D∗D and B∗B (1++) molecules at N2LO from QSSR∗

F. Fanomezanaa, S. Narisonb, A. Rabemananjaraa,∗∗

aInstitute of High Energy Physics of Madagascar (iHEP-MAD), University of Antananarivo, MadagascarbLaboratoire Univers et Particules (LUPM), CNRS-IN2P3 & Universite de Montpellier II,

Case 070, Place Eugene Bataillon, 34095 - Montpellier Cedex 05, France.

Abstract

We use QCD spectral sum rules (QSSR) and the factorization properties of molecule currents to estimate the massesand couplings of the D∗D and B∗B (1++) molecules at N2LO of PT QCD. We include in the OPE the contributionsof non-perturbative condensates up to dimension-eight. With the Laplace sum rules approach (LSR) and in the MS -scheme, we obtain MD∗D = 3738(152) MeV, which agrees within the errors with the newly discovered Zc(3900). Forthe bottom channel, we find MB∗B = 10687(232) MeV in good agreement with the observed Zb(10610). Couplings ofthese states to the currents are also extracted. Our results are improvements of the LO ones in the existing literature.

Keywords: QCD Spectral Sum Rules, molecule states, heavy quarkonia.

1. IntroductionThe recent discovery of the Zc(3900) 1++ by Belle [1]and BESIII [2] from its J/ψπ± decays has motivated dif-ferent theoretical analysis [3]. However, all of the pre-vious analysis like e.g. in [4] from QCD Spectral SumRules (QSSR) [5, 6] have been done at LO of PT QCD.In this paper, we are going to use QSSR to evaluate themass and coupling of the 1++ D∗D and B∗B moleculesat N2LO in the PT series and compare the results withthose obtained at lowest order and with experiments.

2. QCD analysis of spin one molecule

• Current and two-point fonctionThe current for this molecule state is given by:

Jµ ≡ (Qγµq)(qγ5Q) , (1)Q ≡ c, b and q ≡ u, d .

The associated two-point correlation function is:

Πµνmol(q) = i

∫d4x eiq.x〈0|T Jµ(x)Jν†(0)|0〉

= −(q2gµν − qµqν)Πmol(q2)+qµqνΠ(0)

mol(q2) , (2)

∗Talk given at QCD 14 (29 june - 3 july, Montpellier - France).∗∗Speaker.

Email addresses: fanfenos@yahoo.fr (F. Fanomezana),snarison@yahoo.fr (S. Narison), achris 01@yahoo.fr. (A.Rabemananjara)

where Πmol and Π(0)mol are respectively associated to the

spin 1 and 0 molecule states. In the QSSR method andparametrizing the spectral function by one resonanceplus a QCD continuum, the lowest resonance mass MH

and coupling fH normalized as:

〈0|Jµ|H〉 = fH M4Hε

µ , (3)

can be extracted by using the Laplace sum rules (LSR)which gives two well-known sum rules [6]:

M2H =

∫ tc4m2

Qdt t e−tτ 1

πImΠOPE

mol (t)∫ tc4m2

Qdt e−tτ 1

πImΠOPE

mol (t)(4)

and

f 2H =

∫ tc4m2

Qdt e−tτ 1

πImΠOPE

mol (t)

e−τM2H M8

H

(5)

where mQ is the heavy quark mass, τ the sum rule pa-rameter and tc the continuum threshold.

• The QCD two-point function at N2LOTo derive the results at N2LO, we assume factoriza-tion and then use the fact that the two-point functionof molecule state can be written as a convolution of thespectral functions associated to quark bilinear currents.

Preprint submitted to Nuc. Phys. (Proc. Suppl.) February 6, 2020

arX

iv:1

409.

8591

v1 [

hep-

ph]

30

Sep

2014

In the spin one case, we have [7, 8]:

1π

ImΠ(1)mol(t) = θ(t − 4M2

Q)(

14π

)2

t2∫ (

√t−MQ)2

M2Q

dt1 ×∫ (√

t−√

t1)2

M2Q

dt2 λ3/2 1π

ImΠ(1)(t1)1π

ImΠ(0)(t2) (6)

with the phase space factor:

λ =

(1 −

(√

t −√

t1)2

t

) (1 −

(√

t1 +√

t2)2

t

). (7)

ImΠ(1)(t) and ImΠ(0)(t) are respectively the spectralfunction associated to the vector and to the pseudoscalarbilinear currents. The QCD expression of the spectralfunctions for bilinear currents are already known up toorder α2

s and including non-perturbative condensates upto dimension 6. It can be found in [9–12] for the on-shell mass MQ. We shall use the relation between theon-shell MQ and the running mass mQ(ν) to transformthe spectral function into the MS -scheme [13, 14]:

MQ = mQ(ν)[1 +

43

as + (16.2163 − 1.0414nl)a2s

+ ln(ν

MQ

)2 (as + (8.8472 − 0.3611nl)a2

s

)+ ln2

(ν

MQ

)2

(1.7917 − 0.0833nl) a2s

], (8)

where nl = n f − 1 is the number of light flavours andas(ν) = αs(ν)/π at the scale ν.

• QCD parametersThe PT QCD parameters which appear in this analysisare αs, the charm and bottom quark masses mc,b (thelight quark masses have been neglected). We also con-sider non-perturbative condensates from [15] up to di-mension 8 which are the quark condensate 〈qq〉, thetwo-gluon condensate 〈g2G2〉, the mixed condensate〈gqGq〉, the four-quark condensate ρ〈qq〉2, the three-gluon condensate 〈g3G3〉, and the two-quark multiplytwo-gluon condensate ρ〈qq〉〈g2G2〉 where ρ indicatesthe deviation from the four-quark vacuum saturation.Their values are given in Table 1. For the condensates,we shall use these expressions:

〈qq〉(ν) = −µ3q

(Log

ν

Λ

)−2/β1

(9)

〈gqGq〉(ν) = −M20 µ

3q

(Log

ν

Λ

)−1/3β1

(10)

where β1 = −(1/2)(11− 2n f /3) is the first coefficient ofthe β function, µq the renormalization group invariantcondensate and Λ is the QCD scale.

Parameters Values.αs(Mτ) 0.325(8)Λ(n f = 4) (324 ± 15) MeVΛ(n f = 5) (194 ± 10) MeVmc(mc) (1261 ± 24) MeVmb(mb) (4177 ± 22) MeVµq (263 ± 7) MeVM2

0 (0.8 ± 0.2) GeV2

〈αsG2〉 (7 ± 2) × 10−2 GeV4

〈g3G3〉 (8.2 ± 2.0) GeV2 × 〈αsG2〉

ρ = 〈qqqq〉/〈qq〉2 (2 ± 1)

Table 1: QCD input parameters (see e.g. [6, 16] and references therein).

3. Mass of the D∗D(1++) molecule

• τ and tc stabilitiesWe study the behavior of the mass in term of LSR vari-able τ at different values of tc as shown in Fig.1. Weconsider as final and conservative result the one corre-sponding to the beginning of the τ stability for tc=18GeV2 until the one where tc stability is reached for tc '25 GeV2.

0.1 0.2 0.3 0.4 0.5 0.6 0.73.0

3.5

4.0

4.5

5.0

Τ @GeV-2D

MD

*D

@GeV

D

-- 30

-- 27

-- 25

-- 18

tc = 15 GeV2

Ν=4.5, N2LO + dim8

Figure 1: τ-behavior of MD∗D at N2LO for different values of tc and for ν=4.5GeV

• Convergence of the PT seriesAccording to these analysis, we can notice that the τ-stability begins at tc=18 GeV2 and the tc-stability isreached from tc = 25 GeV2. Using these two extremalvalues of tc, we study in Fig. 2 the convergence of thePT series for a given value of ν = 4.5 GeV. We ob-serve that from LO to NLO the mass increases by about+3.5% while from NLO to N2LO, it only increases by+0.5%. This result indicates a good convergence of PTseries which validates the LO result obtained in the lit-erature when the running quark mass is used [4].

• ν-stabilityWe improve our previous results by using different val-ues of ν (Fig. 3). Using the fact that the final result mustbe independent of the arbitrary parameter ν, we con-sider as an optimal result the one at the inflexion point

2

0.2 0.3 0.4 0.5 0.6 0.73.0

3.5

4.0

4.5

5.0

Τ @GeV-2D

MD

*D

@GeV

D

- N2LO

- NLO

25 LO

- N2LO

- NLO

18 LO

tc@GeV2D, Ν=4.5 GeV

Figure 2: τ-behavior of MD∗D for different values of tc=18 and 25 GeV2 andν=4.5 GeV and for different truncation of the PT series

for ν ' (4.0 − 4.5) GeV:

MD∗D = 3738(150)(23) MeV , (11)

where the second error comes from the localisation ofthe inflexion point. This result agrees within the errorswith the observed Zc(3900) candidate.

ò

ò

ò

ò

ò

ò

3 4 5 63000

3200

3400

3600

3800

4000

4200

4400

Ν @GeVD

MD

*D

@MeV

D

Figure 3: ν-behaviour of MD∗D at N2LO

4. Coupling of the D∗D(1++) molecule

We can do the same analysis to derive the decay con-stant fH defined in Eq. (3). Noting that the bilinearpseudoscalar heavy-light current acquires an anomalousdimension, then the decay constant runs as:

fH(ν) = fH

(Log

ν

Λ

)2/−β1

, (12)

where fH is a scale invariant coupling. Taking theLaplace transform of the correlator, this definition willlead us to the expression of the running coupling in Eq.(5). We show in Fig. 4 the τ-behaviour of the runningcoupling fD∗D(ν) for two extremal values of tc where τand tc stabilities are reached. These values are the sameas in the mass determination. One can see in this fig-ure that the αs corrections to the LO term of PT seriesare still small though bigger than in the case of the massdetermination from the ratio of sum rules. It is about+5.13% from LO to NLO and +4.45% from NLO toN2LO. In the Fig. 5, we show the ν behaviour of theinvariant coupling fD∗D. Taking the optimal result at theminimum for ν ' 4 GeV, we obtain in units of MeV:

fD∗D = (7.43 ± 1.40) × 10−2 MeV =⇒

fD∗D(ν) = (11.57 ± 2.17) × 10−2 MeV , (13)

0.1 0.2 0.3 0.4 0.5 0.6

10

12

14

16

18

20

Τ @GeV-2D

f D*

Dx1

05 @GeV

D

- N2LO

- NLO

25 LO

- N2LO

- NLO

18 LO

tc@GeV2D, Ν=4.5 GeV

Figure 4: τ-behavior of the running coupling fD∗D for ν = 4.5 GeV and fortwo extremal values of tc = 18 and 25 GeV2.

òò

òò

òò

3 4 5 656789

10

Ν @GeVD

f`

D*

Dx1

05 @GeV

DFigure 5: ν-behavior of fD∗D at N2LO

which is comparable with the LO result [17]: fXc =

(6.5 ± 1.1) × 10−2 MeV appropriately normalized of theX(3872).

5. Mass and coupling of the B∗B(1++) molecule

We do the same analysis in the case of bottom channel.Fig. 6 shows the τ-behavior of mass for ν = mb(mb)and Fig. 7 shows its variation versus ν. We have chosentwo values of tc which correspond to the beginning ofthe τ-stability (tc = 120 GeV2) and to the beginning oftc stability (tc=150 GeV2). We observe a good conver-gence of PT series (increase of about 0.46% from LOto NLO and of about 0.35% from NLO to N2LO. Con-

0.08 0.10 0.12 0.14 0.16 0.18 0.2010.0

10.5

11.0

11.5

12.0

12.5

Τ @GeV-2D

MB

*B

@GeV

D

- N2LO

- NLO

150 LO

- N2LO

- NLO

120 LO

tc@GeV2D, Ν=mb GeV

Figure 6: τ-behavior of MB∗B with different values of tc for ν = mb(mb) andfor different truncation of the PT series

sidering the one at the minimum in ν = mb(mb) as theoptimal result, we can deduce:

MB∗B = 10687(232)MeV , (14)

3

ò

ò

ò ò

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.510 000

10 500

11 000

11 500

12 000

Ν @GeVD

MB

*B

@MeV

D

Figure 7: ν-behavior of MB∗B mass at N2LO

where one can notice a good agreement with the ob-served Zb(10610) experimental candidate. We show inthe Fig. 8 and Fig. 9 the τ and ν-behavior of the cou-pling for B∗B. Like in the case of the charm channel,we will also have the same tc as in the determinationof the mass. Radiative corrections are more important

0.05 0.10 0.15 0.200

1

2

3

4

Τ @GeV-2D

f B*

Bx1

05 @GeV

D

- N2LO

- NLO

150 LO

- N2LO

- NLO

120 LO

tc@GeV2D, Ν=mb GeV

Figure 8: τ-behavior of the running fB∗B coupling for ν = mb(mb), tc = 120and 160 GeV2 and for different truncations of the PT series.

òò

òò

2 3 4 50.0

0.5

1.0

1.5

2.0

Ν @GeVD

f`

B*

Bx1

05 @GeV

D

Figure 9: ν-behavior of the invariant coupling fB∗B at N2LO.

here than in the case of ratio of moments as expectedwhile the series is slowly convergent. From LO to NLOone has an increase of 10.1% and from NLO to N2LOan increase of about 9.4%. The optimal result for thecoupling is obtained at the minimum for ν = mb(mb):

fB∗B = (0.69 ± 0.29) × 10−2 MeV =⇒

fB∗B(ν) = (1.22 ± 0.51) × 10−2 MeV , (15)

again comparable with the LO result [17]: fXb '

10−2 MeV of the Xb predicted at 10144(107) MeV.

6. Conclusions

We have presented improved predictions of QSSR forthe masses and couplings of the D∗D and B∗B molecule

states at N2LO of PT series and including up to di-mension 8 non-perturbative condensates. Our resultsgiven in Eqs. (11) and (14) for the masses are in goodagreement within the errors with the experimental can-didates Zc(3900) and Zb(10610) suggesting that thesenew states may have large molecule components in theirwave functions. However, if one extrapolate the resultof Ref. [8] for B − B mixing, where the breaking ofthe four-quark factorization is small (about 10% whichshould be explictily checked), one cannot exclude thefour-quark assignement for these states. The couplingsof these states to the corresponding interpolating cur-rents are given in Eqs. (13) and (15) and are comparablewith the ones of the Xc(3872) and Xb(10144) predictedin [17]. The extension of our analysis to some othermolecule states is in progress.

Acknowledgments

F.F. and A.R. would like to thank the CNRS for sup-porting the travel and living expenses and the LUPM-Montpellier for hospitality. We also thank R.M. Albu-querque for many helpful discussions.

References[1] Z. Q. Liu et al. [Belle Collaboration], Phys. Rev. Lett. 110

(2013) 252002.[2] M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 110,

(2013) 252001.[3] For reviews, see e.g.: F. S. Navarra, M. Nielsen, S. H. Lee,

Phys. Rep. 497 (2010) 4; S. L. Zhu, Int. J. Mod. Phys. E17(2008); E.Swanson, Phys. Rep. 429 (2006) 243; N. Brambillaet al., Eur. Phys. J C71 (2011) 1534.

[4] R. Albuquerque et al., Phys. Lett. B715 (2012) 129.[5] M.A. Shifman, A.I. and Vainshtein and V.I. Zakharov, Nucl.

Phys. B147 (1979) 385 .[6] For a review and references to original works, see e.g., S. Nar-

ison, QCD as a theory of hadrons, Cambridge Monogr. Part.Phys. Nucl. Phys. Cosmol. 17 (2002) 1-778 [hep-h/0205006].

[7] A. Pich and E. de Rafael, Phys. Lett. B158 (1985) 477.[8] S. Narison and A. Pivovarov, Phys. Lett. B327 (1994) 341.[9] S. Narison, Phys. Lett. B718 (2013) 1321.

[10] S. Narison, arXiv:1404.6642 [hep-ph].[11] S.C. Generalis, Ph.D. thesis, Open Univ. report, OUT-4102-13

(1982), unpublished.[12] K.G. Chetyrkin and M. Steinhauser, Phys. Lett. B 502 (2001)

104; hep-ph/0108017.[13] S. Narison, Phys. Lett. B197 (1987) 405; S. Narison, Phys.

Lett. B 216 (1989) 191.[14] K.G. Chetyrkin and M. Steinhauser, Nucl. Phys. B 573 (2000)

617; K. Melnikov and T. van Ritbergen, hep-ph/9912391.[15] R. Albuquerque, X. Liu, M. Nielsen, arXiv:1203.6569 [hep-

ph]; R. Albuquerque (private notes and communication).[16] S. Narison, arXiv:1409.8148 (talk given at this conference).[17] R.D. Matheus et al., Phys. Rev. D75 (2007) 014005.

4