QCD AMPLITUDES WITH THE GLUON EXCHANGE AT HIGH ENERGIES (AND GLUON - PowerPoint PPT Presentation

Young Researchers Workshop ”Physics Challenges in the LHC Era” QCD AMPLITUDES WITH THE GLUON EXCHANGE AT HIGH ENERGIES (AND GLUON REGGEIZATION PROOF) contributor : Reznichenko Aleksei co-authors : Fadin Victor, Kozlov Mikhail Frascati, May 2009

1. TERMINOLOGY: REGGEIZATION • Reggeization of any particle assumes s 0 ,k s 1 ,k s 2 ,k s k,n +1 s k,n the signaturized amplitude to acquire the A ′ B ′ s k,l asymptotic behavior like s j ( t ) , when ex- P n +1 P 0 P 1 P 2 P k P l P n changing this particle in the t-channel in . . . . . . . . . Regge limit ( t ≪ s ). Here j ( t ) ≡ 1+ ω ( t ) R e is the Regge trajectory with ω (0) = 0 . • Signature in the channel t l for multi- q 1 , c 1 q 2 , c 2 q k , c k q k +1 , c k +1 q n +1 , c n +1 particle production means the (anti-) A B symmetrization with respect to the sub- stitution s i,j ↔ − s i,j , for i < l ≤ j . Fig.1 The amplitude for the process 2 → n + 2 . • Hypothesis of the gluon reggeization claims that in Regge limit the real part of the NLA-amplitude 2 → n + 2 with negative signature and with octet in all t i -channels has universal form: � n � e ω ( q 2 e ω ( q 2 i )( y i − 1 − y i ) n +1 )( y n − y n +1 ) R e A A ′ B ′ + n = ¯ � γ J i Γ R n +1 Γ R 1 (1) B ′ B , A ′ A AB R i R i +1 q 2 q 2 i ⊥ ( n +1) ⊥ i =1 k + i ) — particle ( P i ) rapidities, and γ J i where y i = 1 R i R i +1 , Γ R 2 ln( i P ′ P — known effective vertices. k −

2. THE PROOF IDEA: BOOTSTRAP RELATIONS AND CONDITIONS • Bootstrap relations: There is infinite number of necessary and sufficient conditions for compatibility of the Regge amplitude form (1) with unitarity : � n +1 k − 1 1 = 1 � � � A A ′ B ′ + n 2( ω ( t k +1 ) − ω ( t k )) R e A A ′ B ′ + n � � R e disc s k,l − disc s l,k (2) AB AB − 2 πi l = k +1 l =0 • Bootstrap conditions: There is finite number of identities making all bootstrap rela- tions fulfilled. These conditions restrict effective vertices and the gluon trajectory. The main goal is to prove them hold true. A ′ B ′ J j − 1 J j J j +1 J 1 J n – Elastic conditions constrain the ker- nel ( ˆ K ) and the effective vertex de- . . . e ˆ e ˆ K Y j +1 K Y n +1 scribing the transition of the initial � �� � � �� � particle (B) to the final one ( B ′ ): . . . | B ′ B � � J j R j | . . . . . . . . . B ′ B � = g Γ R n +1 Γ R 1 | ¯ ¯ B ′ B | R ω ( q B ⊥ ) � , A ′ A J j − 1 γ J 1 γ ˆ ˆ J j +1 J n R j − 1 R j R 1 R 2 ˆ K| R ω ( q ⊥ ) � = ω ( q 2 ⊥ ) | R ω ( q ⊥ ) � , A B – Inelastic conditions appear in ele- Fig.2 s j,n +1 -channel discontinuity calculation via mentary ( 2 → 3 ) inelastic amplitudes. unitarity relation | ¯ J i R i +1 � + ˆ ( i +1) ⊥ = | R ω ( q i ⊥ ) � g γ J i J i | R ω ( q ( i +1) ⊥ ) � g q 2 (3) R i R i +1 ,

3. INELASTIC CONDITION: | ¯ J i R i +1 � — NLO IMPACT-FACTOR • Impact factor of the jet production is the first component of the inelastic bootstrap condition, intrinsically it appears as logarithmically non-enhanced term in the disconti- nuity (in s 0 , 1 or s 1 , 2 channel) for the process 2 → 3 . k, a k, a G k, a G k, a k, a k, a G G G G r 2 , c 2 r 2 , c 2 r 2 , c 2 r 2 , c 2 r 2 , c 2 r 2 , c 2 = + + + + G ′ ( k ′ ) G ′ ( k ′ ) J ( k ′ ) G ′ ( k ′ ) ˆ K r k 1 k 1 k 2 − k 2 q 1 , i r 1 , c 1 q 1 , i r 1 , c 1 q 1 , i r 1 , c 1 q 1 , i r, c r 1 , c 1 q 1 , i r 1 , c 1 q 1 , i r 1 , c 1 (b) (c) (e) (f) (d) (a) Further we consider the most nontrivial NLO impact-factor: J j = G ( k ) � GR 1 |G 1 G 2 � = � GR 1 |G 1 G 2 � v.c. + � GR 1 |G 1 G 2 � loop . (4) � � GR 1 |G 1 G 2 � v.c. = δ ( q 1 ⊥ − k ⊥ − r 1 ⊥ − r 2 ⊥ ) γ G ′ ( C ) R 1 G 1 Γ G 2 ( B ) + γ G ′ ( B ) R 1 G 1 Γ G 2 ( C ) − γ G ′ ( B ) R 1 G 2 Γ G 1 ( C ) − γ G ′ ( C ) R 1 G 2 Γ G 1 ( B ) + γ G ′ ( B ) R 1 G 1 Γ G 2 ( B ) � GG ′ × GG ′ GG ′ GG ′ GG ′ 2 k + G ′ � k 2 �� k 2 − ω (1) ( r 2 ) � 1 � � �� � − γ G ′ ( B ) R 1 G 2 Γ G 1 ( B ) 2( ω (1) ( q 1 ) − ω (1) ( r 1 )) ln ⊥ ⊥ × ln r 1 ↔ r 2 . ( q 1 − r 1 ) 2 r 2 GG ′ 2 ⊥ 2 ⊥ � GR 1 |G 1 G 2 � loop = δ ( q 1 ⊥ − k ⊥ − r 1 ⊥ − r 2 ⊥ ) � � � γ { f } G { f } − γ { f } � R 1 G 1 Γ G 2 R 1 G 2 Γ G 1 { f } − ∆ � GR 1 | ˆ dφ ∆ K B r |G 1 G 2 � B . G { f } 2 k + f =2 g,q ¯ q

4. INELASTIC CONDITION: ONE GLUON PRODUCTION OPERATOR • One gluon production operator is the second component of the inelastic bootstrap condition, describing the transition reggeon-reggeon state to gluon-reggeon-reggeon. G ( k ) G ( k ) G ( k ) G ( k ) G ( k ) q 1 − r 1 q 2 − r q 1 − r 1 q 2 − r q 1 − r 1 q 2 − r q 1 − r 1 q 2 − r q 1 − r 1 q 2 − r = + + + + r 1 r 1 r r r 1 r r 1 r r 1 r ( b ) ( c ) ( e ) ( a ) G ( k ) 2 | ˆ 2 ⊥ )[ γ J i �G ′ 1 G ′ J ∆ i |G 1 G 2 � = δ ( r 1 ⊥ + r 2 ⊥ − k i ⊥ − r ′ 1 ⊥ − r ′ 1 δ ( r 2 ⊥ − r ′ 2 ⊥ ) r 2 2 + q 1 − r 1 q 2 − r 2 ⊥ δ G 2 G ′ G 1 G ′ y i +∆ � dz G G 2 G ′ G 2 G ′ 2(2 π ) D − 1 ( γ { J i G } � + γ J i 2 δ ( r 1 ⊥ − r ′ 1 ⊥ ) r 2 + γ G 1 ] + 2 J i G ) . 2 1 ⊥ δ G 1 G ′ 1 γ 1 γ G 1 G ′ G 2 G ′ G 1 G ′ G G r 1 r y i − ∆ ( f )

5. PREVIOUSLY OBTAINED RESULTS: • By the direct calculation we (V.S., M.G., A.V.) demonstrated that the inelastic bootstrap condition was fulfilled being projected onto the colour octet in the t -channel: � � f ac ′ 1 c ′ = f ac ′ 1 c ′ 2 | ˆ 2 | ¯ 2 | R ω ( q i ⊥ ) � g γ J i �G ′ 1 G ′ J i | R ω ( q ( i +1) ⊥ ) � g q 2 ( i +1) ⊥ + �G ′ 1 G ′ 2 �G ′ 1 G ′ J i R i +1 � R i R i +1 . 2 • We introduced the operator formulation of the bootstrap reggeon formalism: for bootstrap relations and conditions. Further we conjectured that the following condition is valid for arbitrary color representation: ˆ ( i +1) ⊥ + | ¯ J i | R ω ( q ( i +1) ⊥ ) � g q 2 J i R i +1 � = | R ω ( q i ⊥ ) � g γ J i (5) R i R i +1 . • In V.S. Fadin, R. Fiore, M.G. Kozlov, A.V. Reznichenko, Phys. Lett. B 639 (2006) using the direct discontinuity calculation through the unitarity in terms of our components � i l = j � e ω ( q 1 )( y 0 − y 1 ) e ω ( q l )( y l − 1 − y l ) � 2 → n +2 = ¯ � γ J l − 1 − 4 i (2 π ) D − 2 δ ( q ( j +1) ⊥ − q i ⊥ − k l ⊥ ) disc s i,j A S Γ R 1 × A ′ A R l − 1 R l q 2 q 2 1 ⊥ l ⊥ l = i l =2 (6) � j − 1   � n e ω ( q l )( y l − 1 − y l )  e ω ( q n +1 )( y n − y n +1 ) K ( y l − 1 − y l ) ˆ ˆ ˆ � K ( y j − 1 − y j ) | ¯ � Γ R n +1 γ J l × � J i R i | e J l e J j R j +1 � B ′ B ,  R l R l +1 q 2 q 2 l ⊥ ( n +1) ⊥ l = i +1 l = j +1 and applying granted elastic and inelastic NLO bootstrap conditions we proved all boot- strap relations to be fulfilled. So the last millstone on the way of reggeization proof was the validity of (5).

6. DIFFERENT COLOUR REPRESENTATIONS IN T-CHANNEL: This last unproved bootstrap condition ( J i is one gluon) can be present in projected form: 2 | ˆ 2 | ¯ 2 | R ω ( q i ⊥ ) � g γ J i �G ′ 1 G ′ J i | R ω ( q ( i +1) ⊥ ) � g q 2 ( i +1) ⊥ + �G ′ 1 G ′ J i R i +1 � = �G ′ 1 G ′ R i R i +1 , First of all, it was proved for the octet (the most important) in the t-channel. There is no r.h.s. for any other t-channel representations R � = 8 , since from the explicit form of | R ω ( q i ⊥ ) � : P ( R ) c 1 ,c 2 2 | R ω ( q i ⊥ ) � g γ J i �G ′ 1 G ′ R i R i +1 = 0 . (7) c ′ 1 ,c ′ 2 From the explicit form of the effective vertices it is easy to see that there are only THREE 2 | ˆ nontrivial colour structures into the operator of the gluon production �G ′ 1 G ′ J i | R ω ( q ( i +1) ⊥ ) � 2 | ¯ and into the impact-factor �G ′ 1 G ′ J i R i +1 � . The optimal choice is the “trace-based”: Tr[ T c 2 T a T c 1 T i ] , Tr[ T a T c 2 T c 1 T i ] , Tr[ T a T c 1 T c 2 T i ] (8) • The first colour structure is symmetric with respect to c 1 , c 2 and can be reduced: Tr[ T c 2 T a T c 1 T i ] = N 2 2 P (0) + N c +2 2 P (27) + 2 − N c 2 P ( N c > 3) Corresponding coefficient had not c been calculated before. It is the last problem on the reggeization proof way. • The coefficient at second colour structure Tr[ T a T c 2 T c 1 T i ] is very similar to the octet case, and whereby is considered to be calculated. • The last coefficient (at Tr[ T a T c 1 T c 2 T i ] ) can be easily obtained from the previous one.

7. RESULTS AND PLANS: • We formulated the gluon reggeization proof in operating form through the bootstrap approach based on unitarity. • We proved all bootstrap conditions (necessary for reggeization proof) to be correct when projecting on octet colour representation. But it is not sufficient for the final proof! • We proved all bootstrap conditions to be correct for second and third colour structure and for all colour structures in fermionic sector of the inelastic bootstrap condition. • We calculated in the dimensional regularization all components of the last coefficient at the symmetric colour structure for the inelastic bootstrap condition. • For this structure we demonstrated the cancellation of all singular terms (collinear regu- larization, 1 ǫ 2 , and 1 ǫ ), rational, and logarithmic terms. • We are planing to demonstrate the cancellation for dilogarithmic and double logarithmic terms in the inelastic bootstrap condition in the nearest future. THANKS FOR YOUR ATTENTION!