Strong Σ b NB and Σ c ND coupling constants in QCD
aa r X i v : . [ h e p - ph ] S e p Strong Σ b N B and Σ c N D coupling constants in QCD
K. Azizi a ∗ , Y. Sarac b † , H. Sundu c ‡ a Department of Physics, Do˘gu¸s University, Acıbadem-Kadık¨oy, 34722 Istanbul, Turkey b Electrical and Electronics Engineering Department, Atilim University, 06836 Ankara, Turkey c Department of Physics, Kocaeli University, 41380 Izmit, Turkey
Abstract
We study the strong interactions among the heavy bottom spin–1/2 Σ b baryon,nucleon and B meson as well as the heavy charmed spin–1/2 Σ c baryon, nucleon and D meson in the context of QCD sum rules. We calculate the corresponding strongcoupling form factors defining these vertices by using a three point correlation func-tion. We obtain the numerical values of the corresponding strong coupling constantsvia the most prominent structure entering the calculations. PACS number(s): 13.30.-a, 13.30.Eg, 11.55.Hx ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] Introduction
In the recent years, substantial experimental improvements have been made on the spec-troscopic and decay properties of heavy hadrons, which were accompanied by theoreticalstudies on various properties of these hadrons. The mass spectrum of the baryons con-taining heavy quark has been studied using different methods. The necessity of a deeperunderstanding of heavy flavor physics requires a comprehensive study on the processes ofthese baryons such as their radiative, strong and weak decays. For some of related studiesone can refer to references [1–10].The investigation of the strong decays of heavy baryons can help us get valuable in-formation on the perturbative and non-perturbative natures of QCD. The strong couplingconstants defining such decays play important role in describing the strong interactionamong the heavy baryons and other participated particles. Therefore, accurate determina-tion of these coupling constants enhance our understanding on the interactions as well as thenature and structure of the participated particles. The present work is an extension of ourprevious study on the coupling constants g Λ b NB and g Λ c ND [11]. Here, we study the stronginteractions among the heavy bottom spin–1/2 Σ b baryon, nucleon and B meson as well asthe heavy charmed spin–1/2 Σ c baryon, nucleon and D meson in the context of QCD sumrules. In particular, we calculate the strong coupling constants g Σ b NB and g Σ c ND . Thesecoupling constants together with the g Λ b NB and g Λ c ND discussed in our previous work, mayalso be used in the bottom and charmed mesons clouds description of the nucleon whichcan be used to explain the exotic events observed by different Collaborations. In addition,the determination of the properties of the B and D mesons in nuclear medium requiresthe consideration of their interactions with the nucleons from which the Λ b [ c ] and Σ b [ c ] areproduced. Therefore, to determine the modifications on the masses, decay constants andother parameters of the B and D mesons in nuclear medium, one needs to consider thecontributions of the baryons Σ b [ c ] together with the Λ b [ c ] and have the values of the strongcoupling constants g Σ b NB and g Σ c ND besides the couplings g Λ b NB and g Λ c ND [12–15]. Inthe literature, one can unfortunately find only a few works on the strong couplings of theheavy baryons with the nucleon and heavy mesons. One approximate prediction for thestrong coupling g Λ c ND was made at zero transferred momentum squared [4]. The strongcouplings of the charmed baryons with the nucleon and D meson were also discussed in [7]in the framework of light cone QCD sum rules.This paper is organized in three sections as follows. In the next section, we present thedetails of the calculations of the strong coupling form factors among the particles underconsideration. In section 3, the numerical analysis of the obtained sum rules and discussionsabout the results are presented. This section is devoted to the details of the calculations of the strong coupling form factors g Σ b NB ( q ) and g Σ c ND ( q ) from which the strong coupling constants among the participatingparticles are obtained at Q = − q = − m B [ D ] , subsequently. In order to accomplish this1urpose, the following three-point correlation function is used:Π = i Z d x Z d y e − ip · x e ip ′ · y h |T (cid:0) η N ( y ) η B [ D ] (0) ¯ η Σ b [Σ c ] ( x ) (cid:1) | i , (1)whith T being the time ordering operator and q = p − p ′ is the transferred momentum.The currents η N , η B [ D ] and η Σ b [Σ c ] presented in Eq. (1) correspond to the the interpolatingcurrents of the N , B [ D ] and Σ b [ c ] , respectively and their explicit expressions can be givenin terms of the quark field operators as η Σ b [Σ c ] ( x ) = ε ijk (cid:16) u i T ( x ) Cγ µ d j ( x ) (cid:17) γ γ µ b [ c ] k ( x ) ,η N ( y ) = ε ijk (cid:16) u i T ( y ) Cγ µ u j ( y ) (cid:17) γ γ µ d k ( y ) ,η B [ D ] (0) = ¯ u (0) γ b [ c ](0) , (2)where C denotes the charge conjugation operator; and i , j and k are color indices.In the course of calculation of the three-point correlation function one follows two differ-ent ways. The first way is called as OPE side and the calculation is made in deep Euclideanregion in terms of quark and gluon degrees of freedom using the operator product expan-sion. The second way is called as hadronic side and the hadronic degrees of freedoms areconsidered to perform this side of the calculation. The QCD sum rules for the couplingform factors are attained via the match of these two sides. The contributions of the higherstates and continuum are suppressed by a double Borel transformation applied to both sideswith respect to the variables p and p ′ . For the calculation of the OPE side of the correlation function which is done in deepEuclidean region, where p → −∞ and p ′ → −∞ , one puts the interpolating currentsgiven in Eq. (2) into the correlation function, Eq. (1). Possible contractions of all quarkpairs via Wick’s theorem leads toΠ OP E = i Z d x Z d ye − ip · x e ip ′ · y ε abc ε ijℓ × (cid:26) γ γ ν S cjd ( y − x ) γ µ CS bi T u ( y − x ) Cγ ν S ahu ( y ) γ S hℓb [ c ] ( − x ) γ µ γ − γ γ ν S cjd ( y − x ) γ µ CS ai T u ( y − x ) Cγ ν S bhu ( y ) γ S hℓb [ c ] ( − x ) γ µ γ (cid:27) , (3)where S b [ c ] ( x ) and S u [ d ] ( x ) are the heavy and light quark propagators whose explicit formscan be found in Refs. [11, 16].After some straightforward calculations (for details refer to the Ref. [11]), the correlationfunction in OPE side comes out in terms of different Dirac structures asΠ OP E = Π ( q ) γ + Π ( q ) pγ + Π ( q ) q pγ + Π ( q ) qγ . (4)2ach Π i ( q ) function involves the perturbative and non-perturbative parts and is writtenas Π i ( q ) = Z ds Z ds ′ ρ perti ( s, s ′ , q ) + ρ non − perti ( s, s ′ , q )( s − p )( s ′ − p ′ ) . (5)The spectral densities, ρ i ( s, s ′ , q ), appearing in Eq. (5) are obtained from the imaginaryparts of the Π i functions, i.e., ρ i ( s, s ′ , q ) = π Im [Π i ]. Here to provide examples of theexplicit forms of the spectral densities, among the Dirac structures presented above, weonly present the results obtained for the Dirac structure pγ , that is ρ pert ( s, s ′ , q ) and ρ non − pert ( s, s ′ , q ), which are obtained as ρ pert ( s, s ′ , q ) = Z dx Z − x dy π ( x + y − (cid:26) m b [ c ] x (cid:16) x − y − x + 3 xy (cid:17) − m b [ c ] x ( x + y − h − m u ( x + 4 y −
2) + m d (3 x + 6 y − i − q m b [ c ] x h y − y + x (8 y −
1) + x (1 − y + 8 y ) i + m b [ c ] ( x + y − × h sx (8 x − y − x + 8 xy ) + s ′ (cid:16) xy − xy − y − x + 8 x y (cid:17)i − ( x + y − h m d s (cid:16) x + y − y − x + 12 x y + 3 x − xy + 8 xy (cid:17) + 3 m d q y (cid:16) x − x + y − xy (cid:17) + 3 m d s ′ y (cid:16) − x + 4 x − y + 12 xy + 8 y (cid:17)i − m u s h x + y (3 − y ) + x (20 y −
13) + x (9 − y + 16 y ) i + m u s ′ y (cid:16) − x + 4 x − y + 20 xy + 16 y (cid:17) + m u q y (11 x − x + 2 y − xy − (cid:27) Θ h L ( s, s ′ , q ) i , (6)and ρ non − pert ( s, s ′ , q ) = n h u ¯ u i π ( q − m b [ c ] ) (cid:16) m b [ c ] m d − m u m d + 3 m u − q + s − s ′ (cid:17) + h α s G π i m b [ c ] (cid:16) q − m b [ c ] − s ′ (cid:17) π ( q − m b [ c ] ) + m h u ¯ u i π ( q − m b ) (cid:27) Θ h L ( s, s ′ , q ) i − (cid:16) h d ¯ d i − h u ¯ u i (cid:17) Z dx Z − x dy x + 6 y − π Θ h L ( s, s ′ , q ) i , (7)where Θ[ ... ] stands for the unit-step function and L ( s, s ′ , q ) and L ( s, s ′ , q ) are definedas L ( s, s ′ , q ) = s ′ ,L ( s, s ′ , q ) = − m b [ c ] x + sx − sx + s ′ y + q xy − sxy − s ′ xy − s ′ y . (8)3 .2 Hadronic Side On the hadronic side, considering the quantum numbers of the interpolating fields one placethe complete sets of intermediate Σ b [Σ c ], B [ D ] and N hadronic states into the correlationfunction. After carrying out the four-integrals, we getΠ HAD = h | η N | N ( p ′ ) ih | η B [ D ] | B [ D ]( q ) ih Σ b [Σ c ]( p ) | ¯ η Σ b [Σ c ] | i ( p − m b [Σ c ] )( p ′ − m N )( q − m B [ D ] ) × h N ( p ′ ) B [ D ]( q ) | Σ b [Σ c ]( p ) i + · · · . (9)In the above equation, the contributions of the higher states and continuum are denoted by · · · and the matrix elements are represented in terms of the hadronic parameters as follows: h | η N | N ( p ′ ) i = λ N u N ( p ′ , s ′ ) , h Σ b ( p ) | ¯ η Σ b [Σ c ] | i = λ Σ b [Σ c ] ¯ u Σ b [Σ c ] ( p, s ) , h | η B [ D ] | B [ D ]( q ) i = i m B [ D ] f B [ D ] m u + m b [ c ] , h N ( p ′ ) B [ D ]( q ) | Σ b [Σ c ]( p ) i = g Σ b NB [Σ c ND ] ¯ u N ( p ′ , s ′ ) iγ u Σ b [Σ c ] ( p, s ) . (10)Here λ N and λ Σ b [Σ c ] are residues of the N and Σ b [Σ c ] baryons, respectively, f B [ D ] is theleptonic decay constant of B [ D ] meson and g Σ b NB [Σ c ND ] is the strong coupling form factoramong Σ b [Σ c ], N and B [ D ] particles. Using Eq. (10) in Eq. (9) and summing over the spinsof the particles, we obtainΠ HAD = i m B [ D ] f B [ D ] m b [ c ] + m u λ N λ Σ b [Σ c ] g Σ b NB [Σ c ND ] ( p − m b [Σ c ] )( p ′ − m N )( q − m B [ D ] ) × n ( m N m Σ b [Σ c ] − m b [Σ c ] ) γ + ( m Σ b [Σ c ] − m N ) pγ + q pγ − m Σ b [Σ c ] qγ o + · · · . (11)To acquire the final form of the hadronic side of the correlation function we perform thedouble Borel transformation with respect to the initial and final momenta squared, b B Π HAD = i m B [ D ] f B [ D ] m b [ c ] + m u λ N λ Σ b [Σ c ] g Σ b NB [Σ c ND ] ( q − m B [ D ] ) e − m b [Σ c ] M e − m NM ′ × n ( m N m Σ b [Σ c ] − m b [Σ c ] ) γ + ( m Σ b [Σ c ] − m N ) pγ + q pγ − m Σ b [Σ c ] qγ o + · · · , (12)where M and M ′ are Borel mass parameters.As it was already stated, the match of the hadronic and OPE sides of the correlationfunction in Borel scheme provides us with the QCD sum rules for the strong form factors.The consequence of that match for pγ structure leads us to g Σ b NB [Σ c ND ] ( q ) = − e m b [Σ c ] M e m NM ′ ( m b [ c ] + m u )( q − m B [ D ] ) m B [ D ] f B [ D ] λ † Σ b [Σ c ] λ N ( m N m Σ b [Σ c ] − m b [Σ c ] )4 (cid:26) Z s ( m b [ c ] + m u + m d ) ds Z s ′ (2 m u + m d ) ds ′ e − sM e − s ′ M ′ h ρ pert ( s, s ′ , q ) + ρ non − pert ( s, s ′ , q ) i(cid:27) , (13)where s and s ′ are continuum thresholds in Σ b [Σ c ] and N channels, respectively. Having obtained the QCD sum rules for the strong coupling form factors, in this section,we present the numerical analysis of our results and discuss the dependence of the strongcoupling form factors under consideration on Q = − q . To this aim, beside the inputparameters given in table 1, one needs to determine the working intervals of four auxiliaryparameters M , M ′ , s and s ′ . These parameters originate from the double Borel trans-formation and continuum subtraction. The determination of the working regions of themis made on the basis of that the results obtained for the strong coupling form factors beroughly independent of these helping parameters.The continuum thresholds s and s ′ are the parameters related to the beginning of thecontinuum in the initial and final channels. If the ground state masses are given by m and m ′ for the initial and final channels, respectively, to excite the particle to the first excited statehaving the same quantum numbers with them one needs to provide the energies √ s − m and p s ′ − m ′ . For the considered transitions, these quantities can be determined from wellknown excited states of the initial and final states [17] which are roughly in between 0 . . . .
5] GeV ≤ s ≤ . .
6] GeV and 1 .
04 GeV ≤ s ′ ≤ .
99 GeV forthe vertex Σ b N B [Σ c N D ].Here, we shall comment on the selection of the most prominent Dirac structure todetermine the corresponding strong coupling form factors. In principle, one can chooseany structure for determination of these strong coupling form factors. However, we shouldchoose the most reliable one considering the following criteria: • the pole/continuum should be the largest, • the series of sum rule should demonstrate the best convergence, i.e. the perturbativepart should have the largest contribution and the operator with highest dimensionshould have relatively small contribution.Our numerical calculations show that these conditions lead to choose the structure q pγ asthe most prominent structure. In the following, we will use this structure to numericallyanalyze the obtained sum rules.Now, we proceed to present the Borel windows considering the selected structure. Theworking windows for the Borel parameters M and M ′ are determined considering againthe pole dominance and convergence of the OPE. By requirement that the pole contributionexceeds the contributions of the higher states and continuum, and that the contributionof the perturbative part exceeds the non-perturbative contributions we find the windows10[2] GeV ≤ M ≤ and 1 GeV ≤ M ′ ≤ for the Borel mass parameters5arameters Values m b (4 . ± .
03) GeV[17] m c (1 . ± . m d . +0 . − . MeV[17] m u . +0 . − . MeV [17] m B (5279 . ± .
17) MeV [17] m D (1864 . ± .
07) MeV [17] m N (938 . ± . m Σ b (5811 . ± .
9) MeV [17] m Σ c (2452 . ± .
4) MeV [17] f B (248 ± exp ± V ub ) MeV [18] f D (205 . ± . ± .
5) MeV [19] λ N . ± . [20] λ Σ b (0 . ± . [5] λ Σ c (0 . ± . [5] h ¯ uu i (1 GeV ) = h ¯ dd i (1 GeV ) − (0 . ± . GeV [21] h α s G π i (0 . ± . [22] m (1 GeV ) (0 . ± .
2) GeV [22]Table 1: Input parameters used in calculations.of the strong vertex Σ b N B [Σ c N D ]. For these intervals, our results show weak dependenceon the Borel mass parameters (see figures 1-2).
10 12 14 16 18 20691215 g b NB ( Q = )( G e V - ) M (GeV ) structure qp ; M ’2 =2 GeV
10 12 14 16 18 20691215 1.0 1.5 2.0 2.5 3.0691215 g b NB ( Q = )( G e V - ) M’ (GeV ) structure qp ; M =15 GeV Figure 1:
Left: g Σ b NB ( Q = 0) as a function of the Borel mass M at average values ofcontinuum thresholds. Right: g Σ b NB ( Q = 0) as a function of the Borel mass M ′ ataverage values of continuum thresholds.Subsequent to the determination of the auxiliary parameters, their working windowstogether with the other input parameters are used to ascertain the dependency of thestrong coupling form factors on Q . From our analysis we observe that the dependency of6 g c ND ( Q = )( G e V - ) M (GeV ) structure qp ; M ’2 =2 GeV g c ND ( Q = )( G e V - ) M’ (GeV ) structure qp ; M =4 GeV Figure 2: The same as figure 1 but for g Σ c ND ( Q = 0). -30 -20 -10 0 10 20 305101520 g b NB ( G e V - ) Q ( GeV ) QCD sum rules; fit function for structure qp -30 -20 -10 0 10 20 30 -4 -2 0 2 402468 g c ND ( G e V - ) Q ( GeV ) QCD sum rules; fit function for structure qp -4 -2 0 2 4 02468 Figure 3:
Left: g Σ b NB ( Q ) as a function of Q at average values of the continuum thresholdsand Borel mass parameters. Right: g Σ c ND ( Q ) as a function of Q at average values of thecontinuum thresholds and Borel mass parameters.the strong coupling form factors on Q is well characterized by the following fit function: g Σ b NB [Σ c ND ] ( Q ) = c exp h − Q c i + c . (14)The values of the parameters c , c and c for Σ b N B and Σ c N D can be seen in tables 2 and3, respectively. Considering the average values of the continuum thresholds and Borel massparameters we demonstrate the variation of the strong coupling form factors with respectto Q for the QCD sum rules as well as the fitting results in figure 3. The figure indicatesthe truncation of the QCD sum rules at some points at negative values of Q . It can beseen from the figure that there is a good consistency among the results obtained from theQCD sum rules and fit function up to these points. The fit function is used to determinethe value of the strong coupling constant at Q = − m B [ D ] , and the results are presented intable 4. The presented errors in these results originate from the uncertainties of the input7arameters together with the uncertainties coming from the determination of the workingregions of the auxiliary parameters.Parameters c (GeV − ) c (GeV ) c (GeV − )Values 0 . ± .
20 18 . ± .
87 9 . ± . b N B vertex. Parameters c (GeV − ) c (GeV ) c (GeV − )Values − . ± . − . ± .
89 3 . ± . c N D vertex.Coupling Constants g Σ b NB ( Q = − m B )(GeV − ) g Σ c ND ( Q = − m D )(GeV − )Values 12 . ± .
49 3 . ± . g Σ b NB and g Σ c ND .To sum up, in this work, the strong coupling constants among the heavy bottom spin–1/2 Σ b baryon, nucleon and B meson as well as the heavy charmed spin–1/2 Σ c baryon,nucleon and D meson, namely g Σ b NB and g Σ c ND , have been calculated in the framework ofthe three-point QCD sum rules. The obtained results can be applied in the analysis of therelated experimental results at LHC. The predictions can also be used in the bottom andcharmed mesons clouds description of the nucleon that may be applied for the explanationof the exotic events observed by different experiments. These results may also serve thepurpose of analyzing of the results of heavy ion collision experiments like P AN DA at FAIR.The obtained results may also come in handy in the determinations of the changes in themasses, decay constants and other parameters of the B and D mesons in nuclear medium. This work has been supported in part by the Scientific and Technological Research Councilof Turkey (TUBITAK) under the research project 114F018.8 eferenceseferences