1. GASYUKU2002,Kyoto-U ＠ KAGA 1 Computing Feynman Graphs in MSFT Isao Kishimoto (Univ. of Tokyo) based on Collaboration with I.Bars and Y.Matsuo [ hep-th/0211131]

2. GASYUKU2002,Kyoto-U ＠ KAGA 2 Introduction to MSFT Isao Kishimoto (Univ. of Tokyo) Reference: Bars-Matsuo, Phys.Rev. D66 (2002) 066003[hep-th/0204260]

3. GASYUKU2002,Kyoto-U ＠ KAGA 3 Introduction and motivation  Non-commutative field theory boom (1999) (Kawano-Takahashi)  Sen’s conjecture Witten’s SFT(1986) was revival! (Hata-Shinohara,Hata-Teraguchi)  VSFT conjecture (RSZ(2000),GRSZ(2001)) (Kawano-Okuyama,Hata-Kawano,Imamura,Hata-Moriyama,Hata-Kogetsu)

4. GASYUKU2002,Kyoto-U ＠ KAGA 4  Oscillator approach (Gross-Jevicki(1986),…) Formal computations are performed by using algebraic relations among infinite matrices. (Kishimoto) However, we encountered `contradiction’: Twist anomaly! (Hata-Moriyama,Hata-Moriyama-Teraguchi) Neumann matrices are ∞×∞.

5. GASYUKU2002,Kyoto-U ＠ KAGA 5  CFT approach (…,LPP(1989),…) (Takahashi-Tanimoto,Kishimoto-Ohmori,Kishimoto-Takahashi) However, we should take some regularization to treat `Identity state’ I appropriately.

6. GASYUKU2002,Kyoto-U ＠ KAGA 6  Moyal approach (Bars(2001),Bars-Matsuo(2002)) Witten’s ＊  Moyal ★ and regularization fixed!

7. GASYUKU2002,Kyoto-U ＠ KAGA 7 Contents  Introduction and motivation  Half-string formulation  MSFT  Computing Feynman Graphs with Monoid  Summary and Discussion  nazo

8. GASYUKU2002,Kyoto-U ＠ KAGA 8 Half-string formulation  Witten’s ＊ product ～ Matrix product (RSZ, Gross-Taylor, Kawano-Okuyama) Half-string formulation

9. GASYUKU2002,Kyoto-U ＠ KAGA 9  Mode expansion : full string : half string

10. GASYUKU2002,Kyoto-U ＠ KAGA 10  Moyal formulation By Fourier transformation, the above product can be rewritten using Moyal ★ product (I.Bars) :

11. GASYUKU2002,Kyoto-U ＠ KAGA 11  More precisely,… Note: ⇒ String field in MSFT: and Witten’s ＊ :

12. GASYUKU2002,Kyoto-U ＠ KAGA 12 where These matrices and vectors satisfy

13. GASYUKU2002,Kyoto-U ＠ KAGA 13  However, there is a subtlety: associativity anomaly of ∞×∞ matrices. (Bars-Matsuo) This situation causes ambiguity in computation, for example, We need appropriate regularization!

14. GASYUKU2002,Kyoto-U ＠ KAGA 14 MSFT  Setup: For arbitrary define matrices R,T and vectors w,v : In fact, we can solve them explicitly:

15. GASYUKU2002,Kyoto-U ＠ KAGA 15  Some relations: Modified! ※ In the case of these quantities reproduce original ones of Witten’s SFT. We should take this limit at the last stage of computation to avoid subtlety of infinite matrices.

16. GASYUKU2002,Kyoto-U ＠ KAGA 16  Oscillators and perturbative vacuum: Using the transformation from conventional string field, we have the following correspondence

17. GASYUKU2002,Kyoto-U ＠ KAGA 17 We can represent perturbative vacuum as a gaussian: where More generally, external states are given by using gaussian:

18. GASYUKU2002,Kyoto-U ＠ KAGA 18  L 0 and the action in the Siegel gauge

19. GASYUKU2002,Kyoto-U ＠ KAGA 19 Computing Feynman Graphs with Monoid  ξ-basis

20. GASYUKU2002,Kyoto-U ＠ KAGA 20  Example We can evaluate such quantities using formula:

21. GASYUKU2002,Kyoto-U ＠ KAGA 21  Momentum (Fourier transformed) basis: Vertex Simpler than conventional vertex using Neumann coefficients. The same form as ordinary noncommutative field theory. External state From gaussian to gaussian:

22. GASYUKU2002,Kyoto-U ＠ KAGA 22 Propagator Complicated compared to ordinary noncommutative field theory.

23. GASYUKU2002,Kyoto-U ＠ KAGA 23  1-loop vacuum amplitude Correct spectrum! Note: If we take naïve limit in L 0 first, we have wrong result because

24. GASYUKU2002,Kyoto-U ＠ KAGA 24  4-tachyon By including ghost sector contribution and integrating with respect to τ, we should reproduce Veneziano amplitude in the case of

25. GASYUKU2002,Kyoto-U ＠ KAGA 25 Summary and Discussion  We developed the method to compute Feynman diagrams in MSFT: (a) Monoid algebra in noncommutativeξ-space (b) non-diagonal propagator + phase in momentum-space All computations are gaussian integration and can be applied to any frequencies and finite N : Well-defined!

26. GASYUKU2002,Kyoto-U ＠ KAGA 26 Gauge symmetry ? Definition of BRST charge? Non-perturbative vacuum? To get more explicit formulas such as Veneziano amplitude, we should simplify deteminant and inverse matrices which are written in terms of

27. GASYUKU2002,Kyoto-U ＠ KAGA 27  Equation of motion Note: If we ignore γ-term, we can solve the e.o.m. using a projector P which commutes with L 0 : A=-(2L 0 -1) ★ P. But this is not the true vacuum because γ-term is essential to get the correct perturbative spectrum.

28. GASYUKU2002,Kyoto-U ＠ KAGA 28 GASYUKU2002 Theoretical Particle Physics Group, Department of Physics,Kyoto-U  KAGA onsen