The Regge limit in QCD and in N=4 super-yang-mills. Vittorio Del Duca ETH Zürich & INFN LNF

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1 The Regge limit in QCD and in N=4 super-yang-mills Vittorio Del Duca ETH Zürich & INFN LNF ETH 30 May 207

2 Production of two high-energy jets as recorded by CMS at the LHC 207 CERN

3 At the LHC, particles are produced through the head-on collisions of protons, and in particular through the collisions of quarks and gluons within the protons The probability of a collision event is computed through the cross section, which is given as an integral of (squared) scattering amplitudes over the phase space of the produced particles The scattering amplitudes are given as a power series (loop expansion) in the strong and/or electroweak couplings The scattering amplitudes are fundamental objects of particle physics

4 Scattering amplitudes The scattering amplitudes are given as a loop-momentum expansion in the strong and/or electroweak couplings The more terms we know in the loop expansion, the more precisely we can compute the cross section As a matter of fact, we know any amplitude of interest at one loop, several (2 2) amplitudes at two loops, just a couple (2 ) at three loops, and nothing beyond that

5 What outcome do we expect from the loop expansion of an amplitude? From renormalisability and the infrared structure of the amplitude, we expect that the divergent parts are logarithmic functions of the external momenta But, except for unitarity, we have little guidance for the finite parts. Heuristically, we know that: at one loop, logarithmic and dilogarithmic functions of the external momenta occur beyond one loop, higher polylogarithmic functions appear and elliptic functions may appear

6 In the last few years, a lot of progress has been made in understanding the analytic structure of multi-loop amplitudes, in particular on how the polylogarithmic functions appear at any loop level In particular, a lot of progress has been made: in N=4 Super Yang-Mills (SYM) in the Regge limit of QCD in the Regge limit of N=4 SYM

7 One of the most remarkable discoveries in elementary particle physics has been that of the existence of the complex plane incipit The analytic S-matrix Eden Landshoff Olive Polkinghorne 966

8 Regge limit in the leading logarithmic approximation

9 Regge limit of QCD In perturbative QCD, in the Regge limit s» t, any scattering process is dominated by gluon exchange in the t channel For a 4-gluon tree amplitude, we obtain M gg!gg aa 0 bb (s, t) =2g 2 0 s apple (T c ) aa 0C a a 0 (p a,p a 0) s t apple (T c ) bb 0C b b 0 (p b,p b 0) C a a 0 (p a,p a 0) are called impact factors leading logarithms of s/t are obtained by the substitution α(t) is the Regge gluon trajectory, with infrared coefficients t! t (t) s t (t) = s( t, ) 4 () b () K = C A M gg!gg aa 0 bb (s, t) =2g 2 0 s () s ( t, ) + 4 = C A 2 s t 2 (2) + O 3 s (2) = C A apple apple (T c ) aa 0C a a 0 (p a,p a 0) b0 2 + b(2) K " (t) s + t C A 27 s ( t, ) = in the Regge limit, the amplitude is invariant under s u exchange. 2 3 µ 2 s (µ 2 ) 56 + n f 27 To NLL accuracy, the amplitude is given by Fadin Lipatov 993 # (t) apple s (T c ) bb t 0C b b0 (p b,p b 0) t

10 Amplitudes in the Regge limit Regge limit of the gluon-gluon amplitude M gg!gg aa 0 bb (s, t) =2g 2 0 s s t apple (T c ) aa 0C a a 0 (p a,p a 0) strip colour off & expand at one loop m () gg!gg =2g 2 s s t () (t)ln s t +2C gg () (t) " (t) s + t # (t) apple s (T c ) bb t 0C b b0 (p b,p b 0) the Regge gluon trajectory is universal; the one-loop gluon impact factor is a polynomial in t, ε, starting at /ε 2 perform the Regge limit of the quark-quark amplitude get one-loop quark impact factor if factorisation holds, one can obtain the one-loop quark-gluon amplitude by assembling the Regge trajectory and the gluon and quark impact factors the result should match the quark-gluon amplitude in the high-energy limit: it does

11 Balitski Fadin Kuraev Lipatov BFKL is a resummation of multiple gluon radiation out of the gluon exchanged in the t channel the Leading Logarithmic (BFKL ) and Next-to-Leading Logarithmic (Fadin-Lipatov 998) contributions in log(s/ t ) of the radiative corrections to the gluon propagator in the t channel are resummed to all orders in αs the resummation yields an integral (BFKL) equation for the evolution of the gluon propagator in 2-dim transverse momentum space the BFKL equation is obtained in the limit of strong rapidity ordering of the emitted gluons, with no ordering in transverse momentum - multi-regge kinematics (MRK) the solution is a Green s function of the momenta flowing in and out of the gluon ladder exchanged in the t channel

12 Mueller-Navelet jets Mueller Navelet 987 Dijet production cross section with two tagging jets in the forward and backward directions p a = x a P A p b = x b P B incoming parton momenta S: hadron centre-of-mass energy s = xaxbs: parton centre-of-mass energy ETj: jet transverse energies y = y j y j2 'log E Tj E Tj2 is the rapidity interval between the tagging jets gluon radiation is considered in MRK and resummed through the LL BFKL equation s Mueller-Navelet evaluated the inclusive dijet cross section up to 5 loops

13 the cross section for dijet production at large rapidity intervals ŝ with Mueller-Navelet dijet cross section y = y y 2 =ln ŝ = x a x b S, t= t q p 2? p2 2? dˆgg dp 2? dp2 2? d jj = 2 apple CA s p 2? f(~q?,~q 2?, y) apple CA s p 2 2? can be described through the BFKL Green s function f(~q?,~q 2?, y) = (2 ) 2 p q 2? q2 2? +X n= e in Z + d q 2 i? e q 2 2?,n with C A s y and ɸ the angle between q 2 and q2 2 and the LL BFKL eigenvalue,n = 2 E 2 + n 2 + i 2 + n 2 i

14 Mueller-Navelet dijet cross section azimuthal angle distribution (ɸjj = ɸ-π) dˆgg = (C A s ) 2 d jj 2E? 2 " ( jj )+ X k= X n= e in 2 f n,k! k # with f n,k = 2 k! Z d k,n the dijet cross section is ˆgg = (C A s ) 2 2E 2? X f 0,k k k=0 Mueller Navelet 987 with f 0,0 =, f 0, =0, f 0,2 =2 2, f 0,3 = 3 3, f 0,4 = , f 0,5 = 2 ( )

15 N=4 Super Yang Mills maximal supersymmetric theory (without gravity) conformally invariant, β fn. = 0 spin gluon 4 spin /2 gluinos 6 spin 0 real scalars t Hooft limit: Nc with λ = g 2 Nc fixed only planar diagrams AdS/CFT duality Maldacena 97 large-λ limit of 4dim CFT weakly-coupled string theory (aka weak-strong duality)

16 N=4 Super Yang Mills amplitudes in N=4 SYM are much simpler than in Standard Model processes use N=4 SYM as a computational lab: to learn techniques and tools to be used in Standard Model calculations to learn about the bases of special functions which may occur in the scattering processes

17 N=4 Super Yang Mills In the last years, a huge progress has been made in understanding the analytic structure of the S-matrix of N=4 SYM Besides the ordinary conformal symmetry, in the planar limit the S-matrix exhibits a dual conformal symmetry Drummond Henn Smirnov Sokatchev 2006 Accordingly, the analytic structure of the scattering amplitudes is highly constraint 4- and 5-point amplitudes are fixed to all loops by the symmetries in terms of the one-loop amplitudes and the cusp anomalous dimension Anastasiou Bern Dixon Kosower 2003, Bern Dixon Smirnov 2005 Drummond Henn Korchemsky Sokatchev 2007 Beyond 5 points, the finite part of the amplitudes is given in terms of a remainder function R. The symmetries only fix the variables of R (some conformally invariant cross ratios) but not the analytic dependence of R on them

18 for n = 6, the conformally invariant cross ratios are 2 u = x2 3x 2 46 x 2 4 x2 36 u 2 = x2 24x 2 5 x 2 25 x2 4 xi are variables in a dual space s.t. u 3 = x2 35x 2 26 x 2 36 x2 25 p i = x i x i thus x 2 k,k+r =(p k p k+r ) 2 for n points, dual conformal invariance implies dependence on 3n-5 independent cross ratios u i = x2 i+,i+5 x2 i+2,i+4 x 2 i+,i+4 x2 i+2,i+5, u 2i = x2 N,i+3 x2,i+2 x 2 N,i+2 x2,i+3, u 3i = x2 i+4 x2 2,i+3 x 2,i+3 x2 2,i+4

19 Scattering amplitudes Scattering amplitudes of gluons are functions of: the external momenta colour helicities: maximally helicity violating ( ) next-to-maximally helicity violating ( ) and so on

20 Scattering amplitudes in N=4 SYM The progress in understanding the analytic structure of the S-matrix in planar N=4 SYM is also due to an improved understanding of the mathematical structures underlying the scattering amplitudes n-point amplitudes are expected to be written in terms of iterated integrals (on the space of configurations of points in 3-dim projective space Confn(CP 3 ) ) The simplest case of iterated integrals are the iterated integrals over rational functions, i.e. the multiple polylogarithms G(a, w; z) = z 0 dt t a G( w; t), G(a; z) = ln z a Golden Paulos Spradlin Volovich 204 Goncharov 200 G(0, ; z) = It is thought that maximally helicity violating (MHV) and next-to-mhv (NMHV) amplitudes can be expressed in terms of multiple polylogarithms of uniform transcendental weight Li 2 (z) Arkani-Hamed et al. Scattering Amplitudes and the Positive Grassmannian 202

21 Scattering amplitudes in N=4 SYM MHV and NMHV amplitudes feature maximal transcendentality, i.e. L-loop amplitudes are expressed in terms of multiple polylogarithms of weight 2L only MHV amplitudes are pure, i.e. the coefficients of the multiple polylogarithms are (rational) numbers

22 Scattering amplitudes in N=4 SYM 6-pt (N)MHV amplitudes are known analytically up to 5(4) loops Duhr Smirnov VDD 2009 Goncharov Spradlin Vergu Volovich 200 Dixon Drummond Henn 20 Dixon Drummond von Hippel Pennington 203 Dixon Drummond Duhr Pennington 204 Caron-Huot Dixon von Hippel McLeod 206 Dixon Drummond Henn 20 Dixon von Hippel 204 Dixon von Hippel McLeod pt MHV amplitudes are known analytically at two loops Golden Spradlin 204 No analytic result is known beyond 7 points. The cluster algebra which defines the multiple polylogarithms is infinite starting from 8 points Golden Goncharov Spradlin Vergu Volovich 203

23 Multi-Regge limit of N=4 SYM In the Euclidean region (where all Mandelstam invariants are negative), amplitudes in MRK factorise completely in terms of building blocks which are expressed in terms of Regge poles and can be determined to all orders through the 4-pt and 5-pt amplitudes. Thus the remainder functions R vanish at all points Duhr Glover VDD 2008 After analytic continuation to some regions of the Minkowski space, the amplitudes develops cuts which are described by a dispersion relation for octet exchange, which is similar to the singlet BFKL equation in QCD Bartels Lipatov Sabio-Vera 2008 Accordingly, 6-pt amplitudes have been thoroughly examined, both at weak and at strong coupling In particular, 6-pt amplitudes at weak coupling can be expressed in terms of single-valued harmonic polylogarithms Dixon Duhr Pennington 202

24 Regge factorisation of the n-pt amplitude m n (, 2,...,n)=s [gc(p 2,p 3 )] t 2 s 2 s n 3 t n 3 (t2 ) n-pt amplitude in the multi-regge limit (tn 3 ) [gv(q 2,q, )] y 3 y 4 y n ; p 3 p 4... p n s s,s 2,...,s n 3 t, t 2..., t n 3 [gv(q n 3,q n 4, n 4)] (t ) s t [gc(p,p n )] the l-loop n-pt amplitude can be assembled using the l-loop trajectories, vertices and coefficient functions, determined through the l-loop 4-pt and 5-pt amplitudes in Euclidean space, no violation of the BDS ansatz can be found in the multi-regge limit Duhr Glover VDD 2008

25 Moduli space of Riemann spheres in MRK, there is no ordering in transverse momentum, i.e. only the n-2 transverse momenta are non-trivial dual conformal invariance in transverse momentum space implies dependence on n-5 cross ratios of the transverse momenta z i = (x x i+3 )(x i+2 x i+ ) (x x i+ )(x i+2 x i+3 ) = q i+ k i q i k i+ i =,...,n 5 M0,p = space of configurations of p points on the Riemann sphere Because we can fix 3 points at 0,,, its dimension is dim(m0,p)= p-3 M0,n-2 is the space of the MRK, with dim(m0,n-2) = n-5 Its coordinates can be chosen to be the zi s, i.e. the cross ratios of the transverse momenta Drummond Druc Duhr Dulat Marzucca Papathanasiou Verbeek VDD 206

26 Iterated integrals on M0,n-2 on M0,n-2, the singularities are associated to degenerate configurations when two points merge xi xi+ i.e. when momentum pi becomes soft pi 0 iterated integrals on M0,p can be written as multiple polylogarithms Brown 2006 amplitudes in MRK can be written in terms of multiple polylogarithms analytic structure of amplitudes is constrained by unitarity and the optical theorem Disc(M) =imm massless amplitudes may have branch points when Mandelstam invariants vanish sij 0 or become infinite sij, but branch cuts are constrained by unitarity

27 Hopf algebra and the coproduct algebra is a vector space with a product μ: A A A μ(a b) = a b that is associative A A A A A A (a b) c = a (b c) coalgebra is a vector space with a coproduct Δ: B B B that is coassociative B B B B B B μ puts together; Δ decomposes (a) = X i a () i a (2) i a Hopf algebra is an algebra and a coalgebra, such that product and coproduct are compatible Δ(a b) = Δ(a) Δ(b) multiple polylogarithms form a Hopf algebra with a coproduct wx wx (L w )= k,w k(l w )= L k L w k=0 k=0 in particular, on classical polylogarithms k Goncharov 2002 (ln z) = ln z +lnz Li 2 (z) = Li 2 (z) + Li 2 (z) ln( z) ln z the two entries of the coproduct are the discontinuity and the derivative Disc = (Disc Duhr 202

28 then the coproduct of an amplitude is related to unitarity thus, for massless amplitudes (M) =ln(s ij )... in particular, for amplitudes in MRK (M) =ln x i x j 2... except for the soft limit pi 0, in MRK the transverse momenta never vanish x i x j 2 6=0 single-valued functions therefore, n-point amplitudes in MRK of planar N=4 SYM can be written in terms of single-valued iterated integrals on M0,n-2 Drummond Druc Duhr Dulat Marzucca Papathanasiou Verbeek VDD 206 for n=6, iterated integrals on M0,4 are harmonic polylogarithms thus, 6-point amplitudes in MRK of can be written in terms of single-valued harmonic polylogarithms (SVHPL) Dixon Duhr Pennington 202

29 Harmonic polylogarithms classical polylogarithms Li m (z) = Z z 0 dz 0 Li m (z 0 ) z 0 harmonic polylogarithms (HPLs) H(a, ~w; z) = Z z 0 dtf(a; t) H( ~w ; t) f( ; t) = +t, f(0; t) = t, f(; t) = t with {a, ~w} 2 {, 0, } Remiddi Vermaseren 999 HPLs obey the differential equations d dz H 0!(z) = H!(z) z, d dz H!(z) = H!(z) z subject to the constraints H(z) =, H ~ 0 n (z) = n! lnn z, lim z!0 H!6= ~0 n (z) =0 HPLs form a shuffle algebra H! (z) H!2 (z) = X! H! (z) with ω the shuffle of ω and ω2 HPLs are multi-valued functions on the complex plane

30 Single-valued polylogarithms Single-valued functions are real analytic functions on the complex plane Because the discontinuities of the classical polylogarithms are known z Li n (z) =2 i logn (n )! one can build combinations of classical polylogarithms such that all branch cuts cancel on the punctured plane C/{0,} (Riemann sphere with punctures) An example is the Bloch-Wigner dilogarithm D 2 (z) = Im[Li 2 (z)] + arg( z) log z

31 Single-valued harmonic polylogarithms define a function L that is real-analytic and single-valued on and that has the same properties as the HPLs C/{0, } it obeys the L 0!(z) = L!(z) = L!(z) z subject to the constraints L e (z) =, L ~ 0 n (z) = n! lnn z 2 lim z!0 L!6= ~0 n (z) =0 the SVHPLs L ω (z) also form a shuffle algebra L! (z) L!2 (z) = X! L! (z) with ω the shuffle of ω and ω2 SVHPLs can be explicitly expressed as combinations of HPLs such that all the branch cuts cancel Brown 2004 examples L 0 (z) =H 0 (z)+h 0 ( z) =ln z 2 L (z) =H (z)+h ( z) = ln +z 2 L 0, (z) = 2H,0 +2 H,0 +2H 0 H 2 4 H 0 H +2H 0. 2 H 0, = Li 2 (z) Li 2 ( z)+ 2 ln z 2 (ln( z) ln( z))

32 Single-valued multiple polylogarithms Single-valued multiple polylogarithms (SVMPL) can be constructed through a map that to each multiple polylogarithm associates its single-valued version examples of SVMPLs G a (z) =G a (z)+gā( z) =ln G a,b (z) =G a,b (z)+g b,ā ( z)+g b (a)gā( z)+g b(ā)gā( z) G a (b)g b( z)+g a (z)g b( z) z a 2 Gā( b)g b( z) Brown 2004, 203, 205

33 MRK in N=4 SYM In MRK, 6-pt MHV and NMHV amplitudes are known at any number of loops Lipatov Prygarin Dixon Duhr Pennington 202 Pennington 202 Lipatov Prygarin Schnitzer 202 Beyond 6 points, only 2-loop MHV amplitudes were known in MRK at LLA In MRK at LLA, the 2-loop n-pt remainder function Rn (2) can be written as a sum of 2-loop 6-pt remainder functions R6 (2) In MRK at LLA, we can compute all MHV amplitudes at L loops in terms of amplitudes with up to (L+4) points Prygarin Spradlin Vergu Volovich 20 Bartels Kormilitzin Lipatov Prygarin 20 Bargheer Papathanasiou Schomerus 205 Drummond Druc Duhr Dulat Marzucca Papathanasiou Verbeek VDD 206 We showed explicitly that the MHV amplitudes at: 2 loops are determined by the 2-loop 6-pt amplitudes 3 loops are determined by the 6- and 7-pt amplitudes through 3 loops 4 loops are determined by the 6-, 7-, and 8-pt amplitudes through 4 loops 5 loops are determined by the 6-, 7-, 8- and 9-pt amplitudes through 5 loops We computed also all non-mhv amplitudes up 8 points and 4 loops

34 MRK factorisation in N=4 SYM For the helicities h,, hn-4 define the ratio R h,...,h N 4 = apple AN (, +,h,...,h N 4, +, ) A BDS N (, +,...,+, ) MRK, LLA factorisation in MRK at LLA R h,...,h N 4 (,z,..., N 5,z N 5 )! X X NY 5 2 i a i i k! lni k k i=2 i +...+i N 5 =i with τk = function of cross ratios, and with coefficients g (i,...,i N 5 ) h,...,h N 4 (z,...,z N 5 )= ( )N+ 2 h (,n ) k= " NY 5 k= 2 4 Y N 5 j=2 +X n k = zk z k g (i,...,i N 5 ) h,...,h N 4 (z,...,z N 5 ) nk /2 Z + d k 2 z k 2i k E i k 3 k n k # C h j ( j,n j, j,n j ) 5 h N 4 ( N 5,n N 5 ) where: the χ s are the 2 impact factors, the C s are the N-6 central-emission vertices the E s are the N-5 BFKL-like eigenvalues for octet exchange h C h 2 C h N 5 h N 4 i i2 3 in-4 in N-2 N- N + h h2 hn-5 hn-4 +

35 Convolutions we use the Fouries-Mellin (FM) transform F[F (,n)] = X n/2 Z + z z n= d 2 z 2i F (,n) which maps products into convolutions F[F G] =F[F ] F[G] =(f g)(z) = Z d 2 w z w 2 f(w) g w we compute the integral through the residue formula Z d 2 z f(z) =Res z=f (z) X i Res z=ai F (z) Schnetz 203 where F is the antiholomorphic primitive of F = f

36 through the FM transform of the BFKL eigenvalue E(z) =F [E n ] we can write the recursion Convolutions and factorization g (i,...,i k +,...,i N 5 ) (z,...,z N 5 )=E(z k ) g (i,...,i N 5 ) (z,...,z N 5 ) which implies that we can drop all the propagators without a log g (0,...,0,i a,0,...,0,i a2,0,...,0,i ak,0,...,0) (,..., N 5 )=g (i a,i a2,...,i ak ) ( ia, ia2,..., iak ) example for N=7, with h = h2-2 h h - 2 h h - C h 2 h 3 i h2 h3 + - h 3 i 5 6 h 3 + which connects amplitudes with a different number of legs

37 in fact, if all indices are zero except for one g (0,...,0,i a,0,...,0) (,..., N 5 )=g (i a) ++ ( a ) which implies that R (2) = X with appleiapplen 5 g () ++( )= 4 G 0, ( ) ln i g () ++( i ) 4 G,0 ( )+ 2 G, ( ) which shows, as previously stated, that in MRK at LLA, the 2-loop n-pt remainder function Rn (2) can be written as a sum of 2-loop 6-pt amplitudes, in terms of SVHPLs

38 At 3 loops, the n-pt remainder function Rn (3) can be written as a sum of 3-loop 6-pt and 7-pt amplitudes R (3) = 2 X appleiapplen 5 ln 2 i g (2) ++( i )+ X applei<japplen 5 ln i ln j g (,) +++( i, j ) with g (2) ++( )= 8 G 0,0, ( ) 4 G 0,,0 ( )+ 2 G 0,, ( ) + 2 G,0, ( )+ 2 G,,0 ( ) G,, ( ) 8 G,0,0 ( ) g (,) +++(, 2 )= 8 G 0,, 2 ( ) 8 G 0, 2, ( )+ 8 G,, 2 ( ) 8 G, 2,0 ( ) 8 G 2,,0 ( )+ 8 G 2,, ( )+ 4 G, 2, ( ) 4 G ( 2 ) G, 2 ( ) + 8 G ( ) G 0,0 ( 2 ) 8 G 0 ( 2 ) G 0, ( )+ 8 G ( 2 ) G 0, ( ) 8 G 2 ( ) G 0, ( 2 ) + 8 G ( 2 ) G 0, 2 ( ) 8 G 0 ( 2 ) G,0 ( )+ 8 G ( 2 ) G,0 ( )+ 8 G 0 ( 2 ) G, ( ) 8 G ( 2 ) G, ( ) 8 G ( ) G, ( 2 )+ 8 G 2 ( ) G, ( 2 )+ 8 G 0 ( 2 ) G, 2 ( ) + 8 G ( 2 ) G 2,0 ( ) 8 G ( 2 ) G 2, ( ) Note that Rn (3) cannot be written only in terms of SVHPLs, but SVMPLs are necessary

39 At 4 loops, the n-pt remainder function Rn (4) can be written as a sum of 4-loop 6-pt, 7-pt and 8-pt amplitudes, and in general, we can compute all MHV amplitudes at L loops in terms of amplitudes with up to (L+4) points Drummond Druc Duhr Dulat Marzucca Papathanasiou Verbeek VDD 206 We displayed that explicitly up to 5 loops, showing that the n-pt remainder function Rn (5) can be written as a sum of 5-loop 6-, 7-, 8- and 9-pt amplitudes Note also that because the convolutions with pure functions are pure, so are the MHV amplitudes MRK factorisation works also for non-mhv amplitudes, however at each loop the number of building blocks is infinite, and the helicity flips make the non-mhv amplitudes not pure We constructed explicitly all non-mhv amplitudes through 4 loop and 8 points

40 Mueller-Navelet jets and SVHPLs The singlet LL BFKL ladder in QCD, and thus the dijet cross section in the high-energy limit, can also be expressed in terms of SVHPLs, i.e. in terms of single-valued iterated integrals on M0,4 Dixon Duhr Pennington VDD 203 Mueller & Navelet evaluated analytically the inclusive dijet cross section up to 5 loops. We evaluated it analytically up to 3 loops Also, we could evaluate analytically the dijet cross section differential in the jet transverse energies or the azimuthal angle between the jets (up to 6 loops)

41 BFKL Green s function and single-valued functions use complex transverse momentum q k q x k + iq y k and a complex variable z q q 2 the Green's function can be expanded into a power series in µ = µ y f LL (q,q 2, µ )= 2 (2) (q q 2 )+ 2 p q 2 q2 2 X k= k µ k! f k LL (z) where the coefficient functions fk are given by the Fourier-Mellin transform f LL k (z) =F k n = + X n= z z n/2 Z + d 2 z 2i k n the fk have a unique, well-defined value for every ratio of the magnitudes of the two jet transverse momenta and angle between them. So, they are real-analytic functions of w

42 generating functional of SVHPLs to all orders in η the BFKL Green's function can be written in terms of a generating functional of SVHPLs Dixon Duhr Pennington VDD 203 writing the coefficient function fk as f LL k (z) = z 2 z 2 F k(z) we obtain that the first few functions Fk are F (z) =, F 2 (z) =2G (z) G 0 (z), F 3 (z) =6G, (z) 3 G 0, (z) 3 G,0 (z)+g 0,0,0 (z), F 4 (z) = 24 G,, (z)+4g 0,0, (z)+6g 0,,0 (z) 2 G 0,, (z)+4g,0,0 (z) 2 G,0, (z) 2 G,,0 (z) G 0,0,0 (z)+8 3 note that fk has weight k-

43 Azimuthal angle distribution this allows us to write the azimuthal angle distribution as dˆgg = (C A s ) 2 d jj 2E? 2 " ( jj )+ X k= a k ( jj ) where the contribution of the k th loop is k # with A ( jj )= 2 H 0, A 2 ( jj )=H,0, A 3 ( jj )= 2 3 H 0,0,0 2H,, H 0 i 2, A 4 ( jj )= 4 3 H 4 0,0,,0 H 0,,0,0 3 H,0,0,0 +4H,,,0 2 2H 0, H, H 0 + i 2 2 H 2 3, A 5 ( jj )= 46 5 H 0,0,0,0, H 0,0,,,0 +2H 0,,0,,0 +2H 0,,,0, H,0,0,,0 +2H,0,,0, H,,0,0,0 8H,,,,0 2 5 H 20 0,0,0 4H 0,, 4H,0, 3 H,,0 3 2H 0, H, H 0 + i apple 2 H 0 0,0 4H, +4 3 H 3 4 where H i,j,... H i,j,... (e 2i jj ) Dixon Duhr Pennington VDD 203

44 Transverse momentum distribution dˆgg dp 2? dp2 2? = (C A s ) 2 2p 2? p2 2? " (p 2? p 2 2?)+ 2 p p 2? p2 2? b( ; ) # where = w with B ( ) =, B 2 ( ) = 2 H 0 2H, B 3 ( ) = 6 H 0,0 +2H 0, + H,0 +4H,, B 4 ( ) = 24 H 0,0,0 4 3 H 0,0, H 0,,0 4H 0,, 3 H,0,0 4H,0, 2H,,0 8H,, + 3 3, B 5 ( ) = 20 H 0,0,0, H 0,0,0, H 0,0,, H 0,0,, + 3 H 0,,0,0 +4H 0,,0, +2H 0,,,0 +8H 0,,, + 2 H,0,0, H,0,0, +2H,0,,0 +8H,0,, H,,0,0 +8H,,0, +4H,,,0 + 6H,,, H H, where H i,j,... H i,j,... ( 2 ) Dixon Duhr Pennington VDD 203

45 Mueller-Navelet dijet cross section reloaded the MN dijet cross section is the first 5 loops were computed by Mueller-Navelet. We computed it through the 3 loops Dixon Duhr Pennington VDD 203

46 Regge limit in the next-to-leading logarithmic approximation

47 BFKL theory the BFKL equation describes the evolution of the gluon propagator in 2-dim transverse momentum space! f! (q,q 2 )= Z (2) (q q 2 )+ 2 d 2 kk(q,k) f! (k, q 2 ) the solution is given in terms of eigenfunctions Φ νn and an eigenvalue ω νn f! (q,q 2 )= +X n= Z + d!! n n(q ) as a function of rapidity, the solution is n(q 2 ) f(q,q 2,y)= +X Z + n= d n (q ) n(q 2 ) e y! n we expand kernel K, eigenfunctions Φ νn and eigenvalue ω νn in powers of µ = N C S(µ 2 ) K(q,q 2 )= µ At LLA! (0) n = 2 E X l=0 l µ K (l) (q,q 2 )! n = µ X n + 2 n + + i 2 l=0 i l µ! (l) n n(q) = X l=0 l µ (l) n(q) (0) n (q) = 2 (q2 ) /2+i e in note that in N=4 SYM the eigenfunctions and the eigenvalue are the same

48 BFKL eigenvalue at NLLA At NLLA in QCD and in N=4 SYM, the eigenvalue is Fadin Lipatov 998 () n = + n + (2) n 4 4 4! () with one-loop beta function and two-loop cusp anomalous dimension 0 = 3 and with (3) n + (2) K n 2N f 3N c (2) K = n N f 9N c 2 2 Kotikov Lipatov 2000, 2002 Chirilli Kovchegov 203 Duhr Marzucca Verbeek VDD 207 () n 2 n n =! (0) n (2) n = 2 (n, ) 2 (n, ) (3) n = ( 2 + i ) ( apple 2 i ) 2i 2 + i 2 apple n N f 2+3 ( ) Nc 3 (3 2 )( + 2 ) i n 2 + N f Nc 3 ( ) 2(3 2 )( + 2 ) Φ(n,γ) is a sum over linear combinations of ψ functions and γ is a shorthand γ = /2 + iν In blue we labeled the terms which occur only in QCD, in red the ones which occur in QCD and in N=4 SYM

49 BFKL eigenfunctions at NLLA At NLLA in QCD, the eigenfunction is n(q) = (0) n (q) apple + µ 0 8 ln q2 µ P At NLLA, the expansion of the BFKL ladder n + i ln q2 n µ 2 P n Chirilli Kovchegov 203 Duhr Marzucca Verbeek VDD O( 2 µ) f(q,q 2,y)=f LL (q,q 2, µ )+ µ f NLL (q,q 2, µ )+..., µ = µ y f NLL contains the NLO corrections to the eigenvalue and to the eigenfunctions, however if we use the scale of the strong coupling to be the geometric mean of the transverse momenta at the ends of the ladder, then we can use the LO eigenfunctions instead of the NLO ones f(q,q 2,y)= with +X n= µ 2 = s 0 = Z + q q 2 q2 2 d (0) n (q ) (0) n (q 2 ) e y S(s 0 )[! (0) n + S(s 0 )! () n ] +... Duhr Marzucca Verbeek VDD 207

50 Fourier-Mellin transform At NLLA, the BFKL ladder is f NLL (q,q 2, s0 )= 2 p q 2 q2 2 X k= k s 0 k! f NLL k+ (z) s0 = S (s 0 ) y with coefficients given by the Fourier-Mellin transform f NLL k (z) =F h i! n () k n 2 = +X n= z z n/2 Z + using the explicit form of the eigenvalue d 2 z 2i! n () k n 2 n =! n (0) () n = + n + (2) n 4 4 4! () (3) n + (2) K n n the coefficients can be written as f NLL k with (z) = 4 C() k (z)+ 4 C(2) k h i C (i) k (z) =F (i) n k n 2 (z) + 4 C(3) (2) k (z)+ K f LL k (z) 8 0 fk LL (z) fk LL 2(z) the weight of f NLL k is weight(f NLL k)= k k 0 apple w apple k k 2 apple w apple k k k

51 SV functions C () k (z) are SVHPLs of uniform weight k with singularities at z=0 and z= C (3) k (z) are MPLs of type G(a,...,a n ; z ) with a k 2 { i, 0,i} they are SV functions of z because they have no branch cut on the positive real axis, and have weight 0 w k For C (2) k (z) one needs Schnetz generalised SVMPLs with singularities at z = z + z +,,,, 2 C Schnetz 206 then one can show that C (2) k (z) are Schnetz generalised SVMPLs G(a,...,a n ; z) with singularities at a i 2 {, 0,, / z} Duhr Marzucca Verbeek VDD 207 In moment space, the maximal weight of the BFKL eigenvalue and of the anomalous dimensions of the leading twist operators which control the Bjorken scaling violations in QCD is the same as the corresponding quantities in N=4 SYM Kotikov Lipatov 2000, 2002 Kotikov Lipatov Velizhanin 2003 Interestingly, in transverse momentum space at NLLA, the maximal weight of the BFKL ladder in QCD is not the same as the one of the ladder in N=4 SYM Duhr Marzucca Verbeek VDD 207

52 BFKL ladder in a generic SU(Nc) gauge theory one can consider the BFKL eigenvalue at NLLA in a SU(Nc) gauge theory with scalar or fermionic matter in arbitrary representations! () n = 4 () n + 4 (2) n + 4 with 0(ñ f, ñ s )= 3 (3) n (Ñf, Ñs) (2) (ñ f, ñ s ) n 2ñ f 3N c ñ s 6N c (2) (ñ f, ñ s )= (ñ f, ñ s ) 0 ñ f 9N c 2 n 4ñ s 9N c Kotikov Lipatov ñ f = X R n R f T R ñ s = X R n R s T R Tr(T a RT b R)=T R ab T F = 2 ñ s (ñ f )= number of scalars (Weyl fermions) in the representation R (3) n (Ñf, Ñs) = n (3,) (Ñf, X with Ñ x = 2 R (3,2) Ñs)+ n (Ñf, Ñs) n R x T R (2C R N c ), x = f,s Necessary and sufficient conditions for a SU(Nc) gauge theory to have a BFKL ladder of maximal weight are: the one-loop beta function must vanish the two-loop cusp AD must be proportional to ζ2 (3,2) must vanish n 2Ñf = N 2 c + Ñs Duhr Marzucca Verbeek VDD 207 There is no theory whose BFKL ladder has uniform maximal weight which agrees with the maximal weight terms of QCD Duhr Marzucca Verbeek VDD 207

53 Matter in the fundamental and in the adjoint We solve the conditions above for matter in the fundamental F and in the adjoint A representations. We obtain: 2 n F f = n F s 2 n A f =2+n A s which describes the spectrum of a gauge theory with N supersymmetries and n F = nf F chiral multiplets in F and n A = nf A - N chiral multiplets in A There are four solutions to those conditions the first is N=4 SYM the second is N=2 superconformal QCD with Nf = 2Nc hypermultiplets the third is N= superconf. QCD Duhr Marzucca Verbeek VDD 207 because the one-loop beta function is fixed by matter loops in gluon self-energies, we are only sensitive to the matter content of a theory, and not to its details (like scalar potential or Yukawa couplings)

54 Conclusions at LLA n-point amplitudes in the LLA of the Multi-Regge limit of N=4 SYM can be fully expressed in terms of single-valued iterated integrals on M0,n-2 i.e. in terms of single-valued multiple polylogarithms We showed that one can compute all MHV amplitudes at L loops in terms of amplitudes with up to (L+4) points We displayed that explicitly up to 5 loops, showing that the n-pt remainder function Rn (5) can be written as a sum of 5-loop 6-, 7-, 8- and 9-pt amplitudes We constructed explicitly all non-mhv amplitudes through 4 loop and 8 points The singlet BFKL ladder, and thus the dijet cross section in the LLA of the high-energy limit of QCD, can also be expressed in terms of of single-valued iterated integrals on M0,4, i.e. in terms of SVHPLs We computed the Mueller-Navelet dijet cross section through 3 loops We expect that the n-jet cross section in the LLA of the high-energy limit of QCD can be written in terms of single-valued iterated integrals on M0,n+2

55 Conclusions at NLLA The singlet BFKL ladder at NLLA in QCD and in N=4 SYM can also be expressed in terms of SVMPLs, but generalised to non-constant values of the roots In transverse momentum space at NLLA, the maximal weight of the BFKL ladder in QCD is not the same as the one of the ladder in N=4 SYM There is no SU(Nc) gauge theory whose BFKL ladder has uniform maximal weight which agrees with the maximal weight terms of QCD We found four theories with matter in the fundamental and in the adjoint representations, whose BFKL ladder has uniform maximal weight