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Erratum to: “DGLAP and BFKL equations in the N=4 supersymmetric gauge theory”
Nuclear Physics B 685 (2004) 405–407 www.elsevier.com/locate/npe
Erratum
Erratum to: “DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory” [Nucl. Phys. B 661 (2003) 19] ✩ A.V. Kotikov a , L.N. Lipatov b a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia b Theoretical Physics Department, Petersburg Nuclear Physics Institute, Orlova Roscha, Gatchina,
188300 St. Petersburg, Russia Received 16 February 2004
1. In Eq. (69) of the original version of our paper there is a misprint. Namely, the functions Ψ ((j + 1)/2) and Ψ ((j + 1)/2) should be substituted by the expressions Ψ ((j + 2)/2) and Ψ ((j + 2)/2), respectively. Therefore the last expression for Q(j ) looks as follows 3 j +2 1 Q(j ) = ζ (3) − β˜ (j + 1) + Ψ 4 16 2 1 j +2 + Ψ (j + 1) − Ψ (1) ζ(2) − Ψ . 2 2 The substitution does not have any effect on the results of our paper. We are grateful to A. Belitsky, S. Derkachev, G. Korchemsky and A. Manashov for pointing out this misprint.
2. The t-channel partial wave fω in the double-logarithmic approximation for the forward annihilation e+ e− → µ+ µ− is given in Eq. (16) of our paper in a non-consistent way because this expression does not take into account the boundary conditions following from the Bethe–Salpeter equation (15). Although this drawback is not important for our results and the correct expression for fω is well known [1], below we reproduce this expression in another way. The Bethe–Salpeter amplitude A(s, k⊥ ) for an electron colliding with the positron hav2 m2 can be obtained in the double-logarithmic approximation ing the large virtuality k⊥ ✩
doi of original article: 10.1016/S0550-3213(03)00264-5. E-mail address: [email protected] (A.V. Kotikov).
0550-3213/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.032
406
A.V. Kotikov, L.N. Lipatov / Nuclear Physics B 685 (2004) 405–407
(αem ln2 s ∼ 1, αem 1) as a solution of the equation (see [1]) 2 A s, k⊥ =1+λ
s
2/ k 2 ) min(s,sk⊥ ⊥
2 dk⊥ 2 k⊥
m2
2 k⊥
ds 2 A s , k⊥ s
αem . λ= 2π
(1)
It can be transformed after going in the ω-representation 2 A s, k⊥ =
σ+i∞
σ −i∞
ω 2 k⊥ s dω fω (κ) κ = ln 2 2 2πiω k⊥ m
to the form λ fω (κ) = 1 + ω
Λ
dκ e
−ω(κ −κ)
fω (κ ) +
κ
κ
dκ fω (κ ) .
0
2 . It will be pushed to the infinity in We introduced here the ultraviolet cut-off Λ in ln k⊥ the end of calculations. The differentiation of the integral equation gives
e
−ωκ
d fω (κ) = λ dκ
Λ
−ωκ
dκ e
fω (κ )
ωκ d −ωκ d e + λ fω (κ) = 0. and e dκ dκ
κ
One can find the solution of this differential equation in terms of two arbitrary constants aω± ω λ − γω− κ + γω+ κ ± 1± 1−4 2 . fω (κ) = aω e + aω e , γω = (2) 2 ω d From the above expression for e−ωκ dκ fω (κ) we obtain, that −
+
aω− γω− eγω Λ + aω+ γω+ eγω Λ = 0, and from the integral equation aω+ λ aω− . + 1= ω γω− γω+ Therefore
ω Λγω− 1 Λγω− = e e − λ γω+ ω Λγω+ 1 Λγω+ − aω = e e − λ γω− aω+
γω+
γω−2 γω−
γω+2
e
Λγω+
e
Λγω−
−1 ,
−1 .
In particular, for Λ → ∞ we obtain fω (κ) = fω e
γω− κ
,
ω − ω2 λ 1− 1−4 2 , fω = γω = λ 2λ ω
(3)
A.V. Kotikov, L.N. Lipatov / Nuclear Physics B 685 (2004) 405–407
407
where the t-channel partial wave on the mass shell fω satisfies the infrared evolution equation in the form fω = 1 +
λ 2 f . ω2 ω
(4)
Note, that providing that Λ = ∞ from the beginning the condition aω+ = 0 follows from the requirement, that the t-channel partial wave fω (κ) does not grow at ω → ∞ (in the 2 ) to the opposite case we cannot reduce the initial Bethe–Salpeter equation for A(s, k⊥ 2 equation for fω (κ) because the lower limit of integration s = k⊥ is not zero). Thus, the expression for γω− in Eq. (2) coincides with that for γ (ω) in the Section 2 of our paper and the correct expression for fω is given by Eq. (3) in accordance with Ref. [1]. We thank J. Bartels and M. Lyublinsky who drew our attention to this problem.
References [1] G.V. Frolov, V.G. Gorshkov, V.N. Gribov, L.N. Lipatov, Phys. Lett. 22 (1966) 671.
Erratum
Erratum to: “DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory” [Nucl. Phys. B 661 (2003) 19] ✩ A.V. Kotikov a , L.N. Lipatov b a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia b Theoretical Physics Department, Petersburg Nuclear Physics Institute, Orlova Roscha, Gatchina,
188300 St. Petersburg, Russia Received 16 February 2004
1. In Eq. (69) of the original version of our paper there is a misprint. Namely, the functions Ψ ((j + 1)/2) and Ψ ((j + 1)/2) should be substituted by the expressions Ψ ((j + 2)/2) and Ψ ((j + 2)/2), respectively. Therefore the last expression for Q(j ) looks as follows 3 j +2 1 Q(j ) = ζ (3) − β˜ (j + 1) + Ψ 4 16 2 1 j +2 + Ψ (j + 1) − Ψ (1) ζ(2) − Ψ . 2 2 The substitution does not have any effect on the results of our paper. We are grateful to A. Belitsky, S. Derkachev, G. Korchemsky and A. Manashov for pointing out this misprint.
2. The t-channel partial wave fω in the double-logarithmic approximation for the forward annihilation e+ e− → µ+ µ− is given in Eq. (16) of our paper in a non-consistent way because this expression does not take into account the boundary conditions following from the Bethe–Salpeter equation (15). Although this drawback is not important for our results and the correct expression for fω is well known [1], below we reproduce this expression in another way. The Bethe–Salpeter amplitude A(s, k⊥ ) for an electron colliding with the positron hav2 m2 can be obtained in the double-logarithmic approximation ing the large virtuality k⊥ ✩
doi of original article: 10.1016/S0550-3213(03)00264-5. E-mail address: [email protected] (A.V. Kotikov).
0550-3213/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.032
406
A.V. Kotikov, L.N. Lipatov / Nuclear Physics B 685 (2004) 405–407
(αem ln2 s ∼ 1, αem 1) as a solution of the equation (see [1]) 2 A s, k⊥ =1+λ
s
2/ k 2 ) min(s,sk⊥ ⊥
2 dk⊥ 2 k⊥
m2
2 k⊥
ds 2 A s , k⊥ s
αem . λ= 2π
(1)
It can be transformed after going in the ω-representation 2 A s, k⊥ =
σ+i∞
σ −i∞
ω 2 k⊥ s dω fω (κ) κ = ln 2 2 2πiω k⊥ m
to the form λ fω (κ) = 1 + ω
Λ
dκ e
−ω(κ −κ)
fω (κ ) +
κ
κ
dκ fω (κ ) .
0
2 . It will be pushed to the infinity in We introduced here the ultraviolet cut-off Λ in ln k⊥ the end of calculations. The differentiation of the integral equation gives
e
−ωκ
d fω (κ) = λ dκ
Λ
−ωκ
dκ e
fω (κ )
ωκ d −ωκ d e + λ fω (κ) = 0. and e dκ dκ
κ
One can find the solution of this differential equation in terms of two arbitrary constants aω± ω λ − γω− κ + γω+ κ ± 1± 1−4 2 . fω (κ) = aω e + aω e , γω = (2) 2 ω d From the above expression for e−ωκ dκ fω (κ) we obtain, that −
+
aω− γω− eγω Λ + aω+ γω+ eγω Λ = 0, and from the integral equation aω+ λ aω− . + 1= ω γω− γω+ Therefore
ω Λγω− 1 Λγω− = e e − λ γω+ ω Λγω+ 1 Λγω+ − aω = e e − λ γω− aω+
γω+
γω−2 γω−
γω+2
e
Λγω+
e
Λγω−
−1 ,
−1 .
In particular, for Λ → ∞ we obtain fω (κ) = fω e
γω− κ
,
ω − ω2 λ 1− 1−4 2 , fω = γω = λ 2λ ω
(3)
A.V. Kotikov, L.N. Lipatov / Nuclear Physics B 685 (2004) 405–407
407
where the t-channel partial wave on the mass shell fω satisfies the infrared evolution equation in the form fω = 1 +
λ 2 f . ω2 ω
(4)
Note, that providing that Λ = ∞ from the beginning the condition aω+ = 0 follows from the requirement, that the t-channel partial wave fω (κ) does not grow at ω → ∞ (in the 2 ) to the opposite case we cannot reduce the initial Bethe–Salpeter equation for A(s, k⊥ 2 equation for fω (κ) because the lower limit of integration s = k⊥ is not zero). Thus, the expression for γω− in Eq. (2) coincides with that for γ (ω) in the Section 2 of our paper and the correct expression for fω is given by Eq. (3) in accordance with Ref. [1]. We thank J. Bartels and M. Lyublinsky who drew our attention to this problem.
References [1] G.V. Frolov, V.G. Gorshkov, V.N. Gribov, L.N. Lipatov, Phys. Lett. 22 (1966) 671.