Automated Computation of Born Matrix Elements in QCD

Transcription

1 Automated Computation of Born Matrix Elements in QCD Christopher Schwan and Daniel Götz November 4, 2011

2 Outline Motivation Color(-Flow) Decomposition Berends-Giele-Type Recursion Relations (Daniel Götz) Features/Results of an Implementation (Short, Next Seminar)

3 Where are Born Matrix Elements Needed? for LO approximations of observables O, for example jet-rates R or cross sections σ for radiative corrections in NLO calculations, part of virtual correction in subtraction terms for NLO calculations (Catani-Seymour) matrix element generators are part of event generators (e.g. Sherpa, Herwig++, etc.)

4 Wishlist for a Born Matrix Element Generator generic ME generator shall be capable of computing every QCD process ( high jet multiplicities for large s), for example gg gg, gg ggg,......, gg q qq qgggg,... automated INPUT: process + momenta + helicities OUTPUT: squared matrix element fast employ numerical techniques

5 Wishlist for a Born Matrix Element Generator generic ME generator shall be capable of computing every QCD process ( high jet multiplicities for large s), for example gg gg, gg ggg,......, gg q qq qgggg,... automated INPUT: process + momenta + helicities OUTPUT: squared matrix element fast employ numerical techniques How to attack the problem: Feynman-diagrams

6 Wishlist for a Born Matrix Element Generator generic ME generator shall be capable of computing every QCD process ( high jet multiplicities for large s), for example gg gg, gg ggg,......, gg q qq qgggg,... automated INPUT: process + momenta + helicities OUTPUT: squared matrix element fast employ numerical techniques How to attack the problem: Feynman diagrams Color decomposition and recursive relations for QCD (simple and fast)

7 What is Color Decomposition? Idea: Rewrite conventional Feynman diagram method so that in QCD color factorizes from kinematics: A = i C i A i (1) C i : color factor, contains all the information about color A i : partial amplitudes, contains kinematics; sum of diagrams belonging to the same color factor C i How do we do that? Start off with QCD vertices

8 Color-Structure of QCD a, α b, β = gf abc { (p1 p 2 ) α g βγ + (p 2 p 3 ) β g γα + (p 3 p 1 ) γ g αβ } c, γ j a = igt a i j γµ i Different objects are involved: f abc, T a i j

9 Color-Structure of QCD a, α b, β = gf abc { (p1 p 2 ) α g βγ + (p 2 p 3 ) β g γα + (p 3 p 1 ) γ g αβ } c, γ j a = igt a i j γµ a d b c i = ig 2 f abe f cde( g αγ g βδ g αδ g βγ) + f ace f bde( g αβ g γδ g αδ g βγ) + f ade f bce( g αβ g γδ g αγ g βδ) Different objects are involved: f abc, T a i j Color does not factorize in the four-gluon-vertex!

10 Color Decomposition of the Gluon Vertices Rewrite structure constants f abc of SU(3) into traces of generators T a : ( f abc = i Tr T a T b T c T a T c T b) (2) remainder of the three-gluon-vertex is defined as V 3 (p 1, p 2, p 3 ): g µ 1µ 2 (p 1 p 2 ) µ 3 + g µ 2µ 3 (p 2 p 3 ) µ 1 + g µ 3µ 1 (p 3 p 1 ) µ 2 (3) We can now write the full vertex as Tr (T a 1 T a 2 T a 3 ) V µ 1µ 2 µ 3 3 (p 1, p 2, p 3 ) (4) perm of 2,3 For the four-gluon-vertex we apply the same procedure to yield (note the same structure!) Tr (T a 1 T a 2 T a 3 T a 4 ) V µ 1µ 2 µ 3 µ 4 4 (5) perm of 2,3,4

11 Color-Flow Decomposition (CFD) of the Gluon Vertices There are different representations, we choose color-flow decomposition (simple) To derive CFD, make use of trace identity TijT a ji b = δ ab and Fierz identity T a ijt b kl = δ il δ jk 1 3 δ ijδ kl (6) In CFD three-gluon-vertex now reads: δ i1 j 2 δ i2 j 3 δ i3 j 1 V µ 1µ 2 µ 3 3 (p 1, p 2, p 3 ) (7) perm of 2,3 a gluon is now described with two indices i = r, g, b (or 1, 2, 3) and j = r, ḡ, b (or 1, 2, 3) The four-gluon vertex reads: δ i1 j 2 δ i2 j 3 δ i3 j 4 δ i4 j 1 V µ 1µ 2 µ 3 µ4 4 (8) perm of 2,3,4

12 The Three-Gluon Vertex in CFD (Blackboard) δ i1 j 2 δ i2 j 3 δ i3 j 1 V µ 1µ 2 µ 3 3 (p 1, p 2, p 3 ) (9) perm of 2,3 }{{} δ i1 j 2 δ i2 j 3 δ i3 j 1 +δ i1 j 3 δ i3 j 2 δ i2 j 1

13 Color-Flow Decomposition for Gluon Amplitudes A = δ i1 j 2 δ i2 j 3 δ inj 1 A (1, 2,..., n) (10) perm of 2,...,n A (1, 2,..., n): Partial amplitude containing the kinematical information ( Daniel Götz); numbers denote momenta and helicities for external gluons δ i1 j 2 δ i2 j 3 δ inj 1 : Color factor; i k is the color of gluon k, j k is the anti-color of gluon k sum runs over all permutations while keeping 1 fixed Note: Only color-indices of external gluons, internal indices are already contracted!

14 What about Quarks? Let s look at a simple but non-trivial case: dd uu The Feynman diagram for the process is: d, j 2 d, i 1 u, j 4 u, i 3 The color-part of the diagram is: T a j 2 i 1 δ ab T b i 3 j 4 (11)

15 What about Quarks? Let s look at a simple but non-trivial case: dd uu The Feynman diagram for the process is: d, j 2 d, i 1 Rewrite color structure (11) into Kronecker-deltas: δ j2 j 4 δ i1 i }{{ 3 1 } 3 δ i 1 j 2 δ i3 j }{{ 4 } U(3)-gluon U(1)-gluon u, j 4 u, i 3 The color-part of the diagram is: T a j 2 i 1 δ ab T b i 3 j 4 (11)

16 What about Quarks? Let s look at a simple but non-trivial case: dd uu The Feynman diagram for the process is: d, j 2 d, i 1 Rewrite color structure (11) into Kronecker-deltas: δ j2 j 4 δ i1 i }{{ 3 1 } 3 δ i 1 j 2 δ i3 j }{{ 4 } U(3)-gluon U(1)-gluon u, j 4 u, i 3 The color-ordered diagrams are: The color-part of the diagram is: T a j 2 i 1 δ ab T b i 3 j 4 (11)

17 General Algorithm For an arbitrary number of quarks and gluons: A = i C i A i, A = j C j A j (12) Ci : string of Kronecker deltas (process-dependent) i: Special permutations ( 1 fixed, also process-dependent) Summing over color: A A = { } A i C j C i A j = V MV (13) color i,j color V is the amplitude vector, M the color matrix C A 0 0 PC 0 C 0 PC 1 C 0 PC n C V =. M = 1 PC 0 C 1 PC 1 C 1 PC n. A..... n C npc 0 C npc 1 C npc n

18 Recap: Color-Flow Decomposition (CFD) CFD provides us with a formalism where color (C i and M ) separates from kinematics (A i and V ) in matrix elements A: A = C i A i A 2 = V MV (14) i CFD has a simple interpretation: The matrix element is the sum of possible color flows times the partial amplitudes Partial amplitudes A i are gauge invariant; they are the sum of all diagrams belonging to the same color flow They can be efficiently constructed from recursive relations ( Daniel Götz)

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20 Example for an 8-Jet Event Figure: 8-jet event display taken from twiki/bin/view/atlaspublic/eventdisplaypublicresults

21 Berends-Giele Recursion Relations 1) Recursion Relations for Gluons 2) An Example: The Process gg gg 3) Recursion Relations for Quarks? 1 / 11

22 Recursion Relations for Gluons Formulated in terms of off-shell currents: off-shell off-shell off-shell = m 1 j=1 m 2 + m 1 j=1 k=j m 1 j j +1 m 1 j j +1 k k +1 m 2 / 11

23 Recursion Relations for Gluons Formulated in terms of off-shell currents: off-shell off-shell off-shell = m 1 j=1 m 2 + m 1 j=1 k=j m 1 j j +1 m 1 j j +1 k k +1 m Base case: = gluon polarization vector ɛ µ (p i, λ i ) 2 / 11

24 Recursion Relations for Gluons Formulated in terms of off-shell currents: off-shell off-shell off-shell = m 1 j=1 m 2 + m 1 j=1 k=j m 1 j j +1 m 1 j j +1 k k +1 m Base case: = gluon polarization vector ɛ µ (p i, λ i ) Partial amplitude for process with n particles (0, 1,..., n 1): perform Berends-Giele for particles (1, 2,..., n 1); replace off-shell leg with polarization vector of particle 0. 2 / 11

25 An Example: The Process gg gg In terms of Feynman diagrams: s-channel t-channel u-channel 3 / 11

26 An Example: The Process gg gg Now with Berends-Giele: = / 11

27 An Example: The Process gg gg Now with Berends-Giele: = = / 11

28 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

29 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

30 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

31 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

32 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

33 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

34 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

35 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

36 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

37 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

38 An Example: The Process gg gg Matching Berends-Giele with Feynman diagrams Partial amplitude A 1 A 2 A 3 A 4 A 5 A / 11

39 Interlude: Why Recursion is Not as Slow as it Seems Look at one specific splitting which occurs for several permutations: Problem: Right hand sub-current is always the same! Naive recursion recomputes this sub-current for each permutation! Unneccessary waste of computation time! 6 / 11

40 Interlude: Why Recursion is Not as Slow as it Seems Look at one specific splitting which occurs for several permutations: Problem: Right hand sub-current is always the same! Naive recursion recomputes this sub-current for each permutation! Unneccessary waste of computation time! = Solution: Store currents in memory and reuse them! (Example: process gg gggggg sped up by factor 40!) 6 / 11

41 Recursion Relations for Quarks A Short Look [1] Changes compared to the gluonic recursion: Add another term to the recursion which splits a gluon off-shell current into quark and antiquark legs. This results in (anti-)quark off-shell currents which obey their own recursion formulas! 7 / 11

42 Recursion Relations for Quarks A Short Look [1] Changes compared to the gluonic recursion: Add another term to the recursion which splits a gluon off-shell current into quark and antiquark legs. This results in (anti-)quark off-shell currents which obey their own recursion formulas! Additional constraints: quark-gluon vertices have to be flavor conserving; permutations become more complicated due to color structure of the quark-gluon vertex (avoid double counting!) 7 / 11

43 Recursion Relations for Quarks A Short Look [1] Changes compared to the gluonic recursion: Add another term to the recursion which splits a gluon off-shell current into quark and antiquark legs. This results in (anti-)quark off-shell currents which obey their own recursion formulas! Additional constraints: quark-gluon vertices have to be flavor conserving; permutations become more complicated due to color structure of the quark-gluon vertex (avoid double counting!) Most interesting complication: Inclusion of the U(1) gluon. Recall: U(1) gluon only couples to quarks, not to the U(N) part of the gluon! 7 / 11

44 Recursion Relations for Quarks A Short Look [2] Every gluon which only couples to quarks also has to appear as U(1) gluon! Example: each diagram has a different color factor c i ; remember: partial amps are sets of diagrams with the same c i! Each diagrams belongs to different partial amp! 8 / 11

45 Recursion Relations for Quarks A Short Look [3] So: We have to find a way to distinguish partial amplitudes with a U(N) gluon from those with a U(1) gluon! Until now: All neccessary parameters only describe external particles (e.g momentum, helicity, permutation of external particles...) With quarks: Need information on internals structure (i.e. propagators) of the process! = Assign additional property to external particles: Color cluster 9 / 11

46 Recursion Relations for Quarks A Short Look [4] (0, 1) blue: particle ids red: cluster ids (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (7, 1) (8, 1) 10 / 11

47 Recursion Relations for Quarks A Short Look [4] (0, 1) blue: particle ids red: cluster ids (1, 2) (2, 2) (3, 1) (4, 1) (5, 1) (6, 3) (7, 3) (8, 1) 10 / 11

48 Recursion Relations for Quarks A Short Look [4] (0, 1) blue: particle ids red: cluster ids (1, 2) (2, 2) (3, 1) (4, 1) (5, 1) (6, 3) (7, 3) (8, 1) Problem: Finding an algorithm to generate all color cluster combinations! Many subtleties to consider! But: Once algorithm is found, recursion is as simple as in the case of gluons! 10 / 11

49 Summary and Conclusion Recursion formulas together with color decomposition form a simple method to compute scattering amplitudes; Recursion is very fast provided that one remembers currents which have already been calculated; The method can be extended to full QCD (i.e. including quarks). 11 / 11

50 Summary and Conclusion Recursion formulas together with color decomposition form a simple method to compute scattering amplitudes; Recursion is very fast provided that one remembers currents which have already been calculated; The method can be extended to full QCD (i.e. including quarks). So far, we have not shown results of any kind = In two weeks! 11 / 11