The Steinmann Cluster Bootstrap for N = 4 SYM Amplitudes

Transcription

1 The Steinmann Cluster Bootstrap for N = 4 SYM Amplitudes Georgios Papathanasiou w/ Drummond,Spradlin w/ Dixon,Drummond,Harrington,McLeod,Spradlin + in progress w/ Caron-Huot,Dixon,McLeod,von Hippel

2 Outline Motivation: Why Planar N = 4 Amplitudes? The Bootstrap Philosophy Cluster Algebra Upgrade The 3-loop MHV Heptagon Steinmann Upgrade The 3-loop NMHV/4-loop MHV Heptagon New Developments Conclusions & Outlook GP The Steinmann Cluster Bootstrap 2/21

3 Aim: Can we compute scattering amplitudes in SU(N) N = 4 super Yang Mills theory to all loops, for any multiplicity and quantum numbers of the external particles? GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 3/21

4 Aim: Can we compute scattering amplitudes in SU(N) N = 4 super Yang Mills theory to all loops, for any multiplicity and quantum numbers of the external particles? Would amount to solving an interacting 4D gauge theory... GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 3/21

5 Aim: Can we compute scattering amplitudes in SU(N) N = 4 super Yang Mills theory to all loops, for any multiplicity and quantum numbers of the external particles? Would amount to solving an interacting 4D gauge theory... Ambitious, but promising in t Hooft limit, N with λ = g 2 Y M N fixed: GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 3/21

6 Aim: Can we compute scattering amplitudes in SU(N) N = 4 super Yang Mills theory to all loops, for any multiplicity and quantum numbers of the external particles? Would amount to solving an interacting 4D gauge theory... Ambitious, but promising in t Hooft limit, N with λ = g 2 Y M N fixed: Perturbatively, only planar diagrams contribute GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 3/21

7 Aim: Can we compute scattering amplitudes in SU(N) N = 4 super Yang Mills theory to all loops, for any multiplicity and quantum numbers of the external particles? Would amount to solving an interacting 4D gauge theory... Ambitious, but promising in t Hooft limit, N with λ = g 2 Y M N fixed: Perturbatively, only planar diagrams contribute Planar N = 4 SYM Free type IIB superstrings on AdS 5 S 5 strongly coupled weakly coupled GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 3/21

8 Aim: Can we compute scattering amplitudes in SU(N) N = 4 super Yang Mills theory to all loops, for any multiplicity and quantum numbers of the external particles? Would amount to solving an interacting 4D gauge theory... Ambitious, but promising in t Hooft limit, N with λ = g 2 Y M N fixed: k n x 1 x 2 x n A n k 1 k 2 k i = x i+1 x i x 3 Perturbatively, only planar diagrams contribute Planar N = 4 SYM Free type IIB superstrings on AdS 5 S 5 strongly coupled weakly coupled Amplitudes Wilson Loops; Dual Conformal Symmetry [Alday,Maldacena][Drummond,Henn,Korchemsky,Sokatchev][Brandhuber,Heslop,Travaglini] GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 3/21

9 Aim: Can we compute scattering amplitudes in SU(N) N = 4 super Yang Mills theory to all loops, for any multiplicity and quantum numbers of the external particles? Would amount to solving an interacting 4D gauge theory... Ambitious, but promising in t Hooft limit, N with λ = g 2 Y M N fixed: Perturbatively, only planar diagrams contribute Planar N = 4 SYM Free type IIB superstrings on AdS 5 S 5 strongly coupled weakly coupled Amplitudes Wilson Loops; Dual Conformal Symmetry [Alday,Maldacena][Drummond,Henn,Korchemsky,Sokatchev][Brandhuber,Heslop,Travaglini] Integrable structures All loop quantities! [Beisert,Eden,Staudacher] GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 3/21

10 Practical significance Hopefully I ve convinced you that this aim is theoretically interesting and possibly within reach. GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 4/21

11 Practical significance Hopefully I ve convinced you that this aim is theoretically interesting and possibly within reach. Along the way, it is very likely that new computational methods will also be developed, as prompted by earlier successes, GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 4/21

12 Practical significance Hopefully I ve convinced you that this aim is theoretically interesting and possibly within reach. Along the way, it is very likely that new computational methods will also be developed, as prompted by earlier successes, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 4/21

13 Practical significance Hopefully I ve convinced you that this aim is theoretically interesting and possibly within reach. Along the way, it is very likely that new computational methods will also be developed, as prompted by earlier successes, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] Method of Symbols [Goncharov,Spradlin,Vergu,Volovich] GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 4/21

14 Practical significance Hopefully I ve convinced you that this aim is theoretically interesting and possibly within reach. Along the way, it is very likely that new computational methods will also be developed, as prompted by earlier successes, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] Method of Symbols [Goncharov,Spradlin,Vergu,Volovich] leading to significant practical applications! GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 4/21

15 Practical significance Hopefully I ve convinced you that this aim is theoretically interesting and possibly within reach. Along the way, it is very likely that new computational methods will also be developed, as prompted by earlier successes, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] Method of Symbols [Goncharov,Spradlin,Vergu,Volovich] leading to significant practical applications! For example, gg Hg 2 for N 3 LO Higgs cross-section [Anastasiou,Duhr,Dulat,Herzog,Mistlberger] GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 4/21

16 Practical significance Hopefully I ve convinced you that this aim is theoretically interesting and possibly within reach. Along the way, it is very likely that new computational methods will also be developed, as prompted by earlier successes, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] Method of Symbols [Goncharov,Spradlin,Vergu,Volovich] leading to significant practical applications! For example, gg Hg 2 for N 3 LO Higgs cross-section [Anastasiou,Duhr,Dulat,Herzog,Mistlberger] or more recently the 3-loop QCD soft anomalous dimension. [Almelid,Duhr,Gardi,McLeod,White] GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 4/21

17 So which part of this journey are we at? GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 5/21

18 So which part of this journey are we at? Amplitudes with n = 4, 5 particles already known to all loops! Captured by the Bern-Dixon-Smirnov ansatz A BDS n. GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 5/21

19 So which part of this journey are we at? Amplitudes with n = 4, 5 particles already known to all loops! Captured by the Bern-Dixon-Smirnov ansatz A BDS n. More generally, GP The Steinmann Cluster Bootstrap Motivation: Why Planar N = 4 Amplitudes? 5/21

20 The Amplitude Bootstrap GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

21 The Amplitude Bootstrap The most efficient method for computing planar N = 4 amplitudes in general kinematics, at fixed order in the coupling. GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

22 The Amplitude Bootstrap The most efficient method for computing planar N = 4 amplitudes in general kinematics, at fixed order in the coupling. A. Construct an ansatz for the amplitude assuming GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

23 The Amplitude Bootstrap The most efficient method for computing planar N = 4 amplitudes in general kinematics, at fixed order in the coupling. A. Construct an ansatz for the amplitude assuming 1. What the general class of functions that suffices to express it is GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

24 The Amplitude Bootstrap The most efficient method for computing planar N = 4 amplitudes in general kinematics, at fixed order in the coupling. A. Construct an ansatz for the amplitude assuming 1. What the general class of functions that suffices to express it is 2. What the function arguments (encoding the kinematics) are GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

25 The Amplitude Bootstrap The most efficient method for computing planar N = 4 amplitudes in general kinematics, at fixed order in the coupling. A. Construct an ansatz for the amplitude assuming 1. What the general class of functions that suffices to express it is 2. What the function arguments (encoding the kinematics) are B. Fix the coefficients of the ansatz by imposing consistency conditions (e.g. known near-collinear or multi-regge limiting behavior) GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

26 The Amplitude Bootstrap The most efficient method for computing planar N = 4 amplitudes in general kinematics, at fixed order in the coupling. A. Construct an ansatz for the amplitude assuming 1. What the general class of functions that suffices to express it is 2. What the function arguments (encoding the kinematics) are B. Fix the coefficients of the ansatz by imposing consistency conditions (e.g. known near-collinear or multi-regge limiting behavior) First applied very successfully for the first nontrivial, 6-particle amplitude [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel/Duhr,Pennington] through 5 loops. [(Caron-Huot,)Dixon,McLeod,von Hippel] GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

27 The Amplitude Bootstrap The most efficient method for computing planar N = 4 amplitudes in general kinematics, at fixed order in the coupling. A. Construct an ansatz for the amplitude assuming 1. What the general class of functions that suffices to express it is 2. What the function arguments (encoding the kinematics) are B. Fix the coefficients of the ansatz by imposing consistency conditions (e.g. known near-collinear or multi-regge limiting behavior) First applied very successfully for the first nontrivial, 6-particle amplitude [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel/Duhr,Pennington] through 5 loops. [(Caron-Huot,)Dixon,McLeod,von Hippel] Motivated by this progress, we upgraded this procedure for n = 7, with information from the cluster algebra structure of the kinematical space. Surprisingly, more powerful than n = 6! [Drummond,GP,Spradlin] GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 6/21

28 What are the right functions? Multiple polylogarithms (MPLs) GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 7/21

29 What are the right functions? Multiple polylogarithms (MPLs) f k is a MPL of weight k if its differential may be written as a finite linear combination df k = f (α) k 1 d log φ α α over some set of φ α, where f (α) k 1 functions of weight k 1. GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 7/21

30 What are the right functions? Multiple polylogarithms (MPLs) f k is a MPL of weight k if its differential may be written as a finite linear combination df k = f (α) k 1 d log φ α α over some set of φ α, where f (α) k 1 functions of weight k 1. Convenient tool for describing them: The symbol S(f k ) encapsulating recursive application of above definition (on f (α) k 1 etc) S(f k ) = f (α 1,α 2,...,α k ) 0 (φ α1 φ αk ). α 1,...,α k [See Brandhuber s talk] GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 7/21

31 What are the right functions? Multiple polylogarithms (MPLs) f k is a MPL of weight k if its differential may be written as a finite linear combination df k = f (α) k 1 d log φ α α over some set of φ α, where f (α) k 1 functions of weight k 1. Convenient tool for describing them: The symbol S(f k ) encapsulating recursive application of above definition (on f (α) k 1 etc) S(f k ) = f (α 1,α 2,...,α k ) 0 (φ α1 φ αk ). α 1,...,α k Collection of φ α : symbol alphabet f (α 1,...,α k ) 0 rational [See Brandhuber s talk] GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 7/21

32 What are the right functions? Multiple polylogarithms (MPLs) f k is a MPL of weight k if its differential may be written as a finite linear combination df k = f (α) k 1 d log φ α α over some set of φ α, where f (α) k 1 functions of weight k 1. Convenient tool for describing them: The symbol S(f k ) encapsulating recursive application of above definition (on f (α) k 1 etc) S(f k ) = f (α 1,α 2,...,α k ) 0 (φ α1 φ αk ). α 1,...,α k Collection of φ α : symbol alphabet f (α 1,...,α k ) 0 rational Empeirical evidence: L-loop amplitudes=mpls of weight k = 2L [Duhr,Del Duca,Smirnov][Arkani-Hamed,Bourjaily,Cachazo,Goncharov,Postnikov,Trnka][GP] [See Brandhuber s talk] GP The Steinmann Cluster Bootstrap The Bootstrap Philosophy 7/21

33 What are the right variables? GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 8/21

34 What are the right variables? More precisely, what is the symbol alphabet? [See talk by Volovich] GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 8/21

35 What are the right variables? [See talk by Volovich] More precisely, what is the symbol alphabet? For n = 6, 9 letters, motivated by analysis of relevant integrals GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 8/21

36 What are the right variables? [See talk by Volovich] More precisely, what is the symbol alphabet? For n = 6, 9 letters, motivated by analysis of relevant integrals More generally, strong motivation from cluster algebra structure of kinematical configuration space Conf n (P 3 ) [Golden,Goncharov,Spradlin,Vergu,Volovich] GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 8/21

37 What are the right variables? [See talk by Volovich] More precisely, what is the symbol alphabet? For n = 6, 9 letters, motivated by analysis of relevant integrals More generally, strong motivation from cluster algebra structure of kinematical configuration space Conf n (P 3 ) [Golden,Goncharov,Spradlin,Vergu,Volovich] The latter is a collection of n ordered momentum twistors Z i on P 3, (an equivalent way to parametrise massless kinematics), modulo dual [Hodges][See talks by Arkani-Hammed,Bai,Ferro] conformal transformations. x i Z i 1 Z i (x i x j ) 2 ɛ IJKL Z I i 1Z J i Z K j 1Z L j = det(z i 1 Z i Z j 1 Z j ) i 1ij 1j = GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 8/21

38 Cluster algebras [Fomin,Zelevinsky] GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

39 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras with Distinguished set of generators a i, the cluster variables GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

40 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras with Distinguished set of generators a i, the cluster variables Grouped into overlapping subsets {a 1,..., a n } of rank n, the clusters GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

41 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras with Distinguished set of generators a i, the cluster variables Grouped into overlapping subsets {a 1,..., a n } of rank n, the clusters Constructed recursively from initial cluster via mutations GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

42 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras with Distinguished set of generators a i, the cluster variables Grouped into overlapping subsets {a 1,..., a n } of rank n, the clusters Constructed recursively from initial cluster via mutations Can be described by quivers. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

43 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras with Distinguished set of generators a i, the cluster variables Grouped into overlapping subsets {a 1,..., a n } of rank n, the clusters Constructed recursively from initial cluster via mutations Can be described by quivers. Example: A 3 Cluster algebra a 1 a 2 a 3 Initial Cluster GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

44 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras with Distinguished set of generators a i, the cluster variables Grouped into overlapping subsets {a 1,..., a n } of rank n, the clusters Constructed recursively from initial cluster via mutations Can be described by quivers. Example: A 3 Cluster algebra a 1 a 2 a 3 Initial Cluster a 1 a 2 a 3 Mutate a 2 : New cluster General rule for mutation at node k: 1. i k j, add i j, reverse i k j, remove. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

45 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras with Distinguished set of generators a i, the cluster variables Grouped into overlapping subsets {a 1,..., a n } of rank n, the clusters Constructed recursively from initial cluster via mutations Can be described by quivers. Example: A 3 Cluster algebra a 1 a 2 a 3 Initial Cluster a 1 a 2 a 3 Mutate a 2 : New cluster a 2 = (a 1 + a 3 )/a 2 and so on... General rule for mutation at node k: 1. i k j, add i j, reverse i k j, remove. 2. In new quiver/cluster, a k a k = ( a i + arrows i k arrows k j a j )/a k GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 9/21

46 Connection to the kinematic space GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 10/21

47 Connection to the kinematic space The latter is closely related to a Graßmannian: [See talks by Arkani-Hammed... ] Conf n (P 3 ) = Gr(4, n)/(c ) n 1 GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 10/21

48 Connection to the kinematic space The latter is closely related to a Graßmannian: [See talks by Arkani-Hammed... ] Conf n (P 3 ) = Gr(4, n)/(c ) n 1 Graßmannians Gr(k, n) equipped with cluster algebra structure [Scott] GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 10/21

49 Connection to the kinematic space The latter is closely related to a Graßmannian: [See talks by Arkani-Hammed... ] Conf n (P 3 ) = Gr(4, n)/(c ) n 1 Graßmannians Gr(k, n) equipped with cluster algebra structure [Scott] Initial cluster made of a special set of Plücker coordinates i 1... i k GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 10/21

50 Connection to the kinematic space The latter is closely related to a Graßmannian: [See talks by Arkani-Hammed... ] Conf n (P 3 ) = Gr(4, n)/(c ) n 1 Graßmannians Gr(k, n) equipped with cluster algebra structure [Scott] Initial cluster made of a special set of Plücker coordinates i 1... i k Mutations also yield certain homogeneous polynomials of Plücker coordinates GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 10/21

51 Connection to the kinematic space The latter is closely related to a Graßmannian: [See talks by Arkani-Hammed... ] Conf n (P 3 ) = Gr(4, n)/(c ) n 1 Graßmannians Gr(k, n) equipped with cluster algebra structure [Scott] Initial cluster made of a special set of Plücker coordinates i 1... i k Mutations also yield certain homogeneous polynomials of Plücker coordinates Crucial observation: For all known cases, symbol alphabet of n-point amplitudes for n = 6, 7 are Gr(4, n) cluster variables (also known as A-coordinates) [Golden,Goncharov,Spradlin,Vergu,Volovich] GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 10/21

52 Connection to the kinematic space The latter is closely related to a Graßmannian: [See talks by Arkani-Hammed... ] Conf n (P 3 ) = Gr(4, n)/(c ) n 1 Graßmannians Gr(k, n) equipped with cluster algebra structure [Scott] Initial cluster made of a special set of Plücker coordinates i 1... i k Mutations also yield certain homogeneous polynomials of Plücker coordinates Crucial observation: For all known cases, symbol alphabet of n-point amplitudes for n = 6, 7 are Gr(4, n) cluster variables (also known as A-coordinates) [Golden,Goncharov,Spradlin,Vergu,Volovich] Fundamental assumption of cluster bootstrap Symbol alphabet is made of cluster A-coordinates on Conf n (P 3 ). For the heptagon, 42 of them. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 10/21

53 Heptagon Symbol Letters Multiply A-coordinates with suitable powers of i i + 1 i + 2 i + 3 to form conformally invariant cross-ratios, where a 11 = , a 41 = , a 21 = , a 51 = 1(23)(45)(67), a 31 = , a 61 = 1(34)(56)(72), ijkl Z i Z j Z k Z l = det(z i Z j Z k Z l ) a(bc)(de)(fg) abde acfg abfg acde, together with a ij obtained from a i1 by cyclically relabeling Z m Z m+j 1. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 11/21

54 Back to contructing and constraining function space GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 12/21

55 Back to contructing and constraining function space 1. Locality: Amplitudes may only have singularities when intermediate particles go on-shell constrains first symbol entry (7-pts: a 1j ) GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 12/21

56 Back to contructing and constraining function space 1. Locality: Amplitudes may only have singularities when intermediate particles go on-shell constrains first symbol entry (7-pts: a 1j ) 2. Integrability: For given S, ensures function with given symbol f (α 1,α 2,...,α k ) 0 (φ α1 φ αk ) α 1,...,α k omitting φ αj φ αj+1 d log φ αj d log φ αj+1 = 0 j. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 12/21

57 Back to contructing and constraining function space 1. Locality: Amplitudes may only have singularities when intermediate particles go on-shell constrains first symbol entry (7-pts: a 1j ) 2. Integrability: For given S, ensures function with given symbol f (α 1,α 2,...,α k ) 0 (φ α1 φ αk ) α 1,...,α k omitting φ αj φ αj+1 d log φ αj d log φ αj+1 = 0 j. 3. Dual superconformal symmetry constrains last symbol entry of amplitudes (MHV 7-pts: a 2j, a 3j ) [Caron-Huot,He] GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 12/21

58 Back to contructing and constraining function space 1. Locality: Amplitudes may only have singularities when intermediate particles go on-shell constrains first symbol entry (7-pts: a 1j ) 2. Integrability: For given S, ensures function with given symbol f (α 1,α 2,...,α k ) 0 (φ α1 φ αk ) α 1,...,α k omitting φ αj φ αj+1 d log φ αj d log φ αj+1 = 0 j. 3. Dual superconformal symmetry constrains last symbol entry of amplitudes (MHV 7-pts: a 2j, a 3j ) [Caron-Huot,He] 4. Collinear limit: Bern-Dixon-Smirnov ansatz A BDS n contains all IR divergences Constraint on B n A n /A BDS n lim i+1 i B n = B n 1 GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 12/21

59 Back to contructing and constraining function space 1. Locality: Amplitudes may only have singularities when intermediate particles go on-shell constrains first symbol entry (7-pts: a 1j ) 2. Integrability: For given S, ensures function with given symbol f (α 1,α 2,...,α k ) 0 (φ α1 φ αk ) α 1,...,α k omitting φ αj φ αj+1 d log φ αj d log φ αj+1 = 0 j. 3. Dual superconformal symmetry constrains last symbol entry of amplitudes (MHV 7-pts: a 2j, a 3j ) [Caron-Huot,He] 4. Collinear limit: Bern-Dixon-Smirnov ansatz A BDS n contains all IR divergences Constraint on B n A n /A BDS n lim i+1 i B n = B n 1 Define n-gon symbol: A symbol of the corresponding n-gon alphabet, obeying 1 & 2. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 12/21

60 Results [Drummond,GP,Spradlin] Weight k = Number of heptagon symbols ? well-defined in the 7 6 limit ?? which vanish in the 7 6 limit ?? well-defined for all i+1 i ?? with MHV last entries with both of the previous two Table: Heptagon symbols and their properties. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 13/21

61 Results [Drummond,GP,Spradlin] Weight k = Number of heptagon symbols ? well-defined in the 7 6 limit ?? which vanish in the 7 6 limit ?? well-defined for all i+1 i ?? with MHV last entries with both of the previous two Table: Heptagon symbols and their properties. The symbol of the three-loop seven-particle MHV amplitude is the only weight-6 heptagon symbol which satisfies the last-entry condition and which is finite in the 7 6 collinear limit. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 13/21

62 Comparison with the hexagon case Weight k = Number of hexagon symbols well-defined (vanish) in the 6 5 limit well-defined (vanish) for all i+1 i with MHV last entries with both of the previous two Table: Hexagon symbols and their properties. Surprisingly, heptagon bootstrap more powerful than hexagon one! Fact that lim 7 6 R (3) 7 = R (3) 6, as well as discrete symmetries such as cyclic Z i Z i+1, flip Z i Z n+1 i or parity symmetry follow for free, not imposed a priori. GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 14/21

63 [Steinmann][Cahill,Stapp][Bartels,Lipatov,Sabio Vera] Upgrade II: Steinmann Relations GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 15/21

64 [Steinmann][Cahill,Stapp][Bartels,Lipatov,Sabio Vera] Upgrade II: Steinmann Relations Dramatically simplify n-gon function space [Caron-Huot,Dixon,McLeod,von Hippel][Dixon,Drummond,Harrington,McLeod,GP,Spradlin] GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 15/21

65 [Steinmann][Cahill,Stapp][Bartels,Lipatov,Sabio Vera] Upgrade II: Steinmann Relations Dramatically simplify n-gon function space [Caron-Huot,Dixon,McLeod,von Hippel][Dixon,Drummond,Harrington,McLeod,GP,Spradlin] Double discontinuities vanish for any set of overlapping channels GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 15/21

66 [Steinmann][Cahill,Stapp][Bartels,Lipatov,Sabio Vera] Upgrade II: Steinmann Relations Dramatically simplify n-gon function space [Caron-Huot,Dixon,McLeod,von Hippel][Dixon,Drummond,Harrington,McLeod,GP,Spradlin] Double discontinuities vanish for any set of overlapping channels Disc s345 [Disc s234 A] = 0 Channel labelled by Mandelstam invariant we analytically continue GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 15/21

67 [Steinmann][Cahill,Stapp][Bartels,Lipatov,Sabio Vera] Upgrade II: Steinmann Relations Dramatically simplify n-gon function space [Caron-Huot,Dixon,McLeod,von Hippel][Dixon,Drummond,Harrington,McLeod,GP,Spradlin] Double discontinuities vanish for any set of overlapping channels Disc s345 [Disc s234 A] = 0 Channel labelled by Mandelstam invariant we analytically continue Channels overlap if they divide particles in 4 nonempty sets. Here: {2}, {3, 4}, {5}, and {6, 7, 1} GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 15/21

68 [Steinmann][Cahill,Stapp][Bartels,Lipatov,Sabio Vera] Upgrade II: Steinmann Relations Dramatically simplify n-gon function space [Caron-Huot,Dixon,McLeod,von Hippel][Dixon,Drummond,Harrington,McLeod,GP,Spradlin] Double discontinuities vanish for any set of overlapping channels Disc s345 [Disc s234 A] = 0 Channel labelled by Mandelstam invariant we analytically continue Channels overlap if they divide particles in 4 nonempty sets. Here: {2}, {3, 4}, {5}, and {6, 7, 1} Focus on s i 1,i,i+1 a 1i (s i 1i more subtle) GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 15/21

69 [Steinmann][Cahill,Stapp][Bartels,Lipatov,Sabio Vera] Upgrade II: Steinmann Relations Dramatically simplify n-gon function space [Caron-Huot,Dixon,McLeod,von Hippel][Dixon,Drummond,Harrington,McLeod,GP,Spradlin] Double discontinuities vanish for any set of overlapping channels Disc s345 [Disc s234 A] = 0 Channel labelled by Mandelstam invariant we analytically continue Channels overlap if they divide particles in 4 nonempty sets. Here: {2}, {3, 4}, {5}, and {6, 7, 1} Focus on s i 1,i,i+1 a 1i (s i 1i more subtle) Heptagon: No a 1,i±1, a 1,i±2 after a 1,i on second symbol entry GP The Steinmann Cluster Bootstrap Cluster Algebra Upgrade 15/21

70 Results: Steinmann Heptagon symbols Weight k = parity +, flip parity +, flip parity, flip parity, flip Total Table: Number of Steinmann heptagon symbols at weights 1 through 7, and those satisfying the MHV next-to-final entry condition at weight 7. All of them are organized with respect to the discrete symmetries Z i Z i+1, Z i Z 8 i of the MHV amplitude. GP The Steinmann Cluster Bootstrap Steinmann Upgrade 16/21

71 Results: Steinmann Heptagon symbols Weight k = parity +, flip parity +, flip parity, flip parity, flip Total Table: Number of Steinmann heptagon symbols at weights 1 through 7, and those satisfying the MHV next-to-final entry condition at weight 7. All of them are organized with respect to the discrete symmetries Z i Z i+1, Z i Z 8 i of the MHV amplitude. 1. Compare with 7, 42, 237, 1288, 6763 non-steinmann heptagon symbols GP The Steinmann Cluster Bootstrap Steinmann Upgrade 16/21

72 Results: Steinmann Heptagon symbols Weight k = parity +, flip parity +, flip parity, flip parity, flip Total Table: Number of Steinmann heptagon symbols at weights 1 through 7, and those satisfying the MHV next-to-final entry condition at weight 7. All of them are organized with respect to the discrete symmetries Z i Z i+1, Z i Z 8 i of the MHV amplitude. 1. Compare with 7, 42, 237, 1288, 6763 non-steinmann heptagon symbols = 6 9 = 2 3 reduction at weight 2 GP The Steinmann Cluster Bootstrap Steinmann Upgrade 16/21

73 Results: Steinmann Heptagon symbols Weight k = parity +, flip parity +, flip parity, flip parity, flip Total Table: Number of Steinmann heptagon symbols at weights 1 through 7, and those satisfying the MHV next-to-final entry condition at weight 7. All of them are organized with respect to the discrete symmetries Z i Z i+1, Z i Z 8 i of the MHV amplitude. 1. Compare with 7, 42, 237, 1288, 6763 non-steinmann heptagon symbols = 6 9 = 2 3 reduction at weight 2 3. Increase by a factor of 3 instead of 5 at each weight GP The Steinmann Cluster Bootstrap Steinmann Upgrade 16/21

74 Results: Steinmann Heptagon symbols Weight k = parity +, flip parity +, flip parity, flip parity, flip Total Table: Number of Steinmann heptagon symbols at weights 1 through 7, and those satisfying the MHV next-to-final entry condition at weight 7. All of them are organized with respect to the discrete symmetries Z i Z i+1, Z i Z 8 i of the MHV amplitude. 1. Compare with 7, 42, 237, 1288, 6763 non-steinmann heptagon symbols = 6 9 = 2 3 reduction at weight 2 3. Increase by a factor of 3 instead of 5 at each weight 4. E.g. 6-fold reduction already at weight 5! In this manner, obtained 3-loop NMHV and 4-loop MHV heptagon GP The Steinmann Cluster Bootstrap Steinmann Upgrade 16/21

75 New Developments I GP The Steinmann Cluster Bootstrap New Developments 17/21

76 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] GP The Steinmann Cluster Bootstrap New Developments 17/21

77 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: GP The Steinmann Cluster Bootstrap New Developments 17/21

78 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 1. Exorcising Elliptic Beasts GP The Steinmann Cluster Bootstrap New Developments 17/21

79 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 1. Exorcising Elliptic Beasts Elliptic generalizations of MPLs needed starting at 2 loops [See talks by Adams,Broadhurst,Vanhove] GP The Steinmann Cluster Bootstrap New Developments 17/21

80 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 1. Exorcising Elliptic Beasts Elliptic generalizations of MPLs needed starting at 2 loops [See talks by Adams,Broadhurst,Vanhove] By analyzing its cuts, arguments that following integral, potentially contributing to 6-loop NMHV, is [Bourjaily,Parra Martinez] elliptic. GP The Steinmann Cluster Bootstrap New Developments 17/21

81 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 1. Exorcising Elliptic Beasts Elliptic generalizations of MPLs needed starting at 2 loops [See talks by Adams,Broadhurst,Vanhove] By analyzing its cuts, arguments that following integral, potentially contributing to 6-loop NMHV, is [Bourjaily,Parra Martinez] elliptic. Our result is purely MPL, thus lending no support to this claim. GP The Steinmann Cluster Bootstrap New Developments 17/21

82 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 2. Application of heptagon ideas simplifying construction of function bases GP The Steinmann Cluster Bootstrap New Developments 18/21

83 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 2. Application of heptagon ideas simplifying construction of function bases New alphabet: {a, b, c, m u, m v, m w, y u, y v, y w }, where a = u vw, m u = 1 u u, u = , y u = & cyclic GP The Steinmann Cluster Bootstrap New Developments 18/21

84 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 2. Application of heptagon ideas simplifying construction of function bases New alphabet: {a, b, c, m u, m v, m w, y u, y v, y w }, where a = u vw, m u = 1 u u, u = , y u = & cyclic Simplest formulation of Steinmann relations for the amplitude: No b, c can appear after a in 2 nd symbol entry & cyclic GP The Steinmann Cluster Bootstrap New Developments 18/21

85 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 2. Application of heptagon ideas simplifying construction of function bases New alphabet: {a, b, c, m u, m v, m w, y u, y v, y w }, where a = u vw, m u = 1 u u, u = , y u = & cyclic 3. Expose extended Steinmann relations for the amplitude: No b, c can appear after a in any symbol entry & cyclic GP The Steinmann Cluster Bootstrap New Developments 18/21

86 New Developments I The 6-loop, 6-particle N+MHV amplitude [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] Significance: 2. Application of heptagon ideas simplifying construction of function bases New alphabet: {a, b, c, m u, m v, m w, y u, y v, y w }, where a = u vw, m u = 1 u u, u = , y u = & cyclic 3. Expose extended Steinmann relations for the amplitude: No b, c can appear after a in any symbol entry & cyclic Observed empirically at first, must be consequence of original Steinmann holding not just in the Euclidean region, but also on other Riemann sheets. GP The Steinmann Cluster Bootstrap New Developments 18/21

87 New Developments II Double penta-ladders to all orders GP The Steinmann Cluster Bootstrap New Developments 19/21

88 New Developments II Double penta-ladders to all orders Can we construct n-gon function space without solving large linear systems? GP The Steinmann Cluster Bootstrap New Developments 19/21

89 New Developments II Double penta-ladders to all orders Can we construct n-gon function space without solving large linear systems? At least for n = 6 subspace spanned by double penta-ladder integrals, yes! [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] [Arkani-Hamed,Bourjaily,Cachazo,Caron-Huot, Trnka] [Drummond,Henn,Trnka] Ω (L) (u, v, w) GP The Steinmann Cluster Bootstrap New Developments 19/21

90 New Developments II Double penta-ladders to all orders Can we construct n-gon function space without solving large linear systems? At least for n = 6 subspace spanned by double penta-ladder integrals, yes! [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] [Arkani-Hamed,Bourjaily,Cachazo,Caron-Huot, Trnka] [Drummond,Henn,Trnka] Ω (L) (u, v, w) E.g. Ω (2) d 4 Z AB d 4 Z CD (iπ 2 ) 2 AB13 CD AB61 AB12 AB23 AB34 ABCD CD34 CD45 CD56 CD61 GP The Steinmann Cluster Bootstrap New Developments 19/21

91 New Developments II Double penta-ladders to all orders Can we construct n-gon function space without solving large linear systems? At least for n = 6 subspace spanned by double penta-ladder integrals, yes! [Caron-Huot,Dixon,McLeod,GP,von Hippel;to appear] [Arkani-Hamed,Bourjaily,Cachazo,Caron-Huot, Trnka] [Drummond,Henn,Trnka] Ω (L) (u, v, w) E.g. Ω (2) d 4 Z AB d 4 Z CD (iπ 2 ) 2 AB13 CD AB61 AB12 AB23 AB34 ABCD CD34 CD45 CD56 CD61 Can in fact resum Ω λ L Ω (L) in terms of a simple integral. GP The Steinmann Cluster Bootstrap New Developments 19/21

92 Beyond seven particles GP The Steinmann Cluster Bootstrap New Developments 20/21

93 Beyond seven particles For N 8, Gr(N, 8) cluster algebra becomes infinite GP The Steinmann Cluster Bootstrap New Developments 20/21

94 Beyond seven particles For N 8, Gr(N, 8) cluster algebra becomes infinite However, in multi-regge limit, Gr(N, 8) A N 5 A N 5 : finite! [Del Duca,Druc,Drummond,Duhr,Dulat,Marzucca,GP,Verbeek] GP The Steinmann Cluster Bootstrap New Developments 20/21

95 Beyond seven particles For N 8, Gr(N, 8) cluster algebra becomes infinite However, in multi-regge limit, Gr(N, 8) A N 5 A N 5 : finite! [Del Duca,Druc,Drummond,Duhr,Dulat,Marzucca,GP,Verbeek] The two A N 5 factors not independent: Related by single-valuedness Therefore multi-regge limit important stepping stone towards bootstrapping higher-point amplitudes, and also closely related to integrability & collinear OPE limit. [Basso,Caron-Huot,Sever][Drummond,Papathanasiou] GP The Steinmann Cluster Bootstrap New Developments 20/21

96 Conclusions & Outlook In this presentation, we talked about the bootstrap program for constructing N = 4 SYM amplitudes at fixed-order/general kinematics, by exploiting their analytic properties. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 21/21

97 Conclusions & Outlook In this presentation, we talked about the bootstrap program for constructing N = 4 SYM amplitudes at fixed-order/general kinematics, by exploiting their analytic properties. Our improved understanding of the latter has led to two major upgrades: GP The Steinmann Cluster Bootstrap Conclusions & Outlook 21/21

98 Conclusions & Outlook In this presentation, we talked about the bootstrap program for constructing N = 4 SYM amplitudes at fixed-order/general kinematics, by exploiting their analytic properties. Our improved understanding of the latter has led to two major upgrades: Cluster algebras are instrumental in identifying the function space (arguments) in which the amplitude lives GP The Steinmann Cluster Bootstrap Conclusions & Outlook 21/21

99 Conclusions & Outlook In this presentation, we talked about the bootstrap program for constructing N = 4 SYM amplitudes at fixed-order/general kinematics, by exploiting their analytic properties. Our improved understanding of the latter has led to two major upgrades: Cluster algebras are instrumental in identifying the function space (arguments) in which the amplitude lives (Extended) Steinmann relations massively reduce the size of this space much simpler to single it out GP The Steinmann Cluster Bootstrap Conclusions & Outlook 21/21

100 Conclusions & Outlook In this presentation, we talked about the bootstrap program for constructing N = 4 SYM amplitudes at fixed-order/general kinematics, by exploiting their analytic properties. Our improved understanding of the latter has led to two major upgrades: Cluster algebras are instrumental in identifying the function space (arguments) in which the amplitude lives (Extended) Steinmann relations massively reduce the size of this space much simpler to single it out This has led a wealth of results for n = 6, 7 amplitudes, with the power of the method, surprisingly, increasing with n. More to come, n 8, QCD... GP The Steinmann Cluster Bootstrap Conclusions & Outlook 21/21

101 Conclusions & Outlook In this presentation, we talked about the bootstrap program for constructing N = 4 SYM amplitudes at fixed-order/general kinematics, by exploiting their analytic properties. Our improved understanding of the latter has led to two major upgrades: Cluster algebras are instrumental in identifying the function space (arguments) in which the amplitude lives (Extended) Steinmann relations massively reduce the size of this space much simpler to single it out This has led a wealth of results for n = 6, 7 amplitudes, with the power of the method, surprisingly, increasing with n. More to come, n 8, QCD... Ultimately, can the integrability of planar SYM theory, together with a thorough knowledge of the analytic structure of its amplitudes, lead us to the theory s exact S-matrix? GP The Steinmann Cluster Bootstrap Conclusions & Outlook 21/21

102

103 Momentum Twistors Z I [Hodges] GP The Steinmann Cluster Bootstrap Conclusions & Outlook 23/21

104 Momentum Twistors Z I [Hodges] Represent dual space variables x µ R 1,3 as projective null vectors X M R 2,4, X 2 = 0, X λx. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 23/21

105 Momentum Twistors Z I [Hodges] Represent dual space variables x µ R 1,3 as projective null vectors X M R 2,4, X 2 = 0, X λx. Repackage vector X M of SO(2, 4) into antisymmetric representation X IJ = X JI = of SU(2, 2) GP The Steinmann Cluster Bootstrap Conclusions & Outlook 23/21

106 Momentum Twistors Z I [Hodges] Represent dual space variables x µ R 1,3 as projective null vectors X M R 2,4, X 2 = 0, X λx. Repackage vector X M of SO(2, 4) into antisymmetric representation X IJ = X JI = of SU(2, 2) Can build latter from two copies of the fundamental Z I =, X IJ = Z [I ZJ] = (Z I ZJ Z J ZI )/2 or X = Z Z GP The Steinmann Cluster Bootstrap Conclusions & Outlook 23/21

107 Momentum Twistors Z I [Hodges] Represent dual space variables x µ R 1,3 as projective null vectors X M R 2,4, X 2 = 0, X λx. Repackage vector X M of SO(2, 4) into antisymmetric representation X IJ = X JI = of SU(2, 2) Can build latter from two copies of the fundamental Z I =, X IJ = Z [I ZJ] = (Z I ZJ Z J ZI )/2 or X = Z Z After complexifying, Z I transform in SL(4, C). Since Z tz, can be viewed as homogeneous coordinates on P 3. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 23/21

108 Momentum Twistors Z I [Hodges] Represent dual space variables x µ R 1,3 as projective null vectors X M R 2,4, X 2 = 0, X λx. Repackage vector X M of SO(2, 4) into antisymmetric representation X IJ = X JI = of SU(2, 2) Can build latter from two copies of the fundamental Z I =, X IJ = Z [I ZJ] = (Z I ZJ Z J ZI )/2 or X = Z Z After complexifying, Z I transform in SL(4, C). Since Z tz, can be viewed as homogeneous coordinates on P 3. Can show (x x ) 2 2X X = ɛ IJKL Z I ZJ Z K Z L = det(z ZZ Z ) Z ZZ Z GP The Steinmann Cluster Bootstrap Conclusions & Outlook 23/21

109 Momentum Twistors Z I [Hodges] Represent dual space variables x µ R 1,3 as projective null vectors X M R 2,4, X 2 = 0, X λx. Repackage vector X M of SO(2, 4) into antisymmetric representation X IJ = X JI = of SU(2, 2) Can build latter from two copies of the fundamental Z I =, X IJ = Z [I ZJ] = (Z I ZJ Z J ZI )/2 or X = Z Z After complexifying, Z I transform in SL(4, C). Since Z tz, can be viewed as homogeneous coordinates on P 3. Can show (x x ) 2 2X X = ɛ IJKL Z I ZJ Z K Z L = det(z ZZ Z ) Z ZZ Z (x i+i x i ) 2 = 0 X i = Z i 1 Z i GP The Steinmann Cluster Bootstrap Conclusions & Outlook 23/21

110 Conf n (P 3 ) and Graßmannians Can realize Conf n (P 3 ) as 4 n matrix (Z 1 Z 2... Z n ) modulo rescalings of the n columns and SL(4) transformations, which resembles a Graßmannian Gr(4, n). GP The Steinmann Cluster Bootstrap Conclusions & Outlook 24/21

111 Conf n (P 3 ) and Graßmannians Can realize Conf n (P 3 ) as 4 n matrix (Z 1 Z 2... Z n ) modulo rescalings of the n columns and SL(4) transformations, which resembles a Graßmannian Gr(4, n). Gr(k, n): The space of k-dimensional planes passing through the origin in an n-dimensional space. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 24/21

112 Conf n (P 3 ) and Graßmannians Can realize Conf n (P 3 ) as 4 n matrix (Z 1 Z 2... Z n ) modulo rescalings of the n columns and SL(4) transformations, which resembles a Graßmannian Gr(4, n). Gr(k, n): The space of k-dimensional planes passing through the origin in an n-dimensional space. Equivalently the space of k n matrices modulo GL(k) transformations: GP The Steinmann Cluster Bootstrap Conclusions & Outlook 24/21

113 Conf n (P 3 ) and Graßmannians Can realize Conf n (P 3 ) as 4 n matrix (Z 1 Z 2... Z n ) modulo rescalings of the n columns and SL(4) transformations, which resembles a Graßmannian Gr(4, n). Gr(k, n): The space of k-dimensional planes passing through the origin in an n-dimensional space. Equivalently the space of k n matrices modulo GL(k) transformations: k-plane specified by k basis vectors that span it k n matrix GP The Steinmann Cluster Bootstrap Conclusions & Outlook 24/21

114 Conf n (P 3 ) and Graßmannians Can realize Conf n (P 3 ) as 4 n matrix (Z 1 Z 2... Z n ) modulo rescalings of the n columns and SL(4) transformations, which resembles a Graßmannian Gr(4, n). Gr(k, n): The space of k-dimensional planes passing through the origin in an n-dimensional space. Equivalently the space of k n matrices modulo GL(k) transformations: k-plane specified by k basis vectors that span it k n matrix Under GL(k) transformations, basis vectors change, but still span the same plane. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 24/21

115 Conf n (P 3 ) and Graßmannians Can realize Conf n (P 3 ) as 4 n matrix (Z 1 Z 2... Z n ) modulo rescalings of the n columns and SL(4) transformations, which resembles a Graßmannian Gr(4, n). Gr(k, n): The space of k-dimensional planes passing through the origin in an n-dimensional space. Equivalently the space of k n matrices modulo GL(k) transformations: k-plane specified by k basis vectors that span it k n matrix Under GL(k) transformations, basis vectors change, but still span the same plane. Comparing the two matrices, Conf n (P 3 ) = Gr(4, n)/(c ) n 1 GP The Steinmann Cluster Bootstrap Conclusions & Outlook 24/21

116 Imposing Constraints: Integrable Words GP The Steinmann Cluster Bootstrap Conclusions & Outlook 25/21

117 Imposing Constraints: Integrable Words Given a random symbol S of weight k > 1, there does not in general exist any function whose symbol is S. A symbol is said to be integrable, (or, to be an integrable word) if it satisfies f (α 1,α 2,...,α k ) 0 d log φ αj d log φ αj+1 (φ α1 φ αk ) = 0, α 1,...,α k omitting φ αj φ αj+1 j {1,..., k 1}. These are necessary and sufficient conditions for a function f k with symbol S to exist. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 25/21

118 Imposing Constraints: Integrable Words Given a random symbol S of weight k > 1, there does not in general exist any function whose symbol is S. A symbol is said to be integrable, (or, to be an integrable word) if it satisfies f (α 1,α 2,...,α k ) 0 d log φ αj d log φ αj+1 (φ α1 φ αk ) = 0, α 1,...,α k omitting φ αj φ αj+1 j {1,..., k 1}. These are necessary and sufficient conditions for a function f k with symbol S to exist. Example: (1 xy) (1 x) with x, y independent. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 25/21

119 Imposing Constraints: Integrable Words Given a random symbol S of weight k > 1, there does not in general exist any function whose symbol is S. A symbol is said to be integrable, (or, to be an integrable word) if it satisfies f (α 1,α 2,...,α k ) 0 d log φ αj d log φ αj+1 (φ α1 φ αk ) = 0, α 1,...,α k omitting φ αj φ αj+1 j {1,..., k 1}. These are necessary and sufficient conditions for a function f k with symbol S to exist. Example: (1 xy) (1 x) with x, y independent. ydx xdy d log(1 xy) d log(1 x) = 1 xy = dx 1 x x dy dx (1 xy)(1 x) GP The Steinmann Cluster Bootstrap Conclusions & Outlook 25/21

120 Imposing Constraints: Integrable Words Given a random symbol S of weight k > 1, there does not in general exist any function whose symbol is S. A symbol is said to be integrable, (or, to be an integrable word) if it satisfies f (α 1,α 2,...,α k ) 0 d log φ αj d log φ αj+1 (φ α1 φ αk ) = 0, α 1,...,α k omitting φ αj φ αj+1 j {1,..., k 1}. These are necessary and sufficient conditions for a function f k with symbol S to exist. Example: (1 xy) (1 x) with x, y independent. ydx xdy d log(1 xy) d log(1 x) = 1 xy = dx 1 x x dy dx (1 xy)(1 x) Not integrable GP The Steinmann Cluster Bootstrap Conclusions & Outlook 25/21

121 Imposing Constraints: Physical Singularities GP The Steinmann Cluster Bootstrap Conclusions & Outlook 26/21

122 Imposing Constraints: Physical Singularities Locality: Amplitudes may only have singularities when some intermediate particle goes on-shell. GP The Steinmann Cluster Bootstrap Conclusions & Outlook 26/21

123 Imposing Constraints: Physical Singularities Locality: Amplitudes may only have singularities when some intermediate particle goes on-shell. Planar colour-ordered amplitudes in massless theories: Only happens when (p i + p i p j 1 ) 2 = (x j x i ) 2 i 1 i j 1 j 0 GP The Steinmann Cluster Bootstrap Conclusions & Outlook 26/21