Biểu thức Amplitude của Chuỗi Heterotic và Bosonic thông qua Lý thuyết Tương trường

H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].

Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].

Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].

J.J. Carrasco and H. Johansson, Five-Point Amplitudes in N = 4 Super-Yang-Mills Theory and N = 8 Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].

Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].

Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Absence of Three-Loop Four-Point Divergences in N = 4 Supergravity, Phys. Rev. Lett. 108 (2012) 201301 [arXiv:1202.3423] [INSPIRE].

Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].

Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in $$ \mathcal{N}=5 $$ supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].

Z. Bern, J.J. Carrasco, W.-M. Chen, H. Johansson and R. Roiban, Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes, Phys. Rev. Lett. 118 (2017) 181602 [arXiv:1701.02519] [INSPIRE].

H. Johansson, G. Kälin and G. Mogull, Two-loop supersymmetric QCD and half-maximal supergravity amplitudes, JHEP 09 (2017) 019 [arXiv:1706.09381] [INSPIRE].

Z. Bern, J.J.M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev. D 96 (2017) 126012 [arXiv:1708.06807] [INSPIRE].

M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Scattering amplitudes in $$ \mathcal{N}=2 $$ Maxwell-Einstein and Yang-Mills/Einstein supergravity, JHEP 01 (2015) 081 [arXiv:1408.0764] [INSPIRE].

M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies, JHEP 06 (2017) 064 [arXiv:1511.01740] [INSPIRE].

M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy, JHEP 07 (2017) 002 [arXiv:1703.00421] [INSPIRE].

F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].

J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled N ≤ 4 supergravity, JHEP 03 (2013) 056 [arXiv:1212.1146] [INSPIRE].

H. Johansson and A. Ochirov, Pure Gravities via Color-Kinematics Duality for Fundamental Matter, JHEP 11 (2015) 046 [arXiv:1407.4772] [INSPIRE].

M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Complete construction of magical, symmetric and homogeneous N = 2 supergravities as double copies of gaug theories, Phys. Rev. Lett. 117 (2016) 011603 [arXiv:1512.09130] [INSPIRE].

M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Gauged supergravities and spontaneous SUSY breaking from the double copy, Phys. Rev. Lett. 120 (2018) 171601 [arXiv:1710.08796] [INSPIRE].

H. Johansson and J. Nohle, Conformal Gravity from Gauge Theory, arXiv:1707.02965 [INSPIRE].

F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].

F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].

F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].

L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].

E. Casali, Y. Geyer, L. Mason, R. Monteiro and K.A. Roehrig, New Ambitwistor String Theories, JHEP 11 (2015) 038 [arXiv:1506.08771] [INSPIRE].

F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].

F. Cachazo, P. Cha and S. Mizera, Extensions of Theories from Soft Limits, JHEP 06 (2016) 170 [arXiv:1604.03893] [INSPIRE].

D. Nandan, J. Plefka, O. Schlotterer and C. Wen, Einstein-Yang-Mills from pure Yang-Mills amplitudes, JHEP 10 (2016) 070 [arXiv:1607.05701] [INSPIRE].

L. de la Cruz, A. Kniss and S. Weinzierl, Relations for Einstein-Yang-Mills amplitudes from the CHY representation, Phys. Lett. B 767 (2017) 86 [arXiv:1607.06036] [INSPIRE].

F. Teng and B. Feng, Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame, JHEP 05 (2017) 075 [arXiv:1703.01269] [INSPIRE].

Y.-J. Du, B. Feng and F. Teng, Expansion of All Multitrace Tree Level EYM Amplitudes, JHEP 12 (2017) 038 [arXiv:1708.04514] [INSPIRE].

Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].

Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].

Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-Loop Scattering Amplitudes from the Riemann Sphere, Phys. Rev. D 94 (2016) 125029 [arXiv:1607.08887] [INSPIRE].

S. He and O. Schlotterer, New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level, Phys. Rev. Lett. 118 (2017) 161601 [arXiv:1612.00417] [INSPIRE].

S. He, O. Schlotterer and Y. Zhang, New BCJ representations for one-loop amplitudes in gauge theories and gravity, Nucl. Phys. B 930 (2018) 328 [arXiv:1706.00640] [INSPIRE].

N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].

S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].

P. Tourkine and P. Vanhove, Higher-loop amplitude monodromy relations in string and gauge theory, Phys. Rev. Lett. 117 (2016) 211601 [arXiv:1608.01665] [INSPIRE].

S. Hohenegger and S. Stieberger, Monodromy Relations in Higher-Loop String Amplitudes, Nucl. Phys. B 925 (2017) 63 [arXiv:1702.04963] [INSPIRE].

A. Ochirov, P. Tourkine and P. Vanhove, One-loop monodromy relations on single cuts, JHEP 10 (2017) 105 [arXiv:1707.05775] [INSPIRE].

S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].

O. Schlotterer, Amplitude relations in heterotic string theory and Einstein-Yang-Mills, JHEP 11 (2016) 074 [arXiv:1608.00130] [INSPIRE].

C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].

C.R. Mafra and O. Schlotterer, Towards one-loop SYM amplitudes from the pure spinor BRST cohomology, Fortsch. Phys. 63 (2015) 105 [arXiv:1410.0668] [INSPIRE].

S. He, R. Monteiro and O. Schlotterer, String-inspired BCJ numerators for one-loop MHV amplitudes, JHEP 01 (2016) 171 [arXiv:1507.06288] [INSPIRE].

C.R. Mafra and O. Schlotterer, Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, JHEP 10 (2015) 124 [arXiv:1505.02746] [INSPIRE].

M. Berg, I. Buchberger and O. Schlotterer, From maximal to minimal supersymmetry in string loop amplitudes, JHEP 04 (2017) 163 [arXiv:1603.05262] [INSPIRE].

M. Berg, I. Buchberger and O. Schlotterer, String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry, JHEP 07 (2017) 138 [arXiv:1611.03459] [INSPIRE].

C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].

J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].

J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α’-corrections from the open string, JHEP 06 (2017) 093 [arXiv:1608.02569] [INSPIRE].

J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Semi-abelian Z-theory: NLSM+ $$ \phi $$ 3 from the open string, JHEP 08 (2017) 135 [arXiv:1612.06446] [INSPIRE].

C.R. Mafra and O. Schlotterer, Non-abelian Z-theory: Berends-Giele recursion for the α ′ -expansion of disk integrals, JHEP 01 (2017) 031 [arXiv:1609.07078] [INSPIRE].

G. Chen and Y.-J. Du, Amplitude Relations in Non-linear Sigma Model, JHEP 01 (2014) 061 [arXiv:1311.1133] [INSPIRE].

C. Cheung and C.-H. Shen, Symmetry for Flavor-Kinematics Duality from an Action, Phys. Rev. Lett. 118 (2017) 121601 [arXiv:1612.00868] [INSPIRE].

C.R. Mafra and O. Schlotterer, The double-copy structure of one-loop open-string amplitudes, Phys. Rev. Lett. 121 (2018) 011601 [arXiv:1711.09104] [INSPIRE].

Y.-t. Huang, O. Schlotterer and C. Wen, Universality in string interactions, JHEP 09 (2016) 155 [arXiv:1602.01674] [INSPIRE].

O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].

O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].

F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].

S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE].

D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic String Theory. 1. The Free Heterotic String, Nucl. Phys. B 256 (1985) 253 [INSPIRE].

D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic String Theory. 2. The Interacting Heterotic String, Nucl. Phys. B 267 (1986) 75 [INSPIRE].

T. Azevedo and O.T. Engelund, Ambitwistor formulations of R 2 gravity and (DF) 2 gauge theories, JHEP 11 (2017) 052 [arXiv:1707.02192] [INSPIRE].

Y.-t. Huang, W. Siegel and E.Y. Yuan, Factorization of Chiral String Amplitudes, JHEP 09 (2016) 101 [arXiv:1603.02588] [INSPIRE].

N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].

C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, A recursive method for SYM n-point tree amplitudes, Phys. Rev. D 83 (2011) 126012 [arXiv:1012.3981] [INSPIRE].

J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α ′ -expansion of superstring trees from the Drinfeld associator, Phys. Rev. D 89 (2014) 066014 [arXiv:1304.7304] [INSPIRE].

Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].

N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Gravity and Yang-Mills Amplitude Relations, Phys. Rev. D 82 (2010) 107702 [arXiv:1005.4367] [INSPIRE].

N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Proof of Gravity and Yang-Mills Amplitude Relations, JHEP 09 (2010) 067 [arXiv:1007.3111] [INSPIRE].

N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The Momentum Kernel of Gauge and Gravity Theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].

M.B. Green and M. Gutperle, Symmetry breaking at enhanced symmetry points, Nucl. Phys. B 460 (1996) 77 [hep-th/9509171] [INSPIRE].

K. Aomoto, Special values of hyperlogarithms and linear difference schemes, Illinois J. Math. 34 (1990) 191.

T. Terasoma, Selberg integrals and multiple zeta values, Compos. Math. 133 (2002) 1.

F.C.S. Brown, Multiple zeta values and periods of moduli spaces ℳ0,n(ℝ), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].

S. Stieberger, Constraints on Tree-Level Higher Order Gravitational Couplings in Superstring Theory, Phys. Rev. Lett. 106 (2011) 111601 [arXiv:0910.0180] [INSPIRE].

F. Brown, Polylogarithmes multiples uniformes en une variable, C.R. Acad. Sci. Paris Ser. I 338 (2004) 527.

J. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, arXiv:1803.00527 [INSPIRE].

S. Stieberger and T.R. Taylor, Closed String Amplitudes as Single-Valued Open String Amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].

S. He and Y. Zhang, New Formulas for Amplitudes from Higher-Dimensional Operators, JHEP 02 (2017) 019 [arXiv:1608.08448] [INSPIRE].

J. Broedel and L.J. Dixon, Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators, JHEP 10 (2012) 091 [arXiv:1208.0876] [INSPIRE].

N. Berkovits and E. Witten, Conformal supergravity in twistor-string theory, JHEP 08 (2004) 009 [hep-th/0406051] [INSPIRE].

E. Bergshoeff, M. de Roo and B. de Wit, Conformal Supergravity in Ten-dimensions, Nucl. Phys. B 217 (1983) 489 [INSPIRE].

M. de Roo, The R 2 action in d = 10 conformal supergravity, Nucl. Phys. B 372 (1992) 243 [INSPIRE].

W. Siegel, Amplitudes for left-handed strings, arXiv:1512.02569 [INSPIRE].

E. Casali and P. Tourkine, On the null origin of the ambitwistor string, JHEP 11 (2016) 036 [arXiv:1606.05636] [INSPIRE].

K. Lee, S.-J. Rey and J.A. Rosabal, A string theory which isn’t about strings, JHEP 11 (2017) 172 [arXiv:1708.05707] [INSPIRE].

E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].

S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, JHEP 08 (2017) 097 [arXiv:1706.08527] [INSPIRE].

K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math. J. 139 (1995) 67.

R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].

D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].

J. Polchinski and E. Witten, Evidence for heterotic — type I string duality, Nucl. Phys. B 460 (1996) 525 [hep-th/9510169] [INSPIRE].

E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].

R.H. Boels, On the field theory expansion of superstring five point amplitudes, Nucl. Phys. B 876 (2013) 215 [arXiv:1304.7918] [INSPIRE].

G. Puhlfürst and S. Stieberger, Differential Equations, Associators, and Recurrences for Amplitudes, Nucl. Phys. B 902 (2016) 186 [arXiv:1507.01582] [INSPIRE].

D. Oprisa and S. Stieberger, Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums, hep-th/0509042 [INSPIRE].

S. Stieberger and T.R. Taylor, Multi-Gluon Scattering in Open Superstring Theory, Phys. Rev. D 74 (2006) 126007 [hep-th/0609175] [INSPIRE].

S. Stieberger and T.R. Taylor, Supersymmetry Relations and MHV Amplitudes in Superstring Theory, Nucl. Phys. B 793 (2008) 83 [arXiv:0708.0574] [INSPIRE].

J.M. Drummond and É. Ragoucy, Superstring amplitudes and the associator, JHEP 08 (2013) 135 [arXiv:1301.0794] [INSPIRE].

J. Broedel, O. Schlotterer and S. Stieberger, http://mzv.mpp.mpg.de.

N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].

H. Gomez and E.Y. Yuan, N-point tree-level scattering amplitude in the new Berkovits‘ string, JHEP 04 (2014) 046 [arXiv:1312.5485] [INSPIRE].

C.S. Lam and Y.-P. Yao, Evaluation of the Cachazo-He-Yuan gauge amplitude, Phys. Rev. D 93 (2016) 105008 [arXiv:1602.06419] [INSPIRE].

I. Frenkel and M. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math J. 66 (1992) 123.

L. Dolan and P. Goddard, Current Algebra on the Torus, Commun. Math. Phys. 285 (2009) 219 [arXiv:0710.3743] [INSPIRE].