- Email: [email protected]
The Role of Logarithmic Expansions for Nonsinglet QCD Analysis of xF3
Nuclear Physics B (Proc. Suppl.) 174 (2007) 27–30 www.elsevierphysics.com
The Role of Logarithmic Expansions for Nonsinglet QCD Analysis of xF3 Ali N. Khorramiana∗ , S. Atashbar Tehranib and A. Mirjalilic a
Physics Department, Semnan University, Semnan, Iran and Institute for Studies in Theoretical Physics and Mathematics , P.O.Box 19395-5531, Tehran, Iran
b
Physics Department, Persian Gulf University, Boushehr, Iran and Institute for Studies in Theoretical Physics and Mathematics , P.O.Box 19395-5531, Tehran, Iran
c
Physics Department, Yazd University, Yazd, Iran and Institute for Studies in Theoretical Physics and Mathematics , P.O.Box 19395-5531, Tehran, Iran In this paper the analysis of certain logarithmic expansions, which developed for precision studies of the evolution of the QCD parton distributions (pdf) at the Large Hadron Collider, is used. The xF3 data of the CCFR collaboration is used to parameterize the non-singlet parton distributions in this order. In the fitting procedure, Bernstein polynomial method is employed. The results of valence quark distributions in the NLO are compered with exact and truncated solutions for non-singlet case.
1. Introduction Precision studies of some hadronic processes in the perturbative regime are going to be very important in order to confirm the validity of the mechanism of mass generation in the Standard Model at the new collider, the LHC. Studies of radiative corrections for specific processes have been performed by various groups, at a level of accuracy which has reached the next-to-next-toleading order (NNLO) in αs , the QCD coupling constant. The quantification of the impact of these corrections requires the determination of the hard scattering of the partonic cross sections up to order αs3 , with the matrix of the anomalous dimensions of the DGLAP kernels determined at the same perturbative order. Recently Cafarella et al.[1] presented the higher order logarithmic expansions and exact solutions of the DGLAP equations. Mellin methods have been the most popular and have been implemented up to NLO and also at NNLO [2]. In this work we are going to use the NLO logarithmic expansions and exact solutions of the DGLAP equations from n-Space. For parametrization of the non-singlet parton distri∗ e-mail:
[email protected]
0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.08.080
bution functions we need to use the CCFR experimental data [3]. In the fitting procedure, Bernstein polynomial method [4–6] is employed. 2. QCD Background Before we start our analysis it is convenient to introduce the 3-loop evolution of the coupling via its β-function β(αs ) ≡
∂αs (Q2 ) , ∂ log Q2
(1)
and its three-loop expansion is β(αs ) = −
β0 2 β1 3 α − α + O(αs4 ), 4π s 16π 2 s
(2)
where β0
=
β1
=
11 4 NC − Tf , 3 3 34 2 10 N − NC nf − 2CF nf , 3 C 3
(3)
are the coefficients of the beta function. We have set NC = 3,
CF =
Tf = TR nf =
1 nf , 2
NC2 − 1 4 = , 2NC 3 (4)
28
A.N. Khorramian et al. / Nuclear Physics B (Proc. Suppl.) 174 (2007) 27–30
where NC is the number of colors, nf is the number of active flavors, that is fixed by the number of quarks with mq ≤ Q. One can obtain either an exact or an accurate (truncated) solution of this equation. An exact solution includes higher order effects in αs , while a truncated solution retains contributions only up to a given (fixed) order in a certain expansion parameter. Truncated solutions instead can be obtained quite easily, for instance expanding in terms of the logarithm of a specific scale (Λ) 4π 1 β1 log LΛ +O 1− 2 , (5) αs (Q2 ) = β0 LΛ β0 LΛ L3Λ
can be rewritten in the form (αs − α0 ) b1 (0) (1) P −P f (N, αs ) = exp πβ0 2
where
f (N, αs ) =
LΛ = log
Q2 , Λ2M S
(6)
×f LO (N, αs ),
(9)
where f LO (N, αs ) is given by
f
LO
αs (N, αs ) = α0
− 2Pβ(0) 0
f (N, α0 ).
(10)
Expanding (9) to first order around the LO solution we obtain
f LO (N, αs ) × (αs − α0 ) b1 (0) (1) P −P , 1+ πβ0 2 (11)
(n )
and where ΛM fS is unknown value and we will extract from experimental data.
which is the expression of the 1-st truncated solution, accurate at order αs . On the other hand by using the logarithmic 3. Truncated and exact non-singlet soluexpansion in the NLO DGLAP equation from [1], tion we have α
The NLO non-singlet truncated DGLAP equas − β2 P (0) (N ) log α 0 0 f (N, αs ) = e tion in moment space can be written as 4 (1) 2 (0) P (N ) − P (N ) × exp β0 β1 2 (0) ∂f (N, αs ) P (N )+ = − 4πβ0 + αs β1 ∂αs β0 αs log f (N, α0 ), 4πβ0 + α0 β1 αs (Q2 ) b 1 P (1) − P (0) f (N, αs ), (12) 2π 2 (7) According to the reported results in [1] the nonsinglet truncated and exact solution at NNLO in which has the solution Mellin space are also accessible.
− 2Pβ(0) 0 αs f (N, αs ) = × α0 (αs − α0 ) b1 (0) (1) P −P exp × πβ0 2 f (N, α0 ), (8) here b1 = β1 /β0 , αs ≡ α(Q2 ) and α0 ≡ α(Q20 ). Notice that the solution of above equation contains as a factor the LO solution and therefore
4. Non-singlet structure function Let us define the Mellin moments for the νN structure function xF3 (x, Q2 ): 1 2 M(N, Q ) = xN −1 F3 (x, Q2 )dx (13) 0
The solution of the renormalization group equation for non-singlet structure function xF3 can be presented in the following form [7]:
A.N. Khorramian et al. / Nuclear Physics B (Proc. Suppl.) 174 (2007) 27–30
M(N, Q2 ) M(N, Q20 )
= exp −
As (Q2 ) As (Q20 )
(N )
γN S (x) dx β(x)
(N )
×
CN S (As (Q2 )) (N )
CN S (As (Q20 ))
,
(14)
where M(N, Q20 ) is a phenomenological quantity related to the factorization scale dependent factor. The solution of the two loops evolution equation from Eq. (14), is as follows
M(N, Q2 ) = 1 + C (1) As (Q2 ) f (N, Q2 ) . (15) We can now insert the truncated solution and also exact solution from Eqs. (11,12) in above equation. We should notice that f (N, Q20 ) is the valence quark compositions as f (N, Q20 ) = uv (N, Q20 ) + dv (N, Q20 ) .
(16)
To start the parameterizations of the above mentioned valence quark compositions at the input scale of Q20 = 1 we supposed the functional form as √ xuv (x, Q20 ) = Nu xa (1 − x)b (1 + c x + dx) , (17) xdv (x, Q20 ) =
Nd (1 − x)e xuv . Nu
(18)
The motivation for choosing this functional form is, the xa term controls the low-x behavior parton densities, and (1 − x)b,e large values of x. The remaining polynomial factor accounts for additional medium-x values. Normalization constants Nu and Nd are fixed by 1 uv (x)dx = 2 , (19)
29
Using the valence quark distribution functions, the moments of uv (x, Q20 ) and dv (x, Q20 ) distributions can be easily calculated. Now by inserting the Mellin moments of uv and dv valence quark in the Eq. (16), the function of f (N, Q20 ) involves some unknown parameters. 5. Reconstruction of the structure function from moments Although it is relatively easy to compute the nth moment from the structure functions, the inverse process is not obvious. To do this inversion, we adopt a mathematically rigorous and easy method [8] to invert the moments and retrieve the structure functions. The method is based on the fact that for a given value of Q2 , only a limited number of experimental points, covering a partial range of values of x are available. The method devised to deal with this situation is to take averages of the structure function weighted by suitable polynomials. These are the Bernstein polynomials which are defined by Bn,k (x) =
Γ(n + 2) xk (1−x)n−k .(21) Γ(k + 1)Γ(n − k + 1)
Using the binomial expansion, the above equation can be written as Bn,k (x) =
n−k Γ(n + 2) (−1)l xk+l . (22) Γ(k + 1) l!(n − k − l)! l=0
We can now compare the theoretical predictions with the experimental results for the Bernstein averages, which are defined by 1 Fn,k (Q2 )≡ dxBn,k (x)F3 (x, Q2 ) . (23) 0
0
0
1
dv (x)dx = 1 ,
(20)
the above normalizations are very effective to control unknown parameters in Eqs. (17,18) via fitNf =4 ting procedure. The five parameters with ΛQCD will be extracted by using the Bernstain polynomials approach.
Therefore, the integral Eq. (23) represents an average of the function F3 (x, Q2 ) in the region xn,k + 12 Δxn,k . By a suitable x ¯n,k − 12 Δxn,k ≤x≤¯ choice of n, k we manage to adjust the region in where the average is around values which we have experimental data [3]. Substituting Eq. (22) in Eq. (23), it follows that the averages of F3 with Bn,k (x) as weight functions can be obtained in terms of odd and even
30
A.N. Khorramian et al. / Nuclear Physics B (Proc. Suppl.) 174 (2007) 27–30
moments,
n−k l=0
(24) We can only include a Bernstein average, Fn,k , if we have experimental points covering the whole ¯n,k + 12 Δxn,k ] [9]. This range [¯ xn,k − 12 Δxn,k , x means that with the available experimental data we can only use the following 28 averages. The unknown parameters according to Nf . Thus, Eqs. (17,18) will be a, b, c, d, e and ΛQCD there are 6 parameters for each order to be simultaneously fitted to the experimental Fn,k (Q2 ) averages. The minimum χ2 /ndf values for the NLO truncated and exact solutions are 0.624 (4) and 0.519 respectively. Also the value of ΛQCD for the NLO truncated and exact solutions are 259 M eV and 268 M eV respectively. In Fig.1 we plotted the ratio of f (N, αs )/f (N, α0 ) from Eqs. (11,12) as a function of N at Q2 = 20 GeV 2 . The parton densities xuv and xdv at the input scale Q20 = 1.0 GeV 2 by using exact solution(solid line) which compared to results obtained from truncated solution(dashed line) is also presented in Fig.2.
Tru. Exa.
f(N, αs )/f(N, α0 )
0.5
0.4
2
2
Q =20 GeV
0.3
0.2 0
10
20
N
30
0.4
0.6
(−1)l M((k + l) + 1, Q2 ) . l!(n − k − l)!
0.6
0.5 Tru. Exa.
0.7
40
Figure 1. The ratio of f (N, αs )/f (N, α0 ) as a function of N at Q2 = 20 GeV 2 .
0.5
Q20=1 GeV2
0.3
xdv
=
xuv
Fn,k
0.8
(n − k)!Γ(n + 2) × Γ(k + 1)Γ(n − k + 1)
0.4
0.2
0.3 0.2
0.1 0.1 0 -4 10
10-3
10-2
x
10-1
10
0
0 -4 10
10-3
10-2
x
10-1
100
Figure 2. The parton densities xuv and xdv at the input scale Q20 = 1.0 GeV 2 by using exact solution(solid line) compared to results obtained from truncated solution(dashed line).
6. Acknowledgments A.N.K acknowledge the Semnan university for the financial support of this project. REFERENCES 1. A. Cafarella, C. Coriano and M. Guzzi, Nucl. Phys. B 748 (2006) 253. 2. A. Vogt Comput. Phys. Commun. 170 65, (2005). 3. W. g. Seligman et al., Phys. Rev. Lett. 79, 1213 (1997). 4. Ali N. Khorramian, A. Mirjalili, S. Atashbar Tehrani, JHEP 0410 (2004)062, hepph/0411390. 5. S. Atashbar Tehrani, Ali N. Khorramian and A. Mirjalili, Commun. Theor. Phys. 43(2005) 1087. 6. Ali N. Khorramian, A. Mirjalili, S. Atashbar Tehrani, Int. J. Mod. Phys.A 20(2005)1923. 7. A. L. Kataev, G. Parente and A. V. Sidorov, Nucl. Phys. B 573, (2000) 405. 8. F. J. Yndurain, Phys. Lett. B74 (1978) 68. 9. J. Santiago and F. J. Yndurain, Nucl. Phys. B563 (1999) 45; J. Santiago and F. J. Yndurain, Nucl. Phys. B611 (2001) 447.
The Role of Logarithmic Expansions for Nonsinglet QCD Analysis of xF3 Ali N. Khorramiana∗ , S. Atashbar Tehranib and A. Mirjalilic a
Physics Department, Semnan University, Semnan, Iran and Institute for Studies in Theoretical Physics and Mathematics , P.O.Box 19395-5531, Tehran, Iran
b
Physics Department, Persian Gulf University, Boushehr, Iran and Institute for Studies in Theoretical Physics and Mathematics , P.O.Box 19395-5531, Tehran, Iran
c
Physics Department, Yazd University, Yazd, Iran and Institute for Studies in Theoretical Physics and Mathematics , P.O.Box 19395-5531, Tehran, Iran In this paper the analysis of certain logarithmic expansions, which developed for precision studies of the evolution of the QCD parton distributions (pdf) at the Large Hadron Collider, is used. The xF3 data of the CCFR collaboration is used to parameterize the non-singlet parton distributions in this order. In the fitting procedure, Bernstein polynomial method is employed. The results of valence quark distributions in the NLO are compered with exact and truncated solutions for non-singlet case.
1. Introduction Precision studies of some hadronic processes in the perturbative regime are going to be very important in order to confirm the validity of the mechanism of mass generation in the Standard Model at the new collider, the LHC. Studies of radiative corrections for specific processes have been performed by various groups, at a level of accuracy which has reached the next-to-next-toleading order (NNLO) in αs , the QCD coupling constant. The quantification of the impact of these corrections requires the determination of the hard scattering of the partonic cross sections up to order αs3 , with the matrix of the anomalous dimensions of the DGLAP kernels determined at the same perturbative order. Recently Cafarella et al.[1] presented the higher order logarithmic expansions and exact solutions of the DGLAP equations. Mellin methods have been the most popular and have been implemented up to NLO and also at NNLO [2]. In this work we are going to use the NLO logarithmic expansions and exact solutions of the DGLAP equations from n-Space. For parametrization of the non-singlet parton distri∗ e-mail:
[email protected]
0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.08.080
bution functions we need to use the CCFR experimental data [3]. In the fitting procedure, Bernstein polynomial method [4–6] is employed. 2. QCD Background Before we start our analysis it is convenient to introduce the 3-loop evolution of the coupling via its β-function β(αs ) ≡
∂αs (Q2 ) , ∂ log Q2
(1)
and its three-loop expansion is β(αs ) = −
β0 2 β1 3 α − α + O(αs4 ), 4π s 16π 2 s
(2)
where β0
=
β1
=
11 4 NC − Tf , 3 3 34 2 10 N − NC nf − 2CF nf , 3 C 3
(3)
are the coefficients of the beta function. We have set NC = 3,
CF =
Tf = TR nf =
1 nf , 2
NC2 − 1 4 = , 2NC 3 (4)
28
A.N. Khorramian et al. / Nuclear Physics B (Proc. Suppl.) 174 (2007) 27–30
where NC is the number of colors, nf is the number of active flavors, that is fixed by the number of quarks with mq ≤ Q. One can obtain either an exact or an accurate (truncated) solution of this equation. An exact solution includes higher order effects in αs , while a truncated solution retains contributions only up to a given (fixed) order in a certain expansion parameter. Truncated solutions instead can be obtained quite easily, for instance expanding in terms of the logarithm of a specific scale (Λ) 4π 1 β1 log LΛ +O 1− 2 , (5) αs (Q2 ) = β0 LΛ β0 LΛ L3Λ
can be rewritten in the form (αs − α0 ) b1 (0) (1) P −P f (N, αs ) = exp πβ0 2
where
f (N, αs ) =
LΛ = log
Q2 , Λ2M S
(6)
×f LO (N, αs ),
(9)
where f LO (N, αs ) is given by
f
LO
αs (N, αs ) = α0
− 2Pβ(0) 0
f (N, α0 ).
(10)
Expanding (9) to first order around the LO solution we obtain
f LO (N, αs ) × (αs − α0 ) b1 (0) (1) P −P , 1+ πβ0 2 (11)
(n )
and where ΛM fS is unknown value and we will extract from experimental data.
which is the expression of the 1-st truncated solution, accurate at order αs . On the other hand by using the logarithmic 3. Truncated and exact non-singlet soluexpansion in the NLO DGLAP equation from [1], tion we have α
The NLO non-singlet truncated DGLAP equas − β2 P (0) (N ) log α 0 0 f (N, αs ) = e tion in moment space can be written as 4 (1) 2 (0) P (N ) − P (N ) × exp β0 β1 2 (0) ∂f (N, αs ) P (N )+ = − 4πβ0 + αs β1 ∂αs β0 αs log f (N, α0 ), 4πβ0 + α0 β1 αs (Q2 ) b 1 P (1) − P (0) f (N, αs ), (12) 2π 2 (7) According to the reported results in [1] the nonsinglet truncated and exact solution at NNLO in which has the solution Mellin space are also accessible.
− 2Pβ(0) 0 αs f (N, αs ) = × α0 (αs − α0 ) b1 (0) (1) P −P exp × πβ0 2 f (N, α0 ), (8) here b1 = β1 /β0 , αs ≡ α(Q2 ) and α0 ≡ α(Q20 ). Notice that the solution of above equation contains as a factor the LO solution and therefore
4. Non-singlet structure function Let us define the Mellin moments for the νN structure function xF3 (x, Q2 ): 1 2 M(N, Q ) = xN −1 F3 (x, Q2 )dx (13) 0
The solution of the renormalization group equation for non-singlet structure function xF3 can be presented in the following form [7]:
A.N. Khorramian et al. / Nuclear Physics B (Proc. Suppl.) 174 (2007) 27–30
M(N, Q2 ) M(N, Q20 )
= exp −
As (Q2 ) As (Q20 )
(N )
γN S (x) dx β(x)
(N )
×
CN S (As (Q2 )) (N )
CN S (As (Q20 ))
,
(14)
where M(N, Q20 ) is a phenomenological quantity related to the factorization scale dependent factor. The solution of the two loops evolution equation from Eq. (14), is as follows
M(N, Q2 ) = 1 + C (1) As (Q2 ) f (N, Q2 ) . (15) We can now insert the truncated solution and also exact solution from Eqs. (11,12) in above equation. We should notice that f (N, Q20 ) is the valence quark compositions as f (N, Q20 ) = uv (N, Q20 ) + dv (N, Q20 ) .
(16)
To start the parameterizations of the above mentioned valence quark compositions at the input scale of Q20 = 1 we supposed the functional form as √ xuv (x, Q20 ) = Nu xa (1 − x)b (1 + c x + dx) , (17) xdv (x, Q20 ) =
Nd (1 − x)e xuv . Nu
(18)
The motivation for choosing this functional form is, the xa term controls the low-x behavior parton densities, and (1 − x)b,e large values of x. The remaining polynomial factor accounts for additional medium-x values. Normalization constants Nu and Nd are fixed by 1 uv (x)dx = 2 , (19)
29
Using the valence quark distribution functions, the moments of uv (x, Q20 ) and dv (x, Q20 ) distributions can be easily calculated. Now by inserting the Mellin moments of uv and dv valence quark in the Eq. (16), the function of f (N, Q20 ) involves some unknown parameters. 5. Reconstruction of the structure function from moments Although it is relatively easy to compute the nth moment from the structure functions, the inverse process is not obvious. To do this inversion, we adopt a mathematically rigorous and easy method [8] to invert the moments and retrieve the structure functions. The method is based on the fact that for a given value of Q2 , only a limited number of experimental points, covering a partial range of values of x are available. The method devised to deal with this situation is to take averages of the structure function weighted by suitable polynomials. These are the Bernstein polynomials which are defined by Bn,k (x) =
Γ(n + 2) xk (1−x)n−k .(21) Γ(k + 1)Γ(n − k + 1)
Using the binomial expansion, the above equation can be written as Bn,k (x) =
n−k Γ(n + 2) (−1)l xk+l . (22) Γ(k + 1) l!(n − k − l)! l=0
We can now compare the theoretical predictions with the experimental results for the Bernstein averages, which are defined by 1 Fn,k (Q2 )≡ dxBn,k (x)F3 (x, Q2 ) . (23) 0
0
0
1
dv (x)dx = 1 ,
(20)
the above normalizations are very effective to control unknown parameters in Eqs. (17,18) via fitNf =4 ting procedure. The five parameters with ΛQCD will be extracted by using the Bernstain polynomials approach.
Therefore, the integral Eq. (23) represents an average of the function F3 (x, Q2 ) in the region xn,k + 12 Δxn,k . By a suitable x ¯n,k − 12 Δxn,k ≤x≤¯ choice of n, k we manage to adjust the region in where the average is around values which we have experimental data [3]. Substituting Eq. (22) in Eq. (23), it follows that the averages of F3 with Bn,k (x) as weight functions can be obtained in terms of odd and even
30
A.N. Khorramian et al. / Nuclear Physics B (Proc. Suppl.) 174 (2007) 27–30
moments,
n−k l=0
(24) We can only include a Bernstein average, Fn,k , if we have experimental points covering the whole ¯n,k + 12 Δxn,k ] [9]. This range [¯ xn,k − 12 Δxn,k , x means that with the available experimental data we can only use the following 28 averages. The unknown parameters according to Nf . Thus, Eqs. (17,18) will be a, b, c, d, e and ΛQCD there are 6 parameters for each order to be simultaneously fitted to the experimental Fn,k (Q2 ) averages. The minimum χ2 /ndf values for the NLO truncated and exact solutions are 0.624 (4) and 0.519 respectively. Also the value of ΛQCD for the NLO truncated and exact solutions are 259 M eV and 268 M eV respectively. In Fig.1 we plotted the ratio of f (N, αs )/f (N, α0 ) from Eqs. (11,12) as a function of N at Q2 = 20 GeV 2 . The parton densities xuv and xdv at the input scale Q20 = 1.0 GeV 2 by using exact solution(solid line) which compared to results obtained from truncated solution(dashed line) is also presented in Fig.2.
Tru. Exa.
f(N, αs )/f(N, α0 )
0.5
0.4
2
2
Q =20 GeV
0.3
0.2 0
10
20
N
30
0.4
0.6
(−1)l M((k + l) + 1, Q2 ) . l!(n − k − l)!
0.6
0.5 Tru. Exa.
0.7
40
Figure 1. The ratio of f (N, αs )/f (N, α0 ) as a function of N at Q2 = 20 GeV 2 .
0.5
Q20=1 GeV2
0.3
xdv
=
xuv
Fn,k
0.8
(n − k)!Γ(n + 2) × Γ(k + 1)Γ(n − k + 1)
0.4
0.2
0.3 0.2
0.1 0.1 0 -4 10
10-3
10-2
x
10-1
10
0
0 -4 10
10-3
10-2
x
10-1
100
Figure 2. The parton densities xuv and xdv at the input scale Q20 = 1.0 GeV 2 by using exact solution(solid line) compared to results obtained from truncated solution(dashed line).
6. Acknowledgments A.N.K acknowledge the Semnan university for the financial support of this project. REFERENCES 1. A. Cafarella, C. Coriano and M. Guzzi, Nucl. Phys. B 748 (2006) 253. 2. A. Vogt Comput. Phys. Commun. 170 65, (2005). 3. W. g. Seligman et al., Phys. Rev. Lett. 79, 1213 (1997). 4. Ali N. Khorramian, A. Mirjalili, S. Atashbar Tehrani, JHEP 0410 (2004)062, hepph/0411390. 5. S. Atashbar Tehrani, Ali N. Khorramian and A. Mirjalili, Commun. Theor. Phys. 43(2005) 1087. 6. Ali N. Khorramian, A. Mirjalili, S. Atashbar Tehrani, Int. J. Mod. Phys.A 20(2005)1923. 7. A. L. Kataev, G. Parente and A. V. Sidorov, Nucl. Phys. B 573, (2000) 405. 8. F. J. Yndurain, Phys. Lett. B74 (1978) 68. 9. J. Santiago and F. J. Yndurain, Nucl. Phys. B563 (1999) 45; J. Santiago and F. J. Yndurain, Nucl. Phys. B611 (2001) 447.