RRadially excited ψ mesons and the Y enhancements Susana Coito ∗ Institute of Physics, Jan Kochanowski University, 25-406 Kielce, PolandE-mail: [email protected]
While many properties of the vector charmonium first excitations are yet to be measured, en-hancements at unexpected energies are intriguing, alias the Y states. In order to understand thenaturally unquenched mesonic line-shapes, the influence of the most relevant hadronic decaychannels must be taken into account. Within an unitary effective approach we present resultswhere mesonic loops are included in an equivalent manner to coupled-channels. We show resultsfor the ψ ( ) and ψ ( ) systems, where we find the nonperturbative effects of dynamicalgeneration of poles and line-shape distortion. XIII Quark Confinement and the Hadron Spectrum - Confinement201831 July - 6 August 2018Maynooth University, Ireland ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - ph ] N ov mesons and the Y enhancements Susana Coito
1. Introduction
The charmonium vector states, alias the ψ states, can only be considered strictly “quenched”,i.e., almost pure ¯ cc states, if they lay below all the hadronic decay channels involving open-charmmesons, which are allowed by conservation laws. In this category fall only the J / ψ and the ψ ( S ) mesons [1]. All the higher ψ ’s are strongly influenced by their nearby decay channels, not onlythe open ones, but the closed ones as well. Moreover, through the same decay channels, they arealso strongly influenced by the nearest other ψ ’s, below and above. A comprehensive study of thecharmonium vector spectrum should then include all the interferences that are expected from thesimplest interactions, which is an extensive and intricate quest. Some of the phenomena that canresult from such interferences are i) deformation of the line-shapes, ii) generation of poles fromthe continuum, iii) shifting of the mass peak positions in certain decay channels due to specialinterferences between the background and a certain opening threshold.The understanding of some of these interferences should shed light over an apparently differ-ent problem, that is the emergence of Y enhancements, i.e., vectorial structures appearing in thecharmonium energy region in certain non-dominant hadronic decay modes, such as the J / ψπ + π − [2, 3], h c π + π − [4], or ψ ( S ) π + π − [5], with different masses than the ψ states, cf. Fig. 1. Whilesome of the Y signals can actually be mostly independent from the ψ resonances, others can be dueto interferences generated by them, and such possibilities are worthy to be analyzed.We pay attention to the fact that a resonance is not determined by a Breit-Wigner fit to a peakin the mass distribution to a certain decay channel. For instance, a rough look to data in channels DD , DD ∗ , and D ∗ D ∗ , both from BaBar [6] and Belle [7], allows us to see that the peaks and dipsare in different positions in the different channels and therefore, it would be simply wrong to fitwith Breit-Wigners to each peak in each decay channel. Instead, what has been done to determinethe mass of the ψ states, was the analysis of the R distribution, i.e., the ratio of the cross sectionto all hadrons and the cross section to leptons [8, 9]. An equivalent analysis using non-dominanthadronic decay amplitudes only, would be helpful to a better identification of the Y states.In this work we show results of an unitarized effective Lagrangian model that has been pre-viously applied to the light meson sector to generate the dynamical resonances κ and a ( ) [10, 11]. We hypothesize that a similar phenomena could occur to higher energy systems. In-deed, different works have contemplated such possibilities, e.g. in [12] for an open-charm scalar D ∗ ( − ) , or in [13, 14], for a charmonium scalar at 3.7 GeV. A material indication thatthe latter resonance exists may be found in Ref. [15], where Belle has observed different peaks inthe different channels DD , DD ∗ , and D ∗ D ∗ compatible with scalar quantum numbers. Rather thandifferent resonances, one can consider only one pole below DD threshold that generates a tail inthe line-shape of each decay channel, in which case the assignment of a different state X to eachone of the peaks in the different invariant masses is, as discussed above, incorrect.In Refs. [16, 17] we studied in detail the line-shape of the ψ ( ) within the model we brieflydiscuss here, where in addition to a seed pole, a dynamically generated pole has been found. Here,we present some of our main results related to the ψ ( ) . Concerning the higher states, we showline-shape results for the ψ ( ) , following the discussion started in Refs. [18, 19]. Moreover,we refer to the work in Refs. [20, 21], where the ψ ( ) has been treated with the same approach,in which the authors have found a dynamical pole around 4.0 GeV.1 mesons and the Y enhancements Susana Coito −− (PDG 2018) M ( G e V ) ψ ( ) ψ (3770)( ) ψ (4040)( ) ψ (4160)( ) ψ (4415)( ) DDDD ∗ D s D s D ∗ D ∗ D s D ∗ s D ∗ s D ∗ s DD , DD ′ D s D s , D ∗ D D s D ′ s Y (4230) Y (4260) Y (4360) Y (4390) Y (4660) Figure 1:
Vector charmonium spectrum, position of the Y enhancements, and dominant decay modes.
2. The model
We employ an unitarized effective Lagragian model to compute the line-shape and cross sec-tion of production processes of the type e + e − → ψ → m m , where m , is the final meson − anti-meson system. Rather than a simple massive vector in the intermediate state, we consider meson-meson channels in the form of one-loop that obey the “Born” expansion, i.e., the occurrence prob-ability of one one-loop is higher than of two one-loops, etc.. Thus the scalar part of the propagatorseries is convergent and obeys a geometric progression that sums up to ∆ ψ ( s ) = s − m ψ + ∑ Nj Π j ( s ) , (2.1)where m ψ is the bare mass of the ψ , s is the invariant energy squared, N is the number of channels,and Π j ( s ) is the loop function of channel j , that is given by Π j ( s ) = Ω j ( s ) + i √ s Γ j ( s ) , Ω , Γ ∈ ℜ . (2.2)2 mesons and the Y enhancements Susana Coito
The real part of Π j is obtained from the dispersion relations Ω j ( s , m , m ) = PP π (cid:90) ∞ s thj √ s (cid:48) Γ j ( s (cid:48) , m , m ) s (cid:48) − s d s (cid:48) , (2.3)and the imaginary part is given by Γ ψ → ( m m ) j ( s ) = k j ( s , m , m ) π s | M ψ → ( m m ) j | , (2.4)where k j is the final state momentum, and | M ψ → ( m m ) j | = V j ( s , m , m ) f Λ ( q j ) . (2.5)The function V is the vertex amplitude, computed using the Feynmann rules, and f is a vertexform factor that depends on a cutoff parameter Λ , and on the off-shell momentum q j , and it is heredefined by an exponential as f Λ ( (cid:126) q j ) = e − q j / Λ . (2.6)We note that our full vertex function is the product of V with f , and therefore the “form-factor” f is conceptually different than the traditional electromagnetic form-factor that defines the full vertexin a model independent way. The spectral function is given in function of the energy E = √ s by d ψ ( E ) = − E π Im ∆ ψ ( E ) , (2.7)which due to unitarity comes automatically normalized to 1.
3. Results ψ ( ) A detailed study of the ψ ( ) can be found in Refs. [17, 16]. Here we present our mainresults. The interaction Lagrangian density is defined by L ψ ( P P ) j = ig ψ ( P P ) j ψ µ ∑ j (cid:16) ( ∂ µ P P ) j − ( ∂ µ P P ) j (cid:17) , (3.1)where P P are the pseudoscalar mesons D + D − and D ¯ D . The amplitude V comes as V j ( s , P , P ) = g ψ ( P P ) j k j ( s , P , P ) . (3.2)Poles are found when the denominator of the propagator (Eq. (2.1)) is zero on the proper Riemannsheet, defined by the condition Im k j < E > √ s th j in both channels. Only three free parametersare needed, the cutoff Λ , the seed mass m ψ , and the effective coupling for the vertices g ψ ( P P ) ≡ g ψ DD , which is the same for both channels. The three parameters are obtained by the fit to crosssection data in Ref. [22] with χ (cid:39) .
03 and we get Λ (cid:39)
272 MeV, m ψ (cid:39) g ψ DD (cid:39) − i
18 MeV and at 3777 − i
12 MeV, the first one from the continuum,and the second one coming from the seed, as it can be see in Fig. 2. The existence of these twopoles in the same Riemann Sheet is not evident from the line-shape that shows one peak only, butdistorted with relation to a Breit-Wigner shape (cf. Fig. 3 of Ref. [18]).3 mesons and the Y enhancements
Susana Coito E (GeV) I m E ( G e V ) • •↑ ↓ D ¯ D D + D − Figure 2:
Pole trajectories of the ψ ( ) . Left green line: trajectory of the dynamical pole, right blue line:trajectory of the seed pole. The arrows indicate the direction of increasing the coupling at vertices ψ DD .Bullets: pole positions for the fitting parameters (cf. text and Ref. [17]). ψ ( ) In a similar way, we employ the model to the vector ψ ( ) that, as the ψ ( ) , shouldbe dominantly a d -wave. The system is more complex than the ψ ( ) though, since there aremore possibilities of decay (see Fig. 1). For simplicity, we include only the open channels withopen charm, viz. DD , DD ∗ , D ∗ D ∗ , D s D s , D s D ∗ s , and D ∗ s D ∗ s . We consider the coupling strengthsto be flavor independent, and therefore we use the partial decay ratios Γ DD ∗ / Γ D ∗ D ∗ ∼ .
34 and Γ DD / Γ D ∗ D ∗ ∼ .
02 in [1] to compute all the partial couplings involved at vertices ψ ( m m ) j . Therelative amplitudes to each channel are shown in Fig. 3, together with the total amplitude throughthe line-shapes, for a cutoff Λ =
450 MeV. The remaining free parameter is the seed mass whichis set to reproduce the peak mass and width at the values given by PDG [1], i.e. m ∼ Γ ∼
70 MeV. For this cutoff it comes m ψ (cid:39) D s D ∗ s , but its branching ratio has not yetbeen determined in the experiment. We can also verify that the peak position is the same in everychannels, although the line-shape of the partial amplitudes change. In Ref. [19] we discussed theline-shape of the decay ψ ( ) → J / ψ f ( ) with the one-loop effect only, using the six chan-nels above, concluding that the peak position would not shift, therefore no peak for the Y ( ) could be obtained. The fact that in the experiment a peak is not seen at the ψ ( ) position, inthe mode J / ψπ + π − [2, 3], could be justified by a very small coupling to this channel. Yet, giventhe large phase space available, it is unexpected that the ψ ( ) does not manifest in this mode.Instead, one may consider the existence of some other effect that would actually shift the ψ ( ) position, specifically in the J / ψ f ( ) → J / ψπ + π − channel, leading even to the line-shape ofthe Y ( ) . Such idea is developed in Ref. [23].4 mesons and the Y enhancements Susana Coito E (GeV) d ψ ( G e V − ) Total
DDDD ∗ D s D s D ∗ D ∗ D s D ∗ s D ∗ s D ∗ s Figure 3:
Normalized spectral function of the ψ ( ) (see Eq. (2.7)) for Λ =
450 MeV (black), and partialspectral functions. In threshold opening order, DD (red), DD ∗ (green), D s D s (dark blue), D ∗ D ∗ (orange), D s D ∗ s (light blue), and D ∗ s D ∗ s (violet). Finally, in Fig. 4 we study the variation of the total line-shape with the cutoff. For smallercutoff parameters, the distortion with relation to a Breit-Wigner line-shape is more pronounced.This can be understood from Eq. (2.6), from where the exponential falls off quicker for smaller Λ values, thus limiting the range of influence of each loop to smaller energies. Away from theresonance peak, one should include other components such as the ψ ( ) and the ψ ( ) onthe left, and the ψ ( ) on the right. Our result is thus qualitative, so as the value of the cutoff Λ ,which differs from the value obtained for the “cleaner” ψ ( ) system, that came from the fit.
4. Conclusion
In conclusion, we aim to disentangle whether some of the Y enhancements can be manifesta-tions of the ψ states. We presented results concerning the ψ ( ) and ψ ( ) vectors, withinan unitarized effective Lagrangian model, where the most relevant one-loop channels have beenincluded. A distortion from a Breit-Wigner line-shape is visible for both ψ ’s. A dynamically gen-erated companion pole is found for the ψ ( ) , yet not leading to an independent peak in theline-shape. The poles of the ψ ( ) have not been examined here. The cutoff parameter is re-garded as a free parameter, to which one can attribute a physical meaning, related to the size of thesystem, with the possibility that it is also an effective parameter, i.e. that it somehow depends onthe number of variables included in the problem. The one-loop dynamics only leads to an overallshifting from the seed mass, but not to the displacement of the peak position in different channels.Therefore, the Y ( ) does not rise as a manifestation of the ψ ( ) within this model version.Other mechanisms to address the origin of the Y ( ) are under study.5 mesons and the Y enhancements Susana Coito E (GeV) d ψ ( G e V − ) Figure 4:
Variation of the normalized spectral function of the ψ ( ) , with the cutoff Λ . Green line: Λ =
400 MeV, red line: Λ =
450 MeV, blue line: Λ =
500 MeV, black dashed line: Breit-Wigner distribution.
Acknowledgements
The author thanks to F. Giacosa and to K.U. Can for useful discussions. This work was sup-ported by the
Polish National Science Center through the project OPUS no. 2015/17/B/ST2/01625.
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