MSYM amplitudes in the high-energy limit. Vittorio Del Duca INFN LNF

Transcription

1 MSYM amplitudes in the high-energy limit Vittorio Del Duca INFN LNF Gauge theory and String theory Zürich 2 July 2008

2 In principio erat Bern-Dixon-Smirnov ansatz... an ansatz for MHV amplitudes in N=4 SUSY [ ] m n = m (0) n 1+ a L M n (L) (ɛ) = m (0) n exp L=1 [ l=1 ( ) ] a l f (l) (ɛ)m n (1) (lɛ)+const (l) + E n (l) (ɛ) Bern Dixon Smirnov 05 coupling a = λ 8π 2 (4πe γ ) ɛ λ = g 2 sn c t Hooft parameter f (l) (ɛ) = ˆγ(l) K 4 + 2Ĝ(l) l ɛ + f (l) 2 ɛ2 E (l) n (ɛ) =O(ɛ) ˆγ (l) K cusp anomalous dimension, known to all orders of a Korchemsky Radyuskin 86 Beisert Eden Staudacher 06 Ĝ (l) IR function, known through O(a 4 ) Bern Dixon Smirnov 05 Cachazo Spradlin Volovich 07

3 Brief history of BDS ansatz BDS ansatz checked through 3-loop 4-pt amplitude 2-loop 5-pt amplitude Bern Dixon Smirnov 05 Cachazo Spradlin Volovich 06 Bern Czakon Kosower Roiban Smirnov 06 BDS ansatz shown to fail on 2-loop 6-pt amplitude Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08 Hints of break-up also from strong-coupling expansion hexagon Wilson loop multi-regge limit Alday Maldacena 07 Drummond Henn Korchemsky Sokatchev 07 Bartels Lipatov Sabio-Vera 08

4 BDS ansatz and Regge limit 4-pt amplitude p a p b p a p b in the Regge limit s t m 4 = s [g s C(p a,p a )] 1 t ( ) α(t) s [g s C(p b,p b )] t α(t) Regge trajectory C(p a,p a ) coefficient function α(t, ɛ) = l=1 ḡ 2l s (t, ɛ)α (l) (ɛ) ḡ 2 s(t, ɛ) = a 2G(ɛ) ( µ 2 t ) ɛ G(ɛ) = e γɛ Γ(1 2ɛ) Γ(1 + ɛ) Γ 2 (1 ɛ) = 1 + O(ɛ2 ) Because the Regge limit is exponential in the Regge trajectory, one can use (the logarithm of) the BDS ansatz to obtain the Regge trajectory to all loops Naculich Schnitzer 07 Bartels Lipatov Sabio-Vera 08 Glover VDD 08 l-loop Regge trajectory α (l) (ɛ) = 2 l 1 α (1) (lɛ) ( ) ˆγ (l) K 4 + 2Ĝ(l) l ɛ + O(ɛ) α (1) (ɛ) = 2 ɛ

5 the BDS ansatz can also be used to compute (or to derive relations between) the coefficient functions High-energy factorisation is valid also for amplitudes with 5 or more points in generalised Regge limits. The general strategy is to use the modular form of the amplitudes dictated by high-energy factorisation, to obtain information on n-point amplitudes in terms of building blocks derived from m-point amplitudes, with m < n

6 the BDS ansatz can also be used to compute (or to derive relations between) the coefficient functions High-energy factorisation is valid also for amplitudes with 5 or more points in generalised Regge limits. The general strategy is to use the modular form of the amplitudes dictated by high-energy factorisation, to obtain information on n-point amplitudes in terms of building blocks derived from m-point amplitudes, with m < n Because high-energy factorisation is used in the derivation in QCD of the BFKL equation at LL and NLL accuracy, I will start from there with a few slides of a few years ago...

7

8

9

10

11

12

13 maximal trascendentality Kotikov Lipatov 02

14

15 More tree coefficient functions... contributes to NNLL BFKL kernel Frizzo Maltoni VDD 99 Antonov Lipatov Kuraev Cherednikov 05

16 More tree coefficient functions... contributes to NNLL BFKL kernel Frizzo Maltoni VDD 99 Antonov Lipatov Kuraev Cherednikov 05 contributes to NNLO impact factor (boundary condition to NNLL kernel) Frizzo Maltoni VDD 99

17 More tree coefficient functions... contributes to NNLL BFKL kernel Frizzo Maltoni VDD 99 Antonov Lipatov Kuraev Cherednikov 05 contributes to NNLO impact factor (boundary condition to NNLL kernel) Frizzo Maltoni VDD 99 used to compute DGLAP splitting amplitudes for all parton species Frizzo Maltoni VDD 99

18 Tree 4-gluon coefficient function contributes to NNNLO impact factor (boundary condition to NNNLL kernel) Frizzo Maltoni VDD 99

19 Tree 4-gluon coefficient function contributes to NNNLO impact factor (boundary condition to NNNLL kernel) Frizzo Maltoni VDD 99 one may check several kinematic limits

20 Unknown 1-loop coefficient functions, which could be also computed... y 1 y 2 y 3 boundary condition to NNLL kernel

21 Unknown 1-loop coefficient functions, which could be also computed... y 1 y 2 y 3 boundary condition to NNLL kernel y 1 y 2 y 3 y 4 boundary condition to NNNLL kernel y 1 y 2 y 3 y 4 contributes to NNLL kernel In MSYM Bartels Lipatov Sabio-Vera 08

22 as well as 2-loop coefficient functions... y 1 y 2 boundary condition to NNLL kernel

23 as well as 2-loop coefficient functions... y 1 y 2 boundary condition to NNLL kernel The 1-loop and 2-loop coefficient functions I showed in the last two slides have never been computed in QCD. Why? They are building blocks of BFKL kernels or of their boundaries, which, as of now, are unlikely to be built

24 as well as 2-loop coefficient functions... y 1 y 2 boundary condition to NNLL kernel The 1-loop and 2-loop coefficient functions I showed in the last two slides have never been computed in QCD. Why? They are building blocks of BFKL kernels or of their boundaries, which, as of now, are unlikely to be built building blocks of n-point 1-loop or 2-loop amplitudes in particular kinematics, but in QCD we have no clue about the structure of n-point 1-loop or 2-loop amplitudes in arbitrary kinematics (except for 1-loop MHV configurations)

25 N=4 Super Yang-Mills Bern-Dixon-Smirnov computed the 2-loop 4-pt amplitude M4 (2) to O(ε 2 ) and the 3-loop 4-pt amplitude M4 (3) to O(ε 0 ). Those amplitudes can be used to test the high-energy factorisation of the 4-pt amplitude. It is known that the factorisation formula for the QCD colour-dressed amplitude M 4 = s [ ] 1 ig s f aca C(p a,p a ) t [ ( s ) α(t) + t ( ) ] α(t) s [ ig s f bcb C(p b,p b )] t holds only up to NLL accuracy (which was fine for BFKL at NLL) Bern Dixon Smirnov 05 Fadin Lipatov 93

26 N=4 Super Yang-Mills Bern-Dixon-Smirnov computed the 2-loop 4-pt amplitude M4 (2) to O(ε 2 ) and the 3-loop 4-pt amplitude M4 (3) to O(ε 0 ). Those amplitudes can be used to test the high-energy factorisation of the 4-pt amplitude. It is known that the factorisation formula for the QCD colour-dressed amplitude M 4 = s [ ] 1 ig s f aca C(p a,p a ) t [ ( s ) α(t) + t ( ) ] α(t) s [ ig s f bcb C(p b,p b )] t holds only up to NLL accuracy (which was fine for BFKL at NLL) Fadin Lipatov 93 Im M4 (1) contains leading colour structures other than the f s Schmidt VDD 97 In the high-energy limit m (0) 4 ( + +) = m(0) 4 ( ++) which are connected under s u channel crossing. at tree level Clearly, the coefficients of the colour-stripped amplitudes must be the same for the formula above to hold. At n loops, that occurs for the n-th log and for the real part of the (n-1)-th log: that suffices for BFKL at NLL Bern Dixon Smirnov 05

27 natural to use a high-energy factorisation for the colour-stripped amplitude m 4 (,, +, +) m s 4 = s [g s C(p a,p a )] 1 t in the s-channel physical region m 4 (, +,, +) m u 4 = s [g s C(p a,p a )] 1 t in the u-channel physical region ( s t ) α(t) [g s C(p b,p b )] ( ) α(t) s [g s C(p b,p b )] t The formulae above contain the same info: they are related by s u channel crossing

28 natural to use a high-energy factorisation for the colour-stripped amplitude m 4 (,, +, +) m s 4 = s [g s C(p a,p a )] 1 t in the s-channel physical region m 4 (, +,, +) m u 4 = s [g s C(p a,p a )] 1 t in the u-channel physical region ( s t ) α(t) [g s C(p b,p b )] ( ) α(t) s [g s C(p b,p b )] t The formulae above contain the same info: they are related by s u channel crossing Using the high-energy limit of BDS s 2-loop 4-pt amplitude M4 (2) to O(ε 2 ) and 3-loop 4-pt amplitude M4 (3) to O(ε 0 ), one can check that the formulae above hold at 3-loop accuracy Glover VDD 08 Instructive to implement the factorisation formulae with channeldependent coefficient functions. If the test amplitudes are not in the ``right kinematics, the coefficient functions are indeed channel dependent factorisation is broken

29 Factorisation of the 2-loop amplitude 4 = 1 ( α (1)) 2 L 2 ( 2 + α (2) +2C (1) α (1)) L m u(2) + 2C (2) + L = ln (C (1)) 2 ( ) s t

30 Factorisation of the 2-loop amplitude 4 = 1 ( α (1)) 2 L 2 ( 2 + α (2) +2C (1) α (1)) L m u(2) + 2C (2) + L = ln (C (1)) 2 ( ) s t by direct calculation from BDS s 2-loop 4-pt amplitude M4 (2) to O(ε 2 ) we get 2-loop trajectory α (2) MSY M = π2 3ɛ 2ζ 3 4π4 45 ɛ + (6π2 ζ ζ 5 )ɛ 2 + O(ɛ 3 ) 2-loop coefficient function C (2) MSY M = 2 ɛ 4 5π2 6 1 ɛ 2 ζ 3 ɛ π4 + ( ) π 2 6 ζ 3 41ζ 5 ɛ ( 95 2 ζ π6 504 ) ɛ 2 + O(ɛ 3 ) Glover VDD 08

31 BDS ansatz and high-energy factorisation The BDS ansatz implies the 2-loop recursive formula for the 2-loop 4-pt amplitude m4 (2) (rescaled by the tree amplitude) m (2) 4 (ɛ) =1 2 with [ m (1) 4 (ɛ) ] G 2 (ɛ) G(2ɛ) f (2) (ɛ) m (1) 4 (2ɛ) 2 ζ2 2 + O(ɛ) f (2) (ɛ) = ζ 2 ζ 3 ɛ ζ 4 ɛ 2 Anastasiou Bern Dixon Kosower 03 (we use a different normalisation from BDS) G(ɛ) = e γɛ Γ(1 2ɛ) Γ(1 + ɛ) Γ 2 (1 ɛ) = 1 + O(ɛ2 )

32 BDS ansatz and high-energy factorisation The BDS ansatz implies the 2-loop recursive formula for the 2-loop 4-pt amplitude m4 (2) (rescaled by the tree amplitude) m (2) 4 (ɛ) =1 2 with [ m (1) 4 (ɛ) ] G 2 (ɛ) G(2ɛ) f (2) (ɛ) m (1) 4 (2ɛ) 2 ζ2 2 + O(ɛ) f (2) (ɛ) = ζ 2 ζ 3 ɛ ζ 4 ɛ 2 Anastasiou Bern Dixon Kosower 03 (we use a different normalisation from BDS) G(ɛ) = e γɛ Γ(1 2ɛ) Γ(1 + ɛ) Γ 2 (1 ɛ) = 1 + O(ɛ2 ) from the 2-loop recursive formula and high-energy factorisation, we get C (2) MSY M(ɛ) = 1 2 [ ] 2 C (1) 2 G 2 (ɛ) MSY M(ɛ) + G(2ɛ) f (2) (ɛ) C (1) MSY M(2ɛ) ζ2 2 + O(ɛ) Glover VDD 08 one needs C (1) O(ɛ 2 ) MSY M through but we know it to all orders of ε, in QCD C (1) MSY M = ψ(1 + ɛ) 2ψ( ɛ)+ψ(1) ɛ Bern Schmidt VDD 98

33 BDS ansatz and 3-loop high-energy factorisation from BDS s recursive formula for the 3-loop 4-point amplitude and high-energy factorisation, we get a recursive formula for the 3-loop coefficient function C (3) MSY M(ɛ) = 1 [ ] 3 C (1) (1) MSY M(ɛ) + C MSY M(ɛ) C (2) MSY M(ɛ) G3 (ɛ) G(3ɛ) f (3) (ɛ) C (1) MSY M(3ɛ)+4Const (3) + O(ɛ) Glover VDD 08 with f (3) (ɛ) = 11 2 ζ 4 + (6ζ 5 +5ζ 2 ζ 3 )ɛ +(c 1 ζ 6 + c 2 ζ3)ɛ 2 2 ( 341 Const (3) = ) ( 9 c 1 ζ ) 9 c 2 ζ3 2 one needs C (2) O(ɛ 2 ) MSY M through and C (1) MSY M through O(ɛ 4 )

34 Conclusions what s next? once the 2-loop 5-point amplitude in the (quasi)-multi-regge kinematics is known, we can derive the corresponding coefficient functions... work in progress Duhr Glover VDD A bootstrap approach: once we know the coefficient functions from the 2-loop 4-point and 5-point amplitudes, we can use them to build 2-loop amplitudes with 6 or more points, in the multi-regge and quasi-multi-regge kinematics, and thus obtain (hopefully useful) info on the analytic form of 2-loop amplitudes with 6 or more points in arbitrary kinematics