Advertisement

Journal of High Energy Physics

, 2019:85 | Cite as

String correlators: recursive expansion, integration-by-parts and scattering equations

  • Song He
  • Fei TengEmail author
  • Yong Zhang
Open Access
Regular Article - Theoretical Physics
  • 11 Downloads

Abstract

We further elaborate on the general construction proposed in [1], which connects, via tree-level double copy, massless string amplitudes with color-ordered QFT amplitudes that are given by Cachazo-He-Yuan formulas. The current paper serves as a detailed study of the integration-by-parts procedure for any tree-level massless string correlator outlined in the previous letter. We present two new results in the context of heterotic and (compactified) bosonic string theories. First, we find a new recursive expansion of any multitrace mixed correlator in these theories into a logarithmic part corresponding to the CHY integrand for Yang-Mills-scalar amplitudes, plus correlators with the total number of traces and gluons decreased. By iterating the expansion, we systematically reduce string correlators with any number of subcycles to linear combinations of Parke-Taylor factors and similarly for the case with gluons. Based on this, we then derive a CHY formula for the corresponding (DF)2 + YM + ϕ3 amplitudes. It is the first closed-form result for such multitrace amplitudes and thus greatly extends our result for the single-trace case. As a byproduct, it gives a new CHY formula for all Yang-Mills-scalar amplitudes. We also study consistency checks of the formula such as factorizations on massless poles.

Keywords

Scattering Amplitudes Bosonic Strings Gauge Symmetry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    S. He, F. Teng and Y. Zhang, String amplitudes from field-theory amplitudes and vice versa, Phys. Rev. Lett. 122 (2019) 211603 [arXiv:1812.03369] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press, Cambridge U.K. (2016).Google Scholar
  3. [3]
    J.M. Henn and J.C. Plefka, Scattering Amplitudes in Gauge Theories, Lect. Notes Phys. 883 (2014) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press, Cambridge U.K. (2015).Google Scholar
  5. [5]
    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].
  6. [6]
    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].
  7. [7]
    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].
  8. [8]
    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].
  9. [9]
    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].
  10. [10]
    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].
  11. [11]
    H. Johansson, G. Kälin and G. Mogull, Two-loop supersymmetric QCD and half-maximal supergravity amplitudes, JHEP 09 (2017) 019 [arXiv:1706.09381] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Z. Bern, J.J. Carrasco, W.-M. Chen, A. Edison, H. Johansson, J. Parra-Martinez et al., Ultraviolet Properties of \( \mathcal{N} \) = 8 Supergravity at Five Loops, Phys. Rev. D 98 (2018) 086021 [arXiv:1804.09311] [INSPIRE].
  13. [13]
    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].ADSCrossRefGoogle Scholar
  14. [14]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    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].
  16. [16]
    T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].ADSCrossRefGoogle Scholar
  18. [18]
    F. Cachazo, S. He and E.Y. Yuan, One-Loop Corrections from Higher Dimensional Tree Amplitudes, JHEP 08 (2016) 008 [arXiv:1512.05001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    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].
  20. [20]
    Y. Geyer and R. Monteiro, Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere, JHEP 11 (2018) 008 [arXiv:1805.05344] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    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].ADSCrossRefGoogle Scholar
  23. [23]
    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].
  24. [24]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    E. Casali, Y. Geyer, L. Mason, R. Monteiro and K.A. Roehrig, New Ambitwistor String Theories, JHEP 11 (2015) 038 [arXiv:1506.08771] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    W. Siegel, Amplitudes for left-handed strings, arXiv:1512.02569 [INSPIRE].
  27. [27]
    E. Casali and P. Tourkine, On the null origin of the ambitwistor string, JHEP 11 (2016) 036 [arXiv:1606.05636] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    T. Azevedo and R.L. Jusinskas, Connecting the ambitwistor and the sectorized heterotic strings, JHEP 10 (2017) 216 [arXiv:1707.08840] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, P. Tourkine and P. Vanhove, Scattering Equations and String Theory Amplitudes, Phys. Rev. D 90 (2014) 106002 [arXiv:1403.4553] [INSPIRE].
  31. [31]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    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].
  33. [33]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].
  34. [34]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    S. He, R. Monteiro and O. Schlotterer, String-inspired BCJ numerators for one-loop MHV amplitudes, JHEP 01 (2016) 171 [arXiv:1507.06288] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    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].
  37. [37]
    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].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
  39. [39]
    S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].
  40. [40]
    O. Schlotterer, Amplitude relations in heterotic string theory and Einstein-Yang-Mills, JHEP 11 (2016) 074 [arXiv:1608.00130] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Semi-abelian Z-theory: NLSM+ϕ 3 from the open string, JHEP 08 (2017) 135 [arXiv:1612.06446] [INSPIRE].
  45. [45]
    Y.-t. Huang, O. Schlotterer and C. Wen, Universality in string interactions, JHEP 09 (2016) 155 [arXiv:1602.01674] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    T. Azevedo, M. Chiodaroli, H. Johansson and O. Schlotterer, Heterotic and bosonic string amplitudes via field theory, JHEP 10 (2018) 012 [arXiv:1803.05452] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    H. Johansson and J. Nohle, Conformal Gravity from Gauge Theory, arXiv:1707.02965 [INSPIRE].
  48. [48]
    O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
  49. [49]
    S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE].
  50. [50]
    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].
  51. [51]
    O. Schlotterer and O. Schnetz, Closed strings as single-valued open strings: A genus-zero derivation, J. Phys. A 52 (2019) 045401 [arXiv:1808.00713] [INSPIRE].
  52. [52]
    F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, arXiv:1810.07682 [INSPIRE].
  53. [53]
    O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].
  54. [54]
    F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].
  55. [55]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    S. Mizera and G. Zhang, A String Deformation of the Parke-Taylor Factor, Phys. Rev. D 96 (2017) 066016 [arXiv:1705.10323] [INSPIRE].
  57. [57]
    S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, JHEP 08 (2017) 097 [arXiv:1706.08527] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, JHEP 05 (2018) 096 [arXiv:1711.09102] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    F. Teng and B. Feng, Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame, JHEP 05 (2017) 075 [arXiv:1703.01269] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    Y.-J. Du, B. Feng and F. Teng, Expansion of All Multitrace Tree Level EYM Amplitudes, JHEP 12 (2017) 038 [arXiv:1708.04514] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    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].
  63. [63]
    R.P. Stanley, Enumerative Combinatorics: Volume 2, first edition, Cambridge University Press, New York U.S.A. (2001).Google Scholar
  64. [64]
    V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
  65. [65]
    H. Johansson, G. Mogull and F. Teng, Unraveling conformal gravity amplitudes, JHEP 09 (2018) 080 [arXiv:1806.05124] [INSPIRE].
  66. [66]
    S. He and Y. Zhang, New Formulas for Amplitudes from Higher-Dimensional Operators, JHEP 02 (2017) 019 [arXiv:1608.08448] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    L.M. Garozzo, L. Queimada and O. Schlotterer, Berends-Giele currents in Bern-Carrasco-Johansson gauge for F 3 - and F 4 -deformed Yang-Mills amplitudes, JHEP 02 (2019) 078 [arXiv:1809.08103] [INSPIRE].
  68. [68]
    Y.-J. Du and F. Teng, BCJ numerators from reduced Pfaffian, JHEP 04 (2017) 033 [arXiv:1703.05717] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    Y.-J. Du and Y. Zhang, Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes, JHEP 07 (2018) 177 [arXiv:1803.01701] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    L. Hou and Y.-J. Du, A graphic approach to gauge invariance induced identity, JHEP 05 (2019) 012 [arXiv:1811.12653] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization, arXiv:1906.02099 [INSPIRE].
  72. [72]
    P. Vanhove and F. Zerbini, Closed string amplitudes from single-valued correlation functions, arXiv:1812.03018 [INSPIRE].
  73. [73]
    S. He, G. Yan, C. Zhang and Y. Zhang, Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces, JHEP 08 (2018) 040 [arXiv:1803.11302] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  74. [74]
    N. Arkani-Hamed, talk at Amplitudes 2019, Trinity College, Dublin Ireland (2019), https://indico.cern.ch/event/750565/contributions/3439541/attachments/1873668/3084360/Arkani-Hamed.pdf.
  75. [75]
    C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, String-Like Dual Models for Scalar Theories, JHEP 12 (2016) 019 [arXiv:1610.04228] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    A. Tsuchiya, More on One Loop Massless Amplitudes of Superstring Theories, Phys. Rev. D 39 (1989) 1626 [INSPIRE].
  77. [77]
    L. Dolan and P. Goddard, Current Algebra on the Torus, Commun. Math. Phys. 285 (2009) 219 [arXiv:0710.3743] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    C.R. Mafra and O. Schlotterer, Double-Copy Structure of One-Loop Open-String Amplitudes, Phys. Rev. Lett. 121 (2018) 011601 [arXiv:1711.09104] [INSPIRE].
  79. [79]
    C.R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude I: Pure spinors and superfield kinematics, arXiv:1812.10969.
  80. [80]
    C.R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part II. Worldsheet functions and their duality to kinematics, JHEP 08 (2019) 091 [arXiv:1812.10970] [INSPIRE].
  81. [81]
    C.R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part III. One-loop correlators and their double-copy structure, JHEP 08 (2019) 092 [arXiv:1812.10971] [INSPIRE].
  82. [82]
    J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings, JHEP 01 (2019) 052 [arXiv:1811.02548] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    X. Gao, S. He and Y. Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, JHEP 11 (2017) 144 [arXiv:1708.08701] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.CAS Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingChina
  2. 2.School of Physical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.Department of Physics and AstronomyUppsala UniversityUppsalaSweden

Personalised recommendations