Infrared-finite inclusive observables and perturbative unitarity of superstring field theory in four or fewer non-compact dimensions

The Cutkosky-rule program of Pius and Sen (2016), combined with Sen's proofs of the reality of the superstring field theory action and of the decoupling of unphysical and pure-gauge intermediate states, established perturbative unitarity of heterotic and type II superstring field theories in more than four non-compact space-time dimensions. In four or fewer non-compact dimensions, however, the massless supergravity-sector states make the S-matrix infrared divergent, and the unitarity statement must be reformulated for inclusive observables — cross sections summed over energetically degenerate soft final states and averaged over degenerate initial states — in analogy with the Kinoshita–Lee–Nauenberg theorem of quantum field theory. This inclusive reformulation has never been formulated or proven in superstring field theory: the paper that completed the unitarity proof explicitly defers it to future work, and the only concrete result is a two-dimensional type 0B string-model computation (type 0B being a non-supersymmetric cousin of the type II strings) in which the required final-state sum includes rolling-tachyon configurations — time-dependent backgrounds in which the tachyon field rolls down its potential — with no description as finite collections of closed strings. The problem is to define the infrared-safe inclusive observables within superstring field theory and prove their infrared finiteness and unitarity to all orders in perturbation theory.

Background

String field theory (SFT) rewrites the perturbation expansion of string theory as the Feynman-diagram expansion of a space-time quantum field theory whose basic field is the entire string state, so that each interaction vertex is an integral over a bounded region of the moduli space of punctured Riemann surfaces. For closed bosonic strings such a gauge-invariant quantum action was constructed by Zwiebach (1993), and for the heterotic and type II superstrings — in the Ramond–Neveu–Schwarz formulation, the standard world-sheet description with world-sheet supersymmetry in which so-called picture-changing operators (PCOs, operators inserted at the punctures to match the superghost charge between different vertices) must be chosen — a Batalin–Vilkovisky-quantized master action (a gauge-fixing formalism ensuring quantum consistency of theories with gauge symmetry) was constructed by Sen (2015). In the Siegel gauge — the gauge in which the string field is annihilated by the positive ghost modes $b_0$ and $\bar b_0$, so that each propagator is proportional to $(L_0+\bar L_0)^{-1}$, where the Virasoro generator $L_0$ has an eigenvalue containing the space-time mass squared — the propagator has poles exactly at the string spectrum's masses.

A crucial construction device is the choice of stubs: rescaling the local coordinate at each puncture of a vertex by a small factor $\beta$ multiplies the vertex by $\beta^{L_0+\bar L_0}$, which suppresses it exponentially when any external line carries large space-like momentum, making ultraviolet finiteness of each Feynman diagram manifest. The price is that vertices grow exponentially for large time-like momenta, so the loop-energy integrations cannot run along the real axis: they must reach infinity along the imaginary axis and be deformed around propagator poles when the external energies are analytically continued from imaginary (Euclidean) to real (Lorentzian) values. Pius and Sen (2016) analyzed the general class of quantum field theories with these properties and proved that amplitudes defined by this contour prescription satisfy Cutkosky rules to all orders. Cutkosky rules (cutting rules) state that the anti-Hermitian part of the T-matrix, with $S=1-iT$, is a sum of cut diagrams in which each cut propagator is replaced by an on-shell delta function times an energy-flow step function, so that the sum represents matrix elements of $T^\dagger T$ with a complete set of intermediate states inserted.

For a gauge theory such as SFT, cutting rules alone do not imply unitarity, because the Siegel-gauge propagator propagates unphysical, auxiliary and pure-gauge states as well as physical ones. The complete proof of perturbative unitarity therefore required three steps, each carried out in the program to which the source paper belongs: (i) reality of the superstring field theory action — needed to relate amplitudes to their Hermitian conjugates by contour complex conjugation — was proved by Sen (2016) by determining explicit reality conditions on the string fields of the heterotic and type II actions; (ii) the Cutkosky rules under this assumption were proved by Pius and Sen (2016); and (iii) the cancellation of the contributions of unphysical and pure-gauge states in the cut diagrams, using the Ward identities (the gauge-symmetry relations of the quantum theory), was proved by Sen (2016) with full account of mass and wave-function renormalization and of possible shifts of the vacuum. The loop-energy contour prescription was also proved equivalent, to all orders, to the world-sheet $i\varepsilon$-type prescription of Witten — the string-theory analogue of the Feynman $i\varepsilon$ prescription, implemented by deforming the integration cycles in the complexified moduli space of Riemann surfaces — tying the unitary amplitude definition to the standard world-sheet approach, and the analyticity properties of the off-shell momentum-space Green's functions needed in this analysis were proved to all orders by de Lacroix, Erbin and Sen. The 2024 review by Sen and Zwiebach summarizes the outcome: string amplitudes defined by the contour prescription are unitary to all orders in perturbation theory.

This theorem, however, lives in the regime where the S-matrix is infrared finite. The superstring spectrum contains massless states — the graviton and its superpartners — whose exchange produces infrared (low-energy, long-distance) divergences whenever the number $D$ of non-compact space-time dimensions is small. For $D>4$ the integration measure over spatial momenta suppresses the configurations with vanishing spatial momentum that generate the divergences, and the proof goes through. For $D\le4$ the S-matrix itself is ill defined, as in ordinary quantum field theory. The field-theory resolution is the Kinoshita–Lee–Nauenberg (KLN) theorem: although individual amplitudes are infrared divergent, cross sections summed over all final states degenerate in energy with the detected particles (arbitrarily soft massless quanta, and in $D=4$ also collinear configurations), and averaged over degenerate initial states, are finite and respect unitarity. In superstring field theory this inclusive reformulation has never been carried out; the paper that completed the unitarity proof explicitly excludes it, stating that the inclusive treatment "has not yet been worked out in superstring field theory." The only concrete result in this direction is Sen's computation of infrared-finite semi-inclusive cross sections in two-dimensional type 0B string theory (a non-supersymmetric string model), where processes induced by D-instantons — D-branes, extended objects on which open strings can end, wrapped here on all spatial dimensions so that their contributions are non-perturbatively small in the string coupling — produce rolling-tachyon final states (time-dependent configurations in which the tachyon field rolls down its potential) that cannot be regarded as finite linear combinations of closed string states — a warning that the degenerate-state sum in string theory can contain genuinely stringy elements. Since four-dimensional compactifications are the physically interesting superstring vacua, the inclusive statement is the relevant form of perturbative unitarity for them.

Problem Statement

For the superstring field theories of the heterotic and type II strings compactified so that $D\le4$ space-time dimensions remain non-compact, formulate the infrared-safe inclusive observables and prove their perturbative unitarity. Concretely: (1) define, in the string field theory state space, the inclusive cross section in which a fixed set of detected hard string states is accompanied by a sum over all energetically degenerate unobserved states — arbitrary numbers of arbitrarily soft massless string states, including collinear configurations in $D=4$, and any other states degenerate with them in the soft limit — together with the corresponding average over degenerate initial states; (2) prove that this inclusive cross section is infrared finite to all orders in the string field theory perturbation expansion defined by the loop-energy contour prescription of Pius and Sen (or another precisely specified, equivalently defined prescription); and (3) prove that the inclusive rates satisfy the unitarity (optical-theorem) relation that follows from the Cutkosky cutting rules combined with the soft degeneracy sums. The analysis must control the region where massless internal lines carry vanishing spatial momentum — precisely where the fixed-spatial-momentum pinch analysis underlying the cutting rules breaks down — and must remain compatible with the Ward-identity decoupling of unphysical and pure-gauge states after the inclusive sum is imposed.

References

1. Roji Pius and Ashoke Sen, "Cutkosky rules for superstring field theory," JHEP 10 (2016) 024 [erratum: JHEP 09 (2018) 122, doi:10.1007/JHEP09(2018)122], doi:10.1007/JHEP10(2016)024, arXiv:1604.01783, https://arxiv.org/abs/1604.01783
2. Ashoke Sen, "Reality of Superstring Field Theory Action," JHEP 11 (2016) 014, doi:10.1007/JHEP11(2016)014, arXiv:1606.03455, https://arxiv.org/abs/1606.03455
3. Ashoke Sen, "Unitarity of Superstring Field Theory," JHEP 12 (2016) 115, doi:10.1007/JHEP12(2016)115, arXiv:1607.08244, https://arxiv.org/abs/1607.08244
4. Ashoke Sen, "BV Master Action for Heterotic and Type II String Field Theories," JHEP 02 (2016) 087, doi:10.1007/JHEP02(2016)087, arXiv:1508.05387, https://arxiv.org/abs/1508.05387
5. Ashoke Sen, "Equivalence of Two Contour Prescriptions in Superstring Perturbation Theory," JHEP 04 (2017) 025, doi:10.1007/JHEP04(2017)025, arXiv:1610.00443, https://arxiv.org/abs/1610.00443
6. Edward Witten, "The Feynman iε in String Theory," JHEP 04 (2015) 055, doi:10.1007/JHEP04(2015)055, arXiv:1307.5124, https://arxiv.org/abs/1307.5124
7. Corinne de Lacroix, Harold Erbin and Ashoke Sen, "Analyticity and Crossing Symmetry of Superstring Loop Amplitudes," JHEP 05 (2019) 139, doi:10.1007/JHEP05(2019)139, arXiv:1810.07197, https://arxiv.org/abs/1810.07197
8. Ashoke Sen, "Cutkosky Rules and Unitarity (Violation) in D-instanton Amplitudes," JHEP 07 (2021) 205, doi:10.1007/JHEP07(2021)205, arXiv:2012.00041, https://arxiv.org/abs/2012.00041
9. Ashoke Sen, "Infrared finite semi-inclusive cross section in two dimensional type 0B string theory," JHEP 04 (2023) 101, doi:10.1007/JHEP04(2023)101, arXiv:2208.07385, https://arxiv.org/abs/2208.07385
10. Ashoke Sen and Barton Zwiebach, "String Field Theory: A Review," arXiv:2405.19421 (2024), https://arxiv.org/abs/2405.19421
11. Barton Zwiebach, "Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation," Nucl. Phys. B 390 (1993) 33, doi:10.1016/0550-3213(93)90388-6, https://doi.org/10.1016/0550-3213(93)90388-6
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