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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{string scattering amplitude} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry} [[!include supergeometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{Finiteness}{Finiteness}\dotfill \pageref*{Finiteness} \linebreak \noindent\hyperlink{openclosed_scattering_duality_and_klt_relations}{Open-closed scattering duality and KLT relations}\dotfill \pageref*{openclosed_scattering_duality_and_klt_relations} \linebreak \noindent\hyperlink{twistor_string_amplitudes_and_mhv_amplitudes_in_yangmills_theory}{Twistor string amplitudes and MHV amplitudes in Yang-Mills theory}\dotfill \pageref*{twistor_string_amplitudes_and_mhv_amplitudes_in_yangmills_theory} \linebreak \noindent\hyperlink{adic_formulation}{$p$-Adic formulation}\dotfill \pageref*{adic_formulation} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{General}{General}\dotfill \pageref*{General} \linebreak \noindent\hyperlink{superstring_scattering}{Superstring scattering}\dotfill \pageref*{superstring_scattering} \linebreak \noindent\hyperlink{higher_order_corrections}{Higher order corrections}\dotfill \pageref*{higher_order_corrections} \linebreak \noindent\hyperlink{ReferencesOnFiniteness}{On finiteness}\dotfill \pageref*{ReferencesOnFiniteness} \linebreak \noindent\hyperlink{bosonic_string}{Bosonic string}\dotfill \pageref*{bosonic_string} \linebreak \noindent\hyperlink{superstring}{Superstring}\dotfill \pageref*{superstring} \linebreak \noindent\hyperlink{graviton_scattering}{Graviton scattering}\dotfill \pageref*{graviton_scattering} \linebreak \noindent\hyperlink{ReferencesViaAdSCFT}{Via AdS/CFT}\dotfill \pageref*{ReferencesViaAdSCFT} \linebreak \noindent\hyperlink{more}{More}\dotfill \pageref*{more} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In [[perturbative string theory]] [[scattering amplitudes]] are defined as in [[quantum field theory]], but with [[n-point functions]] of 1-dimensional [[worldline theories]] ([[Feynman diagrams]]) replaced by those of [[worldsheet]] [[2d CFTs]]. \begin{quote}% graphics grabbed from \href{https://benjaminjurke.com/content/articles/2010/string-theory/}{Jurke 10} \end{quote} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{Finiteness}{}\subsubsection*{{Finiteness}}\label{Finiteness} The amplitudes are thought (see the commented \hyperlink{References}{references below}) to come out term-wise (for each ``loop order'' hence for each [[genus]] and number of punctures of ([[super Riemann surface|super]]-)[[Riemann surfaces]]) finite (at least UV-finite), hence [[renormalization|renormalized]]: the higher string oscillations may be seen as providing canonical counterterms for the massless excitations in the [[effective field theory]]. In this sense string theory provides a [[UV-completion]] of these effective field theories ([[supergravity]] coupled to [[Yang-Mills theory]]). \begin{remark} \label{}\hypertarget{}{} The full [[perturbation series]] is the [[sum]] of all these (finite) contributions over the [[genus|genera]] of [[Riemann surfaces]] (the ``loop orders''). This \emph{sum} diverges, even if all loop orders are finite. Notice though, that a non-trivial [[perturbation theory|perturbative]] [[QFT]] is not supposed to have a finite [[radius of convergence]] of its scattering amplitudes, since that would imply convergence also for \emph{negative} [[coupling constant]], which is physically unreasonable. (For the [[bosonic string]] the perturbation series has apparently been explicitly shown not to be [[Borel summation|Borel resummable]].) For more on this see at \emph{[[non-perturbative effect]]} and \emph{\href{string+theory+FAQ#NonConvergenceOfPerturbationSeries}{string theory FAQ -- Is the divergence of the pertubation series fatal?}}. \end{remark} \begin{remark} \label{}\hypertarget{}{} A string scattering amplitude is called \emph{UV-finite} at a given loop order ([[genus]] of a [[Riemann surface]] $\Sigma$ with $n$ marked points/string insertions) if the [[correlation function]] $\langle \phi_1, \cdots, \phi_n\rangle_{\Sigma}$ is finite for every single such [[Riemann surface]]. The actual string amplitude at order $(g,n)$ though is the averaging of this over all possible [[conformal structures]] on $\Sigma$, hence the [[integration]] of the [[correlation function]], as a function on the conformal structure, over the compactified [[moduli space of conformal structures]] $\mathcal{M}_{g,n}$ (a [[Deligne-Mumford stack]]). If also this integral is finite, hence if the total [[measure]] on the [[moduli space of conformal structures]] is finite, then one says the amplitude is \emph{IR-finite}. This distinction between UV-finiteness and IR-finiteness is not always highlighted in all of the articles below. All authors argue that the string is \emph{UV-finite} to all order. The IR-finiteness is only discussed much more recently at low loop order. IR non-finiteness is not physically fatal. For instance if a [[perturbation theory|perturbative]] theory of [[quantum gravity]] develops a [[cosmological constant]] perturbatively, then the perturbation series will be IR-divergent, signifying the fact that background spacetime without cosmological constant is no longer a solution to the quantum-corrected [[equations of motion]]. Nevertheless, these potential IR-divergences seem to be absent for the [[superstring]] perturbation series. For the [[cosmological constant]] case this can already be seen from the fact that the [[effective QFT]] of [[type II supergravity]] etc. does not admit a cosmological constant, for that would violate [[supersymmetry]]. \end{remark} \hypertarget{openclosed_scattering_duality_and_klt_relations}{}\subsubsection*{{Open-closed scattering duality and KLT relations}}\label{openclosed_scattering_duality_and_klt_relations} The [[open/closed string duality]] implies certain relations in string scattering amplitudes that in the point-particle limit induces relations between [[scattering amplitudes]] in [[Yang-Mills theory]] and in [[gravity]]. These are the \emph{[[KLT relations]]} in [[QFT]]. See in particular \hyperlink{MafraSchlotterer18a}{Mafra-Schlotterer 18a}, \hyperlink{MafraSchlotterer18b}{Mafra-Schlotterer 18b}, \hyperlink{MafraSchlotterer18c}{Mafra-Schlotterer 18c}. \hypertarget{twistor_string_amplitudes_and_mhv_amplitudes_in_yangmills_theory}{}\subsubsection*{{Twistor string amplitudes and MHV amplitudes in Yang-Mills theory}}\label{twistor_string_amplitudes_and_mhv_amplitudes_in_yangmills_theory} The scattering amplitudes in [[twistor string theory]] induce the MHV amplitudes in ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]. See at \emph{[[string theory results applied elsewhere]]} in the section \emph{\href{string+theory+results+applied+elsewhere#QCDScatteringAmplitudes}{Application to QCD -- Scattering amplitudes}}. \hypertarget{adic_formulation}{}\subsubsection*{{$p$-Adic formulation}}\label{adic_formulation} The [[Veneziano amplitude]] (open bosonic string tree-level scattering) has an equivalent formulation as the inverse product over all [[prime numbers]] $p$ of an amplitude computed not by an integral in the real but in the [[p-adic numbers]]. For other open string amplitudes this holds up to some regularization. This is the topic of \emph{[[p-adic string theory]]}, see there for more details. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item [[Veneziano amplitude]] \item [[Virasoro-Shapiro amplitude]] \item [[vacuum amplitude]] \end{itemize} \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[scattering amplitude]] \item [[perturbation theory]], [[perturbative string theory]] \item [[non-perturbative effect]], [[M-theory]] \item [[string theory FAQ]], [[string theory results applied elsewhere]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{General}{}\subsubsection*{{General}}\label{General} A comprehensive account of the [[superstring]] [[S-matrix]] may be obtained from combining the general idea presented in \begin{itemize}% \item [[Joseph Polchinski]], section 12.5 of vol 2 of \emph{String theory}, Cambridge Monographs on Mathematical Physics (2001) \end{itemize} with the technical details laid out in \begin{itemize}% \item [[Edward Witten]], \emph{Superstring Perturbation Theory Revisited} (\href{https://arxiv.org/abs/1209.5461}{arXiv:1209.5461}) \item [[Edward Witten]], \emph{More On Superstring Perturbation Theory: An Overview Of Superstring Perturbation Theory Via Super Riemann Surfaces} (\href{https://arxiv.org/abs/1304.2832}{arXiv:1304.2832}) \end{itemize} Survey of the tree level string scattering amplitudes includes \begin{itemize}% \item [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], \emph{String Scattering Amplitudes and Low Energy Effective Field Theory}, chapter 16 in \emph{Basic Concepts of String Theory} Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013 (\href{https://link.springer.com/content/pdf/bfm%3A978-3-642-29497-6%2F1.pdf}{TOC pdf}, \href{http://www.springer.com/gp/book/9783642294969}{publisher page}) \item [[Katrin Becker]], [[Melanie Becker]], Ilarion V. Melnikov, Daniel Robbins, Andrew B. Royston, \emph{Some tree-level string amplitudes in the NSR formalism}, JHEP 12 (2015) 010 (\href{http://arxiv.org/abs/1507.02172}{arXiv:1507.02172}) \end{itemize} See also \begin{itemize}% \item [[Gregory Moore]], \emph{Symmetries of the Bosonic String S-Matrix} (\href{https://arxiv.org/abs/hep-th/9310026}{arXiv:hep-th/9310026}) \end{itemize} For more references see also at \emph{[[string theory results applied elsewhere]]}. \hypertarget{superstring_scattering}{}\subsubsection*{{Superstring scattering}}\label{superstring_scattering} A review of superstring scattering amplitudes is in the last section of (\hyperlink{StaessensVernocke10}{Staessens-Vernocke 10}). A general discussion of the problem of superstring amplitudes is in \begin{itemize}% \item [[Eric D'Hoker]] [[Duong Phong]], \emph{Loop amplitudes for the fermionic string}, Nucl. Phys. B 278 (1986) 225; \item [[Greg Moore]], P. Nelson, [[Joseph Polchinski]], \emph{Strings and supermoduli}, Phys. Lett. B 169 (1986) 47-53. \end{itemize} On analycity of the superstring [[S-matrix]]: \begin{itemize}% \item Corinne de Lacroix, Harold Erbin, [[Ashoke Sen]], \emph{Analyticity and Crossing Symmetry of Superstring Loop Amplitudes} (\href{https://arxiv.org/abs/1810.07197}{arXiv:1810.07197}) \end{itemize} Survey of the presence and role of divergences includes \begin{itemize}% \item [[Ashoke Sen]], \emph{Ultraviolet and Infrared Divergences in Superstring Theory} (\href{http://arxiv.org/abs/1512.00026}{arXiv:1512.00026}) \end{itemize} Discussion of superstring scattering amplitudes in terms of [[pure spinors]] with emphasis on [[KLT relations]] is in \begin{itemize}% \item [[Carlos Mafra]], [[Oliver Schlotterer]], \emph{Towards the $n$-point one-loop superstring amplitude I: Pure spinors and superfield kinematics} (\href{https://arxiv.org/abs/1812.10969}{arXiv:1812.10969}) \item [[Carlos Mafra]], [[Oliver Schlotterer]], \emph{Towards the $n$-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics} (\href{https://arxiv.org/abs/1812.10970}{arXiv:1812.10970}) \item [[Carlos Mafra]], [[Oliver Schlotterer]], \emph{Towards the $n$-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure} (\href{https://arxiv.org/abs/1812.10971}{arXiv:1812.10971}) \end{itemize} For more see also at \emph{[[superstring field theory]]}, such as \begin{itemize}% \item Corinne de Lacroix, Harold Erbin, Sitender Pratap Kashyap, [[Ashoke Sen]], Mritunjay Verma, \emph{Closed Superstring Field Theory and its Applications}, International Journal of Modern Physics AVol. 32, No. 28n29, 1730021 (2017) (\href{https://arxiv.org/abs/1703.06410}{arXiv:1703.06410}) \end{itemize} \hypertarget{higher_order_corrections}{}\subsubsection*{{Higher order corrections}}\label{higher_order_corrections} \begin{itemize}% \item [[Christopher Pope]], \emph{Higher-order corrrections in String and M-theory and Generalized Holonomies}, December 2006 (\href{http://www.luth.obspm.fr/IHP06/workshops/cosmo/slides/Pope.pdf}{pdf}) \end{itemize} \hypertarget{ReferencesOnFiniteness}{}\subsubsection*{{On finiteness}}\label{ReferencesOnFiniteness} Here is a commented list of references on the degreewise \hyperlink{Finiteness}{finiteness of string scattering amplitudes}. \hypertarget{bosonic_string}{}\paragraph*{{Bosonic string}}\label{bosonic_string} Introductory lecture notes include \begin{itemize}% \item Wieland Staessens, Bert Vercnocke, \emph{Lectures on Scattering Amplitudes in String Theory}, Lecture notes based on lectures given at the fifth Modave School on Mathematical Physics (August 2009) (\href{http://arxiv.org/abs/1011.0456}{arXiv:1011.0456}) \end{itemize} Discussion of the term-wise finiteness of the bosonic string scattering amplitudes is in \begin{itemize}% \item L. Clavelli, S. T. Jones, \emph{Finiteness of the bosonic string in fewer than 26 dimensions}, Phys. Rev. D 39, 3795--3797 (1989) (\href{http://inspirehep.net/record/24403}{SPIRE}) \end{itemize} There are also arguments in \begin{itemize}% \item Simon Davis, \emph{On the domain of string perturbation theory}, Classical and Quantum Gravity, Volume 6, Issue 12, pp. 1791-1803 (1989) (\href{http://adsabs.harvard.edu/abs/1989CQGra...6.1791D}{web}) \end{itemize} \hypertarget{superstring}{}\paragraph*{{Superstring}}\label{superstring} Finiteness of heterotic and type II superstring $n$-point functions in flat spacetime is argued for in \begin{itemize}% \item Joseph Atick, [[Gregory Moore]], [[Ashoke Sen]], \emph{Catoptrick tadpoles}, Nucl.Phys. B307 (1988) 221 (\href{http://inspirehep.net/record/252598?ln=en}{spire}) \end{itemize} General finiteness of superstring amplitudes is discussed in \begin{itemize}% \item [[Stanley Mandelstam]], \emph{The $n$ Loop String Amplitude: Explicit Formulas, Finiteness and Absence of Ambiguities}, Phys. Lett. B277 (1992) 82. (\href{http://inspirehep.net/record/319501}{inspire}) \end{itemize} which shows that a certain divergence which could appear does not. Also \begin{itemize}% \item A. Restuccia, J. Taylor, \emph{Finiteness of type II superstring amplitudes}, Physics Letters B, Volume 187, Issues 3--4, 26 March 1987, Pages 267--272 \end{itemize} argues finiteness of the superstring amplitudes at each order. Also \begin{itemize}% \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Momentum Analyticity and Finiteness of the 1-Loop Superstring Amplitude}, Phys.Rev.Lett. 70 (1993) 3692-3695 (\href{http://arxiv.org/abs/hep-th/9302003}{arXiv:hep-th/9302003}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{The Box Graph In Superstring Theory}, Nucl.Phys.B440:24-94,1995 (\href{http://arxiv.org/abs/hep-th/9410152}{arXiv:hep-th/9410152}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Dispersion Relations in String Theory}; Nucl.Phys.B440:24-94,1995 (\href{http://arxiv.org/abs/hep-th/9404128}{arXiv:hep-th/9404128}) \end{itemize} The 1-loop amplitudes in [[type II string theory]] have been discussed in \begin{itemize}% \item [[Michael Green]], [[John Schwarz]], \emph{Supersymmetrical string theories}, Phys. Lett. 109 B (1982) 444-448. \end{itemize} and for [[heterotic string theory]] in \begin{itemize}% \item [[David Gross]], J.A. Harvey, E.J. Martinec and R. Rohm, \emph{Heterotic String Theory (II). The interacting heterotic string}, Nucl. Phys. B267 (1986) 75. \end{itemize} The description of 2-loop amplitudes, including the integration over the super-[[moduli space of conformal structures]] in superstring theory is given in the series of articles \begin{itemize}% \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Two-Loop Superstrings I, Main Formulas}, Phys.Lett.B529:241-255,2002 (\href{http://arxiv.org/abs/hep-th/0110247}{arXiv:hep-th/0110247}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Two-Loop Superstrings II, The Chiral Measure on Moduli Space}, Nucl.Phys.B636:3-60,2002 (\href{http://arxiv.org/abs/hep-th/0110283}{arXiv:hep-th/0110283}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities}, Nucl.Phys.B636:61-79,2002 (\href{http://arxiv.org/abs/hep-th/0111016}{arXiv:hep-th/0111016}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms}, Nucl.Phys.B639:129-181,2002 (\href{http://arxiv.org/abs/hep-th/0111040}{arXiv:hep-th/0111040}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function}, Nucl.Phys.B715:91-119,2005 (\href{http://arxiv.org/abs/hep-th/0501196}{arXiv:hep-th/0501196}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function}, Nucl.Phys.B715:3-90,2005 (\href{http://arxiv.org/abs/hep-th/0501197}{arXiv:hep-th/0501197}) \item [[Eric D'Hoker]], [[Duong Phong]], \emph{Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes}, Nucl.Phys.B804:421-506,2008 (\href{http://arxiv.org/abs/0711.4314}{arXiv:0711.4314}) \end{itemize} A review of this work is in \begin{itemize}% \item A.Morozov, \emph{NSR Superstring Measures Revisited}, JHEP0805:086,2008 (\href{http://arxiv.org/abs/0804.3167}{arXiv:0804.3167}) \end{itemize} The technical issue of the [[moduli space]] of [[super Riemann surfaces]] of higher genus (for higher loop string scattering amplitudes) is discussed in \begin{itemize}% \item [[Edward Witten]], \emph{Notes On Super Riemann Surfaces And Their Moduli} (\href{http://arxiv.org/abs/1209.2459}{arXiv:1209.2459}) \item [[Ron Donagi]], [[Edward Witten]], \emph{Supermoduli Space Is Not Projected} (\href{http://arxiv.org/abs/1304.7798}{arXiv:1304.7798}) \end{itemize} Arguments for the finiteness of superstring scattering amplitudes to all loop order based on the [[Berkovits superstring]]-formulation have been promoted in \begin{itemize}% \item [[Nathan Berkovits]], \emph{Multiloop Amplitudes and Vanishing Theorems using the Pure Spinor Formalism for the Superstring}, JHEP 0409:047,2004 (\href{http://arxiv.org/abs/hep-th/0406055}{arXiv:hep-th/0406055}) \end{itemize} (In footnote 2 this article claims that the claimed proofs of the same statement by G.S Danilov in \href{http://arxiv.org/abs/hep-th/9801013}{hep-th/9801013}, \href{http://arxiv.org/abs/hep-th/0312177}{hep-th/0312177} are not in fact proofs.) Discussion of 2-loop amplitudes from holomorphy arguments is in \begin{itemize}% \item [[Edward Witten]], \emph{Notes On Holomorphic String And Superstring Theory Measures Of Low Genus} (\href{http://arxiv.org/abs/1306.3621}{arXiv:1306.3621}) \end{itemize} See also \begin{itemize}% \item [[Ashoke Sen]], \emph{Supersymmetry Restoration in Superstring Perturbation Theory} (\href{http://arxiv.org/abs/1508.02481}{arXiv:1508.02481}) \end{itemize} \hypertarget{graviton_scattering}{}\subsubsection*{{Graviton scattering}}\label{graviton_scattering} Computation of [[graviton]] scattering amplitudes ([[perturbative quantum gravity]]): \begin{itemize}% \item [[Taejin Lee]], \emph{Gravitational Scattering Amplitudes and Closed String Field Theory in the Proper-Time Gauge}, EPJ Web of Conferences 168, 07004 (2018) (\href{https://doi.org/10.1051/epjconf/201816807004}{doi:10.1051/epjconf/201816807004}) \item [[Taejin Lee]], \emph{Four-Graviton Scattering and String Path Integral in the Proper-time gauge} (\href{https://arxiv.org/abs/1806.02702}{arXiv:1806.02702}) \end{itemize} \hypertarget{ReferencesViaAdSCFT}{}\subsubsection*{{Via AdS/CFT}}\label{ReferencesViaAdSCFT} Discussion via [[AdS/CFT]] beyond the [[SCFT]] [[planar limit]], using the [[conformal bootstrap]]: \begin{itemize}% \item [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], \emph{Genus-One String Amplitudes from Conformal Field Theory}, JHEP06(2019) 010 (\href{https://arxiv.org/abs/1809.10670}{arXiv:1809.10670}) \end{itemize} \hypertarget{more}{}\subsubsection*{{More}}\label{more} On point-particle limit via [[tropical geometry]] \begin{itemize}% \item Piotr Tourkine, \emph{Tropical Amplitudes} (\href{http://arxiv.org/abs/1309.3551}{arXiv:1309.3551}) \end{itemize} Scattering amplitudes of highly excited strings: \begin{itemize}% \item Dimitri P. Skliros, Edmund J. Copeland, Paul M. Saffin, \emph{Highly Excited Strings I: Generating Function} (\href{https://arxiv.org/abs/1611.06498}{arxiv:1611.06498}) [[!redirects string scattering amplitudes]] \end{itemize} [[!redirects string scattering amplitudes]] [[!redirects superstring scattering amplitude]] [[!redirects superstring scattering amplitudes]] [[!redirects string scattering]] [[!redirects string perturbation series]] \end{document}