cpcChinese Physics CChin. Phys. C1674-1137Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltdcpc_42_12_12310210.1088/1674-1137/42/12/12310242/12/123102PaperParticles and fieldsWeak decays of doubly heavy baryons: “decay constants”*
Supported by National Natural Science Foundation of China (11575110, 11655002, 11735010), Natural Science Foundation of Shanghai (15DZ2272100, 15ZR1423100), Shanghai Key Laboratory for Particle Physics and Cosmology and Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education
Inspired by the recent observation of the Ξcc++ by the LHCb Collaboration, we explore the “decay constants” of doubly heavy baryons in the framework of QCD sum rules. With the Ξcc,Ξbc,Ξbb, and Ωcc,Ωbc,Ωbb baryons interpolated by three-quark operators, we calculate the correlation functions using the operator product expansion and include the contributions from operators up to dimension six. On the hadron side, we consider both contributions from the lowest-lying states with JP=1/2+ and from negative parity baryons with JP=1/2−. We find that the results are stable and the contaminations from negative parity baryons are not severe. These results are ingredients for the QCD study of weak decays and other properties of doubly-heavy baryons.
doubly heavy baryonsdecay constantQCD sum rules12.39.Hg
Article funded by SCOAP^{3}
arxivppt1711.10289Introduction
For many years, it was widely believed that doubly heavy baryons with two charm and/or bottom quarks exist in reality, but experimental searches for such baryons took some time. The SELEX Collaboration first reported the discovery of Ξcc+ in the Λc+K−π+ final state sixteen years ago ^{[1,2]}, with the mass measured as mΞcc+=(3519±1)MeV^{[1,2]}. However, the SELEX-like Ξcc+ signal was not confirmed by later experiments ^{[3–7]}. In 2017, in the Λc+K−π+π+ final state, the LHCb Collaboration observed the doubly charmed baryon Ξcc++ with the mass ^{[8]}:
mΞcc++=(3621.40±0.72±0.27±0.14)MeV.
In order to decipher the internal nature of doubly heavy baryons and uncover the underlying dynamics in the transition, more experimental investigations of their production and decays are urgently needed. Meanwhile, further theoretical studies on weak decays of doubly heavy baryons will be of great importance ^{[9–30]}, and in particular, there is a need for solid QCD analyses of weak decays and production.
In this work, we present an analysis of the “decay constant” of doubly heavy baryons in the framework of QCD sum rules (QCDSR). QCDSR have been extensively applied to study hadron masses, decay constants and transition form factors, the mixing matrix elements of K-meson and B-meson systems, etc. ^{[31–40]}. In this approach, hadrons are interpolated by the corresponding quark operators. The correlation functions of these operators can be handled using the operator product expansion (OPE), where the short-distance coefficients and long-distance quark-gluon interactions are separated. The former are calculable in QCD perturbation theory, whereas the latter can be parameterized in terms of vacuum condensates. The QCD result is then matched, via a dispersion relation, onto the observable characteristics of hadronic states. Due to various advantages, the QCDSR have been used to calculate the masses of doubly heavy baryons in Refs. [9,25,41–46]. The main motive of this work is to study “decay constants” using the QCDSR. The “decay constants” defined by the interpolating current are necessary inputs for studies of other properties of doubly heavy baryons in QCDSR, for example the heavy-to-light transition form factors.
The rest of the paper is arranged as follows. In section 2, we will present the calculation of correlation function in QCD sum rules, including the explicit expressions of the spectral functions. We include both the contributions from the JP=1/2+ baryons and the contamination from the JP=1/2− baryons. Section 3 is devoted to the numerical results. A summary is presented in the last section.
QCD sum rules study
A doubly heavy baryon is made of two heavy quarks and one light quark. The quantum numbers and quark content of the ground states are given in Table 1. In this work we will study the JP=1/2+ baryons, which can only weakly decay.
Quantum numbers and quark content for the ground state of doubly heavy baryons. The s_{h} denotes the spin of the heavy quark system.
baryon
quark content
shπ
JP
baryon
quark content
shπ
JP
Ξcc
{cc}q
1^{+}
1/2^{+}
Ξbb
{bb}q
1^{+}
1/2^{+}
Ξcc*
{cc}q
1^{+}
3/2^{+}
Ξbb*
{bb}q
1^{+}
3/2^{+}
Ωcc
{cc}s
1^{+}
1/2^{+}
Ωbb
{bb}s
1^{+}
1/2^{+}
Ωcc*
{cc}s
1^{+}
3/2^{+}
Ωbb*
{bb}s
1^{+}
3/2^{+}
Ξbc′
{bc}q
0^{+}
1/2^{+}
Ωbc′
{bc}s
0^{+}
1/2^{+}
Ξbc
{bc}q
1^{+}
1/2^{+}
Ωbc
{bc}s
1^{+}
1/2^{+}
Ξbc*
{bc}q
1^{+}
3/2^{+}
Ωbc*
{bc}s
1^{+}
3/2^{+}
QCD sum rules with only positive parity baryons
The interpolating current for the ΞQQ and ΩQQ is chosen as
JΞQQ=ϵabcQaTCγμQbγμγ5qc,JΩQQ=ϵabcQaTCγμQbγμγ5sc,
where Q = c or Q = b. For the Ξbc and Ωbc, we choose
JΞbc=12ϵabcbaTCγμcb+caTCγμbbγμγ5qc,JΩbc=12ϵabcbaTCγμcb+caTCγμbbγμγ5sc.
In the above equations, we have considered the shπ=1+ baryons only.
The QCDSR analysis starts with the two-point correlator:
Π(q)=i∫d4xeiq·x〈0∣T[J(x),J¯(0)]∣0〉,
where the interpolating current has been given in the above, and J¯ is defined as
J¯=J†γ0.
A Lorentz structure analysis implies that the two-point correlation function has the form:
Π(q)=qΠ1(q2)+Π2(q2).
On the hadronic side, one can insert the complete set of hadronic states into the correlator and then the correlator can be expressed as a dispersion integral over a physical spectral function:
Π(q)=λH2q+mHmH2−q2+1π∫s0∞dsImΠ(s)s−q2,
where H can be a ground-state doubly heavy baryon and m_{H} denotes its mass. In obtaining the above expression, the polarization summation for spinors has been used:
∑su(q,s)u¯(q,s)=q+mH.
The pole residue λH is defined as
〈0∣JH∣H(q,s)〉=λHu(q,s).
The mass dimension for λH is 3, while in analogy with the meson case, it is convenient to use the “decay constant” with the definition
〈0∣JH∣H(q,s)〉=fHmH2u(q,s).
On the OPE side, we will work at leading order in αs in this work and include the condensate contributions up to dimension six. The full propagator for the heavy quark is given as
SijQ(x)=i∫d4k(2π)4e−ik·xδijk−mQ−gsGαβatija4σαβ(k+mQ)+(k+mQ)σαβ(k2−mQ2)2+gsDαGβλntijn(fλβα+fλαβ)3(k2−mQ2)4−gs2(tatb)ijGαβaGμνb(fαβμν+fαμβν+fαμνβ)4(k2−mQ2)5,
with
fλαβ=(k+mQ)γλ(k+mQ)γα(k+mQ)γβ(k+mQ),fαβμν=(k+mQ)γα(k+mQ)γβ(k+mQ)γμ(k+mQ)×γν(k+mQ),
where tn=λn/2 and λn is the Gell-Mann matrix, and the i,j are the color indices. The full propagator for light quarks is given as
Sij(x)=iδijx2π2x4−δij12〈q¯q〉−δijx2〈q¯gsσGq〉192+iδijx2x〈s¯gsσGs〉mq1152−igsGαβtija(xσαβ+σαβx)32π2x2.
With the quark propagators one can express the correlation function in terms of a dispersion relation as:
Πi(q2)=∫(mQ+mQ′)2∞dsρi(s)s−q2,i=1,2,
where the spectral density is given by the imaginary part of the correlation function:
ρi(s)=1πImΠiOPE(s).
After equating the two expressions for Π(q2) based on the quark-hadron duality, and making a Borel transformation, we can write the sum rules as
λH2e−mH2/M2=∫(mQ+mQ′)2s0dsρ1(s)e−s/M2,λH2mHe−mH2/M2=∫(mQ+mQ′)2s0dsρ2(s)e−s/M2.
The spectral functions ρ_{1} and ρ_{2} are given as follows:
ρ1pert(s)=6(2π)4∫αminαmaxdαα∫βmin1−αdββ[αβs−αmQ2−βmQ′2]2+(1−α−β)mQmQ′[αβs−αmQ2−βmQ′2],ρ1(s)=ρ1pert(s)+〈gs2G2〉72mQ2∂3(∂mQ2)3+mQ′2∂3(∂mQ′2)3ρ1pert(s)+4mQmQ′〈gs2G2〉(4π)4∂2(∂mQ2)2+∂2(∂mQ′2)2∫αminαmaxdαα∫βmin1−αdββ(1−α−β)(αβs−αmQ2−βmQ′2)+2〈gs2G2〉(4π)4∂∂mQ2+∂∂mQ′2∫αminαmaxdα∫βmin1−αdβ(3αmQ2+3βmQ′2−mQmQ′−4αβs),ρ2(s)=−〈q¯q〉2π2∫αminαmaxdα(3α(1−α)s−2αmQ2−2(1−α)mQ′2+2mQmQ′)−〈q¯gsσGq〉8π21+sM2A(s)−2〈q¯gsσGq〉(4π)2(αmax−αmin)+12s(αmax−αmin)[αmax(1−αmax)s+αmin(1−αmin)s+4mQmQ′],
with
A(s)=−s3+(mQ2+mQ′2)s2+(mQ2−4mQmQ′+mQ′2)[s(mQ2+mQ′2)−(mQ2−mQ′2)2]2s2(s+mQ2−mQ′2)2−4mQ2s.
The integration limits are given by αmin=[s−mQ2+mQ′2−(s−mQ2+mQ′2)2−4mQ′2s]/(2s), αmax=[s−mQ2+mQ′2+(s−mQ2+mQ′2)2−4mQ′2s]/(2s), and βmin=αmQ2/(sα−mQ′2). For the ΩQQ′, one needs to replace the condensates correspondingly. The integration lower bound (mQ+mQ′)2 is replaced by (mQ+mQ′+ms)2.
In Ref. [41], the authors obtained a similar expression to our Eq. (19):
ρ1(s)=−324π4∫αminαmaxdαα∫βmin1−αdββ[αβs−αmQ2−βmQ′2]2322π4mQmQ′×∫αminαmaxdαα∫βmin1−αdββ(1−α−β)[αβs−αmQ2−βmQ′2]−5mq〈q¯q〉23π2∫αminαmaxdαα(1−α).
A few remarks are in order.
We did not include the mass corrected quark condensate. This might have some impact in the case of Ωcc,bc,bb.
However, the gluon condensate contribution, which was anticipated to be more important, is missing in Eq. (25).
In the massless limit, we have the spectral function:
ρ1(s)=s264π4+2〈gs2G2〉(4π)4.Our result is fully consistent with Ref. [47]:
λH2e−mH2/M2=12(2π)4M61−e−s0/M21+s0M2+12s02M4+〈gs2G2〉4M21−e−s0/M2.
In Ref. [41], the predicted mass mΞcc=(4.26±0.19)GeV is much larger than the experimental result mΞcc++exp=3.621GeV.
QCD sum rules with both positive and negative parity baryons
In the above analysis, only the 1/2^{+} baryons were considered. An interpolating current for the negative parity 1/2^{−} baryon can be defined as
J−≡iγ5J+,
where J+ is given in Eqs. (2-5). When the complete set of hadron states is inserted into the correlation function in Eq. (6), both the positive and the negative parity single-particle states can contribute ^{[48,49]}.
When taking into account the 1/2^{−} single-particle states, Eq. (9) is rewritten as
Π(q)=λ+2q+m+m+2−q2+λ−2q−m−m−2−q2+1π∫s0∞dsImΠ(s)s−q2,
where λ± (m_{±}) stands for the “decay constant” (mass) of positive or negative parity baryons. The λ+ is the “decay constant” λH we have defined in Eq. (11). The λ− is defined as
〈0∣JH+∣H(1/2−,q,s)〉=iγ5λ−u(q,s).
At the hadronic level, one can take the imaginary part of the correlation function as follows:
1πImΠ(s)=λ+2(q+m+)δ(s−m+2)+λ−2(q−m−)δ(s−m−2)+⋯=qρ1had(s)+ρ2had(s),
with
ρ1had(s)=λ+2δ(s−m+2)+λ−2δ(s−m−2)]+⋯,ρ2had(s)=m+λ+2δ(s−m+2)−m−λ−2δ(s−m−2)]+⋯.
Here the ellipsis stand for the contributions from higher resonances and the continuum spectra. Considering the combination sρ1had+ρ2had, and introducing the exponential function exp(−s/M2) to suppress these contributions, one can separate the λ+ contributions:
∫Δs0ds[sρ1had(s)+ρ2had(s)]exp−sM2=2m+λ+2exp−m+2M2,
where s_{0} is the threshold of the continuum states and M2 is the Borel parameter.
On the OPE side, we compute the correlation function Π(q) to obtain the QCD spectral densities
1πImΠ(s)=qρ1OPE(s)+ρ2OPE(s).
Taking the quark-hadron duality below the continuum threshold s_{0}, we arrive at the following QCD sum rule:
2m+λ+2exp−m+2M2=∫Δs0ds[sρ1OPE(s)+ρ2OPE(s)]exp−sM2.
Here Δ is the threshold parameter, Δ=(mQ+mQ′)2 for ΞQQ′, and Δ=(mQ+mQ′+ms)2 for ΩQQ′.
Numerical results
In the numerical analysis, the quark masses used are ^{[50]}: mc=1.35±0.10GeV,mb=4.60±0.10GeV,ms=0.12±0.01GeV, while the u and d quarks are taken as massless. Similar values have been taken in Ref. [42].
The vacuum condensates used are ^{[31,41,51–54]}: 〈q¯q〉=−(0.24±0.01GeV)3, 〈s¯s〉=(0.8±0.1)〈q¯q〉, 〈gs2G2〉=0.88±0.25GeV4, 〈q¯gsσGq〉=m02〈q¯q〉, 〈s¯gsσGs〉=m02〈s¯s〉 and m02=0.8±0.1GeV2 at the energy scale μ = 1 GeV.
The baryon masses used in the analysis of the decay constants are given in Table 2. For the Ξcc++ mass, we adopt the experimental value ^{[8]}, and we use the isospin symmetry for the Ξcc+. For other baryons, we use the lattice QCD results from Ref. [55].
Masses (in units of GeV) of doubly heavy baryons. We adopt the experimental value for the mass of Ξcc^{[8]} and the lattice QCD results from Ref. [55].
baryons
Ξcc
Ωcc
Ξbb
Ωbb
Ξbc
Ωbc
masses
3.621 ^{[8]}
3.738 ^{[55]}
10.143 ^{[55]}
10.273^{[55]}
6.943 ^{[55]}
6.998 ^{[55]}
The continuum threshold s0 used is 0.4∼0.6 GeV higher than the corresponding baryon mass, where we have assumed that the energy gap between the ground state and the first radial excited state is approximately 0.5 GeV ^{[56]}.
Complying with the standard procedure of QCD sum rule analysis, the Borel parameter M2 is varied to find the optimal stability window, in which the perturbative contribution should be larger than the condensate contributions and the pole contribution larger than the continuum contribution.
The sum rule in Eq. (19) will be numerically analyzed since it is expected to have better convergence than the sum rule in Eq. (20).
Masses
Differentiating Eq. (19) or Eq. (35) with respect to −1/M2, one can extract the mass of the doubly heavy baryon as
mH2=∫Δs0dsρ1(s)se−s/M2∫Δs0dsρ1(s)e−s/M2
or
mH2=dd(−1/M2)∫Δs0dssρ1(s)+ρ2(s)e−s/M2∫Δs0dssρ1(s)+ρ2(s)e−s/M2.
The optimal stability window for M2 can be determined as follows. The upper bounds of the Borel parameters M2 can be determined by the requirement that the pole contribution should be larger than the continuum contribution, while the lower bound can be determined by the requirement that the perturbative contribution should be larger than the quark condensate contribution. For the sum rule in Eq. (35), Mmax2=3.3,3.5,8.7,9.5,6.1,6.3GeV2 and Mmin2=2.7,2.0,7.1,5.2,4.7,3.6GeV2 for Ξcc, Ωcc, Ξbb, Ωbb, Ξbc and Ωbc respectively. The optimal windows for M2 can be chosen as [2.7,3.3], [2.9,3.5], [7.3,8.7], [7.5,8.9], [5.1,6.1] and [5.3,6.3] respectively. For the sum rule in Eq. (19), the optimal windows for M2 can be chosen as [2.4,3.0], [2.6,3.2], [6.2,7.6], [6.4,7.8], [4.4,5.4] and [4.6,5.6] respectively.
The dependence of the predicted mass m_{H} on the Borel parameter M2 is shown in Figs. 1 and 2, where the sum rules in Eq. (19) and Eq. (35) are adopted, respectively. Using Eq. (19), we obtain
mΞcc=(3.68±0.08)GeV,
where only the positive parity baryons are taken into account. When the contamination from the 1/2− baryon is considered, we find the mass is slightly changed:
mΞcc=(3.61±0.09)GeV.
Here the uncertainties of the relevant parameters, including M2, s_{0}, the quark masses and the condensates, have been taken into account. It can be seen that our values are consistent with the experimental value when the errors are taken into account. Our results are also consistent with other estimates, for instance Ref. [42]. A collection of the results can be found in Table 3.
(color online) The M2-dependence of the masses of Ξcc, Ωcc (top two figures), Ξbb, Ωbb (middle two figures), Ξbc and Ωbc (bottom two figures). The sum rule in Eq. (19) is considered. The inputs are taken as mc=1.35GeV, mb=4.60GeV, and ms=0.12GeV, and the condensate parameters are taken at μ=1GeV.
(color online) The M2-dependence of the masses of Ξcc, Ωcc (top two figures), Ξbb, Ωbb (middle two figures), Ξbc and Ωbc (bottom two figures). The sum rule in Eq. (35) is considered. The inputs are taken as mc=1.35GeV, mb=4.60GeV, and ms=0.12GeV, and the condensate parameters are taken at μ=1GeV.
Theoretical predictions for the masses (in units of GeV) of the doubly heavy baryons. The results listed under “this work #1” are predicted using Eq. (19) while those under “this work #2” use Eq. (35). The uncertainties of the relavant parameters, including M2, s_{0}, the quark masses and the condensates, have been taken into account. For purposes of comparison, some other QCDSR results from Ref. [41] and Ref. [42] and the lattice QCD results from Ref. [55] are listed. Our results are consistent with Ref. [42] and Ref. [55] but somewhat different from Ref. [41].
baryon
this work #1
this work #2
Ref. [41]
Ref. [42]
Ref. [55]
experiment
Ξcc
3.68 ± 0.08
3.61 ± 0.09
4.26 ± 0.19
3.57 ± 0.14
3.610±0.023±0.022
3.6214 ± 0.0008
Ωcc
3.75 ± 0.08
3.69 ± 0.09
4.25 ± 0.20
3.71 ± 0.14
3.738±0.020±0.020
–
Ξbb
10.16 ± 0.09
10.12 ± 0.10
9.78 ± 0.07
10.17 ± 0.14
10.143±0.030±0.023
–
Ωbb
10.27 ± 0.09
10.19 ± 0.10
9.85 ± 0.07
10.32 ± 0.14
10.273±0.027±0.020
–
Ξbc
6.95 ± 0.09
6.89 ± 0.10
6.75 ± 0.05
—
6.943±0.033±0.028
–
Ωbc
7.01 ± 0.09
6.95 ± 0.09
7.02 ± 0.08
—
6.998±0.027±0.020
–
Decay constants
The dependence of the “decay constants” λH on the Borel parameter M2 is shown in Figs. 3 and 4, where the sum rules in Eq. (19) and Eq. (35) are adopted, respectively. The numerical results for the “decay constants” can be found in Table 4.
(color online) The M2-dependence of the decay constants of Ξcc, Ωcc (top two figures), Ξbb, Ωbb (middle two figures), Ξbc and Ωbc (bottom two figures). The continuum thresholds are taken as s0=4.0∼4.2GeV, s0=4.1∼4.3GeV, s0=10.5∼10.7GeV, s0=10.7∼10.9 GeV, s0=7.3∼7.5GeV and s0=7.4∼7.6 GeV, respectively. The sum rule in Eq. (19) is considered.
(color online) The M2-dependence of the decay constants of Ξcc, Ωcc (top two figures), Ξbb, Ωbb (middle two figures), Ξbc and Ωbc (bottom two figures). The continuum thresholds are taken as s0=4.0∼4.2GeV, s0=4.1∼4.3GeV, s0=10.5∼10.7GeV, s0=10.7∼10.9GeV, s0=7.3∼7.5GeV and s0=7.4∼7.6GeV, respectively. The sum rule in Eq. (35) is considered.
Decay constants λH (in units of GeV3) for the doubly heavy baryons. The results listed under “this work #1” are predicted using Eq. (19) while those under “this work #2” use Eq. (35). The uncertainties of the relevant parameters, including M2, s_{0}, the quark masses, the condensates and the baryon masses, have been taken into account. For purposes of comparison, the QCDSR results from Ref. [42] are listed.
baryon
this work #1
this work #2
Ref. [42]
Ξcc
0.113 ± 0.029
0.109 ± 0.021
0.115 ± 0.027
Ωcc
0.140 ± 0.033
0.123 ± 0.024
0.138 ± 0.030
Ξbb
0.303 ± 0.094
0.281 ± 0.071
0.252 ± 0.064
Ωbb
0.404 ± 0.112
0.347 ± 0.083
0.311 ± 0.077
Ξbc
0.191 ± 0.053
0.176 ± 0.040
–
Ωbc
0.217 ± 0.056
0.188 ± 0.041
–
A few remarks are in order.
When including the contributions from the 1/2− baryons, the threshold parameter might be somewhat higher. In this analysis, we have used approximately the same values.
Comparing the two sets of results in Table 4, one can see that the negative parity baryons do not give significant modifications.
We can see from Table 4 that the decay constants of ΩQQ′ are slightly larger than those of ΞQQ′.
Conclusion
In this work we have calculated the “decay constants” for the doubly heavy baryons Ξcc, Ωcc, Ξbb, Ωbb, Ξbc and Ωbc using QCD sum rules. In the calculation we have included both the positive and negative parity baryons, and found that the 1/2− contamination is not severe. The extracted results for the decay constants are ingredients for the study of weak decays and other properties of doubly heavy baryons, including their lifetimes ^{[57–59]}.
The authors are grateful to Pietro Colangelo, Fulvia De Fazio, Zhi-Gang Wang, Yu-Ming Wang, and Fu-Sheng Yu for useful discussions.