Λ + c fragmentation functions from pQCD approach and the Suzuki model
Mahdi Delpasand, S. Mohammad Moosavi Nejad, Maryam Soleymaninia
aa r X i v : . [ h e p - ph ] J un Λ + c fragmentation functions from pQCD approach and the Suzuki model Mahdi Delpasand a , ∗ S. Mohammad Moosavi Nejad a,b , † and Maryam Soleymaninia b ‡ ( a ) Faculty of Physics, Yazd University, P.O. Box 89195-741, Yazd, Iran and ( b ) School of Particles and Accelerators, Institute for Research inFundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran (Dated: June 16, 2020)Through data analysis, we present new sets of nonperturbative fragmentation functions (FFs) for Λ + c baryon both at leading and next-to-leading order (NLO) and, for the first time, at next-to-next-to-leading order (NNLO) in the minimal subtraction factorization scheme with five massless quarks.The FFs are determined by fitting all available data of inclusive single Λ + c baryon production in e + e − annihilation taken by the OPAL
Collaboration at CERN LEP1 and
Belle
Collaboration at KEKB.We also estimate the uncertainties in the Λ + c FFs as well as in the corresponding observables. Ina completely different approach based on the Suzuki model, we will theoretically calculate the Λ + c FF from charm quark and present our result at leading order perturbative QCD framework. Acomparison confirms a good consistency between both approaches. We will also apply the Λ + c FFsto make theoretical predictions for the energy distribution of Λ + c produced through the top quarkdecay, to be measured at the CERN LHC. I. INTRODUCTION
Study of heavy hadrons properties provides a possi-bility for better understanding the quark-gluon inter-action dynamics in the QCD framework. In this re-gards and due to ongoing experiments there are partic-ular interests in hadron productions at the CERN LHCand the BNL Relativistic Heavy Ion Collider (RHIC). Inthis work, we study the production mechanism of heavybaryons through the fragmentation process. In a gen-eral expression, the fragmentation mechanism describesthe hadronization process where a parton carrying largetransverse momentum decays to form a jet containing theexpected hadron [1]. The hadronization processes are de-scribed by the fragmentation functions (FFs) which referto the probability densities of hadron productions frominitial partons and make the nonperturbative aspects ofhadroproduction processes. These functions along withthe parton distribution functions (PDFs) [2] constructnonperturbative inputs for the calculation of hadropro-duction cross sections. In this work, we focus on theFFs of Λ + c baryon through two different approaches. Inthe first approach which is usually called as the phe-nomenological approach, a specific form including somefree parameters is proposed for the desired FF so thatall these free parameters are extracted through experi-mental data analysis, see, for example, Refs. [3–15]. Inan alternative scheme based on the theoretical models,the heavy hadron FFs might be computed by the virtueof perturbative QCD with limited phenomenological pa-rameters, see, for example, Refs. [16–23] where the heavyhadron FFs are computed by use of the Suzuki model[24]. This elaborate model is related to the perturbative ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
QCD framework where all convenient Feynman diagramsat each order of perturbative QCD are considered for theparton level of hadronization process. In this approach,the nonperturbative aspect of hadronization is emergedin the bound state wave function. Detailed description ispresented in Section III.Independent of approach used to extract the FFs, whenthese functions are computed at the initial scale of frag-mentation they can be evolved to higher scales usingthe timelike Dokshitzer-Gribov-Lipatov-Altarelli-Parisi(DGLAP) evaluation equations [25].In this paper, we first use the phenomenological approachto obtain a set of gluon, charm- and bottom-quark FFsinto the Λ + c baryon through a global QCD fit to all avail-able e + e − single inclusive annihilation (SIA) data mea-sured by OPAL
Collaboration [26, 27] at the CERN LEP1and very recent data from
Belle
Collaboration [28]. InRef. [29], the Λ + c FFs were determined both at leadingand next-to-leading order in the minimal subtraction fac-torization scheme (
M S ) by fitting the fractional energyspectra of Λ + c baryon measured by the OPAL in the e + e − annihilation on the Z-boson resonance. In their work, au-thors have applied the zero-mass variable-flavor-numberscheme (ZM-VFNS) or, in a short expression, the mass-less scheme, where heavy quarks are treated as masslesspartons as well. Recently, KKSS20 Collaboration hasupdated their previous analysis [29] combining the datafor e + e − annihilation from OPAL and the recent one from
Belle , see Ref. [30]. Their strategy for constructing the Λ + c -FFs is the same as their previous work for D-meson[31]. Their work is restricted to the NLO accuracy andthey have not also evaluated the uncertainties for FFsand corresponding observables.In the present work, we focus on the hadronization ofgluon, charm- and bottom-quarks into the Λ + c usingthe massless scheme and provide the first QCD analy-sis of ( g, c, b ) → Λ + c FFs at next-to-next-to-leading order(NNLO). Meanwhile, we go beyond Refs. [29, 30] andperform a full-fledged error estimation for the parton FFsas well as the resulting differential cross sections. In or-der to evaluate the error estimations we apply the well-known Hessian approach [32]. Note that, our analysis isrestricted to only three data sets from single inclusive an-nihilation process due to two reasons: firstly, we are notaware of any other such data from electron-positron anni-hilation, and secondly, due to the lack of other theoreticalpartonic cross sections for the production of partons atNNLO accuracy. Although, among all processes produc-ing baryons, the e + e − annihilation process provides thecleanest environment to calculate the FFs, being devoidof nonperturbative effects beyond fragmentation itself.In the following and in a theoretical approach indepen-dent of data analysis, we compute the initial scale frag-mentation function of charm quark to split into the S-wave Λ + c baryon at lowest-order of perturbative QCD.For this approach, we employ the elaborate Suzuki modelwhich contains most of kinematical and dynamical as-pects of hadroproduction process. Finally, we shall com-pare the initial scale FF of c → Λ + c determined in bothapproaches. Our comparison shows a good consistencybetween both results.In the Standard Model (SM) of particle physics, thetop quark has very short life time so does not haveenough time to form a bound state, then before it decayshadronization takes place. At the lowest order of per-turbative QCD and at the parton level, the decay mode t → bW + followed by b → X + Jets is governed. Here, Xrefers to the detected hadrons in the final state. Thus, atthe CERN LHC a proposed channel to indirect search forthe top quark properties is to study the energy spectrumof produced hadrons through top decays. In this work,as an example of possible applications of extracted FFs,we make the theoretical predictions for the energy distri-butions of Λ + c baryons in top quark decays at LO, NLOand NNLO. This prediction will be compared with theone obtained through using the fit parameters reportedin Ref. [30]. This comparison does also shows a goodconsistency between our analysis and the one performedin Ref. [30].The outline of this paper is as follows: In Section II,we explain the theoretical framework of hadron produc-tion in e + e − annihilation in the massless scheme andintroduce our parametrization of the c/b → Λ + c FF atthe initial scale. We will also describe the minimizationmethod in our analysis and the approach used for deter-mination of error estimation. All applied experimentaldata will be describe in this section and our LO, NLOand NNLO results will be presented and compared withthe data fitted to. In Section III, through the pertur-bative QCD approach we provide a general discussion ofthe fragmentation process for the S -wave heavy baryonand determine the fragmentation distribution of c-quarkto fragment into Λ + c baryon at lowest-order of perturba-tive QCD. In Section. IV, predictions for the normalized-energy distributions of Λ + c baryon produced from topdecay are presented. Our conclusions are listed in Sec-tion V. II. PHENOMENOLOGICAL APPROACH FORDETERMINATION OF Λ + c FFS AND THEIRUNCERTAINTIES
As was mentioned, one of the most common approachto calculate the unpolarized nonperturbative FFs is thephenomenological approach based on the data analysis.In order to present the theoretical predictions for the ob-servables involving cross section of identified hadrons inthe final state, considering different hierarchical featuresis vital which is mentioned in this section. In this regard,we first review the QCD framework including the stan-dard factorization theorem for differential cross sectionin a hadronization process of single inclusive electron-positron annihilation. We shall also introduce the
OPAL and
Belle experimental data as the only data sets for Λ + c production in SIA process. Finally, we indicate ourtheoretical formalisms for determination of Λ + c FFs anddescribe our strategy to determine the uncertainties ofFFs as well as corresponding theoretical cross sections.
A. QCD framework for Λ + c baryon FFs Our analysis depends on the normalized differentialcross section /σ tot × dσ/dx Λ of the annihilation process e + e − → ( γ ∗ , Z ) → Λ + c + X, (1)where, X refers to the unobserved hadrons in the finalstate and Λ + c is the identified hadron. As usual, the scal-ing variable x Λ is defined as x Λ = 2 p Λ · q/q where, p Λ and q refer to the four-momentum of detected baryon andintermediate gauge boson, respectively, so that s = q isthe collision energy. In the center of mass (CM) frame,the scaling variable is simplified as x Λ = 2 E Λ / √ s where E Λ shows the energy of detected baryon.The key implement to divide the perturbative and non-perturbative parts of the e + e − annihilation process (1) isthe factorization theorem in the QCD-improved partonmodel [33]. According to this theorem, the differentialcross section of process (1) is written as the convolu-tion of differential partonic cross sections dσ i ( e + e − → i + X ) /dx i , with the Λ + c -FFs which is denoted by D Λ + c i .Here, i = g, u, ¯ u, . . . , b, ¯ b runs over the active partons sothat the number of active flavors is dependent on theenergy scale. In the ZM-VFNS (or zero-mass scheme)where all light and heavy quarks are considered as mass-less partons, the differential cross section normalized tothe total one is written as [33] σ tot dσdx Λ ( e + e − → Λ + c + X )= X i ˆ x Λ dx i x i σ tot dσ i dx i ( x i , µ R , µ F ) D Λ + c i ( x Λ x i , µ F ) . (2)In the CM frame, the scaling variable x i is also defined as x i = 2 E i / √ s which refers to the energy of produced par-ton i in units of the beam energy. In the relation above,the scales µ F and µ R are the factorization and renormal-ization scales, respectively. Normally, they are arbitraryquantities which appear in each order of perturbationbut in order to omit the ambiguous logarithmic terms ln( s/µ F ) in the partonic cross sections, they are chosento be µ F = µ R = √ s .The experimental data included in our analysis are nor-malized to the total hadronic cross section for the e + e − annihilation. This cross section reads σ tot = 4 πα ( s ) s ( n f X i ˜ e i ( s )) × (1 + α s K (1) QCD + α s K (2) QCD + · · · ) . (3)Here, α s and α are the strong-coupling and fine-structureconstants, respectively, and ˜ e i is the effective electroweakcharge of quark i . The QCD perturbative coefficients K ( i ) QCD are currently known up to NNLO accuracy [34]so that K (1)QCD = 3 C F / (4 π ) , where C F = 4 / , and K (2)QCD ≈ . [35].In Eq. (2), the nonperturbative part of the process (1)related to the transition i → Λ + c is described by the D Λ + c i ( z, µ F ) -FF, where the fragmentation parameter z = x Λ /x i indicates the energy fraction passed on from par-ton i to the Λ + c baryon, i.e., z = E Λ /E i . Since the FFsdepend on the factorization scale µ F , they are evaluatedto the various scales of energies by the DGLAP evolutionequations, i.e. µ dD Λ i dµ ( x Λ , µ ) = X j ˆ x Λ dx i x i P ij ( x Λ x i , α s ( µ )) D Λ j ( x i , µ ) , (4)where, P ij are the splitting functions which have beencomputed up to NNLO accuracy [36–38]. B. Theoretical formalism
According to the phenomenological approach, in orderto extract the nonperturbative FFs, the z -distributionsof i → Λ + c FFs at the starting scale µ are parametrizedfrom the beginning and the free parameters are con-strained from the SIA experimental data. Note that, theselection criterion for the best parametrization form isto score a minimum χ global value as small as possiblewith a set of fit parameters as minimal as possible. The χ global function is defined in our previous works in moredetails [8, 9]. Following Ref. [30], we parametrize the z -dependence of c → Λ + c and b → Λ + c FFs at the start-ing scale as suggested by Bowler [39] while the FFs oflight flavors are assumed to be zero at the starting scale.Moreover, since the data set included in our analysis islimited to the SIA process we can not constrain the gluon FF at the initial scale, thus its corresponding FF is alsoset equal to zero at the initial scale, i.e., D Λ + c i ( z, µ ) = 0 , f or i = u, ¯ u, d, ¯ d, s, ¯ s, g. (5)The FFs of gluon and light flavors will be generated tohigher energy scales via the DGLAP evaluation equa-tions. As was mentioned, for the c → Λ + c and b → Λ + c fragmentation the following Bowler parametrization [39]is considered: D Λ + c c ( z, µ ) = N c (1 − z ) a c z − (1+ b c ) e − b c /z D Λ + c b ( z, µ ) = N b (1 − z ) a b z − (1+ b b ) e − b b /z , (6)which includes six free parameters: N c , a c , b c , N b , a a and b b . It is found that the Bowler parametrization enablesone to do excellent fits at each order of perturbation,i.e. LO, NLO and NNLO. Here, the initial scale is set as µ = 4 . GeV which is a little grater than the bottommass threshold m b = 4 . GeV.Consequently, we have six free parameters which shouldbe extracted from the best QCD fit on the experimentaldata. The optimal values of fit parameters for the charmand bottom FFs are reported in Table I at each order ofperturbation.
Table I: The optimal values for the input parameters of the c → Λ + c and b → Λ + c FFs at the initial scale µ = 18 . GeV determined by QCD analysis of the experimental data listedin Table II.Parameter Best values LO NLO NNLO N c .
183 15210639 .
353 2720317 . a c .
343 2 .
373 2 . b c .
343 3 .
814 3 . N b .
441 323 .
094 318 . a b .
302 9 .
138 9 . b b .
554 1 .
477 1 . Technically, it should be mentioned that for the evo-lution of D Λ + c i ( z, µ ) -FFs as well as for the calculationof SIA cross sections up to NNLO accuracy we employedthe publicly available APFEL package [40] and the free pa-rameters of FFs are determined by minimizing the χ global function using the CERN program MINUIT [41]. Further-more, to estimate the uncertainties of D Λ + c i ( z, µ ) -FFs thedata uncertainties are propagated to the extracted QCDfit parameters using the asymmetric Hessian method (orHessian methodology), as is outlined in [42, 43]. Moredetails can be found in our previous analysis [9]. C. Experimental data and fit results
In our analysis, we applied two data sets measured by
OPAL
Collaboration at the CERN LEP1 Collider [26]. Infact, for the SIA process (1) two different mechanismscontribute with similar rates; direct production through Z → c ¯ c decay followed by the fragmentation c/ ¯ c → Λ + c and the decay Z → b ¯ b (b-tagged events) followed bythe fragmentation of b (or ¯ b ) into the bottom-flavoredhadron H b , i.e., b/ ¯ b → H b , where finally the weak decayof H b into the Λ + c -baryon occurs; H b → Λ + c + X . Con-sequently, the energy spectrum of Λ + c -baryon originatingfrom the decay of H b -hadron is much softer than that dueto primary charm production, as is expected. In order toseparate charmed hadron production through the decayprocess Z → c ¯ c from the decay Z → b ¯ b , the OPAL
Collab-oration investigated the apparent decay length distribu-tions as well as the energy spectra of charmed hadrons.As is expected, the decay lengths of H b hadrons into Λ + c baryon are always longer than those from prompt pro-duction. Note that, the OPAL
Collaboration has presented x Λ -distributions for their Λ + c sample and for the b-tagged( Z → b ¯ b ) subsamples. In addition to the OPAL data whichincludes only 4 points with rather large uncertainties wehave also included a very recent data set measured by
Belle
Collaboration [28] at √ s = 10 . GeV. This newdata set is much more precise and contains more points.
Belle data dose not have contributions from B-mesondecays so that the contribution of b → Λ c is not neededto be taken in the calculation of cross sections. How-ever, the FFs from charm and bottom quarks are cou-pled through the DGLAP evolution equation. FollowingRef. [30], we fix the b → Λ c FF using the values of N b , a b and b b extracted from the OPAL fit. Therefore, the val-ues of N c , a c and b c are yielded from the fit to the Belle data. Here, we describe a technical point over the re-construction of old
OPAL experimental data.
OPAL datasets have been displayed in the form (1 /N had ) dN/dx Λ where N refers to the number of charmed-flavor hadroncandidates which are reconstructed through appropriatedecay chains. Therefore, in order to convert these datainto the convenient cross section (1 /σ tot ) dσ/dx Λ , it isneeded to divide them by the corresponding branchingfractions of decays for the reconstruction of charmed-flavored baryons. In Refs. [26, 27], for the requiredbranching fraction the following value is applied: Br (Λ + c → pK − π + ) = (4 . ± . . (7)Since, in our analysis we are including both the old OPAL data and more recent
Belle data which are based onthe observation of decay Λ + c → pK − π + , then we haveto use the same branching ratio to reconstruct both datasets. Since, the Belle analysis has applied the newestbranching ratio as Br (Λ + c → pK − π + ) = (6 . [44],we therefore rescale the old OPAL data by the factor . / . . .Another point about the Belle data is that we haveto exclude data at small values of the scaling variable z ≤ . , since the theory is not reliable in this rangewithout taking resummation of soft-gluon logarithms intoaccount. Note that, no kinematical cut is taken over the OPAL data. In Table. II, the characteristics of availableexperimental data including the number of data points N datan are presented. We have also listed the individualvalues of χ for inclusive and b -tagged data sets at LO,NLO and NNLO accuracies. Considering the number ofdegrees of freedom (d.o.f), i.e., − , we have alsopresented the total χ divided by the number of d.o.fat LO ( χ /d.o.f = 1 . ), NLO ( χ /d.o.f = 1 . ) andNNLO ( χ /d.o.f = 1 . ). As is seen, these values arearound 1 for individual data sets so this confirms a well-satisfying fit in all three accuracies. From Table. II, itis seen that a reduction in χ /d.o.f occurs when passingfrom LO to NLO accuracy. This behavior is not validwhen NNLO corrections are considered. This is due tothe fact that, the Belle cross sections have been mea-sured at small scale of energy ( √ s = 10 . GeV) wherethe corrections for the finite mass of hadron and partonsare more effective than the higher order corrections. Re-member that we have used the ZM-VFN scheme wherethe hadron and parton masses are set to zero from thebeginning. Nevertheless, the NNLO corrections lead toa reduction in the uncertainties bands of FFs and corre-sponding observables. See Figs. 1-4.In order to show the consistency and goodness betweenthe theoretical prediction and the experimental data usedin the fits, in Fig. 1 we have plotted the inclusive differ-ential cross section, as reported by
Belle , and in Fig. 2we have plotted the b-tagged and total differential crosssections normalized to the total one evaluated with ourrespective Λ + c FFs. Both are compared with the
Belle and
OPAL data sets fitted to. In these figures, the uncer-tainty bands are also plotted using the Hessian approach.Through this approach, we just considered the uncertain-ties due to the experimental data sets so that we ignoredadditional sources of uncertainties. As is seen the qualityof fit is improved when passing to higher order correc-tions. From Fig. 2, it is seen that our theoretical descrip-tions at LO, NLO and NNLO for both b-tagged and totaldifferential cross sections are in good mutual agreement.The consistency seems to be better for the normalizedtotal differential cross section (shown in lower panel) incomparison with the b-tagged one because our theoreti-cal predictions do not go across one of the b-tagged datapoints located at x Λ = 0 . . This is why higher values ofindividual χ occur for b-tagged, see Table. II.In Fig. 2, it is also seen that the behavior of theo-retical predictions in small values of x Λ for the lowestorder accuracy is completely different with the ones atNLO and NNLO. Obviously, in one hand, the theoreticalcross section at LO goes to infinity when x Λ → and, onthe other hand, the LO uncertainty band is anomalouslywider than the NLO and NNLO ones in all ranges of x Λ .Therefore, the LO results are not reliable so that higherorder radiative corrections need to be considered.In order to show the fragmentation contribution of gluon,charm- and bottom-quark to the production of Λ + c , inFig. 3 we have plotted these contributions at the scale √ s = M Z . The total differential cross section at NLO isalso shown which is obtained by the sum of all contribu-tions. As is expected, the contribution of gluon fragmen- Dataset Observable √ s [GeV] N datan χ ( LO ) χ ( NLO ) χ ( NNLO ) Belle
Inclusive 10.52 35 41.419 42.843 55.413
OPAL
Inclusive 91.2 4 1.971 0.444 0.299 b -tagged 91.2 4 4.520 4.524 4.539 TOTAL:
43 47.948 47.812 60.251( χ / d.o.f) 1.296 1.292 1.628Table II: The individual χ values for inclusive and b -tagged cross sections obtained at LO, NLO and NNLO. The total χ and χ /d.o.f fit for Λ + c are also shown. d σ / d z ( e + e - ( Λ + c + c . c ) X )( nb ) z Figure 1: Using our extracted Λ + c -FFs, the theoretical predictions for inclusive differential cross sections at LO (red solid line),NLO (green dashed line) and NNLO (blue dot-dashed line) in √ s = 10 . GeV are compared with
Belle experimental datapoints fitted to. Corresponding uncertainty bands stem from Λ + c FFs are shown as well. tation is very tiny and it increases at small range of x Λ .At large x Λ , the contribution of charm quark (red dashedline) is governed while at small region the contributionof bottom quark (green dot-dashed line) is governed.In Fig. 4, the z -distributions of Λ + c -FFs are plottedat µ F = M Z ; the energy scale of OPAL data sets fittedto. For this purpose, we plotted the ( c, b, g ) → D Λ + c FFsat LO (solid lines), NLO (dashed lines) and NNLO (dot-dashed lines). From this plot, it is seen that the fragmen-tation of charm-quark is peaked at large- z whereas thebottom fragmentation has its maximum at small- z . Thisbehavior is due to the fact that the fragmentation process b → Λ + c contains two-step mechanism. In Fig. 4, the un-certainty bands of Λ + c -FFs are also presented which areneeded to visually quantify the remaining error of analy-sis. Since, the Belle date does not include the contribu-tions from the b → Λ c fragmentation in the calculation of cross sections, then the uncertainties of b -quark FFare considerably much wider than the charm and gluonones in each order of perturbation. Moreover, the er-ror bands of all flavors decrease by increasing the orderof perturbation. In Fig. 4, the NLO KKSS20 results [30](dashed-dashed-dot curves) are also plotted. As is seen,our result for gluon fragmentation is in good agreementwith the one presented by
KKSS20 . In comparison tothe
KKSS20 ’s results, there is a considerable differencebetween the
KKSS20 charm FF and ours in the range z < . . However, the behavior of our bottom-FF is thesame as the KKSS20 one in the whole range of z . Unlikeour procedure in which we set the same scale for the c-and b-quark FFs, the KKSS20 collaboration has selecteddifferent initial scales so that in their work the startingscales for the charm- and bottom-FFs were taken to be µ = m c = 1 . GeV and µ = m b = 5 GeV, respectively. / σ t o t a l d σ / dx Λ ( e + e - ( Λ + c + c . c . ) X ) x Λ / σ t o t a l d σ / dx Λ ( e + e - ( Λ + c + c . c ) X ) x Λ Figure 2: Same as Fig. 1 but for
OPAL experimental data at √ s = M Z . III. THEORETICAL APPROACH FOR Λ + c BARYON FF
As was mentioned in the Introduction, apart from thephenomenological approaches to determine the nonper-turbative FFs there are also some theoretical models tocompute them. In fact, it was fortunately understoodthat for heavy hadron productions these functions canbe analytically calculated by virtue of perturbative QCD(pQCD) including limited phenomenological parameters[45, 46]. The first theoretical effort to illustrate the pro-cedure of heavy hadron production was established by Bjorken [47], so that in the following, Suzuki [24], Ji andAmiri [48] have applied the pQCD approach consideringelaborate models to describe the hadronization process.Since the Suzuki model includes most of the kinemati-cal and dynamical aspects of hadroproduction process itgives us a detailed insight on the hadronization process.Especially, this model is much suitable to consider thespin effects of produced hadron or fragmenting partonwhich is absent in the phenomenological approach, seefor example [49]. It does also enable us to describe thegluon hadronization process which is not well determinedin the phenomenological approach, see Refs. [50–52]. d σ / dx Λ ( e + e - ( Λ + c + c . c . ) X )( nb ) x Λ Figure 3: The NLO contribution of c → Λ + c (red dashed line), b → Λ + c (green dot-dashed line) and g → Λ + c (black dashed-dot-dot line) in inclusive differential cross section at √ s = M Z . The total contribution of all partons (blue solid line) is alsoplotted. In this section, using the Suzuki model we focus on thefragmentation of Λ + c -baryon from charm quark for whichthe respective Feynman diagrams at leading order in α s are shown in Fig. 5. According to this model, the FFfor the production of an S-wave heavy bound state M inhadronization of an initial heavy quark Q might be putin the following general relation [24] D MQ ( z, µ ) =11 + 2 s Q X spincolor ˆ | T M | δ ( X f p f − p Q )Π f d p f , (8)where, p Q and p f are the momenta of the fragmentingquark and the final particles, respectively. The fragmen-tation parameter z is as the one introduced in the phe-nomenological approach (section II.A), i.e., z = E M /E Q which takes the values as ≤ z ≤ . Furthermore, µ stands for the initial fragmentation scale which is in or-der of fragmenting heavy quark mass and s Q refers to thefragmenting quark spin. In the above relation, the quan-tity T M is the probability amplitude for the hadron pro-duction. In the Suzuki model, this amplitude at large mo-mentum transfer is expressed in terms of the hard scat-tering amplitude T H and the process-independent distri-bution amplitude Φ M describing the nonperturbative dy- namics of bound state. In fact, the long-distance ampli-tude Φ M is, in essence, the probability amplitude for con-stituent quarks to be evolved into the final bound state.Therefore, the amplitude T M is expressed as [48, 53] T M = ˆ Π j dx j δ (1 − X j x j ) T H Φ M ( x j , Q ) , (9)where, x j = E j /E M is the energy fraction carried by theconstituent quark j of heavy bound state M .Considering the general definition (8) and the Feynmandiagrams shown in Fig. 5, for the production of S-wave Λ + c -baryon from the initial charm quark, the FF of c → Λ + c is written as D Λ + c c ( z, µ ) = 12 X s,c ˆ | T Λ | × δ ( ¯ P + s ′ + t ′ − p ) d ¯ P d s ′ d t ′ , (10)where, four-momenta are as labeled in Fig. 5, and T Λ ( p, ¯ P , s ′ , t ′ )= ˆ dx dx dx δ (1 − x − x − x ) × T H ( p, ¯ P , s ′ , t ′ , x i )Φ B ( x i , Q ) . (11) LONLONNLOKKSS20 gluonc quark b quark D Λ + ( z , Q = . G e V ) z z Figure 4: The gluon-, charm- and bottom-FFs with their uncertainties obtained at LO (red solid lines), NLO (green dashedlines) and NNLO (blue dot-dashed lines) QCD analyses of Λ + c baryon productions at µ = M Z . The NLO KKSS20 results [30](black dashed-dashed-dot curves) are also plotted. (a) (b)
Figure 5: The lowest-order Feynman diagrams contributing tothe fragmentation of charm-quark into the Λ + c ( ucd ) baryon. The advantage of above scheme is to absorb the soft be-havior of produced bound state into the hard scattering amplitude T H [53]. Ignoring the details, the distribu-tion amplitude Φ B is related to the valence wave func-tion Ψ [53]. Following Ref. [24], we adopt the infinitemomentum frame where the distribution amplitude Φ B ,with neglecting the Fermi motion of constituents, reads[54, 55] Φ B = f B δ ( x i − m i M ) , (12)where, the baryon decay constant f B is related to thenonrelativistic S-wave function Ψ(0) at the origin as f B = p /M | Ψ(0) | .In Eq. (11), the short-distance amplitude T H can becalculated perturbatively considering the Feynman dia-grams shown in Fig. 5, where a charm-quark creates aheavy baryon Λ + c along with two light antiquarks ¯ u and ¯ d . In the old-fashioned noncovariant perturbation the-ory, the hard scattering amplitude T H may be expressedas follows to keep the initial heavy quark on mass shellall the time [56] T H = g s m Λ m c m d m u C F p p ¯ P s ′ t ′ × P i =1 Γ i D , (13)where, D = ¯ P + s ′ + t ′ − p is the energy denominator, C F is the usual color factor and Γ i represents an appro-priate combination of the propagators and the spinorialparts of the amplitude. In the above equation, the am-plitudes Γ i stand for each Feynman diagrams in Fig. 5.Substituting all expressions in Eq. (10) and carrying outthe necessary integrations, we find D Λ + c c ( z, µ ) = N ˆ d s ′ d t ′ t ′ s ′ ˆ P i,j =1 Γ i ¯Γ j ¯ P p D × δ ( ¯ P + s ′ + t ′ − p ) d ¯ P , (14)where, N ∝ ( f B m Λ m c m d m u g s C F ) .In the above relation, using the well-known completenessrelations P spin u ( p )¯ u ( p ) = ( p + m ) and P spin v ( q )¯ v ( q ) =( q − m ) in the unpolarized Dirac string, one has X i,j =1 Γ i ¯Γ j = G T r [( s ′ − m d ) γ µ ( s + m d ) γ ν T µσρν ( t ′ − m u ) γ σ ( t + m u ) γ ρ ] + G T r [( t ′ − m u ) γ σ ( t + m u ) γ ρ T σµνρ ( s ′ − m d ) γ µ ( s + m d ) γ ν ]+2 G G × T r [( s ′ − m d ) γ µ ( s + m d ) γ ν T µσνρ ( t ′ − m u ) γ σ ( t + m u ) γ ρ ] , (15)where, T µσρν = ( p + m c ) γ µ ( q + m c ) γ σ ( r + m c ) γ ρ ( q + m c ) γ ν ,T σµνρ = ( p + m c ) γ σ ( q ′ + m c ) γ µ ( r + m c ) γ ν ( q ′ + m c ) γ ρ ,T µσνρ = ( p + m c ) γ µ ( q ′ + m c ) γ σ ( r + m c ) γ ν ( q + m c ) γ ρ . (16)In the above expressions, after using the Dirac algebraand the traditional trace technique the dot products offour-momenta will appear. To proceed we need to spec-ify our kinematics to determine the relevant dot prod-ucts. Considering the Feynman diagrams shown in Fig. 5,where by ignoring the Fermi motion of quark constituentsthe Λ + c baryon is replaced by its collinear constituents,the relevant four-momenta are set as p µ = [ p , p T , p L ] , t ′ µ = [ t ′ , t ′ T , t ′ L ] ,s ′ µ = [ s ′ , s ′ T , s ′ L ] , r µ = [ r , , r L ] ,t µ = [ t , , t L ] , s µ = [ s , , s L ] , ¯ P µ = [ ¯ P , , ¯ P L ] , (17)where, ¯ P L = r L + t L + s L and we also assumed that theproduced baryon moves along the ˆ z -axes (fragmentationaxes). According to the definition of fragmentation pa-rameter, i.e., z = ¯ P /p , the baryon takes a fraction z of the energy of initial heavy quark (each constituent afraction of x , x and x ) and two antiquarks take theremaining − z (each one with a fraction of x and x ).Thus, the parton energies can be expressed in terms ofthe initial heavy quark energy p , as ¯ P = zp , s = x zp , r = x zp , t = x zp ,s ′ = x (1 − z ) p , t ′ = x (1 − z ) p , (18)where, the condition x + x + x = 1 holds as well as x + x = 1 . Moreover, according to our assumption thatbaryon moves along the ˆ z -axes, the transverse momen-tum of initial quark is carrying by two antiquarks so thatin the infinite momentum frame we have s ′ T = x p T and t ′ T = x p T . With the approximation (12), we are pos-tulating that the contribution of each constituent fromthe baryon energy is proportional to its mass, namely, x i = m i /M where M = m u + m d + m c . We also assumethat x = m d /m ′ and x = m u /m ′ where m ′ = m d + m u .Regarding the kinematics introduced, the dot productsof relevant four-momenta read s ′ · t ′ = s · t = m u m d ,s · r = m c m d , t · r = m c m u ,p · s ′ = m d β , p · t ′ = m u β,p · s = m d η , p · r = m c η,p · t = m u η , s · s ′ = m d α,t · t ′ = m u α , r · t ′ = m u m c α,r · s ′ = m d m c α , s · t ′ = m d m u α (19)where, η = Mz + zm c M (1 + p T m c ) , (20) α = zm ′ M (1 − z ) (1 + p T m ′ ) + (1 − z ) Mzm ′ ,β = m c (1 − z ) m ′ (1 + p T m c ) + m ′ − z (1 + p T m ′ ) − p T m ′ . In Eq. (15), G and G are related to the denominatorof propagators as G = 1 m d m u (2 + α ) ( m d (2 + α ) − η − β ) ,G = 1 m d m u (2 + α ) ( m u (2 + α ) − η − β ) . (21)For the phase space integrations in the relation (14), onehas ˆ d ¯ P δ ( ¯ P + t ′ + s ′ − p )¯ P p D = zG ( z ) , (22)0where, G ( z ) = M − m c − m u − m d − t ′ · s ′ +2 p · t ′ +2 p · s ′ = m ′ (2 m c + β ) , so that for the remaining integrals, accord-ing to the Suzuki model and for simplicity, we replacethe transverse momentum integrations by their averagevalues, e.g., ˆ d t ′ f ( z, t ′ T ) t ′ ≈ m u f ( z, (cid:10) t ′ T (cid:11) ) , (23)where, according our assumption one has t ′ T = x p T =( m u /m ′ ) p T .Substituting all in Eq. (14), the hadronization process c → Λ + c is described by the following function D Λ + c c ( z, µ ) = N z × P i,j =1 Γ i ¯Γ j [2 m c + β ( z )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p T → h p T i , (24)where, N ∝ ( f B m Λ m c m d m u g s C F ) /m ′ but it is de-termined via ´ D Λ + c c ( z, µ ) dz = 1 (normalization condi-tion) [48].In the above relation, P i,j =1 Γ i ¯Γ j is given in (15) whichis simplified in terms of dot products of four-momentaafter using the Dirac algebra. Due to the lengthy andcumbersome expression for this term we ignore to presentanalytical result and just show our numerical analysis.Note that, in the Suzuki model the fragmentation func-tion depends on both the fragmentation parameter z andthe phenomenological parameter (cid:10) p T (cid:11) . Although, the z -dependence of FFs is not yet calculable at each desiredscale, but once they are computed at the initial fragmen-tation scale µ , their µ evolution is determined throughthe DGLAP equations [25]. In the Suzuki model, theinitial scale is the minimum value of the invariant massof the fragmenting parton. Therefore, the FF presentedin Eq. (24) should be regarded as a model for the c → Λ + c transition at the initial scale µ = m Λ + m u + m d .For our numerical analysis, we adopt the input pa-rameters as m c = 1 . GeV , m d = 4 . MeV , m u =2 . MeV , f B = 0 . GeV, and α s ( m c ) = 0 . [57]. Thecolor factor C F is calculated using the simple color linecounting rule so we applied C F = 7 / for our purpose.In Fig. 6, taking (cid:10) p T (cid:11) = 1 GeV our theoretical pre-diction for the D Λ + c c -FF at the starting scale µ is shown(dotted line). In Refs. [16, 22], it is shown that the choiceof (cid:10) p T (cid:11) = 1 GeV is an optimum value for this quantityso that any higher value of this parameter will producethe peak position even at lower-z regions. To check thevalidity of the Suzuki model, using the parameters pre-sented in Table. I we have also plotted the D Λ + c c ( z, µ ) -FF at LO (dashed line), NLO (solid line) and NNLO(dot-dashed line). As is seen, there is a considerable con-sistency between both approaches. This allows one torely on the Suzuki model to determine the heavy quarkFFs. As was mentioned previously, the Suzuki model ismuch suitable to consider the spin effects of producedhadrons or fragmenting partons which is absent in thephenomenological approach. It does also enable us to de-scribe the gluon hadronization process which is not well Suzuki's modelLONLONNLO z D c → Λ ( z , μ ) Figure 6: The D Λ + c c ( z, µ ) -FF at lowest-order in the Suzukimodel (black dotted line) taking (cid:10) p T (cid:11) = 1 GeV . This iscompared with the ones extracted through the phenomeno-logical approach at LO (blue dashed line), NLO (red solidline) and NNLO (green dot-dashed line). Here, we set µ = m Λ + m u + m d . determined in the phenomenological approach. It alsogives one a detailed insight on the hadronization processbecause includes most of the kinematical and dynamicalaspects of hadroproduction process. IV. Λ + c BARYON PRODUCTION BY TOPQUARK DECAY
In this section, as a topical application of our baryonFFs, we study the inclusive single production of Λ + c atthe CERN LHC. Generally, the Λ + c baryon may be pro-duced directly or via the decay process of heavier par-ticles, including the Higgs boson, the Z boson, and thetop quark. At the LHC, the study of energy distributionsof observed hadrons through top quark decays might beconsidered as an indirect channel to search for the prop-erties of top quarks. Since the top quark discovery bythe D and CDF experiments at Fermilab Tevatron [58],the full determination of its properties has not yet beenperformed so it has been one of the main aims in topphysics theories.In the Standard Model of particle physics, the top hasa very short life time ( τ t ≈ . × − s [59]) which ismuch shorter than the typical time to form the QCDbound states, i.e., τ QCD ≈ / Λ QCD ≈ × − s,then the top quark decays rapidly before hadronizationtakes place. Related to the Cabibbo-Kobayashi-Maskawa(CKM) mixing matrix element for which V tb ≈ [60], topquarks almost exclusively decay to bottom quarks via t → bW + so, in the following, produced bottom quarkshadronize by producing final jets. Therefore, a sugges-tion for a new channel to look for top properties at theLHC is to study the inclusive single Λ + c -baryon produc-1tion through the following process t → bW + (+ g ) → Λ + c W + + X, (25)where, X collectively represents any other final-state par-ticles. At the parton level, both the bottom quark andthe gluon may hadronize into the Λ + c -baryon so that thegluon fragmentation contributes to the real radiations atNLO.According to the factorization theorem of QCD-improvedparton model [61], the partial width of the decay process(25) differential in the scaled Λ + c -baryon energy, x Λ , isexpressed as d Γ dx Λ = X i = b,g ˆ x maxi x mini dx i x i d Γ i dx i ( µ R , µ F ) D Λ + c i (cid:18) x Λ x i , µ F (cid:19) , (26)where the factorization and the renormalization scales,i.e., µ F and µ R , are arbitrary but to avoid large loga-rithms appearing in the parton differential decay rates d Γ i /dx i , we set µ R = µ F = m t , as usual. For sim-plicity, we shall work in the top-rest frame in whichthe scaling variables are defined as x Λ = E Λ /E max b and x i = E i /E max b , where E Λ and E i stand for the ener-gies of Λ + c baryon and parton i , respectively. Here, E max b = m t (1 − ω ) / is the maximum energy of the bot-tom quark in the process (25), where ω = ( m W /m t ) . Atpresent, analytic expressions for the Wilson coefficientfunctions d Γ i /dx i are only available at NLO accuracywhich are computed in Refs. [62, 63]. Using our extracted ( b, g ) → Λ + c FFs at LO and NLO, we make our predic-tions for the energy spectrum of Λ + c -baryon producedthrough the unpolarized top quark decay. However, aconsistent analysis is presently restricted to NLO approx-imation, but we also employ the NNLO set of D Λ + c c -FFto probe the possible size of NNLO corrections.Adopting the input parameters as m t = 173 GeV and m W = 80 . GeV, in Fig. 7 we studied the energy dis-tribution of Λ + c -baryon in unpolarized top decays at LO(dotted line), NLO (dashed line) and NNLO (solid line).As is seen, switching from the LO Λ + c -baryon FF set tothe NLO one slightly smoothens the theoretical predic-tion, decreasing it in the peak region and the tail regionthereunder. In Fig. 7, the results for d Γ /dx Λ are alsocompared to the evaluation with the KKSS20 Λ + c -baryonFF set [30]. As is seen, there is a good consistency be-tween both results. In comparison with the KKSS20 spec-trum, the peak position of our results is shifted towardslarger values of x Λ .At the LHC, the study of energy distribution of Λ + c -baryon may be also considered as a new window towardssearches on new physics. Practically, for the energy dis-tribution of produced hadrons any considerable devia-tion from the SM predictions can be assigned to the newphysics. For example, it would be a signal for the exis-tence of charged Higgs bosons produced from t → Λ + c H + in the theories beyond the SM [64–68]. Meanwhile, thestudy of x Λ -distribution in the decay mode (25) will en- NNLO (Ours): t → Λ + +JetsNLO (Ours): t → Λ + +JetsLO (Ours): t → Λ + +JetsNLO (KKSS20): t → Λ + +Jetsx Λ d Γ / dx Λ [ G e V ] Figure 7: The LO (green dotted line), NLO (red dashedline) and NNLO (black solid line) predictions of d Γ( t → Λ + c + Jets ) /dx Λ evaluated with our Λ + c -baryon FF sets. Forcomparison, the evaluation with the NLO KKSS20 Λ + c -FF set[29] is also included (blue dot-dashed line). Here, we set thescale as µ = m t . able us to deepen our understanding of the nonperturba-tive aspects of baryon formation by hadronization. More-over, through studying these distributions the b/g → Λ + c FFs can be also constrained event further.
V. SUMMARY AND CONCLUSIONS
Through this work, we determined the nonperturba-tive unpolarized FFs for the charmed baryon Λ + c in twovarious approaches; phenomenological analysis and the-oretical approach based on the Suzuki model. Employ-ing the theoretical model we computed the Λ + c -FF atlowest-order of perturbative QCD whereas using the phe-nomenological approach we determined the Λ + c -FFs bothat LO, NLO and, for the first time, at NNLO accuracy inthe ZM-VFN scheme by fitting to all available data of in-clusive single Λ + c -baryon production in e + e − annihilationfrom OPAL and
Belle
Collaborations [27, 28]. A compari-son between both approaches showed a good consistencybetween results. Note that, the theoretical frameworkprovided by the ZM-VFN scheme is quite appropriate forour data analysis because the characteristic energy scalesof annihilation process, i.e., M Z , greatly exceeds the c-and b-quark masses, which could thus be neglected. Forour data analysis, we adopted the same functional formfor the parameterization of charm and bottom FFs withthree free parameters, see Eqs. (6). From Fig. 2, it isseen that in the lowest-order approximation the behaviorof theoretical cross section is not acceptable at low- x Λ region but it is reasonably improved when higher order2radiative corrections are considered. Through the follow-ing aspects, our analysis on the Λ + c -baryon FFs improvesa similar analysis in previous works [29, 30]. Firstly, weincreased the precision of calculation to NNLO, howeverdue to few numbers of experimental data for Λ + c -baryonproduction our results showed that the effect of this cor-rection is not considerable. Secondly, we did an accurateestimation of the experimental uncertainties in the Λ + c -FFs using the Hessian approach. The uncertainties bandsof FFs as well as corresponding observables show that theNNLO radiative corrections affect the error band and de-crease them considerably. Meanwhile, we have compared,for the first time, the analytical result obtained for the D Λ + c c ( z, µ ) -FF through the Suzuki model with the oneextracted via the phenomenological analysis. A goodconsistency between both approaches ensure the Suzukimodel, see Fig. 6. This model is suitable to consider thespin effect of produced hadron or/and initial parton onthe corresponding FFs, a subject absent in data analysisapproach. On the other hand, as is well-known, the gluonFF’s play a significant role in hadroproduction but theyare only feebly constrained by e + e − data. But, throughthe Suzuki model it would be possible to determine themanalytically, see Refs. [18, 23, 49].As a topical application of our obtained FFs, we used the LO, NLO and NNLO FFs to make our theoretical pre-dictions for the scaled-energy distributions of Λ + c -baryoninclusively produced in unpolarized top decays. Thischannel is proposed for independent determination of Λ + c -baryon FFs which provides a unique chance to test theiruniversality and DGLAP scaling violations; two impor-tant pillars of the QCD-improved parton model. Further-more, this study provides a new window towards searcheson new physics.For theoretical approach, one can think of other pos-sible improvements including the Fermi motion of con-stituents. This is done by considering the real aspectsof the valence wave function of baryon [53, 69], etc. Re-lated to the phenomenological approaches, improvementsdue to the inclusion of finite quark masses and the re-summation of soft-gluon logarithms would be effective.These effects extend the validity of analysis towards smalland large values of x Λ , respectively. In this regards, thegeneral-mass variable-flavor-number scheme (ZM-VFNS)[70, 71] where the charm- and bottom-quark masses arepreserved from the beginning provides a consistent andnatural finite mass generalization of the ZM-VFNS on thebasis of the MS factorization scheme [33]. The implemen-tation of these improvements reaches beyond the scopeof our present analysis and is left for future researches. [1] E. Braaten and T. C. Yuan, Phys. Rev. Lett. , 1673(1993).[2] M. Salajegheh, S. M. Moosavi Nejad, M. Nejad, H. Khan-pour and S. Atashbar Tehrani, Phys. Rev. C (2018)no.5, 055201.[3] B. A. Kniehl, G. Kramer, I. Schienbein and H. Spies-berger, Eur. Phys. J. C , 2082 (2012).[4] J. Binnewies, B. A. Kniehl and G. Kramer, Phys. Rev.D , 034016 (1998).[5] B. A. Kniehl, G. Kramer, I. Schienbein and H. Spies-berger, Phys. Rev. D , 014011 (2008).[6] M. Salajegheh, S. M. Moosavi Nejad and M. Delpasand,Phys. Rev. D (2019) no.11, 114031.[7] S. M. Moosavi Nejad, M. Soleymaninia and A. Mak-toubian, Eur. Phys. J. A (2016) no.10, 316.[8] M. Soleymaninia, H. Khanpour and S. M. Moosavi Ne-jad, Phys. Rev. D , no. 7, 074014 (2018).[9] M. Salajegheh, S. M. Moosavi Nejad, H. Khanpour,B. A. Kniehl and M. Soleymaninia, Phys. Rev. D (2019) no.11, 114001[10] M. Salajegheh, S. M. Moosavi Nejad, M. Soleymaninia,H. Khanpour and S. Atashbar Tehrani, Eur. Phys. J. C (2019) no.12, 999.[11] M. Soleymaninia, M. Goharipour and H. Khanpour,Phys. Rev. D , no. 7, 074002 (2018).[12] M. Soleymaninia, M. Goharipour and H. Khanpour,Phys. Rev. D , no. 3, 034024 (2019).[13] A. Mohamaditabar, F. Taghavi-Shahri, H. Khanpour andM. Soleymaninia, Eur. Phys. J. A (2019) no.10, 185.[14] D. P. Anderle, F. Ringer and M. Stratmann, Phys. Rev.D (2015) no.11, 114017.[15] V. Bertone et al. [NNPDF Collaboration], Eur. Phys. J. C (2017) no.8, 516.[16] M. A. Gomshi Nobary, J. Phys. G (1994) 65.[17] S. M. Moosavi Nejad, Phys. Rev. D (2017) no.11,114021.[18] Mahdi. Delpasand and S. Mohammad. Moosavi Nejad,Phys. Rev. D (2019), 114028.[19] S. M. Moosavi Nejad and A. Armat, Eur. Phys. J. Plus (2013) 121.[20] S. M. Moosavi Nejad and E. Tajik, Eur. Phys. J. A (2018) no.10, 174.[21] S. Mohammad Moosavi Nejad and A. Armat, Eur. Phys.J. A (2018) no.7, 118.[22] S. M. Moosavi Nejad and P. Sartipi Yarahmadi, Eur.Phys. J. A (2016) no.10, 315.[23] S. M. Moosavi Nejad and D. Mahdi, Int. J. Mod. Phys.A (2015) 1550179.[24] M. Suzuki, Phys. Lett. B (1977) 139.[25] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. (1972) 438 [Yad. Fiz. (1972) 781]; G. Altarelli andG. Parisi, “Asymptotic Freedom in Parton Language,”Nucl. Phys. B (1977) 298.[26] G. Alexander et al. [OPAL Collaboration], Z. Phys. C , 1 (1996).[27] K. Ackerstaff et al. [OPAL Collaboration], Eur. Phys. J.C , 439 (1998).[28] M. Niiyama et al. [Belle], Phys. Rev. D ,no.7, 072005 (2018) doi:10.1103/PhysRevD.97.072005[arXiv:1706.06791 [hep-ex]].[29] B. A. Kniehl and G. Kramer, Phys. Rev. D , 037502(2006).[30] B. Kniehl, G. Kramer, I. Schienbein and H. Spiesberger,[arXiv:2004.04213 [hep-ph]]. [31] T. Kneesch, B. Kniehl, G. Kramer andI. Schienbein, Nucl. Phys. B , 34-59 (2008)doi:10.1016/j.nuclphysb.2008.02.015 [arXiv:0712.0481[hep-ph]].[32] J. Pumplin, D. R. Stump and W. K. Tung, Phys. Rev.D , 014011 (2001).[33] J. C. Collins, Phys. Rev. D , 094002 (1998).[34] S. G. Gorishnii, A. L. Kataev and S. A. Larin, Phys. Lett.B , 144 (1991).[35] K. G. Chetyrkin, A. L. Kataev and F. V. Tkachov, Phys.Lett. , 277 (1979).[36] A. A. Almasy, S. Moch and A. Vogt, Nucl. Phys. B ,133 (2012).[37] S. Moch and A. Vogt, Phys. Lett. B , 290 (2008).[38] A. Mitov, S. Moch and A. Vogt, Phys. Lett. B , 61(2006).[39] M. G. Bowler, Z. Phys. C , 169 (1981).[40] V. Bertone, S. Carrazza and J. Rojo, Comput. Phys.Commun. (2014) 1647.[41] F. James and M. Roos, Comput. Phys. Commun. , 343(1975).[42] A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt,Eur. Phys. J. C , 189 (2009).[43] D. Stump, J. Pumplin, R. Brock, D. Casey, J. Huston,J. Kalk, H. L. Lai and W. K. Tung, Phys. Rev. D ,014012 (2001).[44] C. Patrignani et al. [Particle Data Group], Chin. Phys.C (2016) 100001.[45] J. P. Ma, Nucl. Phys. B , 329 (1997).[46] E. Braaten, K. m. Cheung and T. C. Yuan, Phys. Rev.D , 4230 (1993).[47] J. D. Bjorken, Phys. Rev. D (1978) 171.[48] F. Amiri and C. -R. Ji, Phys. Lett. B (1987) 593.[49] S. M. Moosavi Nejad and M. Delpasand, Int. J. Mod.Phys. A (2015) no.32, 1550179.[50] S. M. Moosavi Nejad, Eur. Phys. J. Plus (2015) no.7,136.[51] M. Delpasand and S. M. Moosavi Nejad, Eur. Phys. J. A (2020) no.2, 56. [52] M. Delpasand and S. M. Moosavi Nejad, Phys. Rev. D (2019) no.11, 114028.[53] S. J. Brodsky and C. R. Ji, Phys. Rev. Lett. (1985)2257.[54] S. M. Moosavi Nejad, Eur. Phys. J. Plus (2018) no.1,25.[55] S. M. Moosavi Nejad, Eur. Phys. J. A (2016) no.5,127.[56] M. Suzuki, Phys. Rev. D (1986) 676.[57] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D , no. 3, 030001 (2018).[58] Tevatron Electroweak Working Group [CDF and D0 Col-laborations], “Combination of CDF and D0 Results onthe Mass of the Top Quark,” arXiv:0903.2503 [hep-ex].[59] K. G. Chetyrkin, R. Harlander, T. Seidensticker andM. Steinhauser, Phys. Rev. D , 114015 (1999).[60] N. Cabibbo, Phys. Rev. Lett. , 531 (1963);M. Kobayashi and T. Maskawa, Prog. Theor. Phys. ,652 (1973).[61] P. V. Pobylitsa, Phys. Rev. D , 094002 (2002).[62] S. M. M. Nejad, Phys. Rev. D (2013) no.9, 094011;[63] B. A. Kniehl, G. Kramer and S. M. Moosavi Nejad, Nucl.Phys. B (2012) 720;[64] S. M. Moosavi Nejad, Eur. Phys. J. C (2012) 2224;[65] S. M. Moosavi Nejad, S. Abbaspour and R. Farashahian,Phys. Rev. D (2019) no.9, 095012.[66] S. M. Moosavi Nejad and S. Abbaspour, Nucl. Phys. B (2017) 86.[67] S. M. Moosavi Nejad and S. Abbaspour, JHEP (2017) 051.[68] S. M. Moosavi Nejad, Phys. Rev. D (2012) 054010.[69] S. M. Moosavi Nejad, M. Roknabady and M. Delpasand,Nucl. Phys. B (2020) 115036.[70] S. M. Moosavi Nejad and M. Balali, Eur. Phys. J. C (2016) no.3, 173;[71] S. Abbaspour and S. M. Moosavi Nejad, Nucl. Phys. B930