Introduction to the BRST Quantization
BRST Invariant String Path Integral
In the first quantization, the geometry of the target spacetime is given, and the string represents the oscillations propagating on this background. The first term is a manifestation of the conformal anomaly and is related to the Tz-z component.
Scattering Amplitude on the Punctured Riemann Surface
We explicitly showed the top operator coupling constants κχ to emphasize that they are different for different particle states. A normal ordered vertex operator corresponds to a string wave function defined on the edge of a small hole, which is then shrunk to a point to obtain a local operator perturbation [18]. At gth order in the string perturbation expansion, the world sheet has g handles and N punctures associated with insertions of the vertex operator.
This formulation treats the insertion points of the vertex operator za on par with ra⅛,s, and in this way we will find that all BRST anomalies can be traced to the same source: the limit of moduli space √Ma,jv∙. Cotangent space conjugate one-forms are defined by scalar products. and together with conjugate en-forms associated with module variables miandm∙. the volume form of the module space Λ4<7jjv is written as. The dilaton was found to interact with the ghosts through a derivative coupling as in Eq. 2.2.28), and the action does not say anything whether the dilaton is condensed or not.
Due to the ghost number anomaly ratio eq. 2.2.14) must take into account the insertions of bzz and cz zero modes. But this means that the path integrals of eq. 2.3.20) is multiplied by and correctly reproduces the coupling constant multiplication factor κ~λ'9.
Example - Calculation of the Dilaton Tadpoles
The inner circles of C2 and Λ∕2 are parameterized by t, and the off-shell state is described by Xμ(t). 2.4.8) The vectors na and ta denote respectively the inward normal vector and the tangent vector on the boundary ∂M. 2.4.9) The Weil-Petersson measure and the Faddeev-Popov ghost determinant can be. The factor ∣ again comes from the order of the mapping class group of the Möbius strip in unoriented open string theory. 2.4.30) Renormalizing the puncture wave function is the dilaton tail toad amplitude of RP2.
External zero-momentum tachyon tadpoles have also been calculated [26] and the out-of-shell prescription automatically ensures that the tachyon tadpole is finite. It is evident that the curvature of the world plate or, equivalently, the insertion of mechanical flow contributes to the Gell-Mann-Low ^-function of the dilaton. This essentially ensures the separation of unphysical states in the boundary on the shell, as no irregular contribution to the worldsheet symmetries can appear and the longitudinal degrees of freedom give at most surface terms that vanish as the boundary runs to infinity.
Alternatively, a semi-off-shell amplitude with open Dirichlet boundaries can be reduced to an on-shell scattering amplitude without the boundaries. Both arguments prove that unphysical states, such as the track of the graviton, never appear in the limit of the open boundary of the Dirichlet.
Degeneration of One Loop Scattering Amplitudes
We will unwrap the BRST current contour around V⅛v+ι and move it to other regions [3] using the analytical worldsheet property of the integrand in Eq. This gives a total derivative with respect to the modulus zA, which is the source of the BRST anomaly. These configurations are conformally equivalent to those obtained by the following transformation* of the complex za-moduli coordinates.
In fact, we need a more careful treatment of the coordinate transformation, since the world needs a specification of the local coordinates around the pinching point. In the degeneracy configuration, we try to deform the BRST current around the (Ar + l)-th hole. The singularity mentioned above as a surface term is an obstacle to the removal of the BRST current, and thus generates a BRST anomaly.
This arises because the conformal invariance is broken, and the amplitude involved in "each" separate Vj insertion depends explicitly on the choice of the wj position on the world sheet. One should not consider the BRST anomalies as arising from a short-range physics in spacetime.
Structure of BRST Anomalies
Finally, we look at k > 2 configurations. 3.1.17) and a discussion afterwards, we see that the BRST anomalies occur for all intermediate states for which Δ∕ ≤ 0. In general, a sufficiently timelike configuration of intermediate momentum pi will generate a large number of divergent anomalous BRST terms. The usual argument to get around this is to start with a sufficiently space-like configuration of pj so that there is no BRST anomaly at all (i.e. p2 → oo).
This configuration corresponds to the scattering amplitude in S2 with operators N vertices plus two intermediate state holes, linked together by a suitable propagator. This limit shows an infinity due to the existence of the tachyon in the bosonic string spectrum. In fact, we will find that there is no local counterterm, which can cancel out this kind of anomaly.
This problem is another indication that bosonic string theory is internally inconsistent; it certainly does not arise in superstring theories with consistent spatiotemporal origins.
General Strategy for Local Counterterms
It is also related to potential chiral anomalies in superstring theories [6]: gauge, gravitational, local Lorentz and supersymmetry anomalies. A realistic superstring must yield chiral fermions with the non-trivial local symmetry charge assignments that we observe in the low-energy world. On the other hand, these charge assignments are strongly constrained by the condition that chiral anomalies are absent in order to have gauge invariances intact at the quantum level.
The chiral anomalies mean that non-physical longitudinal components of the measurement fields propagate in the scattering amplitudes. However, we have seen that the unphysical longitudinal states of gauge fields are elements of ImQ5ps21, as discussed in Section 2.1, and that their propagation should reoccur at the limits of moduli space, by the same argument as the BRST anomalies. This can also be seen by seeing that the longitudinal components of measurement fields can be written as .
This is a total derivative of the puncture module, and we need to take care of its worldsheet short-range singularities. In fact, the BRST anomalies, the gauge anomalies, and the infinities arising to the string scattering amplitudes are all generated by the unphysical states IîïiQbrst ⅛ the indeterminate metric Hilbert space.
Fischler-Susskind Mechanism as Tadpole Counterterm
The correlation function on Srz is exactly the same as the amplitude with N + 1 node operators of the original set we had on the torus, eq. Therefore, the only source of local counterterms would be to modify the node operator ½(z) in an explicitly anomalous BRST way. The node operator Wzz = czczV(z} is BRST invariant only if. 4.3.5) Therefore, the antiterm implies a displacement of the mass shell condition of Eq.
4.4.4) is characteristic of contact interaction, which means point interaction at the boundary of module space. For example, in quantum electrodynamics, Ward's identity is satisfied only after including the contribution of the seagull term [12]. When the energy transfer beyond the degeneracy point is below the tachyon mass threshold, we cannot create any particles and must have a regular behavior of the string scattering amplitudes.
5.1.9) after a change of variables from t° zN+2∙ Now part of the potential BRST anomaly comes from the second-order processes of the one-loop anomalies in the form of "one-particle reducible" diagrams [3]. In this section, we provide a general inductive argument to extend the proof of the BRST anomaly and its cancellation to any finite order of string loop perturbation theory. Since the counter concept is local on the worldsheet, there is no special role for the basis worldsheet manifold of genus g with vertex operators.
5.3.9) is actually 'renormalized' to give a BRST invariant amplitude for all orders of the string loop expansion.
Mass Renormalization as On-Shell Counterterm
Contact Interaction as Intermediate Cut Counterterm
However, we need to identify all the overlapping divergences and the one-particle reduction diagrams in advance. One-particle reducible diagrams are those that contain strings of subdiagrams connected by a single-particle state propagator. Analogous to ordinary field theory, one particle reduction diagrams are not counted for higher order corrections.
However, in the same order, we have con. tributes coming from lower-order local counterterms. First consider reducible one-particle one- or two-loop diagrams. They have abnormal structures. We checked the cancellation for all possible degeneracy limits and confirmed the notion of irreducible one-particle diagrams in a systematic renormalization scheme.
Similarly, we can introduce the local counterterm to the other BRST anomalies we classified in Chapter 3. After subtracting them, the 'one-particle irreducible' BRST anomaly can again be canceled by a two-ring local counterterm from the Fischler mechanism - Susskind.
Scattering Amplitudes at Two Loops
Structure of Two Loop BRST Anomalies and Counterterms
Higher Loop Generalization
One-loop and two-loop analysis of BRST anomalies and local counterterms revealed that only three types of anomalies are significant: the tadpole, the two-point amplitude on the shell, and the intermediate unphysical cut. Considering the one-loop BRST anomalies from Chapter 3, we introduced zero-loop local counterpart terms (tree-level diagram) to restore BRST invariance. If there is no node operator on the one-loop part of the amplitude, it is a BRST tadpole anomaly.
The analysis in the previous two sections shows that all subleading anomalies are automatically canceled by one loop counterterms. The structure of counterterms, which we denote as Cg for g loop order counterterm in the string loop perturbation expansion, is as follows:. Note that we have introduced the local antiterms order by order in string loop perturbative expansion.
We can now proceed again with an inductive argument to show that the right-hand side of Eq. Also, this argument can be used to establish an exact non-renormalization theorem (for all orders of string loop expansion) [6j to the distribution amplitude of massless particles.
