Paper
0
Chiral Closed strings: Four massless states scattering amplitude
Authors
Marcelo M. Leite,Warren Siegel
Olivier Ragueneau,Olesya Vasyukova,Vlastimil Krivan,Ferdinando Gliozzi,Marco Fabbrichesi,Alberto Casas González,Aibin Zhan,Andrey Poliakov,Friedlinde Goetz-Neunhoeffer,Kwan Cheng,Domenec Espriu,Stephen Naculich,Alex van Heezik,Michael Flohr,Ana Martinez-Pastor,Ben Bellows,Vladimir Orlov,Stephen DiFazio,Jason J. Venkiteswaran,John Lord,Simone Antonio De Rose,José Pablo Mora Gomez,Martin Goodchild,Trevor Glasbey,Marjorie Wonham,Hartmut Gimpel,Cynthia Keeler,Srinivasan Jagannathan,Nenad Đurđević,Frédéric Dreyer,Dejan Matić,Milivoje Lapčević,Veljko Vlašković,Zoran Vuković,Zoran Jovanović,Radoje Brković,Miodrag Mićović,Emilija Stanković,Milena Petrović,Jelena Vučković,Srđan Vladetić,Jovan Vujičić,Sveto Puríć,Dragica Živojinović, Dj,Dragan Vujisić,Milan Palević,Božin Vlašković,Nina Planojević,Milan Rapajić,Siniša Varga,Violeta Todorović,Srdjan Đorđević,Michele Portolan,Torben Gaedt,Nishita Desai,Jasmina Labudović Stanković,Tim Sykes,Milan Dana,Alain Jaccard,Carolyn Dzierba,Erdal Can Alkoclar,Marc Torrens,Benjamin Bellenie,Anne Co,Bartomeu Fiol,Andrew Degnan,Andy Van Brocklin,marilia rippe,Anatoly Dymarsky,Ksenija Vlašković,Digamber Porob,David Van Wie,Andrej Micovic,Jose Endrino,Marie Connett,Antonio Amores Arrocha,John Joseph Carrasco,Bulent Basol,Diogo Aurélio,Thorsten Alexander Kern,Wai-Ching Lam,Martin Köhne,Bo Feng,Dan Andersen,José Pedro Viegas Barros,Ana Mª Roldán Gómez,Víctor Manuel Palacios Macías,Kimmo Tuominen,Charles Fox,Martin Ward,Dirk Lennarz,Dominique Bonnet ,Katherine Saul,Jacqueline Cole,Sonja Jovanović,Rajko Bukvic,Tatiana Batura,Pei-Ming Ho,Peter Søgaard Jørgensen,Parinya Karndumri,Bruno Dufour,YU-TIN HUANG,Dimitrios Lykomitros,Marina Petri,Koenraad Schalm,Nilanjan Brahma,Luis Pessoa,Noah Goldberg,Johanna Knapp,Valentina Giangreco Puletti,Joao Penedones,Tatsuhiro Misumi,Boris Pioline,Jan Pieter van der Schaar,James Gainer,Chaitali Sengupta,Shankha Banerjee,Gerrit Meixner,Sai Putcha,Nathan Seiberg,Manuel Masip,Cristiane WOSNIAK,Dorothea Bedigian,Marinei Lopes Pedralli,Christopher Fedo,Eoin O'Cearbhaill,Per Gustavsson,Jesus M Corral-Santana,ALESSANDRA BORGES,Andrzej Bałaban,Mariam Tortola,Agustin Sabio Vera,Dirk Pandel,Tomislav Marketin,Gianluca Giavitto,Agata Wawrzyniak,Gernot Maier,Barbara Wąsikowska,Greg Stachowski,Scott Griffiths,Vincent Rouessac,Guy Moore,Joan Simon,Ranjan Laha,Spencer Chang,Darek HAFTOR,Kathryn Zurek,Peter Arnold,Cheng-Wei Chiang,Helena Barroso,Licia Verde,Francisco Torrentí,Bruno Breitmeyer,Alain Hébert,HSIN-YI LIN,Emanuel Malek,Tommi Alanne,Yuji Tachikawa,Karl Landsteiner,Alexander Rothkopf,Joao Eduardo Coin Carvalho,Calin Iuliu Lazaroiu,Karen D McCoy,Søren E. Larsen,Gavin Salam,Sylvain Ribault,Seodong Shin,Chethan Krishnan,Gerhard Kresbach,Gabriella D'Andrea-Carlino,Hanna Pawlowska,Emilio Bellini,Kazumi Okuyama,Elena Bichenkova,Michael King,Ralf Johannes Radlanski,Josser Delgado Almandoz,Xiaohui Yang,Poulose Poulose,Sergio Giardino,Carolyn Copenheaver,Gabriel Lopes Cardoso,alberto zaffaroni,Wendy Flavell,Martina Paatela-Nieminen,Attila Gido,Morten Christensen,Kai Sun,Daniel Waldram,Marek Adamkiewicz,Agnieszka Bieńkowska,Victor Rivelles,Karen Salomé Caballero Mora,Jakub Scholtz,ALBERTO IBORT,Naoto Yokoi,Oliver Rudzick,Olga Goulko,Riccardo Rattazzi,Klaus Spohr,Margarita Samudio,Syksy Räsänen,Adem Zeynev,Mattia Bruno,Tali Lee,Robert Lauer,MICHELE PIANA,Marc Ribó,Xue-Mei Li,Abd Al Karim Haj Ismail,Giuseppe Dibitetto,Lorenzo Vitale,KAZUO HOSOMICHI,Periklis Pantazis,Christopher Verhaaren,Kenneth Park,Zhaoxing Meng,Michael McCulloch, PhD MPH LAc,Pascal Anastasopoulos,Ernesto López Orendain,Phillip Stanley-Marbell,Moni Nader,Yousuke Yamashita,isabella paola CARUCCI,Justin Hart,Pankaj Kalore,Marco Serone,Paulina Olechowska,Jan Troost,Jeppe Andersen,Carlos Hoyos Badajoz,Estela Mary Peralta de Aguayo,Ayad Mahmood,Chia-Yi Liu,patrizia vitale,Jarosław Szczechowicz,Ewa Janus,Hubert Piquet,Maresuke Shiraishi,George Craig,Krystyna Szczechowicz,Fedele Lizzi,Maureen Cribb,Alessandro Tomasiello,Geraldine Allen,Chris Klingenberg,David Fenard,Gerasimos Cassis,Diego Di Martire,Marianne Okal,Tomohiro Fujita,Ian Brooks,Harry H. Hendon,Ramalingam Saravanan,Jeffrey Conner,Robert Rauber,Bogdan Palosz,N/A ,Joseph McHugh,Ewa Kaptur,Frank Ferrari,Kari Enqvist,Richard Rood,Antony Lewis,Duccio Pappadopulo,Marek Biszczanik,Daniel Francis,Sebastian Garcia-Saenz,JiJi Fan,Carl Mueller,María Josefa Tarifa Castilla,Shu Niu,Nik Ding,Marcos Roberto de Freitas,Wilfrid Kendall,PIETRO PALA,Cezary Purwin,Gerardo Aboy Carlés,Josef Kainz,Sandra Yuter,Fernando Barros,Colin Braithwaite,William Donaldson,David MacLean,Niels Obers,Marton Kormos,David González Cruz,Yong-Zhuang Li,Junjie Rao,CONSTANTINOS PALLIS,Charles Marteau,Grigorios Chachamis,Leyla Neyzi,Branka Drljača Margić,Masumi Yamagishi,Tomasz Stebel,Miguel S Costa,Yusuke Yamada,Gregory Soyez,kirtiman ,Ennio Salvioni,Lynsey Spillman,KHELIFA ADEL,Nadav Drukker,Illan Halpern,Paolo Benincasa,Tomasz Daszkiewicz,Fernanda Marcondes,Manuel Pérez-Victoria Moreno De Barreda,Jeong-Hyuck Park,Yi Pang,De'an Sun,Pabitra Pradhan,Alexander Krikun,Isabel Maria Fernandes,Pere Munar-Adrover,Alicia López Oramas,María Antonia Frías Sagardoy,Karina Maria Salvatore Freitas,Michele Caselle,Carlos Eduardo Santos Bonfim,Shahrokh Kazempour-Osaloo,Mariana Grana,Vygandas Rutkunas,Gherardo Chirici,Eva Kernbauer,Muhammad Ayub,Carlo Maccaferri,Ville Vaskonen,Roberto Reis,Utkov Pavel,David Battisti,Sergey Martynovich,Bill Page,Junchen Rong,aritra banerjee,ERIC CHEN-HUA YU,SHU-CHIN WANG,棟樑 廖,CHUN-I CHEN,Takashi Yokota,Hugues Beaufrere,Jelle Hartong,ANTONIO SIQUEIRA,Leonid G. Rybnikov,Maxim Zabzine,David Schaich,Barbara Carlson,Joy Bose,Simon Rogerson,Paul Humphries,Shin'ichiro Ando,Maria Valaste,Amr ALHOSSARY,Fabrizio Flores,Renata Sathler,Theodore Erler,Axel Kaehne,Stefan Prohazka,Toru Masuda,Thais Fernandes,Yashar Akrami,Ihor Rushchenko,Fabricio Valarelli,Evangelos Sfakianakis,Mohamed Meguid,Lauri Mehtätalo,Gonzalo Palma,Stephen Frasier,Christopher Lucas,Leszek Belzyt,Tomas Kucera,Sven Englund,Pushpa Jha,Dilaila Botas,Ірина Востоцька,Peter Poór,Charles Rabideau,Rasmus Sloth Lundkvist,Xian Hou,Julien Cucherousset,Lawrence Kawchuk,Christopher Hull,Viktor Goliáš,Albion Lawrence,Kim Wan YOUNG,Bernard Legras,Shun Zhou,Anna Mikolon,Annika Peter,DONG CHUAN WU,Ahmed Zidan,Sergey Sibiryakov,Marcio Rodrigues de Almeida,Diego Blas,Haifeng Wang ,Oriol Pujolas,Tong Wan,Dorit Naot,Taiga Kunishima,Heitor S. Ramos,Charles OKOLIE,Silvia Caporale-Bizzini,Mariana Marquezan,Marja Kokko,Linda Vahdat,Tânia Luisa Koltermann da Silva,José Luís Brandão,Joaquim Pais-Barbosa,Alfred Botha,Juris Prikulis,Simon Foreman,Alexey Inozemtsev,Kumari Neelam,Yolande Serra,Timofey Tolmachev,DAVID KINGSMILL,Henar Pizarro Llorente,Gursharn Randhawa,Jarcilene Silva de Almeida,David Normando,Heiko Wittmer,Pernilla Ingelsson,Min Hyuk Kim,Evangelina Salazar Röling,Makoto Inoue,Miglena Drumeva (Todorova),Naoufel Werghi,Iranloye Yetunde,Maria Patricia Reséndiz Ramos,Jonathan Taylor,Pier Paolo Monticone,Isidro Arceo,Américo Ariel Rubin de Celis Vidal,Maryna Tkachenko,Ksenia Frolova,Nicusor Cristian Pop,Tiffany Troso-Sandoval,Denise Brown,Zuraifah Asrah binti Mohamad,Emma Cockburn,Shafika Isaacs,Sandra C. Díaz Bello,Madhusudhan B N,Rehan Pradhan,Sven Stadtmüller,Tosho Rii,Andrea Marzeddu,Huazhong Guo,Christine Lindstrøm,mauricio plaza,Tom Denison,Зоя Киреева,Peterson Padilha,Celestino Joanguete,Wolfgang Fiedler,Tiyamike Ngonda,Ekaterina Alekseeva,Josep Maria Casas Tuset,Victor Godet,Francesco Coradeschi,Anatoly Polyanskiy,Matheus Alves Siqueira de Assunção,浩二 津村,Anna Puchkova,Ettore de Lacerda Arpini,Hanteng Wang,Armando Arteaga Rosero,Debora Machado Fracalossi,Alexandr Zegzhda,Joanny Martins,VANDO RONDELLI,Herschel Chawdhry,Zoltán Magyar,Alexander Chudnovskiy,Paulo Gonzales,Teri Yohn,Olena Yelizarova,Bárbara Calegari,sid ahmed besbas,Soumya Bhattacharya,Hayati Koknaroglu,Stanisław Milewski,Marco Sanjuan,rafea elgorashe,Наталья Волох,Bruna Kelly Fehlberg,Dominique Gagnon,Sávio Torres Melo,John Ristau,Mikhail Kazakov,Nikolaos D. Karakousis,Anderson Domingues,Alicia Grasby,Alexes Mes,José Javier Álvaro,和嗣 野呂,John Pring,hainv gao,Elizabeth Kaschalk Woods,Arrojo Maria Jose,Елена Данзанова,Núria Carrera,Espen Sagvolden,Steven Platnick,Yvete Flavio da Costa,LILIANA DAMARIS PABÓN GIRALDO,Mustafa Berker Yurtseven,Dmitry Khochenkov,Sreemanti Chakraborti,David DELMAIL,Fazlollah Hajkarim,Ada Ferrer Fuertes,Sebastian Felix Rauh,Jeffrey Lorch,Kristian Allee,Maristella Gatto,TETIANA STANKEVYCH,Daniela S. Rivera,Zhiyun Guo,Akinyomade Owolabi,Zeng Yanhua,Daniel Magues,Alberto Uribe-Jongbloed,晨光 李,STEFANIA MIGLIORI,Krzysztof Grygierek,Chris Gale,Oghenerobor Akpor,Phil Hayes,Ricardo Cesar Gobbi de Oliveira,Rosa Borge,Fernando Daniel Alfaro Ayllon,Petar Curkovic,Rossana Villa-Rojas,YADUNATH SINGH,Renata Cristina Gobbi de Oliveira,Christopher Crosby,Андрій Бузун,Francisco J. Garces-Vega,Rachel Harding,Rola Naeb,Olexandr Cherepok,Sumit Bhowmik,Hugo Perales,Locif Redouani,Wojciech Bujalski,Jian Lin,BARBARA SŁOMKA-SŁUPIK,ZHANG LIANDONG ,Francisco Carrapiço,Wijitha thero Koggalle,Davide Carnevali,Fuzhan Song,Domenico Torretta,Abiola Olaniran,MARCHAND Frederic,Yevgen Strokov,Atri Sengupta,Maria Lopes,Hongli Jiang,Igor Basso,Andrew Ramey,Joanna Ferdyn-Grygierek,Hideyuki Kamimera,Anton Getsov,Wenhu Guo,ANDRES ADOLFO NAVARRO NEWBALL,Amir Poursamad,Katarzyna Kowalska,Gina Passante,Zachary Anesbury,Leonardo eiji Kawamoto,Scott Herron,Christopher Kirby,Daniel ,Hao Ran,Irina Lobzhanidze,Gláucia Diniz,ghaffar raeisi,Emma Wu Dowd,Ricardo Mendes Junior,Zengjian LOU,Michal Berec,Melanie Paschke,Zenon Tański,Brandt Meixell,Miguel Montero,Алексей Валерьевич Шерстюк,Igor Silva de Sá,Federico Righi,Michael Böttinger,Wenhan Zeng,Hui-Chen Chen,Yaqing Liu,Eli Galanti,Iris Munhoz Costa,Greice Ayra Franco-Assis,Jorge Valadares,Andrew Schonebaum,Alejandro Rivera Baro,Mark Lampi,Martinez-Sañudo Isabel,Hans Volkert,kiinging wong,Karthik A Balasubramaniam,Joaquim Campuzano,Daniela Bessa,Jadwiga Wichman,Dorthe Kristensen,Dalia Varanka,Artur Hideyuki Tomita,Elena Broaddus-Shea,Andrew Hill,Adrian Bowman,Nicolas Latte,Daniel Larson,Elsa Sanchez-Alonso,Paul Calder,Todd Thompson,Riccardo Martino,andrew ackerman,Dorota Kubiak,Mehdi Veiskarami,Norma Isela Rodríguez Arévalo,Aleksandra Ledwoń,Consolación Fernández Mellén,John W. Castro,Jinzhu Peng,Timothy Laurence Burnett,Volodymyr Shekhovtsov,Jian Ping Li,Andreas Och,Thomas Klee,Stefan Scheiner,Luca Mazzon,Paolo Pacciolla,MONICA BUSTAMANTE,Colin O'Dowd ,Pascal LABROUSSE,Serkan Yuksel,V.S. Saravanan ,Ronald Gooding,José Palmeira,PEDRO ERNESTO MOREIRA GREGORI,Rozalia Małgorzata Ligus,Teppei Okumura,Clara Ulrich,Julia Kuhn, ELENA LARO-GONZÁLEZ,Tânia Mara Silva,Marston Hillier,Carlo Rodrigues,Marina Solnyshkina,Pasi Rautio,Rafał Winarski,Paolo Scopece,Aleksandra Niewiara,Mandy Wan,Maria da Felicidade Araújo Morais,Malgorzata Natonek-Wisniewska,katya abrantes,Jessica Pérez-Luzardo Díaz,Natalie Pang,Ahmad Nezami,Dominic Mills-Howell,Oliver Henrich,CARMEN JOVER RAMIREZ,Irina Petrova,Lynn Russell,Hernan Herrera-Echeverri,Hatem Galal A Ibrahim,Steven Zarit,George Bryan,Bruno Salvatore MESSINA,Konstantinos Papadopoulos,Biswarup Basu,John Millis,Atsuo Okazaki,Dirk Frederik de Lange,Annarita Mariotti,María Belén Lorente Molina,Aliaa Youssif,Bourlaye Fofana,PASCALE BESSE,Andrea Mojica,Pat Scott,Arun Das,Elina Staaf,PABITRA KUMAR PAUL,Yoshiyuki Kobayashi,Ana Margarida Pereira,Fadekemi Oginni,Adam Stecyk,Roman Akasov,Irene Ylönen,Arjun Balar,Lukáš Holata,Andrey Elchaninov,Larry Stillman,steen hansen,Kenichi Kashimada,Jaime Hernandez-Sanchez,Annemarie Bastrup-Birk,Licinio Esmeraldo Silva,Ioannis Polyzos,Angel Vega-Zamanillo,Stephanie Fay Stokes,Alessandra Palma,Pavel Kral,Xavier Cornejo,Jawali Maharudrappa Mallikarjuna,Vagner Queiroz,Mercedes Sanz-Gil,ANTHONY ADENEKAN,Lukasz Niedzwiecki,Martinho Soares,Thanh Ngo-Duc,Antonio Fernando Morais de Oliveira,AMANDA VALOIS CÓRDOVA CAMPOS,Esau João ,Mia Hamilton Jee,Zainab Ngaini,Věra Strnadová,José Álvarez Marcos,Adriano TOMASSINI,Silvio Andrés Mosquera-Sánchez,Basilio Carrasco,Franck Laloë,Sebastian Abades,Silvia Profili,Marian Wong,Viktoria Robertson,Willemijn Elkhuizen,James Hunt,Gui-Ying Yang,Tanumoy Mandal,Wojciech Kosior,Luis Pinheiro,Edward Abbott-Halpin,KT Vaughan,fakher frikha,Lu Wang,Chia-Lin Li,Katarzyna Tworek,Veerle Van den Eynden,Walter Mauricio Villa Acevedo,ofelia vargas,Matteo Martinelli,Bo Christiansen,Gustavo Lobos,Zhikun Zheng,Takahiro Tsutsumi,Firat Tigrek,Agustín Wallace Ruiz,Xiaoming Sun,JIAN LI,Andrew Heidinger,Maria Guilhermina Martins Moutinho,Ambresh Shivaji,Marek Kočiško,Dori Cabrera,Mikhail Ivanov,David Giles,Mila Gasco-Hernandez,Erin L Cortus,Víctor-M Giménez-Bertomeu,Ruth Carmichael,FEVZİ ÖZGÖKÇE,Nuno da Silva,Alla Generalova,DEBABRATA GHOSH DASTIDAR,Aitana Rodríguez-Pérez,John Okeniyi,João Paulo Silva,Lisa Rubin,Pasquale Massimo Pinto,Ann Viera,takuma akimoto,Xinguo Hong,Jason Masuda,Martin Sparre,Nickolay Gapanovich-Kajdalov,Galina Bolshakova,martin lindhardt,Katrin Huber,Christoph Mandl,Alexander V. Melnikov,Bruce Ebanks,Timothy Hall,Francesc Vilardell,Jacob Østergaard Madsen,Alexander Andrianov,Andrew Bell,Marcus Lehnert,Sjouke Mauw,Erez Lapid,Konstantin Pfrang,Pedro Mogorrón Huerta,María Teresa Ponte Iglesias,Romina Cavatassi,Albu Camelia,Ludmila Kučerová,Claude Duee Zoghbi,Miguel-Ángel Franco-García,Ingrid Fonseca,Peter Godolphin,Bradley Thuro,Carsten Brühl,Ludek Berec,José Rubens Pirani,César Augusto Henao Botero,Jaroslaw Milewski,Lotte Huniche,Jun Xiang,Cathrine McIntyre,Gyu-bong Cho,Ronit Pinkas-Kramarski,Witold W. Kilarski,Fernando Arosa,Pablo Díaz Rodríguez,Satya Prakash,Ivo Viscor,David Weir,Abubakr Idris,Katerina Kubalcikova,Adrian Matthews,Lorenzo Magnea,Graham Moore,Paula Ferreira,Gerard Watts,Ana Cláudia Conti,Diego Luis Durango Restrepo,Sergio Lario,Sherif Abdelwahed,Nikola Paunkovic,Satoshi Ono,Elizabeth Aitken,Kirsten Drotner,Miguel Angel Perez Angon,Brian Walker,Fabiana Zanol Araujo,Jacek Łazeczko,Zachary Kallenborn,Geoffrey Lloyd,Erin Jessee,Юрий Прокопенко,Paulus Kruger,Sveva Grande,Michael Glickman,MARGARITA CANTERA MONTENEGRO,Mark Holzer,Oscar Jaime Restrepo-Baena,Maria Ahlholm,Grzegorz Gmiterek,Boris Polyakov,Lech Borowiec,Anne Schukraft,Subrata Borgohain Gogoi,Karen Gram-Skjoldager,Denys Malyshev,Nishanthini Jegatheesan,Massimiliano Demata,Timur Fatkhudinov,Craig Bishop,Luke Middelburg,K. M. Pandey,Xavier Bonnet,Fátima Lopes Alves,Maria Larrondo Petrie,Michela Costa,Дмитрий / Dmitro Савельев / Saveliev,Agata Walczak-Niewiadomska,Filip Plesinger,Dieter Brillert,Claudio Blanco,Syed Arshad Hussain,Daniel G. Blackburn,Therese Poland,Fernando Vaz-de-Mello,Ecaterina Balica,Piyushkumar Dholariya,Trenton Garner,Brendan McCane,Phillip Gibson,Sissades Tongsima,Lorenz Hilty,Graeme Earl,MTeresa Caceres-Lorenzo,Pikka Jokelainen,Stephan Borrmann,Yun Hsiao,Matthew Wheeler,Craig Storey,Haibing Zhou,Sherifa Hamed,Mariam Bouhmadi Lopez,Amber Griffioen,Peter Hoonakker,Hans Josef Dohmen,María Luisa Cerdeño,Ursula S. Hofstoetter,Lucas Alves,Tomasz Jeż,Dirk Heinen,David Heinrichs,Jayson Semmens,FRANCISCO ALCARAZ,Gerardo Salazar,Paul Edelman,Shu-Xu Yi,Elena Poddubskaya,DİLEK URAL,Dylan Fraser,Damiano Zanotto,Simone Blomberg,Hsi-Te Shih,Satoshi Nakahira,Bernd Kärcher,Thomas Bregnballe,María Dolores Cervilla Garzón,Marek Andrzejewski,Mª Elena Sánchez Trigo,Gregory Jackson,Yamama Naciri,Mariyana Schoultz,Kevin Chih-Yang Huang,Anna Mirovska,GABRIELE LORIS BECCARO,Anis Saad,Ovidiu Florin Caltun,Kuan-Man Xu,Michael Kurgansky,Robert Arnold,Júnia Bicalho Duarte Rosa,Martin ,Lars Mohrmann,Steven Mackinson,Marc St-Arnaud,Sam Brown,Konstantinos Dimopoulos,Reina Maruyama,Alexander Patselov,Carlos José Hernando Sánchez,xuechuan hong,Oleh Koshovyi,Justin Kirby,Ricardo Biondi,Stuart Rosen,Rickard Ström,William Hunter,Jenni Romaniuk,Sherry Cox,Bo Kristian Holm,Renata Melo,Paul Field,Maria Helena Santos,Andrea Trombettoni,Andrey Svalov,Yusup Subagio Sutanto,Matthew Michalska-Smith,Amanda Franklin,Alessandro Ridolfi,Anna Pollmann,marie harder,Maria L Lozano,Bassam Gamal Nasser Muthanna,Carlos Tavares da Silva,Dr. Muhammad Irfan,Fabian Jankowski,Lane Baker,Dr. Abdul Qahar Sarwari,Paul McNamara,Angelo Carfì,Jonathan Miller,Sushanta Banerjee,Masayuki Shimojo,Masahiro Kino-oka,Steven Stovitz,Ori Friedman,Jason Kennington,Francien Dechesne,Vassil Skumryev,Frauke von Versen-Höynck,Chang-Chun Hsiao,Klaus Schröer,Pablo Álvarez Pérez,Igor Florinsky
+988 authors
,Petya KaraivanovaJournal
Published
Oct 6, 2016
Show more
Save
Tip0
TipSave
arXiv:1610.02052v3 [hep-th] 12 Jan 2017
YITP-SB-16-36
Chiral Closed strings: Four massless states scattering amplitude
Marcelo M. Leite∗ and Warren Siegel†
C N Yang Institute for Theoretical Physics,
State University of New York, Stony Brook, NY 11794-3840
Abstract
We compute the scattering amplitudes of four massless states for chiral (closed) bosonic and
type II superstrings using the Kawai-Lewellen-Tye (KLT ) factorization method. The amplitude in
the chiral bosonic case is identical to a field theory amplitude corresponding to the spin-2 tachyon,
massless gravitational sector and massive spin-2 tardyon states of the spectrum. Chiral type II
superstrings amplitude only possess poles associated with the massless gravitational sector. We
briefly discuss the extension of the calculation to heterotic superstrings.
Keywords: Conformal Field Models in String Theory, Scattering Amplitudes, String Duality, Gauge-gravity
correspondence
∗ Permanent Address: Laborat´orio de F´ısica Te´orica e Computacional, Departamento de F´ısica, Universi-
dade Federal de Pernambuco, 50670-901, Recife, PE, Brazil (mleite@df.ufpe.br)
† siegel@insti.physics.sunysb.edu
1
YITP-SB-16-36
Chiral Closed strings: Four massless states scattering amplitude
Marcelo M. Leite∗ and Warren Siegel†
C N Yang Institute for Theoretical Physics,
State University of New York, Stony Brook, NY 11794-3840
Abstract
We compute the scattering amplitudes of four massless states for chiral (closed) bosonic and
type II superstrings using the Kawai-Lewellen-Tye (KLT ) factorization method. The amplitude in
the chiral bosonic case is identical to a field theory amplitude corresponding to the spin-2 tachyon,
massless gravitational sector and massive spin-2 tardyon states of the spectrum. Chiral type II
superstrings amplitude only possess poles associated with the massless gravitational sector. We
briefly discuss the extension of the calculation to heterotic superstrings.
Keywords: Conformal Field Models in String Theory, Scattering Amplitudes, String Duality, Gauge-gravity
correspondence
∗ Permanent Address: Laborat´orio de F´ısica Te´orica e Computacional, Departamento de F´ısica, Universi-
dade Federal de Pernambuco, 50670-901, Recife, PE, Brazil (mleite@df.ufpe.br)
† siegel@insti.physics.sunysb.edu
1
I. INTRODUCTION
The connection among field-theoretic scattering amplitudes and string theory amplitudes
proposed in a series of papers by Cachazo, He and Yuan (CHY ) [1–4] shed further light in
the mathematical properties of scattering amplitudes. Within this setting, supersymmetric
gauge theories, gravity and also field theory with arbitrary spin in arbitrary dimensions
might be better understood in principle, since the final form of the amplitude with just
few simple terms eliminate the necessity of dealing with a lot of cancelations involved in
intermediate steps.
In the string-theoretic context, the analogue of the approach proposed by CHY includes
the scattering equations [5] as constraints implemented inside the amplitude through the
insertion of Dirac delta functions in the vertex operators (see also e. g. [6]). A second alter-
native to stringy scattering amplitudes proposes a new gauge condition on the worldsheet
which effectively yields the scattering equations after integration over the worldsheet vari-
able ¯z, without the use of delta functions [7]. Recently, a combination of the two methods
was proposed. The modification of the boundary conditions for the worldsheet fields along
with the choice of conformal gauge originated the “chiral” string in its different versions:
bosonic, type II and heterotic strings [8].
It is noteworthy to point out that the new chiral strings described in the present work dif-
fer from other chiral strings already discussed in the literature. It is important to emphasize
their difference and similarities. Whereas the usual standard string theories have towers of
massive states with higher angular momentum/spin, a curious aspect of chiral string theories
is the finite number of states in the spectrum. Inspired in the work involving tensionless
strings [9], the first analysis which indicated only two massive states with the same absolute
value but with different signs for the masses in the spectrum of the bosonic string (as well
as three massless states) was carried out in [10] in the context of BRST approach, but the
model was discarded due to inconsistencies: those states have negative norm and decouple
of physical scattering amplitudes.
More recently, Ambitwistors chiral strings were developed [11]. In this chiral theory
only massless states survive in the spectrum due to the absence of normal ordering. This
happens because the OP E < XX > is trivial such that important operators like eik.X do not
acquire anomalous conformal weight, therefore not being able to compensate the anomalous
2
The connection among field-theoretic scattering amplitudes and string theory amplitudes
proposed in a series of papers by Cachazo, He and Yuan (CHY ) [1–4] shed further light in
the mathematical properties of scattering amplitudes. Within this setting, supersymmetric
gauge theories, gravity and also field theory with arbitrary spin in arbitrary dimensions
might be better understood in principle, since the final form of the amplitude with just
few simple terms eliminate the necessity of dealing with a lot of cancelations involved in
intermediate steps.
In the string-theoretic context, the analogue of the approach proposed by CHY includes
the scattering equations [5] as constraints implemented inside the amplitude through the
insertion of Dirac delta functions in the vertex operators (see also e. g. [6]). A second alter-
native to stringy scattering amplitudes proposes a new gauge condition on the worldsheet
which effectively yields the scattering equations after integration over the worldsheet vari-
able ¯z, without the use of delta functions [7]. Recently, a combination of the two methods
was proposed. The modification of the boundary conditions for the worldsheet fields along
with the choice of conformal gauge originated the “chiral” string in its different versions:
bosonic, type II and heterotic strings [8].
It is noteworthy to point out that the new chiral strings described in the present work dif-
fer from other chiral strings already discussed in the literature. It is important to emphasize
their difference and similarities. Whereas the usual standard string theories have towers of
massive states with higher angular momentum/spin, a curious aspect of chiral string theories
is the finite number of states in the spectrum. Inspired in the work involving tensionless
strings [9], the first analysis which indicated only two massive states with the same absolute
value but with different signs for the masses in the spectrum of the bosonic string (as well
as three massless states) was carried out in [10] in the context of BRST approach, but the
model was discarded due to inconsistencies: those states have negative norm and decouple
of physical scattering amplitudes.
More recently, Ambitwistors chiral strings were developed [11]. In this chiral theory
only massless states survive in the spectrum due to the absence of normal ordering. This
happens because the OP E < XX > is trivial such that important operators like eik.X do not
acquire anomalous conformal weight, therefore not being able to compensate the anomalous
2
conformal weight of a generic polynomial, say, in ∂Xµ for the values of k2 on the mass-
shell. In addition, all the ghosts and fields of the various Ambitwistor strings are purely
holomorphic. An interesting connection among the Ambitwistor strings with tensionless and
tensionful strings was assessed very recently in Ref. [12]. From now on we switch to the
chiral string theory that is going to be described below with ghosts and fields possessing
holomorphic and antiholomorphic components and with nontrivial OPES; see below.
As mentioned above, one of the amazing features of these new closed string theories is the
finite number of states in their spectrum. The chiral bosonic string possesses only three mass
levels: a spin-2 tachyon, massless (“gravitational”) sector and a massive spin-2 “tardyon”.
The chiral type II superstring only contains the massless states, whereas the chiral heterotic
superstring can have either tachyon or tardyon (never both) beyond the massless sector.
In the present work we compute the scattering amplitude of four massless chiral string
states for the bosonic and type II superstrings using the Kawai-Lewellen-Tye [13] factor-
ization of closed string amplitudes as the product of open string amplitudes. Remarkably,
their form indicates that they can be expressed entirely as amplitudes obtained using field
theory techniques. Furthermore, we use the factorization property to prove that they can
be obtained as the product of two three-point amplitudes involving two external massless
and one intermediate arbitrary state of the corresponding string spectrum. Using common
aspects of bosonic, type II and heterotic strings, we construct the scattering amplitude of
four massless states of the heterotic superstring at the massless gravitational sector.
II. REVIEW OF VERTEX OPERATORS AND COMPUTATION OF 3-POINT
SCATTERING AMPLITUDES
A. Closed Chiral Bosonic String
The chiral bosonic string has only the matter fields Xµ(z, ¯z) and conformal ghosts
(b(z), c(z), ˜b(¯z), ˜c(¯z)). The operator product expansion of Xµ is modified (due to a Bo-
goliubov transformation in the oscillators) when compared to the standard bosonic string
to
< Xµ(zi, ¯zi)Xν (zj , ¯zj ) >= α′
2 ηµν ln
( ¯zij
zij
)
, (1)
3
shell. In addition, all the ghosts and fields of the various Ambitwistor strings are purely
holomorphic. An interesting connection among the Ambitwistor strings with tensionless and
tensionful strings was assessed very recently in Ref. [12]. From now on we switch to the
chiral string theory that is going to be described below with ghosts and fields possessing
holomorphic and antiholomorphic components and with nontrivial OPES; see below.
As mentioned above, one of the amazing features of these new closed string theories is the
finite number of states in their spectrum. The chiral bosonic string possesses only three mass
levels: a spin-2 tachyon, massless (“gravitational”) sector and a massive spin-2 “tardyon”.
The chiral type II superstring only contains the massless states, whereas the chiral heterotic
superstring can have either tachyon or tardyon (never both) beyond the massless sector.
In the present work we compute the scattering amplitude of four massless chiral string
states for the bosonic and type II superstrings using the Kawai-Lewellen-Tye [13] factor-
ization of closed string amplitudes as the product of open string amplitudes. Remarkably,
their form indicates that they can be expressed entirely as amplitudes obtained using field
theory techniques. Furthermore, we use the factorization property to prove that they can
be obtained as the product of two three-point amplitudes involving two external massless
and one intermediate arbitrary state of the corresponding string spectrum. Using common
aspects of bosonic, type II and heterotic strings, we construct the scattering amplitude of
four massless states of the heterotic superstring at the massless gravitational sector.
II. REVIEW OF VERTEX OPERATORS AND COMPUTATION OF 3-POINT
SCATTERING AMPLITUDES
A. Closed Chiral Bosonic String
The chiral bosonic string has only the matter fields Xµ(z, ¯z) and conformal ghosts
(b(z), c(z), ˜b(¯z), ˜c(¯z)). The operator product expansion of Xµ is modified (due to a Bo-
goliubov transformation in the oscillators) when compared to the standard bosonic string
to
< Xµ(zi, ¯zi)Xν (zj , ¯zj ) >= α′
2 ηµν ln
( ¯zij
zij
)
, (1)
3
where zij = zi − zj . (The OPE for the ghosts also presents a factor of (−1) in the right
mover sector, namely < ˜b(¯zi), ˜c(¯zj ) >= − 1
¯zij ). The modification in comparison with the
usual bosonic string occurs only in the right-mover sector characterized by the coordinates
¯zi. This can be interpreted as a change of sign (“flipping”) in the metric for all right-mover
fields. For instance, the operator eik.X(z,¯z) for the chiral string has conformal dimension
(α′k2
4 , −α′k2
4 ). The spectrum has only 3 mass levels: a spin-2 tachyon (M2 = − 4
α′ ), a massless
state (composed by a graviton, a massless two-form field and a massless scalar dilaton) and
a spin-2 tardyon (M2 = 4
α′ ).
The spectrum can be classified in terms of vertex operators as follows.
1. Tachyonic sector (M2 = − 4
α′ ) whose vertex operator is given by
V− = ¯ǫµν ¯∂Xµ ¯∂Xν eik.X(z,¯z). (2)
2. Massless gravitational sector (includes graviton, dilaton and antisymetric tensor), with
V0 = eµν ∂Xµ ¯∂Xν eik.X(z,¯z), (3)
eµν = eµ ˜eν . (4)
3. Tardyonic sector (M2 = 4
α′ ) vertex operator reads
V+ = ǫµν ∂Xµ∂Xν eik.X(z,¯z). (5)
Observe that the polarization tensor of the gravitational sector was decomposed into
the direct product of two vector polarizations. This will help to present the scattering
amplitudes in their factorized form. The location of three vertex operators in a generic tree-
level amplitude have to be specified in the scattering of, say, N states. For bosonic strings
this specification consists in attaching the ghost factor c(z)˜c(¯z) in the exact position of the
vertex operator. For N > 3, in addition to the above insertions, the remaining (N − 3)
coordinates are integrated over (z, ¯z).
Before proceeding in the evaluation of scattering amplitudes, we anticipate that the tachy-
onic and tardyonic states have ghost-like nature. We shall unveil this feature later, when
performing the comparison using the KLT computation of the four massless external states
with the product of two three-point scattering amplitudes involving two external massless
states (gravitational sector) and one internal tachyon or tardyon. The agreement among
these results only occurs exactly if the propagator for the tachyon or tardyon includes a
4
mover sector, namely < ˜b(¯zi), ˜c(¯zj ) >= − 1
¯zij ). The modification in comparison with the
usual bosonic string occurs only in the right-mover sector characterized by the coordinates
¯zi. This can be interpreted as a change of sign (“flipping”) in the metric for all right-mover
fields. For instance, the operator eik.X(z,¯z) for the chiral string has conformal dimension
(α′k2
4 , −α′k2
4 ). The spectrum has only 3 mass levels: a spin-2 tachyon (M2 = − 4
α′ ), a massless
state (composed by a graviton, a massless two-form field and a massless scalar dilaton) and
a spin-2 tardyon (M2 = 4
α′ ).
The spectrum can be classified in terms of vertex operators as follows.
1. Tachyonic sector (M2 = − 4
α′ ) whose vertex operator is given by
V− = ¯ǫµν ¯∂Xµ ¯∂Xν eik.X(z,¯z). (2)
2. Massless gravitational sector (includes graviton, dilaton and antisymetric tensor), with
V0 = eµν ∂Xµ ¯∂Xν eik.X(z,¯z), (3)
eµν = eµ ˜eν . (4)
3. Tardyonic sector (M2 = 4
α′ ) vertex operator reads
V+ = ǫµν ∂Xµ∂Xν eik.X(z,¯z). (5)
Observe that the polarization tensor of the gravitational sector was decomposed into
the direct product of two vector polarizations. This will help to present the scattering
amplitudes in their factorized form. The location of three vertex operators in a generic tree-
level amplitude have to be specified in the scattering of, say, N states. For bosonic strings
this specification consists in attaching the ghost factor c(z)˜c(¯z) in the exact position of the
vertex operator. For N > 3, in addition to the above insertions, the remaining (N − 3)
coordinates are integrated over (z, ¯z).
Before proceeding in the evaluation of scattering amplitudes, we anticipate that the tachy-
onic and tardyonic states have ghost-like nature. We shall unveil this feature later, when
performing the comparison using the KLT computation of the four massless external states
with the product of two three-point scattering amplitudes involving two external massless
states (gravitational sector) and one internal tachyon or tardyon. The agreement among
these results only occurs exactly if the propagator for the tachyon or tardyon includes a
4
100%
