Ψ(σ) in terms of P (σ) (so the ψn’s are the αn’s of (7.1.7a)), and
the conformal transformations (9.2.7a) (cf. (8.1.6)) are determined by the conformal weights w (= 1 for ˇ
P ). The ρ → z map is the map from the above figure to the
upper-half plane. The ζ → ρ map is the map from the free-string σ ∈ [0, π] to the rth interacting-string σ ∈ [π r−1 p
p
s=1
+s, π
rs=1 +s]. All σ integrals from −π to π become
contour integrals in the z plane. (The upper-half z plane corresponds to σ ∈ [0, π], while the lower half is σ ∈ [−π, 0].) Since the string (including its extension to [−π, 0]) is mapped to the entire z plane without boundaries, (9.2.6) gets contributions from only the ends, represented by zr. We work directly with P (σ), rather than X(σ),
since P depends only on ρ, while X depends on both ρ and ¯
ρ. ( ˆ
X has a cut in the
z plane, since pσ isn’t periodic in σ.) The open string results can also be applied directly to the closed string, which has separate operators which depend on ¯
z instead
of z (i.e., + and − modes, in the notation of sect. 6.2). Actually, there is a bit of a cheat, since P (σ) doesn’t contain the zero-mode x. However, this zero-mode needs
special care in any method: Extra factors of 2 appeared in the path-integral approach because of the boundary conditions along the real z-axis and at infinity. In fact, we’ll see that the lost zero-mode terms can be found from the same calculation generally
182
9. GRAPHS
used in both operator and path integral approaches to find the integration measure
of step 3 above, and thus requires no extra effort.
We therefore look directly for a propagator e∆ that gives
ˇ
Ψ(z) = e∆ ˇ
Ψr(z)e−∆ = ˇ
Ψr(z) + [∆, ˇ
Ψr(z)] + · · · ,
(9.2.8)
where ˇ
Ψr corresponds to a free in-field for the rth string in the interaction picture,
and ˇ
Ψ to the interacting field. To do this, we first find a ˜
∆ for which
[ ˜
∆, ˇ
Ψr] = ˇ
Ψ .
(9.2.9)
We next subtract out the free part of ˜
∆ (external-line amputation):
˜
∆ = ∆ + ∆free ,
[∆free, ˇ
Ψr] = ˇ
Ψr .
(9.2.10)
This gives a ∆ which is quadratic in operators, but contains no annihilation operators (which are irrelevant anyway, since the |0 will kill them). As a result, there are no terms with multiple commutators in the expansion of the exponential. We therefore
obtain (9.2.8). When we subtract out free parts below, we will include the parts of the external-line amputation which compensate for the fact that z and zr are not at the same time.
We first consider applying this method to operators of arbitrary conformal weight,
as in (8.1.23). The desired form of ∆ which gives (9.2.8) for both f and δ/δf is
dz
dz′
1
δ
∆
ˇ
0 = −
f r(z)
− free-string terms ,
(9.2.11a)
r,s
z
2πi
2πi z
r
zs
− z′
δ ˇ
f s(z′)
where
δ
δ
, ˇ
fs(z2) = 2πiδrsδ(z2 − z1)
↔
, ˆ
f s(σ2) = 2πδrsδ(σ2 − σ1) ,
δ ˇ
f r(z1)
δ ˆ
f r(σ1)
(9.2.11b)
as follows from the fact that the conformal transformations preserve the commutation relations of f and δ/δf . The integration contours are oriented so that
dρ
πp+r
dσ
=
= 1 .
(9.2.12)
ρ 2πip
2πp
r
+r
−πp+r
+r
(9.2.11) can easily be shown to satisfy (9.2.8). The value of τ r (→ ±∞) for the
integration contour is fixed, so the δ in ζ in these commutation relations is really just a δ in σ of that contour. The free-string terms are subtracted as explained above.
9.2. Trees
183
(In fact, they are poorly defined, since the integration contours for r = s fall on top of each other.)
Unfortunately, ∆ will prove difficult to evaluate in this form. We therefore rewrite it by expressing the 1/(z − z′) as the derivative with respect to either z or z′ of a ln and perform an integration by parts. The net result can be written as
dz
dz′ ′
∆( ˇ
Ψ1, ˇ
Ψ2) =
ln(z − z′) ˇ
Ψ1r(z) ˇ
Ψ2s(z′)
r,s
z
2πi
2πi
r
zs
− free-string terms
+ (zero-mode)2 terms , (9.2.13a)
[ ˇ
Ψ2r(z2) , ˇ
Ψ1s(z1)} = −2πiδrsδ′(z2 − z1) ,
(9.2.13b)
The ′ on the contour integration is because the integration is poorly defined due
to the cut for the ln: We therefore define it by integration by parts with respect to either z or z′, dropping surface terms. This also kills the constant part of the ln which contributes the (zero-mode)2 terms, which we therefore add back in. Actually, (9.2.11) has no (zero-mode)2 terms, but in the case Ψ1 = Ψ2 = P , these terms determine
the evolution of the zero-mode x, which doesn’t appear in P , and thus could not be determined by (9.2.8) anyway. (x does appear in X and ˆ
X, but they’re less convenient
to work with, as explained above.) These (quadratic-in-)zero-mode contributions are most easily calculated separately by considering the case when all external states are ground states of nonvanishing momentum (see below). In order for the commutation
relations (9.2.13b) to be preserved by the conformal transformations, it’s necessary that the conformal weights w1 and w2 of Ψ1 and Ψ2 both be 1. In that case, ˇ
Ψ can
be replaced with ˜
Ψ in (9.2.13a) while replacing dz with dρ. However, one important
use of this equation is for the evaluation of vertices (S-matrices with no internal propagators). Since these vertices are just δ functionals in the string field coordinates (see below), and δ functionals are independent of conformal weight except for the
ζ → ρ transformation (since that transformation appears explicitly in the argument
of the δ functionals), we can write this result, for the cases of S-matrices with w1 =
w2 = 1 or vertices (with w1 + w2 = 2) as
dρ
dρ′ ′
∆( ˇ
Ψ1, ˇ
Ψ2) =
ln(z − z′) ˇ
Ψ1r(ρ) ˇ
Ψ2s(ρ′)
r,s
ρ 2πi
2πi
r
ρs
− free-string terms
+ (zero-mode)2 terms , (9.2.14a)
[Ψ2r(ζ2) , Ψ1s(ζ1)} = −2πiδrsδ′(ζ2 − ζ1) ,
(9.2.14b)
where the appropriate ∆ for X (w1 = w2 = 1) is
∆ = 1 ∆( ˇ
P , ˇ
P ) .
(9.2.15)
2
184
9. GRAPHS
These ′ and free-string corrections may seem awkward, but they will automatically
be fixed by the same method which gives a simple evaluation of the contour integrals: i.e., the terms which are difficult to evaluate are exactly those which we don’t want.
(For non-vertex S-matrices with w1 = 1 = w2, (9.2.11) or (9.2.13) can be used, but
their evaluated forms are much more complicated in the general case.) In general
(for covariant quantization, supersymmetry, etc.) we also need extra factors which
are evaluated at infinitesimal separation from the interaction (splitting) points, which follow from applying the conformal transformation (9.2.7), and (9.2.6) with z = zINT .
Only creation operators contribute.
The contour-integral form (9.2.14) can also be derived from the path-integral form
(9.2.4). For the open string [9.5], these contour integrals can be obtained by either combining integrals over semicircles in the upper-half z plane (σ integrals from 0 to π) with their reflections [9.6], or more directly by reformulating the open string as a closed string with modes of one handedness only, and with interactions associated with just the points σ = 0, π rather than all σ.
For the second step, Grsmn unfortunately is hard to calculate in general. For
open-string tree graphs, we perform the following conformal mapping to the upper-
half complex plane [9.4], where ∆ is easy to calculate:
N
ρ =
p+rln(z − zr) .
(9.2.16)
r=2
The boundary of the (interacting) string is now the real z axis, and the interior is the upper half of the complex z-plane. (The branches in the ln’s in (9.2.16) are thus chosen to run down into the lower-half plane. When we use the whole plane for
contour integrals below, we’ll avoid integrals with contours with cuts inside them.) As a result, operators such as P , which were periodic in σ, are now meromorphic
at zr, so the contour integrals are easy to evaluate. Also, extending σ from [0, π]
to [−π, π] extends the upper-half plane to the whole complex plane, so there are no boundary conditions to worry about. To evaluate (9.2.14), we note that, since we are neglecting (zero-modes)2 and free string terms (r = s, m = −n), we can replace
1
∂
∂
ln(z − z′)e−mρ/p+r−nρ′/p+s
→
+
ln(z
m
− z′) e−mρ/p+r−nρ′/p+s
+ n
∂ρ
∂ρ′
p+r
p+s
(9.2.17)
by integration by parts. We then use the identity, for the case of (9.2.16),
∂
∂
N
∂
∂
+
ln(z − z′) =
p
ln(z − z
ln(z′ − z
∂ρ
∂ρ′
+r
∂ρ
r )
∂ρ′
r )
.
(9.2.18)
r=2
9.2. Trees
185
Then we can trivially change completely to z coordinates by using dρ∂/∂ρ = dz∂/∂z,
and converting the ρ exponentials into products of powers of z monomials. Differen-
tiating the ln’s gives (products of) single-variable contour integrals which can easily be evaluated as multiple derivatives:
1
N
∆ =
ψ1rmψ2sn(p+r)1−w1(p+s)1−w2
p+tArtmAstn + (zero-mode)2 ,
rsmn
np+r + mp+s t=2
−m
dz
1
r−1
N
A
rtm =
(z − zr)
(zs − z)p+s/p+r
(z − zs)p+s/p+r
.
z
2πi z
r
− zt
s=2
s=r+1
(9.2.19)
For the third step, for open-string trees, we also need the Jacobian from
dτ i →
( dzi)V (z), which for trees can easily be calculated by considering the graph where all external states are tachyons and all but 2 strings (one incoming and one outgoing) have infinitesimal length. We can also restrict all transverse momenta to vanish, and determine dependence on them at the end of the calculation by the requirement of
Lorentz covariance. (Alternatively, we could complicate the calculation by including transverse momenta, and get a calculation more similar to that of (9.1.9).) We then have the amplitude (from nonrelativistic-style quantum mechanical arguments, or
specializing (9.2.2))
N−1
N−1
A = gN−2f (p
p−rτr
+r)
dτ i e− r=2
,
(9.2.20)
i=3
where f is a function to be determined by Lorentz covariance, the τ ’s are the interaction points, the strings 1 and N are those whose length is not infinitesimal, and we also choose z1 = ∞, zN = 0 in the transformation (9.2.16). We then solve for τr
(= −Re(ρr)), in terms of zr (in this approximation of infinitesimal lengths for all but 2 of the strings), as the finite values of ρ where the boundary “turns around”:
∂ρ
= 0
→
∂z ρr
p
N−1
p
2
ρ
+r
+r
r = p+Nln zr + p+r
ln −
z
p
.
p
r
− 1 +
+sln(zr − zs) + O
+N
p
s=2,s=r
+N
(9.2.21)
We then find, using the mass-shell condition p− = 1/p+ for the tachyon (p− is −H in nonrelativistic-style calculations)
N−1 p
N−1
N
A = gN−2 f p
−1
+r
+N
dz
(z
e
i
z2
r − zs)p+r/p+s+p+s/p+r ·
r=2
i=3
s>r=2
186
9. GRAPHS
·e pirnrs(p+,z)pis ,
(9.2.22a)
where we have now included the transverse-momentum factor nrs, whose exponential
form follows from previous arguments. Its explicit value, as well as that of f , can now be determined by the manifest covariance of the tachyonic amplitude. However,
(9.2.22a) is also the correct measure for the z-integration to be applied to (9.2.1) (or (9.2.4)), using ∆ from (9.2.19) (which is expressed in terms of the same transformation (9.2.16)). At this point we can see that Lorentz covariance determines f to be such that the factor in brackets is a constant. We then note that, using p− = −H =
−(1p 2 + N
p 2 + N
2 i
− 1)/p+, we have pr · ps = pirpis − [(p+r/p+s)(12 is
− 1) + r ↔ s].
This determines the choice of nrs which makes the amplitude manifestly covariant for tachyons:
N−1
N
1
2
1
2
dz V (z) = gN−2
dz
2 ps −1)−(p+s/p+r)( 2 pr −1)
i
z2
(zr − zs)pr·ps−(p+r/p+s)(
.
i=3
s>r=2
(9.2.22b)
Note that p
2
− dependence cancels, so pr · ps and pr
can be chosen to be the covariant
ones. Taking N=1 to compare with the tachyonic particle theory, we see this agrees
with the result (9.1.15) (after choosing also z2 = 1). It also gives the (zero-mode)2
terms which were omitted in our evaluation of ∆. (That is, we have determined both
of these factors by considering this special case.) For the case of the tachyon, we could have obtained the covariant result (9.2.22b) more directly by using covariant amputation factors, i.e., by using p− as an independent momentum instead of as the
hamiltonian (see sect. 2.5). However, the result loses its manifest covariance, even on shell, for excited states because of the usual 1/p+ interactions which result in the light-cone formalism after eliminating auxiliary fields.
As mentioned in sect. 8.1, there is an Sp(2) invariance of free string theory. In
terms of the tree graphs, which were calculated by performing a conformal map to the upper-half complex plane, it corresponds to the fact that this is the subgroup of the conformal group which takes the upper-half complex plane to itself. Explicitly, the transformation is z → (m11z + m12)/(m21z + m22), where the matrix mij is real, and
without loss of generality can be chosen to have determinant 1. This transformation also takes the real line to itself, and when combined with (9.2.16) modifies it only by adding a constant and changing the values of the zr (but not their order). In
particular, the 3 arbitrary parameters allow arbitrary values (subject to ordering) for z1, z2, and zN, which were ∞, 1, and 0. (This adds a term for z1 to (9.2.16) which was previously dropped as an infinite constant. ρ → ∞ as z → ∞ in (9.2.16) corresponds
9.2. Trees
187
to the end of the first string.) Because of the Sp(2) invariance, (9.2.22) can be
rewritten in a form with all z’s treated symmetrically: The tachyonic amplitude is
then
N
A = gN−2
r=1 dzr (z
(z
dz
i − zj )(zi − zk)(zj − zk)
r − zs)pr·ps
, (9.2.22c)
idzj dzk
1≤r<s≤N
where zi, zj, zk are any 3 z’s, which are not integrated over, and with all z’s cyclically ordered as in (9.1.14).
Closed-string trees are similar, but whereas open-string interaction points occur
anywhere on the boundary, closed-string interaction points occur anywhere on the
surface. (Light-cone coordinates are chosen so that these points always occur for
those values of τ where the strings split or join.) Thus, for closed strings there are also integrations over the σ’s of the interaction points. The amplitude corresponding to (9.1.15) or (9.2.22) for the closed string, since it has both clockwise and counterclockwise modes, has the product of the integrand for the open string (for one set
of modes) with its complex conjugate (for the modes propagating in the opposite
direction), and the integral is over both z’s and ¯
z’s (i.e., both τ ’s and σ’s). (There
are also additional factors of 1/4 in the exponents due to the different normalization of the zero modes.) However, whereas the integral in (9.1.15) for the open string is restricted by (9.1.14) so that the z’s (interaction points) lie on the boundary (the real axis), and are ordered, in the closed string case the z’s are anyplace on the surface (arbitrary complex).
We next consider the evaluation of the open-string 3-point function, which will be
needed below as the vertex in the field theory action. The 3-string vertex for the open string can be written in functional form as a δ-functional equating the coordinates of a string to those of the strings into which it splits. In the case of general string coordinates Z:
SINT = g
dτ D3Z d3p+ δ
p+ δ[˜
Z1(σ) − ˜
Z3(σ)]δ[˜
Z2(σ) − ˜
Z3(σ)]Φ[1]Φ[2]Φ[3] ,
(9.2.23)
with ˜
Z as in (9.2.7). We now use (9.2.6-16) for N= 3, with zr = ∞, 1, 0 in (9.2.16),
and p+1 with the opposite sign to p+2 and p+3. The splitting point is
∂ρ
p
= 0
→
z = z
+3
,
ρ = τ
∂z
0 = − p
0 + iπp+2
,
(9.2.24a)
+1
τ
2
0 = 1
p
) .
(9.2.24b)
2
+ ln (p+
188
9. GRAPHS
For (9.2.19), we use the integral
dz
1 dn
u(u − 1) · · · (u − n + 1)
u
z−n−1(z + 1)u =
(z + 1)u|z=0 =
=
,
0 2πi
n! dzn
n!
n
−u + n − 1
u
= (−1)n
(9.2.25)
n
n
to evaluate
1
−mp+,r+1
m > 0 :
A
p+r
r2m = p+3N rm
,
Ar3m = −p+2N rm ; N rm =
;
p+,r+1
m
m = 0 :
Art0 = δrt − δr1 .
(9.2.26)
The result is then (see, e.g., [9.4]):
1
1
p2 + M2
∆( ˇ
Ψ1, ˇ
Ψ2) = −ψ1Nψ2 − ψ1N ψ