A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon

Transcription

1 A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon Georgios Papathanasiou Laboratoire d Annecy-le-Vieux de Physique Théorique & CERN Humboldt-Universität zu Berlin March 6, [hep-th] with Drummond & Spradlin work in progress

2 Outline Motivation: Why N = 4 SYM? Scattering Ampitudes, Wilson Loop OPE and Integrability The Amplitude Bootstrap and its Cluster Algebra Upgrade A Symbol of Uniqueness: The 3-loop MHV Heptagon Conclusions & Outlook GP A Symbol of Uniqueness 2/34

3 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

4 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories N = 4 SU(N) SYM Type IIB superstring theory on AdS 5 S 5. GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

5 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories N = 4 SU(N) SYM Type IIB superstring theory on AdS 5 S 5. strongly coupled weakly coupled GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

6 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories N = 4 SU(N) SYM Type IIB superstring theory on AdS 5 S 5. strongly coupled weakly coupled In the t Hooft limit, N with λ = g 2 Y M N fixed: Integrable structures All loop, interpolating quantities! [Beisert,Eden,Staudacher] GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

7 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories N = 4 SU(N) SYM Type IIB superstring theory on AdS 5 S 5. strongly coupled weakly coupled In the t Hooft limit, N with λ = g 2 Y M N fixed: Integrable structures All loop, interpolating quantities! [Beisert,Eden,Staudacher] Ideal theoretical playground for developing new computational tools, GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

8 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories N = 4 SU(N) SYM Type IIB superstring theory on AdS 5 S 5. strongly coupled weakly coupled In the t Hooft limit, N with λ = g 2 Y M N fixed: Integrable structures All loop, interpolating quantities! [Beisert,Eden,Staudacher] Ideal theoretical playground for developing new computational tools, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

9 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories N = 4 SU(N) SYM Type IIB superstring theory on AdS 5 S 5. strongly coupled weakly coupled In the t Hooft limit, N with λ = g 2 Y M N fixed: Integrable structures All loop, interpolating quantities! [Beisert,Eden,Staudacher] Ideal theoretical playground for developing new computational tools, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] Method of Symbols [Goncharov,Spradlin,Vergu,Volovich] GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

10 N = 4 Super Yang Mills Theory & Why Should We Care Unique possibility for the nonperturbative investigation of gauge theories N = 4 SU(N) SYM Type IIB superstring theory on AdS 5 S 5. strongly coupled weakly coupled In the t Hooft limit, N with λ = g 2 Y M N fixed: Integrable structures All loop, interpolating quantities! [Beisert,Eden,Staudacher] Ideal theoretical playground for developing new computational tools, Generalised Unitarity [Bern,Dixon,Dunbar,Kosower... ] Method of Symbols [Goncharov,Spradlin,Vergu,Volovich] Then apply to QCD, e.g. gg Hg 2 for N 3 LO Higgs cross-section! [Anastasiou,Duhr,Dulat,Herzog,Mistlberger] GP A Symbol of Uniqueness Motivation: Why N = 4 SYM? 3/34

11 Scattering Amplitudes: dσ A 2 For N = 4, all fields massless and in adjoint of gauge group SU(N). GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 4/34

12 Scattering Amplitudes: dσ A 2 For N = 4, all fields massless and in adjoint of gauge group SU(N). Can thus use helicity h = S ˆp to classify on-shell particle content, h 1 1/2 0 1/2 1 G Q1 ΓA Q2 Φ AB Q 3 Γ A Q 4 G + For the gluons G ±, the gluinos Γ, Γ, and the scalars Φ. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 4/34

13 Scattering Amplitudes: dσ A 2 For N = 4, all fields massless and in adjoint of gauge group SU(N). Can thus use helicity h = S ˆp to classify on-shell particle content, h 1 1/2 0 1/2 1 G Q1 ΓA Q2 Φ AB Q 3 Γ A Q 4 G + For the gluons G ±, the gluinos Γ, Γ, and the scalars Φ. For n gluons, A L loop n ({k i, h i, a i }) = σ S n/z n Tr(T a σ(1) T a σ(n) ) A (L) n (σ(1 h 1 ),..., σ(n hn )) +multitrace terms, subleading by powers of 1/N 2. A (L) n : color-ordered amplitude, all color factors removed. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 4/34

14 Maximally Hellicity Violating (MHV) Amplitudes These are the simplest amplitudes: A (L) n (1 +,..., i,..., j,..., n + ) GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 5/34

15 Maximally Hellicity Violating (MHV) Amplitudes These are the simplest amplitudes: A (L) n (1 +,..., i,..., j,..., n + ) They also have remarkable properties, namely they are dual to null polygonal Wilson loops. [Alday,Maldacena][Drummond,Korchemsky,Sokatchev][Brandhuber,Heslop,Travaglini] x n k n k i x i+1 x i x i+1,i, x 1 A n k 1 x 2 x 3 k 2 k 2 i = x 2 i+1,i = 0 k i = 0 log W n = log AMHV n A MHV n,tree automatically satisfied + O(ɛ) k 3 GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 5/34

16 Maximally Hellicity Violating (MHV) Amplitudes These are the simplest amplitudes: A (L) n (1 +,..., i,..., j,..., n + ) They also have remarkable properties, namely they are dual to null polygonal Wilson loops. [Alday,Maldacena][Drummond,Korchemsky,Sokatchev][Brandhuber,Heslop,Travaglini] x n k n k i x i+1 x i x i+1,i, x 1 A n k 1 x 2 x 3 k 2 k 2 i = x 2 i+1,i = 0 k i = 0 log W n = log AMHV n A MHV n,tree automatically satisfied + O(ɛ) k 3 exhibit (formally) dual conformal invariance (DCI) under x µ i xµ i x 2 i GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 5/34

17 MHV Scattering Amplitudes In reality DCI broken by divergences, (IR in massless N = 4/UV in cusped WL). Breaking controlled by conformal Ward identity. [Drummond,Henn,Korchemsky,Sokatchev] GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 6/34

18 MHV Scattering Amplitudes In reality DCI broken by divergences, (IR in massless N = 4/UV in cusped WL). Breaking controlled by conformal Ward identity. [Drummond,Henn,Korchemsky,Sokatchev] For n = 4, 5, the latter uniquely determines the dimensionally regularized A n /W n to all loops! Given by ansatz W BDS of [Anastasiou,Bern,Dixon,Kosower][Bern,Dixon,Smirnov] n GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 6/34

19 MHV Scattering Amplitudes In reality DCI broken by divergences, (IR in massless N = 4/UV in cusped WL). Breaking controlled by conformal Ward identity. [Drummond,Henn,Korchemsky,Sokatchev] For n = 4, 5, the latter uniquely determines the dimensionally regularized A n /W n to all loops! Given by ansatz W BDS of [Anastasiou,Bern,Dixon,Kosower][Bern,Dixon,Smirnov] For n 6, W n = W BDS n e Rn(u 1,...,u m) where the remainder function R n is conformally invariant, and thus a function of conformal cross ratios, e.g u = x2 46 x2 13. x 2 36 x2 14 n GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 6/34

20 MHV Scattering Amplitudes In reality DCI broken by divergences, (IR in massless N = 4/UV in cusped WL). Breaking controlled by conformal Ward identity. [Drummond,Henn,Korchemsky,Sokatchev] For n = 4, 5, the latter uniquely determines the dimensionally regularized A n /W n to all loops! Given by ansatz W BDS of [Anastasiou,Bern,Dixon,Kosower][Bern,Dixon,Smirnov] For n 6, W n = W BDS n e Rn(u 1,...,u m) where the remainder function R n is conformally invariant, and thus a function of conformal cross ratios, e.g u = x2 46 x2 13. x 2 36 x2 14 # of independent u i : m = 4n n 15 = 3n 15 For the moment, focus on R 6 (u 1, u 2, u 3 ). n GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 6/34

21 Nonperturbative Definition via the Collinear Limit Kinematics GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 7/34

22 Nonperturbative Definition via the Collinear Limit Kinematics Pick 2 non-intersecting segments (PO), (SF), form square by connecting them with another 2 null segments (PS), (OF). GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 7/34

23 Nonperturbative Definition via the Collinear Limit Kinematics Pick 2 non-intersecting segments (PO), (SF), form square by connecting them with another 2 null segments (PS), (OF). Fix all of its 16-4 coordinates by conformal transformations (15): In fact, each square invariant under subset of 3 transformations! GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 7/34

24 Nonperturbative Definition via the Collinear Limit Kinematics Pick 2 non-intersecting segments (PO), (SF), form square by connecting them with another 2 null segments (PS), (OF). Fix all of its 16-4 coordinates by conformal transformations (15): In fact, each square invariant under subset of 3 transformations! Convenient to put cusps at origin O, spacelike and null (past+future) infinity S,P,F in (x 0, x 1 ) plane. Symmetries generated by dilatations D, boosts M 01, and rotations on (x 2, x 3 ) plane M 23. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 7/34

25 Nonperturbative Definition via the Collinear Limit Kinematics Pick 2 non-intersecting segments (PO), (SF), form square by connecting them with another 2 null segments (PS), (OF). Fix all of its 16-4 coordinates by conformal transformations (15): In fact, each square invariant under subset of 3 transformations! Convenient to put cusps at origin O, spacelike and null (past+future) infinity S,P,F in (x 0, x 1 ) plane. Symmetries generated by dilatations D, boosts M 01, and rotations on (x 2, x 3 ) plane M 23. Collinear limit: Act with e τ(d M 01) on A and B, and take τ. Parametrize u 1, u 2, u 3 by group coordinates τ, σ, φ. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 7/34

26 Nonperturbative Definition via the Collinear Limit Dynamics. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 8/34

27 Nonperturbative Definition via the Collinear Limit Dynamics Can think of (P O),(SF ) as a color-electric flux tube sourced by a quark-antiquark pair moving at the speed of light, and decompose the Wilson loop with respect to all possible excitations ψ i of this flux tube. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 8/34

28 Nonperturbative Definition via the Collinear Limit Dynamics Can think of (P O),(SF ) as a color-electric flux tube sourced by a quark-antiquark pair moving at the speed of light, and decompose the Wilson loop with respect to all possible excitations ψ i of this flux tube. Schematically, W = ψ i e τe i+ip i +im i φ P(0 ψ i )P(ψ i 0) GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 8/34

29 Nonperturbative Definition via the Collinear Limit Dynamics Can think of (P O),(SF ) as a color-electric flux tube sourced by a quark-antiquark pair moving at the speed of light, and decompose the Wilson loop with respect to all possible excitations ψ i of this flux tube. Schematically, W = ψ i e τe i+ip i +im i φ P(0 ψ i )P(ψ i 0) Propagation of square eigenstates GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 8/34

30 Nonperturbative Definition via the Collinear Limit Dynamics Can think of (P O),(SF ) as a color-electric flux tube sourced by a quark-antiquark pair moving at the speed of light, and decompose the Wilson loop with respect to all possible excitations ψ i of this flux tube. Schematically, W = ψ i e τe i+ip i +im i φ P(0 ψ i )P(ψ i 0) Propagation of square eigenstates Transition between squares GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 8/34

31 Nonperturbative Definition via the Collinear Limit Dynamics Can think of (P O),(SF ) as a color-electric flux tube sourced by a quark-antiquark pair moving at the speed of light, and decompose the Wilson loop with respect to all possible excitations ψ i of this flux tube. Schematically, W = ψ i e τe i+ip i +im i φ P(0 ψ i )P(ψ i 0) Propagation of square eigenstates Transition between squares WL Operator Product Expansion (OPE) [Alday,Gaiotto,Maldacena,Sever,Vieira] GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 8/34

32 Wilson Loop OPE & Integrability In N = 4 SYM, flux tube excitations in 1-1 correspondence with excitations of an integrable spin chain with hamiltonian D M 01. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 9/34

33 Wilson Loop OPE & Integrability In N = 4 SYM, flux tube excitations in 1-1 correspondence with excitations of an integrable spin chain with hamiltonian D M 01. Lightest excitations made of 6 scalars φ, 4+4 fermions ψ, ψ and 1+1 gluons F, F of the theory, with classical S = 1, over the vacuum = tr (ZD S +Z), Z = φ 1 + iφ 2, D + = D 0 + D 1 GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 9/34

34 Wilson Loop OPE & Integrability In N = 4 SYM, flux tube excitations in 1-1 correspondence with excitations of an integrable spin chain with hamiltonian D M 01. Lightest excitations made of 6 scalars φ, 4+4 fermions ψ, ψ and 1+1 gluons F, F of the theory, with classical S = 1, over the vacuum = tr (ZD S +Z), Z = φ 1 + iφ 2, D + = D 0 + D 1 Integrability enables the calculation of the excitation energies E(p), E(p) = ( S) 1 ( S) vac = 1 + λ l E (l) (p) to all loops, and implies that for M excitations E M = M + O(λ). [Basso] l=1 GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 9/34

35 Wilson Loop OPE & Integrability In N = 4 SYM, flux tube excitations in 1-1 correspondence with excitations of an integrable spin chain with hamiltonian D M 01. Lightest excitations made of 6 scalars φ, 4+4 fermions ψ, ψ and 1+1 gluons F, F of the theory, with classical S = 1, over the vacuum = tr (ZD S +Z), Z = φ 1 + iφ 2, D + = D 0 + D 1 Integrability enables the calculation of the excitation energies E(p), E(p) = ( S) 1 ( S) vac = 1 + λ l E (l) (p) to all loops, and implies that for M excitations E M = M + O(λ). [Basso] Thus, weak coupling WL OPE=expansion in terms e τm, M = 1, 2... l=1 GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 9/34

36 The Proposal of Basso,Sever,Vieira To complete WL OPE description, also emission/absorption form factors or pentagon transitions P(0 ψ 1 ), P(ψ 1 0) needed. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 10/34

37 The Proposal of Basso,Sever,Vieira To complete WL OPE description, also emission/absorption form factors or pentagon transitions P(0 ψ 1 ), P(ψ 1 0) needed. Obtained for all M = 1, 2 particle excitations, and all arbitrary M gluonic excitations by exploiting once more the power of integrability. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 10/34

38 The Proposal of Basso,Sever,Vieira To complete WL OPE description, also emission/absorption form factors or pentagon transitions P(0 ψ 1 ), P(ψ 1 0) needed. Obtained for all M = 1, 2 particle excitations, and all arbitrary M gluonic excitations by exploiting once more the power of integrability. Yield all-loop integral expressions for individual terms in collinear limit expansion. Can compute at weak coupling. [GP 13][GP 14] GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 10/34

39 The Proposal of Basso,Sever,Vieira To complete WL OPE description, also emission/absorption form factors or pentagon transitions P(0 ψ 1 ), P(ψ 1 0) needed. Obtained for all M = 1, 2 particle excitations, and all arbitrary M gluonic excitations by exploiting once more the power of integrability. Yield all-loop integral expressions for individual terms in collinear limit expansion. Can compute at weak coupling. [GP 13][GP 14] Also very interesting integrability-based methods for constructing Yangian invariants relevant to scattering amplitudes. [Arkani-Hamed,Beisert,Bourjaily,Broedel,Cachazo,Caron-Huot,Chicherin,Derkachov,Ferro,Frassek,Goncharov Kanning,Kirchner,Ko,Leeuw,Lukowski,Menenghelli,Plefka,Postnikov,Rosso,Staudacher,Trnka] GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 10/34

40 The Proposal of Basso,Sever,Vieira To complete WL OPE description, also emission/absorption form factors or pentagon transitions P(0 ψ 1 ), P(ψ 1 0) needed. Obtained for all M = 1, 2 particle excitations, and all arbitrary M gluonic excitations by exploiting once more the power of integrability. Yield all-loop integral expressions for individual terms in collinear limit expansion. Can compute at weak coupling. [GP 13][GP 14] Also very interesting integrability-based methods for constructing Yangian invariants relevant to scattering amplitudes. [Arkani-Hamed,Beisert,Bourjaily,Broedel,Cachazo,Caron-Huot,Chicherin,Derkachov,Ferro,Frassek,Goncharov Kanning,Kirchner,Ko,Leeuw,Lukowski,Menenghelli,Plefka,Postnikov,Rosso,Staudacher,Trnka] Spectral Problem Wisdom If exact S-matrix within reach, look at many data points at weak/strong coupling to extract its general pattern. GP A Symbol of Uniqueness Scattering Ampitudes, Wilson Loop OPE and Integrability 10/34

41 How do we compute R n (L) in general kinematics? GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

42 How do we compute R n (L) in general kinematics? For n = 6, very successful amplitude bootstrap up to L = 4 loops. [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel,Pennington] [Dixon,Drummond,Duhr,Pennington] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

43 How do we compute R n (L) in general kinematics? For n = 6, very successful amplitude bootstrap up to L = 4 loops. [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel,Pennington] [Dixon,Drummond,Duhr,Pennington] A. Construct an ansatz assuming GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

44 How do we compute R n (L) in general kinematics? For n = 6, very successful amplitude bootstrap up to L = 4 loops. [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel,Pennington] [Dixon,Drummond,Duhr,Pennington] A. Construct an ansatz assuming 1. What the general class of functions that suffices to express R n (L) is GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

45 How do we compute R n (L) in general kinematics? For n = 6, very successful amplitude bootstrap up to L = 4 loops. [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel,Pennington] [Dixon,Drummond,Duhr,Pennington] A. Construct an ansatz assuming 1. What the general class of functions that suffices to express R n (L) 2. What the function arguments (encoding the kinematics) are is GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

46 How do we compute R n (L) in general kinematics? For n = 6, very successful amplitude bootstrap up to L = 4 loops. [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel,Pennington] [Dixon,Drummond,Duhr,Pennington] A. Construct an ansatz assuming 1. What the general class of functions that suffices to express R n (L) is 2. What the function arguments (encoding the kinematics) are B. Fix the coefficients of the ansatz by imposing consistency conditions (e.g. collinear data we described in previous part of talk) GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

47 How do we compute R n (L) in general kinematics? For n = 6, very successful amplitude bootstrap up to L = 4 loops. [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel,Pennington] [Dixon,Drummond,Duhr,Pennington] A. Construct an ansatz assuming 1. What the general class of functions that suffices to express R n (L) is 2. What the function arguments (encoding the kinematics) are B. Fix the coefficients of the ansatz by imposing consistency conditions (e.g. collinear data we described in previous part of talk) Motivated by this progress, we upgraded this procedure for n = 7, with information from the cluster algebra structure of the kinematical space. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

48 How do we compute R n (L) in general kinematics? For n = 6, very successful amplitude bootstrap up to L = 4 loops. [Dixon,Drummond,Henn] [Dixon,Drummond,Hippel,Pennington] [Dixon,Drummond,Duhr,Pennington] A. Construct an ansatz assuming 1. What the general class of functions that suffices to express R n (L) is 2. What the function arguments (encoding the kinematics) are B. Fix the coefficients of the ansatz by imposing consistency conditions (e.g. collinear data we described in previous part of talk) Motivated by this progress, we upgraded this procedure for n = 7, with information from the cluster algebra structure of the kinematical space. Surprisingly, we found that heptagon bootstrap is more powerful than the hexagon one! Obtained the symbol of R (3) 7 from very little input. [Drummond,GP,Spradlin] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 11/34

49 What are the right functions? Generalised polylogarithms (GPLs) GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 12/34

50 What are the right functions? Generalised polylogarithms (GPLs) f k is a GPL 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 12/34

51 What are the right functions? Generalised polylogarithms (GPLs) f k is a GPL 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. Very 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 GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 12/34

52 What are the right functions? Generalised polylogarithms (GPLs) f k is a GPL 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. Very 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 GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 12/34

53 What are the right functions? Generalised polylogarithms (GPLs) f k is a GPL 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. Very 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=gpls of weight k = 2L [Duhr,Del Duca,Smirnov][Arkani-Hamed...][GP] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 12/34

54 What are the right variables? GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 13/34

55 What are the right variables? More precisely, what is the symbol alphabet? GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 13/34

56 What are the right variables? More precisely, what is the symbol alphabet? For n = 6, 9 letters, motivated by analysis of relevant integrals GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 13/34

57 What are the right variables? 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 13/34

58 What are the right variables? 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 conformal transformations. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 13/34

59 Momentum Twistors Z I [Hodges] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 14/34

60 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 14/34

61 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 14/34

62 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 14/34

63 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 14/34

64 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 14/34

65 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 14/34

66 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 15/34

67 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 15/34

68 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 15/34

69 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 15/34

70 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 15/34

71 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 15/34

72 Cluster algebras [Fomin,Zelevinsky] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

73 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras equipped with a distinguished set of generators (= cluster variables), grouped into overlapping subsets (= clusters) with the same number of elements (= the rank of the algebra). Constructed from an initial cluster by an iterative process (= mutation). GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

74 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras equipped with a distinguished set of generators (= cluster variables), grouped into overlapping subsets (= clusters) with the same number of elements (= the rank of the algebra). Constructed from an initial cluster by an iterative process (= mutation). Example: A 2 Cluster algebra GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

75 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras equipped with a distinguished set of generators (= cluster variables), grouped into overlapping subsets (= clusters) with the same number of elements (= the rank of the algebra). Constructed from an initial cluster by an iterative process (= mutation). Example: A 2 Cluster algebra Cluster variables: a m, m Z GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

76 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras equipped with a distinguished set of generators (= cluster variables), grouped into overlapping subsets (= clusters) with the same number of elements (= the rank of the algebra). Constructed from an initial cluster by an iterative process (= mutation). Example: A 2 Cluster algebra Cluster variables: a m, m Z Initial cluster: {a 1, a 2 } GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

77 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras equipped with a distinguished set of generators (= cluster variables), grouped into overlapping subsets (= clusters) with the same number of elements (= the rank of the algebra). Constructed from an initial cluster by an iterative process (= mutation). Example: A 2 Cluster algebra Cluster variables: a m, m Z Initial cluster: {a 1, a 2 } Clusters: {a m, a m+1 }, m Z GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

78 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras equipped with a distinguished set of generators (= cluster variables), grouped into overlapping subsets (= clusters) with the same number of elements (= the rank of the algebra). Constructed from an initial cluster by an iterative process (= mutation). Example: A 2 Cluster algebra Cluster variables: a m, m Z Initial cluster: {a 1, a 2 } Clusters: {a m, a m+1 }, m Z Mutation: {a m 1, a m } {a m, a m+1 } with a m 1 a m+1 = 1+am a m 1 GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

79 Cluster algebras [Fomin,Zelevinsky] They are commutative algebras equipped with a distinguished set of generators (= cluster variables), grouped into overlapping subsets (= clusters) with the same number of elements (= the rank of the algebra). Constructed from an initial cluster by an iterative process (= mutation). Example: A 2 Cluster algebra Cluster variables: a m, m Z Initial cluster: {a 1, a 2 } Clusters: {a m, a m+1 }, m Z Mutation: {a m 1, a m } {a m, a m+1 } with a m 1 a m+1 = 1+am a m 1 Here, finite number of cluster variables: a 3 = 1 + a 2 a 1, a 4 = 1 + a 1 + a 2 a 1 a 2, a 5 = 1 + a 1 a 2, a 6 = a 1, a 7 = a 2 GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 16/34

80 Cluster algebras (cont d) For our purposes, can be described by quivers, where each variable a k of a cluster corresponds to node k. Example: A 2 Cluster algebra Initial cluster: {a 1, a 2 }: 1 2 GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 17/34

81 Cluster algebras (cont d) For our purposes, can be described by quivers, where each variable a k of a cluster corresponds to node k. Mutation at node k: i k j, add arrow i j, reverse all arrows to/from k, remove and. Example: A 2 Cluster algebra Initial cluster: {a 1, a 2 }: 1 2 Mutate at 1: 1 2 GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 17/34

82 Cluster algebras (cont d) For our purposes, can be described by quivers, where each variable a k of a cluster corresponds to node k. Mutation at node k: i k j, add arrow i j, reverse all arrows to/from k, remove and. In this manner, obtain new quiver/cluster where a k a k = 1 a k arrows i k a i + arrows k j a j Example: A 2 Cluster algebra Initial cluster: {a 1, a 2 }: 1 2 Mutate at 1: 1 2 Leads to new cluster {a 2, a 3 } with a 3 = a 1 = 1+a 2 a 1 and so on GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 17/34

83 Connection with Conf n (P 3 ) = Gr(4, n)/(c ) n 1 GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 18/34

84 Connection with Conf n (P 3 ) = Gr(4, n)/(c ) n 1 Graßmannians Gr(k, n) equipped with cluster algebra structure [Scott] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 18/34

85 Connection with 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 18/34

86 Connection with 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 18/34

87 Connection with 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 18/34

88 Connection with 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 18/34

89 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 19/34

90 Imposing Constraints: Integrable Words GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 20/34

91 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 20/34

92 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 20/34

93 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 20/34

94 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 20/34

95 Imposing Constraints: Physical Singularities GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 21/34

96 Imposing Constraints: Physical Singularities Locality: Amplitudes may only have singularities when some intermediate particle goes on-shell. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 21/34

97 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 21/34

98 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 Singularities of generalised polylogarithm functions are encoded in the first entry of their symbols. First-entry condition: Only i 1 i j 1 j allowed in the first entry of S GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 21/34

99 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 Singularities of generalised polylogarithm functions are encoded in the first entry of their symbols. First-entry condition: Only i 1 i j 1 j allowed in the first entry of S Particularly for n = 7, this restricts letters of the first entry to a 1j. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 21/34

100 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 Singularities of generalised polylogarithm functions are encoded in the first entry of their symbols. First-entry condition: Only i 1 i j 1 j allowed in the first entry of S Particularly for n = 7, this restricts letters of the first entry to a 1j. Define a heptagon symbol: An integrable symbol with alphabet a ij that obeys first-entry condition. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 21/34

101 MHV Constraints: Yangian anomaly equations GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 22/34

102 MHV Constraints: Yangian anomaly equations Tree-level amplitudes exhibit (usual + dual) superconformal symmetry [Drummond,Henn,Korchemsky,Sokatchev] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 22/34

103 MHV Constraints: Yangian anomaly equations Tree-level amplitudes exhibit (usual + dual) superconformal symmetry [Drummond,Henn,Korchemsky,Sokatchev] Combination of two symmetries gives rise to a Yangian [Drummond,Henn,Plefka][Drummond,Ferro] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 22/34

104 MHV Constraints: Yangian anomaly equations Tree-level amplitudes exhibit (usual + dual) superconformal symmetry [Drummond,Henn,Korchemsky,Sokatchev] Combination of two symmetries gives rise to a Yangian [Drummond,Henn,Plefka][Drummond,Ferro] Although broken at loop level by IR divergences, Yangian anomaly equations governing this breaking have been proposed [Caron-Huot,He] Consequence for MHV amplitudes: Their differential is a linear combination of d log i j 1 j j+1, which implies Last-entry condition: Only i j 1 j j+1 may appear in the last entry of the symbol of any MHV amplitude. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 22/34

105 MHV Constraints: Yangian anomaly equations Tree-level amplitudes exhibit (usual + dual) superconformal symmetry [Drummond,Henn,Korchemsky,Sokatchev] Combination of two symmetries gives rise to a Yangian [Drummond,Henn,Plefka][Drummond,Ferro] Although broken at loop level by IR divergences, Yangian anomaly equations governing this breaking have been proposed [Caron-Huot,He] Consequence for MHV amplitudes: Their differential is a linear combination of d log i j 1 j j+1, which implies Last-entry condition: Only i j 1 j j+1 may appear in the last entry of the symbol of any MHV amplitude. Particularly here: Only the 14 letters a 2j and a 3j may appear in the last symbol entry of R 7. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 22/34

106 Imposing Constraints: The Collinear Limit It is baked into the definition of the BDS-subtracted n-particle L-loop MHV remainder function that it should smoothly approach the corresponding (n 1)-particle function in any simple collinear limit: lim i+1 i R(L) n = R (L) n 1. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 23/34

107 Imposing Constraints: The Collinear Limit It is baked into the definition of the BDS-subtracted n-particle L-loop MHV remainder function that it should smoothly approach the corresponding (n 1)-particle function in any simple collinear limit: lim i+1 i R(L) n = R (L) n 1. For n = 7, taking this limit in the most general manner reduces the 42-letter heptagon symbol alphabet to 9-letter hexagon symbol alphabet, plus nine additional letters. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 23/34

108 Imposing Constraints: The Collinear Limit It is baked into the definition of the BDS-subtracted n-particle L-loop MHV remainder function that it should smoothly approach the corresponding (n 1)-particle function in any simple collinear limit: lim i+1 i R(L) n = R (L) n 1. For n = 7, taking this limit in the most general manner reduces the 42-letter heptagon symbol alphabet to 9-letter hexagon symbol alphabet, plus nine additional letters. A function has a well-defined i+1 i limit only if its symbol is independent of all nine of these letters. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 23/34

109 Computing (MHV) Heptagon Symbols GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 24/34

110 Computing (MHV) Heptagon Symbols Step 1 (Straightforward) Form linear combination of all length-k symbols made of a ij obeying initial (+final) entry conditions, with unknown coefficients grouped in vector X. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 24/34

111 Computing (MHV) Heptagon Symbols Step 1 (Straightforward) Form linear combination of all length-k symbols made of a ij obeying initial (+final) entry conditions, with unknown coefficients grouped in vector X. Step 2 (Challenging) Solve integrability constraints, which take the form A X = 0. Namely all weight-k (MHV) heptagon functions will be the right nullspace of rational matrix A. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 24/34

112 Computing (MHV) Heptagon Symbols Step 1 (Straightforward) Form linear combination of all length-k symbols made of a ij obeying initial (+final) entry conditions, with unknown coefficients grouped in vector X. Step 2 (Challenging) Solve integrability constraints, which take the form A X = 0. Namely all weight-k (MHV) heptagon functions will be the right nullspace of rational matrix A. Just linear algebra, however for e.g. 3-loop MHV hexagon A boils down to a size of Tackled with fraction-free variants of Gaussian elimination that bound the size of intermediate expressions. [Storjohann] GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 24/34

113 Results 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 A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 25/34

114 Results 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 two-loop seven-particle MHV remainder function R (2) 7 is the only weight-4 heptagon symbol which is well-defined in all i+1 i collinear limits. GP A Symbol of Uniqueness The Amplitude Bootstrap and its Cluster Algebra Upgrade 25/34