AdS black disk model for small-x DIS Miguel S. Costa Faculdade de Ciências da Universidade do Porto 0911.0043 [hep-th], 1001.1157 [hep-ph] Work with. Cornalba and J. Penedones Rencontres de Moriond, March 2010
Motivation QCD is approximately conformal at high energies, for very small probes with r probe Λ QCD 1 Regge limit of high center of mass energy with other kinematic invariants fixed corresponds to low Bjorken - x in DIS.
Motivation QCD is approximately conformal at high energies, for very small probes with r probe Λ QCD 1 Regge limit of high center of mass energy with other kinematic invariants fixed corresponds to low Bjorken - x in DIS. Use conformal symmetry and AdS/CFT to make general predictions for conformal limit of DIS in QCD (weak coupling).
Deep Inelastic Scattering e γ e Q 2 = q 2 Q>1 Gev α s small X P
Deep Inelastic Scattering e γ e Q 2 = q 2 Q>1 Gev α s small X P Optical theorem γ 2 γ γ s = (q + P ) 2 Q2 x X P X = Im P P Regge limit of large s small x
Hadronic tensor structure functions F i ( x,q 2 ) = 1 2π Im Π i T ab = i d 4 ye iq y P T { j a (y 1 ) j b (y 3 ) } P (η ab qa q b ) (p a + qa )(p b + qb ) T ab = q 2 Π 1 (x,q 2 ) 2x Q 2 2x 2x Π 2 (x,q 2 )
Hadronic tensor structure functions F i ( x,q 2 ) = 1 2π Im Π i T ab = i d 4 ye iq y P T { j a (y 1 ) j b (y 3 ) } P (η ab qa q b ) (p a + qa )(p b + qb ) T ab = q 2 Π 1 (x,q 2 ) 2x Q 2 2x 2x Π 2 (x,q 2 ) Simulate proton with scalar operator O of dimension and off-shellness p 2 = M 2 Λ 2 QCD ( kj ) (2π) d δ T ab (k j )= j a (k 1 ) O(k 2 ) j b (k 3 ) O(k 4 ) j a with dimension ξ =3
Regge limit in CFTs F.T. T ab (k i ) A ab (y i ) C.T. B mn (p, p) A mn (x, x) F.T.
Regge limit in CFTs F.T. Amplitude depends on 2 cross ratios T ab (k i ) A ab (y i ) B mn = B mn (S, ) C.T. S =4 p p B mn (p, p) F.T. A mn (x, x) H 3 p p + cosh = p p p p Regge limit is S large, fixed
Regge limit in CFTs F.T. Amplitude depends on 2 cross ratios T ab (k i ) A ab (y i ) B mn = B mn (S, ) C.T. S =4 p p B mn (p, p) F.T. A mn (x, x) H 3 p p + cosh = p p p p Regge limit is S large, fixed Current conservation p m B mn =0 B i j matrix on H 3 polarization space
Regge limit in CFTs F.T. Amplitude depends on 2 cross ratios T ab (k i ) A ab (y i ) B mn = B mn (S, ) C.T. S =4 p p B mn (p, p) F.T. A mn (x, x) H 3 p p + cosh = p p p p Regge limit is S large, fixed Current conservation p m B mn =0 B i j matrix on H 3 polarization space ( )] Regge theory B i j 2πi dν S j(ν) 1 [β 1 (ν) δ i j + β 4 (ν) i j 1 3 δi j Ω iν ()
Regge limit in CFTs F.T. Amplitude depends on 2 cross ratios T ab (k i ) A ab (y i ) B mn = B mn (S, )? C.T. S =4 p p B mn (p, p) F.T. A mn (x, x) H 3 p p + cosh = p p p p Regge limit is S large, fixed Current conservation p m B mn =0 B i j matrix on H 3 polarization space ( )] Regge theory B i j 2πi dν S j(ν) 1 [β 1 (ν) δ i j + β 4 (ν) i j 1 3 δi j Ω iν ()
AdS/CFT in Regge limit T ab (k j ) 2is d 2 l e iq l dr r 3 d r r 3 (ψ in) a i(r)(ψ out ) bj (r) φ( r) φ ( r) B i j(s, )
[1 e iδ(s,l ) ] ab AdS/CFT in Regge limit T ab (k j ) 2is d 2 l e iq l dr r 3 d r r 3 (ψ in) a i(r)(ψ out ) bj (r) φ( r) φ ( r) B i j(s, )
[1 e iδ(s,l ) ] ab AdS/CFT in Regge limit T ab (k j ) 2is d 2 l e iq l dr r 3 d r r 3 (ψ in) a i(r)(ψ out ) bj (r) φ( r) φ ( r) B i j(s, ) r l S = r rs, AdS energy squared r R 2 H 3 cosh = r2 + r 2 + l 2 2r r, impact parameter
[1 e iδ(s,l ) ] ab AdS/CFT in Regge limit T ab (k j ) 2is r d 2 l e iq l dr r 3 d r r 3 (ψ in) a i(r)(ψ out ) bj (r) φ( r) φ ( r) B i j(s, ) AdS gauge field wave function l S = r rs, AdS energy squared r R 2 H 3 cosh = r2 + r 2 + l 2 2r r, impact parameter
AdS/CFT in Regge limit T ab (k j ) 2is r d 2 l e iq l dr r 3 d r AdS gauge field wave function AdS scalar wave function r 3 (ψ in) a i(r)(ψ out ) bj (r) φ( r) φ ( r) B i j(s, ) [1 e iδ(s,l ) ] ab l S = r rs, AdS energy squared r R 2 H 3 cosh = r2 + r 2 + l 2 2r r, impact parameter
AdS/CFT in Regge limit T ab (k j ) 2is r d 2 l e iq l dr r 3 d r AdS gauge field wave function AdS scalar wave function r 3 (ψ in) a i(r)(ψ out ) bj (r) φ( r) φ ( r) B i j(s, ) [1 e iδ(s,l ) ] ab l S = r rs, AdS energy squared r R 2 H 3 cosh = r2 + r 2 + l 2 2r r, impact parameter CFT impact parameter representation (AdS phase shift)
AdS/CFT in Regge limit T ab (k j ) 2is r d 2 l e iq l dr r 3 d r AdS gauge field wave function AdS scalar wave function r 3 (ψ in) a i(r)(ψ out ) bj (r) φ( r) φ ( r) B i j(s, ) [1 e iδ(s,l ) ] ab l S = r rs, AdS energy squared r R 2 H 3 cosh = r2 + r 2 + l 2 2r r, impact parameter CFT impact parameter representation (AdS phase shift) Graviton/Pomeron trajectory [Brower, Polchinski, Strassler, Tan 06] j = j(ν, ᾱ s ), β = β(ν, ᾱ s )
AdS black disk model for small-x DIS T ab (k j ) = 2is d 2 l e iq l [ 1 e iδ(s,l ) ] ab
AdS black disk model for small-x DIS T ab (k j ) = 2is d 2 l e iq l [ 1 e iδ(s,l ) ] ab r [ ] 1 e iδ(s,l ab ) dr d r = r 3 r 3 Ψab j (r) Φ( r) i [1 e iχ(s,)] i j l R 2 H 3 r S = r rs cosh = r2 + r 2 + l 2 2r r
AdS black disk model for small-x DIS T ab (k j ) = 2is d 2 l e iq l [ 1 e iδ(s,l ) ] ab r [ ] 1 e iδ(s,l ab ) dr d r = r 3 r 3 Ψab j (r) Φ( r) i [1 e iχ(s,)] i j l R 2 H 3 r Ψ ab j i (r) =(ψ in ) a i(r)(ψ out ) bj (r) non-normalizable, localized around 0 < r < 1/Q S = r rs cosh = r2 + r 2 + l 2 2r r
AdS black disk model for small-x DIS T ab (k j ) = 2is d 2 l e iq l [ 1 e iδ(s,l ) ] ab r [ ] 1 e iδ(s,l ab ) dr d r = r 3 r 3 Ψab j (r) Φ( r) i [1 e iχ(s,)] i j l R 2 H 3 r Ψ ab j i (r) =(ψ in ) a i(r)(ψ out ) bj (r) non-normalizable, localized around 0 < r < 1/Q S = r rs cosh = r2 + r 2 + l 2 2r r Φ( r) = φ( r) 2 normalizable, localized around r 1/M
AdS black disk model for small-x DIS T ab (k j ) = 2is d 2 l e iq l [ 1 e iδ(s,l ) ] ab r [ ] 1 e iδ(s,l ab ) dr d r = r 3 r 3 Ψab j (r) Φ( r) i [1 e iχ(s,)] i j l R 2 H 3 r Ψ ab j i (r) =(ψ in ) a i(r)(ψ out ) bj (r) non-normalizable, localized around 0 < r < 1/Q S = r rs cosh = r2 + r 2 + l 2 2r r Φ( r) = φ( r) 2 normalizable, localized around r 1/M S Q x ln Q M
AdS black disk model for small-x DIS T ab (k j ) = 2is d 2 l e iq l [ 1 e iδ(s,l ) ] ab r [ ] 1 e iδ(s,l ab ) dr d r = r 3 r 3 Ψab j (r) Φ( r) i [1 e iχ(s,)] i j l R 2 H 3 r Ψ ab j i (r) =(ψ in ) a i(r)(ψ out ) bj (r) non-normalizable, localized around 0 < r < 1/Q S = r rs cosh = r2 + r 2 + l 2 2r r Φ( r) = φ( r) 2 normalizable, localized around r 1/M S Q x ln Q M At zero momentum transfer d 2 l 2πr r ln r r d sinh
At weak coupling T ab =4πs dr d r r 3 r 3 Ψab j (r) Φ( r) i ln r r d sinh χ i j(s, )
At weak coupling T ab =4πs dr d r r 3 r 3 Ψab j (r) Φ( r) i ln r r d sinh χ i j(s, ) Can estimate growth with linear pomeron exchange. Saddle point gives χ i j 1 N 2 dνβ(ν) e (j(ν) 1) ln S (1+iν) j (ν) ln S = i
At weak coupling T ab =4πs dr d r r 3 r 3 Ψab j (r) Φ( r) i ln r r d sinh χ i j(s, ) Can estimate growth with linear pomeron exchange. Saddle point gives χ i j 1 N 2 dνβ(ν) e (j(ν) 1) ln S (1+iν) j (ν) ln S = i Non-linear effects become important for Im χ i j(s, ) 1 This happens when phase in saddle vanishes ln S S Q x s (S) ω ln S ln Q M s (S) ω ln S B B
At weak coupling T ab =4πs dr d r r 3 r 3 Ψab j (r) Φ( r) i ln r r d sinh χ i j(s, ) Can estimate growth with linear pomeron exchange. Saddle point gives χ i j 1 N 2 dνβ(ν) e (j(ν) 1) ln S (1+iν) j (ν) ln S = i Non-linear effects become important for Im χ i j(s, ) 1 This happens when phase in saddle vanishes ln S S Q x s (S) ω ln S ln Q M s (S) ω ln S inear pomeron gives ω = ij (ν s ) = 2.44ᾱ s B B
At weak coupling T ab =4πs dr d r r 3 r 3 Ψab j (r) Φ( r) i ln r r d sinh χ i j(s, ) Can estimate growth with linear pomeron exchange. Saddle point gives χ i j 1 N 2 dνβ(ν) e (j(ν) 1) ln S (1+iν) j (ν) ln S = i Non-linear effects become important for Im χ i j(s, ) 1 This happens when phase in saddle vanishes ln S S Q x s (S) ω ln S ln Q M s (S) ω ln S inear pomeron gives ω = ij (ν s ) = 2.44ᾱ s From DIS geometric scaling ω 0.14 B B
AdS black disk Im χ i j(s, ) 1 Black disk in AdS (or in conformal QCD) [1 e iχ(s,)] i j = Θ ( s (S) ) δ i j
AdS black disk Im χ i j(s, ) 1 Black disk in AdS (or in conformal QCD) [1 e iχ(s,)] i j = Θ ( s (S) ) δ i j T ab 4πis dr d r r 2 r 2 Ψab j (r) Φ( r) i s ln r r d sinh δ i j Black disk ln S S Q x 2πis dr r 2 d r r 2 Ψab i i (r) Φ( r) [ (sr r) ω +(sr r) ω r r r r ] B B
AdS black disk Im χ i j(s, ) 1 Black disk in AdS (or in conformal QCD) [1 e iχ(s,)] i j = Θ ( s (S) ) δ i j T ab 4πis dr d r r 2 r 2 Ψab j (r) Φ( r) i s ln r r d sinh δ i j Black disk ln S S Q x 2πis dr r 2 d r r 2 Ψab i i (r) Φ( r) [ (sr r) ω +(sr r) ω r r r r ] dominant for low x B B
AdS black disk Im χ i j(s, ) 1 Black disk in AdS (or in conformal QCD) [1 e iχ(s,)] i j = Θ ( s (S) ) δ i j T ab 4πis dr d r r 2 r 2 Ψab j (r) Φ( r) i s ln r r d sinh δ i j Black disk ln S S Q x 2πis dr r 2 d r r 2 Ψab i i (r) Φ( r) [ (sr r) ω +(sr r) ω r r r r ] dominant for low x B B It is all AdS (or CFT) kinematics. Only dynamical information is: - on-set of black disk region - the target wave function ( integrals) r d r h(ω) Φ( r) = r 2 ω M 1+ω
Universal ratio from kinematics deep inside disk T ab 2πis 1+ω dr i Ψab r2 ω i (r) d r Φ( r) r 2 ω ln S S Q x T îĵ = δîĵ Π 1, T ++ 1 2x 2 (Π 2 2xΠ 1 ) B B
Universal ratio from kinematics deep inside disk T ab 2πis 1+ω dr i Ψab r2 ω i (r) d r Φ( r) r 2 ω ln S S Q x T îĵ = δîĵ Π 1, T ++ 1 2x 2 (Π 2 2xΠ 1 ) F 2 2xF 1 x ω (Q/M) 1+ω π 5 2 Γ 3 ( ) 3+ω Ch(ω) ( 2 ), 3Γ 4+ω 2xF 1 x ω (Q/M) 1+ω π 5 ( 2 Γ 5+ω 2 2 ) Γ ( 3+ω 2 12 Γ ( 4+ω 2 ) ( Γ 1+ω ) 2 ) Ch(ω) B B
Universal ratio from kinematics deep inside disk T ab 2πis 1+ω dr i Ψab r2 ω i (r) d r Φ( r) r 2 ω ln S S Q x T îĵ = δîĵ Π 1, T ++ 1 2x 2 (Π 2 2xΠ 1 ) F 2 2xF 1 x ω (Q/M) 1+ω π 5 2 Γ 3 ( ) 3+ω Ch(ω) ( 2 ), 3Γ 4+ω 2xF 1 x ω (Q/M) 1+ω π 5 ( 2 Γ 5+ω 2 2 ) Γ ( 3+ω 2 12 Γ ( 4+ω 2 ) ( Γ 1+ω ) 2 ) Ch(ω) Q/x GeV 10 5 10 4 10 3 B B 10 2 10 1 10-1 1 10 10 2 GeV Q
Universal ratio from kinematics deep inside disk T ab 2πis 1+ω dr i Ψab r2 ω i (r) d r Φ( r) r 2 ω ln S S Q x T îĵ = δîĵ Π 1, T ++ 1 2x 2 (Π 2 2xΠ 1 ) F 2 2xF 1 x ω (Q/M) 1+ω π 5 2 Γ 3 ( ) 3+ω Ch(ω) ( 2 ), 3Γ 4+ω 2xF 1 x ω (Q/M) 1+ω π 5 ( 2 Γ 5+ω 2 Ratio is fixed by power in 1/x 2 ) Γ ( 3+ω 2 12 Γ ( 4+ω 2 ) ( Γ 1+ω ) 2 ) Ch(ω) Q/x GeV 10 5 10 4 10 3 B B 10 2 F F 2xF 2 1 1+ω F T 2xF 1 3+ω 10 1 10-1 1 10 10 2 GeV Q
Final Comments Is deeply saturated DIS showing us a black disk in AdS (or in conformal QCD)?
Final Comments Is deeply saturated DIS showing us a black disk in AdS (or in conformal QCD)? Non-conformal extension Understand AdS unitarization
Final Comments Is deeply saturated DIS showing us a black disk in AdS (or in conformal QCD)? Non-conformal extension Understand AdS unitarization THANK YOU
ADDITIONA SIDES
Use different Poincaré patches to cover each operator P 4 P 3 P 4 P 3 x 4 x 3 x 1 x 2 P 1 P 2 P 1 P 2
Cross ratios P 4 P 3 Translations in or Special Conformal in P 1,P 3 P 2,P 4 x 1,3 x 1,3 + a, x 2,4 x 2,4 + O(a 2 ) x 4 x 3 Translations in P 2,P 4 or Special Conformal in P 1,P 3 x 1 x 2 x 1,3 x 1,3 + O(b 2 ), x 2,4 x 2,4 + b P 1 P 2 x x 1 x 3, x x 2 x 4 Invariants orentz Transformations x Λx, x Λ x Dilatations x λx, x 1 λ x σ 2 = x 2 x 2, cosh ρ = x x x x Residual transverse conformal group SO(1, 1) SO(3, 1) Regge limit σ 0 fixed ρ
j n O Single t-channel conformal partial wave gives in Regge limit j m (J, d2 + iν ) O A mn σ 1 J [ E(ρ) η mn + F (ρ) xm x n x 2 + G(ρ) xm x n x 2 + H(ρ) xm x n + x m x n x x ] Sum over spins and dimensions gives for a Regge pole j(ν) [Cornalba 07] 4 A mn 2πi dνσ 1 j(ν) α k (ν) D mn k Ω iν (ρ) k=1 Ω iν (ρ) harmonic functions on H 3 ( H3 + ν 2 +1 ) Ω iν (ρ) =0 Ω iν (ρ) = ν sin νρ 4π 2 sinh ρ D mn 1 = η mn xm x n x 2 D mn 2 = xm x n x 2 D mn 3 = x m n + x n m D mn 4 = x 2 m n +(x m n + x n m ) 1 3 (η mn xm x n x 2 ) x 2 x
Graviton/Pomeron Regge trajectory [Brower, Polchinski, Strassler, Tan 06] Trajectory depends on t Hooft coupling j = j(ν, ᾱ s ) α k = α k (ν, ᾱ s ) Strong coupling (N=4 graviton trajectory) Weak coupling (BFK Pomeron trajectory) ) )) j 2 4+ν2 4π ᾱ s j 1 + ᾱ s (2Ψ(1) Ψ ( 1+iν 2 Ψ ( 1 iν 2 α k real j a j b α k imaginary j a j b From now on weak coupling O O O O
Geometric scaling F 2 /Q 2 10 [Stasto et al] 0.0001 0.001 0.01 0.1 1 10 100 1 0.1 ln S 0.01 B s ω ln S 0.001 τ B B BFK pomeron gives ω =2.44 ᾱ s ω fixed experimentally by geometric scaling to be ω =0.138 ± 0.021 [Gelis et al]
Real Data Compare with Data F 2 /Q 0.8 0.6 0.4 Y = 5.0 Y = 4.8 Y = 4.6 With 4 parameters can reproduce available data 0.2 0.8 Y = 4.4 Y = 4.2 Y = 4.0 0.6 0.4 0.2 0.8 Y = 3.8 Y = 3.6 GeV 0.6 10 5 0.4 Q/x 10 4 10 3 0.2 10 2 0.5 1.0 2.0 4.0 0.5 1.0 2.0 4.0 Q [GeV] 10 1 10-1 1 10 10 2 GeV Q