88
3 Worldsheet Path Integral: Scattering Amplitudes
Since the on-shell characters (3.73) of the BRST states and of the BRST
symmetry are intimately related to the construction of the worldsheet integral, one
can expect difficulty for going off-shell.
3.3
Summary
In this chapter, we derived general formulas for string scattering amplitudes. The
general BRST formalism has been summarized. Moreover, we gave general motivations for restricting the absolute cohomology to the smaller relative cohomology. In
Chap. 8, a more precise derivation of the BRST cohomology is worked out. It also
includes a proof of the no-ghost theorem: the ghosts and the negative-norm states (in
Minkowski signature) are unphysical particles and should not be part of the physical
states. This theorem asserts that it is indeed the case. It will also be the occasion to
recover the details of the spectrum in various cases.
3.4
Suggested Readings
• The delta function approach to the gauge fixing is described in [16, sec. 3.3, 13,
sec. 15.3.2], with a more direct computation in [12].
• The most complete references for scattering amplitudes in the path integral
formalism are [4, 16].
• Computation of the tree-level 2-point amplitude [8,18] (for discussions of 2-point
function, see [4, p. 936–7, 3, 5, 6, 17, p. 863–4]).
• The BRST quantization of string theory is discussed in [2, 14, 16, chap. 4]. For
a general discussion, see [10, 20, 23]. The use of an auxiliary field is considered
in [24, sec. 3.2].
References
1. J. Collins, A New Approach to the LSZ Reduction Formula (2019). arXiv: 1904.10923
2. B. Craps, K. Skenderis, Comments on BRST quantization of strings. J. High Energy Phys.
2005(05), 001 (2005). https://doi.org/10.1088/1126-6708/2005/05/001. arXiv: hep-th/0503038
3. P. Deligne, P. Etingof, D.S. Freed, L.C. Jeffrey, D. Kazhdan, J.W. Morgan, D.R. Morrison, E.
Witten (edd.), Quantum Fields and Strings: A Course for Mathematicians. Volume 2 (American
Mathematical Society, Providence, 1999)
4. E. D’Hoker, D.H. Phong, The geometry of string perturbation theory. Rev. Modern Phys. 60(4),
917–1065 (1988). https://doi.org/10.1103/RevModPhys.60.917
5. H. Dorn, H.-J. Otto, Two and three-point functions in Liouville theory. Nuclear Phys. B 429(2),
375–388 (1994). https://doi.org/10.1016/0550-3213(94)00352-1. arXiv: hep-th/9403141
6. H. Dorn, H.-J. Otto, Some Conclusions for Noncritical String Theory Drawn from Two- and
Three-Point Functions in the Liouville Sector (1995). arXiv: hep-th/9501019
7. A. Duncan, The Conceptual Framework of Quantum Field Theory, 1st edn. (Oxford University
Press, Oxford, 2012)
3 Worldsheet Path Integral: Scattering Amplitudes
Since the on-shell characters (3.73) of the BRST states and of the BRST
symmetry are intimately related to the construction of the worldsheet integral, one
can expect difficulty for going off-shell.
3.3
Summary
In this chapter, we derived general formulas for string scattering amplitudes. The
general BRST formalism has been summarized. Moreover, we gave general motivations for restricting the absolute cohomology to the smaller relative cohomology. In
Chap. 8, a more precise derivation of the BRST cohomology is worked out. It also
includes a proof of the no-ghost theorem: the ghosts and the negative-norm states (in
Minkowski signature) are unphysical particles and should not be part of the physical
states. This theorem asserts that it is indeed the case. It will also be the occasion to
recover the details of the spectrum in various cases.
3.4
Suggested Readings
• The delta function approach to the gauge fixing is described in [16, sec. 3.3, 13,
sec. 15.3.2], with a more direct computation in [12].
• The most complete references for scattering amplitudes in the path integral
formalism are [4, 16].
• Computation of the tree-level 2-point amplitude [8,18] (for discussions of 2-point
function, see [4, p. 936–7, 3, 5, 6, 17, p. 863–4]).
• The BRST quantization of string theory is discussed in [2, 14, 16, chap. 4]. For
a general discussion, see [10, 20, 23]. The use of an auxiliary field is considered
in [24, sec. 3.2].
References
1. J. Collins, A New Approach to the LSZ Reduction Formula (2019). arXiv: 1904.10923
2. B. Craps, K. Skenderis, Comments on BRST quantization of strings. J. High Energy Phys.
2005(05), 001 (2005). https://doi.org/10.1088/1126-6708/2005/05/001. arXiv: hep-th/0503038
3. P. Deligne, P. Etingof, D.S. Freed, L.C. Jeffrey, D. Kazhdan, J.W. Morgan, D.R. Morrison, E.
Witten (edd.), Quantum Fields and Strings: A Course for Mathematicians. Volume 2 (American
Mathematical Society, Providence, 1999)
4. E. D’Hoker, D.H. Phong, The geometry of string perturbation theory. Rev. Modern Phys. 60(4),
917–1065 (1988). https://doi.org/10.1103/RevModPhys.60.917
5. H. Dorn, H.-J. Otto, Two and three-point functions in Liouville theory. Nuclear Phys. B 429(2),
375–388 (1994). https://doi.org/10.1016/0550-3213(94)00352-1. arXiv: hep-th/9403141
6. H. Dorn, H.-J. Otto, Some Conclusions for Noncritical String Theory Drawn from Two- and
Three-Point Functions in the Liouville Sector (1995). arXiv: hep-th/9501019
7. A. Duncan, The Conceptual Framework of Quantum Field Theory, 1st edn. (Oxford University
Press, Oxford, 2012)