Radcor-Loopfest 2015

Transcription

1 E H U N I V E R S I T T Y O H F E D I N B U R G Radcor-Loopfest 205 Long-distance singularities in multi-leg scattering amplitudes Einan Gardi Higgs Centre for theoretical physics, Edinburgh! new 3-loop result for the soft anomalous dimension - work with Øyvind Almelid and Claude Duhr 9 June, 205

2 long-distance singularities in multi-leg scattering amplitudes Plan of the talk! Soft singularities: fixed-angle factorization, Wilson lines, rescaling symmetry.! The soft anomalous dimension for massless partons: the dipole formula.! Quadrupole interaction at 3-loop.! Special kinematics: Regge limit, collinear limit.

3 The soft (Eikonal) approximation and rescaling symmetry Eikonal Feynman rules:! k p Assuming such that all components of k are small: ū(p) ig s T (a) µ i(p/ + k/ + m) (p + k) 2 m 2 +i p T p+k k<<p ū(p)g s T (a) p µ p k +i Rescaling invariance: only the direction and the colour charge of the emitter matter. equivalent to emission from a Wilson line: g s T (a) p µ p k +i" = g st (a) (, 0) P i exp ig s Z 0 µ k +i" d A( ) This symmetry is realised differently for lightlike and massive Wilson lines.

4 IR Singularities from Wilson lines Factorization at fixed angles: all kinematic invariants are simultaneously taken large p i p j = Q 2 i j 2 Soft singularities factorise to all orders & computed from a product of Wilson lines: p 5"hard"gluon"amplitude"" p 5 5"Wilson"line"amplitude"" 5 p 4 4 p 2 p M J (p i, IR )= X K S JK ( ij, IR ) H K (p i ) S is a product of Wilson lines: S = h 2... n i Due to rescaling symmetry it only depends on angles: ij = 2 i q 2 i j 2 j

5 IR Singularities for AmplitudEs with massless legs Exponentiation through solution of renormaliaztion-group equations: M pi µ, s, =exp ( 2 Z µ 2 0 d /s ij, s( 2, ) ) H pi µ, s The Dipole Formula: simple ansatz for the singularity structure of multi-leg massless amplitudes (, s )= 4 b K ( s ) X (i,j) Dip. ln 2 s ij T i T j + nx i= J i ( s ) Becher & Neubert, EG & Magnea (2009) Complete two-loop calculation by Dixon, Mert-Aybat and Sterman in 2006! (confirming Catani s predictions from 998).!! Generalization to all orders motivated by constraints based on soft/jet factorisation and rescaling symmetry.!! Jet Jet 2 Soft Jet 5 Jet 3 Jet 4

6 Factorization of AmplitudEs with massless legs Fixed angle scattering p i p j = Q 2 i j 2 p 2 i =0 with lightlike partons! IR singularities can be factorised - all originate in soft and collinear regions M N (p i /µ, ) = X L Lightlike Wilson lines! S NL ( i ny i= 2pi p j j, ) H L µ 2, (2p i n i ) 2 n 2 i µ2 J i (2pi n i ) 2 n 2 i µ2, J i 2( i n i ) 2 n 2 i Jets (colour singlet) Double counting of soft-collinear region is removed by dividing by eikonal jets.! Kinematic dependence of the soft function is now on i j, violating rescaling symmetry. This collinear anomaly is restored by the eikonal jets. This implies an all-order constraint on the soft function, leading to the Dipole Formula., Becher & Neubert, EG & Magnea (2009)

7 ! Corrections to the dipole formula First possible corrections to the Dipole Formula: Functions of conformally-invariant cross ratios at 3-loops, 4 legs: = Dip. + ( ijkl ) ijkl = (p i p j )(p k p l ) (p i p k )(p j p l ) ( ijkl ) is highly constrained by: Non-Abelian exponentiation! Bose symmetry Transcendental weight Collinear limits Regge limit if abe if cde T a T b 2 T c 3 T d 4 EG & Magnea, Becher & Neubert (2009) Dixon, EG & Magnea (200) Del Duca, Duhr, EG, Magnea & White (20) Ahrens & Neubert &Vernazza (202) Caron-Huot (203)

8 The structure of the Soft Anomalous dimension: Massless vs. Massive partons 2 3 cusp Dip. Tri Quad. 4 / planar T T 2 if abc T a T b 2 T c 3 if abe if cde T a T b 2 T c 3 T d 4 massive 3-loop* 2-loops 3-loop 3-loop - in progress massless 3-loop forbidden to all loops 3-loop done! ** * Grozin, Henn, Korchemsky & Marquard, Phys. Rev. Lett. 4, (205) **Almelid, Duhr, EG - to appear

9 Computing IR Singularities at 3-loops Classes of three-loop webs connecting four Wilson lines Single connected subgraph Each web depends on all six angles - can form conformally-invariant cross ratios (cicrs) Two connected subgraphs Depends on 4, 23, 24, 34 only. Cannot form cicrs - yields products of logs for near lightlike kinematics 2 Three connected subgraphs (multiple gluon exchanges) Depends on 3 angles only! Cannot form cicrs - yields products of logs for near lightlike kinematics

10 dual momentum Box integral W 4g = z Parametrise the positions along the Wilson lines by x µ i = µ i s i Define auxiliary momenta p i = x i x i The z integral is a 4-mass Box(p,p 2,p 3,p 4 ) C 4g =T a T b 2 T c 3 T d 4 h i f abe f cde ( )+f ade f bce ( )+f ace f bde ( ) Z W 4g =gs 6 N 4 C 4g ds ds 2 ds 3 ds 4 Box(x x 4,x 2 x,x 3 x 2,x 4 x 3 ) 0 s s 2 s 3 s 4 C A = 0 0 ca c( a) ( c)b ( c)( b) C A Integration over yields an overall / UV pole.! Remaining integrations can be done in 4 dimensions. Ø. Almelid, C. Duhr, EG

11 Connected three-loop webs with two 3-gluon vertices A similar mapping - but with a diagonal box Ø. Almelid, C. Duhr, EG W (3g) 2 =! may have non-trivial kinematic dependence in the limit! ij kl!! ijkl =!!! = ( i j)( k l) ( i k)( j l)! W 4g and W (3g) 2 ik jl 234 = z z 432 =( z)( z) 2 i! 0! We extract the asymptotic near-lightlike behaviour using the Mellin-Barnes technique. The remaining MB integral is three-fold, and can be converted into! an iterated parameter integral and be expressed in terms of polylogarithms.

12 Connected webs: results and Bose symmetry (3, ) s 3 w 4g = T a 4 T b 2T c 3T4h d f abe f cde (z z z z) i + f ade f bce ( z z)+f ace f bde ( z z) z z g (z, z,{ ij }) 2 ijkl = ij kl ik jl = ( i j)( k l) ( i k)( j l) 234 = z z 432 =( z)( z) 4 3 The permutation symmetry of the colour factors is mapped onto the kinematics g (z, z,{ ij }) is symmetric under these transformations 2

13 summing the Connected webs results s 3 T a 4 T b 2T c 3T4h d (3, ) w 4g = f abe f cde (z z z z) i + f ade f bce ( z z)+f ace f bde ( z z) z z g (z, z,{ ij }) (3, ) w (2)(34) = s 3 T a 4 T b 2T c 3T d 4 f abe f cdeh g 0 (z, z,{ ij }) z z z z z z 3 i g (z, z,{ ij }) cancels in the sum! Pure function of uniform weight 5 (N=4 SYM property) Symbol alphabet {z, z, z, z} relating to collinear/regge limits

14 From the connected webs to the full quadrupole term in the soft anon. dim. After applying Jacobi Identity one finds (3, ) w con. = s 3 i T a 4 T b 2T c 3T4h d f ade f bce F con. (z, z,{ ij })+f abe f cde F con. 2 (z, z,{ ij }) and the functions separate into a polylogarithmic function of depending only on conformally invariant cross ratios via {z, z}, and a function involving purely logarithmic dependence on individual cusp angles: F n con. (z, z,{ ij })=F n con. (z, z)+q con. n ({log( ij )}) Rescaling symmetry implies that the quadrupole contribution to the light-like soft anomalous dimension would depend exclusively on {z, z}! Indeed, we so far put aside all non-connected webs These, in the light-like asymptotics, only involve logarithms, similarly to Q con. n ({log( ij )}). These must cancel any dependence on ln( ij ) which isn t rescaling invariant.!! One can infer the final, rescaling-invariant answer from connected webs alone!!

15 Regge limit constraints The high-energy limit is dominated by t-channel exchange. Large Logs of (-t/s) are summed through Reggeization: t! t (t) s t Long-distance singularities of the Regge trajectory are controlled by (t) = 4 (T 2 + T 3 ) 2 Z 0 t d 2 2 b K( s ( 2, )) Korchemskaya and Korchemsky (996) Del Duca, Duhr, EG, Magnea & White (20) Reggeization is proven through NLL, and the structure of the singularities (in the real part of the amplitude) is fully explained by the dipole formula. Therefore the quadrupole term can start contributing at i 3 s log 2 ( t/s) or 3 s log( t/s)

16 The quadrupole function (z, z) = 6 " s 3 4 f abe f cde (F ( /z) F (/z)) f ace f bde (F (z) F ( z)) + f ade f bce (F (/( z)) F ( /( z))) # F (z) =L 00 (z)+2 2 L 00 (z)+l 00 (z) +6 4 L (z) Ø. Almelid, C. Duhr, EG L 0... (z) are the single-valued harmonic polylogarithms introduced by Francis Brown in They are defined in the region where! z = z

17 the collinear limit M L (p,p 2, {p j }; µ) k2! Sp (p,p 2 ; µ) M L (P, {p j }; µ) In particular, IR singularities of the splitting amplitude are those present in L parton scattering (with 2) and not in L- parton scattering:!! Sp = L L

18 At 3-loops (beyond the planar limit c.f. Kosower 999) the splitting amplitude resolves the colour and directions of the rest of the process!!! Surprise in the collinear limit Previous expectation: the splitting amplitude should not depend on the rest of the process, thus (z, z) should vanish in any collinear limit.! Becher & Neubert (2009) EG & Magnea & Dixon (200) We find, however, for p! xp p 2! ( x)p (z, z) k2! s 3 X j,k6={,2} 3 4 ln P 2 2x( x) p j p k (P p j )(P p k ) [[T T 2,T T j ],T 2 T k ]

19 conclusions IR singularities of massless scattering amplitudes are now known to 3-loops.! The first correction to the dipole formula takes the form of a quadrupole interaction simultaneously correlating colour and kinematics of 4 patrons.! The quadrupole term is expressed in terms of single-valued harmonic polylogarithms of weight 5, depending on {z, z}. These variables are simple algebraic functions of conformally-invariant cross ratios, and they manifest the symmetries and analytic structure of the quadruple interaction.! Contrary to previous expectations, 3-loop splitting amplitudes acquire sensitivity to the colour flow and directions of the remaining partons. Many thanks to my students and collaborators!

20 Analytic structure explored by taking the Regge limit Euclidean Scattering momentum conserving Regge limit s 2 = analytic continuation s 34 = se i s 23 = s 4 = t>0 s 3 = s 4 = u = s + t>0 z = z = s/(s + t) z = z! Im(z) 2 z z = ( s 2)( s 34 ) ( s 3 )( s 24 ) = s2 e i2 (s + t) 2 ( z)( z) = ( s 4)( s 23 ) ( s 3 )( s 24 ) = ( t)2 (s + t) Re(z) Regge limits in different channels correspond to discontinuities around z =0,, - -2 ln(z z)! ln(z z)+2 i

21 ! multiple gluon exchange webs The combinations of diagrams appearing in the exponent (subtracted webs) are much simpler than individual diagrams: (n) 3 w (n, ) s n Z ny = Ci,i 4 2,...i n+ dx dx 2...dx n p 0 (x k, k ) k= G n x,x 2,...,x n ; q(x, ),q(x 2, 2 ),...q(x n, n ) " 2 p 0 (x, ij )= ˆi ˆj (x Conjecture: ˆi ( x) ˆj) = r( ij) 2 x () is made exclusively of powers of logs and Heaviside functions.! G n ij x + ij ij (2) w (n, ) is a sum of products of polylogs each depending on a single! each having a Symbol with alphabet!, 2 EG (arxiv: ) JHEP 404 (204) 044 A restricted basis of harmonic polylogarithmic functions was constructed, which is conjectured to be sufficient for all multiple gluon exchange webs! Falcioni, EG, Harley, Magnea, White (arxiv: ) JHEP 40 (204) 0 #

22 Computing IR Singularities Using non-lightlike Wilson lines Product of non-lightlike Wilson lines is multiplicatively renormalizable Arefeva, Dotsenko-Vergeles, Brandt-Neri-Sato (8) IR - UV relation: S = h 2... n i Korchemsky-Radyushkin (87) S ( ij, IR )=S UV+IR {z } = Z ( ij, UV ) To determine the renormalization Z of a product of Wilson lines we put an exponential IR cutoff along the Wilson lines and compute the UV singularity in dimensional regularisation,! D =4 2, > 0 dz Define the soft anomalous dimension! d ln µ = Z The anomalous dimension is determined by the coefficient of the single / UV pole of Z - it is independent of the IR regularisation!

23 Diagrammatic SOft gluon exponentiation Webs are the diagrams contributing to w, the exponent of the corrector S =expw = h 2... n i Abelian case (96) Only connected diagrams contribute to the exponent! Non- abelian, colour singlet (two- line) case (983) Only irreducible diagrams contribute to the exponent Yennie-Frautchi-Suura Gatheral Frenkel & Taylor Non- abelian, mul<- line case (200) Also reducible diagrams contribute - webs are formed by sets of diagrams. EG, Leanen, Stavenga & White Mitov, Sterman & Sung Theorem: all colour structures in the exponent correspond to connected graphs EG, Smillie & White (203)

24 Three-loop web: example (a) (b) (c) (d) The entire web contributes: = 0 F(3a) F(3b) F(3c) F(3d) C A T CB A@ C(3a) C(3b) C(3c) C(3d) C A Kinematics Web mixing matrix R Colour

25 Three-loop web: example = 0 (a) (b) (c) (d) F(3a) F(3b) F(3c) F(3d) C A T = F(3a) 2F(3b) 2F(3c)+F(3d) 6 {z } subdivergences cancel 0 CB A@ C(3a) C(3b) C(3c) C(3d) C A C(3a) C(3b) C(3c)+C(3d) {z }

26 three-loop web result w (3) (a) (b) (c) (d) Very simple structure: sum of products of polylogs of individual angles! 22 = f abe f cde T a T2 b T3 c T4 d s 3 r( 2 )r( 23 )r( 34 ) ij + = 2 i j q " 4 ij 2 2 i j 8U 2 ( 2 )ln 23 ln 34 8U 2 ( 34 )ln 2 ln U 2 ( 23 ) 2 2 ( 23 ) ln 2 ln 34 2ln 2 U ( 23 ) U ( 34 ) 2ln 34 U ( 2 ) U ( 23 )+4ln 23 U ( 2 ) U ( 34 ) # = ij S [U ( )] = 4 2 S [U 2 ( )] = S [ 2 ( )] = 2 special points: =0 = = lightlike limit EG (arxiv: ) JHEP 404 (204) 044 production at threshold straight line