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Open Charm and Beauty Chiral Multiplets in QCD

Stephan Narison

Laboratoire de Physique Mathématique et Théorique, UM2, Place Eugène Bataillon, 34095 Montpellier
Cedex 05, France
Email: narison@lpm.univ-montp2.fr

We study the dynamics of the the spin zero open charm and beauty mesons using QCD spectral sum rules
(QSSR), where we observe the important rôle of the chiral condensate 〈ψ̄ψ〉 in the mass-splittings between
the scalar-pseudoscalar mesons. Fixing the sum rule parameters for reproducing the well-known D(0−) and
Ds(0

−) masses, we re-obtain the running charm quark mass: m̄c(mc) = 1.13
+0.08
−0.04 GeV, which confirms

our recent estimate from this channel [1]. Therefore, using sum rules with no-free parameters, we deduce
MDs(0+) ≃ (2297 ± 113) MeV, which is consistent with the observed Ds(2317) meson, while a small SU(3)
breaking of about 25 MeV for the Ds(0

+)−D(0+) mass-difference has been obtained. We extend our analysis
to the B-system and find MB(0+) − MB(0−) ≃ (422 ± 196) MeV confirming our old result from moment sum
rules [2]. Assuming an approximate (heavy and light) flavour and spin symmetries of the mass-splittings
as indicated by our results, we also deduce MD∗

s
(1+) ≃ (2440 ± 113) MeV in agreement with the observed

DsJ (2457). We also get: fD(0+) ≃ (217 ± 25) MeV much bigger than fπ=130.6 MeV, while the size of the
SU(3) breaking ratio fDs(0+)/fD(0+) ≃ 0.93 ± 0.02 is opposite to the one of the 0− channel of about 1.14.

1 Introduction

The recent BABAR, BELLE and CLEO observations of two new states DsJ (2317) and DsJ (2457) [3] in the
Dsπ, Dsγ and Dsπγ final states have stimulated a renewed interest in the spectroscopy of open charm states
which one can notice from different recent theoretical attempts to identify their nature [3]. In this paper,
we will try to provide the answer to this question from QCD spectral sum rules à la Shifman-Vainshtein-
Zakharov [4]. In fact, we have already addressed a similar question in the past [2], where we have predicted
using QSSR the mass splitting of the 0+ − 0− and 1− − 1+ b̄u mesons using double ratio of moments sum
rules based on an expansion in the inverse of the b quark mass. We found that the value of the mass-
splittings between the chiral multiplets were about the same and approximately independent of the spin of
these mesons signaling an heavy quark-type approximate symmetry:

MB(0+) − MB(0−) ≈ MB∗(1+) − MB∗(1−) ≈ (417 ± 212) MeV . (1)

The effect and errors on the mass-splittings are mainly due to the chiral condensate 〈ψ̄ψ〉 and to the value
of the b quark mass. In this paper, we shall use an analogous approach to the open charm states. However,
the same method in terms of the 1/mc expansion and some other nonrelativistic sum rules will be dangerous
here due to the relatively light value of the charm quark mass. Instead, we shall work with relativistic
exponential sum rules used successfully in the light quark channels for predicting the meson masses and
QCD parameters [5] and in the D and B channels for predicting the (famous) decay constants fD,B [1, 5, 6]
and the charm and bottom quark masses [1, 5, 6, 7, 8].

2 The QCD spectral sum rules

We shall work here with the (pseudo)scalar two-point correlators:

ψP/S (q
2) ≡ i

∫

d4x eiqx 〈0|T JP/S (x)J†P/S (0)|0〉, (2)

built from the (pseudo)scalar and (axial)-vector heavy-light quark currents:

JP/S (x) = (mQ ± mq)Q̄(iγ5)q, JµV /A = Q̄γ
µ(γ5)q . (3)



If we fix Q ≡ c and q ≡ s, the corresponding mesons have the quantum numbers of the Ds(0−), Ds(0+)
mesons. mQ and ms are the running quark masses. In the (pseudo)scalar channels, the relevant sum rules
for our problem are the Laplace transform sum rules:

LHP/S (τ) =
∫ ∞

t≤

dt e−tτ
1

π
ImψHP/S (t), and RHP/S (τ) ≡ −

d

dτ
log LHP/S (τ), (4)

where t≤ is the hadronic threshold, and H denotes the corresponding meson. The latter sum rule, or its
slight modification, is useful, as it is equal to the resonance mass squared, in the simple duality ansatz
parametrization of the spectral function:

1

π
ImψHP (t) ≃ f2DM4Dδ(t − M2D) + “QCD continuum”Θ(t − tc), (5)

where the “QCD continuum comes from the discontinuity of the QCD diagrams, which is expected to give
a good smearing of the different radial excitations 1. The decay constant fD is analogous to fπ = 130.6
MeV; tc is the QCD continuum threshold, which is, like the sum rule variable τ, an (a priori) arbitrary
parameter. In this paper, we shall impose the τ- and tc-stability criteria for extracting our optimal results.
The corresponding tc value also agrees with the FESR duality constraints [9, 5] and very roughly indicates
the position of the next radial excitations. However, in order to have a conservative result, we take a largest
range of tc from the beginning of τ- to the one of tc-stabilities.
The QCD expression of the correlator is well-known to two-loop accuracy (see e.g. [5] and the explicit
expressions given in [1, 6]), in terms of the perturbative pole mass MQ, and including the non-perturbative
condensates of dimensions less than or equal to six 2. For a pedagocial presentation, we write the sum
rule in the chiral limit (ms = 0), where the expression is more compact. In this way, one can understand
qualitatively the source of the mass splittings. The sum rule reads to order αs:

LHP/S (τ) = M
2
Q

{

∫ ∞

M2
Q

dt e−tτ
1

8π2

[

3t(1 − x)2
[

1 +
4

3

(ᾱs
π

)

f(x)
]

+ O(α2s)
]

+ C4〈O4〉P/S + τC6〈O6〉P/S e−M
2
Qτ

}

, (6)

The different terms are 3:

x ≡ M2Q/t,

f(x) =
9

4
+ 2Li2(x) + log x log(1 − x) −

3

2
log(1/x − 1)

− log(1 − x) + x log(1/x − 1) − (x/(1 − x)) log x,

C4〈O4〉P/S = ∓MQ〈d̄d〉 e−M
2
Qτ + 〈αsG2〉

(

3

2
− M2Qτ

)

/12π

C6〈O6〉P/S = ∓
MQ
2

(

1 −
M2Qτ

2

)

g〈d̄σµν
λa
2
Gµνa d〉 (7)

−
(

8π

27

)

(

2 −
M2Qτ

2
−
M4Qτ

2

6

)

ραs〈ψ̄ψ〉2 , (8)

where we have used the contribution of the gluon condensate given in [11], which is IR finite when letting
mq → 0 4. The previous sum rules can be expressed in terms of the running mass m̄Q(ν) through the
perturbative two-loop relation [13]:

MQ = m̄Q(p
2)

[

1 +

(

4

3
+ ln

p2

M2Q

)

(ᾱs
π

)

+ O(α2s)
]

, (9)

where MQ is the pole mass. Throughout this paper we shall use the values of the QCD parameters given in
Table 1.

1At the optimization scale, its effect is negligible, such that a more involved parametrization is not necessary.
2We shall include the negligible contribution from the dimension six four-quark condensates, while we shall neglect an

eventual tachyonic gluon mass correction term found to be negligible in some other channels [10].
3Notice a missprint in the expression given in [2, 1] which does not affect the result obtained there.
4The numerical change is negligible compared with the original expression obtained in [12].



Table 1: QCD input parameters used in the analysis.

Parameters References

Λ4 = (325 ± 43) MeV [5]
Λ5 = (225 ± 30) MeV [5]
m̄b(mb) = (4.24 ± 0.06) GeV [5, 8, 1, 14]
m̄s(2 GeV) = (111 ± 22) MeV [5, 8, 15, 16]
〈d̄d〉1/3(2 GeV)=−(243 ± 14) MeV [5, 8, 17]
〈s̄s〉/〈d̄d〉 = 0.8 ± 0.1 [5, 18]
〈αsG2〉 = (0.07 ± 0.01) GeV4 [5, 19]
M20 = (0.8 ± 0.1) GeV2 [5, 2]
αs〈ψ̄ψ〉2 = (5.8 ± 2.4) × 10−4 GeV6 [5, 19, 20]

We have used for the mixed condensate the parametrization:

g〈d̄σµν
λa
2
Gµνa d〉 = M20 〈d̄d〉, (10)

and deduced the value of the QCD scale Λ from the value of αs(MZ ) = (0.1184 ± 0.031) [21, 22]. We have
taken the mean value of ms from recent reviews [5, 8, 16].

3 Calibration of the sum rule from the D(0−) and Ds(0
−) masses

and re-estimate of m̄c(mc)

Figure 1: τ in GeV−2-dependence of the a) MD(0
−) in GeV for m̄c(mc) = 1.11 GeV and b) MDs(0−) in

GeV for m̄c(mc) = 1.15 GeV at a given value of tc = 7.5 GeV
2. The dashed line is the result including the

leading 〈ψ̄ψ〉 contribution. The full line is the one including non-perturbative effects up to dimension-six.

This analysis has been already done in previous papers to order αs and α
2
s and has served to fix the running

charm quark mass. We repeat this analysis here to order αs for a pedagogical purpose. We show in Fig.
1a), the τ-dependence of the D(0−) and in Fig 1b) the one of the Ds(0

−) masses for a given value of tc,
which is the central value of the range:

tc = (7.5 ± 1.5) GeV2 , (11)

where the lowest value corresponds to the beginning of τ-stablity and the highest one to the beginning of
tc stability obtained by [1, 6, 5] in the analysis of fD and fDs . This range of tc-values covers the different
choices of tc used in the sum rule literature. As mentioned previously, the one of the beginning of tc-stability
cöıncides, in general, with the value obtained from FESR local duality constraints [9, 5]. Using the input



values of QCD parameters in Table 1, the best fits of D(0−) (resp. Ds(0
−)) masses for a given value of

tc = 7.5 GeV
2 correspond to a value of m̄c(mc) of 1.11 (resp. 1.15) GeV. Taking the mean value as an

estimate, one can deduce:

m̄c(m
2
c ) = (1.13

+0.07
−0.02 ± 0.02 ± 0.02 ± 0.02) GeV , (12)

where the errors come respectively from tc, 〈ψ̄ψ〉, Λ and the mean value of mc required from fitting the
D(0−) and Ds(0

−) masses. This value is perfectly consistent with the one obtained in [1, 6] obtained to
the same order and to order α2s, indicating that, though the α

2
s corrections are both large in the two-point

function and mc [23], it does not affect much the final result from the sum rule analysis. In fact, higher
corrections tend mainly to shift the position of the stability regions but affect slightly the output value of
mc. This value of mc is in the range of the current average value (1.23 ± 0.05) GeV reviewed in [5, 8, 21].
However, it does not favour higher values of mc allowed in some other channels and by some non relativistic
sum rules and approaches. However, these non relativistic approaches might be quite inaccurate due to the
relative smallness of the charm quark mass. Higher values of mc would lead to an overestimate of the D(0

−)
and Ds(0

−) masses. In the following analysis, we shall use the central value m̄c(mc) = 1.11 (resp. 1.15)
GeV for the non-strange (resp. strange) meson channels.

4 The 0+ − 0− meson mass-splittings

Figure 2: Similar to Fig. 1 but τ-behaviour of MDs(0+) for given values of tc = 7.5 GeV
2 and mc(mc) = 1.15

GeV.

• We study in Fig. 2), the τ-dependence of the Ds(0+) mass at the values of tc and mc obtained
previously. In this way, we obtain:

MDs(0+) ≃ (2297
+81+63
−98−70 ± 11 ± 2 ± 11) MeV =⇒ MDs(0+) −MDs(0−) = (328 ± 113) MeV , (13)

where the errors come respectively from tc, mc, 〈ψ̄ψ〉, ms, and Λ. We have used the experimental value
of MDs(0−). The reduction of the theoretical error needs precise values of the continuum threshold

5

and of the charm quark mass which are not within the present reach of the estimate of these quantities.
Further discoveries of the continuum states will reduce the present error in the splitting. One should
also notice that in the ratio of sum rules with which we are working, we expect that perturbative
radiative corrections are minimized though individually large in the expression of the correlator and
of the quark mass.

• The value of the mass-splittings obtained previously is comparable with the one of the B(0+)-B(0−)
given in Eq. (1), and suggests an approximate heavy-flavour symmetry of this observable.

• We also derive the result in the limit of SU(3)F symmetry where the strange quark mass is put to
zero, and where the 〈ψ̄ψ〉 condensate is chirally symmetric (〈s̄s〉 = 〈d̄d〉). In this case, one can predict
an approximate degenerate mass within the errors:

MDs(0+) − MD(0+) ≃ 25 MeV , (14)
5The range of tc values 6-9 GeV

2 obtained previously for the D(0−) mesons cöıncides a posteriori with the corresponding
range for the D(0+) meson if one assumes that the splitting between the radial excitations is the same as the one between the
ground states, i.e about 300 MeV. We have cheked during the analysis that this effect is unimportant and is inside the large
error induced by the range of tc used.



which indicates that the mass-splitting between the strange and non-strange 0+ open charm mesons
is almost not affected by SU(3) breakings, contrary to the case of the 0− mesons with a splitting of
about 100 MeV.

• We extend the analysis to the case of the B(0+) meson. Here, it is more informative to predict the
ratio of the 0+ over the 0− masses as the prediction on the absolute values though presenting stability
in τ tend to overestimate the value of MB. We obtain:

MB(0+)
MB(0−)

≃ (1.08 ± 0.03 ± 0.03 ± 0.02 ± 0.02) =⇒ MB(0+) −MB(0−) ≃ (422 ± 196) MeV , (15)

where the errors come respectively from tc taken in the range 43− 60 GeV2, mb, 〈ψ̄ψ〉, and τ. It agrees
with the result in Eq. (1) obtained from moment sum rules [2]. We have used the value of m̄b(mb)
given in Table 1.

5 The 1+ − 1− meson mass-splitting
Our previous results in Eqs. (1), (13) to (15) suggest that the mass-splittings are approximately (heavy
and light) flavour and spin independent. Therefore, one can write to a good approximation the empirical
relation:

MDs(0+)−MDs(0−) ≈ MD(0+)−MD(0−) ≈ MB(0+)−MB(0−) ≈ MB(1+)−MB(1−) ≈ MD∗s (1+)−MD∗s (1−) . (16)

Using the most precise number given in Eq. (13), one can deduce:

MD∗s (1+) − MD∗s (1−) ≃ (328 ± 113) MeV =⇒ MD∗s (1+) ≃ (2440 ± 113) MeV . (17)

This result is consistent with the 1+ assignement of the c̄s meson DsJ (2457) discovered recently [3]. In a
future work, we plan to study in details this spin one channel using QSSR.

6 The D(0+) and Ds(0
+) decay constants

Figure 3: Similar to Fig. 1 but τ-behaviour of fD(0
+) for given values of tc = 7.5 GeV

2 and m̄c(mc) = 1.11
GeV.

For completing our analysis, we estimate the decay constant fD(0+) analogue to fπ = 130.6 MeV. We show
the behaviour of fD(0+) versus τ, where a goood stablity is obtained. Adopting the range of tc-values

obtained previously and using m̄c(mc) = 1.11
+0.08
−0.04 GeV required for a best fit of the non strange D(0

+)
meson mass, we deduce to two-loop accuracy:

fD(0+) = (217
+5+15
−15−19 ± 10 ± 10) MeV , (18)

where the errors come respectively from the values of tc, mc, 〈ψ̄ψ〉 and Λ. We have fixed MD(0+) to be about
2272 MeV from our previous fit. It is informative to compare this result with the one of fD = (205 ± 20)
MeV, where the main difference can be attributed by the sign flip of the quark condensate contribution in



the QCD expression of the corresponding correlators. A numerical study of the SU(3) breaking effect leads
to:

rs ≡
fDs(0+)

fD(0+)
≃ 0.93 ± 0.02 , (19)

which is reverse to the analogous ratio in the pseudoscalar channel fDs/fD ≃ 1.14 ± 0.04 given semi-
analytically in [24]. In order to understand this result, we give a semi-analytic parametrization of this SU(3)
breaking ratio. Keeping the leading term in ms and 〈ψ̄ψ〉, one obtains:

rs ≃
(

1 − ms
mc

)

[

1 − 7.5〈s̄s − d̄d〉
]1/2

(

MDs(0+)

MD(0+)

)2

≃ 0.9 , (20)

where the main effect comes from the negative sign of the ms contribution in the overall normalization of
the scalar current, while the meson mass ratio does not compensate this effect because of the almost equal
mass of Ds(0

+) and D(0+) obtained in previous analysis. This feature is opposite to the case of fD(0−).

7 Summary and conclusions

Due to the experimental recent discovery of the DsJ (2317) and D
∗
sJ (2457), we have analyzed using QSSR

the dynamics of the 0± and 1± open charm and beauty meson channels:

• We have re-estimated the running charm quark mass from the D and Ds mesons. The result in Eq.
(12) confirms earlier results obtained to two- and three-loop accuracies [6, 1].

• We have studied the mass-splittings of the 0+-0− in the D systems using QSSR. Our result in the (0+)
channel given in Eq. (13) agrees with the recent experimental findings of the DsJ (2317) suggesting
that this state is a good candidate for being a c̄s 0+ meson.

• We also found, in Eq. (14), that the SU(3) breaking responsible of the mass-splitting between the
Ds(0

+) and D(0−) is small of about 25 MeV contrary to the case of the pseudoscalar Ds-D mesons of
about 100 MeV.

• We have extended our analysis to the B-system. Our results in Eqs. (13), (15) and (1) suggest an
approximate (light and heavy) flavour and spin symmetries of the meson mass-splittings. We use this
result to get the mass of the c̄s D∗s (1

+) meson in Eq. (17), which is in (surprising) good agreement
with the observed D∗sJ (2457).

• Using QSSR, we have also determined the decay constants of the 0+ mesons and compare them with
the ones of the 0− states. The result in Eq. (18), which is similar to the pseudoscalar decay constant
fD ≃ 205 MeV, suggests a huge violation of the heavy quark symmetry 1/

√
MD scaling law. Finally,

our results in Eqs. (19) and (20) indicate that the SU(3) breaking act in an opposite way compared
to the case of the 0− channels.

Experimental or/and lattice measurements of the previous predictions are useful for testing the validity of
the results obtained to two-loop accuracy in this paper from QCD spectral sum rules. However, a complete
confirmation of the nature of these new states needs a detail study of their production and decays. We plan
to come back to these points in a future work.

During the editorial preparation of this paper, there appears in the literature a paper using sum rules
method to the same channel [25]. Instead of our result in Eq. (13): MDs(0+) = (2297 ± 113) MeV, the
authors obtain MDs(0+) = (2480 ± 30) MeV which is higher than the BABAR result Ds(2317) by about 160
MeV. Considering this deviation as significant, the authors conclude that the experimental candidate cannot
be a c̄s state contrary to the conclusion reached in the present paper. However, by scrutinizing the analysis
of Ref. [25], we find that the true errors of the analysis have been underestimated:
• As one can see from their figures, the quoted error of 30 MeV only takes into account the one due to the
choice of the QCD contiuum threshold taken in a smaller range 8.1-9.3 GeV2, than the more conservative
value 6-9 GeV2 used in the present paper, which induces a larger error of 80-98 MeV.
• The high central value obtained in [25] is related to a higher choice of the (ill-defined) charm quark pole
mass of 1.46 GeV compared to the value 1.3 GeV which would have been deduced from its relation with the
running charm quark mass 1.15 GeV in Eq. (9) used in this paper to reproduce the Ds(0

−) mass. As the
analysis is very sensitive to mc, its induced error should be included in the final number. With mc in Eq.
(12), this uncertainty is about 70 MeV, and might be bigger if one considers all range of mc given in the



literature. In this paper, we have used MD andMDs as alternative channels for extracting mc.
• Taking properly the different sources of errors including the running of condensate and the effects of some
other anomalous dimensions, one then leads to a consistency within the errors of both theoretical estimates
with the present data.

References

[1] S. Narison, Phys. Lett. B 520 (2001) 115.

[2] S. Narison, Phys. Lett. B 210 (1988) 238.

[3] For a review see e.g., M. Bondioli, for the BABAR Collaboration, Nucl. Phys. (Proc. Suppl.) B 133
(2004) 133 and references therein.

[4] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385, 448.

[5] For a review and references to original works, see e.g., S. Narison, QCD as a theory of hadrons,
Cambridge Monogr. Part. Phys. Nucl. Phys. Cosmol. 17 (2002) 1-778 [hep-ph 0205006]; QCD spectral
sum rules , World Sci. Lect. Notes Phys. 26 (1989) 1-527; Acta Phys. Pol. 26 (1995) 687; Riv. Nuov.
Cim. 10N2 (1987) 1; Phys. Rep. 84 (1982).

[6] see e.g.: S. Narison, Phys. Lett. B 198 (1987) 104; Phys. Lett. B 322 (1994) 247; Nucl. Phys. (Proc.
Suppl.) B 74 (1999) 304.

[7] see e.g.: S. Narison, Phys. Lett. B 341 (1994) 73; B 216 (1989) 191.

[8] see e.g., S. Narison, [hep-ph 0202200]; Nucl. Phys. (Proc. Suppl.) B 86 (2000) 242.

[9] R.A. Bertlmann, G. Launer and E. de Rafael, Nucl. Phys. B 250 (1985) 61.

[10] F.V. Gubarev, M.I. Polikarpov and V.I. Zakharov, Nucl. Phys. (Proc. Suppl.) B 86 (2000) 457; V.I.
Zakharov, Nucl. Phys. (Proc. Suppl.) B 74 (1999) 392; K. Chetyrkin, S. Narison and V.I. Zakharov,
Nucl. Phys. B 550 (1999) 353.

[11] S. Generalis, J. Phys. G 16 (1990) 367; M. Jamin and M. Münz, Z. Phys. C 60 (1993) 569.

[12] V.A. Novikov et al, Neutrino 1978 Conference, Purdue Univ. 1978.

[13] R. Tarrach, Nucl. Phys. B 183 (1981) 384;

[14] S. Narison, Phys. Lett. B 197 (1987) 405; B 216 (1989) 191.

[15] S. Narison, Phys. Lett. B 466 (1999) 345; B 358 (1995) 113.

[16] see e.g. M. Jamin, talk given at QCD 03, 2-9th July 2003, Montpellier-FR.

[17] H.G. Dosch and S. Narison, Phys. Lett. B 417 (1998) 173.

[18] S. Narison, Phys. Lett. B 216 (1989) 191; S. Narison et al., Nuovo. Cim A 74 (1983) 347.

[19] S. Narison, Phys. Lett. B 361 (1995) 121; B 387 (1996) 162.

[20] G. Launer et al., Z. Phys. C 26 (1984) 433.

[21] PDG 2002, K. Hagiwara et al., Phys. Rev. D 66 (2002) 010001.

[22] S. Bethke, Nucl. Phys. (Proc. Suppl.) B, A 54 (1997); hep-ex/0004021.

[23] K.G. Chetyrkin and M. Steinhauser, Eur. Phys. J. C 21 (2001) 319 and references therein; K. Melnikov
and T. van Ritbergen, Phys. Lett. B 482 (2000) 99.

[24] S. Narison, Phys. Lett. B 322 (1994) 247.

[25] A. Hayashigaki and K. Terasaki, hep-ph/0411285.

http://arXiv.org/abs/hep-ph/0205006
http://arXiv.org/abs/hep-ph/0202200
http://arXiv.org/abs/hep-ex/0004021
http://arXiv.org/abs/hep-ph/0411285

	Introduction
	The QCD spectral sum rules
	Calibration of the sum rule from the D(0-) and Ds(0-) masses and re-estimate of "7016mc(mc)
	The 0+-0- meson mass-splittings
	The 1+-1- meson mass-splitting
	The D(0+) and Ds(0+) decay constants
	Summary and conclusions