Gravitational Scattering Amplitudes and Closed String Field Theory in the Proper-Time Gauge

Home , String (physics), String field theory

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.

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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.

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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 = hΦ|KΦi + hΦ|Φ ◦ Φi + hΦ ◦ Φ|Φi . 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 √ 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 |P i = exp n −n + −n −n − −n −n − n −n |0i. 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), ∆ = |ξ − ξ0|, 0 0 GC (z, z ) = ln |z − z | −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 = ∞.

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 |zr − z s | ( −n∆ ) X e 0 0 = −δ ein(ηs −ηr ) + e−in(ηs −ηr ) − max(ξ, ξ0) rs 2n n=1 0 0 X ¯ rs |n|ξr +|m|ξs 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 |Zr − Zs |, r 6= s, ¯ rr X αi 1 (r) C00 = − ln |Zr − Zi | + 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 |n|ξr +|m|ξs + Cnme P−n · P−m n,m ! X (r) = hP| exp ξr L0 |V [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 |Zr − Zs |, r 6= s, ¯ rr ¯ rr X αi 1 C00 = N00 = − ln |Zr − Zi | + 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] = gh{k(r)}| ( (r)† (r)† (r)† (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 (  !2 ) X 1 1 p(r) exp τ − 1 0 α 2 2  r r

( (r)† (r)† (r)† (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 (  !2 ) X 1 1 p(r) exp τ − 1 |0i. 0 α 2 2  r r

Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 17 / 26 Factorization of Three-Closed-String Scattering Amplitude

Factorization of three-closed-string scattering amplitude

Aclosed[1, 2, 3] = Aopen[1, 2, 3] Aopen[1, 2, 3].

Scattering amplitude of three closed strings can be completely factorized into those of three open strings except for the zero modes. Question: Can we factorize general closed string scattering amplitudes into those of open string theory?

Realization of the Kawai-Lewellen-Tye (KLT) relations in the framework of the second quantized string theory .

Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 18 / 26 Three-Graviton Scattering Amplitude

Decomposition of the spin-2 ﬁeld into graviton, anti-symmetric tensor, and scalar ﬁeld n1 1 o n1 o n 1 o h = (h + h ) − η hσ + (h − h ) + η hσ . µν 2 µν νµ µν d σ 2 µν νµ µν d σ We choose the covariant gauge condition

µ ∂ hµν = 0, which becomes de Donder gauge condition for the graviton 1 ∂µh − ∂ hσ = 0. µν d − 2 ν σ For three-graviton scattering, we choose the external string state as

3 Y n r (r)µ (r)νo |Ψ3G i = hµν(p )α−1 α˜−1 |0i. r=1

Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 19 / 26 Three-Graviton Scattering Amplitude

Three-graviton scattering amplitude

3 3 ! Z Y X 2g A = dp(r)δ p(r) hΨ |E Closed[1, 2, 3]|0i [3−graviton] 3 3G [3] r=1 r=1 3 3 ! P3 1 Z 2g −2τ0 Y (i) X (i) = e r=1 αr dp δ p 3 i=1 i=1 ( 3 )  3  Y (i)µ (i)ν 1 X (r)† (s)† h0| h (p(i))a · ˜a N¯ rs a · a µν 1 1 25  11 1 1  i=1 r,s=1

3 !  3  X (t)† 1 X (l)† (m)† N¯ t a · p N¯ lmã · ã 1 1 25  11 1 1  t=1 l,m=1 3 ! X ¯ n (n)† N1 ã1 · p |0i. n=1

Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 20 / 26 Three-Graviton Scattering Amplitude

We note that A[3−graviton] can be written also as

3 3 ! 2g 1 Z Y X A = dp(i)δ p(i) [3−graviton] 3 28 i=1 i=1 ( 3 ) Y (i) (i)µ (i)ν Open ˜ Open h0| hµν(p )a1 · ˜a1 E[3−Gauge] E[3−Gauge]|0i. i=1 Making use of the Neumann functions of the open string 1 1 N¯ 11 = , N¯ 22 = , N¯ 33 = 22, 11 24 11 24 11 1 1 1 N¯ 12 = N¯ 21 = , N¯ 23 = N¯ 32 = , N¯ 31 = N¯ 13 = , 11 11 24 11 11 2 11 11 2 1 N¯ 1 = N¯ 2 = , N¯ 3 = −1, 1 1 4 1

Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 21 / 26 Three-Graviton Scattering Amplitude

We are able evaluate the three-graviton interaction term as follows

2 3 3 ! 2g 1 Z Y X A = 26 dp(i)δ p(i) [3−graviton] 3 25 i=1 i=1 (1) (2) (3) hµ1ν1 (p )hµ2ν2 (p )hµ3ν3 (p ) n 1 1 1 − ηµ1µ2 pµ3 + ηµ1µ3 pµ2 + ηµ2µ3 pµ1 24 23 23 1 1 1 o − ηµ2µ1 pµ3 + ηµ3µ1 pµ2 + ηµ3µ2 pµ1 24 23 23 n 1 1 1 − ην1ν2 pν3 + ην1ν3 pν2 + ην2ν3 pν1 24 23 23 1 1 1 o − ην2ν1 pν3 + ην3ν1 pν2 + ην3ν2 pν1 . 24 23 23

Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 22 / 26 Three-Graviton Scattering Amplitude

A[3−graviton] is precisely the three-graviton interaction term which may be obtained from the Einstein’s gravity action.

Z 3 3 ! Y (i) X (i) A[3−graviton] = κ dp δ p i=1 i=1 (1) (2) (3) hµ1ν1 (p )hµ2ν2 (p )hµ3ν3 (p ) n o ηµ1µ2 p(1)µ3 + ηµ2µ3 p(2)µ1 + ηµ3µ1 p(3)µ2 n o ην1ν2 p(1)ν3 + ην2ν3 p(2)ν1 + ην3ν1 p(3)ν2

g √ where κ = 27·3 = 32πG10.

Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 23 / 26 Scattering Amplitude of Four Strings

~~Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 24 / 26 Shapiro-Virasoro Amplitude~~

~~Scattering of four closed string tachyons~~

~~Z (1) (2) (2) (3) 2 2 2 p · p 2 p · p ATachyon[1, 2, 3, 4] = g d Z |Z| 2 2 |1 − Z| 2 2 Z 2 2 2 − s −2 2 − u −2 = g d Z |Z| ( 8 )|1 − Z| ( 8 )~~

~~s t u 2 Γ −1 − 8 Γ −1 − 8 Γ −1 − 8 = 2πg s t u . Γ 2 + 8 Γ 2 + 8 Γ 2 + 8 Mandelstam variables:~~

~~2 2 2 s = −(p1 + p2) , t = −(p1 + p3) , u = −(p1 + p4) .~~

~~The Koba-Nielsen variables:~~

~~Z1 = 0, Z2 = Z, Z3 = 1, Z4 = ∞.~~

~~Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 25 / 26 Conclusions and Discussions~~

1 Construction of covariant closed string ﬁeld theory in the proper-time gauge. 2 Neuman functions and Fock space representations of closed string vertices. 3 Scattering amplitude of three closed strings. 4 Complete factorization of three-closed-string amplitudes. 5 Three-graviton scattering and its relation to the three-gauge-particle scattering. 6 Generalized Kawai-Lewellen-Tye (KLT) relations.

~~Taejin Lee (KNU) 4th Conf. of Polish Soc. on Relativity Sep. 26, 2017 26 / 26~~