Coaction structure for Feynman amplitudes and a small graphs - PowerPoint PPT Presentation

Overview Coaction conjecture Motivic amplitudes Coaction structure for Feynman amplitudes and a small graphs principle Francis Brown, IH´ ES-CNRS Member IAS, Princeton New geometric structures in scattering amplitudes, Oxford University, 23rd September 2014 1 / 31

Overview Coaction conjecture Motivic amplitudes Overview and goals The main goals: 1 Formulate O. Schnetz’ coaction conjecture for scalar massless amplitudes. Explain its remarkable predictive power for high-loop amplitudes. 2 Define motivic amplitudes. This a vast generalisation of the notion of ‘symbol’, but contains more information. 3 Prove a version of the coaction conjecture. The small graphs principle allows one to deduce all-order results in perturbation theory from a finite computation. Point (3) states that there is a hidden recursive structure in the amplitudes of quantum field theories: information about low-loop amplitudes propagates to all higher loop orders. 2 / 31

Overview Coaction conjecture Motivic amplitudes Overview and goals The main goals: 1 Formulate O. Schnetz’ coaction conjecture for scalar massless amplitudes. Explain its remarkable predictive power for high-loop amplitudes. 2 Define motivic amplitudes. This a vast generalisation of the notion of ‘symbol’, but contains more information. 3 Prove a version of the coaction conjecture. The small graphs principle allows one to deduce all-order results in perturbation theory from a finite computation. Point (3) states that there is a hidden recursive structure in the amplitudes of quantum field theories: information about low-loop amplitudes propagates to all higher loop orders. 2 / 31

Overview Coaction conjecture Motivic amplitudes Overview and goals The main goals: 1 Formulate O. Schnetz’ coaction conjecture for scalar massless amplitudes. Explain its remarkable predictive power for high-loop amplitudes. 2 Define motivic amplitudes. This a vast generalisation of the notion of ‘symbol’, but contains more information. 3 Prove a version of the coaction conjecture. The small graphs principle allows one to deduce all-order results in perturbation theory from a finite computation. Point (3) states that there is a hidden recursive structure in the amplitudes of quantum field theories: information about low-loop amplitudes propagates to all higher loop orders. 2 / 31

Overview Coaction conjecture Motivic amplitudes Overview and goals The main goals: 1 Formulate O. Schnetz’ coaction conjecture for scalar massless amplitudes. Explain its remarkable predictive power for high-loop amplitudes. 2 Define motivic amplitudes. This a vast generalisation of the notion of ‘symbol’, but contains more information. 3 Prove a version of the coaction conjecture. The small graphs principle allows one to deduce all-order results in perturbation theory from a finite computation. Point (3) states that there is a hidden recursive structure in the amplitudes of quantum field theories: information about low-loop amplitudes propagates to all higher loop orders. 2 / 31

Overview Coaction conjecture Motivic amplitudes A simple analogy An analogy is Erastosthenes’ sieve. Suppose that we have a set S of natural numbers with the following property: If n ∈ S , and m is a divisor of n , then m ∈ S . Write the natural numbers in a table: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Now suppose that we have some low-order information: 2 / ∈ S . Cross off all multiples of 2 3 / ∈ S . Cross off all multiples of 3 The fact that S has few low-order elements means that S is full of holes at all orders. 3 / 31

Overview Coaction conjecture Motivic amplitudes A simple analogy An analogy is Erastosthenes’ sieve. Suppose that we have a set S of natural numbers with the following property: If n ∈ S , and m is a divisor of n , then m ∈ S . Write the natural numbers in a table: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Now suppose that we have some low-order information: 2 / ∈ S . Cross off all multiples of 2 3 / ∈ S . Cross off all multiples of 3 The fact that S has few low-order elements means that S is full of holes at all orders. 3 / 31

Overview Coaction conjecture Motivic amplitudes A simple analogy An analogy is Erastosthenes’ sieve. Suppose that we have a set S of natural numbers with the following property: If n ∈ S , and m is a divisor of n , then m ∈ S . Write the natural numbers in a table: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Now suppose that we have some low-order information: 2 / ∈ S . Cross off all multiples of 2 3 / ∈ S . Cross off all multiples of 3 The fact that S has few low-order elements means that S is full of holes at all orders. 3 / 31

Overview Coaction conjecture Motivic amplitudes A simple analogy An analogy is Erastosthenes’ sieve. Suppose that we have a set S of natural numbers with the following property: If n ∈ S , and m is a divisor of n , then m ∈ S . Write the natural numbers in a table: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Now suppose that we have some low-order information: 2 / ∈ S . Cross off all multiples of 2 3 / ∈ S . Cross off all multiples of 3 The fact that S has few low-order elements means that S is full of holes at all orders. 3 / 31

Overview Coaction conjecture Motivic amplitudes A simple analogy An analogy is Erastosthenes’ sieve. Suppose that we have a set S of natural numbers with the following property: If n ∈ S , and m is a divisor of n , then m ∈ S . Write the natural numbers in a table: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Now suppose that we have some low-order information: 2 / ∈ S . Cross off all multiples of 2 3 / ∈ S . Cross off all multiples of 3 The fact that S has few low-order elements means that S is full of holes at all orders. 3 / 31

Overview Coaction conjecture Motivic amplitudes A simple analogy An analogy is Erastosthenes’ sieve. Suppose that we have a set S of natural numbers with the following property: If n ∈ S , and m is a divisor of n , then m ∈ S . Write the natural numbers in a table: 1 5 7 11 13 17 19 23 25 29 Now suppose that we have some low-order information: 2 / ∈ S . Cross off all multiples of 2 3 / ∈ S . Cross off all multiples of 3 The fact that S has few low-order elements means that S is full of holes at all orders. 3 / 31

Overview Coaction conjecture Motivic amplitudes A simple analogy An analogy is Erastosthenes’ sieve. Suppose that we have a set S of natural numbers with the following property: If n ∈ S , and m is a divisor of n , then m ∈ S . Write the natural numbers in a table: 1 5 7 11 13 17 19 23 25 29 Now suppose that we have some low-order information: 2 / ∈ S . Cross off all multiples of 2 3 / ∈ S . Cross off all multiples of 3 The fact that S has few low-order elements means that S is full of holes at all orders. 3 / 31

Overview Coaction conjecture Motivic amplitudes What happens for amplitudes? Let P be the vector space of amplitudes of, e.g. massless φ 4 . The coaction conjecture predicts the following property for amplitudes. If ξ ∈ P , and ξ ′ is a Galois conjugate of ξ , then ξ ′ ∈ P . At low loop orders, the amplitudes are multiple zeta values. Write a basis for multiple zeta values in a table. ζ (2) 2 ζ (3) 2 1 ζ (2) ζ (3) ζ (5) ζ (7) ζ (3 , 5) ζ (2) 3 ζ (3) 2 ζ (2) ζ (3) ζ (2) ζ (5) ζ (2) ζ (3) ζ (2) 2 : . Now look at amplitudes of small graphs (with ≤ 4 loops). There are very few of them. We see that: ζ (2) / ∈ P . Cross off all linear terms in ζ (2) ζ (2) 2 / ∈ P . Cross off all quadratic terms in ζ (2) A finite calculation leads to constraints at all higher loop orders . 4 / 31

Overview Coaction conjecture Motivic amplitudes What happens for amplitudes? Let P be the vector space of amplitudes of, e.g. massless φ 4 . The coaction conjecture predicts the following property for amplitudes. If ξ ∈ P , and ξ ′ is a Galois conjugate of ξ , then ξ ′ ∈ P . At low loop orders, the amplitudes are multiple zeta values. Write a basis for multiple zeta values in a table. ζ (2) 2 ζ (3) 2 1 ζ (2) ζ (3) ζ (5) ζ (7) ζ (3 , 5) ζ (2) 3 ζ (3) 2 ζ (2) ζ (3) ζ (2) ζ (5) ζ (2) ζ (3) ζ (2) 2 : . Now look at amplitudes of small graphs (with ≤ 4 loops). There are very few of them. We see that: ζ (2) / ∈ P . Cross off all linear terms in ζ (2) ζ (2) 2 / ∈ P . Cross off all quadratic terms in ζ (2) A finite calculation leads to constraints at all higher loop orders . 4 / 31

Overview Coaction conjecture Motivic amplitudes What happens for amplitudes? Let P be the vector space of amplitudes of, e.g. massless φ 4 . The coaction conjecture predicts the following property for amplitudes. If ξ ∈ P , and ξ ′ is a Galois conjugate of ξ , then ξ ′ ∈ P . At low loop orders, the amplitudes are multiple zeta values. Write a basis for multiple zeta values in a table. ζ (2) 2 ζ (3) 2 1 ζ (2) ζ (3) ζ (5) ζ (7) ζ (3 , 5) ζ (2) 3 ζ (3) 2 ζ (2) ζ (3) ζ (2) ζ (5) ζ (2) ζ (3) ζ (2) 2 : . Now look at amplitudes of small graphs (with ≤ 4 loops). There are very few of them. We see that: ζ (2) / ∈ P . Cross off all linear terms in ζ (2) ζ (2) 2 / ∈ P . Cross off all quadratic terms in ζ (2) A finite calculation leads to constraints at all higher loop orders . 4 / 31