D*D coupling constant from the QCD sum rules

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1 Journal of Phsis: Conferene Series PAPER OPEN ACCESS A stu of the gη * oupling onstant from the QC sum rules To ite this artile: B Osório Rorigues et al 215 J. Phs.: Conf. Ser Relate ontent - Harons in AS/QC moels Henrique Boshi-Filho - Aitional strange resonanes in Lattie QC Miha Marzenko - Unraveling the organization of the QC tapestr J Papavassiliou View the artile online for upates an enhanements. This ontent was ownloae from IP aress on 18/6/218 at 19:24

2 A stu of the g η oupling onstant from the QC sum rules B Osório Rorigues 1,3, M E Brao 2 an C M Zanetti 2 1 Instituto e Físia, Universiae o Estao o Rio e Janeiro, Rua São Franiso Xavier 524, 255-9, Rio e Janeiro, RJ, Brazil 2 Faulae e Tenologia, Universiae o Estao o Rio e Janeiro, Ro. Presiente utra Km 298, Pólo Inustrial, , Resene, RJ, Brazil bosorio@bpf.br Abstrat. In this work, we alulate the form fator an the oupling onstant of the harme verte η using the QC sum rules (QCSR) for the ase where the meson is onsiere off-shell. This is aomplishe b using not onl the perturbative ontribution of the OPE series but also the non-perturbative effets of the gluon onensates. The form fator of this verte presents two tensor strutures, both of whih were analse. The oupling onstant is obtaine b the etrapolation of the eponentiall parametrize form fator urve to the pole of the meson. 1. Introution Our QC sum rules (QCSR) group has been working etensivel for the past ears in the harm setor, with the alulation of a vast number of oupling onstants as the π 1], ρ 2], J/ψ ( ) ( ) 3, 4] an more reentl, the J/ψ s s 5]. Continuing this line of researh, we starte the stu of harme verties that inlue the pseuo-salar meson η, in this partiular work, we fous our attention on the η verte. This verte is relate to phsial problems of urrent interest, as the η ea in two light vetor mesons (η V V ). Eperimental ata iniate that this tpe of ea, as the η ρρ an η K (892) K (892), is among the most ontributing ea hannels for the η, with branhing ratios 6] of (1.8 ±.5) 1 2 an (6.8 ± 1.3) 1 3 respetivel for the mentione hannels. However, these eas shoul be suppresse b the heliit seletion rule (HSR). This is an intriguing problem that shares some similarities with the notorious ρπ puzzle, in whih the ea J/ψ ρπ also presents a branhing ratio of the orer 1 2 even though being a suppresse ea aoring to the OZI rule. One wa to irumvent the HSR suppression is b onsiering the η V V ea with an intermeiate step through 7] the mesons an, i.e. η V V. In this piture, the oupling onstant g η will be neessar to the alulation of these ea amplitues, therefore, the results of this work an be useful to this analsis. 3 Present aress: Centro Brasileiro e Pesquisas Físias, Rua Xavier Sigau 15, , Rio e Janeiro, RJ, Brazil. Content from this work ma be use uner the terms of the Creative Commons Attribution 3. liene. An further istribution of this work must maintain attribution to the author(s) an the title of the work, journal itation an OI. Publishe uner liene b Lt 1

3 Following our previous works, we alulate the g η oupling onstant using the QCSR framework for the ase where the meson is off-shell. In the following setions, we present our evelopment, results an onlusion for this work. 2. evelopment The starting point of the QCSR alulation is the three-point orrelation funtion 8]. For the ase where the meson is onsiere off-shell, the orrelation funtion an be written as: Π ( ) µ (p, p ) = 4 4 e ip e iq T {j η 5 ()j + µ ()j 5 ()}, (1) where q = p p is the transferre momentum, T is the time orere prout, jµ M is the interpolating urrent of the meson M an is the non-perturbative QC vauum. Eah orrelation funtion an be stuie in two ifferent was: using quark egrees of freeom, alle the OPE sie or using haron egrees of freeom, alle the phenomenologial sie. Both representations are equivalent an an be equate invoking the quark-haron ualit after appling a ouble Borel transform on eah sie. Then, the form fator an the oupling onstant an be etrate. The OPE sie orrespons to a Wilson s operator prout epansion an it an be obtaine from Eq. 1 b using the interpolating urrents written in terms of the quark fiels: j η 5 = i γ 5, j + µ = γ µ an j5 = γ 5. The OPE sie an be written as a sum of the perturbative ontribution an the non-perturbative ontributions to the orrelation funtion: Π OP E( ) µ = Π pert( ) µ + Π non-pert( ) µ, (2) where the non-perturbative term of Eq. 2 inlues all the ontributions from the QC onensates ( qq, g 2 G 2, qgσgq,...). In this off-shell alulation, we will onl alulate non-perturbative ontributions from gluon onensates: Π OP E( ) µ = Π pert( ) µ + Π g2 G 2 ( ) µ. (3) This is motivate b the later use of the ouble Borel transform, whih will suppress the ontributions of quark onensates an mie quark-gluon onensates when onsiering an OPE series up to imension five operators. Calulating Eq. 3 orrespons to the alulation of the sum of the seven iagrams in Fig. 1, where the first iagram is the perturbative term an the following si iagrams are the g 2 G 2 ontributions to the orrelation funtion. The perturbative ontribution of Fig. 1a an be alulate b using ispersion relations an the Cutkosk rule, whih leas to: ] Π pert( ) µ (p, p ) = 1 F pµ (s, u, t) + F p µ (s, u, t)p µ 4π 2 su s inf u inf (s p 2 )(u p 2, (4) ) where s = p 2, u = p 2 an t = q 2 are the Manelstam variables an F pµ (s, u, t) an F p µ (s, u, t) are given b F pµ (s, u, t) = 3su(s u t) + um2 (s + u t)] (s 2 + u 2 + t 2, (5) 2tu 2st 2su) 3/2 F p µ (s, u, t) = 3su(u s t) + m2 (2tu + 2st s 2 u 2 t 2 )] (s 2 + u 2 + t 2 2tu 2st 2su) 3/2, (6) 2

4 (a) η (b) η () η () η η η η (e) (f) (g) Figure 1: OPE iagrams up to imension 5 operators for off-shell. The iagram (a) orrespons to the perturbative OPE term, while iagrams (b g) orrespon to the gluon onensate ontributions to the OPE series. where the limit m has been use. Making the variable hanges p 2 P 2, p 2 P 2, q 2 Q 2 an appling the ouble Borel transform, we have the final epression for the QCSR ontribution of the perturbative term: B M 2B M 2 Π pert( ) µ ] = 1 4π 2 su e s/m 2 e u/m 2 s inf u inf ] F pµ (s, u, t) + F p µ (s, u, t)p µ, (7) where M 2 an M 2 are new variables introue b the ouble Borel transform alle Borel masses. In this work, the Borel masses are relate to eah other b the following ansatz: M 2 = m2 η m 2 M 2, (8) with m η an m being the masses of the mesons η an respetivel. 3

5 After making the same variable hanges an appling the ouble Borel transform, the gluon onensate ontribution an be epresse as: ] BB Π g2 G 2 g 2 G 2 ( ) αtm 2 t µ = 96π 2 M 2 +M 2 αm2 M 2 (1+α 2 M 2 ) αm 2 M 2 M 2 + p p µ µ, (9) αe 1 M 2 where the full epression for the funtion is presente in appeni A. Having alulate the OPE sie, the net step is to alulate the orrelation funtion of the phenomenologial sie. This is ahieve b epaning Eq. 1 using haron egrees of freeom. Suh epansion lea to the following orrelation funtion: Π phen µ = f η f f m 2 η m m 2 ɛ µ(q, λ)γ(p, p ) 2m 2 (p 2 m 2 )(p 2 m 2 η )(q 2 m 2 + h.r., (1) ) where h.r. stans for the ontributions of higher resonanes, ɛ µ is the polarization vetor an f η, f an f are the ea onstants of the mesons η, an respetivel. The Γ(p, p ) term in Eq. 1 is relate to the Fenman s sattering amplitue, is iretl proportional to the form fator an an be obtaine from the effetive Lagrangian: L η = ig ( ) η (Q2 ) ±α ( ±η α α η ), where g ( ) η (Q2 ) is the form fator for the ase where the meson is off-shell an is relate to the oupling onstant b the limit g η = The ouble Borel transform of the phenomenologial sie is: BB Π phen] ) = g( η (Q2 )f η f f m 2 η m 2 2m 2 m (Q 2 + m 2 ) lim g ( ) Q 2 m 2 η (Q2 ). (11) M 2 e m 2 η M 2 e m 2 (m 2 + m 2 η m 2 ) + (m 2 + m2 m 2 η )p ] µ + BB h.r.], (12) where one more the variable hanges p 2 P 2, p 2 P 2, q 2 Q 2 were performe. Before obtaining the QCSR epression for g ( ) η (Q2 ), we nee to remove the BB h.r.] ontribution from Eq. 12. This an be ahieve b the introution of the ontinuum threshol parameters s = (m + s ) 2 an u = (m η + u ) 2 in Eq. 7 as the upper limits of the ouble integration. Finall, appling the QCSR, we are able to obtain the off-shell form fator of the verte η : ] Π g2 G 2 an 1 s u 4π g ( ) 2 s inf u inf su e s/m 2 e u/m 2 F pµ (s, u, t) + BB η (Q2 ) pµ = f η f f m 2 η m2 2m 2 m (Q 2 +m 2 ) e m2 M 2 e m2 η M 2 (m 2 + m 2 η m 2 ), (13) ] 1 s u 4π g ( ) 2 s inf u inf su e s/m 2 e u/m 2 F p µ (s, u, t) + BB Π g2 G 2 p µ η (Q2 ) p µ = f η f f m 2 η m2 2m 2 m (Q 2 +m 2 ) e m2 M 2 e m2 η M 2 (m 2 + m 2 m2 η ) where F pµ an F p µ are given b eqs. 5 an 6 respetivel an BB, (14) Π g2 G 2 ] is given b Eq. 9. 4

6 3. Results In orer to obtain numerial results from eqs. 13 an 14, it is neessar to know the numerial values of all the quantities presente in these equations. The values of the masses (harons an quarks) an ea onstants are presente in table 1. The value of the gluon onensate is g 2 G 2 =.88 GeV 4 9]. Table 1: Numerial values neessar to the g ( ) η (Q2 ) QCSR alulation. Partile Quantit η m (MeV) 6] f (MeV) ] 24211] ] The Borel mass an ontinuum threshol parameters are quantities introue b the QCSR an o not have a speifi value, however, eah of these quantities is onfine to a partiular range. In the ase of the Borel mass, it is allowe to have an value that is ontaine in the so alle Borel winow. The Borel winow is a region limite b two onstrains regaring the orrelation funtion: the pole ontribution must be bigger than the ontinuum ontribution an the perturbative ontribution must be bigger than the sum of the non-perturbative ontributions. Sine our latest work 5], we o not use an arbitrar value for the Borel mass. Instea, we use the average of the form fator on the Borel mass within their allowe values of the Borel winow. The values of the ontinuum threshol parameters s an u are etermine b using the stabilit riteria, where the preferre ombination of s an u is the one that presents the most stable form fator regaring the Borel winow. The possible values of s an u are onfine in the ranges m 2 < (m + s ) 2 < m 2 an m 2 η < (m η + u ) 2 < m 2 η respetivel, where m an m η are the masses of the first raial resonanes of the mesons. From eqs. 13 an 14, it is evient that the g ( ) η (Q2 ) form fator an be alulate from two ifferent sum rules, one for eah tensor struture available ( an p µ), but in this work, just the struture presente a vali an stable Borel winow. In this ase, the most stable Borel winow is M 2 = ] GeV 2, obtaine with the ontinuum threshols s =.6 GeV an u =.4 GeV in the Q 2 = 1. 3.] GeV 2 winow. Using these parameters, we obtain the g ( ) η (Q2 ) QCSR form fator, represente b the squares in Fig. 2. This form fator is well fitte b an eponential urve (full line in Fig. 2): g ( ) η (Q2 ) = 1.79e.282Q2. (15) Using Eq. 11, we etrapolate Eq. 15 to the pole of the meson an obtain the g η oupling onstant (blak irle in Fig. 2): g η = 5.6. (16) 4. Conlusion In this work, we use the QCSR to alulate the form fator an the oupling onstant of the η verte for the off-shell ase. The OPE series use in this QCSR onsiere operators 5

7 7 6 Eponential fit QCSR form fator Coupling onstant 5 g ( ) η (Q2 ) Q 2 (GeV 2 ) Figure 2: The QCSR form fator for the off-shell ase of the η verte. up to imension 5 with non-perturbative ontribution elusivel from the gluon onensates while quark onensates an mie quark-gluon onensates were suppresse b the use of a ouble Borel transform. The g ( ) η (Q2 ) form fator an be alulate b two ifferent QCSR, given b eqs. 13 an 14, one for eah tensor struture presente in the orrelation funtion. However, just the struture oul be use in this work as the p µ i not presente a vali Borel winow. The result for this QCSR is an eponential form fator for the off-shell ase (Eq. 15) whih leas to the oupling onstant g η = 5.6 (Eq. 16). The net step in the stu of the η verte is to alulate the two other possible form fators: g () η (Q2 ) an g (η) η (Q2 ). From Eq. 11, we an epet the oupling onstant obtaine from eah of these two form fators to be lose to the value obtaine for the off-shell ase. This fat will be use in the analsis of the unertainties of our QCSR alulation. Aknowlegments This work has been partiall supporte b CNPq an CAPES. Appeni A. Full epressions for an p are funtions of α µ s use in Eq. 9 for the ontribution of the gluon onensates to the orrelation funtion. These funtions an be efine as:,p µ = m,p µ (),p µ () + m,p µ (e),p µ (f),p µ (g) (A.1) where (), with = g, is relate to eah gluon onensate iagram in figs.1 1g. The term m,p µ (b) that orrespon to the iagram of Fig.1b is omitte from Eq. A.1 as we are 6

8 using the limit m. We will also omit full epressions for the p funtion as the µ struture p µ was not use in this QCSR. In the net lines, we will show the full epressions for the () funtions. () = (m M 2 (α 5 M 8 M 6 Q 2 2α 4 M 6 M 6 Q 2 +α 3 M 4 M 6 Q 2 +2α 4 M 8 M 4 Q 2 4α 3 M 6 M 4 Q 2 +2α 2 M 4 M 4 Q 2 +α 3 M 8 M 2 Q 2 2α 2 M 6 M 2 Q 2 +αm 4 M 2 Q 2 α 4 M 6 M 8 +3α 3 M 4 M 8 α 3 m 2 M 2 M 8 3α 2 M 2 M 8 +αm 8 2α 3 m 2 M 4 M 6 +2α 2 M 4 M 6 2α 2 m 2 M 2 M 6 4αM 2 M 6 +2M 6 +α 4 M 1 M 4 3α 3 M 8 M 4 α 3 m 2 M 6 M 4 +7α 2 M 6 M 4 4α 2 m 2 M 4 M 4 9αM 4 M 4 αm 2 M 2 M 4 +4M 2 M 4 + 2α 2 M 8 M 2 2α 2 m 2 M 6 M 2 4αM 6 M 2 2αm 2 M 4 M 2 + 2M 4 M 2 αm 2 M 6 )) /((αm 2 1) 2 (M 2 + M 2 ) 4 ). (A.2) () = (M 2 (α 4 M 8 M 6 Q 2 3α 3 M 6 M 6 Q 2 +3α 2 M 4 M 6 Q 2 αm 2 M 6 Q 2 α 4 M 1 M 4 Q 2 + 2α 3 M 8 M 4 Q 2 α 2 M 6 M 4 Q 2 α 3 M 1 M 2 Q 2 + 2α 2 M 8 M 2 Q 2 αm 6 M 2 Q 2 2α 3 M 6 M 8 3α 3 m 2 M 4 M 8 + 4α 2 M 4 M 8 + 2α 2 m 2 M 2 M 8 3αM 2 M 8 + M 8 + α 3 M 8 M 6 5α 3 m 2 M 6 M 6 α 2 M 6 M 6 + α 2 m 2 M 4 M 6 3αM 4 M 6 + αm 2 M 2 M 6 + 3M 2 M 6 + 4α 3 M 1 M 4 α 3 m 2 M 8 M 4 6α 2 M 8 M 4 3α 2 m 2 M 6 M 4 αm 6 M 4 +αm 2 M 4 M 4 +3M 4 M 4 +α 3 M 1 2M 2 +α 3 m 2 M 1 M 2 α 2 M 1 M 2 α 2 m 2 M 8 M 2 αm 8 M 2 αm 2 M 6 M 2 + M 6 M 2 + α 2 m 2 M 1 αm 2 M 8 )) /(α(αm 2 1)M 2 (M 2 + M 2 ) 4 ) (A.3) (e) = (αm M 4 (α 3 M 4 M 4 Q 2 2α 2 M 2 M 4 Q 2 + αm 4 Q 2 + α 2 M 4 M 2 Q 2 2αM 2 M 2 Q 2 +M 2 Q 2 +2α 2 M 2 M 6 α 2 m 2 M 6 2αM 6 +3α 2 M 4 M 4 2α 2 m 2 M 2 M 4 2αM 2 M 4 αm 2 M 4 M 4 + α 2 M 6 M 2 α 2 m 2 M 4 M 2 2αm 2 M 2 M 2 M 2 M 2 αm 2 M 4 )) /((αm 2 1) 4 M 4 (M 2 + M 2 )) (A.4) (f) = (M 2 (4α 4 M 6 M 6 Q 2 7α 3 M 4 M 6 Q 2 + 2α 2 M 2 M 6 Q 2 + αm 6 Q 2 + 6α 3 M 6 M 4 Q 2 12α 2 M 4 M 4 Q 2 + 6αM 2 M 4 Q 2 + α 3 M 8 M 2 Q 2 2α 2 M 6 M 2 Q 2 + αm 4 M 2 Q 2 4α 3 M 4 M 8 12α 3 m 2 M 2 M 8 +8α 2 M 2 M 8 +6α 2 m 2 M 8 4αM 8 24α 3 m 2 M 4 M 6 2α 2 M 4 M 6 2α 2 m 2 M 2 M 6 + 4αM 2 M 6 + 6αm 2 M 6 2M 6 + 4α 3 M 8 M 4 12α 3 m 2 M 6 M 4 8α 2 M 6 M 4 22α 2 m 2 M 4 M 4 +4αM 4 M 4 +1αm 2 M 2 M 4 +2α 2 M 8 M 2 14α 2 m 2 M 6 M 2 4αM 6 M 2 +2αm 2 M 4 M 2 +2M 4 M 2 2αm 2 M 6 ))/(2α(αM 2 1) 2 M 2 (M 2 + M 2 ) 3 ) (A.5) 7

9 (g) = (M 2 (2α 4 M 8 M 4 Q 2 5α 3 M 6 M 4 Q 2 +4α 2 M 4 M 4 Q 2 αm 2 M 4 Q 2 +α 3 M 8 M 2 Q 2 2α 2 M 6 M 2 Q 2 + αm 4 M 2 Q 2 4α 3 M 6 M 6 3α 3 m 2 M 4 M 6 + 9α 2 M 4 M 6 + 2α 2 m 2 M 2 M 6 6αM 2 M 6 + M 6 4α 3 M 8 M 4 6α 3 m 2 M 6 M 4 + 8α 2 M 6 M 4 + 2α 2 m 2 M 4 M 4 4αM 4 M 4 +αm 2 M 2 M 4 3α 3 m 2 M 8 M 2 α 2 M 8 M 2 2α 2 m 2 M 6 M 2 +2αM 6 M 2 +2αm 2 M 4 M 2 M 4 M 2 2α 2 m 2 M 8 + αm 2 M 6 ))/(α(αm 2 1) 2 M 2 (M 2 + M 2 ) 3 ) Referenes 1] Navarra F S, Nielsen M, Brao M E, Chiapparini M an Shat C L 2 Phs. Lett. B ] Rorigues B O, Brao M E, Nielsen M an Navarra F S 211 Nul. Phs. A ] Brao M E, Chiapparini M, Navarra F S an Nielsen M 25 Phs. Lett. B ] Matheus R, Navarra F S, Nielsen M an Rorigues a Silva R 25 Int. J. Mo. Phs. A ] Osório Rorigues B., Brao M E an Chiapparini M 214 Nul. Phs. A ] Olive K A et al 214 Chin. Phs. C ] Wang Q, Liu X an Zhao Q 212 Phs. Lett. B ] Brao M E, Chiapparini M, Navarra F S an Nielsen M 212 Prog. Part. Nul. Phs ] Narison S 212 Phs. Lett. B ] avies C T H, MNeile C, Follana E, Lepage G P, Na H an Shigemitsu J 21 Phs. Rev ] Gelhausen P, Khojamirian A, Pivovarov A A an Rosenthal 213 Phs. Rev (A.6) 8

arxiv: v1 [hep-ph] 7 Jul 2017

arxiv: v1 [hep-ph] 7 Jul 2017 A QCD sum rules alulation of the η D D and η D sd s form fators and strong oupling onstants B. Osório Rodrigues a, M. E. Brao b, and C. M. Zanetti b a Instituto de Apliação Fernando Rodrigues da Silveira,