Determination of QCD running coupling constant in NNLO approximation, using complete RG-improvment

1. MENU 2004 International Symposium August 29 - Sept 4, 2004•Beijing,China Determination of QCD running coupling constant in NNLO approximation, using complete RG-improvment A Talk Given By Abolfazl Mirjalili

2. Introduction • Bernstein averages • Moments in GORGI approach • Phenomenological results • Discussions and Conclusion

3. We employed Complete Renormalization Group Improvement (CORGI) approach for moments of structure function to extract the transmuted QCD parameter from CCFR experimental data for scattering. Introduction The recent measurements of the CCFR collaboration provide the most precise up to now experimental results for the structure functions of the deep inelastic scattering (DIS) of neutrino and antineutrinos on nucleons . We reformulated the moment of SF in CORGI approach and by using the Bernstein polynomials as weight functions, reconstructed the SF in terms of moments and by fitting our theoretical formalism with experimental Bernstein averages, we did a phenomenological task and extract .

4. (1) where is the finite integral Bernstein averages Mellin moments for the non singlet (NS) structure function In a theoretical view using and functions, the factorized form for the moments can be written as

5. Heredue to existence, for each value of Q2, of only measurements of xF3 (x, Q2) for a limited range of x, one does not compare with experiment the values of themselves insteadly we compare theoretical predictions and experimental results for the Bernstein averages which is defined by stands for a(µ) and a for a(M). M and µ are factorization and renormalisation scales. (2) where Pnk(x) are modified Bernstein polynomials, (3)

6. and are normalized to unity, .Therefore the integral (3) represents an average of the function F3(x). Using the binomial expansion in (3), it follows that the averages of F3 with Pnk(x) as weight functions, can be obtained in terms of odd moments, (4) Using Eq.1 then (5) This then evaluates theoretically and the results compare to the experimentally evaluated averages which are obtained by substituting experimental structure function F3(exp)(x,Q2) in Eq.2.

7. Moments in CORGI approach For one single scale case, a perturbative series for observable R can be represented by R(i)= +X2(n) 3+ X3(n) 4+… Xi(n) i+1 , (6) Where Xi(n)are representing the RG-unpredictable part of ri(n)whichare Q-independent and RS-invariant. Here = (0,0,0,..)is the coupling in this scheme and it satisfies ( 7) ,

8. In fact the solution of this equation can be written in closed form in terms of the Lambert-W function, defined implicitly by (8) In the moment problem, with the integral vanishes , so that finally the sum of all RG-predictable terms for the moment problem at NiLOwill be

9. Substituting for a0 in terms of the Lambert- W function using Eq.(8) we then obtain where X2(n) is expressed by X2(n) is a FRS-invariant and can be computed in any scheme, for instance, in MS scheme. A(n) and LMS should be fitted simultaneously to the data for Mn(Q2) and for our case in fact Fnk(Q2) as Bernstein averages

10. Phenomenological results We only include an average with pnk if we have experimental points covering the whole range is the average quantity of and is the spread of . ]. [ By choosing n,k , we manage to adjust the region where the average is peaked to that in which we have experimental data and considering the limitation that the are only known for the odd numbers n= 1, 3,..., 13 then we can use only the 10 averages :

11. F21(exp)(Q2), F31(exp)(Q2), F32(exp)(Q2), F41(exp)(Q2), F42(exp)(Q2), F51(exp)(Q2), F52(exp)(Q2), F61(exp)(Q2), F62(exp)(Q2), F62(exp)(Q2) . We use for each value of Q2, a phenomenological expression for xF3 as Using Eq.(5) and substituting the relating values for n and k, we will get the following expressions for the chosen Fn k in terms of odd moments M3(Q2), … , M13(Q2) , (12)

13. Using these expressions for Fnk , we can do fitting and get the numerical values for seven parameters A(n) and LMS . LMis QCD transmuted parameter in CORGI approach which can be converted to LMS in MS scheme by (13) In Figure.1 we plot the fit result of xF3 with Bernstein average using Eqs.(13) where we are in a region with 5 active flavor and have seven fit parameters, LMS and six A(n)'s. The errors of Fnk(Q2) has obtained by varying xF3(exp) (x,Q2) within its statistic and systematic errors bars.

15. (14) By defining a c2 which covers all data experimental data points of Figure.1 and minimizing it to get the goodness of fit, we obtained the following value for LMS, the minimum c2 is acceptable (c2 /d.o.f = 1.2905/43) and estimating an error by allowing c2 within 1 of the minimum gives, which is correspond to the following value of the coupling at the Z-mass, (15)

16. Table1 : Numerical values of fitting parameters, resulted from Figure1.

17. Conclusions • Here we have focussed on extracting LMS from xF3 CCFR experimental data by employing the CORGI approach, where we could further resume to all-orders, the complete set of ultraviolet logarithms to avoid any dependence on arbitrary renormalization and factorization scale. • Since for a given value of Q2, only a limited number of experimental points, covering a partial range of values of x, are available, we can not use the moments directly. Instead we compare theoretical predictions and experimental results for the Bernstein averages. • In doing the fit of experimental Bernstein averages and their theoretical predictions, we had to find seven parameters, LMS and A(3), ..., A(13). We could do this fitting while in all part of calculations we used from the closed analytical form of Lambert-W function. By allowing the c2 within 1 of the minimum while we had seven parameters, we could obtain the error of LMS and other six fitting parameters.