Theory of the New

Ds Resonances

William A. Bardeen

Fermilab

Abstract

> I discuss theoretical issues
>
> associated with the new
>
> Ds resonances.

• Introduction.

Quantum chromodynamics, QCD, has been established as the dynamical
theory of the strong interactions.

Perturbative QCD has been directly applied to the physics of hard
scattering processes.

However, many aspects of strong dynamics including spectroscopy and
hadronic decays require knowledge of nonperturbative aspects of QCD.

• Models have played an important role in understanding the physics of
hadrons.

Potential models have been very successful in describing the physics of
quarkonia, the boundstates of heavy quarks.

Constituent quark models have also been quite useful in describing the
physics of the light quarks and baryons.

Many other models have been constructed to apply to the physics of
hadrons in specific situations.

• Symmetries also play an important role in our fundamental
understanding of hadrons and hadronic processes.

SU(3) favor symmetry is an obvious example which is related to the light
quark substructure of QCD.

Chiral symmetries have also played an essential but less obvious role in
developing our understanding of hadronic physics.

Finally, new symmetries emerge for heavy quarks as the heavy quark spin
becomes a flavor symmetry.

• The physics of heavy-light bound states.

Bound states involving a heavy quark and a light antiquark are somewhat
complicated.

Simplifications occur in the heavy quark limit where states related by
the spin of the heavy quark become degenerate.

The physics of the light quarks and gluons are clearly nonperturbative
but the states must reflect the flavor and chiral symmetries of the
light quark.

Charmed hadrons are particularly difficult to describe because the charm
quark may not be sufficiently heavy to justify the use of heavy quark
symmetries and the light quark physics is not accessible to direct
analytic analysis.

Nevertheless, we have learned that there may be many "dual" descriptions
of QCD where different models may be abstracted that contain much of the
relevant physics.

• The new Ds states.

The discovery of the new narrow Ds resonances by BaBar, Cleo, and Belle
are novel in a number of ways.

• BaBar Collaboration, arXiv:hep-ex/0304021

• Cleo Collaboration, arXiv:hep-ex/0305017

• Belle Collaboration, arXiv:hep-ex/0307041

Potential models had predicted additional J~light~=1/2 p-wave resonances
should exist but they were expected to be heavy and rapidly decay to a
D-meson and a K-meson.

The new states are much lighter than the potential model predictions,
the Ds(2317) meson being more than 40 MeV below the D-K threshold.

However, the narrow width can now be understood as the leading strong
decays are forbidden by phase space.

Many models or other descriptions of the new states have been recalled
or proposed since their discovery.

• Modified potential models

• Heavy quark and chiral symmetries

• DK Molecules

• 4 quark models, hybrids

• Unitarized chiral meson bound states

• Bag models

• QCD sum rules

• Phenomenological string models

• Lattice field theory

• Potential Models.

Potential Models were developed to describe the physics of quarkonia,
the heavy quark bound states.

Successful phenomenological potentials for quarkonia combine a
vectorlike coulomb interaction with a long-range scalar confining
potential.

![](media/image1.pct){width="6.497222222222222in"
height="8.151388888888889in"}

Predictions for heavy-light systems use these potentials to describe the
interactions between the heavy quark and a light constituent quark.

• S. Godfrey and N. Isgur, Phys. Rev. D32,189(1985)

• S. Godfrey and R. Kokoski, Phys. Rev. D43,1679(1991)

• M. Di Pierro and E. Eichten, Phys. Rev. D64,114004(2002)

Potential models have been successful in describing many aspects of the
D-meson and B-meson spectrum.

The models clearly fail to describe the new Ds states.

Adhoc modifications to the potential model were proposed shortly after
the discovery of the new states but are less predictive and are not
completely successful in describing the new states.

• R.N. Cahn and J.D. Jackson, arXiv:hep-ph/0305012

• P. Bicudo and G.M. Marques, arXiv:hep-ph/0309231

• Heavy quark symmetry and chiral symmetry.

Since D-mesons are bound states of a heavy charm quark and a light u,d,s
quark, the meson spectrum and interactions should reflect both
symmetries.

Bardeen, Hill and Eichten have argued that the heavy quark and chiral
symmetries suggest a parity-doubled spectrum for the heavy-light mesons
where the new Ds states are identified the parity partners of the
(0-,1-) ground states.

• W.A. Bardeen, E. Eichten, C.T. Hill, arXiv:hep-ph/0305049

Heavy quark and chiral symmetry would require a parity-doubled spectrum
of states if the chiral symmetry were not spontaneously broken in the
QCD ground state.

The (0+,1+) "p-wave" states form a supermultiplet with the (0-,1-)
"s-wave" states having total J~light~=1/2.

The mass splitting between the two multiplets occurs when the light
quark symmetry is spontaneously broken in the true QCD ground state.

An effective field theory can be constructed to describe the dynamics of
this chiral supermultiplet similar to the linear sigma model for nucleon
states.

The mass splitting between the chiral multiplets arises through
couplings to the chiral sigma field in a manner similar the nucleon.

![](media/image2.pct){width="5.263888888888889in"
height="2.2020833333333334in"}

This effective field theory had actually been constructed 10 years ago
using a truncated version of QCD where the mass splitting between
multiplets is related to the effective mass of the constituent quark.

• M.A. Nowak, M. Rho and I. Zahed, Phys. Rev. D48,4370(1993)

• W.A. Bardeen and C.T. Hill, Phys. Rev. D49, 409(1994)

This truncated QCD replaces the gluon exchange interaction by an
attractive four-fermion interaction that preserves both heavy quark
symmetry and light quark chiral symmetry. The model produces four bound
states corresponding to J~light~=1/2 states of the potential model.
Contrary to full QCD, the four-fermion coupling constant can be adjusted
to allow for a chiral broken or a chiral symmetric ground state. In the
chiral symmetric phase, the four bound states form a degenerate
supermultiplet transforming as linear representations of
SU(3)~L~XSU(3)~R~ chiral symmetry. In the phase with spontaneous chiral
symmetry breaking, the splitting between the s- and p-wave states is
generated.

A model independent description of this dynamics can be written using a
nonlinear sigma model with appropriate couplings to the heavy meson
fields.

![](media/image3.pct){width="5.180555555555555in"
height="4.490972222222222in"}

Note that G~A~ is introduced as a model independent parameter that is
expected to be close to 1, the value in the linear model.

Deviations of G~A~ from 1 can occur from mixing with higher states or
through higher order terms in the effective field theory. G~A~ should be
the determined from the model-independent analysis like the other chiral
parameters.

Pionic transitions between the (0+,1+) and the (0-,1-) multiplets are
governed by a Goldberger-Treiman relation and the mass splitting between
the multiplets.

$$G_{p}\text{\textasciitilde}DM$$

The finite charm quark mass will break the heavy quark symmetry
generating spin splittings. In the chiral symmetric phase, these spin
splittings must be common to both (0-,1-) and (0+,1+) multiplets.

This analysis implies a common mass splitting between the respective s-
and p-wave states.

The prediction, $DM(0 + ,0 - ) = DM(1 + ,1 - )$, is seen to be good to a
few MeV experimentally.

Predictions for the decays of the new Ds states have been made using
these chiral Lagrangians and a constituent quark model for the
electromagnetic transitions. The predictions for masses and decay widths
for both D-mesons and B-mesons are tabulated.

The table indicates the parameter sensitivity for each decay. Widths are
also sensitive to the actual masses.

![](media/image4.pct){width="6.496527777777778in"
height="5.970138888888889in"}

Most radiative decays of the new Ds states are suppressed by an
"accidental" cancellation between processes involving the charm quark
and the strange constituent quark \[r~cs~ \~ 1-2m~s~/m~c~ \~ 0.37\].
This cancel-lation is not present in the static limit or for Bs states.

• W. Bardeen, E. Eichten and C.T. Hill, arXiv:hep-ph/0305049

see also (for decays)

• S. Godfrey, arXiv:hep-ph/0305122(v1,v2) \[G~A~ \~ O(g~A~)\]

• P. Colangelo et al, arXiv:hep-ph/0305140 \[G~A~ \~ 0.6 (QCD SR)\]

and (for masses)

• A. Deandrea, G. Nardulli and A. Polosa, arXiv:0305049

• M.A. Nowak, M. Rho and I. Zahed, arXiv:hep-ph/0307102

In the chiral symmetric phase of QCD, parity doubling is a feature of
all states not just the ground states. This implies a shift in the
intercepts of Regge trajectories for the meson states. \[from di Pierro
and Eichten\]

![](media/image5.pct){width="5.792361111111111in"
height="5.044444444444444in"}

• Lattice field theory

Lattice field theory provides an almost unique path to study the
nonperturbative aspects of QCD.

Methods have been developed to address various problems associated with
realistic simulations of QCD including improved gauge actions, heavy
quark actions, chiral fermion actions, the treatment of sea quarks and
the extrapolation to the continuum limit.

Predictions for the Ds(0+)-Ds(0-) mass splitting cover a wide range of
values and sophistication.

Before the recent discoveries, predictions for the splitting included
Hein et al (500(80)MeV)and Lewis et al (530(20)MeV) ) using NRQCD at
fixed lattice spacing, consistent with the potential model predictions.

• Hein et al, Phys. Rev. D62,074503(2000), arXiv:hep-ph/0003130

• R. Lewis et al, Phys Rev. D62,114507, arXiv:hep-lat/0003011

Following the discovery, Bali studied the continuum limit of a theory
with a static heavy quark as well as dynamical light quarks and
concluded that his results predict a splitting that is too large to be
consistent with the masses observed for the new states.

• G.S. Bali, arXiv:hep-ph/0305209

The UKQCD Collaboration analyzed the continuum limit of quenched QCD and
finds a (0+,0-) splitting of (389(47)MeV) and a (1+,0-) splitting of
(500(62)MeV) consistent with the masses of the new states.

• A. Dougall et al, arXiv:hep-lat/0307001

Using MILC unquenched gauge configurations, a calculation with almost
realistic sea quark masses yields values of (384(45)MeV) for the (0+,0-)
splitting and (414(31)MeV) for the (1+,1-) splitting.

• M. Di Pierro et al, 2003 Lattice Conference

While the lattice analyses do show a considerable spread in their
predictions, the more realistic analyses seem to give values for the Ds
splittings that are coming close to the experimental observations.

![](media/image6.pct){width="6.496527777777778in"
height="8.180555555555555in"}

The results are preliminary and we can look forward to more realistic
and complete analyses in the future. The lattice field theory methods
are focused on complete calculations in full QCD and, therefore, do not
distinquish between the various QCD inspired models.

• D-K Molecule States

The new Ds states lie close to the D-K threshold in the case of the (0+)
state and the D\*-K threshold in the case of the 1+ state.

Molecule states could form through residual attractive interactions
between the two mesons.

These loosely bound mesons are suggested to be the analogue of the
deuteron and, perhaps, the scalar meson states located near K+K-
threshold at 1 GeV.

Molecule states should be distinquished by soft form factors in the
production process and by decay patterns that differ from those expected
of a normal diquark state.

Large isospin mixing may be expected for these states which could
explain the dominance of the pionic isospin violating decays over the
competing radiative transitions.

• N. Isgur and H.J. Lipkin, Phys. Lett., 151(1981)

• T. Barnes, F.E. Close, H.J. Lipkin, arXiv:hep-ph/0305025

• T.D. Cohen, B.A. Gelman and S. Nussinov, arXiv:hpe-ph/0309210

• Four quark states, hybrids.

Proposals also include models based on 4-quark states with a different
quark structure that arises due to an attractive force between diquarks
in a 3\* configuration.

These mesons should also be distinguished by their production and decay
patterns.

• K. Terasaki, Phys. Rev. D68,011501(2003), arXiv:hep=ph/0305213

• H.Y. Chen and W.S. Hou, Phys. Lett. B566,193(2003)

• C.-H. Chen and H.-N. Li, arXiv:hep-ph/0306155

• Unitarized chiral meson bound states.

The new states are close to the thresholds for an s-wave decay to a
D-meson and a K-meson.

The observed narrow Ds states could arise by a strong mixing of the
diquark state with four quark states induced by the presence of the
nearby thresholds. In this scenario the largely diquark state has its
mass suppressed below threshold and a broad four-quark state should be
present above threshold.

• T.E. Browder, S. Pakvasa and A.A. Petrov, arXiv:hep-ph/0307054

A separate argument suggests that higher mass diquark resonances may
trigger the formation of lighter dimeson bound states.

• E. Van Beveren and G. Rupp, arXiv:hep-ph/0305035

A chiral bound state model relies on the unitarization of coupled
channel meson scattering amplitudes generated by an appropriate
nonlinear meson chiral Lagrangian.

• E.E. Kolomeitsev and M.F.M Lutz, arXiv:hep-ph/0307133

• J. Hoffmann and M.F.M. Lutz, arXiv:hep-ph/0308263

• Bag model.

• M Sadziikowski, arXiv:hep-ph/0307084

• QCD sum rules.

• Y.-B. Dai, C.-S. Huang, C. Liu and S.L. Zhu, arXiv:hep-ph/0306274

• QCD String Model.

String models differ from potential models as much of the energy and
angular momentum is associated with the string (or flux tube) and not
the quarks at the end of the string.

Kalashnikova et al apply a version of the QCD string model to the Ds
states and find a mass spectrum consistent with potential models.

•Y.S. Kalashnikova, A.V. Nefediev and Y.A. Simonov, Phys. Rev.

D64,014037(2001)

On the other hand Bardeen finds that a semiclassical string with proper
treatment of the light quark spin can reproduce the chiral structure of
the Ds states.

• W. Bardeen, unpublished

In the absence of spontaneously broken chiral symmetry this string model
predicts degenerate parity-doubled supermultiplets in accord with our
previous analysis.

The mass splitting between Ds multiplets depends on the constituent
quark mass generated by explicit and spontaneous chiral symmetry
breaking and vanishes as the effective constituent quark mass vanishes.

![](media/image7.pct){width="6.493055555555555in"
height="7.645833333333333in"}![](media/image8.pct){width="6.49375in"
height="7.8805555555555555in"}

• Conclusions.

The new Ds states discovered by BaBar, Cleo and Belle provide an
intriguing challenge for QCD.

It is likely that these states can be identified with the missing
J~light~=1/2 p-states.

The main argument against this identification is the mass shift from the
values expected in phenomenological potential models.

However, we have stressed that QCD is not equivalent to a potential
model and additional degrees of freedom or the inclusion of new effects
may be essential.

Symmetry arguments have shown that the J~light~=1/2 p-states and the
J~light~=1/2 s-states can be viewed as a chiral supermultiplet which
would be degenerate in mass in a phase where the light quark chiral
symmetry remains unbroken. Spontaneous chiral symmetry breaking
generates the splitting within the supermultiplet.

Chiral constituent quark models make detailed predictions for decay
modes involving both pions and photons. The pattern of the decays helps
to establish nature of these new states and can be used to distinguish
between the various models discussed above.

Lattice field theory is making progress toward the application of the
full structure of QCD to many nonperturbative problems in hadron physics
including the spectrum of Ds mesons.