High energy scattering in QCD and in quantum gravity

Transcription

1 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 of the BFKL dynamics 4. Spectrum of Pomerons in QCD and new physics 5. Effective action for high energy QCD 6. Pomeron in N=4 SUSY and graviton in the AdS space 7. High energy amplitudes in gravity 8. Effective action for high energy gravity 9. Double-logarithmic asymptotics in supergravity

2 Pomeron in hadron interactions High energy kinematics s = 4E t) = q m Pomeron contribution and Pomeranchuck theorem A P s, t) is γ t) s ωt), ωt) = + α t, σ pp = σ p p s Mandelstam cut contribution d k A Mand s, t) = is π) Φ k, q k)s ω k ) s ω q k) ) t-channel partial wave and unitarity relations a+i dω As, t) = is a i πi sω f ω t), I t f ω t) dω n f ω n) n

3 S = Gribov Pomeron calculus Ordering of particle clusters in rapidities y r = ln k + m r + k k + m r k, << y r y r << ln s Non-relativistic Pomeron propagator G 0 = E + k m, E = ω, α = m Gribov effective action dy d ρ ϕ y )ϕ + ) m µϕ ) + iλϕ ϕ + ϕ ϕ ) +... Weak coupling solution and analogy with the graviton exchange = 0, γ ii 0) = const, γ ir 0) = 0 3

4 3 Gluon reggeization in QCD QCD Born amplitude at high energies s t M A B AB Born = s g T c A A δ λa λ A t g T c B B δ λb λ B Leading Logarithmic Approximation Ms, t) = M Born s ωt), α s ln s, α s = g 4π ω q ) = α sn c 4π Gluon trajectory in LLA d k q k q k α sn c π ln q λ Bootstrap equation ωf = + ω k ) + ω q k ) + ˆK ) 8 f, f = ω ω q ) 4

5 4 Amplitudes in multi-regge kinematics BF KL M +n sω q gt d c c Cq, q ) sω q...gt d n ω r = α sn c π ln q r λ, Cq, q ) = q q q q c n+ c n Cq n+, q n ) sωn+ q n+,, σ t = n dγ n M +n 5

6 5 BFKL equation 975) Balitsky-Fadin-Kuraev-Lipatov equation E Ψ ρ, ρ ) = H Ψ ρ, ρ ), σ t s, = α sn c π E 0 Hamiltonian for the Pomeron wave function H = p p ln ρ )p p + p p ln ρ )p p + ln p p 4ψ), ρ k aρ k + b cρ k + d ρ = ρ ρ, ρ r = x r + iy r Möbius invariance and Pomeron intercept, m = γ + n/, m = γ n/, γ = / + iν, E = ϵ m + ϵ m, ϵ m = ψm) + ψ m) ψ), = g N c π ln > 0 6

7 6 Integrability of the BFKL dynamics Holomorphic separability of BKP hamiltonian at N c L.) H = n h+h ), h = lnp k p k+ ) + ) ln ρ k,k+ ρ k,k ) p k ψ) ) p k k= Holomorphic factorization and duality symmetry Ψ ρ, ρ,..., ρ n ) = r,s a r,s Ψ r ρ,..., ρ n ) Ψ s ρ,..., ρ n), ρ r p r ρ r+ Monodromy matrix and Yang-Baxter equation L. 993)) tu) = L L...L n = Au) Bu), L k = u + ρ k p k p k Cu) Du) ρ k p k u ρ k p k, t s r u) t s r v) l r r r r v u) = l s s s s v u) t s r v) t s r u), ˆl = u ˆ + i ˆP 7

8 7 Spectrum of Pomerons in QCD α s k ) π = Asymptotic freedom β 0 ln k Λ QCD, β 0 = 3 n f Eigenfunctions of the BFKL hamiltonian ) i ν k f n k) = d ν Λ e iψ) Γ + iν) QCD Γ iν) ) β 0 ωn Pomeron intercepts and parton distributions at HERA KLR) 0.5 ω n n, gx, k ) = c n x ω n f n k) n= Essential transverse momenta and physics BSM KLR) k n Λ QCD e 4n 8

9 8 High energy effective action in QCD Locality in the rapidity space y = ln ϵ k + k ϵ k k, y y 0 < η, η << ln s Gluon and reggeized gluon fields v µ x) = it a v a µx), A ± x) = it a A a ±x), δa ± x) = 0 Effective action for the reggeon interactions L., 995) S = d 4 x L QCD + T rv + µa + V µa + ) ), V + = g + P exp g x + v + x )dx ) + ) = v + gv + + v

10 9 Production amplitude in N=4 SUSY Remainder factor R = A 4 /A BDS 4 L. 009), F.,L. 0)) R e iπδ = cos πω ab +i a ) n e iϕn w iν ) ων,n) dν Φν, n), n= ν + n u u 4 3 u = ss, u = s t 3, u 3 = s 3t, w = u, cos ϕ = u u u 3 s 0 s 3 s 0 t s 3 t u 3, u u 3 δ = γ K 8 ln w + w 4, ω ab = γ K 8 ln w, Φ = a ων, n) = ae ν,n a ϵ F νn L +3ζ3)), E νn = n / ν + n 4 Next-to-leading correction to ω F.,L. 0) ϵ F L νn = R ψ + iν + n ) iνψ + iν + n ν + n 0 4 E νn n /ν + n 4 ) + ζ) ) ) +Rψ+iν+ n ) ψ) ζ) E νn 4 n ν n ν + n 4 4 ) 3 ) )

11 0 Pomeron and graviton in N=4 SUSY Eigenvalue of the BFKL kernel in a diffusion approximation j = ν, γ = + j + iν AdS/CFT relation for the graviton Regge trajectry j = + α t, t = E /R, α = R Large coupling asymptotics for γ and KLOV, BPST) γ = + j )/, = λ, λ = g N c Exact expression for the slope of γ at j = KLOV, Basso) γ ) = λ 4 + λ 4 5 λ λ λ = λ 4 I 3 λ) I λ)

12 High energy amplitudes in gravity Production amplitudes in LLA L.L. 98)) A n = s Γ µ ν µν Γ µµ s ωq ) q s ωq Γ ρ σ q ) Graviton-graviton-reggeon vertex Γ µ ν µν = κ 4 Γ µµ Γ νν + Γ µν Γ νµ ) Γ ρ σ...γ ρ σ ρσ Gluon-gluon-reggeized gluon vertex = δ µµ + pa µ p B µ + p A µ p B µ p A p B + q p B µ p B µ p A p B ) Reggeon reggeon-graviton vertex Γ ρσ = κ ) p 4 C ρc σ N ρ N σ ), N = q A q kp A pb kp B

13 Graviton trajectory in supergravity Graviton Regge trajectory L. 98)) ωq ) = α q d k κ π k fk, q), α = q k) 8π, ) fk, q) = k, q k) k + q k) q + N k, q k) Gravitino action S 3/ = N ϵµνρσ d 4 x ψ µγ r 5 γ ν ρ ψσ r r= Divergencies of the graviton Regge trajectory ωq ) = α q ln q λ + N 4 ) ln Λ q 3

14 3 High energy action in gravity Locality in the rapidity space y = ln ϵ k + k ϵ k k, y y 0 < η, η << ln s Reggeized graviton fields δa ++ x) = δa x) = 0, + A ++ x) = A x) = 0 Effective action for the high energy gravity L. 0) S = g d 4 x R + + j κ µa ++ + j + µa )) Hamilton-Jacobi equation for effective currents j ± = x ± ω ± g µν µ ω ± ν ω ± = 0, ± j = h ±± h ρ± ) ρ h ±± +... ± 4

15 4 Effective currents for shock waves Aichelburg - Sexl metric ds) = η µν dx µ dx ν +a ln x δx ) dx ), a = 8 G µ, z = a x x j + = a µ Effective current for the shock wave ln x + ln fz) ) z 4 f, fz) = + z z) + Perturbative expansion j + = a ln x + a xσ x ) a3 x µ x Variational principle for j + µ ) xσ +... x j + = x g ++ y, ρ y )) + ρ ) ), δj + δ ρ y + ) = 0 5

16 5 Double-logarithmic asymptotics Mellin representation for the scattering amplitude As, t) = A Born s α q ln q a+i ) ω d ω s λ πi ω q f ω, α = κ 8π a i Infrared evolution equation for supergravity BLS 0)) f N) ω ω = 6 N f ω = + b d d ω f ω ω b N 6 f ω ω, b = α q Solution in terms of parabolic cylinder function b d d x ln dn) x), d N) x) = e x 4 D 6 N x), x = ω b Perturbative expansion of scattering amplitudes in ξ = α t ln As, t) = A Born s α q ln q λ N 4 ξ + N 4)N 3) s q ξ 4! +... ) 6

17 6 Production amplitudes in DLA A N) 3 = ABorn r N) = Multi-Regge kinematics s s i = k i + k i ) k r, n+ r= s r = s n r= k r One graviton production amplitude in DLA BLS) q µ 3 s α q ln s α q ln +i i dx dx πi) ex ρ +x ρ dn) ϕ N) x, x ) = 6 N q µ r N) ρ, ρ ), ρ r = α ln s r / q r, q q x ) d N) q q x ) )) d N) q q x d N) q q x d N) x ) d N) x ) ) q q x )) d N) q q x d N) q q x q q x ) ϕ N) x, x ), 7

18 7 Discussion. Locality in the rapidity space and Gribov calculus.. BFKL Pomeron as a composite state of reggeized gluons. 3. High energy effective action in QCD. 4. Integrability of Hamiltonian for gluon composite states. 5. Pomeron-graviton duality in N = 4 SUSY. 6. High energy effective vertices in gravity. 7. Graviton Regge trajectory at supergravity. 8. Effective action for high energy gravity. 9. Hamilton-Jacobi equation for effective currents. 0. Double-logarithmic scattering amplitudes at supergravity. 8