Lectures on String Theory

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Recall that for tachyons on shell we have ki 2 = 1 . Performing now the Wick rotation
α ′
τ → −iτ and changing the integration variable τ for x = e −τ we find
A =
∫ 1
0
dx x 2α′ k 3 k 4
(1 − x) 2α′ k 2 k 3
,
where for the sake of simplicity we omitted the δ-function which encodes the conservation
law of the momenta. Introducing the Mandelstam variables s = (k 1 + k 2 ) 2
and t = (k 2 + k 3 ) 2 the last formula can be cast in the form
A =
∫ 1
0
dx x α′ s−2 (1 − x) α′ t−2 = Γ(α′ s − 1)Γ(α ′ t − 1)
Γ(α ′ (s + t) − 2)
In fact this function is known as the Euler beta-function. One of the interesting
properties of the representation of A in terms of the Euler beta-function is that the
latter is explicitly symmetric under the interchange of s and t. Search of amplitudes
with this symmetry property led Veneziano in 1960’s to this amplitude which was
the starting point of modern string theory.
It is interesting to analyze the Veneziano formula in more detail. The Γ-function has
poles at non-positive integers with residues
Γ(x) → (−1)n
n!
1
x + n
as x → −n , n ≥ 0 .
Thus, when α ′ s → 1 − n, n = 0, 1, . . . the amplitude behaves as
A(s, t) → (−1)n
n!
1 Γ(α ′ t − 1)
α ′ s − 1 + n Γ(α ′ t − 1 − n)
Here the dependents of the variable t is polynomial because for n > 0 we have
Γ(α ′ t − 1) Γ(w + n)
Γ(α ′ = = (w + n − 1) · · · (w + 1)w
}
t −
{{
1 − n
}
) Γ(w)
w
= (α ′ t − 2)(α ′ t − 3) · · · (α ′ t − n − 1) ≡ P n (α ′ t) ,
i.e. the r.h.s. is a polynomial of degree n. Thus, the scattering amplitude can be
essentially written as
∞∑ (−1) n P n (α ′ t)
A(s, t) =
n! n − 1 + α ′ s , P 0(α t) = 1 .
n=0
In scattering theory resonances (or simply poles) of the scattering amplitude are
interpreted as an exchange by intermediate particles whose masses are obtained from
the condition of having poles. In our case we see that the poles arise due to exchange
by hypothetical particles whose masses are quantized as
Mn 2 = −s = 1 (n − 1) .
α
′
.