Covariant Gauges in String Field Theory

Transcription

1 Covariant Gauges in String Field Theory Mitsuhiro RIKEN symposium SFT07 In collaboration with Masako Asano (Osaka P.U.) New covariant gauges in string field theory PTP 117 (2007) 569, Level truncated tachyon potential in various gauges JHEP 0701 (2007) 028, and the work under study.

2 Introduction Motivation and what we do 1

3 SFT has a huge gauge symmetry. Needs gauge-fixing Siegel gauge has been a unique choice of covariant gauge since 1984 (even before the gauge-inv. action was found.) In ordinary gauge theory (e.g. Abelian case) [ S gauge = d D x 1 4 F µνf µν + B µ A µ + α ] 2 B2 + i c µ µ c α is a gauge parameter α = 1 α = 0 Feynman gauge Landau gauge Siegel gauge gives Feynman gauge for massless gauge mode. Then what about Landau and other gauges? 2

4 Tachyon potential in Siegel gauge showed singular behavior. V phi V phi -200 Alternative gauges have been desired To distinguish the physical and gauge artifact (or truncation-scheme artifact,) To obtain reliable results for wide range (Must for quantum analysis.) 3

5 In this talk, Proposal of new covariant gauge A single parameter family of gauges which naturally corresponds to the covariant gauges in ordinary gauge theory. Application to tachyon potential The identification of physical branch. Obtaining smooth potential. 4

6 SFT and Siegel gauge Notations and Fundamentals 5

7 String field Φ 1 [X µ (σ), c(σ), b(σ)] = d 26 x s s, x ψ s (x) = d 26 p (2π) 26 d 26 p = (2π) 26 s, p ψ s (p) s [ φ(p) + Aµ (p)α µ 1 + iχ(p)c 0b 1 + ] 0, p X µ (σ) = x µ + n 0 α µ n n cos(nσ) string coordinate (gh# = 0) c(σ) = n b(σ) = n c n e inσ worldsheet ghost (gh# = 1) b n e inσ worldsheet anti-ghost (gh# = 1) [x µ, p ν ] = iη µν, [α µ n, α ν m] = η µν nδ n+m,0, {c n, b m } = δ n+m,0 Base vectors s, p = f, p or c 0 f, p f, p = α µ 1 n 1 α µ i n i c l1 c lj b m1 b mk 0, p; 6

8 Action S = 1 2 Φ 1, QΦ 1 g 3 Φ 1, Φ 1 Φ 1 where Q is BRST operator (first quantized BRST charge) Q = c 0 L 0 + b 0 M + Q (Φ Ψ)[X(σ)] Φ, Ψ = bpz(φ) Ψ DY DZ Φ[Y (σ )] Ψ[Z(σ )] δ[x(σ), Y (σ ), Z(σ )] Support of δ[x(σ), Y (σ ), Z(σ )] X(σ) Z(σ ) Y (σ ) 7

9 Gauge invariance Action S is invariant under the gauge transformations δφ 1 = QΛ 0 + g(φ 1 Λ 0 Λ 0 Φ 1 ) where Λ 0 = Λ 0 [X µ (σ), c(σ), b(σ)] is gh# 0 string field. Λ 0 = d 26 p (2π) 26 [ λ(p)b 1 + ] 0, p δa µ (p) = ip µ λ(p) 8

10 SU(1,1) Q = c 0 L 0 + b 0 M + Q M together with and M = 2 n>0 nc n c n M = n>0 M z = 1 Ñ g = constitute SU(1, 1) algebra n>0 1 2n b nb n (c n b n b n c n ) [M, M ] = 2M z, [M z, M] = M, [M z, M ] = M 9

11 Isomorphism F = ( Ñ g = FÑ g + c 0 FÑ g). f F n M n f F n, M n f = 0 f = 0 f F n g F n s.t. f = M n g There exists W n : F n F n W n M n f = f for any f F n M n W n g = g for any g F n F n W n M n F n 10

12 Gauge invariant decomposition Expand string field in c 0 Φ 1 = φ (0) + c 0 ω ( 1) In terms of φ (0) and ω ( 1), S 2 = 1 2 ( φ (0), c 0 L 0 φ (0) + 2 Qφ (0), c 0 ω ( 1) + Mω ( 1), c 0 ω ( 1) ) = 1 2 (φ (0) 1L0 Qω ( 1) ), c 0 L 0 ( φ (0) 1 L 0 Qω ( 1) ) 11

13 Note that ζ (1) = Qφ (0) + Mω ( 1) is gauge invariant for g = 0. Using ζ (1) instead of ω ( 1), we have S 2 = 1 2 ( φ (0), c 0 L 0 φ (0) Qφ (0), c 0 W 1 ( Qφ (0) ) + ζ (1), c 0 W 1 ζ (1) ) Cf. 1 4 F µνf µν = 1 2 (A µ A µ A A) 12

14 Siegel gauge b 0 Φ 1 = 0 Introducing ghost Φ 0, anti-ghost Φ 2 and Nakanishi-Lautrup field B 3, gauge-fixed action is S = 1 2 Φ 1, QΦ 1 g 3 Φ 1, Φ 1 Φ 1 Φ 2, QΦ 0 + gφ 1 Φ 0 + b 0 B 3, Φ 1 Now we have another gauge symmetry δφ 0 = QΛ 1 + g(φ 0 Λ 1 + Λ 1 Φ 0 ) 13

15 Ghost (for ghost) n Again we fix b 0 Φ 0 = 0 introducing ghost for ghost Φ 1 (and Φ 3, B 4 ) Again we have another gauge symmetry Again gauge-fixing with ghost for ghost for ghost Φ 0, Φ 1, Φ 2, Φ 2, Φ 3, Φ 4, B 3, B 4, B 5, 14

16 Procedure 1 Extend to all ghost number (field gh# + ws gh# = 1) Φ 1 Φ = n Φ n 2 Find gauge conditions for all sectors b 0 Φ n = 0 s.t. (a) any string field can be transformed to satisfy them (b) no residual gauge symmetry... shown for g = 0 3 Write gauge-fixed action S Siegel = 1 2 Φ, QΦ g 3 Φ, Φ Φ + b 0B, Φ 4 Check BRS invariance δ B S Siegel = 0 15

17 BRS transformation for Siegel gauge δ B Φ n = η b 0 B n+1 (n > 1), δ B Φ n = η (QΦ n 1 + g δ B B n = 0 η is a grassmann odd parameter. k= (Φ n k Φ k )) (n 1), 16

18 New covariant gauges b 0 (M + a c 0 Q)Φ 1 = 0 17

19 Gauge condition for all ghost number sector For n 2 ( b0 M n 1 + a b 0 c 0 M n 2 Q ) Φ 3 n = 0 ( b0 W n 2 + a b 0 c 0 W n 1 Q ) Φ n = 0 a = 1 is prohibited for g = 0, since (b 0 M + a b 0 c Q)Φ 0 1 = 0 Mω ( 1) + a Qφ (0) = 0 L.H.S. reduces to (g = 0) gauge invariant combination at a = 1 ζ (1) = Qφ (0) + Mω ( 1) 18

20 Feynman-Siegel point (a = 0) b 0 M n 1 Φ 3 n = 0 b 0 Φ 3 n = 0 b 0 W n 2 Φ n = 0 b 0 Φ n = 0 (n 2) Because of the isomorphism between F n and F n b 0 M n 1 ω (1 n) = 0 b 0 ω (1 n) = 0 b 0 W n 2 ω (n 2) = 0 b 0 ω (n 2) = 0 Φ n = φ (n 1) + c 0 ω (n 2) 19

21 Gauge fixed action where S a = 1 2 n= + n=2 Φn, QΦ n+2 g 3 l+m+n=3 Φ l, Φ m Φ n ( (Oa B) n+3, Φ n + (Oa B) n, Φ n+3 ) (O a B) n = ( b 0 M n 1 + ac 0 b 0 M n 2 Q ) B 3 n, (O a B) n+3 = ( b 0 W n 2 + ac 0 b 0 W n 1 Q ) B n. 20

22 S a is invariant under the BRS transformation δ B Φ n = η(o a B) n (n > 1) δ B Φ n = η(qφ n 1 + g δ B B n = 0 k= provided (O a B) n, (O a B) n+3 = 0. (Φ n k Φ k )) (n 1) 21

23 Landau point a = b 0 c 0 QΦ 1 = 0 ( Qφ (0) = 0) S = 1 2 ( φ (0), c 0 L 0 φ (0) + Mω ( 1), c 0 ω ( 1) ) g 3 ( φ (0), φ (0) φ (0) + 3 φ (0), φ (0) c 0 ω ( 1) + 3 φ (0), c 0 ω ( 1) c 0 ω ( 1) ) Remark: Banks-Peskin L BP = 1 2 Φ(L 0 1)P Φ with projector P onto L n Φ = 0 (n = 1, 2,...) Cf. IKKO (1985) 22

24 Level 1 (massless) fields φ N=1 = ω N=1 = d 26 p (2π) 26 1 α ( γ(p)b 1 + A µ (p)α µ 1 + i γ(p)c ) 1 0, p;, d 26 p (2π) ( iχ(p)b 1 + u µ (p)α µ 1 + v(p)c 1 ) 0, p;. B N=1 ω = d 26 p 1 ( 2 iβχ (2π) 26 (p)b 1 + β uµ (p)α µ 1 + β ) v(p)c 1 0, p;. SN=1 quad = d 26 x [ 1 4 F µνf µν 1 2 (χ + µa µ ) 2 i γ µ µ γ iu µ µ γ +β χ (χ + a µ A µ ) β u µ (u µ a µ γ) β vv ] 23

25 By use of field redefinitions B = (a 1)β χ c = (a 1) γ c = γ χ = χ + µ A µ β χ ũ µ = u µ a µ γ β uµ = β uµ + 2i µ γ, the above action can be written into the well-known form plus decoupled auxiliary fields term SN=1 quad [ = d 26 x 1 4 F µνf µν + B µ A µ + α 2 B2 + i c µ µ c 1 2 χ β uµ ũ µ + 1 ] 4 β vv with α = 1 (a 1) 2 24

26 Gauge independence of on-shell amplitude Propagator a = b 0 L 0 a 1 a ( Q b 0c 0 W 1 + c 0b 0 W 1 Q L 0 L 0 ) + a(2 a) (1 a) 2Qb 0W 1 L 0 2 Q Siegel = b 0 L 0 Landau = b 0 L 0 (1 P 0 ) + c 0 W 1 25

27 Theorem (I) QΦ i = 0, Φ i : odd (i = 1,, n) j {1,, n} s.t. Φ j = QZ (Z: even) = A tree cyclic (Φ 1,, Φ n ) = 0 (II) QΦ i = 0, Φ i : odd (i = 1,, n) = A tree cyclic (Φ 1,, Φ n ) a = A tree cyclic (Φ 1,, Φ n ) 0 26

28 Application to tachyon condensation Examining gauge (in-)dependence of tachyon potential 27

29 Sen s conjecture V φ φ V (φ ) = T 25 Potential depth of the vacuum is equal to D25-brane tension T 25. No open string excitation around the tachyon vacuum. Lower dimensional D-branes are solitonic solutions around the vacuum. 28

30 Tachyon potential in level truncation We take twist-even, N g = 1, scalar fields up tp level L Φ (L) 1 = φ + i Level truncated potential φ i f i + c 0 ω j g j V (L,3L) (φ, {φ i }, {ω j }) = S ( j Φ (L) 1 ). Gauge fixing we substitute ω j = ω j ({φ i }) for a < or φ j for a = where {φ j } is a set of solutions to Qφ (0) = 0 29

31 Example Fields up to Level 2: Φ (L=2) 1 = φ + φ 1 (α 1 α 1 ) + φ 2 b 1 c 1 + ω 1 c 0 b 2. Gauge condition: Level (2,6) truncated potential: 4ω 1 + a(26φ 1 + 3φ 2 ) = 0 V a (2,6) (φ, φ 1, φ 2 ) = V (2,6) (φ, φ 1, φ 2, ω 1 =a(26φ 1 + 3φ 2 )/4), V (2,6) (φ, φ 1, ω 1 ) = V (2,6) (φ, φ 1, φ 2 = 26φ 1 /3, ω 1 ). 30

32 Tachyon potential at varying a V 0 V phi phi V 0 V phi phi V 0 V phi phi 31

33 Dangerous zone Fields up to Level 2: Φ (L=2) 1 = φ + φ 1 (α 1 α 1 ) + φ 2 b 1 c 1 + ω 1 c 0 b 2. Gauge transformation ( δφ = g κλ 16 9 φ φ φ ) 81 ω 1, δφ 1 = λ ( g κλ 243 φ φ φ ) 2187 ω 1, ( δφ 2 = 3λ + g κλ φ φ φ ω 1 ( 224 δω 1 = λ + g κλ 81 φ φ φ ) 729 ω 1, where κ = 1 3 ( ) 3 At around a = 1.85 gauge slice is parallel to the gauge flow through the vacuum of gauge-unfixed action. (cf. a = 1 for g = 0) ) 32,

34 Branch structure V phi Level(0,0) black Level(2,6) a = (Landau) orange Level(2,6) a = 0 (Feynman-Siegel) blue 34

35 Higher level V phi a = Level(0,0) black Level(2,6) orange Level(4,12) blue Level(6,18) rose 35

36 V phi a = Level(0,0) black Level(2,6) orange Level(4,12) blue Level(6,18) rose 36

37 Vacuum solution Level(2,6) sky-blue Level(4,12) blue Level(6,18) rose E_vacuum a

38 E_vacuum a Level(2,6) sky-blue Level(4,12) blue Level(6,18) rose 38

39 Level (0,0) (2,6) (4,12) (6,18) a E vac /T 25 E vac /T 25 E vac /T 25 E vac /T

40 Conclusions 40

41 Summary New covariant gauges are proposed A single parameter family of gauges which naturally corresponds to the covariant gauges in ordinary gauge theory. Action is simplified in Landau point. Applied to tachyon potential The identification of physical branch. Branching behavior in Siegel gauge is gauge-artifact. 41

42 Outlook Analysis of space(-time) dependent solutions may be also simplified in Landau point (both numerically and analytically.) Absence of open string mode on tachyon vacuum: Kishimoto-Takahashi (2002), Imbimbo (hep-th/ ) Ellwood-Schnabl (hep-th/ ) Non-perturbative gauge fixing ambiguity Relation to Feng-Siegel (hep-th/ ) 42