Structure of the sigma meson and the softening
aa r X i v : . [ h e p - ph ] J u l October 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8 Structure of the σ meson and the softening T. HYODO ∗ Department of Physics, Tokyo Institute of Technology,Meguro, Tokyo 152-8551, Japan ∗ E-mail: [email protected]
D. JIDO
Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606–8502, Japan
T. KUNIHIRO
Department of Physics, Kyoto University,Kyoto 606–8502, Japan
We study the structure of the σ meson, the lowest-lying resonance of the ππ scattering in the scalar-isoscalar channel, through the softening phenomena as-sociated with the partial restoration of chiral symmetry. We build dynamicalchiral models to describe the ππ scattering amplitude, in which the σ me-son is described either as the chiral partner of the pion or as the dynamicallygenerated resonance through the ππ attraction. It is shown that the internalstructure is reflected in the softening phenomena; the softening pattern of thedynamically generated σ meson is qualitatively different from the behavior ofthe chiral partner of the pion. On the other hand, in the symmetry restora-tion limit, the dynamically generated σ meson behaves similarly to the chiralpartner. Keywords : σ meson, chiral dynamics, softening, hadronic molecule
1. Introduction
The study of the structure of hadron resonances is one of the central is-sues in modern hadron spectroscopy. Among others, the structure of the σ meson has been intensively studied, since the σ meson is considered toplay an important role in various aspects of hadron and nuclear physics.For instance, since the scalar-isoscalar excitation of QCD vacuum can beregarded as the amplitude fluctuation of the chiral order parameter h ¯ qq i ,the nature of the σ meson is crucial to understand the dynamical chiral ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8 symmetry breaking in QCD. While the existence of the scalar-isoscalarresonance in the ππ scattering is established by recent analyses (see e.g.Ref. 2), its internal structure is still controversial. Thus, it is our aim hereto investigate the structure of the σ meson.There are many proposals for the structure of the σ meson based onvarious effective models: collective ¯ qq excitation, tetraquark structurewith strong diquark correlation, dynamically generated ππ resonance, andso on. Conventionally, the validity of certain structure has been tested bycomparing the model prediction with experimental data, for instance, massspectrum and the decay properties.It is also possible to investigate the structure of the resonance from theresponse to the change of the internal/external parameter of the models.For instance, the study of the N c scaling is successful to disentangle the¯ qq and other structures for meson resonances. Following this philosophy,we would like to study the spectral change of the σ meson when the chiralsymmetry is partially restored, in order to discriminate the different internalstructures.In association with chiral symmetry restoration, the softening of the σ meson has been discussed. In the linear realization of chiral symme-try, the σ meson forms a chiral four-vector together with the pion, andhence they are chiral partners. The partial restoration of chiral symmetryinduces the softening of the σ spectrum, which results in the enhancementof the ππ cross section in the scalar-isoscalar channel near threshold. Itwas shown later that the threshold enhancement in the I = J = 0 channeltakes place also in the nonlinear realization of chiral symmetry without thebare σ field where the σ meson is expressed as a dynamically generatedresonance from the attractive ππ interaction. Although similar thresholdenhancement of the cross section is observed in both cases, the mechanismwhich causes the softening is quite different. Based on these observations,we demonstrate that the softening phenomena reflects the structure of the σ meson, paying attention to the nature of the s -wave resonance.When the symmetry is completely restored, the chiral partners emergeas the pair of particles with a degenerate mass. If we regard the massdegeneracy as the condition for the chiral partner, we can extend the notionof the chiral partner for the dynamically generated σ meson in the nonlinearrealization. We will study the structure of the ππ scattering amplitude ofour model in the restoration limit, to discuss the chiral partner of the pion. ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8
2. Formulation
Here we describe the ππ scattering amplitude with the σ resonance in the I = J = 0 channel. We consider the low energy behavior of the ampli-tude based on chiral effective Lagrangian, and then introduce the unitaritycondition to extend the applicability of the model to the resonance energyregion.We start from the Lagrangian of two-flavor linear sigma model to derivethe ππ scattering amplitudes: L = 14 Tr (cid:20) ∂M ∂M † − µ M M † − λ
4! (
M M † ) + h ( M + M † ) (cid:21) , (1)where M = σ + i τ · π . In this Lagrangian, the σ meson is treated as thechiral partner of the pion. For negative µ , chiral symmetry is spontaneouslybroken and three parameters in the Lagrangian µ , λ , and h are related tothe chiral condensate (pion decay constant) h σ i = f π , the mass of the pion m π , and the mass of the σ meson m σ in the mean-field level.Crossing symmetry enables us to express the general ππ scattering am-plitude as T tree ( s, t, u ) = A ( s, t, u ) δ ab δ cd + A ( t, s, u ) δ ac δ bd + A ( u, t, s ) δ ad δ bc ,where the invariant amplitude A is given, from the Lagrangian (1) at treelevel, by A ( s ) = s − m π h σ i − ( s − m π ) h σ i s − m σ . In this expression, the first (second) term can be regarded as the leading(higher) order contribution in the chiral perturbation theory. The coefficientof the leading order term is fixed by the low energy theorem, while thelow energy constant for the higher order terms is not constrained by thesymmetry and should be determined by experiments. Thus, we introduce aparameter x to express the general amplitude A ( s ; x ) = s − m π h σ i − x ( s − m π ) h σ i s − m σ . (2)By choosing x = 1, we recover the result of the original Lagrangian (1). Ifwe take x = 0, then we are left with the leading order interaction, whichalso corresponds to the heavy m σ limit. In this way, we can smoothly con-nect the original linear sigma model ( x = 1) and the leading order chiralperturbation theory without the bare σ field ( x = 0). Projecting Eq. (2)onto the I = J = 0 channel, we obtain the tree-level amplitude for the ππ ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8 scattering as T tree ( s ; x ) = m σ − m π h σ i " s − m π m σ − m π (1 − x ) − x − x m σ − m π s − m σ − x m σ − m π s − m π ln (cid:18) m σ m σ + s − m π (cid:19) , (3)in the center-of-mass frame.Next we consider the unitarity condition Im T − ( s ) = − Θ( s ) / s > m π , with the two-body phase space function Θ( s ) = (16 π ) − p − m π /s .Based on the N/D method, we write down the general expression of the uni-tary scattering amplitude T ( s ; x ). Matching the chiral interaction T tree ( s ; x )with the loop expansion of the full amplitude T ( s ; x ), we obtain the ampli-tude which is consistent with both chiral low energy theorem and unitarityas T ( s ; x ) = 1 T − ( s ; x ) + G ( s ) ,G ( s ) = 12 1(4 π ) a ( µ ) + ln m π µ + r − m π s ln q − m π s + 1 q − m π s − , where a ( µ ) is the subtraction constant at the subtraction point µ . We deter-mine the subtraction constant by excluding the nontrivial CDD pole (stateswhich does not originate in the two-body dynamics) in the amplitude G ( s ) = 0 at s = m π , (4)which leads to a ( m π ) = − π/ √
3. With this subtraction constant, the fullscattering amplitude T reduces into the tree level one T tree at s = m π .The full amplitude T ( s ; x ) corresponds to the nonperturbative resum-mation of the s-channel loop diagrams up to infinite order. For the x = 1case, the bare σ pole in the Lagrangian acquire the finite width through thecoupling to the ππ state, and the full scattering amplitude exhibits a polein the complex energy plane. On the other hand, for the x = 0 case withoutthe bare σ pole, a resonance can be dynamically generated as the pole ofthe amplitude, if the two-body interaction is sufficiently attractive. Indeed,it is shown that the resummation of the leading order interaction generatesthe σ meson dynamically. In the following we compare the properties ofthese σ states: one originating in the chiral partner of the pion in the linear σ model ( x = 1), and another generated dynamically from the attractive ππ interaction ( x = 0). ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8
3. Chiral symmetry restoration3.1.
Prescription for the symmetry restoration
Now we introduce the effect of chiral symmetry restoration through themodification of the model parameters. It is known that the chiral condensateshould decrease with the chiral symmetry restoration, so we parametrizethe condensate by h σ i = Φ h σ i , where h σ i is the condensate in vacuum and we vary the parameter Φ fromone to zero to express the symmetry restoration. The NJL model indicatesthat m π hardly changes as symmetry restoration, so we assume that it isa constant: m π = const.The bare mass of the σ should be degenerated with pion when the symmetryis restored. This can be achieved by the mean-field relation of Eq. (1) m σ = r λ h σ i m π , with λ and m π being fixed. Behavior of the amplitude in the restoration limit
It is instructive to study the behavior of the ππ scattering amplitude in thelimit h σ i →
0. Here we analytically derive the pole of the amplitude in therestoration limit. In subsection 4.2, we will numerically demonstrate thatthe obtained result corresponds to the asymptotic behavior of the σ polefor h σ i → x = 1 case, where the σ meson is the chiral partnerof the pion. To make the h σ i dependence in m σ explicit, we rewrite the treelevel amplitude as T tree ( s ; 1) = − λ − λ h σ i s − m π − λ h σ i − λ h σ i s − m π ln m π + λ h σ i s − m π + λ h σ i . The second term represents the bare pole of the σ meson. As h σ i → T tree ( s ; 1) coincides with the full amplitude T ( s ; 1) at s = m π , so the ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8 full amplitude T ( s ; 1) also has a pole as in the same way with T tree ( s ; 1).Approximating the amplitude by the Breit-Wigner form around the pole, T ( s ; 1) ∼ − g s − M , we extract the mass of the state M pole and the coupling to the scatteringstate g . In the present case, we find g → , M pole → m π for h σ i → . This is what we anticipate for the properties of the chiral partner; the massdegeneracy with the pion and the vanishing of the coupling constant to the ππ state.Next we consider the x = 0 case without the bare σ pole. In this case, h σ i dependence of the tree-level amplitude (3) exclusively stems from theoverall factor, T tree ( s ; x ) ∝ h σ i . Taking the restoration limit h σ i →
0, this term diverges, and therefore thefull amplitude is solely determined by the loop function G ( s ): T ( s ; x ) = 1 T − ( s, x ) + G ( s ) → G ( s ) for h σ i → . Thus, the pole of the amplitude in the restoration limit is given by the zeroof G ( s ). The present renormalization scheme requires G ( s ) = 0 for s = m π ,which indicates the existence of a pole at √ s = m π in the σ channel. Thecoupling constant g can be obtained by calculating the residue of the pole: g | h σ i→ =(4 π ) (cid:18) π √ − (cid:19) − m π . (5)Thus, for the dynamically generated σ meson, the amplitude has a pole atthe pion mass with the coupling constant which is proportional to m π for h σ i → m π → × SU(2) symmetry is exact in the Wigner phase.In the chiral limit, Eq. (5) indicates g | h σ i→ = 0, so the asymptotic valueof the mass and coupling constant of the dynamically generated σ mesonis exactly the same with the chiral partner case. Namely, the dynamicallygenerated σ meson behaves as if it is the chiral partner of the pion, for m π → ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8
4. Numerical study for the softening phenomena4.1.
Structure of the σ meson in vacuum We first show the description of the scattering amplitude in vacuum. Wechoose canonical values of the parameters as h σ i = 93 MeV, m π = 140MeV, and m σ = 550 MeV. By taking the parameter x = 1, the σ meson isdescribed as the chiral partner of the pion (chiral σ ), and we refer to thiscase as “model A”. Choosing x = 0, we obtain the dynamically generated σ meson in the amplitude. This case is called as “model B”.To check the agreement with the physical amplitude, we study the prop-erties of these models without symmetry restoration. We calculate the scat-tering length a = π T (4 m π ) (in units of m − π ) in these models. The resultsare shown in Table 1, together with the pole position of the amplitude inthe complex energy plane. We find a qualitative agreement with the recentanalyses of experimental data, a exp ∼ .
216 and z = 441 − i MeV.
Softening of the σ meson We then study the variation of the scattering amplitude in the σ channelalong with partial restoration of chiral symmetry. We plot the reduced crosssection ¯ σ = | T | /s and the trace of the pole position as functions of thetotal center-of-mass energy √ s , by changing the parameter Φ from 1 to 0.The invariant mass spectra and the pole trajectory of model A are shownin Fig. 1, where the softening of σ is clearly observed. As the symmetry isrestored, the σ pole moves toward the ππ threshold and the spectrum getsnarrow and enhanced around threshold. In this case, since the σ meson istreated as the chiral partner of pion, the softening phenomena is driven bythe decrease of the bare mass of the σ and its consequence of the reductionof the phase space for the decay. In the limit h σ i →
0, the pole approachesthe mass of the pion, as indicated by the analysis in section 3.2.We show the results of model B in Fig. 2. In this case, although thethreshold enhancement takes place, the change of the spectrum as well asthe trace of the pole are qualitatively different from those of model A.
Table 1. Properties of the models: value of parameter x , possible origin ofthe pole, the scattering length a in units of m − π , and the pole position of theamplitude in vacuum. x origin a [ m − π ] pole position [MeV]model A 1 chiral partner 0.244 423 − i model B 0 dynamically generated 0.174 364 − i ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8 | T | / s [ / M e V ] [MeV] Φ -400-300-200-1000 I m s / [ M e V ] [MeV] Φ = 1 Fig. 1. Mass spectra of the σ meson (left) and the trace of the pole positions (right) inmodel A ( x = 1). The poles on the first (second) Riemann sheet are plotted by triangles(crosses). Symbols are marked with each 0.1 step of Φ. Arrow indicates the direction ofthe movement of the pole as the parameter Φ is decreased from 1 to 0. Dotted (dashed)line represents the energy corresponds to the threshold (mass of pion). | T | / s [ / M e V ] [MeV] Φ -400-300-200-1000 I m s / [ M e V ] [MeV] Φ = 1 Fig. 2. Mass spectra of the σ meson (left) and the trace of the pole positions (right) inmodel B ( x = 0). The poles on the first (second) Riemann sheet are plotted by triangles(crosses). Symbols are marked with each 0.1 step of Φ. Arrows indicate the direction ofthe movement of the pole as the parameter Φ is decreased from 1 to 0. Dotted (dashed)line represents the energy corresponds to the threshold (mass of pion). Especially, the pole in the complex energy plane goes to the subthresholdenergy region, keeping the width finite. This is a peculiar feature of thedynamically generated σ meson. As a consequence, the strong enhancementof the cross section occurs at much later stage of the symmetry restoration,compared with the model A.Let us discuss how this structure appears in model B. The mechanismof the threshold enhancement in the nonlinear σ model has been studied inRef. 9; the partial restoration of chiral symmetry induces the reduction ofthe pion decay constant. Since the low energy interaction is proportional to1 / h σ i , the symmetry restoration effectively increases the attractive inter-action. Thus, the dynamically generated σ resonance will eventually turnsinto a ππ bound state when the interaction becomes sufficiently attractive. ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8 At this stage, however, it is important to recall that the σ is in s -wave. Inthis case, when the attraction is increased, the resonance first becomes the virtual state which is the pole on the second Riemann sheet of the energyplane but lies below the threshold. The peculiar pole trajectory in Fig. 2is due to the appearance of the virtual state.It is worth mentioning that the finite pion mass is important for theappearance of the virtual state. If m π = 0, the ππ threshold lies at √ s = 0,so there is no region where the virtual state appears. Indeed, the virtualstate was not seen in the analysis of the dynamically generated σ meson inRef. 10, studied in the chiral limit. In addition, the s -wave nature of the σ is also essential for the virtual state. Therefore, the behavior of the ρ mesonin p -wave amplitude will not exhibit such a structure (see also Ref. 16).As we further restore the symmetry, the σ meson becomes the boundstate. In the limit h σ i →
0, the pole moves toward the pion mass. This isagain in agreement with the result in section 3.2: the appearance of thepole at the pion mass in the restoration limit.
5. Summary
We have studied the properties of the σ meson in the ππ scattering associ-ated with the restoration of chiral symmetry. Two models are constructedbased on chiral low energy interaction and unitarity of the scattering am-plitude: one describes the σ meson as the chiral partner of the pion, andthe other treats the σ meson as dynamically generated resonance.For the dynamically generated σ meson, we find the qualitative dif-ference from the chiral partner σ in the softening behavior, namely, themovement of the pole of the amplitude and its consequence of the changeof the spectrum. The difference stems from the mechanism which drivesthe softening, and it is the appearance of the virtual state that leads to thedistinct behavior of the dynamically generated σ .We also study the asymptotic properties of the σ pole in the restorationlimit. For the σ meson as the chiral partner, we find that the mass of the σ pole approaches the pion mass and the coupling to the ππ state vanishes.For the dynamically generated σ , we also find the mass degeneracy withthe pion, and the coupling strength vanishes in the chiral limit m π → σ pole in the restorationlimit is essentially the same with what we expect for the chiral partner.This is a nontrivial result which urge us to speculate the possibility of thedynamically generated σ meson as the chiral partner of the pion.In this way, through the comparison of two models, we draw two con- ctober 30, 2018 4:57 WSPC - Proceedings Trim Size: 9in x 6in hyodo8 clusions: (i) with partial restoration of chiral symmetry, the difference ofthe internal structures is reflected in the spectral change of the σ channel,and (ii) in the symmetry restoration limit, the difference of the structureis reduced and we obtain essentially the same behavior of the σ pole for m π →
0. More comprehensive analysis, including the case with the σ mesonas the CDD pole, is now underway. Acknowledgments
The authors are grateful to M. Oka and Y. Kanada-En’yo for useful dis-cussion. This work is partly supported by the Global Center of ExcellenceProgram by MEXT, Japan through the Nanoscience and Quantum PhysicsProject of the Tokyo Institute of Technology, the Grant-in-Aid for Scien-tific Research from MEXT and JSPS (Nos. 21840026, 20540273, 2054026,and 20028004), and the Grant-in-Aid for the Global COE Program “TheNext Generation of Physics, Spun from Universality and Emergence” fromMEXT of Japan. This work was done in part under the Yukawa Interna-tional Program for Quark-hadron Sciences (YIPQS).
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