Generalized Prescriptive Unitarity. Jacob Bourjaily University of Copenhagen QCD Meets Gravity 2017 University of California, Los Angeles
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1 Generalized Prescriptive Unitarity Jacob Bourjaily University of Copenhagen QCD Meets Gravity 2017 University of California, Los Angeles
2 Organization and Outline Overview: Generalized & Prescriptive Unitarity Constructing Integrands Constructively Formalism: tensor reduction & integrand bases Power counting: notation (and subtleties); stratifications Illustration: one-loop unitarity redux Generalized Prescriptive Unitarity The Badness of Box-Power Counting (in 4d) The Triviality of Prescriptivity: the Triangularity of Cuts A New Strategy for Supergravity? 2
3 Overview: Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; ] Integrands are rational functions so may be expanded into an arbitrary (complete) basis: 3
4 Overview: Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; ] Integrands are rational functions so may be expanded into an arbitrary (complete) basis: with coefficients determined by cuts (which may be gauge invariant or not) 3
5 Overview: Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; ] Integrands are rational functions so may be expanded into an arbitrary (complete) basis: with coefficients determined by cuts (which may be gauge invariant or not) A repreresentation is prescriptive if all the coefficients are individual field-theory cuts [JB, Herrmann, Trnka] 3
6 Overview: Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; ] Integrands are rational functions so may be expanded into an arbitrary (complete) basis: with coefficients determined by cuts (which may be gauge invariant or not) A repreresentation is prescriptive if all the coefficients are individual field-theory cuts [JB, Herrmann, Trnka] X A L=2 n = f L L 3
7 Overview: Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; ] Integrands are rational functions so may be expanded into an arbitrary (complete) basis: with coefficients determined by cuts (which may be gauge invariant or not) A repreresentation is prescriptive if all the coefficients are individual field-theory cuts A L=2 n = X L f L 2 f L 2 8 >< >: 8 >< >: X L,,,, [JB, Herrmann, Trnka] 9 >= >; 9 >= >; 3
8 Overview: Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; ] Integrands are rational functions so may be expanded into an arbitrary (complete) basis: with coefficients determined by cuts (which may be gauge invariant or not) A repreresentation is prescriptive if all the coefficients are individual field-theory cuts X [JB, Herrmann, Trnka] A L=3 n = f W + X L f L W here the integrals span all contact-term topologies of those drawn above, 3
9 Punchlines of this Talk Any basis with worse than -gon power-counting in dimensions is (essentially) trivially prescriptive 4
10 Punchlines of this Talk Any basis with worse than -gon power-counting in dimensions is (essentially) trivially prescriptive Integrand bases are stratified by power-counting, so amplitudes can be constructed iteratively 4
11 Punchlines of this Talk Any basis with worse than -gon power-counting in dimensions is (essentially) trivially prescriptive Integrand bases are stratified by power-counting, so amplitudes can be constructed iteratively The maximal-(and next-to-maximal-)weight parts of any amplitude are always cut-constructible 4
12 Punchlines of this Talk Any basis with worse than -gon power-counting in dimensions is (essentially) trivially prescriptive Integrand bases are stratified by power-counting, so amplitudes can be constructed iteratively The maximal-(and next-to-maximal-)weight parts of any amplitude are always cut-constructible The maximal-weight part of SUGRA has better than BCJ power-counting for high enough loops 4
13 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: 5
14 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: A fixed spacetime dimension (or ) 5
15 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: A fixed spacetime dimension (or ) A bound on the (loop-dependent) polynomial degrees power-counting 5
16 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: A fixed spacetime dimension (or ) A bound on the (loop-dependent) polynomial degrees power-counting A Notational Triviality: (WLOG) use inverse propagators for everything! 5
17 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: A fixed spacetime dimension (or ) A bound on the (loop-dependent) polynomial degrees power-counting A Notational Triviality: (WLOG) use inverse propagators for everything! 5
18 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: A fixed spacetime dimension (or ) A bound on the (loop-dependent) polynomial degrees power-counting A Notational Triviality: (WLOG) use inverse propagators for everything! 5
19 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: A fixed spacetime dimension (or ) A bound on the (loop-dependent) polynomial degrees power-counting A Notational Triviality: (WLOG) use inverse propagators for everything! 5
20 Building Bases of Loop Integrands In order to define a (finite-dimensional) basis of loop integrands, two things must be specified: A fixed spacetime dimension (or ) A bound on the (loop-dependent) polynomial degrees power-counting A Notational Triviality: (WLOG) use inverse propagators for everything! 5
21 Basics of Basic Basis Reduction [Passarino, Veltman; van Neerven, Vermaseren] Consider one-loop integrands in d dimensions A classic result (of P-V) is that all integrands with more than propagators are reducible: 6
22 Basics of Basic Basis Reduction [Passarino, Veltman; van Neerven, Vermaseren] Consider one-loop integrands in d dimensions A classic result (of P-V) is that all integrands with more than propagators are reducible: 6
23 Basics of Basic Basis Reduction [Passarino, Veltman; van Neerven, Vermaseren] Consider one-loop integrands in d dimensions A classic result (of P-V) is that all integrands with more than propagators are reducible: 6
24 Basics of Basic Basis Reduction [Passarino, Veltman; van Neerven, Vermaseren] Consider one-loop integrands in d dimensions A classic result (of P-V) is that all integrands with more than propagators are reducible: Moreover, the only independent integrands with propagators can be chosen to be parity-odd 6
25 Power-Counting & Contructibility An integrand has -gon power-counting if: 7
26 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) 7
27 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) Let denote a complete basis of integrands with p-gon power-counting 7
28 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) Let denote a complete basis of integrands with p-gon power-counting (because, ) 7
29 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) Let denote a complete basis of integrands with p-gon power-counting (because, ) 7
30 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) Let denote a complete basis of integrands with p-gon power-counting (because, ) 7
31 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) Let denote a complete basis of integrands with p-gon power-counting (because, ) An amplitude is p-gon constructible if: 7
32 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) Let denote a complete basis of integrands with p-gon power-counting (because, ) An amplitude is p-gon constructible if: (nota bene: this may be loop-order (L) dependent!) 7
33 Power-Counting & Contructibility An integrand has -gon power-counting if: (much less* ambiguous for integrand bases than amplitudes) Let denote a complete basis of integrands with p-gon power-counting (because, ) An amplitude is p-gon constructible if: (nota bene: this may be loop-order (L) dependent!) 7
34 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: 8
35 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: 8
36 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: 8
37 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: 8
38 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: Something interesting for worse than d-gon powercounting in d dimensions: e.g. in d=4, 8
39 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: Something interesting for worse than d-gon powercounting in d dimensions: e.g. in d=4, 8
40 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: Something interesting for worse than d-gon powercounting in d dimensions: e.g. in d=4, 8
41 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: Something interesting for worse than d-gon powercounting in d dimensions: e.g. in d=4, 8
42 One Loop Integrand Bases At one loop, consists of scalar p-gons, & tensors: Something interesting for worse than d-gon powercounting in d dimensions: e.g. in d=4, 8
43 One Loop Unitarity Redux (4d) Re-considering one-loop unitarity in 4 dimensions [Ossola, Papadopoulos, Pittau; Forde, Kosower] 9
44 One Loop Unitarity Redux (4d) Re-considering one-loop unitarity in 4 dimensions [Ossola, Papadopoulos, Pittau; Forde, Kosower] 9
45 One Loop Unitarity Redux (4d) Re-considering one-loop unitarity in 4 dimensions [Ossola, Papadopoulos, Pittau; Forde, Kosower] 9
46 One Loop Unitarity Redux (4d) Re-considering one-loop unitarity in 4 dimensions [Ossola, Papadopoulos, Pittau; Forde, Kosower]
47 One Loop Unitarity Redux (4d) Re-considering one-loop unitarity in 4 dimensions [Ossola, Papadopoulos, Pittau; Forde, Kosower]
48 One Loop Unitarity Redux (4d) Re-considering one-loop unitarity in 4 dimensions [Ossola, Papadopoulos, Pittau; Forde, Kosower]
49 One Loop Unitarity Redux (4d) Re-considering one-loop unitarity in 4 dimensions [Ossola, Papadopoulos, Pittau; Forde, Kosower]
50 Triangularity & Diagonalization Consider a theory that is bubble constructible (such as SYM) 10
51 Triangularity & Diagonalization Consider a theory that is bubble constructible (such as SYM) 10
52 Triangularity & Diagonalization Consider a theory that is bubble constructible (such as SYM) 10
53 Triangularity & Diagonalization Consider a theory that is bubble constructible (such as SYM) 10
54 Triangularity & Diagonalization Consider a theory that is bubble constructible (such as SYM) 10
55 Triangularity & Diagonalization Consider a theory that is bubble constructible (such as SYM) 10
56 Triangularity & Diagonalization Consider a theory that is bubble constructible (such as SYM) 10
57 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut 11
58 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] 11
59 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] 11
60 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] 11
61 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
62 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
63 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
64 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
65 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
66 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
67 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
68 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
69 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
70 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
71 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
72 Using Prescriptive Unitarity Rather than starting from an arbitrary basis of loop integrands, tailor each to manifestly match one cut one loop: [JB, Caron-Huot, Trnka] two loop: [JB, Trnka] 11
73 Extending the Prescriptive Reach 12
74 Extending the Prescriptive Reach This procedure continues to work at three loops: [JB, Herrmann, Trnka] 12
75 Extending the Prescriptive Reach This procedure continues to work at three loops: [JB, Herrmann, Trnka] 12
76 Extending the Prescriptive Reach This procedure continues to work at three loops: [JB, Herrmann, Trnka] Application to non-planar amplitudes is also now known to all orders(!) [JB, Herrmann, McLeod, Stankowicz, Trnka (in prep)] 12
77 The Badness of Boxes The overcompleteness of integrands with box powercounting is increasingly severe at high loop orders and beyond the planar limit 13
78 The Badness of Boxes The overcompleteness of integrands with box powercounting is increasingly severe at high loop orders and beyond the planar limit This even affects planar four-point amplitudes: 13
79 The Badness of Boxes The overcompleteness of integrands with box powercounting is increasingly severe at high loop orders and beyond the planar limit This even affects planar four-point amplitudes: box power-counting: over-complete for L>2 13
80 The Badness of Boxes The overcompleteness of integrands with box powercounting is increasingly severe at high loop orders and beyond the planar limit This even affects planar four-point amplitudes: box power-counting: over-complete for L>2 & DCI: over-complete for L>5 13
81 The Badness of Boxes The overcompleteness of integrands with box powercounting is increasingly severe at high loop orders and beyond the planar limit This even affects planar four-point amplitudes: box power-counting: over-complete for L>2 & DCI: over-complete for L>5 & dihedral: UV divergent for L>7 13
82 The Badness of Boxes The overcompleteness of integrands with box powercounting is increasingly severe at high loop orders and beyond the planar limit This even affects planar four-point amplitudes: box power-counting: over-complete for L>2 & DCI: over-complete for L>5 & dihedral: UV divergent for L>7 13
83 Beyond Planar Prescriptivity Abandoning box power-counting, however, immediately allows for prescriptive bases [Feng, Huang] 14
84 Beyond Planar Prescriptivity Abandoning box power-counting, however, immediately allows for prescriptive bases [Feng, Huang] [JB, Herrmann, McLeod, Stankowicz, Trnka (in prep)] 14
85 Beyond Planar Prescriptivity Abandoning box power-counting, however, immediately allows for prescriptive bases [Feng, Huang] [JB, Herrmann, McLeod, Stankowicz, Trnka (in prep)] I 1 I 2 I 3 I 4 I 5 I (8+221) [`1] 3 [`2] [`1][`2] 3 [`1] 2 [`2] 2 [`1+ `2] (4+160) [`1] 4 [`1] 2 [`2] 2 [`1] 3 [`2][`1+ `2] (4+116) [`1] 3 [`2] (4+32) [`1] 2 [`2] (10+45) [`1] 3 [`1][`2] 2 [`1] 2 [`2][`1+ `2] (8+28) [`1] 2 [`2] 2 I 7 I 8 I 9 I 10 I 11 I 12 I 13 I a (2+18) [`1] 3 [`2] I b 14 I (2+4) [`1] 3 [`2] I a (6+6) [`1] 2 [`2] 2 [`1][`2][`1+ `2] I b (3+3) [`1] 2 [`2] I (1+0) [`1][`2] I a (1+0) [`1][`2] I b [`1] 4 I [`1] 3 I [`1] [`1] [`1] [`1] [`1] 5+1 [`1] 4+2 [`1]
86 No-Bubble Hypothesis for N=8 Theories with leading transcendentality should be triangle-constructible thus fully representable prescriptively to all orders 15
87 No-Bubble Hypothesis for N=8 Theories with leading transcendentality should be triangle-constructible thus fully representable prescriptively to all orders Even if a theory were not triangle-constructible, the leading-weight part of any amplitude would be represented in the same basis merely with different (theory-specific) coefficients 15
88 No-Bubble Hypothesis for N=8 Theories with leading transcendentality should be triangle-constructible thus fully representable prescriptively to all orders Even if a theory were not triangle-constructible, the leading-weight part of any amplitude would be represented in the same basis merely with different (theory-specific) coefficients Is supergravity triangle-constructible? 15
89 No-Bubble Hypothesis for N=8 Theories with leading transcendentality should be triangle-constructible thus fully representable prescriptively to all orders Even if a theory were not triangle-constructible, the leading-weight part of any amplitude would be represented in the same basis merely with different (theory-specific) coefficients Is supergravity triangle-constructible? can be tested constructively, without integration 15
90 No-Bubble Hypothesis for N=8 Theories with leading transcendentality should be triangle-constructible thus fully representable prescriptively to all orders Even if a theory were not triangle-constructible, the leading-weight part of any amplitude would be represented in the same basis merely with different (theory-specific) coefficients Is supergravity triangle-constructible? can be tested constructively, without integration If so, a triangle power-counting representation would be (asymptotically, arbitrarily) better than BCJ in the UV! 15
91 Questions?
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