New Developments in Heavy-Light Systems

Nuclear Physics B (Proc. Suppl.) 142 (2005) 242–246 www.elsevierphysics.com

New Developments in Heavy-Light Systems E. Eichtena∗ a

Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA

The study of the narrow excited states in the (c¯ s) heavy-light system provides new insight into QCD dynamics. These effects involve the light quarks and are unaccounted for in relativistic potential models.

1. INTRODUCTION Last year the BABAR[1] experiment found evidence for a new narrow charmed-strange meson in the decay mode, Ds (2317) → Ds π 0 . Shortly thereafter, this state was confirmed by CLEO[2] and a second narrow state was found in the mode Ds (2459) → Ds∗ π 0 . Both these states were confirmed by Belle[3]. At this conference, we have heard of the observation of another narrow state in the charmed-strange system Ds (2632) in two modes by the SELEX[4] collaboration. Powerful tools exist in QCD to study heavylight systems: HQET[5] for the heavy quark and chiral symmetry for the light quark degrees of freedom[6]. In the heavy quark limit, the excitation spectrum is independent of the heavy quark mass, mQ , and spin, sQ and is classified by the total angular momentum of the light quark, jl = L + sl , and parity P . Each state is doubly degenerate, with J = jl ± sQ . Furthermore, hadronic transitions between these doublet states are related by symmetry. 2. RELATIVISTIC POTENTIALS In general the QCD Hamiltonian for heavylight systems can be expanded in powers of m−1 Q :

1 1 H(1) + 2 H(2) . H = H(0) + mQ mQ

(1)

The leading term H(0) contains the interactions of the light quark and gluons in the static source ∗ Work performed at Fermi National Accelerator Laboratory, which is operated by University Research Association, Inc., under contract DE-AC02-76CHO3000.

0920-5632/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2005.01.044

of the heavy quark. The next term in the m−1 Q expansion, H(1) , contains the leading hyperfine splitting interaction between the heavy and light quark as well as correction terms for the spinorbit interaction and the kinetic energy term due to heavy quark motion. Heavy Quark Symmetry (HQS) provides many rigorous relations but some dynamical quantities still need explicit nonperturbative calculation. Although Lattice QCD numerical results are available in some cases, for the most part only phenomenological models are available. In the case of heavy-light systems the Relativistic Quark Model[7] and modern variants [8] have been used extensively. Within the relativistic potential approach, in leading order, the light quark satisfies a Dirac equation with a potential V(R) representing the QCD dynamical interaction with the heavy quark that acts a static source. Explicitly, H(0) = −i α · ∂ + γ 0 mq + V (R)

(2)

with the potential V (R) of the same form as used in quarkonium phenomenology V (R) = γ 0 MQ + γ 0 Vs (R) + Vv (R).

(3)

Explicit forms for the scalar potential Vs (R) = bR and the vector potential, Vv (R): Vv (R) = −

4 αs erf(λR) 3R

(4)

have been used for heavy-light systems [8,9]. Fitting the well-known heavy-light states determines the parameters [9]: the QCD coupling αs , a short distance cutoff λ, the slope parameter b, quark

E. Eichten / Nuclear Physics B (Proc. Suppl.) 142 (2005) 242–246

masses mu = md , ms , mc and mb , and meson mass parameters Mc and Mb . Using these potentials and explicit forms for the H(1) interactions [9], the masses and wavefunctions for heavy-light states can be calculated. Using ideas from the chiral quark model [12], Goity and Roberts [8] and DiPierro and Eichten [9] also computed the amplitudes for pseudoscalar transitions between excited states in heavy-light systems. Every thing was is reasonable agreement with the known properties of heavy-light states circa 2002. 3. Ds (0+ ) AND Ds (1+ ) In April 2003, BABAR [1] discovered a new narrow state Ds (2317) decaying to Ds π 0 . Soon thereafter, CLEO [2] discovered a second narrow state, Ds (2459) decaying into Ds∗ π 0 . Initially there was speculation that these states were DK molecules [10]. It now seems clear that these are + s states with J P = 0+ the usual jqP = 12 P-wave c¯ P + and J = 1 respectively. Narrow widths are expected since the states are below threshold for Zweig allowed strong decays. Lattice calculations still have large errors but are generally consistent with the observed masses [13]. However, their masses are significantly different from those expected from the Relativistic Potential models. The masses are more than 100 MeV lower expected [9]. The failure of relativistic potential models was surprising. Attempts to add simple modifications to reduce the disagreement were not successful. Modifications that improved the agreement on masses tended to make the agreement for decay + widths of the jlP = 32 P-wave (c¯ s) states worse [14].

4. CHIRAL DYNAMICS 4.1. What’s missing ? For heavy quark-antiquark systems the gluonic degrees of freedom adjust quickly relative to the heavy quark motion. This leads to an effective static energy between the heavy quark and antiquark, which can be determined in Lattice QCD [11]. However as one quark becomes light, this

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difference of time scales disappears. The light quark and gluonic degrees of freedom interact with the heavy quark on the same time scale. The QCD “string” degrees of freedom need to be included in the dynamics in a more complete way. The string can carry not just energy but also angular momentum [15]. Furthermore, the effects of light quark pairs cannot be ignored. To remedy the defects of potential models in this situation, it is useful to begin from an alternative approach. Imagine chiral symmetry and confinement could coexist. Then, ignoring explicit chiral symmetry breaking terms, the light hadrons could become massless but heavy-light systems are still forced to remain massive because of the heavy quark. The result is that heavylight states would be parity doubled [19]. The basic question is “How far away is QCD from this point?” 4.2. Chiral supermultiplets For unbroken chiral symmetry, the heavy-light states form chiral supermultiplets (CMS). The ground state contains four degenerate states: the − jlP = 12 J P = (0− , 1− ) 1S states (HP S , HV ) and + the jlP = 12 J P = (0+ , 1+ ) 1P states (HS , HA ). This jj = 12 multiplet can be represented in heavy quark notation as:

/ µ 1+v ]( √ ).(5) H = [γ 5 HP S + γµ HVµ + HS + γ 5 γµ HA 2 2

The two lowest sets of excited states are: (1) A “radially” excited multiplet with jl = 12 , H , and the same form as the ground state multiplet. (2) An “orbitally” excited multiplet with jl = 32 + consisting of the jlP = 32 J P = (1+ , 2+ ) 1P − states (PAµ , KTµν ) and the jlP = 32 J P = (1− , 2− ) µ µν 1D states (PV , KT 5 ) given by:

Pµ

=


1 1 [ 3/2(δνµ − γν γ µ + γν v µ ) 3 3

(6)

1 + v/ (γ 5 PAν + PVν ) + (KTµν + iγ 5 KTµν5 )]( √ ) 2 2

4.3. Effective chiral lagrangians The effective chiral lagrangian for the heavylight meson systems is now easy to construct. The

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E. Eichten / Nuclear Physics B (Proc. Suppl.) 142 (2005) 242–246

3− − 2− 2− 1

1− 0−

2+ 1+ 1++ 0

m2 (mu − md ) 0 π η. Llight = ... + √π 3(mu + md )

1− 0−

L=0

usual chiral lagrangian is used for the light pseudoscalar meson octet (“pions”) self-interactions. The expansion of the mass term in the meson fields to quadratic order in that lagrangian yields an isospin violating π 0 η mixing term:

J L=1

1− 2

It was emphasized by Cho and Wise [21] that this term allows the isospin violating decays of excited Ds mesons via π 0 transitions. Chiral projections of the supermultiplet HL,R = (1 ± γ 5 )H/2 transform as (3, 1) and (1, 3) under SU (3)L × SU (3)R . For the ground state supermultiplet, H, the effective Lagrangian through the first order interaction with the chiral field Σ (which breaks CMS) and first order in the HQS breaking (1/mQ ) corrections is [20]:

P

L=2

HQS

5− 2 3− 2

3+ 2 1+ 2

−iTr(H† v · ∂H) − iTr(H† v/ ∆0 H) + . . .  gA  Tr(H† (∂/ Σ)H) + . . . −gπ Tr(H† v/ ΣH) + i fπ  c2  † µν (8) Tr(H σ Hσµν ) + . . . + mQ

Leff

1− 2

j
P CMS 1 2

1 2

(7)

5 2

3 2

j


Figure 1. Successively increasing symmetry in the heavy-light excitation spectrum. The top figure shows a typical physical spectrum. The HQS spectrum ignores all 1/mQ corrections to the heavy quark limit. The CMS spectrum also ignores breaking of chiral supermultiplet symmetry.

=

where (for spontaneously broken chiral symmetry) Σ = fπ exp(iπ a ·λa /fπ ). This Lagrangian encodes all of the same information found in Eq.(12) of [16] in a compact form. Fig. 1 illustrates the change of excitation spectrum as the CMS breaking and the HQS breaking m−1 Q interaction terms are included. 4.4. Predictions There are two immediate consequences of the chiral multiplet approach for the Ds excitation spectrum. First, M (1+ ) − M (0+ ) = M (1− ) − M (0− ) because the m−1 Q hyperfine splitting term in Eq.(8) is independent of the parity of the heavy quark multiplet. Second, the splitting between the (0− , 1− ) and (0+ , 1+ ) states, ∆M , due to the CMS breaking of the gπ Tr[H† v/ ΣH] term in Eq.(8) is given by a Goldberger-Treiman relation ∆M = gπ fπ [16]. Using the measured mass difference between the parity multiplets, 349 MeV and fπ = 93.3 MeV, one obtains gπ = 3.74. The parity changing π 0 transitions between the + + (0 , 1 ) states and the (0− , 1− ) states are determined by the interaction term gπ H† v/ γ 5 π · λH.

E. Eichten / Nuclear Physics B (Proc. Suppl.) 142 (2005) 242–246

This term can be rewritten in the usual nonlinear chiral Lagrangian form as an axial current term, iGA Tr[H† γ 5 v · AH], where the the axial currents is Aµ = −∂µ π ˜ /2fπ + ... GA is introduced as a phenomenological parameter but can be far from unity if the overall picture is valid. There are also parity-conserving one pion transitions between 1± and 0± states. They are described by the operators, gA Tr[H† γ 5 γµ Aµ H]. The value of gA can be extracted from a fit to the D+∗ total width. The hadronic decays of Ds excited states contained in the H supermultiplet proceed by emitting a virtual η “pion” which then mixes with the π 0 through isospin-violating effects of Eq.(7). All these rates as well as the small rate for Ds (2459) → Ds + 2π were predicted by Bardeen, Eichten and Hill [16]. The radiative transitions were also computed. The most important point here is that the corrections to the HQS limit for the c¯ s system are large and suppress the photon transition rates. Defining rQ¯q = 1 −

eQ mq¯ mQ eq¯

(9)

the E1 and M1 rates depend on r2 . For the c¯ s system rc¯s = 0.38. For the Bs analog states the suppression is absent (r¯bs = 1.10) and the photon transitions dominate. The comparison of all these predictions with the present experimental data is shown in Table 1. The agreement is quite encouraging.

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5.2. Possible interpretations Although this state is not confirmed by BABAR [22] and so its status remains uncertain, it is interesting to speculate on possible interpretations. The most likely interpretation of a narrow Ds (2632) state is the jqP = 1− radial excitation of the Ds∗ state. The observation of a DK decay mode suggests J P = 1− . In this case, the mass is 100-175 MeV below Relativistic Potential model predictions. Correcting for phase space, the decay modes DK:D∗ K: Ds η would have been expected to be in the approximate ratio 8:2:1, respectively. The observed Ds η mode is much larger than predicted. This might be due to cancellations in the decay amplitude associated with the radial node in the Ds (2632) wavefunction [9]. The HQS partner state J P = 0− , the radial excitation of the Ds state, would be expected to be about 100 MeV below the Ds (2632), very near the observed + Ds (2535) jqP = 12 P-wave J P = 1+ state. This HQS partner state would be very narrow and its dominant decay could be Ds + 2π. An alternative interpretation of the Ds (2632) would be the lowest D-wave excitation of the Ds system. The quantum numbers would be − J P = 1− and jqP = 32 . However the expected mass for this D-state more than 250 MeV above the observed state mass. The suppression of the DK decays would be unexplained. Here the HQS partner state with J P = 2− would be expected with a mass 40 MeV above the observed state. This state would decay primarily into D∗ K with Ds∗ η less than 10%.

5. NARROW Ds STATE AT 2632 MeV?

6. SUMMARY

5.1. Observation The SELEX collaboration has reported evidence for a new narrow state in the c¯ s system observed in two decay modes [4]. The decay mode Ds+ η is seen with a significance of 7.2σ at a mass of 2635.9 ± 2.9 MeV/c2 . While the second decay mode D0 K + is seen with a significance of 5.3σ at 2631.5 ± 1.9 MeV/c2 . The decay width of this state is less than 17 MeV at 90% confidence level. The relative branching fraction Γ(D0 K + )/Γ(Ds+ η) = 0.16 ± 0.06.

The narrow Ds (0+ ) and Ds (1+ ) states provide insight into heavy-light dynamics. Potential models are inadequate. New degrees of freedom are required. The jl = 12 S and P wave states can be viewed as a chiral supermultiplet. Spontaneous chiral symmetry and HQS symmetry breaking account for splittings within the supermultiplet. Testable predictions result for decay modes involving pions and photons. If the SELEX narrow Ds (2632) state is confirmed, it would likely be a J P = 1− state. Possi-

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Table 1 The comparison of theory and experiments for ratios of branching ratios. For experiment results first error is statistical and second is systematic.

BR ratio [Ds (2317) → + γ]/[Ds (2317) → Ds + π 0 ] [Ds (2459) → + γ]/[Ds (2459) → Ds∗ + π 0 ] [Ds (2459) → Ds + γ]/[Ds (2459) → Ds∗ + π 0 ]

Theory [16] 0.081 0.217 0.236

[Ds (2459) → Ds + π + π − ]/[Ds (2459) → Ds∗ + π 0 ]

0.088

[Ds (2459) → Ds (2317) + γ]/[Ds (2459) → Ds∗ + π 0 ] [Ds (2459) → Ds (2317) + π 0 ]/[Ds (2459) → Ds∗ + π 0 ]

0.127 0.0004

Ds∗ Ds∗

−

−

ble assignments are jlP = 12 or jlP = 32 . In both cases an observable narrow HQS partner state must exist nearby.

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