Completing the linear-dilaton regularization of light-cone gauge superstring amplitudes: extension to heterotic and type I strings, and infrared divergences of the critical limit

Light-cone gauge closed string field theory, built from Mandelstam's interacting-string picture, defines multiloop superstring amplitudes using only cubic string interactions, but its Feynman amplitudes diverge where worldsheet supercurrent insertions collide at the interaction points. Ishibashi and Murakami (JHEP 01 (2017) 034) regularized these contact-term divergences by keeping ten spacetime dimensions but placing the transverse worldsheet theory in a linear dilaton background with dilaton profile $\Phi_{\rm dilaton}=-iQX^1$ and $Q^2>10$, proving that the regularized multiloop amplitudes coincide with the covariant first-quantized amplitudes under analytic continuation $Q\to0$ for even spin structures and (NS,NS) external states. Follow-up work by the same group (Ishibashi and Murakami on odd spin structures; Ishibashi alone on type II strings) subsequently resolved the odd spin structures and, for type II strings, Ramond-sector external lines. What remains open in making this regularization scheme complete is: (i) the extension to heterotic and type I superstrings, where for heterotic strings the light-cone gauge integrand fails to holomorphically factorize; and (ii) establishing under what conditions the critical limit $Q\to0$ exists, i.e. the treatment and classification of the infrared divergences produced by degenerating Riemann surfaces at the boundaries of moduli space, including the conditionally convergent integrals they generate.

Background

In perturbative string theory, scattering amplitudes at genus (number of handles) $g$ are integrals over the moduli space of Riemann surfaces of genus $g$ with $n$ punctures, where the moduli space is the space of complex structures on the surface. Two equivalent-looking routes define these integrals. In the covariant first-quantized approach one integrates correlation functions of vertex operators (operators encoding external string states inserted at the punctures). For superstrings this route must be handled carefully: the standard computational device is the picture-changing operator (PCO), an operator that converts vertex operators between equivalent representations called pictures, and whose insertion positions must be patched across overlapping regions of moduli space to avoid spurious singularities; the Sen-Witten prescription makes this patching precise and defines the reference amplitudes used below. In the second-quantized approach, string field theory (SFT) promotes the full tower of string states to a field; in light-cone gauge one fixes the worldsheet coordinate parametrization using the light-cone variables $X^\pm=(X^0\pm X^{d-1})/\sqrt{2}$, and, following Mandelstam's interacting-string picture, the closed-string field theory action contains only cubic interaction vertices. Its Feynman amplitudes integrate over a decomposition of moduli space into cells associated with light-cone string diagrams.

The superstring is usually described by the Neveu-Schwarz-Ramond (NSR) formalism, in which the worldsheet theory is a superconformal field theory (a 2D quantum field theory invariant under supersymmetric extensions of conformal transformations) of bosons $X^\mu$ and their worldsheet fermion partners. Bosonic spacetime states come from the Neveu-Schwarz (NS) sector and spacetime fermions from the Ramond (R) sector; on a genus-$g$ surface the worldsheet fermions have $2^{2g}$ consistent sign choices around the cycles, called spin structures, which are separated into even and odd classes. Light-cone gauge NSR string field theory amplitudes suffer from contact term divergences: short-distance singularities where supercurrent insertions collide at the interaction points of the string diagram (Greensite-Klinkhamer; Green-Seiberg).

A long program initiated by Baba, Ishibashi and Murakami regularizes these divergences by dimensional regularization: one continues the worldsheet theory away from the critical dimension (10 for superstrings), where the mismatch of central charges (the constant controlling the conformal anomaly of the worldsheet theory) spoils conformal invariance of the longitudinal variables, but a specially constructed nonstandard worldsheet theory for the longitudinal variables restores a BRST-invariant (i.e. invariant under the Becchi-Rouet-Stora-Tyutin charge encoding gauge symmetry) description. The refined scheme of Ishibashi and Murakami keeps ten spacetime dimensions and instead moves the transverse worldsheet theory into a linear dilaton background $\Phi_{\rm dilaton}=-iQX^1$ (a background in which the string coupling varies linearly along a spacelike direction $X^1$), which lowers the central charge of the transverse theory from its critical value $12$ to $12-12Q^2$; with the Feynman $i\varepsilon$ prescription and $Q^2>10$ the central charge is large and negative, and all genus-$g$ amplitudes are finite. The amplitudes are defined as analytic functions of $Q^2$ and analytically continued back to $Q\to0$ to recover the critical-dimension results. Proven so far: multiloop bosonic amplitudes (2013); NSR superstring amplitudes for even spin structures with (NS,NS) external lines (2017); odd spin structures, where the matching covariant expression requires two vertex operators in nonstandard pictures (2018); and type II superstrings including Ramond-sector external lines (2018). In all cases the regularized light-cone amplitudes $\mathcal{A}^{(g)}_N(Q)$ equal the Sen-Witten amplitudes $\mathcal{A}^{(g)\,{\rm SW}}_N(0)$ in the limit $Q\to0$, provided that limit exists.

Two gaps remain in this program. First, heterotic strings (whose left-moving sector is a superstring and right-moving sector a bosonic string) and type I open-plus-closed superstrings are not covered: for heterotic strings the integrand should factorize into a holomorphic superstring part times an antiholomorphic bosonic part, but in the light-cone formulation the uniformizing $\rho$-coordinate of the string diagram depends on the antiholomorphic moduli, so holomorphic factorization fails and no candidate integrand is known. Second, the $Q\to0$ equality holds only when the critical amplitude is well-defined: regions of moduli space where the surface degenerates (long thin handles or tubes, i.e. propagation of nearly on-shell states; a separating degeneration pinches the surface into two disconnected components, a nonseparating one only lengthens a handle) can produce infrared divergences and conditionally convergent integrals that the scheme regularizes at $Q\neq0$ but does not classify or control at $Q=0$.

Problem Statement

Within the program of defining multiloop superstring perturbation theory by light-cone gauge string field theory regularized with the linear dilaton background $\Phi_{\rm dilaton}=-iQX^1$ (amplitudes finite for $Q^2>10$, analytically continued to $Q\to0$), the problem is to close the two remaining gaps that separate the scheme from a complete definition equivalent to the covariant first-quantized formulation. The two requested results serve one common objective, namely that every genus-$g$ multiloop amplitude of each ten-dimensional superstring theory be well-defined by this scheme and equal to the corresponding Sen-Witten first-quantized amplitude, with controlled infrared behavior at the critical limit:

(1) Heterotic and type I extension. Construct the multiloop light-cone gauge amplitudes for heterotic and type I superstrings within the linear-dilaton regularization, for arbitrary genus, all spin structures, and arbitrary NS and Ramond external states, and prove (i) finiteness for $Q^2>10$ and (ii) equality, under analytic continuation in $Q^2$, with the first-quantized amplitudes in the $Q\to0$ limit, provided that limit exists. The concrete identified obstruction is holomorphic factorization: the heterotic integrand should be the product of the holomorphic superstring part and the antiholomorphic bosonic part, but the uniformizing $\rho$-coordinate of the light-cone diagram depends on the antiholomorphic moduli.

(2) Infrared divergences of the critical limit. Determine under what conditions on external states, genus, and vacuum the limit $\lim_{Q\to0}\mathcal{A}_N^{(g)}(Q)$ exists; identify and classify the divergences arising from boundary regions of moduli space (separating and nonseparating degenerations, i.e. long tubes carrying nearly on-shell states, including massless tadpole configurations) and the conditionally convergent integrals they generate; and state how the linear-dilaton regularization treats each class, in particular what remains divergent at $Q=0$ and how this compares with the covariant super-Riemann-surface analysis.

The two parts are posed as one objective: both the heterotic and type I extension and the analysis of the $Q\to0$ divergences are needed for the scheme to be complete. A fully gauge-invariant string field theory to which the method applies (motivated by the unbounded cubic light-cone Hamiltonian and the singular $p^+=0$ region) remains open as a separate programmatic direction and is not a required deliverable of this problem.

References

1. N. Ishibashi and K. Murakami, "Multiloop Amplitudes of Light-cone Gauge NSR String Field Theory in Noncritical Dimensions," J. High Energy Phys. 2017 (2017) 034, DOI 10.1007/jhep01(2017)034, arXiv:1611.06340, https://arxiv.org/abs/1611.06340
2. N. Ishibashi and K. Murakami, "Multiloop Amplitudes of Light-cone Gauge Superstring Field Theory: Odd Spin Structure Contributions," J. High Energy Phys. 2018 (2018) 063, DOI 10.1007/JHEP03(2018)063, arXiv:1712.09049, https://arxiv.org/abs/1712.09049
3. N. Ishibashi, "Multiloop Amplitudes of Light-cone Gauge String Field Theory for Type II Superstrings," 2018, arXiv:1810.03801 (unpublished preprint), https://arxiv.org/abs/1810.03801
4. N. Ishibashi and K. Murakami, "Multiloop Amplitudes of Light-cone Gauge Bosonic String Field Theory in Noncritical Dimensions," J. High Energy Phys. 2013 (2013) 053, DOI 10.1007/jhep09(2013)053, arXiv:1307.6001, https://arxiv.org/abs/1307.6001
5. Y. Baba, N. Ishibashi and K. Murakami, "Light-Cone Gauge Superstring Field Theory and Dimensional Regularization," J. High Energy Phys. 2009 (2009) 035, DOI 10.1088/1126-6708/2009/10/035, arXiv:0906.3577, https://arxiv.org/abs/0906.3577
6. N. Ishibashi and K. Murakami, "Light-cone Gauge NSR Strings in Noncritical Dimensions II -- Ramond Sector," J. High Energy Phys. 2011 (2011) 008, DOI 10.1007/JHEP01(2011)008, arXiv:1011.0112, https://arxiv.org/abs/1011.0112
7. N. Ishibashi, "Light-cone gauge superstring field theory in linear dilaton background," Prog. Theor. Exp. Phys. 2017 (2017) 033B01, DOI 10.1093/ptep/ptx012, arXiv:1605.04666, https://arxiv.org/abs/1605.04666
8. N. Ishibashi and K. Murakami, "Worldsheet theory of light-cone gauge noncritical strings on higher genus Riemann surfaces," J. High Energy Phys. 2016 (2016) 087, DOI 10.1007/JHEP06(2016)087, arXiv:1603.08337, https://arxiv.org/abs/1603.08337
9. A. Sen and E. Witten, "Filling the gaps with PCO's," J. High Energy Phys. 2015 (2015) 004, DOI 10.1007/JHEP09(2015)004, arXiv:1504.00609, https://arxiv.org/abs/1504.00609
10. A. Sen, "Gauge Invariant 1PI Effective Action for Superstring Field Theory," J. High Energy Phys. 2015 (2015) 022, DOI 10.1007/JHEP06(2015)022, arXiv:1411.7478, https://arxiv.org/abs/1411.7478
11. C. de Lacroix, H. Erbin, S. P. Kashyap, A. Sen and M. Verma, "Closed Superstring Field Theory and its Applications," Int. J. Mod. Phys. A 32 (2017) 1730021, DOI 10.1142/S0217751X17300216, arXiv:1703.06410, https://arxiv.org/abs/1703.06410
12. A. Sen, "Supersymmetry Restoration in Superstring Perturbation Theory," J. High Energy Phys. 2015 (2015) 075, DOI 10.1007/JHEP12(2015)075, arXiv:1508.02481, https://arxiv.org/abs/1508.02481
13. E. Witten, "More On Superstring Perturbation Theory: An Overview Of Superstring Perturbation Theory Via Super Riemann Surfaces," 2013, arXiv:1304.2832, https://arxiv.org/abs/1304.2832
14. M. B. Green and N. Seiberg, "Contact Interactions in Superstring Theory," Nucl. Phys. B 299 (1988) 559, DOI 10.1016/0550-3213(88)90549-4, https://doi.org/10.1016/0550-3213(88)90549-4
15. G. Giribet, N. Labranche and J. La Madrid, "Remarks on the two-point string amplitudes," Phys. Rev. D 107 (2023) 106021, DOI 10.1103/PhysRevD.107.106021, arXiv:2303.15658, https://arxiv.org/abs/2303.15658
16. N. Ishibashi, "The Fokker-Planck formalism for closed bosonic strings," Prog. Theor. Exp. Phys. 2023 (2023) 023B05, DOI 10.1093/ptep/ptad014, arXiv:2210.04134, https://arxiv.org/abs/2210.04134
17. N. Ishibashi, "Strebel Differentials and String Field Theory," Prog. Theor. Exp. Phys. 2024 (2024) 073B02, DOI 10.1093/ptep/ptae099, arXiv:2402.09641, https://arxiv.org/abs/2402.09641
18. S. Mandelstam, "Interacting String Picture of the Neveu-Schwarz-Ramond Model," Nucl. Phys. B 69 (1974) 77, DOI 10.1016/0550-3213(74)90127-8, https://doi.org/10.1016/0550-3213(74)90127-8
19. J. Greensite and F. R. Klinkhamer, "New Interactions for Superstrings," Nucl. Phys. B 281 (1987) 269, DOI 10.1016/0550-3213(87)90256-2, https://doi.org/10.1016/0550-3213(87)90256-2