Analysis o ｆ tree level 5-point amplitudes in open superstring field theory

1. Analysis oｆ tree level 5-point amplitudes in open superstring field theory Yoji Michishita March 12, 2007 KEK workshop

2. Contents • Introduction to the superstring field theory • A problem in gauge fixing procedure • Computation of 5-point amplitudes • 5 bosons • 4 fermions and 1 boson • 2 fermions and 3 bosons (order FFBBB) • 2 fermions and 3 bosons (order FBFBB)

3. Berkovits’ open superstring field theory • based on NSR formalism • Star products: the same ones as Witten’s • String fields have ghost numbers and picture numbers • state space: large Hilbert space • 2 gauge symmetries • action: and insertions in interaction terms • infinitely many higher order terms

4. Gauge fixing in Witten’s bosonic string field theory To compute amplitudes, we have to fix the gauge symmetry. gauge fixing condition: Siegel gauge (Other conditions: Asano-Kato (2006), Schnabl gauge (2005), modified Schnabl gauge (Nakayama-Fuji-Suzuki 2006) ) Ordinary Faddeev-Popov procedure does not work: Introduction of a ghost-antighost pair does not fix the gauge completely.) ghost of ghost, ghost of ghost of ghost……

5. “fields” “antifields” Application of Batalin-Vilkovisky (BV) formalism original field spectrum: as in the 1st quantized formulation Physics must be independent of gauge fixing condition. )(classical) master equation result action: original one with ghost number restriction removed

6. Gauge fixing in Berkovits’ superstring field theory gauge fixing condition: Analogy to the bosonic case: original field picture number fixed )1st quantized spectrum reproduced action: original one with ghost number restriction removed? “antifileds” “fields”

7. Then… cubic terms reproduce 3-point on-shell 1st quantized amplitudes. seems quite natural, and no other choice. However, cubic contribution to the classical master equation ) BV formalism does not work ! different way of applying BV formalism? or other way of gauge fixing? At present we have no justification of the “gauge fixed” system.

8. But…. People do not care: Calculation of 4-point amplitudes Berkovits-Echevarria(1999), Berkovits-Schnabl(2003), Y.M. (2004), Fuji-Nakayama-Suzuki(2006) )1st quantized on-shell amplitudes reproduced ! It seems that we are on the right track. Further check: calculation of 5-point amplitudes

9. Calculation of on-shell amplitudes: Generalities • Bosonic string Expression expected to be reproduced Propagator = : inserting a strip with the length :Beltrami differential insertion

10. UHP

11. Theorem: Schwinger parameters for Riemann surfaces made of the propagators and Witten type 3-point vertices form a single cover of the moduli space (Zwiebach) )1st quantized on-shell amplitudes reproduced • Superstring case Expression expected to be reproduced: • Picture numbers adjusted by • superfluous removed by Propagator: Infinitely many interaction vertices with and insertions

12. and should be relocated to adjust picture numbers. Those may hit and remove Number of propagators reduced )correspond to zero length propagators )should be canceled by diagrams with higher order vertices. This is the reason why the superstring field theory has higher order vertices.

13. 5-point amplitude: 5 bosons • 1st quantized amplitude Let us see if our “gauge fixed” system reproduce this.

14. 1 5 2 1 3 2 4 3 5 4 2 4 3 5 4 1 5 2 1 3 3 4 5 1 2 1 5 2 1 3 2 4 3 2 3 2 3 4 5 1 3 4 4 5 5 1 1 2 5 4 1 5 2 4 3 • Diagrams with 2 propagators • Diagrams with 1 propagator • Diagrams with no propagator

15. 1 5 2 4 3 We expect that • Each gives one of terms in and extra terms with 1 propagator through relocating and . • Those extra terms and are combined into terms with no propagator. • Those terms cancel Let us compute first.

16. expected term terms with 1 propagator

17. 1 5 2 3 4 • Other : cyclic permutations in should sum up to terms with no propagator.

18. The sum is

19. Then 1st quantized amplitude reproduced !

20. How to describe R-sector • Classical description (Y.M. 2004) R-sector string field + additional field Action + constraint • Naïve inference of Feynman Rules • Ghost number restriction removed • Propagator • External on-shell replaced by • Interaction vertices: higher terms in the action

21. We have to impose the constraint on the interaction vertices. Here we naively impose the linearized constraint i.e. and is replaced by Then… Correctness of cubic and quartic vertices has been confirmed by calculating 4-point amplitudes. 5-point vertices correct?

22. 5-point amplitude: 4 fermions and 1 boson • 1st quantized amplitude

23. 1 5 2 1 3 2 5 4 2 4 3 5 4 1 1 3 3 4 5 2 4 3 • Diagrams with 2 propagators • Diagrams with 1 propagator None • Diagrams with no propagator None Used vertices: 5 2 1

24. 1 5 2 4 3 • Expectation • Each diagram gives one of terms in the 1st quantized amplitude and extra terms with 1 propagator. • Those extra terms cancel each other. expected term Terms with 1 propagator

25. As is expected, terms with 1 propagator cancel, and remaining terms reproduce the 1st quantized amplitude ! Therefore seems correct.

26. 5-point amplitude: 2 fermions and 3 bosonsin the order FFBBB • 1st quantized amplitude

27. 1 5 2 1 3 2 4 3 5 4 2 4 3 5 4 1 5 2 1 3 3 4 5 1 2 1 5 4 2 3 1 5 2 4 3 • Diagrams with 2 propagators • Diagram with 1 propagator • Diagram with no propagator

28. Expectation • Each gives one of terms in and extra terms with 1 propagator through relocating and . • Those extra terms and are combined into terms with no propagator. • Those terms cancel

29. 1 5 2 4 3 expected term Terms with 1 propagator

30. 1 5 4 2 3 Terms with 1 propagator in should sum up to terms with no propagator. The sum is as expected. However, Should be modified to ???

31. 5-point amplitude: 2 fermions and 3 bosonsin the order FBFBB • 1st quantized amplitude

32. 1 5 2 1 3 2 4 3 5 4 2 4 3 5 4 1 5 2 1 3 3 4 5 1 2 1 5 2 4 3 • Diagrams with 2 propagators • Diagrams with 1 propagator None • Diagrams with no propagator

33. Expectation • Each gives one of terms in the 1st quantized amplitude and extra terms with 1 propagator through relocating and . • Those extra terms sum up to terms with no propagator. • Those cancel .

34. 1 5 2 4 3 expected term Terms with 1 propagator

35. Terms with 1 propagator in should sum up to terms with no propagator. The sum is as expected. However, Should be modified to ???

36. Summary • Naïve gauge fixing procedure using the unfixed action with ghost number restriction removed, does not fit BV formalism. • Nevertheless it reproduces the 1st quantized amplitude with 5 bosons through many nontrivial cancellations. • For 5-point amplitudes with fermions, we have found some discrepancies, and those are remedied by changing coefficients of 5-point vertices.

37. Discussion • Do we have to consider another way of applying BV formalism, or have to invent new gauge fixing procedure to justify our system? How can we define observables? • Should on-shell amplitudes coincide with those of the 1st quantization at all? • Discrepancies in amplitudes with fermions: Does it indicate failure of our gauge fixing procedure, or subtlety of imposing constraints to the vertices? • Loop contributions: quantum corrected vertices needed to satisfy quantum master equation?