In this work, an Effective Field Theory (EFT) incorporating light SU(3)-flavour and heavy quark spin symmetries is used to describe charmed meson-antimeson bound states. At Lowest Order (LO), this means that only contact range interactions among the heavy meson and antimeson fields are involved. Besides, the isospin violating decays of the X(3872) will be used to constrain the interaction between the D and a (D) over bar* mesons in the isovector channel. Finally, assuming that the X(3915) and Y(4140) resonances are D* (D) over bar* and D-s* (D) over bar (s)* molecular states, we can determine the four Low Energy Constants (LECs) of the EFT that appear at LO and, therefore, the full spectrum of molecular states with isospin I = 0, 1/2 and 1

AlFiky

C. Hidalgo-Duque

Choi

Fleming

Gamermann

Hanhart

Hidalgo-Duque

J. Nieves

M. Pavón Valderrama

Nieves

Valderrama

Nuclear Physics A

arXiv

International audienceIn this work, an Effective Field Theory (EFT) incorporating light SU(3)-flavour and heavy quark spin symmetries is used to describe charmed meson-antimeson bound states. At Lowest Order (LO), this means that only contact range interactions among the heavy meson and antimeson fields are involved. Besides, the isospin violating decays of the X(3872) will be used to constrain the interaction between the D and a View the MathML source mesons in the isovector channel. Finally, assuming that the X(3915) and Y(4140) resonances are View the MathML source and View the MathML source molecular states, we can determine the four Low Energy Constants (LECs) of the EFT that appear at LO and, therefore, the full spectrum of molecular states with isospin I=0, View the MathML source and 1

Hidalgo-Duque, C.

Nieves, J.

Pavón Valderrama, M.

HAL-IN2P3

Heavy quark spin symmetry and SU(3)-flavour partners of the X(3872)

Crossref

Heavy quark spin symmetry and SU(3)-flavour partners of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3872</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>

In this work, an Effective Field Theory (EFT) incorporating light SU(3)-flavour and heavy quark spin symmetries is used to describe charmed meson-antimeson bound states. At Lowest Order (LO), this means that only contact range interactions among the heavy meson and antimeson fields are involved. Besides, the isospin violating decays of the X(3872) will be used to constrain the interaction between the D and a (D) over bar* mesons in the isovector channel. Finally, assuming that the X(3915) and Y(4140) resonances are D* (D) over bar* and D-s* (D) over bar (s)* molecular states, we can determine the four Low Energy Constants (LECs) of the EFT that appear at LO and, therefore, the full spectrum of molecular states with isospin I = 0, 1/2 and 1.This research was supported by DGI and FEDER funds, under contract FIS2011-28853-C02-02, and the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), by Generalitat Valenciana under contract PROMETEO/20090090 and by the EU HadronPhysics2 project, grant agreement No. 227431. CHD acknowledges support from CSIC JAE program.Peer reviewe

Hidalgo Duque, Carlos

Nieves, Juan Miguel

Pavón Valderrama, Manuel

Digital.CSIC

arXiv:1211.7004v1  [hep-ph]  29 Nov 2012Heavy Quark Spin Symmetry and SU(3)-Flavour Partners of the X(3872)C. Hidalgo-Duquea, J. Nievesa, M. Pavo´n ValderramabaInstituto de F´ısica Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, Institutos de Investigacio´n dePaterna, Aptd. 22085, E-46071 Valencia, Spain.bInstitut de Physique Nucle´aire, Universite´ Paris-Sud, IN2P3/CNRS, F-91406 Orsay Cedex, FranceAbstractIn this work, an Effective Field Theory (EFT) incorporating light SU(3)-flavour and heavy quark spinsymmetry is used to describe charmed meson-antimeson bound states. At Lowest Order (LO), this meansthat only contact range interactions among the heavy meson and antimeson fields are involved. Besides, theisospin violating decays of the X(3872) will be used to constrain the interaction between the D and a D¯∗mesons in the isovector channel. Finally, assuming that the X(3915) and Y (4140) resonances are D∗D¯∗ andD∗sD¯∗s molecular states, we can determine the four Low Energy Constants (LECs) of the EFT that appearat LO and, therefore, the full spectrum of molecular states with isospin I = 0, 12 and 1.Keywords: Heavy Quark Spin and Flavour Symmetries, Hidden charm molecules, XYZ states.1. IntroductionSince the discovery of the X(3872) resonance [1],many theoretical approaches have been used tryingto describe it. Even though it contains a cc¯ pair, itdoes not seem to fit in the charmonium spectrum.Because of that disagreement, other more exoticproposals have been made. Among these, the inter-pretation of the X(3872) as a hadronic molecule isthe most likely so far. Within this assumption, theX(3872) would be a bound state of both a charmmeson and antimeson. However, this hypothesisstrongly depends on the JPC quantum numbers ofthe resonance, which have not been experimentallyestablished. They are either 1++ or 2−+ (analysisdone in Ref. [2]), of which only 1++ is compatiblewith a low-lying S-wave hadronic molecule.Apart from the X(3872), many other experimen-tal hidden charm resonances have been observed:the XYZ states. Many of these states can be newcandidates for heavy meson-antimeson molecules.These resonances are near their threshold. Thismeans that the meson and the antimeson are notso close to be sensitive to find the details of the in-teraction at short distances. So, both of them arepreserving their individuality and will not probethe specific details of the short range interactionresponsible of their binding. Hence, a scale sepa-ration exists and the interaction between these twomesons can be described by different EFT’s [3, 4].Besides, the presence of a heavy quark in the heavymesons imposes that our EFT should be consistentwith Heavy Quark Spin Symmetry (HQSS), whichimplies certain constraints to the heavy meson-antimeson interaction [5]. On the other hand, thelight quark content of the heavy mesons (q = u, d, sin this work) imposes SU(3) flavour symmetry and,because of that, molecular states should be classi-fied into SU(3) multiplets.As a consequence of the symmetries discussed(HQSS and SU(3)) flavour symmetry) only fourparameters are enough to describe the molecularstates at leading order in the EFT we use. That is,we need four data points to predict the full molec-ular spectrum. For this purpose we will assume themolecular nature of certain XYZ states such as theX(3872), X(3915) and Y(4140) and the fourth as-sumption will be derived from the analysis of theisospin violating branching ratio of the X(3872) de-cays into J/ψω and J/ψρ.2. The EFT Description at Lowest OrderIn this section, the EFT used in this work todescribe the heavy meson molecules is briefly pre-sented. The EFT description must involve pionsPreprint submitted to Nuclear Physics A November 30, 2012and heavy meson/antimeson fields and local inter-actions among these degrees of freedom that arecompatible with the known low energy symmetries,most notably HQSS and chiral symmetry. Never-theless, according to [6]: pion exchanges are weakerthan naively expected and only enter as a perturba-tion at subleading orders. Similarly, coupled chan-nel effects turn out to be also sufficiently small to beignored at LO. Hence, at LO, the EFT consists ofheavy mesons and antimesons interacting througha contact range potential, similar to the one in [5].Then, once we have determined our potential, thewave functions and observables will be calculatedin the standard quantum mechanical fashion. Forexample, we can generate bound states by iteratingthe EFT potential in the Schro¨dinger / Lippmann-Schwinger equation, as previously done in [6]. Seedetails in [7].3. Isospin Symmetry Violation in theX(3872)The first step is to solve the bound state equationfor the X(3872). We consider that the X(3872) is aDD¯∗ molecule with quantum numbers JPC = 1++,where we distinguish between the neutral (D0D¯∗0)and charged (D+D∗−) components of the wavefunction. That is, there are two channels in thebound state equation. And we will regularize theLO potential (here, and in the following cases whennot specified) with a Gaussian regulator functionwith a cut-off Λ varying in the 0.5− 1.0GeV range.Moreover, the Belle collaboration reportedthe decays of the X(3872) into the (isoscalar)J/Ψpi+pi−pi0 and the (isovector) J/Ψ pi+pi− chan-nels. The latest measurements yield to the isospinviolating ratio [8]:BX = Γ(X(3872)→ J/Ψ pi+pi−pi0)Γ(X(3872)→ J/Ψ pi+pi−) = 0.8± 0.3(1)which is difficult to accomodate from the theoreticalpoint of view.In this work, we will assume that this isospin vi-olation is caused by the isospin breaking generatedby the mass difference of the neutral (D0D¯0∗) andcharged (D+D∗−) channels in the X(3872), whichwould not have a definite isospin. In this picture, atshortDD¯∗ distances, theX(3872) would be a linearcombination of I = 0 and I = 1 components whilstthe J/Ψ3pi and J/Ψ2pi decays would be describedby an isospin invariant coupling via an intermediateρ and ω meson (as suggested in [9]).The analysis performed by Hanhart et al. in [2],leads to the following branching ratioRX =M(X → J/Ψ ρ)M(X → J/Ψω) = 0.26+0.08−0.05 , (2)where BX is translated into a ratio of the decay am-plitudes of the X(3872) instead of its correspondingdecay widths. In our model, this ratio RX can berewritten as (details in [7])RX =gρgωΨˆX0 − ΨˆX1ΨˆX0 + ΨˆX1. (3)being gV = MV [DD¯∗(I = 0, 1) → J/Ψ V ] withV = ρ/ω and ΨˆX0 y ΨˆX1 an average of the neutraland charged D∗D¯ wave function components in thevicinities of the origin.So, with the experimental determination of RXand the binding energy of the X(3872) we coulddetermine the contact range potential that bindsthe X(3872) if we knew the gρ/gω ratio,. But thegρ/gω ratio can be determined from the SU(3) re-lation gρ − gω = −√2 gφ , and, as the OZI (thestrange quark pair creation is suppressed) rule im-plies gρ, gω ≫ gφ , that is, we are ignoring theX(3872) decay into J/ψφ. Thus, we can approx-imately consider gρ/gω ≃ 1, so the RX ratio in ourmodel isRX =ΨˆX0 − ΨˆX1ΨˆX0 + ΨˆX1. (4)which only depends on the wave function compo-nents in the vicinities of the origin.4. The SU(3) and HQSS Partners of theX(3872)If we determine the value of the counterterms ofthe LO EFT, we will be able to calculate the loca-tion of the molecular partners of the X(3872).There are four unknown LEC’s. We fix two ofthem from the location of the X(3872) resonanceand its isospin breaking branching ratio, as ex-plained in the previous section. The remainingtwo require the identification of two partners of theX(3872): the X(3915) as a 0++ D∗D¯∗ moleculeand the Y (4140) as a 0++ D∗sD¯∗s molecule, guidedby its apparently dominant decay into J/Ψφ.Apart from the RX errors, there is an extra errorsource which has to be taken into account: the ap-proximate nature of HQSS. In this EFT, we expand2JPC HH¯ E (Λ = 0.5 GeV) E (Λ = 1 GeV)0++ DD¯ 3709± 10 3715+12−151++ D∗D¯ Input Input1+− D∗D¯ 3815± 17 3821+23−260++ D∗D¯∗ Input Input1+− D∗D¯∗ 3955± 17 3958+24−272++ D∗D¯∗ 4013††−9 4013††−12Table 1: Predicted masses (in MeV) of the SU(2) isoscalar HQSSpartners of the X(3872) resonance for two different values of the Gaus-sian cutoff. Errors in the predicted masses are obtained by adding inquadratures the uncertainties stemming from the two sources of sys-tematic errors discussed at the end in Subsect. 4. †† : see discussion in[7]JPC HH¯ E (Λ = 0.5 GeV) E (Λ = 1 GeV)0+ D+s D¯− 3835.8+2.3−3.9 3837.7+0.4−4.31+ DsD¯∗ 3949± 13 3957+14−190+ D∗sD¯∗ 4056± 22 4061+29−331+ D∗sD¯∗ 4091+13−14 4097+15−202+ D∗sD¯∗ − −Table 2: Predicted masses (in MeV) of the isospinor(I = 12) HQSS partners of the X(3872) resonance,for two different values of the Gaussian cutoff. Themeaning of the quoted errors in the table is the sameas in Table 1.the QCD lagrangian into powers of (ΛQCD/mQ) sothatV LO(mQ=mc) = VLO(mQ→∞)(1 ± ΛQCDmc) , (5)Taking an approximate value of mc ≃ 1.5 GeVfor the charm quark mass and ΛQCD ∼ 200 MeV,we should expect a 15% violation of HQSS for theLO contact range potentials. Since the two errorsources are independent, the total error will be com-puted by adding in quadratures the partial errors.4.1. The SU(2) Isoscalar (I = 0) PartnersIn this sector the hidden strange componentsare ignored. We do not take into account particlecoupled channel effects as they are subleading un-less the mass gap between the neutral and chargedchannels is similar in size to the binding energy inthe isospin symmetric limit. Thus, in the 1++ and2++ channels, we are using a coupled channel po-tential. The analysis of this sector was previouslydone in Ref. [6] without including strangeness intothe analysis (SU(2) light quark flavour symmetry)and, as there is almost no difference in the reso-nances predicted, we conclude that the effect of theisospin violation and the inclusion of the quark smust be small. The spectrum of molecular states ispresented in Table 1.4.2. The Isospinor (I = 12) PartnersIn this sector, the C-parity of the molecules isnot well defined, as they are not bound states of aheavy meson and its antimeson. However, the for-malism is identical to the one in the previous case,except for the 1+ DsD¯∗ and DD¯∗s molecules. TheDsD¯∗ and DD¯∗s thresholds are separated by only2MeV and require a coupled channel treatment. Inthis case, we obtain just one single bound state.This resonance and its other isospinor partners areshown in Table 2, where we have considered onlythe strangeness one states (it would be the samefor strangeness S = −1).4.3. The Isovector (I = 1) PartnersThe potential in the isovector and isospinor sec-tor is the same (except in the 1++ and 2++molecules owing to the isospin violation). For thisreason, the spectrum in the isovector sector wouldbe similar to the isospinor one. The four molecularstates obtained are listed in Table 3. The other twopossible states correspond to the isovector partnersof the X(3872) and X(4012) resonances where asingle bound state is obtained.4.4. The Hidden Strange PartnersIn this sector, the resonances must also con-tain a ss¯ quark-antiquark pair. The potential de-rived for this latter case is the arithmetic mean ofthe isoscalar and isovector one. So if there wasa bound state in the isoscalar and isovector sec-tor, there would probably exist its hidden strangepartner. On the other hand, considering that theX(3872) and X(4012) molecules have no isovectorpartners, it is very likely that there will not behidden strange partners either with those quantumnumbers. Therefore, the four states obtained arelisted in Table 4.3JPC HH¯ E (Λ = 0.5 GeV) E (Λ = 1 GeV)0++ D+D¯0 3732.5+2.0−3.9 3734.3+0.1−3.61++ D∗D¯ − −1+− D∗D¯ 3848+12−11 3857+13−180++ D∗D¯∗ 3953± 22 3960+28−321+− D∗D¯∗ 3988± 13 3995+14−192++ D∗D¯∗ − −Table 3: Predicted masses (in MeV) of the SU(2) isovectorHQSS partners of the X(3872) resonance for two different val-ues of the Gaussian cutoff. The meaning of the quoted errorsin the table is the same as in Table 1.JPC HH¯ E (Λ = 0.5 GeV) E (Λ = 1 GeV)0++ DsD¯s 3924+6−7 3928+7−111++ D∗sD¯s − −1+− D∗sD¯s 4035± 15 4040+20−240++ D∗sD¯∗s Input Input1+− D∗sD¯∗s 4177± 16 4180+21−242++ D∗sD¯∗s − −Table 4: Predicted masses (in MeV) of the hidden strangeisoscalar HQSS partners of the X(3872) resonance for two dif-ferent values of the Gaussian cutoff. The meaning of the quotederrors is the same as in Table 1.5. ConclusionsAs a summary, we have established the locationof 15 molecular partners of the X(3915), Y (4140)and X(3872) states using HQSS and SU(3) flavoursymmetry, see Tables 1-4. These predictions havea series of uncertainties, being the most importantthe approximate nature of HQSS. According to theestimations of these uncertainties, the spectrum ofthese states should be stable but their location canvary up to a few tens of MeV in certain cases.However, the family of D(∗)D¯(∗) states we the-orize depends on the assumptions we made: themolecular description of the X(3872), X(3915) andY (4140) resonances. Whilst there is a consensus onthe molecular nature of the X(3872) resonance, andits existence is well established, the situation for theX(3915) and Y (4140) are more dubious. Thus notall the predictions are probable alike. Predictionsderived from the X(3872) are supposed to be moresolid than those depending on the X(3915), whichin turn are less hypothetical that the ones obtainedfrom the Y (4140). In this sense, as mentioned inRef. [6] too, the 2++ D∗D¯∗ isoscalar partner of theX(3872) is still the most reliable prediction of thepresent work. So, if any new analysis of the XYZresonances provides us with a better molecular can-didate than one of our assumptions, it can easily beincluded in this scheme. Thus, we will be able to re-place one of the doubtful molecular states we haveassumed with this new resonance so the new pre-dictions are more robust. On the other way, if even-tually any of the predictions we have established isdetected, it could be a proof to the molecular natureof these resonances.ACKNOWLEDGMENTSThis research was supported by DGI and FEDERfunds, under contract FIS2011-28853-C02-02, andthe Spanish Consolider-Ingenio 2010 ProgrammeCPAN (CSD2007-00042), by Generalitat Valen-ciana under contract PROMETEO/20090090and by the EU HadronPhysics2 project, grantagreement no. 227431.References[1] S. K. Choi et al. [Belle Collaboration], Phys. Rev. Lett.91 (2003) 262001 [hep-ex/0309032].[2] C. Hanhart, Y. .S. Kalashnikova, A. E. Kudryavtsevand A. V. Nefediev, Phys. Rev. D 85 (2012) 011501[3] S. Fleming, M. Kusunoki, T. Mehen and U. van Kolck,Phys. Rev. D 76 (2007) 034006[4] M. P. Valderrama, Phys. Rev. D 85 (2012) 114037[5] M. T. AlFiky, F. Gabbiani and A. A. Petrov, Phys.Lett. B 640 (2006) 238[6] J. Nieves and M. P. Valderrama, Phys. Rev. D 86 (2012)056004[7] C. Hidalgo-Duque, J. Nieves and M. P. Valderrama,arXiv:1210.5431 [hep-ph].[8] S. -K. Choi, S. L. Olsen, K. Trabelsi, I. Adachi, H. Ai-hara, K. Arinstein, D. M. Asner and T. Aushev et al.,Phys. Rev. D 84 (2011) 052004[9] D. Gamermann, J. Nieves, E. Oset and E. Ruiz Arriola,Phys. Rev. D 81 (2010) 0140294

Heavy quark spin symmetry and SU(3)-flavour partners of the X (3872)

Nieves Pamplona, Juan Miguel

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

Heavy quark spin symmetry and SU(3)-flavour partners of the X (3872)

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