We review our work on the rho and sigma resonances derived from the Inverse
Amplitude Method. In particular, we study the leading 1/N_c behavior of the
resonances masses and widths and their evolution with changing m_pi. The 1/N_c
expansion gives a clear definition of quark-antiquark states, which is neatly
satisfied by the rho but not by the sigma, showing that its dominant component
is not quark-antiquark. The m_pi dependence of the resonance properties is
relevant to connect with lattice studies. We show that our predictions compare
well with some lattice results and we find that the rho-pi-pi coupling constant
is m_pi independent, in contrast with the sigma-pi-pi coupling, that shows a
strong m_pi dependence.Comment: To appear in the Proceedings of the Chiral10 Workshop, June 21-24
  2010. Valencia, Spai

Hanhart, C.

Pelaez, J. R.

Rios, G.

English

arXiv

G. Ríos

C. Hanhart

J. R. Peláez

Juan M. Nieves

Eulogio Oset

Manuel J. Vicente-Vacas

Crossref

N[sub c] and m[sub π] dependence of ρ and σ mesons from unitarized Chiral Perturbation Theory

©Amer Inst Physics.
ISSN: 0094-243X
International Workshop on Chiral Symmetry in Hadrons and Nuclei (2010. Valencia).We review our work on the ρ and σ resonances derived from the Inverse Amplitude Method. In particular, we study the leading 1/N_c behavior of the resonances masses and widths and their evolution with changing m_π. The 1/Nc expansion gives a clear definition of ¯qq states, which is neatly satisfied by the ρ but not by the σ, showing that its dominant component is not ¯qq. The m_π dependence of the resonance properties is relevant to connect with lattice studies. We show that our predictions compare well with some lattice results and we find that the ρππ coupling constant is mπ independent, in contrast with the σππ coupling, that shows a strong m_π dependence.Depto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Peláez Sagredo, José Ramón

Docta Complutense

Nc and mπ dependence of ρ and σ mesons fromunitarized Chiral Perturbation TheoryG. Ríos∗, C. Hanhart† and J. R. Peláez∗∗Departamento de Física Teórica II. Universidad Complutense de Madrid.†Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungzentrum Jülich GmbH.Abstract. We review our work on theρ and σ resonances derived from the Inverse AmplitudeMethod. In particular, we study the leading 1/Nc behavior of the resonances masses and widths andtheir evolution with changingmπ . The 1/Nc expansion gives a clear definition of ¯qq states, whichis neatly satisfied by theρ but not by theσ , showing that itsdominant componentis not q̄q. Themπ dependence of the resonance properties is relevant to connet with lattice studies. We show thatour predictions compare well with some lattice results and we find that theρππ coupling constantis mπ independent, in contrast with theσππ coupling, that shows a strongmπ dependence.Keywords: Scalar mesons, chiral lagrangians, 1/Nc expansionPACS: 14.40.Cs, 12.39.Fe, 13.75.Lb,11.15.PgLight hadron spectroscopy lies beyond the realm of perturbaive QCD. At low ener-gies, however, one can use the QCD low energy effective theory, named Chiral Perturba-tion Theory (ChPT) [1], to describe the dynamics of the lightest mesons. ChPT describesthe interactions of the Goldstone bosons of the QCD chiral symmetry breaking, namely,the pions, by means of a effective lagrangian compatible with all QCD symmetries in-volving only the pion field. The infinite tower of terms in thislagrangian is organizedas a low energy expansion in powers ofp2/Λ2χ , wherep stands either for derivatives,momenta or masses, andΛχ ≃ 4π fπ , where fπ denotes the pion decay constant. ChPTis renormalized order by order by absorbing loop divergences in the renormalization ofhigher order parameters, known as low energy constants (LECs), that parametrize thehigh energy QCD dynamics andcarry no energy or mass dependence. They depend ona regularization scaleµ but after renormalization the observables are independentof thisscale. The value of the LECs depend on the underlying QCD dynamics and are deter-mined from experiment. Up to the desired order, the ChPT expansion provides asys-tematic and model independentdescription of how observables depend on some QCDparameters like the light quark mass ˆm= (mu +md/2) or the number of colors,Nc [2].The use of ChPT is limited to low energies and masses, neverthel ss, combinedwith dispersion relations and elastic unitarity it leads toa successful description ofmeson dynamics up to energies around 1 GeV, generating resonant states not originallypresent in the lagrangian, without any a priori assumption on their existence or nature.In particular, we find theρ and σ resonances as poles on the second Riemann sheetof ππ elastic scattering amplitudes. With this approach we can the study some ofthese resonances properties, like their spectroscopic nature through their mass and widthdependence onNc, or their dependence on the pion mass in order to connect withlatticestudies. In the following sections we review this “unitarized ChPT” approach, named452the Inverse Amplitude Method (IAM) [3, 4, 5], and then apply it to study the leading1/NC behavior and the chiral extrapolation of theρ andσ mesons.Theρ andσ resonances appear as poles on the second Riemann sheet of the(I ,J) =(1,1) and(I ,J) = (0,0) ππ scattering partial waves of definite isospin,I and angularmomentumJ, respectively. Elastic unitarity implies for these partial waves,t(s), andphysical values ofs below inelastic thresholds, thatIm t(s) = σ(s)|t(s)|2 ⇒ Im1/t(s) = −σ(s), with σ(s) = 2p/√s, (1)wheres is the Mandelstam variable andp is the center of mass momentum. Conse-quently, the imaginary part of 1/t is known exactly. However, ChPT amplitudes, beingan expansiont ≃ t2 + t4 + · · · , with tk = O(pk), can only satisfy Eq. (1) perturbativelyIm t2(s) = 0, Im t4(s) = σ(s)t22(s) . . . (2)The resonance region lies beyond the reach of standard ChPT. Tis region however, canbe reached combining ChPT with dispersion theory through theIAM [3, 4, 5].The analytic structure of theππ scattering amplitudet(s), consisting on a right cutextending fromsth = 4m2π to ∞, and a left cut from−∞ to 0, allows to write a dispersionrelation for the auxiliary functionG(s) ≡ t22(s)/t(s)G(s) = G(0)+G′(0)s+ 12G′′(0)s2 +s3π∫ ∞sthds′ImG(s′)s′3(s′−s− iε) +LC(G)+PC, (3)where the integral over the left cut has been abbreviated asLC(G) andPC stands fromthe pole contributions in the scalar wave corresponding to the Adler zero. The terms inEq.(3) are evaluated using unitarity and ChPT as follows: Theright cut (RC) isexactlyevaluated considering the elastic unitarity conditions Eq. (1), (2): ImG(s′) =−Im t4(s′).The subtraction constants only involve the amplitude and its derivatives evaluated ats = 0, so they can be safely approximated with ChPT:G(0) ≃ t2(0)− t4(0), G′(0) ≃t ′2(0)− t ′4(0), G′′(0)≃−t ′′4(0). The LC, being suppressed by 1/s′3(s′−s), is weighted atlow energies, so it is appropriate to approximate it with ChPT:LC(G) ≃ −LC(t4). ThePC countsO(p6), it has been calculated explicitly [6] and it is numericallynegligibleexcept near the Adler zero, away from the physical region.NeglectingPC for the moment, taking into account thatt2(s) is just a first orderpolynomial ins, and that a dispersion relation can be also written fort4, we can writeEq.(3) asG(s) ≃ t2(0)− t ′2(0)s− t4(0)− t ′4(0)s− 12t′′4(0)s2−RC(t4)−LC(t4) = t2(s)− t4(s), (4)which immediately leads to the IAM formulat IAM(s) = t22(s)t2(s)−t4(s) . The IAM formulasatisfies exact elastic unitarity and, when reexpanded at low energies, reproduces theChPT expansion up to the order used to approximate the subtraction constants and theleft cut. Here we have presented anO(p4) IAM but it can be generalized to higherchiral orders. Note that in the IAM derivation ChPT has been always used at lowenergies, to evaluate parts of a dispersion relation whose ela tic unitarity cut has been453 0 0.5 1 1.5 2 2.5 3 4  6  8  10  12  14  16  18  20NcMρ / MρphysΓρ / ΓρphysMσ / MσphysΓσ / Γσphysq-q scalingρσ 0 0.5 1 1.5 2 2.5 3 4  6  8  10  12  14  16  18  20NcMσ / MσphysΓσ / Γσphysq-q scalingFIGURE 1. Left: ρ andσ 1/Nc scalingO(p4). Right: σ 1/Nc scalingO(p6).taken into account exactly. Thus, there are no additional model dependencies in theapproach, which is reliable up to energies where inelasticities become important. Takingthe pole contribution into account leads to a modified IAM formula [6] which is almostindistinguishable from the ordinary one except in the Adlerz o region, where themodified formula should be used. Actually, we use the modifiedIAM in this work since,as it will be shown below, one amplitude pole gets near the Adler zero region.This simple IAM formula is able to reproduceππ scattering data up to roughly 1 GeVand generates theρ andσ poles with values of the LECs compatible with standard ChPT[5]. The 1/Nc expansion is implemented in ChPT through the LECs, whose leading 1/Ncscaling is known from QCD. Also, themπ dependence of IAM agrees with ChPT up tothe order used. Hence, it is straightforward to study the leading 1/NC behavior and them̂dependence of the resonances generated with the IAM.The QCD 1/Nc expansion [2] provides a clear definition of ¯qq bound states: theirmasses and widths scale asO(1) and O(1/Nc) respectively. The QCD leading 1/Ncbehavior of the ChPT parameters (fπ , mπ and the LECs) is well known. Hence, byscaling withNc the ChPT parameters in the IAM, theNc dependence of theρ andσmesons mass and width has been determined [7, 8]. They are define from the polepositions as√spole= M− iΓ. Note that we should not take too largeNC values, since theNc → ∞ is a weakly interacting limit, where the IAM approach is lessreliable [9]. Also,for very largeNc, a tiny admixture of ¯qq in the physical state could become dominant,but this does not give any information about the physical state dominant component.Fig. 1 (left) shows theρ andσ mass and widthNc scaling. It can be seen that theρfollows remarkably well the expected behavior of a ¯qqstate, confirming that the methodobtains the correctNc behavior of well known ¯qqstates. In contrast,theσ does not followthatq̄qpattern, allowing us to conclude that itsdominant component is notq̄q.Loop contributions play an important role in determining the σ pole position. Sincethey are 1/Nc suppressed compared to tree level terms, it may happen that for largerNc they become comparable to tree levelO(p6) terms, which are subdominant in theChPT series, but notNc suppressed. Thus we checked theO(p4) results with anO(p6)IAM calculation [8]. We defined aχ2-like function to measure how close a resonanceis from aq̄q behavior. First, we used it atO(p4) to show that it is not possible to find aset of LECs that makes theσ to behave predominantly as a ¯qq state. Next, we obtainedanO(p6) data fit where theρ q̄q behavior was imposed. Figure 1 (right) shows theMσandΓσ Nc scaling obtained from that fit. Note that bothMσ andΓσ grow near Nc = 3,4540 1 2 3 4 5Re (√s / mπ)-1.5-1-0.500.511.5Im (√s / mπ)ππ thresholdAdler zero 0.7 0.8 0.9 1 1.1 0  0.1  0.2  0.3  0.4  0.5  0.6Mρ (GeV)mπ (GeV)IAMMILCETMCRBC-UKQCDQCDSFCP-PACS+AdelaidePDG0 1 2 3mπ / mπphys0.511.52σ vs. ρMass of0 1 2 3mπ / mπphys00.511,5σ vs. ρWidth of 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0  50  100  150  200  250  300  350  400  450|g / gphys |mπ (MeV)ρππσππ (upper)σππ (lower)FIGURE 2. Top Left: Movement of theσ (dashed lines) andρ (dotted lines) poles for increasingmπ on the second sheet. The filled (open) boxes denote theσ (ρ) pole positions atmπ = 1, 2, and3×mphysπ , respectively.Top Right: Comparison the IAMMρ dependence onmπ with some recent latticeresults[14].Bottom Left: Comparison of theρ (light) andσ (dark) mass dependence onmπ . BottomCenter: Comparison of theρ (light) andσ (dark) width dependence onmπ . The dotted (ρ) dot-dashed(σ ) lines show the decrease due to only phase space assuming a cost nt coupling toππ. Bottom Right:ρ andσ couplings calculated from the pole residue. In all panels, the bands cover the LECs uncertainty.confirming theO(p4) result of a non ¯qqdominant component. However, forNc between8 and 15, where we still trust the IAM,Mσ becomes constant andΓσ starts decreasing.This may hint to asubdominant̄qq component, arising as loops become suppressed asNc grows. Finally, by forcing theσ to behave as a ¯qq, we found that in the best case thissubdominant component could become dominant aroundNc > 6−8, but always with anNc → ∞ mass above 1 GeV instead of its physical∼ 450 MeV value. This supports theemerging picture of two low energy scalar nonets, one of exotic nature below 1 GeV andanother of ordinary ¯qqnature above 1 GeV.ChPT also provides an expansion ofmπ in terms ofm̂ (at leading orderm2π ∼ m̂).Thus, by changingmπ in the amplitudes we see how the IAM poles depend on ˆm. Wereport here our analysis of theρ andσ properties dependence onmπ [10].The values ofmπ considered should fall within the ChPT applicability range and llowfor some elastic regime belowKK̄, that would almost disappear ifmπ > 500, whichwould be the most optimistic applicability range. We expecthigher order corrections tobe more relevant asmπ increases. Thus, our results become less reliable asmπ grows.Fig. 2 (top left) shows the evolution of theσ andρ pole positions asmπ is increased. Inorder to see the pole movements relative to theππ threshold, which is also increasing, weuse units ofmπ , so the threshold is fixed at√s= 2. Both poles move closer to thresholdand they approach the real axis. Theρ poles reach the real axis at the same time that theycross threshold. One of them jumps into the first sheet and becom s a bound state, whileits conjugate partner remains on the second sheet practically at the very same position455as that in the first. In contrast, theσ poles go below threshold with a finite imaginarypart before they meet in the real axis, still on the second sheet, b coming virtual states.As mπ increases, one pole moves toward threshold and jumps througthe branch pointto the first sheet staying in the real axis below threshold. The ot erσ pole moves downin energies away from threshold and remains on the second sheet. Similar movementswere found within quark models [12] and a finite density analysis [13].Fig. 2 (top right) shows our results forMρ dependence onmπ compared with somelattice results [14] and theMρ PDG value. In view of the incompatibilities betweendifferent lattice collaborations, we find a qualitative good agreement with lattice results.TheMρ dependence onmπ agrees also with estimations for the two first coefficients ofits chiral expansion [15].In Fig. 2 (bottom left) we compare themπ dependence ofMρ andMσ , normalized totheir physical values. The bands cover the LECs uncertainties. Both masses grow withmπ , butMσ grows faster thanMρ . Above 2.4mphysπ , we show two bands since the twoσpoles lie on the real axis with two different masses.In the bottom center panel of Fig. 2 we compare themπ dependence ofΓρ andΓσnormalized to their physical values: note that both widths become smaller. We comparethis decrease with the expected phase space reduction as resonances approach theππthreshold. We find thatΓρ follows very well this expected behavior, which implies thatthe ρππ coupling is almostmπ independent. In contrast,Γσ deviates from the phasespace reduction expectation. This suggests a strongmπ dependence of theσ couplingto two pions, which we confirm with a explicit calculation of the resonances couplingsfrom the pole residues as shown in the bottom left panel.REFERENCES1. S. Weinberg, PhysicaA96 (1979) 327. J. Gasser and H. Leutwyler, Annals Phys.158 (1984) 142;2. G. ’t Hooft, Nucl. Phys. B72, 461 (1974). E. Witten, Annals Phys.128, 363 (1980).3. T. N. Truong, Phys. Rev. Lett.61 (1988) 2526. Phys. Rev. Lett.67, (1991) 2260; A. Dobado et al.,Phys. Lett.B235 (1990) 134.4. A. Dobado and J. R. Peláez, Phys. Rev. D47 (1993) 4883; Phys. Rev. D56 (1997) 3057.5. F. Guerrero and J. A. Oller, Nucl. Phys. B537 (1999) 459 [Erratum-ibid. B602 (2001) 641].J. R. Peláez, Mod. Phys. Lett. A19, 2879 (2004) A. Gómez Nicola and J. R. Peláez, Phys. Rev. D65 (2002) 054009 and AIP Conf. 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C40 (2005) 97.456Copyright of AIP Conference Proceedings is the property of American Institute of Physics and its content maynot be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express writtenpermission. However, users may print, download, or email articles for individual use.

N_c and m_π dependence of ρ and σ mesons from unitarized Chiral Perturbation Theory

N_c and m_pi dependence of rho and sigma mesons from unitarized Chiral
  Perturbation Theory

N_c and m_pi dependence of rho and sigma mesons from unitarized Chiral Perturbation Theory

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