Tree-level amplitudes from the pure spinor superstring
2025-05-06
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2.77MB
202 页
10玖币
侵权投诉
arXiv:2210.14241v2 [hep-th] 9 Apr 2023
UUITP-44/22
Tree-level amplitudes from the pure spinor superstring
Carlos R. Mafraa, Oliver Schlottererb
aSTAG Research Centre and Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK
bDepartment of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden
Abstract
We give a comprehensive review of recent developments on using the pure spinor formalism to compute
massless superstring scattering amplitudes at tree level. The main results of the pure spinor computations
are placed into the context of related topics including the color-kinematics duality in field theory and the
mathematical structure of α′-corrections.
Keywords: Pure spinors, scattering amplitudes, multiparticle superfields
Contents
1 Introduction 6
1.1 Summary of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Basics of the pure spinor formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 The prescription for disk amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.3 The multiparticle formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 SYM tree-level amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.5 Superstring disk amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.6 Color-kinematics duality and double copy . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.7 α′-expansions of open- and closed-superstring tree amplitudes . . . . . . . . . . . . . . 10
1.1.8 A web of field-theory double copies for string amplitudes . . . . . . . . . . . . . . . . . 11
1.1.9 Example of a possible shortcut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Related topics beyond the scope of this review . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 The CHY formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 String field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 The hybrid formalism and strings in AdS spacetimes . . . . . . . . . . . . . . . . . . 13
1.3 Conventions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Super Yang–Mills in ten dimensions 14
2.1 Ten-dimensional SYM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Linearized superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 θ-expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Superfields of higher mass dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Equations of motion at higher mass dimension . . . . . . . . . . . . . . . . . . . . . . 17
Email addresses: c.r.mafra@soton.ac.uk (Carlos R. Mafra), oliver.schlotterer@physics.uu.se (Oliver Schlotterer)
Preprint submitted to Elsevier April 11, 2023
3 Pure spinor formalism and disk amplitudes 18
3.1 Difficulties with the covariant quantization of the Green-Schwarz string . . . . . . . . . . . . 19
3.2 Siegel’s reformulation of the Green–Schwarz formulation . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Difficulties with Siegel’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Fundamentals of the pure spinor formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 U(5) decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 The pure spinor ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.3 The action of the pure spinor formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.4 Operator product expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Scattering amplitudes on the disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Massless vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Scattering-amplitude prescription at genus zero . . . . . . . . . . . . . . . . . . . . . . 31
3.4.3 The field-theory limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.4 Pure spinor superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.5 Component expansion from pure spinor superspace . . . . . . . . . . . . . . . . . . . . 36
3.4.6 Preview of higher-point SYM amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Multiparticle SYM in ten dimensions 38
4.1 Local superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 The contact-term map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 Multiparticle superfield in the Lorenz gauge . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.3 Equations of motion of local multiparticle superfields in Lorenz gauge . . . . . . . . . 45
4.1.4 Local multiparticle superfields in the BCJ gauge . . . . . . . . . . . . . . . . . . . . . 46
4.1.5 Generalized Jacobi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.6 Multiparticle superfields in the BCJ gauge . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.7 Equations of motion of multiparticle superfields in the BCJ gauge . . . . . . . . . . . 51
4.2 Non-local superfields and Berends–Giele currents . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Non-linear wave equations and Berends–Giele currents . . . . . . . . . . . . . . . . . . 54
4.2.2 Perturbiner solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.3 Equations of motion of Berends–Giele currents . . . . . . . . . . . . . . . . . . . . . . 56
4.2.4 Symmetry properties of Berends–Giele currents . . . . . . . . . . . . . . . . . . . . . . 57
4.2.5 Berends–Giele currents and finite gauge transformations . . . . . . . . . . . . . . . . . 57
4.2.6 The multiparticle Berends–Giele polarizations . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Combinatorial framework of Berends–Giele currents . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 Planar binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Berends–Giele currents from planar binary trees . . . . . . . . . . . . . . . . . . . . . 61
4.3.3 The S bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.4 The contact-term map and the Sbracket . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.5 The KLT map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 SYM tree amplitudes from the cohomology of pure spinor superspace 70
5.1 Berends–Giele recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.1 Kleiss–Kuijf amplitude relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 The pure spinor superspace formula for SYM tree amplitudes . . . . . . . . . . . . . . . . . . 72
5.2.1 Manifesting cyclic symmetry via BRST integration by parts . . . . . . . . . . . . . . . 75
5.2.2 Component expansion of the pure spinor SYM tree amplitude . . . . . . . . . . . . . . 76
5.2.3 Equivalence with the gluonic Berends–Giele recursion . . . . . . . . . . . . . . . . . . 77
5.2.4 Kleiss–Kuijf amplitude relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.5 Bern–Carrasco–Johansson amplitude relations . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 The generating series of tree-level amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2
6 Superstring disk amplitudes with the pure spinor formalism 82
6.1 CFT analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1.1 Double poles versus logarithmic singularities . . . . . . . . . . . . . . . . . . . . . . . 83
6.1.2 Lie-polynomial structure of the correlator . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Local form of the disk correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.1 Four-point example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.2 Five-point example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.3 Six-point example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Non-local form of the disk correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3.1 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.2 The trading identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.3 The n-point disk amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 The open superstring as a field-theory double copy . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4.1 Parke–Taylor factors and Z-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4.2 Open superstrings as a KLT formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4.3 KK and BCJ relations of Z-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4.4 Bi-adjoint scalars from the field-theory limit of Z-integrals . . . . . . . . . . . . . . . . 97
6.5 The field-theory limit of the superstring disk amplitudes . . . . . . . . . . . . . . . . . . . . . 100
7 String and field-theory amplitude relations 102
7.1 Color-kinematics duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.1.1 Review of the color-kinematics duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.1.2 DDM form of YM and bi-adjoint scalar amplitudes . . . . . . . . . . . . . . . . . . . . 106
7.1.3 Local BCJ numerators from disk amplitudes . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.4 Component expansion of BCJ numerators . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.5 The M¨obius product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.1.6 Local BCJ numerators from the M¨obius product . . . . . . . . . . . . . . . . . . . . . 110
7.1.7 The field-theory limit of the superstring disk amplitude for arbitrary orderings . . . . 111
7.1.8 Local BCJ numerators from finite gauge transformations . . . . . . . . . . . . . . . . . 112
7.1.9 An explicit solution to BCJ relations in KLT form . . . . . . . . . . . . . . . . . . . . 113
7.2 String-theory KLT relations and the double-copy form of gravity numerators . . . . . . . . . 114
7.2.1 String-theory KLT relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2.2 Sphere integrals and their field-theory limit . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2.3 The local form of the gravitational double copy . . . . . . . . . . . . . . . . . . . . . . 117
7.2.4 Consistency check with the field-theory KLT relation . . . . . . . . . . . . . . . . . . . 118
7.3 Monodromy relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3.1 The (n−2)! form of color-dressed open-string amplitudes . . . . . . . . . . . . . . . . . 119
7.3.2 The (n−3)! form of color-dressed open-string amplitudes . . . . . . . . . . . . . . . . . 120
7.4 Double copies beyond gravity from string amplitudes . . . . . . . . . . . . . . . . . . . . . . . 121
7.4.1 Born–Infeld and NLSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.4.2 BCJ numerators of the NLSM from disk integrals . . . . . . . . . . . . . . . . . . . . . 122
7.4.3 Coupling NLSM to bi-adjoint scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.5 Heterotic strings and Einstein–Yang–Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.5.1 Basics of heterotic-string amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.5.2 Heterotic double copy and Einstein–Yang–Mills . . . . . . . . . . . . . . . . . . . . . . 126
7.5.3 Heterotic strings as a field-theory double copy . . . . . . . . . . . . . . . . . . . . . . . 127
7.5.4 Einstein–Yang–Mills amplitude relations from string theories . . . . . . . . . . . . . . 128
7.5.5 Reducing heterotic-string amplitudes to the single-trace sector . . . . . . . . . . . . . 130
3
8α′-expansion of superstring tree-level amplitudes 131
8.1 Basics of α′-expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.1.1 Four-point α′-expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.1.2 Low-energy effective actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.1.3 On the scope of four-point amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.1.4 Manifestly supersymmetric approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2 Multiple zeta values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2.1 Motivic MZVs and the f alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2.2 The Drinfeld associator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.2.3 Single-valued multiple zeta values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.4 Comments on conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.3 Patterns in the α′-expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.3.1 The pattern in terms of MZVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.3.2 The pattern in the falphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3.3 Coaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.4 KK-like and BCJ relations within the α′-expansion . . . . . . . . . . . . . . . . . . . . . . . . 142
8.4.1 BCJ relations at all orders in α′..............................143
8.4.2 KK-like relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4.3 Berends–Giele idempotents and BRST-invariant permutations . . . . . . . . . . . . . . 145
8.4.4 BCJ and KK relations of Z-theory amplitudes . . . . . . . . . . . . . . . . . . . . . . 146
8.5 String corrections from the Drinfeld associator . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.5.1 Construction of the matrices e0, e1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.5.2 Uniform transcendentality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.5.3 Regularized boundary values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.5.4 Connection with twisted deRham theory and outlook . . . . . . . . . . . . . . . . . . 150
8.6 Berends–Giele recursion for disk integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.6.1 Extending the field-theory limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.6.2 Planar binary trees and α′-corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.7 Closed strings as single-valued open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.7.1 From the KLT formula to the single-valued map . . . . . . . . . . . . . . . . . . . . . 156
8.7.2 Closed-string amplitudes as a field-theory double copy . . . . . . . . . . . . . . . . . . 157
8.7.3 Sphere integrals as single-valued disk integrals . . . . . . . . . . . . . . . . . . . . . . 157
8.7.4 The web of field-theory double copies for string amplitudes . . . . . . . . . . . . . . . 158
8.7.5 Twisted KLT relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9 Conclusion and outlook 160
9.1 Loop amplitudes in the pure spinor formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.2 Worldsheet integrals in loop-level string amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 163
Appendix A Gamma matrices 164
Appendix A.1 The Clifford Algebra in R1,9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Appendix A.2 Fierz decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Appendix A.3 Duality properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Appendix A.4 Traces of gamma matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Appendix A.5 Products of gamma matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Appendix A.6 Gamma matrix identities and pure spinors . . . . . . . . . . . . . . . . . . . . . 169
Appendix B The U(5) decomposition of SO(10) 169
Appendix B.1 The Clifford algebra in R10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Appendix B.2 Vectors and Lorentz generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Appendix B.3 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4
Appendix C Combinatorics on words 175
Appendix C.1 The dual Lie polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Appendix D Dynkin labels of SO(10) 177
Appendix E Pure spinor superspace correlators 178
Appendix F θ-expansion of SYM superfields 180
Appendix G Redefinitions from the Lorenz gauge to the BCJ gauge 182
Appendix H The contact-term map is nilpotent 184
Appendix I Different representation of multiparticle vertex 185
Appendix J BRST-invariant permutations at low multiplicities 187
5
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arXiv:2210.14241v2[hep-th]9Apr2023UUITP-44/22Tree-levelamplitudesfromthepurespinorsuperstringCarlosR.Mafraa,OliverSchlottererbaSTAGResearchCentreandMathematicalSciences,UniversityofSouthampton,Highfield,SouthamptonSO171BJ,UKbDepartmentofPhysicsandAstronomy,UppsalaUniversity,75108Uppsala,SwedenAbstrac...
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