4
3 2 1
6
5 →
226
12. GAUGE-INVARIANT INTERACTIONS
and thus, relabeling r → r + 1 (or performing an equivalent rotation of the ˆλ circle) 5
ρ =
p+rln(ˆ
z − zr) − ln 3 ,
r=1
√
1
1
√
p+r = (−1)r+1 , zr = ( 3, √ , 0, −√ , − 3, ∞) .
(12.2.22)
3
3
We can then use the same procedure as the light cone. However, it turns out to be
more convenient to evaluate the contour integrals in terms of ζ rather than ˆ
z. Also,
instead of applying (9.2.18) to (12.2.22), we apply it to the corresponding expression for ˆ
λ:
5
ρ =
αrln(ˆλ − λr) − 1iπ ,
4
r=1
p+r = (−1)r , λr = e−iπ(r−2)/3 .
(12.2.23)
Reexpressing (9.2.18) in terms of ˆ
λ, we find
∂
∂
1
1
+
ln(ˆ
λ − ˆλ′) = 1 (ˆλ3 + ˆλ′3) + (ˆλˆλ′2 + ˆλ2ˆλ′) +
+
.
∂ρ
∂ρ′
6
ˆ
λˆ
λ′2
ˆ
λ2ˆ
λ′ (12.2.24)
Using the conservation laws, the first set of terms can be dropped. Since it’s actually λ = ˆ
λ2 (or z), and not ˆ
λ, for which the string is mapped to the complex plane
(ˆ
λ describes a 6-string vertex, and thus double counts), the ln we actually want to
evaluate is
ln(λ − λ′) = ln(ˆλ − ˆλ′) + ln(ˆλ + ˆλ′) .
(12.2.25)
This just says that the general coefficients Nrs in ∆ multiplying oscillators from
string r times those from string s is related to the corresponding fictitious 6-string coefficients Nrs by
Nrs = Nrs + N r,s+3 .
(12.2.26)
The contour integrals can now be evaluated over ζ in terms of
1 + x 1/3
∞
1 + x 2/3
∞
=
a
=
b
1 − x
nxn
,
1
nxn
,
(12.2.27)
0
− x
0
These coefficients satisfy the recursion relations
(n + 1)an+1 = 2 a
b
3 n + (n − 1)an−1
,
(n + 1)bn+1 = 43 n + (n − 1)bn−1 , (12.2.28)
which can be derived by appropriate manipulations of the corresponding contour
integrals: e.g.,
dx 1
1 + x 1/3
an = 0 2πix xn 1 − x
dx 1
1 + x 1/3 ′
=
3 (1
.
(12.2.29)
2
− x2)
0 2πix xn
1 − x
12.2. Midpoint interaction
227
Because of i’s relative to (12.2.27) appearing in the actual contour integrals, we use instead the coefficients
(−1)n/2
(n even)
An = an ·
,
(12.2.30)
(−1)(n−1)/2
(n odd)
and similarly for Bn. We finally obtain an expression similar to (9.2.27), except that we must use (12.2.26), and
1
Nrsmn =
M
m + n
rsmn
,
p+r
p+s
Mr,r+t,mn = 1c
3 mnt AmBn + (−1)m+n+tBmAn
,
(−1)mRe(eit2π/3)
(m + n even)
cmnt =
.
(12.2.31)
Im(eit2π/3)
(m + n odd)
The terms for n = m = 0 or n = 0 = m can be evaluated by taking the appropriate
limit (n → m or n → 0). m = n = 0 can then be evaluated separately, using
(9.2.22b), (12.2.22), and (12.2.26). The final result is
33
∆( ˇ
Ψ1, ˇ
Ψ2) = −
′ψ1Nψ2 − 1ln
p2
,
(12.2.32)
4
24
where
′ is over r, s = 1, 2, 3 and m, n = 0, 1, . . . , ∞ except for the term m = n = 0.
As for (9.2.27), ψ refers to all sets of oscillators, with ψ replaced with p 1−w
+
ψ for
oscillators of weight w. In this case we use (12.2.21), and the p+’s are all ±1, so for the ghosts there is an extra sign factor p+rp+s for N rsmn.
There are a number of problems to resolve for this formalism: (1) In calculating
S-matrix elements, the 4-point function is considerably more difficult to calculate than in the light-cone formalism [12.9], and the conformal maps are so complicated that
it’s not yet known how to derive even the 5-point function for tachyons, although arguments have been given for equivalence to the light-cone/external-field result [12.10].
(2) It doesn’t seem possible to derive an external-field approach to interactions, since the string lengths are all fixed to be π. In the light-cone formalism the external-field approach follows from choosing the Lorentz frame where all but 2 of the string lengths (i.e., p+’s) vanish. (Thus, e.g., in the 3-string vertex 1 string reduces to a point on the boundary, reducing to a vertex as in sect. 9.1.) This is related to the fact that I of (12.2.11) is just the harmonic oscillator ground state at vanishing momentum (and length) for the light-cone formalism, but for this formalism it’s ∼ δ[X(σ)−X(π −σ)].
(3) The fact that the gauge-invariant vertex is so different from the light-cone vertex indicates that gauge-fixing to the light-cone gauge should be difficult. Furthermore,
228
12. GAUGE-INVARIANT INTERACTIONS
the light-cone formalism requires a 4-point interaction in the action, whereas this covariant formalism doesn’t. Perhaps a formalism with a larger gauge invariance exists such that these 2 formalisms are found by 2 different types of gauge choices. (4)
There is some difficulty in extending the discussion of sect. 11.1 for the closed string to the interacting case, since the usual form of the physical-state vertex requires that the vertex be related to the product of open-string vertices for the clockwise and
counterclockwise states, multiplied by certain vertex factors which don’t exist in this formalism (although they would in a formalism more similar to the light-cone one,
since the light-cone formalism has more zero-mode conservation laws). This is particularly confusing since open strings generate closed ones at the 1-loop level. However, some progress in understanding these closed strings has been made [12.11]. Also, a
general analysis has been made of some properties of the 3-point closed-string vertex required by consistency of the 1-loop tadpole and 4-string tree graphs [12.12], using techniques which are applicable to vertices more general than δ-functionals [12.13].
The gauge-fixing of this formalism with a BRST algebra that closes on shell has
been studied [12.14]. It has been shown both in the formalism of light-cone-like closed string theory [12.15] and for the midpoint-interaction open string theory [12.16] that the kinetic term can be obtained from an action with just the cubic term by giving
an appropriate vacuum value to the string field. However, whereas in the former case (barring difficulties in loops mentioned above) this vacuum value is natural because of the vacuum value of the covariant metric field for the graviton, in the latter case there is no classical graviton in the open string theory, so the existence (or usefulness) of such a mechanism is somewhat confusing.
The midpoint-interaction formulation of the open superstring (as a truncated
spinning string) has also been developed [11.3,12.17]. The supersymmetry algebra
closes only on shell, and the action apparently also needs (at least) 4-point interactions to cancel divergences in 4-point amplitudes due to coincidence of vertex operator
factors (both of which occur at the midpoint) [12.18]. Such interactions might be of the same type needed in the light-cone formulation (chapt. 10).
Exercises
(1) Check the BRST invariance of (12.2.18).
(2) Find the transformation of ln(z−z′) under the projective transformation z → az+b cz+d
(and similarly for z′). Use the conservation law
r ψ0r = 0 to show that (9.2.14)
is unaffected.
Exercises
229
(3) Derive the last term of (12.2.32).
230
REFERENCES
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