QCD sum rules study of $QQ-\bar{u}\bar{d}$ mesons

Abstract

We use QCD sum rules to study the possible existence of $QQ-\bar{u}\bar{d}$ mesons, assumed to be a state with $J^{P}=1^{+}$. For definiteness, we work with a current with an axial heavy diquark and a scalar light antidiquark, at leading order in $\alpha_s$. We consider the contributions of condensates up to dimension eight. For the $b$-quark, we predict $M_{T_{bb}}= (10.2\pm 0.3) {\rm GeV}$, which is below the $\bar{B}\bar{B}^*$ threshold. For the $c$-quark, we predict $M_{T_{cc}}= (4.0\pm 0.2) {\rm GeV}$, in agreement with quark model predictions.

Full text

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arXiv:hep-ph/0703071v2 10 Apr 2007
QCD sum rules study of QQ − ¯ u¯d mesons
Fernando S. Navarra∗and Marina Nielsen†
Instituto de F´ ısica, Universidade de S˜ ao Paulo,
C.P. 66318, 05389-970 S˜ ao Paulo, SP, Brazil
Su Houng Lee‡
Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea
We use QCD sum rules to study the possible existence of QQ − ¯ u¯d mesons, assumed to be a
state with JP= 1+. For definiteness, we work with a current with an axial heavy diquark and a
scalar light antidiquark, at leading order in αs. We consider the contributions of condensates up
to dimension eight. For the b-quark, we predict MTbb= (10.2 ± 0.3) GeV, which is below the¯B¯B∗
threshold. For the c-quark, we predict MTcc= (4.0 ± 0.2) GeV, in agreement with quark model
predictions.
PACS numbers:11.55.Hx, 12.38.Lg , 12.39.-x
The general idea of possible stable heavy tetraquarks has been first suggested by Jaffe [1]. The case
of a tetraquark QQ¯ u¯d with quantum numbers I = 0, J = 1 and P = +1 which, following ref.[2], we call
TQQ, is especially interesting. As already noted previously [2, 3], the Tbb and Tccstates cannot decay
strongly or electromagnetically into two¯B or two D mesons in the S wave due to angular momentum
conservation nor in P wave due to parity conservation. If their masses are below the¯B¯
thresholds, these decays are also forbidden. Moreover, in the large mQlimit, the light degrees of freedom
cannot resolve the closely bound QQ system. This results in bound states similar to the¯ΛQstates, with
QQ playing the role of the heavy antiquark [4]. Therefore, the stability of¯ΛQimplies that QQ¯ u¯d is also
safe from decaying through QQ¯ u¯d → QQq + ¯ q¯ u¯d . As a result, TQQ is stable with respect to strong
interactions and must decay weakly.
There are some predictions for the masses of the TQQstates. In ref. [5] the authors use a color-magnetic
interaction, with flavor symmetry breaking corrections, to study heavy tetraquarks. They assume that
the Belle resonance, X(3872), is a cq¯ c¯ q tetraquark, and use its mass as input to determine the mass of
other tetraquark states. They get MTcc= 3966 MeV and MTbb= 10372 MeV. In ref. [2], the authors
use one-gluon exchange potentials and two different spatial configurations to study the mesons Tccand
Tbb. They get MTcc= 3876 − 3905 MeV and MTbb= 10519 − 10651 MeV. There are also calculations
using expansion in the harmonic oscillator basis [6], and variational method [7].
In this work we use QCD sum rules (QCDSR) [8, 9, 10], to study the two-point functions of the state
TQQ. There are several reasons, why it is interesting to investigate this channel. First of all, having
two heavy quarks, it is an explicit exotic state. The experimental observation would already prove the
existence of the tetraquark state without any theoretical extrapolation. Moreover, from a technical point
of view, this means that there are no contributions from the disconnected diagrams, which are technically
very difficulty to estimate in QCD sum rules or in lattice gauge theory calculation.
In previous calculations, the QCDSR approach was used to study the light scalar mesons [11, 12, 13,
14, 15] the D+
sJ(2317) meson [16, 17] and the X(3872) meson [18], considered as four-quark states and a
good agreement with the experimental masses was obtained. However, the tests were not decisive as the
usual quark–antiquark assignments also provide predictions consistent with data [10, 12, 19, 20].
Considering TQQas an axial diquark-antidiquark state, a possible current describing such state is given
by:
B∗and DD∗
jµ= i[QT
aCγµQb][¯ uaγ5C¯dT
b] ,(1)
where a, b are color indices, C is the charge conjugation matrix and Q denotes the heavy quark.
In general, one should consider all possible combinations of different 1+four-quark operators, as was
done in [21] for the 0++light mesons. However, the current in Eq.(1) well represents the most attractive
∗Electronic address: navarra@if.usp.br
†Electronic address: mnielsen@if.usp.br
‡Electronic address: suhoung@phya.yonsei.ac.kr

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configuration expected with two heavy quarks. This is so because the most attractive light antidiquark
is expected to be the in the color triplet, flavor anti-symmetric and spin 0 channel [22, 23, 24]. This
is also expected quite naturally from the color magnetic interaction, which can be phenomenologically
parameterized as,
Vij= −
C
mimjλi· λjσi· σj. (2)
Here, m,λ,σ are the mass, color and spin of the constituent quark i,j. Eq.(2) favors the anti-diquark to
be in the color triplet and spin 0 channel. The flavor anti-symmetric condition then follows from requiring
anti-symmetric wave function of the anti-diquark. Similarly, since the anti-diquark is in the color triplet
state the remaining QQ should be in the color anti-triplet spin 1 state. Although the spin 1 configuration
is repulsive, its strength is much smaller than that for the light diquark due to the heavy charm quark
mass. Therefore a constituent quark picture for TQQwould be a light anti-diquark in color triplet, flavor
anti-symmetric and spin 0 (ǫabc[¯ ubγ5C¯dT
The simplest choice for the current to have a non zero overlap with such a TQQconfiguration is given
in Eq. (1). While a similar configuration Tss is also possible [25], we believe that the repulsion in the
strange diquark with spin 1 will be larger and hence energetically less favorable. As discussed above,
since the quantum number is 1+, the decay into DD or¯B¯B would be forbidden and the allowed decay
into DD∗or¯B¯B∗would have a smaller phase space, and the tetraquark state might have a small width,
or may even be bound.
The QCDSR is constructed from the two-point correlation function
c]) combined with a heavy diquark of spin 1 (ǫaef[QT
eCγµQf]).
Πµν(q) = i
?
d4x eiq.x?0|T[jµ(x)j†
ν(0)]|0? = −Π1(q2)(gµν−qµqν
q2) + Π0(q2)qµqν
q2. (3)
Since the axial vector current is not conserved, the two functions, Π1and Π0, appearing in Eq. (3) are
independent and have respectively the quantum numbers of the spin 1 and 0 mesons.
The calculation of the phenomenological side proceeds by inserting intermediate states for the meson
TQQ. Parametrizing the coupling of the axial vector meson 1+, to the current, jµ, in Eq. (1) in terms of
the meson decay constant fT and the meson mass MT as:
?0|jµ|TQQ? =
√2fTM4
Tǫµ, (4)
the phenomenological side of Eq. (3) can be written as
Πphen
µν
(q2) =
2f2
M2
TM8
T− q2
T
?
−gµν+qµqν
M2
T
?
+ ··· , (5)
where the Lorentz structure gµν gets contributions only from the 1+state. The dots denote higher
axial-vector resonance contributions that will be parametrized, as usual, through the introduction of a
continuum threshold parameter s0[26].
On the OPE side, we work at leading order in αsand consider the contributions of condensates up to
dimension eight. To keep the charm quark mass finite, we use the momentum-space expression for the
charm quark propagator. We follow ref. [27] and calculate the light quark part of the correlation function
in the coordinate-space, which is then Fourier transformed to the momentum space in D dimensions. The
resulting light-quark part is combined with the charm quark part before it is dimensionally regularized
at D = 4.
The correlation function, Π1, in the OPE side can be written as a dispersion relation:
ΠOPE
1
(q2) =
?∞
4m2
Q
dsρ(s)
s − q2,(6)
where the spectral density is given by the imaginary part of the correlation function:
Im[ΠOPE
1
(s)]. After making a Borel transform of both sides, and transferring the continuum contri-
bution to the OPE side, the sum rule for the axial vector meson TQQup to dimension-eight condensates
can be written as:
πρ(s) =
2f2
TM8
Te−M2
T/M2=
?s0
4m2
Q
ds e−s/M2ρ(s) + Πmix?¯ qq?
1
(M2) , (7)

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where
ρ(s) = ρpert(s) + ρ?¯ qq?(s) + ρ?G2?(s) + ρmix(s) + ρ?¯ qq?2(s) + ρmix?¯ qq?(s) ,(8)
with
ρpert(s) =
1
29π6
αmax
?
αmin
dα
α3
1−α
?
βmin
dβ
β3(1 − α − β)?(α + β)m2
Q− αβs?3
×
ρ?¯ qq?(s) = 0,
?1 + α + β
4
?(α + β)m2
Q− αβs?− m2
Q(1 − α − β)
?
,
ρ?G2?(s) = −?g2G2?
210π6


−1
4
αmax
?
αmin
dα
α(1 − α)(m2
Q− α(1 − α)s)2
+
αmax
?
αmin
m2
3α2(1 − α − β)
1
48αβ2(1 − α − β)?(α + β)m2
ρmix(s) = 0,
ρ?¯ qq?2(s) =?¯ qq?2
24π2s
dα
α
1−α
?
βmin
dβ
?
(α + β)m2
Q− αβs
4β
?(α + β)m2
Q− αβs + 2m2
Q
?
+
Q
?
m2
Q(1 − α − β) +?(α + β)m2
Q− αβs?2(5 − α − β)
Q− αβs??
−4 − α − β +3
β(1 − α)
??
+
??
,
?
1 − 4m2
Q/s. (9)
where the integration limits are given by αmin = (1 −
and βmin= αm2
is assumed to be suppressed by the loop factor 1/16π2. We have included, for completeness, a part of the
dimension-8 condensate contributions. We should note that a complete evaluation of these contributions
require more involved analysis including a non-trivial choice of the factorization assumption basis [28]
?
1 − 4m2
Q/s)/2, αmax = (1 +
?
1 − 4m2
Q/s)/2
Q/(sα − m2
Q). The contribution of dimension-six condensates ?g3G3? is neglected, since it
ρmix?¯ qq?(s) = −?¯ qgσ.Gq??¯ qq?
(M2) = −m2
26π2
Q?¯ qgσ.Gq??¯ qq?
253π2
?
1 − 4m2
Q/s,
Πmix?¯ qq?
1
?1
0
dα
?
4 −
m2
Q
α(1 − α)M2
?
exp
?
−
m2
Q
α(1 − α)M2
?
.(10)
In order to extract the mass MT without worrying about the value of the decay constant fT, we take
the derivative of Eq. (7) with respect to 1/M2, divide the result by Eq. (7) and obtain:
M2
T=
?s0
?s0
4m2
Qds e−s/M2s ρ(s)
4m2
Qds e−s/M2ρ(s)
.(11)
This quantity has the advantage to be less sensitive to the perturbative radiative corrections than the
individual moments. Therefore, we expect that our results obtained to leading order in αswill be quite
accurate.
In the numerical analysis of the sum rules, the values used for the quark masses and condensates are (see
e.g. [10, 29]): mc(mc) = (1.23± 0.05) GeV, mb(mb) = (4.24 ± 0.06) GeV, ?¯ qq? = −(0.23± 0.03)3GeV3,
?¯ qgσ.Gq? = m2
We start with the double charmed meson Tcc. We evaluate the sum rules in the range 2.0 ≤ M2≤
4GeV2for s0in the range: 4.6 ≤√s0≤ 5.0 GeV.
Comparing the relative contribution of each term in Eqs. (9) to (10), to the right hand side of Eq. (7)
we obtain a quite good OPE convergence (the perturbative contribution is at least 50% of the total)
0?¯ qq? with m2
0= 0.8 GeV2, ?g2G2? = 0.88 GeV4.

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2 2.2 2.4 2.62.83 3.23.4 3.63.84
M
2(GeV
2)
0
0.2
0.4
0.6
0.8
1
1.2
OPE convergence
FIG. 1: The relative OPE convergence in the region 2.0 ≤ M2≤ 4.0 GeV2for√s0 = 4.8 GeV. We start with the
perturbative contribution divided by the total (long-dashed line) and each subsequent line represents the addition
of one extra condensate dimension in the expansion: +?g2G2? (dot-dashed line), +?¯ qq?2(dotted-line), +m2
(solid line).
0?¯ qq?2
for M2> 2.5 GeV2, as can be seen in Fig. 1. This analysis allows us to determine the lower limit
constraint for M2in the sum rules window. This figure also shows that, although there is a change of
sign between dimension-six and dimension-eight condensates contributions, the contribution of the latter
is very small, where, we have assumed, in Fig. 1 to Fig. 4, the validity of the vacuum saturation for
these condensates. The relatively small contribution of the dimension-eight condensates may justify the
validity of our approximation, unlike in the case of the 5-quark current correlator, as noticed in [30].
2 2.22.42.6 2.833.23.4 3.63.84
M
2(GeV
2)
0
0.2
0.4
0.6
0.8
1
pole X continuum (%)
FIG. 2: The solid line shows the relative pole contribution (the pole contribution divided by the total, pole plus
continuum, contribution) and the dashed line shows the relative continuum contribution for√s0 = 4.8 GeV.
We get an upper limit constraint for M2by imposing the rigorous constraint that the QCD continuum
contribution should be smaller than the pole contribution. The maximum value of M2for which this
constraint is satisfied depends on the value of s0. The comparison between pole and continuum contri-
butions for√s0= 4.8 GeV is shown in Fig. 2. The same analysis for the other values of the continuum
threshold gives M2≤ 3.1 GeV2for√s0= 4.6 GeV and M2≤ 3.6 GeV2for√s0= 5.0 GeV.
In Fig. 3, we show the Tccmeson mass obtained from Eq. (11), in the relevant sum rules window, with

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2 2.22.42.62.83 3.2 3.43.63.84
M
2(GeV
2)
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
MTcc (GeV)
|
|
|
|
FIG. 3: The Tcc meson mass as a function of the sum rule parameter (M2) for different values of the continuum
threshold:√s0 = 4.6 GeV (dotted line) and√s0 = 5.0 GeV (solid line). The bars indicate the region allowed
for the sum rules: the lower limit (cut below 2.5 GeV2) is given by OPE convergence requirement and the upper
limit by the dominance of the QCD pole contribution.
the upper and lower validity limits indicated. From Fig. 3 we see that the results are reasonably stable
as a function of M2. In our numerical analysis, we shall then consider the range of M2values from 2.5
GeV2until the one allowed by the sum rule window criteria as can be deduced from Fig. 3 for each value
of s0.
We found that our results are not very sensitive to the value of the charm quark mass, neither to
the value of the condensates. The most important source of uncertainty is the value of the continuum
threshod and the Borel interval. Using the QCD parameters given above, the QCDSR predictions for the
Tccmesons mass is:
MTcc= (4.0 ± 0.2) GeV, (12)
in a very good agreement with the predictions in refs. [2] and [5].
One can also evaluate the decay constant, defined in Eq. (4), to leading order in αs:
fTcc= (5.95 ± 0.65) × 10−5GeV , (13)
which can be more affected by radiative corrections than MTcc.
In the case of the double-beauty meson Tbb, using consistently the perturbative MS-mass mb(mb) =
(4.24±0.6) GeV, and the continuum threshold in the range 11.3 ≤√s0≤ 11.7 GeV, we find a good OPE
convergence for M2> 7.5 GeV2. We also find that the pole contribution is bigger than the continuum
contribution for M2< 9.6 GeV2for√s0< 11.3 GeV, and for M2< 11.2 GeV2for√s0< 11.7 GeV.
In Fig. 4 we show the Tbbmeson mass obtained from Eq. (11), in the relevant sum rules window, with
the upper and lower validity limits indicated. From Fig. 4 we see that the results are very stable as a
function of M2in the allowed region. Taking into account the variation of M2and varying s0and mbin
the regions indicated above, we arrive at the prediction:
MTbb= (10.2 ± 0.3) GeV ,(14)
also in a very good agreement with the results in refs. [2], [5] and [7]. For completeness, we predict the
corresponding value of the decay constant to leading order in αs:
fTbb= (10.4 ± 2.8) × 10−6GeV .(15)
We have presented a QCDSR analysis of the two-point functions of the double heavy-quark axial meson,
TQQ, considered as a four quark state. We find that the sum rules results for the masses of Tccand Tbb