anomalous dimension matrix for the twist-2 operators. The BFKL equation for the gluon distributions at small x d dln (1/x). Ïg(x, k2. â¥) = â« d2kâ¥K(kâ¥,kâ¥)Ïg(x, ...
DGLAP and BFKL equations in supersymmetric gauge theories. A.V. Kotikov, L.N. Lipatov, V.N. Velizhanin St. Petersburg Nuclear Physics Institute
Content
1. Introduction.
2. Remarkable properties of the BFKL dynamics in LLA.
3. Next-to-leading corrections to the BFKL equation in SUSY.
4. Analitic continuation of the BFKL eigenvalue to negative |n|.
5. Next-to-leading corrections to anomalous dimension matrix.
6. Relation between the DGLAP and BFKL equations.
7. AdS/CFT correspondence and the perturbation theory.
10. Conclusion.
0.1
Introduction
Integrated and unintegrated parton distributions 2
fa (x, Q ) =
Z 2 ,
˜ µa ,...,µ |h > , dx xj−1 ∆fa (x, Q2 ) = < h|˜ nµ1 ... n ˜ µj O 1 j
a = (q, g, ϕ) , a = (q, g) . (1)
The vector n ˜ µ ∼ q µ + xpµ is lightlike n ˜ 2 = 0.
Generally one can introduce the mixed projections for this tensor σ
σ1 n ˜ µ1 ...˜ nµ1+ω Oµa1 ,...,µ1+ω ,σ1 ,...,σ|n| l⊥ ...l⊥|n| , (l⊥ , p) = (l⊥ , q) = 0,
where |n| is conformal spin and ω = j − 1 is an eigenvalue of the BFKL kernel 1 n 1 n ω = ω(ν, |n|) , m = + iν + , m ˜ = + iν − 2 2 2 2 1
with the M¨obius-group invariant eigenfunction < 0|φ(ρ~1 )φ(ρ~2 )O|n|,ν (ρ~0 )|0 >= (
ρ12 m ρ∗12 m˜ ) ( ∗ ∗ ) . ρ10 ρ20 ρ10 ρ20
The anomalous dimensions γ do not depend on various projections of Oµ1 ,...,µJ .
2
0.2
Remarkable properties of the BFKL dynamics in LLA
1. M¨obius invariance (L.L. (1986)) ρk →
aρk + b cρk + dk
Eigenvalues of two Casimir operators: m(1 − m) and m( ˜ m ˜ − 1) with m=
1 n 1 n + iν + , m ˜ = + iν − 2 2 2 2
for the principal series of the unitary representations.
2. Holomorphic separability of the BFKL hamiltonian (L.L (1989)) c K 12 = −
h12 = ln p1 + ln p2 +
g2 H12 , H12 = h12 + h∗12 , 2 8π
1 1 ∗ + ln ρ p p ln ρ12 p∗1 p2 + 2γE , ρ12 = ρ1 − ρ2 12 1 2 ∗ ∗ p1 p2 p1 p2
3. Holomorphic factorization of the n-gluon wave function at large Nc (L.L. (1989)) Ψ(~ ρ1 , ρ~2 , ..., ρ~n ) =
X r,s
ar,s Ψr (ρ1 , ..., ρn ) Ψs (ρ∗1 , ..., ρ∗2 ),
n X h + h∗ , h= hr,r+1 . H= 2 r=1
4. Duality symmetry in multi-colour QCD (L.L. (1999)) ρr,r+1 → pr → ρr−1,r ,
5. Integrability at large Nc (L.L. (1993)). Integrals of motion 3
[qr , qs ] = [qr , h] = 0 , qr =
X k1
