Solution of the NLO BFKL Equation from Perturbative Eigenfunctions
|
|
- David Egbert Thompson
- 5 years ago
- Views:
Transcription
1 Solution of the NLO BFKL Equation from Perturbative Eigenfunctions Giovanni Antonio Chirilli The Ohio State University JLAB - Newport News - VA 0 December, 013 G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
2 Outline Leading-Log-Approximation at high-energy: general case. BFKL in N =4 SYM theory. Solution of the NLO BFKL equation in QCD. General Form of the Solution of Higher-Order BFKL equation. DGLAP anomalous dimension from solution of BFKL. NLO photon impact factor high-energy OPE γ -γ cross section. Conclusions. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, 013 / 8
3 Linear Evolution equations in QCD DGLAP evolution equation Resum α s ln Q Λ QCD Eigenfunctions at any order: Powers of x B G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
4 Linear Evolution equations in QCD DGLAP evolution equation Resum α s ln Q Λ QCD Eigenfunctions at any order: Powers of x B BFKL evolution equation LO: resum (α s ln s) n. NLO: resum α s (α s ln s) n Eigenfunctions: LO: (k ) γ 1 NLO and higher order: Perturbative eigenfunctions (G.A.C. and Kovchegov). G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
5 Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
6 Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) Resum (α s Y) n LO BFKL eq. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
7 Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) Resum (α s Y) n LO BFKL eq. Resum α s (α s Y) n NLO BFKL eq. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
8 Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) Resum (α s Y) n LO BFKL eq. Resum α s (α s Y) n NLO BFKL eq. The kernel is real and symmetric: K(k, k ) = K(k, k) K(k, k ) is Hermitian and the eigenvalues are real. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
9 LO BFKL equation Y G(k, k, Y) = d q K LO (k, q) G(q, k, Y) d q K LO (k, q) (q ) 1+γ = ᾱ µ χ 0 (γ)(k ) 1+γ ᾱ µ α µn c π (k ) 1+γ are eigenfunctions. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
10 LO BFKL equation Y G(k, k, Y) = d q K LO (k, q) G(q, k, Y) d q K LO (k, q) (q ) 1+γ = ᾱ µ χ 0 (γ)(k ) 1+γ ᾱ µ α µn c π (k ) 1+γ are eigenfunctions. For γ = 1 + iν and ν real parameter (k ) 1+γ form a complete set. LO eigenvalues χ 0 (ν) = ψ(1) ψ( 1 + iν) ψ( 1 iν) are real and sym. ν ν LO BFKL is Conformal invariant. G(k, k, Y) = dν π k k ( k k ) iν eᾱµχ 0(ν) Y G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
11 BFKL equation in the N =4 SYM case In N = 4 SYM theory the coupling constant does not run. (k ) 1 +iν are eigenfunctions at any order. K(q, k) = ᾱ µ K LO (q, k) + ᾱ µ KNLO (q, k) +... G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
12 BFKL equation in the N =4 SYM case In N = 4 SYM theory the coupling constant does not run. (k ) 1 +iν are eigenfunctions at any order. K(q, k) = ᾱ µ K LO (q, k) + ᾱ µ KNLO (q, k) +... d q K(q, k)(q ) 1 +iν = [α s χ 0 (ν) + α s χ 1(ν)... ](k ) 1 +iν G(k, k, Y) = dν π k k e[αsχ 0(ν)+α s χ 1(ν)...] ( k k ) iν The eigenvalues ᾱ µ χ 0 (ν) + ᾱ µ χ 1 (ν) +... are real and symmetric for ν ν. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
13 NLO Amplitude in N =4 SYM I. Balitsky and G.A.C. (009) Factorization in rapidity and composite conformal operator x y Y A Y B <Y< Y A Y B x y G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
14 NLO Amplitude in N =4 SYM I. Balitsky and G.A.C. (009) Factorization in rapidity and composite conformal operator x y x y x y Y A Y B <Y< Y A Y B x y x y x y G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
15 NLO Amplitude in N =4 SYM I. Balitsky and G.A.C. (009) Factorization in rapidity and composite conformal operator x y x y x y Y A Y B <Y< Y A Y B x y x y x y (x y) 4 (x y ) 4 T{Ô(x)Ô (y)ô(x )Ô (y )} = d z 1 d z d z 1 d z (x, y; z IFa0 1, z )[DD] a0,b0 (z 1, z ; z 1, z ) (x, y ; z IFb0 1, z ) a 0 = x+ y + (x y), b 0 = x y (x y ) impact factors do not scale with energy all energy dependence is contained in [DD] a0,b0 G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
16 BFKL equation at NLO in QCD K LO+NLO (k, q) ᾱ µ K LO (k, q) + ᾱ µ KNLO (k, q) d q K LO+NLO (k, q) q γ = [ᾱ µ χ 0 (γ) ᾱ µ β χ 0 (γ) ln k µ + ᾱ µ ] δ(γ) k γ 4 ᾱ µ = α µn c π, β = 11 N c N f 1 N c G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
17 BFKL equation at NLO in QCD K LO+NLO (k, q) ᾱ µ K LO (k, q) + ᾱ µ KNLO (k, q) d q K LO+NLO (k, q) q γ = [ᾱ µ χ 0 (γ) ᾱ µ β χ 0 (γ) ln k µ + ᾱ µ ] δ(γ) k γ 4 ᾱ µ = α µn c π, β = 11 N c N f 1 N c ᾱ µ β χ 0 (γ) ln k µ 1-loop running coupling. δ(γ) = β χ 0 (γ) + 4χ 1(γ) NLO Conformal terms Fadin-Lipatov (1998) χ 1 (γ) Real and symmetric in γ 1 γ γ = 1 + iν. d dγ χ 0(γ) χ 0 (γ) imaginary and not symmetric. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
18 NLO BFKL eigenfunctions G.A.C. and Yu. Kovchegov Perturbative eigenfunctions H γ (k) = k γ + ᾱ µ F γ (k) we have to determine F γ (k) so that d q K LO+NLO (k, q) H γ (q) = (γ) H γ (k) (γ) eigenvalues to be determined. F γ (k) has to satisfy: χ 0 (γ) F γ (k) + d q K LO (k, q)f γ (q) β χ 0 (γ) H γ (k) ln k µ = c(γ) H γ(k) For some function c(γ). G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
19 NLO eigenfunctions ansatz At this order we can write the condition for F γ (k) as: χ 0 (γ)f γ (k) + d q K LO (k, q)f γ (q) β χ 0 (γ) k γ ln k = c(γ) kγ µ Ansatz [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ + c 3(γ) ln 3 k µ +... G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
20 NLO eigenfunctions Truncate the series at n = [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
21 NLO eigenfunctions Truncate the series at n = [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ For c (γ) = β χ 0 (γ) χ 0 (γ) and for any c 0 (γ) and c 1 (γ) the eigenfunction is [ ( )] H γ (k) = k γ β χ 0 (γ) k 1 + ᾱ µ ln (γ) µ + c 1(γ) ln k µ + c 0(γ) and eigenvalues (γ) = ᾱ µ χ 0 (γ) + ᾱ µ χ 0 ( 1 β χ 0 (γ) + χ 1(γ) + c 1 (γ)χ 0 (γ) + β ) χ 0 (γ) χ 0 (γ) G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
22 NLO eigenfunctions Truncate the series at n = [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ For c (γ) = β χ 0 (γ) χ 0 (γ) and for any c 0 (γ) and c 1 (γ) the eigenfunction is [ ( )] H γ (k) = k γ β χ 0 (γ) k 1 + ᾱ µ ln (γ) µ + c 1(γ) ln k µ + c 0(γ) and eigenvalues (γ) = ᾱ µ χ 0 (γ) + ᾱ µ χ 0 ( 1 β χ 0(γ) + χ 1 (γ) + c 1 (γ)χ 0(γ) + β ) χ 0 (γ) χ 0 (γ) Next step is to impose Completeness relation. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
23 Completeness of the NLO eigenfunctions σ+i σ i dγ πi H γ(k) H γ (k ) = δ(k k ) Completeness relation has to be satisfied order-by-order in α µ. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
24 Completeness of the NLO eigenfunctions σ+i σ i dγ πi H γ(k) H γ (k ) = δ(k k ) Completeness relation has to be satisfied order-by-order in α µ. Completeness relation at LO is satisfied with σ = 1 γ = 1 + iν with ν real parameter. we have to impose α µ -order to be 0 and Re[c 1 (ν)] = β χ 0 (ν) ν χ 0 (ν) ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
25 Completeness of the NLO eigenfunctions σ+i σ i dγ πi H γ(k) H γ (k ) = δ(k k ) Completeness relation has to be satisfied order-by-order in α µ. Completeness relation at LO is satisfied with σ = 1 γ = 1 + iν with ν real parameter. we have to impose α µ -order to be 0 and Re[c 1 (ν)] = β χ 0 (ν) ν χ 0 (ν) ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 If completeness relation is satisfied then orthogonality relation is also satisfied. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
26 Completeness of the NLO H-eigenfunctions provided that H 1 +iν(k) = k 1+iν ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 [1 + ᾱ µ ( i β χ 0 (ν) k χ ln 0 (ν) µ + β ( χ 0 (ν) ν χ 0 (ν) the eigenfunctions are ) )] ln k µ + iim[c 1(ν)] ln k µ + Re[c 0(ν)] G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
27 Completeness of the NLO H-eigenfunctions provided that H 1 +iν(k) = k 1+iν ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 [1 + ᾱ µ ( i β χ 0 (ν) k χ ln 0 (ν) µ + β ( χ 0 (ν) ν χ 0 (ν) the eigenfunctions are ) )] ln k µ + iim[c 1(ν)] ln k µ + Re[c 0(ν)] and the eigenvalues are real (Im[c 1 (ν)]χ 0 (ν) is real function of ν) (ν) = ᾱ µ χ 0 (ν) + ᾱ µ [ ] χ 1 (ν) + Im[c 1 (ν)]χ 0(ν) G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
28 Completeness of the NLO H-eigenfunctions provided that H 1 +iν(k) = k 1+iν ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 [1 + ᾱ µ ( i β χ 0 (ν) k χ ln 0 (ν) µ + β ( χ 0 (ν) ν χ 0 (ν) the eigenfunctions are ) )] ln k µ + iim[c 1(ν)] ln k µ + Re[c 0(ν)] and the eigenvalues are real (Im[c 1 (ν)]χ 0 (ν) is real function of ν) (ν) = ᾱ µ χ 0 (ν) + ᾱ µ [ ] χ 1 (ν) + Im[c 1 (ν)]χ 0(ν) The imaginary term β ᾱ µχ 0 (γ) has canceled. We have freedom to chose Re[c 0 (ν)] and Im[c 1 (ν)] This freedom will not affect the solution. It is just an artifact of the phase and ν-reparametrization freedom of the H 1 +iν function. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
29 Phase freedom and ν-reparametrization Phase freedom of the H-functions: H 1 +iν(k) e iᾱµ(im[c 0(ν)] Im[c 1 (ν)] ln µ ) H 1 +iν(k) ν-reparametrization: ν = ν + ᾱ µ Im[c 1 (ν)] The Phase freedom and ν-reparametrization allow us to put c 0 (ν) = 0 and Im[c 1 (ν)] = 0. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
30 Solution of NLO BFKL equation NLO eigenfunctions: perturbation around the conformal LO eigenfunctions ( i ν H 1 +i ν(k) = k 1+ [1 + ᾱ µ β i χ 0 (ν) k ln (ν) µ + 1 χ 0 ( ν ) )] χ 0 (ν) χ 0 (ν) ln k µ NLO eigenvalues (ν) = ᾱ µ χ 0 (ν) + ᾱ µ χ 1(ν) G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
31 Solution of NLO BFKL equation NLO eigenfunctions: perturbation around the conformal LO eigenfunctions ( i ν H 1 +i ν(k) = k 1+ [1 + ᾱ µ β i χ 0 (ν) k ln (ν) µ + 1 χ 0 ( ν ) )] χ 0 (ν) χ 0 (ν) ln k µ NLO eigenvalues (ν) = ᾱ µ χ 0 (ν) + ᾱ µ χ 1(ν) Solution of NLO BFKL equation G.A.C. and Yu. Kovchegov G(k, k, Y) = dν [ ] χ 0(ν)+ᾱ π e[ᾱµ µ χ 1(ν)] Y H 1 +i ν(k) H 1 +i ν(k ) The perturbative expansion is in both the exponent and in the eigenfunctions (contrary to DGLAP case and N =4 BFKL). G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
32 µ-independence of the NLO solution The H γ (k) eigenfunctions diagonalize the LO+NLO BFKL kernel ᾱ µ K LO (k, q) + ᾱ µ KNLO (k, q) = 1 +i 1 i dγ π i (γ) H γ(k)h γ (q) LO+NLO BFKL kernel is µ-independent up to O(α 3 µ ) So is its diagonalization through H γ (k) eigenfunctions. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
33 µ-independence of the NLO solution G(k, k, Y) = = dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1 (ν)] Y H γ (k)h γ (q) dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1(ν)] Y ( k G(k, k, Y) is µ-independent up to order O(α 3 µ). k ) i ν ) (1 ᾱ µ β χ 0 (ν) Y ln kk µ G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
34 µ-independence of the NLO solution G(k, k, Y) = = dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1 (ν)] Y H γ (k)h γ (q) dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1(ν)] Y ( k G(k, k, Y) is µ-independent up to order O(α 3 µ). At NLO we may write the solution as k ) i ν ) (1 ᾱ µ β χ 0 (ν) Y ln kk µ G(k, k, Y) = dν π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s (k k ) χ 1 (ν)] Y ( k k ) i ν At this order the scale ᾱ λ s (k ) ᾱ λ s (k )ᾱs 1 λ (k k ) (for real λ) works as well. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
35 General Form of the Solution of All-Order BFKL equation Ansatz G(k, k, Y) = dν π e[ᾱs(k k ) χ 0(ν)+ᾱ s (k k ) χ 1(ν)+ᾱ 3 s (k k ) χ (ν)+...] Y χ (ν) and higher-order coefficients indicated by the ellipsis in the exponent are the scale-invariant (conformal) (ν ν)-even (real-valued) parts of the prefactor function generated by the action of the next-to-next-to-leading-order (NNLO) (and higher-order) kernels on the LO eigenfunctions. ( k k ) i ν G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
36 General Form of the Solution of All-Order BFKL equation Ansatz G(k, k, Y) = dν π e[ᾱs(k k ) χ 0(ν)+ᾱ s (k k ) χ 1(ν)+ᾱ 3 s (k k ) χ (ν)+...] Y To check the ansatz we have to plug it in the evolution eq. and we need the two-loop beta-function β 3 : ( k k ) i ν and µ dᾱ µ dµ = β ᾱ µ + β 3ᾱ 3 µ { d q K LO+NLO+NNLO (k, q) q 1+iν = ᾱ µ χ 0 (ν) [1 ᾱ µ β ln k +ᾱ µ [ i β χ 0 (ν) + χ 1(ν) ] [1 ᾱ µ β ln k µ k ln µ + ᾱ µ β 3 ln k µ } µ + ᾱ µ β ] + ᾱ 3 µ [χ (ν) + iδ (ν)] ] k 1+iν G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
37 General Form of the Solution of All-Order BFKL equation The ansatz does not work, but it allows us to recover the structure of the NNLO solution G(k, k, Y) = dν π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s (k k )χ 1 (ν)+ᾱ 3 s (k k ) χ (ν)]y {1 + (ᾱ µ β ) [ 1 4 (ᾱ µy) 3 χ 0 (ν) χ 0(ν) (ᾱ µy) χ 0 (ν) provided that the imaginary part iδ (γ) is ( k k ( χ 0 (ν) ) i ν χ 0 (ν) χ 0(ν) ) + ᾱ µ Y χ 0 (ν) 4 ] } iδ (ν) = i χ 0 (ν)β 3 + iχ 1 (ν)β This solution satisfies also the initial condition: the solution is unique so it is the right one. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
38 General Form of the Solution of All-Order BFKL equation The ansatz does not work, but it allows us to recover the structure of the NNLO solution G(k, k, Y) = dν π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s (k k )χ 1 (ν)+ᾱ 3 s (k k ) χ (ν)]y {1 + (ᾱ µ β ) [ 1 4 (ᾱ µy) 3 χ 0 (ν) χ 0(ν) (ᾱ µy) χ 0 (ν) provided that the imaginary part iδ (γ) is ( k k ( χ 0 (ν) ) i ν χ 0 (ν) χ 0(ν) ) + ᾱ µ Y χ 0 (ν) 4 ] } iδ (ν) = i χ 0 (ν)β 3 + iχ 1 (ν)β This solution satisfies also the initial condition: the solution is unique so it is the right one. The imaginary part iδ (ν) has to be confirmed from the explicit calculation of the NNLO eigenfunction. So, let us calculate explicitly the NNLO eigenfunction. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
39 General Form of the Solution of All-Order BFKL equation The eigenfunction of the NNLO BFKL equation is ( i ν χ 0 (ν) k H 1 +i ν(k) = k 1+ [1 + ᾱ µ β i χ ln 0 (ν) µ + 1 ( ) ] k + ᾱ µ f µ,ν +... ( ν ) ) χ 0 (ν) χ 0 (ν) ln k µ The function f (k/µ,ν) denotes the NNLO corrections to the eigenfunctions. Ansatz for f (k/µ,ν): f (k/µ,ν) = c () 0 k (ν) + c() 1 (ν) ln µ + c() k (ν) ln µ + c() 3 (ν) ln3 k µ +... G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
40 NNLO BFKL Solution: the NNLO eigenfunction This time we truncated the series up to c () 4 and proceeding similarly to the NLO case and we get ( Re[c () χ 1 ] = β 1 (ν)) χ 0 (ν) χ 1(ν)χ 0 (ν) χ 0 (ν) χ 1 (ν)χ 0 (ν) χ 0 (ν) + χ ) 0(ν)χ 1 (ν)χ 0 (ν) χ 3 0 ( (ν) 1 β 3 χ 0(ν)χ 0 (ν) ) χ 0 (ν) c () = β ( 5 χ 0 (ν)χ 0 (ν) 8 χ 4 0 (ν) χ0(ν)χ 0 (ν) 4χ 0 (ν) χ 0 (ν)χ 0 (ν) 4χ 3 0 (ν) 1 ) 8 ) χ 0 (ν) iβ 3 χ 0 (ν) ( χ1 (ν) +iβ χ 0 (ν) χ 0(ν)χ 1 (ν) χ 0 (ν) ( c () 3 = iβ 5 χ 0 (ν)χ 0 (ν) 1 χ 3 0 (ν) + χ ) 0(ν) 4χ 0 (ν) 4 = β χ 0 (ν) 8χ (ν) The Im[c () (ν)] is fixed to be 0 using again the ν-reparametrization. c () 1 G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
41 Structure of the NNLO BFKL Solution From explicit calculation of f (γ) we not only confirm the imaginary part iδ (ν) = i χ 0 (ν)β 3 + iχ 1 (ν)β but also confirm the structure of the NNLO BFKL solution obtained above in an indirect way ( ) G(k, k dν, Y) = π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s(k k ) χ 1 (ν)+ᾱ 3 s(k k ) χ (ν)] Y k i ν k [ {1 + (ᾱ µ β ) 1 4 (ᾱ µ Y) 3 χ 0 (ν) χ 0 (ν) + 1 ( χ 4 (ᾱ µ Y) χ 0 (ν) 0 (ν) ) χ 0 (ν) χ 0 (ν) + ᾱ µ Y χ 0 (ν) ] } 4 It looks like QCD is not just conformal part and running coupling contributions. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, 013 / 8
42 BFKL at NNLO and higher orders VS. At NNLO there are Pomeron loop contributions which are inevitable in the symmetric case of γ -γ case a simple generalization of BFKL at NNLO (and higher) does not exist. In the asymmetric case of DIS, there are no pomeron loops. Linearization (with large N c limit) of the Balitsky-Kovchegov equation at any order provides a systematic procedure to consistently define a BFKL type of evolution equation at any order. From the Leading Log Approximation point of view is not easy to define a BFKL type of evolution equation at any order for the asymmetric case of DIS. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
43 BFKL equation in DIS case K BFKL is k k symmetric Y sym = ln s k k G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
44 BFKL equation in DIS case VS. K BFKL is k k symmetric Y sym = ln s k k NLO K DIS BFKL Y DIS = ln s k ln 1 x B is not Symmetric G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
45 BFKL equation in DIS case VS. K BFKL is k k symmetric Y sym = ln s k k KNLO DIS = Ksym NLO 1 d D q K LO (q 1, q ) ln q q K LO(q, q ) NLO K DIS BFKL Y DIS = ln s k ln 1 x B is not Symmetric Fadin Lipatov(1998) KBFKL DIS is not symmetric eigenvalues not γ 1 γ symmetric. Eigenvalues get an extra term: DIS (γ) = sym (γ) 1 ᾱ µ χ 0 (γ)χ (γ) Reproduced lower order and predicted (and later confirmed) the 3-loop DGLAP anomalous dimension (Fadin-Lipatov (1998)) G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
46 DGLAP anomalous dimension from NLO BFKL solution G(k, k, Y) = dν [ ] χ 0(ν)+ᾱ π e[ᾱµ µ χ 1(ν)] (Y DIS +ln k k ) H 1 +i ν(k) H 1 +i ν(k ) dν [ ] χ 0(ν)+ᾱ π e[ᾱµ µ χ 1(ν)] Y DIS H 1 +i ν(k) H 1 +i ν(k ) (1 + ᾱ µ χ 0 (ν) ln k ) k perform partial integration and exponentiate the Y DIS -dependent terms [ )] G(k, k, Y DIS dν [ ] ) = π e ᾱ µ χ 0 (ν)+ᾱ µ (χ 1 (ν)+ i χ 0 (ν)χ 0 (ν) Y DIS H 1 +i ν(k) H 1 +i ν(k ) ( 1 + i ) ᾱµ χ 0(ν) DIS (γ) = ᾱ µ χ 0 (γ) + ᾱ µ χ 1 (γ) 1 ᾱ µ χ 0 (γ)χ 0 (γ) Agrees with DGLAP 3-loop anomalous dimension. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
47 γ -γ scattering cross-section at NLO: work in progress x y Y A Y B <Y< Y A Y B x y Using high-energy Operator Product Expansion in composite Wilson line operator we get NLO Impact Factor that does non scale with energy. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
48 γ -γ scattering cross-section at NLO: work in progress x y x y x y Y A Y B <Y< Y A Y B x y x y x y Using high-energy Operator Product Expansion in composite Wilson line operator we get NLO Impact Factor that does non scale with energy. G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
49 Photon Impact Factor for BFKL pomeron in momentum space k T -factorization form I. Balitsky and G.A.C. d 4 x e iq x p T{ĵ µ (x)ĵ ν (0)} p = s d k I µν (q, k )V a=xb (k ) k where the evolution of the dipole gluon distribution at NLO is a d da V a(k) = α sn c π ( 1 + α sb 4π [ Va (k ) (k k ) ln (k k ) + α sn c 4π d k {[ V a (k ) (k k ) (k, k )V a (k) ] k (k k ) [ ln µ k + N c(67 b 9 π 3 10n f ) ]) bα s 9N c 4π k V a (k) k k (k k ) ln (k k ) ] k [ ln (k /k ) (k k ) + F(k, k ) + Φ(k, k ) ] } V a (k ) + 3 α s N c π ζ(3)v a(k) V(k ) d z e i(k,z) V(z ) V(z ) = [1 1 N c Tr{U(z )U (0)}] G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
50 Photon Impact Factor for BFKL pomeron in momentum space k T -factorization form I. Balitsky and G.A.C. I µν (q, k ) = N c dν sinh πν ( k ) 1 iν{[( 9 3 πν (1 + ν ) cosh πν Q 4 + ν)( 1 + α s π + α sn c ( ν)( 1 + α s π + α ) ] sn c π F (ν) P µν ν ( k 1 + α s π + α ) sn c π F 3(ν) [ P µν k + P µν k ]} ) π F 1(ν) P µν 1 P µν 1 = g µν q µq ν q P µν = 1 ( q q µ pµ q )( q ν pν q ) q p q p P µν = ( g µ1 ig µ p µ q )( g ν1 ig ν p ν q ) q p q p P µν = ( g µ1 + ig µ p µ q )( g ν1 + ig ν p ν q ) q p q p G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
51 Photon Impact Factor for BFKL pomeron in momentum space k T -factorization form I. Balitsky and G.A.C. ( F 1() (ν) = Φ 1() (ν) + χ γ Ψ(ν), F 3 (ν) = F 6 (ν) + χ γ 1 ) Ψ(ν), γγ Ψ(ν) ψ( γ) + ψ( γ) ψ(4 γ) ψ( + γ), F 6 (γ) = F(γ) C γγ 1 γ γ 31 + χ γ 1 γ γ, + γγ Φ 1 (ν) = F(γ) + 3χ γ + γγ ( γ) + 1 γ 1 γ 7 18(1 + γ) (1 + γ) Φ (ν) = F(γ) + 3χ γ + γγ γγ 7 ( + 3 γγ) + χ γ 1 + γ + χ γ(1 + 3γ) + 3 γγ, F(γ) = π 3 π sin πγ Cχ γ + χ γ γγ γ = 1 + iν G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
52 Conclusions The NLO BFKL eigenfunctions have been constructed: they satisfy completeness and orthogonality condition NLO BFKL solution. NNLO Eigenfunctions has also been presented The structure of the NNLO solution has been found up to the still unknown conformal contribution χ (ν). Procedure to construct the solution of the BFKL equation to any order is now available. With the shift Y sym Y DIS + ln k k one can obtain the solution with non-symmetric kernel for DIS from the symmetric one. Using the High-Energy Product Expansion we have calculated the NLO photon impact factor for pomeron exchange in k T -factorized form (Linear case) and for DIS off a large nucleus (non-linear case for Electron Ion Collider). G. A. Chirilli (The Ohio State Uni.) Solution of the NLO BFKL Equation JLAB - 0 December, / 8
Rapidity evolution of Wilson lines
Rapidity evolution of Wilson lines I. Balitsky JLAB & ODU QCD evolution 014 13 May 014 QCD evolution 014 13 May 014 1 / Outline 1 High-energy scattering and Wilson lines High-energy scattering and Wilson
More informationHigh-energy amplitudes in N=4 SYM
High-energy amplitudes in N=4 SYM I. Balitsky JLAB & ODU Regensburg 15 July 014 I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July 014 1 / 50 Outline Regge limit in a conformal
More informationImproving the kinematics in BK/BFKL to resum the dominant part of higher orders
Improving the kinematics in BK/BFKL to resum the dominant part of higher orders Guillaume Beuf Brookhaven National Laboratory QCD Evolution Workshop: from collinear to non collinear case Jefferson Lab,
More informationHigh energy scattering in QCD and in quantum gravity
High energy scattering in QCD and in quantum gravity. Gribov Pomeron calculus. BFKL equation L. N. Lipatov Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg, Russia Content 3. Integrability
More informationarxiv:hep-ph/ v1 29 Oct 2005
1 Electroproduction of two light vector mesons in next-to-leading BFKL D.Yu. Ivanov 1 and A. Papa arxiv:hep-ph/0510397v1 9 Oct 005 1 Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia Dipartimento
More informationAzimuthal angle decorrelation of Mueller Navelet jets at NLO
Azimuthal angle decorrelation of Mueller Navelet jets at NLO Physics Department, Theory Division, CERN, CH- Geneva 3, Switzerland E-mail: Agustin.Sabio.Vera@cern.ch F. Schwennsen II. Institut für Theoretische
More informationA Matrix Formulation for Small-x RG Improved Evolution
A Matrix Formulation for Small-x RG Improved Evolution Marcello Ciafaloni ciafaloni@fi.infn.it University of Florence and INFN Florence (Italy) In collaboration with: D. Colferai G.P. Salam A.M. Staśto
More informationBFKL pomeron in the external field of the nucleus in (2 + 1)-dimensional QCD. Mihail Braun, Andrey Tarasov
BFKL pomeron in the external field of the nucleus in (2 + 1)-dimensional QCD Mihail Braun, Andrey Tarasov QCD at high energies Strong interaction at high energies is mediated by the exchange of BFKL pomerons
More informationPhoton-Photon Diffractive Interaction at High Energies
Photon-Photon Diffractive Interaction at High Energies Cong-Feng Qiao Graduate University Chinese Academy of Sciences December 17,2007 1 Contents Brief About Diffractive Interaction Leading Order Photon
More informationSymmetric and Non-Symmetric Saturation. Overview
Symmetric and Non-Symmetric Saturation Leszek Motyka DESY Theory, Hamburg & Jagellonian University, Kraków motyka@mail.desy.de Overview BFKL Pomeron and Balitsky Kovchegov formalism Field theoretical formulation
More informationRapidity veto effects in the NLO BFKL equation arxiv:hep-ph/ v1 17 Mar 1999
CERN-TH/99-64 MC-TH-99-4 SHEP-99/2 March 999 Rapidity veto effects in the NLO BFKL equation arxiv:hep-ph/99339v 7 Mar 999 J.R. Forshaw, D.A. Ross 2 and A. Sabio Vera 3 Theory Division, CERN, 2 Geneva 23,
More informationExact anomalous dimensions of 4-dimensional N=4 super-yang-mills theory in 'thooft limit
PACIFIC 2015 UCLA Symposium on Particle Astrophysics and Cosmology Including Fundamental Interactions September 12-19, 2015, Moorea, French Polynesia Exact anomalous dimensions of 4-dimensional N=4 super-yang-mills
More informationarxiv:hep-ph/ v1 15 Jul 1998
The DLLA limit of BFKL in the Dipole Picture arxiv:hep-ph/9807389v1 15 Jul 1998 M. B. Gay Ducati and V. P. Gonçalves Instituto de Física, Univ. Federal do Rio Grande do Sul Caixa Postal 15051, 91501-970
More informationTHE PROPERTY OF MAXIMAL TRANSCENDENTALITY IN THE N =4SYM A. V. Kotikov
ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ 200.. 4.. 6 THE PROPERTY OF MAXIMAL TRANSCENDENTALITY IN THE N =4SYM A. V. Kotikov Joint Institute for Nuclear Research, Dubna We show results for the versal anomalous dimension γ j of
More informationHigh Energy QCD and Pomeron Loops
High Energy QCD and Pomeron Loops Dionysis Triantafyllopoulos Saclay (CEA/SPhT) Based on : E. Iancu, D.N.T., Nucl. Phys. A 756 (2005) 419, Phys. Lett. B 610 (2005) 253 J.-P. Blaizot, E. Iancu, K. Itakura,
More informationFOLLOWING PINO - THROUGH THE CUSPS AND BEYOND THE PLANAR LANDS. Lorenzo Magnea. University of Torino - INFN Torino. Pino Day, Cortona, 29/05/12
FOLLOWING PINO - THROUGH THE CUSPS AND BEYOND THE PLANAR LANDS Lorenzo Magnea University of Torino - INFN Torino Pino Day, Cortona, 29/05/12 Outline Crossing paths with Pino Cusps, Wilson lines and Factorization
More informationarxiv: v2 [hep-th] 29 May 2014
Prepared for submission to JHEP DESY 14-057 Wilson loop OPE, analytic continuation and multi-regge limit arxiv:1404.6506v hep-th] 9 May 014 Yasuyuki Hatsuda DESY Theory Group, DESY Hamburg, Notkestrasse
More informationRenormalon approach to Higher Twist Distribution Amplitudes
Renormalon approach to Higher Twist Distribution Amplitudes Einan Gardi (Cambridge) plan conformal expansion: why, why not cancellation of ambiguities in the OPE example: quadratic UV divergence of twist
More informationUsing HERA Data to Determine the Infrared Behaviour of the BFKL Amplitude
Using HERA Data to Determine the Infrared Behaviour of the BFKL Amplitude with L.N. Lipatov, D.A. Ross and G. Watt arxiv 1005.0355 Short overview of HERA data - evidence for Pomeron Discreet Pomeron of
More informationHigh-energy amplitudes and impact factors at next-to-leading order
High-energ amplitudes and impact factors at net-to-leading order CPHT, École Poltechnique, CNRS, 9118 Palaiseau Cede, France & LPT, Université Paris-Sud, CNRS, 91405 Orsa, France E-mail: giovanni.chirilli@cpht.poltechnique.fr
More informationarxiv: v1 [hep-ph] 27 Dec 2018
Rare fluctuations of the S-matrix at NLO in QCD arxiv:8.739v hep-ph] 7 Dec 8 Wenchang Xiang,,, Yanbing Cai,, Mengliang Wang,, and Daicui Zhou 3, Guizhou Key Laboratory in Physics and Related Areas, Guizhou
More informationQCD saturation predictions in momentum space: heavy quarks at HERA
QCD saturation predictions in momentum space: heavy quarks at HERA J. T. S. Amaral, E. A. F. Basso and M. B. Gay Ducati thiago.amaral@ufrgs.br, andre.basso@if.ufrgs.br, beatriz.gay@ufrgs.br High Energy
More informationThe growth with energy of vector meson photo-production cross-sections and low x evolution
The growth with energy of vector meson photo-production cross-sections and low x evolution Departamento de Actuaria, Física y Matemáticas, Universidad de las Américas Puebla, Santa Catarina Martir, 78,
More informationIntegrability of Conformal Fishnet Theory
Integrability of Conformal Fishnet Theory Gregory Korchemsky IPhT, Saclay In collaboration with David Grabner, Nikolay Gromov, Vladimir Kazakov arxiv:1711.04786 15th Workshop on Non-Perturbative QCD, June
More informationFactorization in high energy nucleus-nucleus collisions
Factorization in high energy nucleus-nucleus collisions ISMD, Kielce, September 2012 François Gelis IPhT, Saclay 1 / 30 Outline 1 Color Glass Condensate 2 Factorization in Deep Inelastic Scattering 3 Factorization
More informationGluon density and gluon saturation
Gluon density and gluon saturation Narodowe Centrum Nauki Krzysztof Kutak Supported by NCN with Sonata BIS grant Based on: Small-x dynamics in forward-central dijet decorrelations at the LHC A. van Hameren,
More informationN=4 Super-Yang-Mills in t Hooft limit as Exactly Solvable 4D Conformal Field Theory
PACIFIC 2014 UCLA Symposium on Particle Astrophysics and Cosmology Including Fundamental Interactions September 15-20, 2014, Moorea, French Polynesia N=4 Super-Yang-Mills in t Hooft limit as Exactly Solvable
More informationarxiv: v1 [hep-ph] 3 May 2010
DESY 1-61 SHEP-1-4 CERN-PH-TH/21-91 arxiv:15.355v1 [hep-ph] 3 May 21 Using HERA Data to Determine the Infrared Behaviour of the BFKL Amplitude H. Kowalski 1, L.N. Lipatov 2,3, D.A. Ross 4, and G. Watt
More informationarxiv:hep-ph/ v1 5 Jun 1998
Highlights of the Theory arxiv:hep-ph/9806285v1 5 Jun 1998 Boris Z. Kopeliovich 1,2 and Robert Peschanski 3 1 Max-Planck-Institut für Kernphysik Postfach 30980, 69029 Heidelberg, Germany 2 Joint Institute
More informationOpportunities with diffraction
Opportunities with diffraction Krzysztof Golec-Biernat Institute of Nuclear Physics in Kraków IWHSS17, Cortona, 2 5 April 2017 Krzysztof Golec-Biernat Opportunities with diffraction 1 / 29 Plan Diffraction
More informationHigh energy factorization in Nucleus-Nucleus collisions
High energy factorization in Nucleus-Nucleus collisions François Gelis CERN and CEA/Saclay François Gelis 2008 Symposium on Fundamental Problems in Hot and/or Dense QCD, YITP, Kyoto, March 2008 - p. 1
More informationHigh Energy Scattering in QCD: from small to large distances
High Energy Scattering in QCD: from small to large distances Functional Methods in Hadron and Nuclear Physics, ECT*,Trento, Aug.21-26, 2017 Collaboration with C.Contreras and G.P.Vacca Introduction: the
More informationJIMWLK evolution: from principles to NLO
JIMWLK evolution: from principles to NLO Alex Kovner University of Connecticut, Storrs, CT JLab, March 3, 2014 with... Misha Lublinsky, Yair Mulian Alex Kovner (UConn) JIMWLK evolution: from principles
More informationarxiv:hep-ph/ v1 1 Mar 2000
1 SPbU-IP-00-06 arxiv:hep-ph/0003004v1 1 Mar 2000 Nucleus-nucleus scattering in perturbative QCD with N c M.Braun Department of High Energy physics, University of S.Petersburg, 198904 S.Petersburg, Russia
More informationIntroduction to chiral perturbation theory II Higher orders, loops, applications
Introduction to chiral perturbation theory II Higher orders, loops, applications Gilberto Colangelo Zuoz 18. July 06 Outline Introduction Why loops? Loops and unitarity Renormalization of loops Applications
More informationIntroduction to Saturation Physics
Introduction to Saturation Physics Introduction to Saturation Physics April 4th, 2016 1 / 32 Bibliography F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, Ann. Rev. Nucl. Part. Sci. 60, 463 (2010)
More informationTriumvirate of Running Couplings in Small-x Evolution
Triumvirate of Running Couplings in Small-x Evolution arxiv:hep-ph/699v1 8 Sep 6 Yuri V. Kovchegov and Heribert Weigert Department of Physics, The Ohio State University Columbus, OH 431,USA September 6
More informationVirtuality Distributions and γγ π 0 Transition at Handbag Level
and γγ π Transition at Handbag Level A.V. Radyushkin form hard Physics Department, Old Dominion University & Theory Center, Jefferson Lab May 16, 214, QCD Evolution 214, Santa Fe Transverse Momentum form
More informationarxiv:hep-ph/ v1 22 Dec 1999
DTP/99/4 DAMTP-999-79 Cavendish-HEP-99/9 BFKL Dynamics at Hadron Colliders arxiv:hep-ph/992469v 22 Dec 999 Carlo Ewerz a,b,, Lynne H. Orr c,2, W. James Stirling d,e,3 and Bryan R. Webber a,f,4 a Cavendish
More informationInclusive B decay Spectra by Dressed Gluon Exponentiation. Einan Gardi (Cambridge)
Inclusive B decay Spectra by Dressed Gluon Exponentiation Plan of the talk Einan Gardi (Cambridge) Inclusive Decay Spectra Why do we need to compute decay spectra? Kinematics, the endpoint region and the
More informationSmall-x Scattering and Gauge/Gravity Duality
Small-x Scattering and Gauge/Gravity Duality Marko Djurić University of Porto work with Miguel S. Costa and Nick Evans [Vector Meson Production] Miguel S. Costa [DVCS] Richard C. Brower, Ina Sarcevic and
More informationResummation at small x
Resummation at small Anna Stasto Penn State University & RIKEN BNL & INP Cracow 09/13/0, INT Workshop, Seattle Outline Limitations of the collinear framework and the fied order (DGLAP) calculations. Signals
More informationMueller Navelet jets at LHC: a clean test of QCD resummation effects at high energy?
Mueller Navelet jets at LHC: a clean test of QCD resummation effects at high energy? LPT, Université Paris-Sud, CNRS, 945, Orsay, France E-mail: Bertrand.Ducloue@th.u-psud.fr Lech Szymanowski National
More informationHigh energy factorization in Nucleus-Nucleus collisions
High energy factorization in Nucleus-Nucleus collisions François Gelis CERN and CEA/Saclay François Gelis 2008 Workshop on Hot and dense matter in the RHIC-LHC era, TIFR, Mumbai, February 2008 - p. 1 Outline
More informationarxiv: v1 [hep-th] 25 Apr 2011
DESY--05 Collinear and Regge behavior of 4 MHV amplitude in N = 4 super Yang-Mills theory J. Bartels, L. N. Lipatov, and A. Prygarin arxiv:04.4709v [hep-th] 5 Apr 0 II. Institute of Theoretical Physics,
More informationMSYM amplitudes in the high-energy limit. Vittorio Del Duca INFN LNF
MSYM amplitudes in the high-energy limit Vittorio Del Duca INFN LNF Gauge theory and String theory Zürich 2 July 2008 In principio erat Bern-Dixon-Smirnov ansatz... an ansatz for MHV amplitudes in N=4
More informationarxiv:hep-ph/ v1 31 Aug 1999
The AGL Equation from the Dipole Picture arxiv:hep-ph/9908528v1 31 Aug 1999 M. B. Gay Ducati and V. P. Gonçalves Instituto de Física, Univ. Federal do Rio Grande do Sul Caixa Postal 15051, 91501-970 Porto
More informationFunctional RG for few-body physics
Functional RG for few-body physics Michael C Birse The University of Manchester Review of results from: Schmidt and Moroz, arxiv:0910.4586 Krippa, Walet and Birse, arxiv:0911.4608 Krippa, Walet and Birse,
More informationarxiv:hep-ph/ v1 19 Mar 1998
DFF302/0398 Energy Scale(s) and Next-to-leading BFKL Equation 1 arxiv:hep-ph/9803389v1 19 Mar 1998 Marcello Ciafaloni and Gianni Camici Dipartimento di Fisica dell Università, Firenze and INFN, Sezione
More informationPrecision theoretical predictions for hadron colliders
Precision theoretical predictions for hadron colliders giuseppe bozzi Università degli Studi di Milano and INFN, Sezione di Milano IPN Lyon 25.02.2010 giuseppe bozzi (Uni Milano) Precision theoretical
More informationSpiky strings, light-like Wilson loops and a pp-wave anomaly
Spiky strings, light-like Wilson loops and a pp-wave anomaly M. Kruczenski Purdue University Based on: arxiv:080.039 A. Tseytlin, M.K. arxiv:0804.3438 R. Ishizeki, A. Tirziu, M.K. Summary Introduction
More informationAnalytical properties of NLL-BK equation
Guillaume Beuf Brookhaven National Laboratory Institute for Nuclear Theory, Seattle, September 28, 2010 Outline 1 Introduction 2 Traveling wave solutions of BK with fixed coupling 3 Asymptotic behavior
More informationRadiative transitions and the quarkonium magnetic moment
Radiative transitions and the quarkonium magnetic moment Antonio Vairo based on Nora Brambilla, Yu Jia and Antonio Vairo Model-independent study of magnetic dipole transitions in quarkonium PRD 73 054005
More informationQCD and Rescattering in Nuclear Targets Lecture 2
QCD and Rescattering in Nuclear Targets Lecture Jianwei Qiu Iowa State University The 1 st Annual Hampton University Graduate Studies Program (HUGS 006) June 5-3, 006 Jefferson Lab, Newport News, Virginia
More informationHiggs-charm Couplings
Higgs-charm Couplings Wai Kin Lai TUM December 21, 2017 MLL Colloquium, TUM OUTLINE Higgs physics at the LHC OUTLINE Higgs physics at the LHC H J/ψ + γ as a key channel to measure Hc c coupling CHRISTMAS
More informationJets at the LHC. New Possibilities. Jeppe R. Andersen in collaboration with Jennifer M. Smillie. Blois workshop, Dec
Jets at the LHC New Possibilities Jeppe R. Andersen in collaboration with Jennifer M. Smillie Blois workshop, Dec 17 2011 Jeppe R. Andersen (CP 3 -Origins) Jets at the LHC Blois Workshop, Dec. 17 2011
More informationInclusive di-hadron production at 7 and 13 TeV LHC in the full NLA BFKL approach
Inclusive di-hadron production at 7 and 3 TeV LHC in the full NLA BFKL approach Francesco Giovanni Celiberto francescogiovanni.celiberto@fis.unical.it Università della Calabria - Dipartimento di Fisica
More informationMatthias Strohmaier. given at QCD Evolution , JLAB. University of Regensburg. Introduction Method Analysis Conclusion
Three-loop evolution equation for non-singlet leading-twist operator based on: V. M. Braun, A. N. Manashov, S. Moch, M. S., not yet published, arxiv:1703.09532 Matthias Strohmaier University of Regensburg
More informationFactorization, Evolution and Soft factors
Factorization, Evolution and Soft factors Jianwei Qiu Brookhaven National Laboratory INT Workshop: Perturbative and nonperturbative aspects of QCD at collider energies University of Washington, Seattle,
More informationMultiplicity distribution of colour dipoles. at small x. G.P. Salam. Cavendish Laboratory, Cambridge University, Madingley Road, Cambridge CB3 0HE, UK
Cavendish-HEP-95/01 hep-ph/9504284 April 1995 Multiplicity distribution of colour dipoles at small x G.P. Salam Cavendish Laboratory, Cambridge University, Madingley Road, Cambridge CB3 0HE, UK e-mail:
More informationResumming large collinear logarithms in the non-linear QCD evolution at high energy
Resumming large collinear logarithms in the non-linear QCD evolution at high energy Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France E-mail: edmond.iancu@cea.fr
More informationDecay constants and masses of light tensor mesons (J P = 2 + )
Decay constants and masses of light tensor mesons (J P = + ) R. Khosravi, D. Hatami Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran Abstract We calculate the masses and
More informationarxiv: v2 [hep-th] 26 Feb 2008
CERN-PH-TH/008-07 DESY-08-05 BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes J. Bartels, L. N. Lipatov,, A. Sabio Vera 3 arxiv:080.065v [hep-th] 6 Feb 008 II. Institut Theoretical Physics,
More informationEvolution equations beyond one loop from conformal symmetry. V. M. Braun
Evolution equations beyond one loop from conformal symmetry V. M. Braun University of Regensburg based on V. Braun, A. Manashov, Eur. Phys. J. C 73 2013 2544 [arxiv:1306.5644 [hep-th V. Braun, A. Manashov,
More informationTop Quarks, Unstable Particles... and NRQCD
Top Quarks, Unstable Particles... and NRQCD André H. Hoang Max-Planck-Institute for Physics Munich (Thanks to A. Manohar, I. Stewart, T. Teubner, C. Farrell, C. Reisser, M. Stahlhofen ) INT Workshop, May
More informationMeasurements of Proton Structure at Low Q 2 at HERA
Measurements of Proton Structure at Low Q 2 at HERA Victor Lendermann Kirchhoff-Institut für Physik, Universität Heidelberg Im Neuenheimer Feld 227, 69120 Heidelberg Germany Abstract. Inclusive ep scattering
More informationarxiv:hep-ph/ v4 24 Feb 1999
NUC-MN-99/-T TPI-MINN-99/5 Small x F Structure Function of a Nucleus Including Multiple Pomeron Exchanges arxiv:hep-ph/998v4 4 Feb 999 Yuri V. Kovchegov School of Physics and stronomy, University of Minnesota,
More informationNonperturbative QCD in pp scattering at the LHC
Nonperturbative QCD in pp scattering at the LHC IX Simpósio Latino Americano de Física de Altas Energias SILAFAE Jochen Bartels, Hamburg University and Universidad Tecnica Federico Santa Maria Introduction:
More informationDi-hadron Angular Correlations as a Probe of Saturation Dynamics
Di-hadron Angular Correlations as a Probe of Saturation Dynamics Jamal Jalilian-Marian Baruch College Hard Probes 2012, Cagliari, Italy Many-body dynamics of universal gluonic matter How does this happen?
More informationarxiv:hep-ph/ v1 8 Jul 1996
Resummation at Large Q and at Small x CCUTH-96-04 arxiv:hep-ph/960756v1 8 Jul 1996 Hsiang-nan Li Department of Physics, National Chung-Cheng University, Chia-Yi, Taiwan, Republic of China PACS numbers:
More informationCTEQ6.6 pdf s etc. J. Huston Michigan State University
CTEQ6.6 pdf s etc J. Huston Michigan State University 1 Parton distribution functions and global fits Calculation of production cross sections at the LHC relies upon knowledge of pdf s in the relevant
More informationParticles and Deep Inelastic Scattering
Particles and Deep Inelastic Scattering University HUGS - JLab - June 2010 June 2010 HUGS 1 Sum rules You can integrate the structure functions and recover quantities like the net number of quarks. Momentum
More informationFall and rise of the gluon splitting function (at small x)
Fall and rie of the gluon plitting function (at mall ) Gavin Salam LPTHE Univ. Pari VI & VII and CNRS In collaboration with M. Ciafaloni, D. Colferai and A. Staśto 8 th workhop on non-perturbative QCD
More informationarxiv:hep-ph/ v1 13 Jan 2003
Preprint typeset in JHEP style - HYPER VERSION Cavendish HEP 2002 21 DAMTP 2002 154 IPPP/02/80 DCPT/02/160 arxiv:hep-ph/0301081v1 13 Jan 2003 Energy Consumption and Jet Multiplicity from the Leading Log
More informationA pnrqcd approach to t t near threshold
LoopFest V, SLAC, 20. June 2006 A pnrqcd approach to t t near threshold Adrian Signer IPPP, Durham University BASED ON WORK DONE IN COLLABORATION WITH A. PINEDA AND M. BENEKE, V. SMIRNOV LoopFest V p.
More informationProton structure functions at small x
Journal of Physics: Conference Series PAPER OPEN ACCESS Proton structure functions at small To cite this article: Martin Hentschinski 5 J. Phys.: Conf. Ser. 65 Related content - ON VARIABLE 39 IN IC 63
More informationViolation of a simple factorized form of QCD amplitudes and Regge cuts
Violation of a simple factorized form of QCD amplitudes and Regge cuts Author affiliation Budker Institute of Nuclear Physics of SD RAS, 630090 Novosibirsk Russia Novosibirsk State University, 630090 Novosibirsk,
More informationarxiv:hep-ph/ v1 27 Jul 2006
xxx-xxx-xxx Event generation from effective field theory Christian W. Bauer 1 and Matthew D. Schwartz 1 1 Ernest Orlando Lawrence Berkeley National Laboratory and University of California, Berkeley, CA
More informationIntroduction to Operator Product Expansion
Introduction to Operator Product Expansion (Effective Hamiltonians, Wilson coefficients and all that... ) Thorsten Feldmann Neckarzimmern, March 2008 Th. Feldmann (Uni Siegen) Introduction to OPE March
More informationThe Uses of Conformal Symmetry in QCD
The Uses of Conformal Symmetry in QCD V. M. Braun University of Regensburg DESY, 23 September 2006 based on a review V.M. Braun, G.P. Korchemsky, D. Müller, Prog. Part. Nucl. Phys. 51 (2003) 311 Outline
More informationIntroduction to Perturbative QCD
Introduction to Perturbative QCD Lecture 3 Jianwei Qiu Iowa State University/Argonne National Laboratory PHENIX Spinfest at RIKEN 007 June 11 - July 7, 007 RIKEN Wako Campus, Wako, Japan June 6, 007 1
More informationDiffusive scaling and the high energy limit of DDIS
RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA E-mail: hatta@quark.phy.bnl.gov After reviewing the recent developments on the high energy evolution equation beyond the
More informationFactorization and Fully Unintegrated Factorization
Factorization and Fully Unintegrated Factorization Ted C. Rogers Vrije Universiteit Amsterdam trogers@few.vu.nl Factorization and unintegrated PDFs: open problems Fully unintegrated factorization INT 09
More informationarxiv:hep-ph/ v1 5 Jan 1996
SLAC PUB 95 763 December 1995 Renormalization Scale Setting for Evolution Equation of Non-Singlet Structure Functions and Their Moments Wing Kai Wong arxiv:hep-ph/961215v1 5 Jan 1996 Stanford Linear Accelerator
More informationarxiv: v2 [hep-ph] 17 May 2010
Heavy χ Q tensor mesons in QCD arxiv:00.767v [hep-ph] 7 May 00 T. M. Aliev a, K. Azizi b, M. Savcı a a Physics Department, Middle East Technical University, 0653 Ankara, Turkey b Physics Division, Faculty
More informationTHE RENORMALIZATION GROUP AND WEYL-INVARIANCE
THE RENORMALIZATION GROUP AND WEYL-INVARIANCE Asymptotic Safety Seminar Series 2012 [ Based on arxiv:1210.3284 ] GIULIO D ODORICO with A. CODELLO, C. PAGANI and R. PERCACCI 1 Outline Weyl-invariance &
More informationNONLINEAR EVOLUTION EQUATIONS IN QCD
Vol. 35 (24) ACTA PHYSICA POLONICA B No 2 NONLINEAR EVOLUTION EQUATIONS IN QCD Anna M. Staśto Physics Department, Brookhaven National Laboratory Upton, NY 973, USA and H. Niewodniczański Institute of Nuclear
More informationarxiv: v1 [hep-ph] 28 May 2012
Evidence for the higher twists effects in diffractive DIS at HERA M. Sadzikowski, L. Motyka, W. S lomiński Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 3-59 Kraków, Poland We
More informationarxiv: v1 [hep-ph] 3 Apr 2015
IPMU 15-0038 KANAZAWA-15-0 FTPI-MINN-15/16 QCD Effects on Direct Detection of Wino Dark Matter arxiv:1504.00915v1 hep-ph] 3 Apr 015 Junji Hisano a,b,c), Koji Ishiwata d), Natsumi Nagata c,e) a) Kobayashi-Maskawa
More informationFall and rise of the gluon splitting function (at small x)
Fall and rie of the gluon plitting function (at mall ) Gavin Salam LPTHE Univ. Pari VI & VII and CNRS In collaboration with M. Ciafaloni, D. Colferai and A. Staśto DIS 24 Štrbké Pleo 16 April 24 Fall and
More informationUnitarity, Dispersion Relations, Cutkosky s Cutting Rules
Unitarity, Dispersion Relations, Cutkosky s Cutting Rules 04.06.0 For more information about unitarity, dispersion relations, and Cutkosky s cutting rules, consult Peskin& Schröder, or rather Le Bellac.
More informationIntroduction to Perturbative QCD
Introduction to Perturbative QCD Lecture Jianwei Qiu Iowa State University/Argonne National Laboratory PHENIX Spinfest at RIKEN 007 June 11 - July 7, 007 RIKEN Wako Campus, Wako, Japan June 5, 007 1 Infrared
More informationarxiv:hep-th/ v1 2 Jul 1998
α-representation for QCD Richard Hong Tuan arxiv:hep-th/9807021v1 2 Jul 1998 Laboratoire de Physique Théorique et Hautes Energies 1 Université de Paris XI, Bâtiment 210, F-91405 Orsay Cedex, France Abstract
More informationQuantum Physics III (8.06) Spring 2008 Final Exam Solutions
Quantum Physics III (8.6) Spring 8 Final Exam Solutions May 19, 8 1. Short answer questions (35 points) (a) ( points) α 4 mc (b) ( points) µ B B, where µ B = e m (c) (3 points) In the variational ansatz,
More informationPDF from Hadronic Tensor on the Lattice and Connected Sea Evolution
PDF from Hadronic Tensor on the Lattice and Connected Sea Evolution Path-integral Formulation of Hadronic Tensor in DIS Parton Degrees of Freedom Numerical Challenges Evolution of Connected Sea Partons
More informationHigh Energy Scattering in Quantum Chromodynamics
High Energy Scattering in Quantum Chromodynamics University of Jyväskylä, August 2012 François Gelis IPhT, Saclay t z Goal : understand as much as we can of hadronic collisions in terms of the microscopic
More informationMueller-Navelet jets at LHC: the first complete NLL BFKL study
Mueller-Navelet jets at LHC: the first complete NLL BFKL study B. Ducloué LPT, Université Paris-Sud, CNRS, 945, Orsay, France E-mail: Bertrand.Ducloue@th.u-psud.fr L. Szymanowski National Center for Nuclear
More informationMueller-Navelet jets at LHC: an observable to reveal high energy resummation effects?
Mueller-Navelet jets at LHC: an observable to reveal high energy resummation effects? LPT, Université Paris-Sud, CNRS, 945, Orsay, France E-mail: Bertrand.Ducloue@th.u-psud.fr Lech Szymanowski National
More informationStudies of TMD resummation and evolution
Studies of TMD resummation and evolution Werner Vogelsang Univ. Tübingen INT, 0/7/014 Outline: Resummation for color-singlet processes Contact with TMD evolution Phenomenology Conclusions Earlier work
More informationFour-point functions in the AdS/CFT correspondence: Old and new facts. Emery Sokatchev
Four-point functions in the AdS/CFT correspondence: Old and new facts Emery Sokatchev Laboratoire d Annecy-le-Vieux de Physique Théorique LAPTH Annecy, France 1 I. The early AdS/CFT correspondence The
More information