Symmetries and Feynman Rules for Ramond Sector in WZW-type Superstring Field Theories

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1 Symmetries and Feynman Rules for Ramond Sector in WZW-type Superstring Field Theories Hiroshi Kunitomo YITP Women s U. Nara arxiv: , 153.xxxx Y TP YUKAWA INSTITUTE FOR THEORETICAL PHYSICS 1

2 1. Introduction WZW-type Superstring field theory A formulation with no explicit picture changing operator and working well for NS sector: Open [Berkovits 1995] / Hetetrotic [Okawa and Zwiebach 24, Berkovits, Okawa and Zwiebach 24] Difficult to construct an action for R sector EOM [Berkovits 21] [HK 214] A pseudo-action [Michishita 25] [HK 214] We can construct the self-dual Feynman rules reproducing the on-shell four point amplitudes. However, these rules do not reproduce the five-point amplitudes [Michishita 27], or have an ambiguity [HK 213]. Q: Can we construct consistent Feynman rules? 2

3 Plan of the Talk 1. Introduction 2. New Feynman rules in the WZW-type open SSFT 2.1 EOM and the pseudo-action 2.2 Gauge fixing and the self-dual Feynman rules 2.3 Gauge symmetries and the new Feynman rules 3. Amplitudes with the external fermions 4. Summary and discussion 3

4 2. New Feynman rules in the WZW-type Open SSFT 2.1 EOM [Berkovits] and the pseudo-action [Michishita] String fields : Grassmann even, NS Φ H NS large G, P =, and, 1/2, respectively. NS action : S NS = 1 2 e Φ Qe Φ e Φ ηe Φ 1 and R Ψ HR large with dte tφ t e tφ {e tφ Qe tφ, e tφ ηe tφ }. 1 EOM including the R sector : ηe Φ Qe Φ + ηψ 2 =, Q ηψ =, 2 where Q A = QA + Ae Φ Qe Φ 1 A e Φ Qe Φ A. Gauge symmetry : e Φ δe Φ = Q Λ + ηλ 1 ηψλ 12 Λ 12 ηψ, δψ = Q Λ 12 + ηλ 32 + ΨηΛ 1 ηλ 1 Ψ. 3a 3b 4

5 Auxiliary field : Grassmann even, Ξ H R large with G, P =, 1/2. The pseudo-action : The variation of S = S NS + S R yields S R = 1 2 QΞeΦ ηψe Φ. 4 ηe Φ Qe Φ {ηψ, Q Ξ } =, Q ηψ =, ηq Ξ =. 5 where Ξ = e Φ Ξe Φ. If we eliminate Ξ by imposing a constraint Q Ξ = ηψ, 5 become EOM 2. Since this is not the true action we cannot derive the Feynman rules. 5

6 2.2 Gauge fixing and the self-dual Feynman rules Quadratic part of the action has gauge symmetries S = 1 2 QΦηΦ 1 QΞηΨ, 6 2 δφ = QΛ + ηλ 1, δψ = QΛ 12 + ηλ 32, δξ = QΛ η Λ Fix them by the simplest gauge conditions b Φ = ξ Φ =, b Ψ = ξ Ψ =, b Ξ = ξ Ξ =. 8 The propagators become ΦΦ Π = ξ b L = dτξ b e τl, ΨΞ = ΞΨ = 2 ξ b L = 2Π. 9 In order to take into account the constraint, we replace ηψ and QΞ by their linearized self-dual part ω = ηψ + QΞ/2 in vertices: the self-dual rules. 6

7 Propagator : NS : ΦΦ = ξ b L, R : ωω = 1 2 Q ξ b ξ b η + η L L Q diagonal Vertices :... on-shell external fermions : ω Restrict them by the linearized constraint QΞ = ηψ, Then ω = ηψ 1 which automatically implies the on-shell condition QηΨ =. 7

8 The self-dual Feynman rules give the well-known on-shell 4-point amplitudes but cannot reproduce the 5-point amplitudes. [Michishita] 2.3 Gauge symmetries and the new Feynman rules Gauge symmetries : The pseudo- action S = S NS + S R is invariant under e Φ δe Φ = Q Λ + ηλ 1, 11a δψ = ηλ 32 + [Ψ, ηλ 1 ], δξ = QΛ [Q, Λ ]. 11b These are compatible with the self-dual/anti-self-dual decomposition. δq Ξ ± ηψ = [Q Ξ ± ηψ, Λ 1 ], 12 and so respected by the self-dual rules. However, these symmetries are not enough to gauge away all the un-physical states. They do not contain all the symmetries of the quadratic action 7. 8

9 The missing symmetries generated by Λ1 2 and Λ1 2 can be extended to e Φ δe Φ = 1 2 {Q Ξ, Λ 12 } {ηψ, Λ 12 }, 13 δψ = Q Λ 12, δξ = e Φ η Λ 12 e Φ. 14 The variation of the action under these transformations becomes δs = 1 4 Λ 1 2 [Q Ξ 2, Q Ξ ηψ] Λ 12 [ηψ 2, Q Ξ ηψ], 15 which vanishes under the constraint. These additional symmetries are not compatible with the self-dual/andti-selfdual decomposition. The self-dual rules do not respect them. Claim : The Feynman rules have to respect these symmetries too. 9

10 The new Feynman rules Propagator : NS : ΦΦ = ξ b L, R : ΨΞ = ΞΨ = 2 ξ b L off-diagonal Vertices :... 1

11 on-shell external fermions : Add two possibilities, Ψ and Ξ. Restrict them by the linearized constraint QΞ = ηψ. 16 We can show that all the on-shell 4- and 5 point amplitudes are correctly reproduced by the new Feynman rules. 11

12 3. Amplitudes with the external fermions Four-point amplitudes are correctly reproduced as in the self-dual rules. Difference between two rules comes from from the diagram with either i at least two fermion propagators, A QΠ R η + ηπ R Q QΠ R η + ηπ R Q W, 17 in the self-dual rules and A QΠ R η QΠ R η W + ηπ R Q ηπ R Q W, 18 in the new rules. Or, ii two-fermion-even-boson interaction at least one of whose fermions, e.g. S 4 R = 1 4 Φ 2 QΞηΨ Φ 2 ηψqξ ΦQΞΦηΨ ΦηΨΦQΞ. 19 These differences improve the discrepancy in the five-point amplitudes! 12

13 3.2 Five-point amplitudes For example, ordering FFBBB Five two-propagator 2P diagrams A E B A C B B D C E D A C a D b E c D C E D E B A C A d B e 13

14 A E B D C a A 2Aa F F BBB = 1 dτ 1 dτ 2 8 QΞ A ηψ B + ηψ A QΞ B ξ c1 b c1 QΦ C ηξ c2 b c2 QΦ D ηφ E + ηφ D QΦ E W + QΞ A ηψ B + ηψ A QΞ B ξ c1 b c1 ηφ C Qξ c2 b c2 QΦ D ηφ E + ηφ D QΦ E W Witten diagram 14

15 Using One can rewrite it as A 2Aa F F BBB = dτ{q, b }e τl = d 2 τ QΞ A ηψ B + ηψ A QΞ B dτ QΞ A ηψ B + ηψ A QΞ B dτ τ e τl, 21 ξ c1 b c1 QΦ C b c2 QΦ D ηφ E W ξ c b c Φ C QΦ D ηφ E + ηφ D QΦ E W 2 QΞ A ηψ B + ηψ A QΞ B ξ c b c QΦ C ηφ D Φ E W QΦ D ηφ E + ηφ D QΦ E ξ c b c QΞ A ηψ B + ηψ A QΞ B Φ C W 2 ηφ D Φ E ξ c b c QΞ A ηψ B + ηψ A QΞ B QΦ C W, 22 15

16 B A C E D b A 2Pb F F BBB = 1 2 dτ 1 dτ 2 QΞ B Φ C ηξ c1 b c1 Q Φ D ηξ c2 b c2 Q Φ E ηψ A W + ηψ B Φ C Qξ c1 b c1 η Φ D Qξ c2 b c2 η Φ E QΞ A W 23 16

17 = 1 2 d 2 τ QΞ B QΦ C ξ c1 b c1 QΦ D b c2 ηφ E ηψ A W + ηψ B QΦ C ξ c1 b c1 QΦ D b c2 ηφ E QΞ A W 1 2 dτ QΞ B Φ C ξ c b c ηφ D QΦ E ηψ A W QΞ B QΦ C ξ c b c ηφ D Φ E ηψ A W QΦ E ηψ A ξ c b c QΞ B ηφ C Φ D W + ηφ E QΞ A ξ c b c ηψ B QΦ C Φ D W + ηφ E ηψ A ξ c b c QΞ B QΦ C Φ D W Φ E ηψ A ξ c b c QΞ B QΦ C ηφ D W

18 A 2Pc F F BBB =1 2 d 2 τ QΦ C QΦ D ξ c1 b c1 ηφ E b c2 QΞ A ηψ B + ηψ A QΞ B W + 1 dτ 2 Φ C QΦ D ξ c b c ηφ E QΞ A ηψ B + ηψ A QΞ B W 8 + QΦ C ηφ D ηφ C QΦ D ξ c b c Φ E QΞ A ηψ B + ηψ A QΞ B W 2 QΦ C ηφ D ξ c b c Φ E QΞ A ηψ B + ηψ A QΞ B W 2 QΦ C Φ D ξ c b c ηφ E QΞ A ηψ B + ηψ A QΞ B W 2 QΞ A ηψ B + ηψ A QΞ B ξ c b c Φ C QΦ D ηφ E W QΞ A ηψ B + ηψ A QΞ B ξ c b c QΦ C ηφ D ηφ C QΦ D Φ E W + 2 QΞ A ηψ B + ηψ A QΞ B ξ c b c QΦ C ηφ D Φ E W, 25 18

19 A 2Pd F F BBB = d 2 τ QΦ D ηφ E ξ c1 b c1 QΞ A b c2 ηψ B + ηψ A b c2 QΞ B dτ QΦ D ηφ E + ηφ D QΦ E ξ c b c ηψ A QΞ B Φ C W + ηφ D Φ E ξ c b c QΞ A ηψ B + ηψ A QΞ B QΦ C W + QΞ B Φ C ξ c b c QΦ D ηφ E + ηφ D QΦ E ηψ A W QΞ B QΦ C ξ c b c ηφ D Φ E ηψ A W QΦ C W ηψ B QΦ C ξ c b c ηφ D Φ E QΞ A W, 26 19

20 A 2Pe F F BBB =1 2 d 2 τ ηφ E QΞ A ξ c1 b c1 ηψ B b c2 QΦ C QΦ D W + ηφ E ηψ A ξ c1 b c1 QΞ B b c2 QΦ C QΦ D W 1 4 dτ Φ E ηψ A ξ c b c QΞ B QΦ C ηφ D + ηφ C QΦ D W ηφ E QΞ A ξ c b c ηψ B QΦ C Φ D Φ C QΦ D W ηφ E ηψ A ξ c b c QΞ B QΦ C Φ D Φ C QΦ D W QΦ C ηφ D + ηφ C QΦ D ξ c b c Φ E ηψ A QΞ B W QΦ C Φ D Φ C QΦ D ξ c b c ηφ E QΞ A ηψ B + ηψ A QΞ B W, 27 2

21 five one-propagator diagrams A E B A C B D E A B a C C b D D c E D C E D B C E d A A e B 21

22 Similarly, A 1Pa F F BBB = 1 24 dτ QΞ A ηψ B + ηψ A QΞ B ξ c b c Φ C QΦ D ηφ E ηφ D QΦ E W 2 QΞ A ηψ B + ηψ A QΞ B ξ c b c QΦ C Φ D ηφ E ηφ C Φ D QΦ E + QΞ A ηψ B + ηψ A QΞ B W ξ c b c QΦ C ηφ D ηφ C QΦ D Φ E W, 28 22

23 A 1Pb F F BBB =1 4 dτ ηψ B ηφ C ξ c b c QΦ D Φ E QΞ A W QΞ B ηφ C ξ c b c QΦ D Φ E ηψ A W ηψ B Φ C ξ c b c QΦ D ηφ E ηφ D QΦ E QΞ A W A 1Pc F F BBB = QΞ A ηψ B Φ C Φ D Φ E W, 29 dτ QΦ C ηφ D + ηφ C QΦ D ξ c b c Φ E QΞ A ηψ B ηψ A QΞ B W, 3 A 1Pd F F BBB =1 8 dτ QΦ D ηφ E + ηφ D QΦ E ξ c b c QΞ A ηψ B ηψ A QΞ B Φ C W, 31 23

24 A 1Pe F F BBB =1 4 dτ Φ E ηψ A ξ c b c QΞ B QΦ C ηφ D ηφ C QΦ D W + ηφ E QΞ A ξ c b c ηψ B ηψ A ξ c b c QΞ B QΦ C Φ D W 1 4 ηψ A QΞ B Φ C Φ D Φ E W. 32 No-propagator diagram : A E 5 B 3 C D A NP F F BBB = 1 12 QΞ A ηψ B + ηψ A QΞ B Φ C Φ D Φ E W

25 A F F BBB = = e i=a A 2Pi F F BBB + d 2 τ e i=a A 1Pi F F BBB + ANP F F BBB QΞ A ηψ B + ηψ A QΞ B ξ c1 b c1 QΦ C b c2 QΦ D ηφ E W + ηψ B QΦ C ξ c1 b c1 QΦ D b c2 ηφ E QΞ A W + QΞ B QΦ C ξ c1 b c1 QΦ D b c2 ηφ E ηψ A W + QΦ C QΦ D ξ c1 b c1 ηφ E b c2 QΞ A ηψ B + ηψ A QΞ B W + QΦ D ηφ E ξ c1 b c1 QΞ A b c2 ηψ B + ηψ A b c2 QΞ B QΦ C W + ηφ E QΞ A ξ c1 b c1 ηψ B b c2 QΦ C QΦ D W + ηφ E ηψ A ξ c1 b c1 QΞ B b c2 QΦ C QΦ D W 25

26 = d 2 τ ηψ A ηψ B ξ c1 b c1 QΦ C b c2 QΦ D ηφ E W + ηψ B QΦ C ξ c1 b c1 QΦ D b c2 ηφ E ηψ A W + QΦ C QΦ D ξ c1 b c1 ηφ E b c2 ηψ A ηψ B W + QΦ D ηφ E ξ c1 b c1 ηψ A b c2 ηψ B QΦ C W + ηφ E ηψ A ξ c1 b c1 ηψ B b c2 QΦ C QΦ D W, 34 after eliminating Ξ by QΞ = ηψ. Since the final expression has the same form up to ξ as the bosonic SFT with ηψ = V 1/2, QΦ = V A = V, we can conclude that this is equal to the well-known amplitude. Similarly, we can compute the ampolitudes with FFFFB and FBFBB, and show that the results are equal to the well-known amplitues. 26

27 4. Summary and discussion We have found that the missing gauge-symmetries are realized as those under which the variation of the pseudo-action is propotional to the constraint. We have proposed the new Feynman rules for the open SSFT respecting all the gauge- symmetries. We have shown that the correct on-shell four- and five-point amplitudes are reproduced by the new rules. We can apply the similar argument to the heterotic string field theory, and obtain the new Feynmnan rules which reproduce the correct four- and five-point amplitudes without any ambiguity. 27

28 To clarify whether the new Feynman rules reproduce all the on-shell amplitudes at the tree level. general theory of the pseudo-action and its symmetries To extend the Feynman rules to those applicable beyond the tree level. gauge fixing by means of the BV method extra factor 1/2 for each fermion loop? 28