Four-particle tree-level scattering amplitudes in string theory are magically consistent with unitarity, reflected in the non-trivial fact that beneath the critical dimension, the residues of the amplitudes on massive poles can be expanded in partial waves with all positive coefficients. While this follows (rather indirectly) from the no-ghost theorem, the simplicity of the statement and its fundamental importance for the physical consistency of string theory begs for a more direct and elementary understanding. In this note we take a step in this direction by presenting a new expression for the partial wave coefficients of string amplitudes, given by surprisingly simple double/triple contour integrals for open/closed strings. This representation allows us to directly prove unitarity of all superstring theories in $D \leq 6$ spacetime dimensions, and can also be used to determine various asymptotics of the partial waves at large mass levels.
On the unitarity of gauged non-compact WZNW strings
本文研究规范非紧WZNW弦的么正性，即用G/H' WZNW模型表示的弦论，其中G是非紧群.这些模型代表了具有一个类时分量的非平凡弯曲时空上的弦论。我们将证明，一类模型连接到埃尔米特对称空间，和一个自然的一组离散的最高权表示，BRST制定，其中的规范是通过BRST条件定义，产生酉性。幺正性要求Dynkin指数的水平乘以是整数，以及整数值的最高权w. r. t。紧子代数我们还将表明，BRST配方是不等同于传统的GKO陪集配方，通过施加一个最高权条件w.r.t. H'.后者导致了非幺正的物理弦态，据我们所知，这是两种表述之间根本区别的第一个例子。
我们研究弦的幺正性的非平凡时空背景的基础上规范WZNW模型涉及SO（n，2），规范SO（n，1）子群。作为几何陪集空间，它们是Anti de Sitter空间。我们现在的模型比以前考虑的模型更复杂，例如那些连接到厄米对称空间的模型。在本例中，类时场分量不是极大紧致子代数的中心，导致了几个额外的复杂性。假设离散表示，它将只可能考虑n偶数，导致奇数维空间。我们将证明这样的模型是免费的鬼一类离散表示。
In perturbative amplitudes in quantum field theory and string field theory, Cutkosky rule expresses the anti-hermitian part of a Feynman diagram in terms of sum over all its cut diagrams, and this in turn is used to prove unitarity of the theory. For D-instanton contribution to a string theory amplitude, the cutting rule needed for the proof of unitarity is somewhat different; we need to sum over only those cut diagrams for which all the world-sheet boundaries ending on some particular D-instanton lie on the same side of the cut. By working with the closed string effective action, obtained aft…
Higher-order effects of CPT and Lorentz violation within the SME effective framework including Myers-Pospelov dimension-five operator terms are studied. The model is canonically quantized by giving special attention to the arising of indefinite-metric states or ghosts in an indefinite Fock space. As is well-known, without a perturbative treatment that avoids the propagation of ghost modes or any other approximation, one has to face the question of whether unitarity and microcausality are preserved. In this work, we study both possible issues. We found that microcausality is preserved due to th…
We show that in the quadratic curvature theory of gravity, or simply $R_{μν} ^2$ gravity, the tree-level unitariy bound (tree unitarity) is violated in the UV region but an analog for $S$-matrix unitarity ($SS^{\dagger} = 1$) is satisfied. This theory is renormalizable, and hence the failure of tree unitarity is a counter example of Llewellyn Smith's conjecture on the relation between them. We have recently proposed a new conjecture that $S$-matrix unitarity gives the same conditions as renormalizability. We verify that $S$-matrix unitarity holds in the matter-graviton scattering at tree level…
We study tree-unitarity and renormalizability in Lifshitz-scaling theory, which is characterized by an anisotropic scaling between the spacial and time directions. Due to the lack of the Lorentz symmetry, the conditions for both unitarity and renormalizability are modified from those in relativistic theories. For renormalizability, the conventional discussion of the power counting conditions has to be extended. Because of the dependence of $S$-matrix elements on the reference frame, unitarity requires stronger conditions than those in relativistic cases. We show that the conditions for unitari…
We study the three-particle and four-particle scattering amplitudes for an arbitrary, finite number of massive scalars, spinors and vectors by employing the on-shell massive spinor formalism. We consider the most general three-particle amplitudes with energy growing behavior at most of $\mathcal{O} (E)$. This is the special case of the requirement of tree unitarity, which states that the $N$-particle scattering amplitudes at tree-level should grow at most as $\mathcal{O} (E^{4-N})$ in the high energy hard scattering limit, i.e. at fixed non-zero angles. Then the factorizable parts of the four-…
We compute the imaginary parts of genus-one string scattering amplitudes. Following Witten's $i\varepsilon$ prescription for the integration contour on the moduli space of worldsheets, we give a general algorithm for computing unitarity cuts of the annulus, Möbius strip, and torus topologies exactly in $α'$. With the help of tropical analysis, we show how the intricate pattern of thresholds (normal and anomalous) opening up arises from the worldsheet computation. The result is a manifestly-convergent representation of the imaginary parts of amplitudes, which has the analytic form expected from…
We calculate the tree-level amplitudes for electrically neutral $2\to2$ scattering for the Standard Model Higgs doublet non-minimally coupled to the Ricci scalar. We consider both the metric and the Palatini formulation of gravity. We find the partial wave unitarity limit for a general background field value. Our results are in agreement with previous work. In the electroweak vacuum, tree-level unitarity is violated at $\sim M_\text{Planck}/ξ$ in the metric formulation, and at $\sim M_\text{Planck}/\sqrtξ$ in the Palatini formulation. In the inflationary large field background, the unitarity l…
Generalized unitarity and the worldsheet S matrix in AdS_n x S^n x M^(10-2n)
与AdS（n）/CFT（n-1）对偶相关的弦论的基于可积性的解依赖于世界表S矩阵。利用广义么正性，我们构造了一个一般可积世界单理论中弦的世界单S矩阵中的一个和两个圈阶对外动量具有对数依赖性的项。我们还讨论了在更高的订单计算方面。S-矩阵元表示为系数以树级世界面四点散射振幅形式给出的积分之和。单圈有理函数不是由二维么正性切割决定的，而是由对称性考虑确定的。它们在确定双回路对数贡献中起着重要作用。我们通过计算AdS_5 x S^5、AdS_4 x CP^3、AdS_3 x S^3 x S^3 x S^1和AdS_3 x S^3 x T^4中弦论大质量世界面扇区中单圈和双圈四粒子S矩阵元的对数项来说明一般分析。我们探索的S矩阵的结构，并提供明确的证据，没有高阶的矩阵和一个循环敷料阶段的指数。
We use the S-matrix bootstrap to carve out the space of unitary, crossing symmetric and supersymmetric graviton scattering amplitudes in ten dimensions. We focus on the leading Wilson coefficient $α$ controlling the leading correction to maximal supergravity. The negative region $α<0$ is excluded by a simple dual argument based on linearized unitarity (the desert). A whole semi-infinite region $α\gtrsim 0.14$ is allowed by the primal bootstrap (the garden). A finite intermediate region is excluded by non-perturbative unitarity (the swamp). Remarkably, string theory seems to cover all (or at le…
The Coon amplitude is a deformation of the Veneziano amplitude with logarithmic Regge trajectories and an accumulation point in the spectrum, which interpolates between string theory and field theory. With string theory, it is the only other solution to duality constraints explicitly known and it constitutes an important data point in the modern S-matrix bootstrap. Yet, its basics properties are essentially unknown. In this paper we fill this gap and derive the conditions of positivity and the low energy expansion of the amplitude. On the positivity side, we discover that the amplitude switche…
