 0.77 −1.4 + 0.21i 1.4 −1.1− 0.12i 1.1KΞ −0.45− 0.41i 0.61 0.11− 0.33i 0.35 3.4 + 0.14i 3.5We observe that the second resonance with I = 0 couples strongly to K¯Nchannel, while the first resonance couples more strongly to piΣ.3. Influence of the poles on the physical observablesIn a given reaction the Λ(1405) resonance is always seen in piΣ mass distribution.However, the Λ(1405) can be produced through any of the channels in the Table.Hence, it is clear that, should there be a reaction which forces this initial channelto be K¯N , then this would give more weight to the second resonance, R2, andhence produce a distribution with a shape corresponding to an effective resonancenarrower than the nominal one and at higher energy. Such a case indeed occursin the reaction K−p → Λ(1405)γ studied theoretically in Ref. 11. It was shownthere that since the K−p system has a larger energy than the resonance, one hasto lose energy emitting a photon prior to the creation of the resonance and this iseffectively done by the Bremsstrahlung from the original K− or the proton. Hencethe resonance is initiated from the K−p channel. This is also the case in the reactionγp → K∗Λ(1405) which has also the Λ(1405) initiated by K¯N through the vertexγ → K∗K 12.August 4, 2013 8:17 WSPC/INSTRUCTION FILE menu6 Authors’ Names4. The interaction of the decuplet of baryons with the octet ofmesonsGiven the success of the chiral unitary approach in generating dynamically lowenergy resonances from the interaction of the octets of stable baryons and thepseudoscalar mesons, in 13 the interaction of the decuplet of 3/2+ with the octetof pseudoscalar mesons was studied and shown to lead to many states which wereassociated to experimentally well established 3/2− resonances.The lowest order chiral Lagrangian for the interaction of the baryon decupletwith the octet of pseudoscalar mesons is given by 14L = iT¯ µDνγνTµ −mT T¯ µTµ (6)where T µabc is the spin decuplet field and Dν the covariant derivative given by in 14.The identification of the physical decuplet states with the T µabc can be seen in15,where a detailed study of this interaction and the resonances generated can be seen.The study is done along the lines of the former sections, looking for poles in thesecond Riemann sheet of the complex plane, the coupling of the resonances to thedifferent channels and the stability of the results with respect to variations of theinput parameter, which in our case is just the subtraction constant,a. This allowsthe association of the resonances found to existing states of the particle data book,and the prediction of new ones. A detail of the results obtained can be seen in Fig.2.Another interesting result is the generation of an exotic state of S = 1 and I = 1which is generated by the interaction of the ∆K channels and stands as a ∆Kresonance. The pole appears in a Riemann sheet below threshold when also the signof the momentum is changed, but it leads to a ∆K amplitude which accumulatesstrength close to threshold and produces a broad peak in the cross section, see Fig.3,in contrast to the I = 2 cross section which is much smaller and very smooth.In order to investigate this interaction we propose the study of the followingreactions: 1) pp→ Λ∆+K+, 2) pp→ Σ−∆++K+, 3) pp→ Σ0∆++K0. In the firstcase the ∆+K+ state produced has necessarily I = 1. In the second case the ∆++K+state has I = 2. In the third case the ∆++K0 state has mostly an I = 1 component.The study of the ∆K invariant mass distribution in these reactions would provideexperimental information to eventually prove the prediction made here, and thusgiving support to this new exotic baryonic state, which, although in the quark modelwould require at least five quarks and could qualify as a pentaquark, finds a muchsimpler interpretation as a resonant ∆K state.5. AcknowledgmentsThis work is partly supported by DGICYT contract number BFM2003-00856, CI-CYT (Spain) Grant Nos. FPA2002-03265 and FPA2004-03470, the E.U. EURIDICEnetwork contract no. HPRN-CT-2002-00311 and the Research Cooperation programof the JSPS and the CSIC.August 4, 2013 8:17 WSPC/INSTRUCTION FILE menuInstructions for Typing Manuscripts (Paper’s Title) 71300 1500 1700 1900 2100ECM (MeV)00.010.020.030.040.050.06|T|2  (1/MeV2 )∆piΣ*K∆pi (with width)Σ*K (with width)(S,I)=(0,1/2)N*(1520)1300 1500 1700 1900 2100ECM (MeV)00.0010.0020.0030.0040.005|T|2  (1/MeV2 )∆piΣ*K∆η(S,I)=(0,3/2)∆(1700)∆ (?)1300 1500 1700 1900 2100ECM (MeV)00.0050.010.0150.020.025|T|2  (1/MeV2 )Σ∗piΞ*K(S,I)=(−1,0)Λ(1520)1300 1500 1700 1900 2100ECM (MeV)00.20.40.60.81|T|2  (1/MeV2 )∆KΣ∗piΣ∗ηΞ*K(S,I)=(−1,1) −Σ(1670)Σ(1940)1300 1500 1700 1900 2100ECM (MeV)00.050.10.15|T|2  (1/MeV2 )Σ*KΞ∗piΞ∗ηΩK(S,I)=(−2,1/2) −Ξ(1820)Ξ (?)1300 1500 1700 1900 2100ECM (MeV)00.020.040.060.08|T|2  (1/MeV2 )Ξ*KΩpi(S,I)=(−3,0)−Ω (?)Fig. 2. Resonances obtained from the interaction of the octet of mesons with the decuplet ofbaryons.August 4, 2013 8:17 WSPC/INSTRUCTION FILE menu8 Authors’ Names1400 1500 1600 1700 1800 1900 2000 2100C. M. Energy (MeV)−0.07−0.05−0.03−0.010.010.030.05∆K AmplitudesRe tIm tRe t (with width)Im t (with width)|t|2|t|2 (with width)S=1I=1X 10Fig. 3. Amplitudes for ∆K → ∆K for I = 1References1. G. Ecker, Prog. Part. Nucl. Phys. 35 (1995) 1; V. Bernard, N. Kaiser and U.-G. Meißner, Int. J. Mod. Phys. E 4 (1995) 193.2. J. A. Oller and U.-G. Meißner, Phys. Lett. B 500 (2001) 263.3. E. Oset, A. Ramos and C. Bennhold, Phys. Lett. B 527 (2002) 99 [Erratum-ibid. B530 (2002) 260].4. E. Oset and A. Ramos, Nucl. Phys. A 635 (1998) 99.5. A. Ramos, E. Oset and C. Bennhold, Phys. Rev. Lett. 89 (2002) 252001.6. N. Kaiser, P. B. Siegel and W. Weise, Nucl. Phys. A 594 (1995) 3257. T. Inoue, E. Oset and M. J. Vicente Vacas, Phys. Rev. C 65 (2002) 035204.8. D. Jido, J.A. Oller, E. Oset, A. Ramos, U.G. Meissner, Nucl. Phys. A 725 (2003) 181.9. J. C. Nacher, E. Oset, H. Toki and A. Ramos, Phys. Lett. B 455, 55 (1999).10. J. K. Ahn et al., Proceedigns of the XVI Particles and Nuclei International Conference(PANIC02), Eds. K. Imai, T. Kishimodo and H. Toki, Nucl. Phys. A721 (2003) 715-718.11. J. C. Nacher, E. Oset, H. Toki and A. Ramos, Phys. Lett. B 461 (1999) 299.12. T. Hyodo, A. Hosaka, M.J. Vicente Vacas and E. Oset, Phys. Lett. B593 (2004)75-81.13. E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B 585 (2004) 243.14. E. Jenkins and A. V. Manohar, Phys. Lett. B 259 (1991) 353.15. S. Sarkar, E. Oset and M. J. Vicente Vacas, arXiv:nucl-th/0407025.SLS 4mpi2

Dynamically generated resonances in the chiral unitary approach to meson baryon interaction

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DYNAMICALLY GENERATED RESONANCES IN THE CHIRAL UNITARY APPROACH TO MESON BARYON INTERACTION

Dynamically Generated Resonances in the Chiral Unitary Approach to Meson
  Baryon Interaction

Dynamically Generated Resonances in the Chiral Unitary Approach to Meson Baryon Interaction

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