Gravitational Scattering Amplitudes and Closed String Field Theory in the Proper-Time Gauge Taejin Lee Kangwon National University KOREA [email protected] Sep. 26, 2017 Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 1 / 26 Contents 1 Quantum Gravity and String Field Theory 2 String Field Theory in the Proper-Time Gauge 3 Scattering Amplitudes of Closed String Field Theory 4 Neumann Functions for Closed String 5 Scattering Amplitudes of Three Closed Strings 6 Three Graviton Scattering Amplitude 7 Scattering Amplitudes of Four Closed Strings 8 Kawai-Lewellen-Tye (KLT) Relations 9 Discussions and Conclusions Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 2 / 26 Quantum Gravity and String Field Theory Classical General Relativity Derived from Quantum Gravity Boulware and Deser, Ann. Phys. 89 (1975): \A quantum particle description of local (noncosmological) gravitational phenomena necessarily leads to a classical limit which is just a metric theory of gravity. As long as only helicity ±2 gravitons are included, the theory is precisely Einsteins general relativity." Closed String Field Theory Closed string theory contains massless spin 2 particles in its spectrum. The low energy limit of the covariant interacting closed string field theory must be the Einstein's general relativity. The closed string field theory may provide a consistent framework to describe a finite quantum theory of the spin 2 particles, the gravitons. We need to examine the graviton scattering amplitudes of the covariant string field theory and compare them with those of the perturbation theory of the gravity in the low energy region. Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 3 / 26 String Field Theory in the Proper-Time Gauge I. Open string field theory in the proper-time gauge We constructed a covariant string field theory on Dp-branes, and calculated three-string scattering amplitude and the four-string scattering amplitude in the low energy limit. TL, Phys. Lett. B 768, 248 (2017); arXiv:1609.01473; arXiv:1703.06402, to be published in PLB. S. H. Lai, J. C. Lee, Y. Yang and TL, arXiv:1706.08025, to be published in JHEP. (2) (2) (3) (3) (1) (1) (4) Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 4 / 26 The Proper-Time Gauge String in the proper-time gauge The length parameters are fixed by using the reparametrization invariance: For three-string vertex, α1 = α2 = 1, α2 = −2 The length parameters are written in trms of the two dimensional metric on the string world sheet. The BRST invariance guarantees that the physical S-matrix does not depend on the length parameters. TL, String theory in covariant gauge, unpublished (1987) ; Ann. Phys. 183, 191 (1988) . Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 5 / 26 Closed String Field Theory in the Proper-Time Gauge II. Closed string field theory in the proper-time gauge We will construct a covariant string field theory and calculate three-string scattering amplitude and the four-string scattering amplitude in the low energy limit by extending the previous works on the open string field theory. (2) (2) (3) (3) (1) (1) (4) Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 6 / 26 Fock Space Representation of the Closed String Field Theory in the Proper-Time Gauge Closed String Field Theory in the Proper-Time Gauge ! g S = hΦjKΦi + hΦjΦ ◦ Φi + hΦ ◦ ΦjΦi : 3 The closed string field theory in the proper-time gauge generates the string scattering diagrams, which can be represented by the Polyakov string path integrals: Z p µ ν 1 αβ @X @X SP = − 0 dτdσ −hh α β ηµν; µ, ν = 0;:::; d − 1: 4πα M @σ @σ String Scattering Amplitudes Z Z AM = D[X ]D[h] exp −i dτdσL : M Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 7 / 26 String Scattering Amplitudes of Closed String Field Theory and Polyakov String Path Integral Strategy of Calculation of String Scattering Amplitudes 1 Construct the covariant closed string field theory 2 Rewrite the scattering amplitudes generated by the closed string field theory by using the Polyakov string path integral 3 Re-express the Polakov string path integrals in terms of the oscillator operators 4 Identify the Fock space (operator) representations of the string field theory vertices 5 Choose appropriate external string states, corresponding to the various particles and evaluate the scattering amplitudes. Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 8 / 26 Closed String Theory: Review Free String Theory 1 Z S dτdσ @X · @X : = 4πα0 Decomposition of X in terms of left-movers and right-movers X (τ; σ) = XL(τ + σ) + XR (τ − σ): Mode expansions r r α0 α0 X 1 X (τ; σ) = x + p (τ + σ) + i α e−in(τ+σ); L L 2 L 2 n n n6=0 r r α0 α0 X 1 X (τ; σ) = x + p (τ − σ) + i α~ e−in(τ−σ); R R 2 R 2 n n n6=0 where x = xL + xR . Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 9 / 26 Closed String Theory: Review Canonical commutation relations r α0 [x ; p ] = [x ; p ] = i ; L L R R 2 [αm; αn] = [~αm; α~n] = mδ(m + n): Momentum eigentate with eigenvalue Pn; n 6= 0: r 1 P α P α~ α α~ P · P jP i = exp n −n + −n −n − −n −n − n −n j0i: n πn n n n 4n Mapping from cylindrical surface onto the complex plane z = eρ = eξ+iη; − π ≤ η ≤ π: Green's function on complex plane (ξ > ξ0), ∆ = jξ − ξ0j, 0 0 GC (z; z ) = ln jz − z j −n∆ 1 X e 0 0 = max(ξ; ξ0) − ein(η −η) + e−in(η −η) : 2 n n=1 Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 10 / 26 Closed String Interaction in the Proper Time Gauge CS mapping from the world sheet of three closed string scattering onto the complex plane. For the three-string vertex in the proper-time gauge, ρ = ln(z − 1) + ln z: The local coo vrdinates ζr = ξr + iηr , r = 1; 2; 3 defined on invidual string world sheet patches are related to z as follows: 1 e−ζ1 = eτ0 ; z(z − 1) 1 e−ζ2 = −eτ0 ; z(z − 1) −ζ − τ0 p e 3 = −e 2 z(z − 1): For convenience we choose, by using SL(2; C) invariance Z1 = 0; Z2 = 1; Z3 = 1: Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 11 / 26 Local Coordinates ρ = τ + iσ ρ = τ + iσ 2π −2π −π 0 π 0 η η 2 ζ 2 ζ 2 2 0 0 σ η3 σ η3 π ζ 3 −π ζ 3 η1 ζ η1 ζ 1 1 π 0 −π 0 0 0 τ0 τ0 τ τ Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 12 / 26 Neumann Functions of the Closed String Vertices Fourier components of the Green's function on complex plane 0 0 GC (ρr ; ρs ) = ln jzr − z s j ( −n∆ ) X e 0 0 = −δ ein(ηs −ηr ) + e−in(ηs −ηr ) − max(ξ; ξ0) rs 2n n=1 0 0 X ¯ rs jnjξr +jmjξs inηr imηs + Cnme e e : n;m Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 13 / 26 ¯ rs Integral Formulas for Cnm ¯ rs C00 = ln jZr − Zs j; r 6= s; ¯ rr X αi 1 (r) C00 = − ln jZr − Zi j + τ0 αr αr i6=r 1 I dz 1 ¯ rs ¯ sr −nζr (z) Cn0 = C0n = e ; n ≥ 1; 2n Zr 2πi z − Zs 0 1 I dz I dz 1 0 0 ¯ rs −nζr (z)−mζs (z ) Cnm = 0 2 e ; n; m ≥ 1: 2nm Zr 2πi Zs 2πi (z − z ) Reality conditions of the Green's function ¯ rs ¯ ∗rs ¯ rs Cnm = C−n−m; C−nm = 0; n; m ≥ 1: Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 14 / 26 Scattering Amplitude of Three Strings Scattering amplitude M ! Z X Z Z W = DX exp i Pr (σ) · X (τr ; σ)dσ − dτdσL r=1 ( ( 1 X X X 1 (r) (r) = [det ∆]−d=2 exp ξ P2 − P · P 4 r 0 n n −n r r n=1 ) 0 (r) (s) X ¯ rs jnjξr +jmjξs + Cnme P−n · P−m n;m ! X (r) = hPj exp ξr L0 jV [N]i: r Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 15 / 26 Neumann Functions of Three-Closed-String Vertex Neumann functions of three-closed string vertex and those of three-open string vertex: ¯ rs ¯ rs C00 = N00 = ln jZr − Zs j; r 6= s; ¯ rr ¯ rr X αi 1 C00 = N00 = − ln jZr − Zi j + τ0; αr αr i6=r 1 1 I dz 1 ¯ rs ¯ rs ¯ rs −nζr (z) Cn0 = C−n0 = Nn0 = e ; n ≥ 1; 2 2n Zr 2πi z − Zs 1 C¯ rs = C¯ rs = N¯ rs nm −n−m 2 mn 0 1 I dz I dz 1 0 0 −nζr (z)−mζs (z ) = 0 2 e ; n; m ≥ 1; 2nm Zr 2πi Zs 2πi (z − z ) ¯ rs ¯ rs Cn−m = C−nm = 0: Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 16 / 26 Factorization of Three-Closed-String Scattering Amplitude A[1; 2; 3] = ghfk(r)gj ( (r)y (r)y (r)y (s) ) X X 1 αn αm X αn p exp N¯ rs · + N¯ rs · 2 nm 2 2 n0 2 2 r;s n;m≥1 n≥1 ( 0 !2 1) X 1 1 p(r) exp τ − 1 0 α @2 2 A r r ( (r)y (r)y (r)y (s) ) X X 1 α~n α~m X α~n p exp N¯ rs · + N¯ rs · 2 nm 2 2 n0 2 2 r;s n;m≥1 n≥1 ( 0 !2 1) X 1 1 p(r) exp τ − 1 j0i: 0 α @2 2 A r r Taejin Lee (KNU) 4th Conf.