Lectures on String Theory

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where we assume that τ 1 > τ 2 . By using the formula (4.12) we find
〈V (k 1 , τ 1 )V (k 2 , τ 2 )〉 = e iα′ k 2 1 τ 1+iα ′ k 2 2 τ 2
e k 1 k 2
πT log(eiτ 1−e iτ 2) 〈0| : V (k 1 , τ 1 )V (k 2 , τ 2 ) : |0〉 .
The vacuum expectation value of the normal-ordered expression on the right hand
side reduces to the contribution of the zero modes only and we get
〈V (k 1 , τ 1 )V (k 2 , τ 2 )〉 = e iα′ k1 2τ 1+iα ′ k2 2τ 2
e 2α′ k 1 k 2 log(e iτ 1−e iτ2) 〈0|e i(kµ 1 +kµ 2 )x µ
|0〉 .
} {{ }
δ(k 1 +k 2 )
Thus, the two-point function is non-zero only if k 1 = k = −k 2 . In this case we get
We thus find that
Four-tachyon scattering amplitude
〈V (k, τ 1 )V (−k, τ 2 )〉 = eiα′ k 2 (τ 1 +τ 2 )
(e iτ 1 − e
iτ 2) 2α ′ k 2 .
〈V (k, τ 1 )V (−k, τ 2 )〉 = ei∆(τ 1+τ 2 )
(e iτ 1 − e
iτ 2) 2∆ .
Here we compute the scattering amplitude of four tachyonic particles. This is the
famous Veneziano amplitude which subsequently led to discovery of string theory.
The amplitude is defines as
A =
∫ ∞
0
dτ 〈k 4 |V (k 3 , τ)V (k 2 , 0)|k 1 〉
Here 〈k 4 | is understood in an unusual way 〈k 4 | = 〈0|e ikµ 4 x µ
. Using the definition of
the tachyonic vertex operators and the formula (4.12) one finds
V (k 3 , τ)V (k 2 , 0) = e iα′ k 2 3 τ : V (k 3 , τ) :: V (k 2 , 0) :
= e iα′ k 2 3 τ e −kµ 3 kν 2 〈X µ(τ)X ν (0)〉 : V (k 3 , τ)V (k 2 , 0) :
Recalling the open string propagator at σ = σ ′ = 0:
( )
〈X(τ)X(0)〉 = − ηµν
πT log e iτ − 1 .
Thus we find
A =
∫ ∞
Further simplification gives
A =
=
=
0
∫ ∞
0
∫ ∞
0
∫ ∞
0
dτ e iα′ k 2 3 τ e 2α′ (k 2 k 3 ) log(e iτ −1) 〈k 4 |e i(kµ 2 +kµ 3 )x µ
e i2α′ k µ 3 p µτ |k 1 〉
dτ e iα′ k3 2τ (e iτ − 1) 2α′ (k 2 k 3 ) e 2iα′ (k 3 k 1 )τ δ ( ∑
4 )
k i
i=1
dτ e iα′ k3 2τ e 2iα′ (k 1 +k 2 )k 3 τ (1 − e −iτ ) 2α′ (k 2 k 3 ) δ ( ∑
4 )
k i
i=1
dτ e −iα′ k3 2τ e −2iα′ (k 3 k 4 )τ (1 − e −iτ ) 2α′ (k 2 k 3 ) δ ( ∑
4 )
k i .
i=1