Radiative Q Qγ and Q Q γ transitions in light cone QCD

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1 Eur. Phys. J. C :14 DOI 1.114/epj/s Regular Artile - Theoretial Physis Radiative Q Qγ and Q Q γ transitions in light one QCD T. M. Aliev 1,2,a, K. Azizi 3,, H. Sundu 4, 1 Institute of Physis, Baku, Azeraijan 2 Department of Physis, Middle East Tehnial University, 6531 Ankara, Turkey 3 Department of Physis, Doğuş University, Aıadem-Kadıköy, Istanul, Turkey 4 Department of Physis, Koaeli University, 4138 Izmit, Turkey Reeived: 29 Septemer 214 / Aepted: 8 Deemer 214 / Pulished online: 14 January 215 The Authors 214. This artile is pulished with open aess at Springerlink.om Astrat We alulate the magneti dipole and eletri quadrupole moments assoiated with the radiative Q Q γ and Q Qγ transitions with Q = or in the framework of light one QCD sum rules. It is found that the orresponding quadrupole moments are negligily small, while the magneti dipole moments are onsideraly large. A omparison of the results of the onsidered multipole moments as well as orresponding deay widths with the preditions of the vetor dominane model is performed. 1 Introdution In the reent years, there has een signifiant experimental progress on hadron spetrosopy. Many new aryons ontaining heavy ottom and harm quarks as well as many new harmonium like states are oserved. Now, all heavy aryons with single heavy quark have een disovered in the experiments exept the aryon with spin 3/2. In the ase of douly heavy aryons only the douly harmed aryon has een disovered y SELEX Collaoration [1,2] utthe experimental attempts on the identifiation of other memers of the douly aryons as well as triply heavy aryons predited y quark model are ontinued. Considering this progress and the failities of experiments speially at LHC, it would e possile to study the deay properties of heavy aryons in the near future. Theoretial studies on eletromagneti, weak, and strong deays of heavy aryons reeive speial attention in the light of the experimental results. In the present work we alulate the eletromagneti form fators of the radiative Q Qγ and Q Q γ trana taliev@metu.edu.tr kazizi@dogus.edu.tr; kazem.azizi@ern.h hayriye.sundu@koaeli.edu.tr sitions in the framework of the light one QCD sum rules as one of the est appliale non-perturative tools to study hadron physis. Here, aryons with orrespond to spin 3/2, while those without are spin-1/2 aryons. Using the eletromagneti form fators at the stati limit q 2 =, we otain the magneti dipole and eletri quadrupole moments as well as the deay widths of the onsidered radiative deays. We ompare our results with the preditions of the vetor meson dominane model VDM [3], whih uses the values of the strong oupling onstants etween spin-3/2 and spin- 1/2 heavy aryons with vetor mesons [4] to alulate the magneti dipole and eletri quadrupole moments of the transitions under onsideration. The eletromagneti multipole moments of heavy aryons an give valuale information on their internal struture as well as their geometri shapes. Note that other possile radiative transitions among heavy spin-3/2 and spin-1/2 aryons with a single heavy quark, namely Q Qγ, Q Qγ, and Q Qγ,have een investigated in [5] using the same framework. Some of these radiative transitions have also een previously studied using hiral perturation theory [6], heavy quark and hiral symmetries [7,8], the relativisti quark model [9], and light one QCD sum rules at leading order in HQET in [1]. The outline of the paper is as follows. In the next setion, QCD sum rules for the eletromagneti form fators of the transitions under onsideration are alulated. In the last setion, we numerially analyze the otained sum rules. This setion also inludes a omparison of our results with the preditions of VDM on the multipole moments as well as the orresponding deay widths. 2 Theoretial framework The aim of this setion is to otain light one QCD sum rules LCQSR for the eletromagneti form fators defining the

2 14 Page 2 of 1 Eur. Phys. J. C :14 radiative Q Qγ and Q Qγ transitions. For this goal we use the following two-point orrelation funtion in the presene of an external photon field: μ p, q = i d 4 xe ip x T {ηx η μ } γ, 1 where η and η μ are the interpolating urrents of the heavy flavored aryons with spin 1/2 and 3/2, respetively. The main task in the following is to alulate this orrelation funtion one in terms of hadroni parameters alled the hadroni side and in terms of photon distriution amplitudes DAs with inreasing twist with the help of operator produt expansion OPE. By equating the oeffiients of appropriate strutures from the hadroni to the OPE side, we otain LCQSR for the transition form fators. To suppress the ontriution of the higher states and ontinuum, we apply Borel transformations with respet to the momentum squared of the initial and final aryoni states. For further pushing down those ontriutions, we also apply a ontinuum sutration to oth sides of the LCQSRs otained. 2.1 Hadroni side To otain the hadroni representation, we insert omplete sets of intermediate states having the same quantum numers as the interpolating urrents into the aove orrelation funtion. As a result of this we get μ p, q = η 2p, s p 2 m 2 2p, s 1p + q, s γ 2 1p +q, s η μ p + q 2 m , 2 1 where the dots indiate the ontriutions of the higher states and ontinuum and q is the photon s momentum. In the aove equation, 1p + q, s and 2p, s denote the heavy spin- 3/2 and spin-1/2 states and m 1 and m 2 are their masses, respetively. To proeed, we need to know the matrix elements of the interpolating urrents etween the vauum and the aryoni states. They are defined in terms of spinors and residues as 1p + q, s η μ =λ 1 ū μ p + q, s, η 2p, s =λ 2 up, s, where u μ p, s is the Rarita Shwinger spinor; and λ 1 and λ 2 are the residues of the heavy aryons with spin 3/2 and 1/2, respetively whih are alulated in [5]. The matrix element 2p, s 1p + q, s γ is also defined as [11,12] 3 2p, s 1p + q, s γ =eūp, s {G 1 q μ ε ε μ q+g 2 [Pεq μ Pqε μ ]γ 5 +G 3 [qεq μ q 2 ε μ ]γ 5 }u μ p + q, s, 4 where the G i are eletromagneti form fators, ε μ is the photon s polarization vetor and P = p+p+q 2. In the aove equation, the term proportional to G 3 is zero for the real photon whih we onsider in the present study. At q 2 =, the transition magneti dipole moment G M and the eletri quadrupole moment G E are defined in terms of the remaining eletromagneti form fators as [ G M = 3m 1 + m 2 G 1 G E = m 1 m 2 + m 1 m 2 G 2 m 1 [ ] G1 m2 + G 2 m 1 3. ] m2 3, Now, we use Eqs. 4 and 3 ineq.2 and perform a summation over the spins of the Dira and Rarita Shwinger spinors. In the ase of spin 3/2 this summation is written as s u μ p, sū ν p, s = p + m 2m 2p μ p ν 3m 2 { g μν γ μγ ν 5 } p μγ ν p ν γ μ 3m. 6 Using Eqs. 3 6, in priniple, one an straightforwardly alulate the hadroni side of the orrelation funtion. But here appear two unwanted prolems: There is pollution from spin-1/2 aryons, sine the interpolating urrent η μ ouples with spin-1/2 aryons also. All Lorentz strutures are not independent. In order to solve the first prolem, let us write the orresponding matrix element of the urrent η μ etween vauum and J = 1/2 states, whih an e parameterized as η μ 1p + q, s =[αγ μ + βp + q μ ]up + q, s. 7 Multiplying oth sides of this equation y γ μ and using γ μ η μ = as well as the Dira equation we get ] η μ 1p + q, s =α [γ μ 4m2 p + q μ up + q, s. From this expression it follows that ontriutions of spin-1/2 states are either proportional to the γ μ at the end or p +q μ. Taking into aount this fat, from Eq. 6 it follows that only terms proportional to g μν ontain ontriutions oming 8

3 Eur. Phys. J. C :14 Page 3 of 1 14 only from spin-3/2 states. This oservation shows how spin- 1/2 states ontriutions oupled to η μ an e removed. The seond prolem an e solved if one orders the Dira matries in an appropriate way. In this work, we hoose the ordering ε q pγ μ. After some alulations, for the hadroni side of the orrelation funtion we get 1 1 μ = eλ 1 λ 2 p 2 m 2 2 p + q 2 m 2 1 [ [εμ pq εpq μ ]{ 2G 1 m 1 G 2 m 1 m 2 + G 2 p + q 2 +[2G 1 G 2 m 1 m 2 ] p + m 2 G 2 q G 2 q p}γ 5 +[q μ ε ε μ q]{g 1 p 2 + m 1 m 2 G 1 m 1 + m 2 p}γ 5 + 2G 1 [ εpq qεp]q μ γ 5 G 1 ε qm 2 + pq μ γ 5 + other strutures with γ μ at the end or whih are proportional to p + q μ ], 9 where we need two invariant strutures to alulate the form fators G 1 and G 2. In the present work, we selet the strutures ε pγ 5 q μ and q pγ 5 εpq μ for G 1 and G 2, respetively. The advantage of these strutures is that these terms do not reeive ontriutions from ontat terms. 2.2 OPE side On the OPE side, the aforementioned orrelation funtion is alulated in terms of the QCD degrees of freedom and photon DAs. To this aim, we sustitute the expliit forms of the interpolating urrents of the heavy aryons into the orrelation funtion in Eq. 1 and use Wik s theorem to otain the orrelation in terms of the quark propagators. The interpolating urrents for spin-3/2 aryons are taken as { η μ = Aɛ a q1 a Cγ μq2 Q + q2 a Cγ μq q1 } + Q a Cγ μ q1 q2, 1 where q 1 and q 2 stand for light quarks; a,, and are olor indies and C is the harge onjugation operator. The normalization fator A and light quark ontent of the heavy spin-3/2 aryons are presented in Tale 1. Tale 1 The normalization fator A and light quark ontent of heavy spin-3/2 aryons Heavy spin-3/2 aryons A q 1 q 2 + 1/ 3 s s 2/3 s u 2/3 s d Tale 2 The onstant B and light quark ontent of the heavy spin-1/2 aryons under onsideration Heavy spin-3/2 aryons B q 1 q 2 1/ 2 s s + 1 s u 1 s d The general form of the interpolating urrents for the heavy spin-1/2 aryons under onsideration an e written as see for instane [13] η = B { ɛ a q1 at CQ γ 5 q2 + β q1 at Cγ 5Q q2 2 [ Q at Cq2 γ 5 q 1 + β Q at Cγ 5 q 2 q 1 ]}, 11 where β is an aritrary parameter and β = 1orresponds to the Ioffe urrent. The onstant B and quark fields q 1 and q 2 for the orresponding heavy spin-1/2 aryons are given in Tale 2. The orrelation funtion on the OPE side reeives three different ontriutions: 1 perturative ontriutions, 2 mixed ontriutions at whih the photon is radiated from short distanes and at least one of the quarks forms a ondensate, and 3 non-perturative ontriutions where a photon is radiated at long distanes. The last ontriution is parameterized y the matrix element γq 1 Ɣ 2, whih is expanded in terms of photon DAs with definite twists. Here Ɣ is the full set of Dira matries Ɣ j = {1, γ 5,γ α, iγ 5 γ α,σ αβ / 2}. The perturative ontriution at whih the photon interats with the quarks perturatively is otained y replaing the orresponding free quark propagator y { } a Sαβ a d 4 ys free x y AS free y, 12 where the free light and heavy quark propagators are given as Sq free = i x 2π 2 x 4 m q 4π 2 x 2, S free Q K 1 m Q x 2 4π 2 x 2 = m2 Q αβ i m2 Q x 4π 2 x 2 K 2 m Q x 2, 13 with K i eing the Bessel funtions. The non-perturative ontriutions are otained y replaing one of the light quark propagators that emits a photon y S a αβ 1 4 qa Ɣ j q Ɣ j αβ, 14 where a sum over j is applied, and the remaining ontriutions y full quark propagators involving the perturative as

4 14 Page 4 of 1 Eur. Phys. J. C :14 well as the non-perturative parts. The full heavy and light quark propagators whih we use in the present work are see [14,15] S Q x = SQ free x ig s 1 [ k + m Q dv 1 + m 2 Q vx μg μν γ ν k2 d 4 k 2π 4 e ikx m 2 Q k2 2 Gμν vxσ μν ], 1 i m q 4 x S q x = Sq free x m q 4π 2 x 2 qq 12 x2 192 m2 qq 1 i m q 6 x 1 [ x ig s du 16π 2 x 2 G μνuxσ μν ux μ G μν uxγ ν i 4π 2 x 2 m q x i 32π 2 G μνσ μν 2 2 ln 4 + 2γ E ], 15 where is the sale parameter; we hoose it at the fatorization sale =.5 1 GeV [16,17]. In order to alulate the non-perturative ontriutions, we need the matrix elements γq qɣ i q. These matrix elements are determined in terms of the photon DAs as [18] γq σ μν q = ie q qqε μ q ν ε ν q μ 1 due iū χϕ γ u + x2 16 Au i 2 e q qq [x ν ε μ q μ ] 1 x μ ε ν q ν due iū h γ u, γq γ μ q 1 = e q f 3γ ε μ q μ due iū ψ v u, γq γ μ γ 5 q = e q f 3γ ɛ μναβ ε ν q α x β due iū ψ a u, γq g s G μν vxq = ie q qq ε μ q ν ε ν q μ Dα i e iα q +vα g Sα i, γq g s G μν iγ 5 vxq = ie q qq ε μ q ν ε ν q μ Dα i e iα q +vα g Sα i, γq g s G μν vxγ α γ 5 q = e q f 3γ q α ε μ q ν ε ν q μ Dα i e iα q +vα g Aα i, γq g s G μν vxiγ α q = e q f 3γ q α ε μ q ν ε ν q μ Dα i e iα q +vα g Vα i, γq σ αβ g s G μν vxq {[ = e q qq ε μ q μ ε μ q μ ε ν q ν ε ν q ν q.x g αν 1 g βν 1 g αμ 1 q αx ν +q ν x α q β x ν + q ν x β q α q αx μ + q μ x α q β ] q α g βμ 1 q β x μ + q μ x β Dα i e iα q +vα g T 1 α i [ ε α q α g μβ 1 ε α q α ε β q β ε β q β g νβ 1 g μα 1 q μx β + q β x μ q ν q ν x β + q β x ν q μ q μx α + q α x μ q ν ] q μ g να 1 q ν x α + q α x ν Dα i e iα q +vα g T 2 α i + 1 q μx ν q ν x μ ε α q β ε β q α Dα i e iα q +vα g T 3 α i + 1 } q αx β q β x α ε μ q ν ε ν q μ Dα i e iα q +vα g T 4 α i, q β 16 where ϕ γ u is the leading twist 2, ψ v u, ψ a u, A, and V are the twist 3; and h γ u, A, and T i i = 1, 2, 3, 4 are the twist 4 photon DAs [18]. Here χ is the magneti suseptiility of the quarks. The measure Dα i is defined as Dα i = dα q dα q dα g δ1 α q α q α g. 17 In order to otain the sum rules for the form fators G 1 and G 2, we equate the oeffiients of the strutures ε pγ 5 q μ and q pγ 5 εpq μ from oth hadroni and OPE representations of the same orrelation funtion. We apply the Borel transformations with respet to the variales p 2 and p +q 2 as well as ontinuum sutration to suppress the ontriutions of the higher states and ontinuum. Finally, we otain the following shematially written sum rules for the eletromagneti form fators G 1 and G 2 : 1 G 1 = λ 1 λ 2 m 1 + m 2 e +e Q 1 ] m 2 1 m 2 M e M 2 2 [e q1 1 + e q2 1 q 1 q 2

5 Eur. Phys. J. C :14 Page 5 of 1 14 G 2 = 1 e λ 1 λ 2 m 2 1 m 2 M e M 2 2 [e q1 2 + e q2 2 q 1 q 2 + e Q 2 ], where the funtions i [ i ] an e written as i [ i ]= s m 2 Q 18 e s m2 M 2 ρ i s[ρ i s]ds + e Q M 2 Ɣ i [Ɣ i ], 19 where s is the ontinuum threshold and we take M1 2 = M2 2 = 2M2 sine the masses of the initial and final aryons are lose to eah other. The expressions for the spetral densities ρ i s[ρ i s] and the funtions Ɣ i[ɣ i ] are very lengthy; hene, we do not present these expliit expressions here. 3 Numerial results In this part, we numerially analyze the sum rules for the magneti dipole G M and eletri quadrupole G E otained in the previous setion. To this aim, we use the input parameters ūu 1 GeV = dd 1 GeV = GeV 3, ss 1 GeV =.8 ūu 1 GeV, m 2 1GeV =.8 ±.2 GeV 2 [19], and f 3γ =.39 GeV 2 [18]. The values of the magneti suseptiility are alulated in [2 22]. Here we use the value χ1gev = 4.4 GeV 2 [22] for this quantity. The LCQSR for the magneti dipole and eletri quadrupole moments also inlude the photon DAs [18], whose expressions are given as 3 2 ϕ γ u = 6uū1 + ϕ 2 μc2 u ū, ψ v u = 332u wV γ 5w γ A 3 32u u 1 4, ψ a u = 1 2u u wv γ 3 16 w γ A, Aα i = 36α q α q αg wγ A 1 2 7α g 3, Vα i = 54wγ V α q α q α q α q αg 2, h γ u = κ C2 u ū, Au = 4u 2 ū 2 3κ κ ζ 2 + 3ζ 2[uū2 + 13uū +2u u + 6u 2 lnu +2ū ū + 6ū 2 lnū], T 1 α i = 123ζ 2 + ζ 2 + α q α q α q α q α g, T 2 α i = 3αg 2 α q α q κ κ + + ζ 1 ζ α g +ζ 2 3 4α g, T 3 α i = 123ζ 2 ζ 2 + α q α q α q α q α g, T 4 α i = 3αg 2 α q α q κ + κ + + ζ 1 + ζ α g +ζ 2 3 4α g, Sα i = 3αg 2 {κ +κ+ 1 α g + ζ 1 +ζ α g1 2α g +ζ 2 [3α q α q 2 α g 1 α g ]}, Sα i = 3αg 2 {κ κ+ 1 α g +ζ 1 ζ α g1 2α g +ζ 2 [3α q α q 2 α g 1 α g ]}, 2 where the onstants inside the DAs are given y ϕ 2 1GeV =, w V γ = 3.8 ± 1.8, w A γ = 2.1 ± 1., κ =.2, κ+ =, ζ 1 =.4, ζ 2 =.3, ζ + 1 =, and ζ + 2 = [18]. The sum rules for the eletromagneti form fators ontain three more auxiliary parameters: the Borel mass parameter M 2, the ontinuum threshold s, and the aritrary parameter β entering the expressions of the interpolating urrents of the heavy spin-1/2 aryons. Any physial quantities, like the magneti dipole and eletri quadrupole moments, should e independent of these auxiliary parameters. Therefore, we try to find working regions for these auxiliary parameters suh that in these regions G M and G E are pratially independent of these parameters. The upper and lower ands for M 2 are found requiring that not only the ontriutions of the higher states and ontinuum are less than the ground state ontriution, ut also the ontriutions of the higher twists are less ompared to the leading twists. By these requirements, the working regions of Borel mass parameter are otained as 15 GeV 2 M 2 3 GeV 2 and 6 GeV 2 M 2 12 GeV 2 for aryons ontaining and quarks, respetively. The ontinuum threshold s is the energy square whih haraterizes the eginning of the ontinuum. If we denote the ground state mass y m, the quantity s m is the energy needed to exite the partile to its first exited state with the same quantum numers. The s m is not well known for the aryons under onsideration, ut it should lie etween.3 GeV and.8 GeV. The dependene of the magneti dipole moment G M and eletri quadrupole moment G E on the Borel mass parameter at different fixed values of the ontinuum threshold and general parameter β are depited in Figs. 1, 2, 3, 4, 5, and 6 for the radiative transitions under onsideration. Note that, in all figures, we plot the asolute values of the physial quantities under study sine it is not possile to predit the signs of the residues from the mass sum rules. From these figures, we see that the results weakly depend on the M 2 and s in their working regions. To determine the working regions for the general parameter β at different radiative hannels, we depit the dependene of the results on this parameter at different fixed values of the Borel mass parameter and ontinuum threshold in Figs. 7, 8, 9, 1, 11, and 12. Note that instead of β we use os θ, where β = tan θ. The interval 1 os θ 1 orresponds to β etween to +, whih we shall onsider in our alulations. The numerial results show that the values of G E are negligily small and therefore we onsider only the dependene of G M on β, in order to find its working region.

6 14 Page 6 of 1 Eur. Phys. J. C :14 Fig. 1 Left: The dependene of the magneti dipole moment G M for γ transition on the Borel mass parameter M2. Right: The dependene of the eletri quadrupole moment G E for γ transition on the Borel mass parameter M2 Fig. 2 ThesameasFig.1, utfor γ Fig. 3 ThesameasFig.1, utfor γ From Figs. 7, 8, 9, 1, 11, and 12, we otain the region.25 os θ.5 ommon for all radiative transitions under onsideration, at whih the dependene of the G M on os θ is relatively weak. In most of the figures related to the magneti dipole moment, the Ioffe urrent whih orresponds to os θ.71 remains out of the reliale region. Considering the working regions for the auxiliary parameters, the photon DAs, and other input parameters, we extrat the values of the magneti dipole moment G M and the eletri quadrupole moment G E orresponding to the onsidered radiative transitions as presented in Tale 3. For omparison, we also present the preditions of VDM [3]onG M and G E in this tale. From this tale we see that, onsidering the errors

7 Eur. Phys. J. C :14 Page 7 of 1 14 Fig. 4 ThesameasFig.1, utfor + + γ Fig. 5 ThesameasFig.1, utfor γ Fig. 6 ThesameasFig.1, utfor γ in our results, our preditions are omparale with those of the VDM on the magneti dipole moment G M for all transitions exept that γ, for whih our result is onsideraly small ompared to that of VDM. In oth models, the values of G E are negligily small for all onsidered hannels. At the end of this setion we would like to present the deay width for the radiative transitions under onsideration. Considering the transition matrix element in Eq. 4 and definitions of the magneti dipole and eletri quadrupole moments in terms of the form fators G 1 and G 2, we get the following formula for the widths of the orresponding

8 14 Page 8 of 1 Eur. Phys. J. C :14 Fig. 7 Left: The dependene of the magneti dipole moment G M for moment f G E for γ on os θ γ on os θ. Right: The dependene of the eletri quadrupole Fig. 8 ThesameasFig.7, utfor γ Fig. 9 ThesameasFig.7, utfor γ transitions: Ɣ = 3 α m 2 1 m m 3 G 2 1 m2 M + 3G2 E Using the numerial values for the magneti dipole and eletri quadrupole moments as well as the QCD sum rules preditions for the aryon masses, viz. = 6.17±.15 GeV, = 2.79 ±.19 GeV, = 6.2 ±.17 GeV, = 2.65 ±.2 GeV, = 6.11 ±.16 GeV, = 2.7 ±.2 GeV, = 5.96 ±.17 GeV, and = 2.56 ±.22 GeV [23,24], we get the values for the widths as presented in Tale 4. For omparison, we also depit the existing preditions from the VDM in the same tale. Looking at this tale we see that our results are overall omparale in orders of magnitudes with the results of [3] exept for the γ hannel, at whih our

9 Eur. Phys. J. C :14 Page 9 of 1 14 Fig. 1 ThesameasFig.7, utfor + + γ Fig. 11 ThesameasFig.7, utfor γ Fig. 12 ThesameasFig.7, utfor γ result is roughly one order of magnitude smaller ompared to that of [3]. When we ompare our results with those of [25,26], we see onsiderale differenes in the orders of magnitudes etween the two models preditions exept for the γ and + + γ hannels where our preditions are in the same orders of magnitude as those of [25,26]. The ig differenes etween our results, [3], and [25,26] may e attriuted to the different aryon masses that are used sine the width in Eq. 19 is very sensitive to the masses of the initial and final aryons. In summary, we have alulated the transition magneti dipole moment G M and eletri quadrupole moment G E as well as the deay width for the radiative Q Qγ and Q Qγ transitions within the light one QCD sum rule approah and ompared the results with the preditions of the VDM. Considering the reent progress on the identifiation

10 14 Page 1 of 1 Eur. Phys. J. C :14 Tale 3 The asolute values of the magneti dipole moment G M and eletri quadrupole moment G E for the orresponding radiative deays in units of the natural magneton. PW means present work and VDM refers to the vetor dominane model G M PW G E PW G M VDM [3] G E VDM [3] γ ± γ ± γ 2.3 ± γ.688 ± γ 3.37 ± γ ± Tale 4 Widths of the orresponding radiative transitions in KeV Ɣ PW Ɣ VDM [3] Ɣ VDM [25,26] γ γ γ γ γ γ and spetrosopy of the heavy aryons, we hope it will e possile to study these radiative deay hannels experimentally in the near future. Aknowledgments K. A. and H. S. would like to thank TUBITAK for their partial finanial support through the projet 114F18. Open Aess This artile is distriuted under the terms of the Creative Commons Attriution Liense whih permits any use, distriution, and reprodution in any medium, provided the original authors and the soure are redited. Funded y SCOAP 3 /LienseVersionCCBY4.. Referenes 1. M. Mattson et al., SELEX Collaoration, Phys. Rev. Lett. 89, A. Oherashvili et al., SELEX Collaoration, Phys. Lett. B 628, T.M. Aliev, M. Savi, V.S. Zamiralov, Mod. Phys. Lett. A 27, T.M. Aliev, K. Azizi, M. Savi, V.S. Zamiralov, Phys. Rev. D 83, T.M. Aliev, K. Azizi, A. Ozpinei, Phys. Rev. D 79, M. Banuls, A. Pih, I. Simemi, Phys. Rev. D 61, H.Y. Cheng et al., Phys. Rev. D 47, S. Tawfig, J.G. Koerner, P.J. O Donnel, Phys. Rev. D 63, M.A. Ivanov et al., Phys. Rev. D 6, S.L. Zhu, Y.B. Dai, Phys. Rev. D 59, H.F. Joens, M.D. Sadron, Ann. Phys. 81, R.C.E. Devenish, T.S. Eisenshitz, J.G. Korner, Phys. Rev. D 14, E. Bagan, M. Chaa, H.G. Dosh, S. Narison, Phys. Lett. B 278, I.I. Balitsky, V.M. Braun, Nul. Phys. B 311, V.M. Braun, I.E. Filyanov, Z. Phys. C 48, K.G. Chetyrkin, A. Khodjamirian, A.A. Pivovarov, Phys. Lett. B 661, I.I. Balitsky, V.M. Braun, A.V. Kolesnihenko, Nul. Phys. B 312, P. Ball, V.M. Braun, N. Kivel, Nul. Phys. B 649, V.M. Belyaev, B.L. Ioffe, JETP 56, J. Rohrwild, JHEP 79, I.I. Balitsky, A.V. Kolesnihenko, A.V. Yung, Yad. Fiz. 41, V.M. Belyaev, I.I. Kogan, Yad. Fiz. 4, Z.-G. Wang, Eur. Phys. J. C 68, Z.-G. Wang, Phys. Lett. B 685, Z.-G. Wang, Eur. Phys. J. A 44, Z.-G. Wang, Phys. Rev. D 81,