The 7th International Conference on Hyperons, Charm And Beauty Hadrons

1. QCD analysis of heavy quarks production in hadronic collisions Abolfazl Mirjalili Phys Dept., Yazd Univ. and IPM-Iran The 7th International Conference on Hyperons, Charm And Beauty Hadrons 2nd to 8th July 2006 University of Lancaster, England

2. Introduction One of the clean test of perturbative QCD is heavy quark production in hard collisions of hadrons, leptons and photons. Recent data on b-b Bar production in p-p bar collisions at the Tevatron, γp collision in HERA and γγ collisions in LEP2 indicates that they lie systematically by a factor of about 2-4 above the median of current theoretical calculations. The QCD calculations for above cases depend on a number of inputs for instance αs, parton distribution functions(PDF) of colliding hadrons or photons, … and finally the choice of renonrmalization (RS) and factorization (FS) scales µ and M. We are going to investigate in detail the dependence of existing fixed order (LO and NLO)QCD calculation of the total cross section in p-p Bar collisions on the choice of the renormalization and factorization scales and to indicate if we do a resummation on all ultraviolet terms which involve these two scales then they will be avoided from our calculations and finally we expect to have a theoretical result which are in more consistence with available experimental data.

3. Basic definitions The basic quantity of perturbative QCD calculations, the renormalized color coupling αs(µ), depends on the renormalization scale µ which is governed by where for nf massless quarks β0=11-(2/3)nf and β1=102-(38/3)nf. The solutions of above equation depend beside µ also on renormalization scheme (RS). At NLO calculation this RS can be specified, for instance, via the parameter ΛRs corresponding to the renormalization scale for which αs diverges. At two loop calculations the coupling α(µ) is then given as the solution of the equation

4. For hadrons the factorization scale dependence of PDF (M) is determined by the system of evolution equations for quark singlet, non-singlet and gluon distribution functions where The cross symbol in above indicates the convolution

5. The spiliting functions admit expansion in powers of αs(M) where Pij(0)(0) are uniqe, where as all higher splitting functions Pkl(j), j>0 depend On The choice of the factorization scheme (FS). Conversely, they can be taken as defining the FS. RS and FS dependence of coefficients For a single case of a dimensional observable R(Q) with perterbative series The RS can be labeled by the non-universal coefficients of the beta-function and ΛRS.

6. Self-consistency of perturbation theory that is the derivative of N-th order approximant R(n)With respect to ln(µ) is of higher order than the approximant itself, will yields Is a renormalization scale and scheme invariant. For other coefficients we will arrive first on following partial derivatives for r2 and for r3,r4,… we will have

7. on integration one finds and finally General structure is as follows Where Xn are Q-independent and RS-invariantandare unknown unless a complete NnLO calculation has been performed.

8. Given a NLO calculation of r1, the sum of all-orders of the RG-predictable term simplifies to a geometric progression : which is completely independent of renormalization µ scale Complete RG Improvement From pervious results forr1 , r2, … At NLO calculation r1 is known but X2,X3,…are unknown. Thus the complete subset of known terms in above at NLO is It is RS-invariant. Choose r1=0, c2=c3=…=cn=0 we obtain a0=a.

9. At NNLO calculation X2 is unknown. Further infinite subset of terms are known and And can be resummed to all orders. Finally we will arrive at where a0=a(0,0,0, …) is the coupling in this scheme and satisfies The solution of this transcendental equation can be written in closed form in terms of the Lambert-W function where Lambert-W function is defined by

10. Other approaches in investigating total cress section of heavy pair production In addition to previous approach to deal perturbative QCD we can have the following approaches: I) Principle of Minimal Sensitivity (PMS)II (The effective charge (EC) PMS: Here the emphasize is put on the stability of the results with respect to variable µ. In absence of information on higher perturbative terms, the PMS approach is natural As it selects the point µ where the truncated perturbative expansion is most stable and has thus locally the property possessed by the all other result globally. EC: It is based on the criterion of apparent convergent of perturbation expansions and The renormalization scale µ is chosen in such a way that all higher order contributions Vanish i.e. demanding R(n)=R(k) for all k. Complete RG Improvement: We explained about this approach in more details in last part. Just remind that this approach suggest that ay any given order of Feynman diagram calculation for QCD observable, all renormalization group (RG) predictable terms should be resummed to all orders. This approach serves to separate the perturbation series into infinite subset of terms which when summed are renormalization scheme (RS) invariant. Crucially all ultraviolet logarithms involving the dimensional parameter Q, on which the observable depends, are resummed thereby building the correct Q-dependence. Due to these properties of this approach in which RS invariant quantities we expect to have more consistent result in comparison with experimental data, specially for total cross section of heavy quark pair production.

11. General form of σtot(q-q bar) According to the factorization theorem the total cross section of heavy quark pair Production in p-p Bar collision at the center of mass energy √s has the form where partonic cross section σij as well asPDF of the beam particles depend on the factorization scale M. The above expression is basically of non perturbative nature. Fixed order perturbation theory enters if we we insert in above equation the solution of evolution equations with The splitting function Pij and calculate σij (S,M) as power expansion in the coupling αs(µ), taking at the renormalization scale µ, which as emphasiazed is in general unrelated to the factorization scale M. Summed to all orders of αs , the r,h.s of above equation will be independent of µ whereas all order sum of this equation depends on the factorization scale M.

12. The latter dependence is cancelled by that of PDF provided the splitting function Pij in the evolution equation re taken to all orders. At NLO, i.e. taking into accont the first two terms in related perturbative series For σij and Pij we get where here the sum is over nf quarks and antiquarks and the relation between PDF of protons and antiprotons was taken into account. In dealing to Complete RG improvement approach, what we need to do is first to extend the perturbative expressions to any desired order and then to calculate the partial derivatives of σij (x,M)expression with respect to µ and M, using the Beta function and evolution functions of PDF and finally demanding consistency principle. This help us to find the strict dependence of coefficient function (σij,...) to µ and M and in general case to other parameters. At this step The structure of FRS invariants X2, X3,… will be obtained. Final step will be to fo to case where all coefficient of β and related function in σ’s are equal zero where ususal coupling constant should be converted to a0 which can be expressed in terms of Lambert-W.

13. By knowing the expresion X2 , X3, … the comparison of calculated result at each Specified order to experimental data for σbbBar(total) can be done and compared to other theoretical models like PMS and EC models.

14. Total X-section (nb) CORGI approach Standard approach Pt(min)