In a series of recently published papers we reanalyzed the existing treatments of Veneziano and Veneziano-like amplitudes and the models associated with these amplitudes. In this work we demonstrate that the already obtained new partition function for these amplitudes can be exactly mapped into that for the Polychronakos-Frahm (P-F) spin chain model. This observation allows us to recover many of the existing string-theoretic models, including the most recent ones.
String configurations with nonzero winding number describe soliton string states. We compute the Veneziano amplitude for the scattering of arbitrary winding states and show that in the large radius limit the strings always scatter trivially and with no change in the individual winding numbers of the strings. In this limit, then, these states scatter as true solitons.
In this note we demonstrate how one can compute the Veneziano amplitude for bosonic string theory using the BCFW method. We use an educated ansatz for the cubic amplitude of two tachyons and an arbitrary level string state.
We have proposed a new type of string field theory. The main point of the present article is to cure some technical troubles: missing two out three terms in veneziano amplitude. Our novel string field theory, describes a theory with many strings in terms of "objects",which are not exactly, but close to Charles Thorn's string bits. The new point is that the objects in terms of which the universe states are constructed,and which have an essentially 26-momentum variable called J^μ, can have essentially the energy J^0 be also negative.We get a long way in deriving in this model the veneziano model…
We propose a number system covariance principle between $p$-adic and Archimedean frameworks. We use it to derive several closed-form expressions for the five-point open string tachyon scattering amplitude.
We detail the properties of the Veneziano, Virasoro, and Coon amplitudes. These tree-level four-point scattering amplitudes may be written as infinite products with an infinite sequence of simple poles. Our approach for the Coon amplitude uses the mathematical theory of $q$-analysis. We interpret the Coon amplitude as a $q$-deformation of the Veneziano amplitude for all $q \geq 0$ and discover a new transcendental structure in its low-energy expansion. We show that there is no analogous $q$-deformation of the Virasoro amplitude.
Bosonic string theory with the possibility for an arbitrary number of strings - i.e. a string field theory - is formulated by a Hilbert space (a Fock space), which is just that for massless noninteracting scalars. We earlier presented this novel type of string field theory, but now we show that it leads to scattering just given by the Veneziano model amplitude. Generalization to strings with fermion modes would presumably be rather easy. It is characteristic for our formulation /model that: 1) We have thrown away some null set of information compared to usual string field theory, 2)Formulated …
We consider theories of weakly interacting higher spin particles in flat spacetime. We focus on the four-point scattering amplitude at high energies and imaginary scattering angles. The leading asymptotic of the amplitude in this regime is universal and equal to the corresponding limit of the Veneziano amplitude. In this paper, we find that the first sub-leading correction to this asymptotic is universal as well. We compute the correction using a model of relativistic strings with massive endpoints. We argue that it is unique using holography, effective theory of long strings and bootstrap tec…
The Veneziano amplitude is derived from fixing one degree of freedom of $PSL(2,\mathbb{R})$ symmetry by the insertion of a mostly BRST exact operator. Evaluating the five-point function which consists of four open string tachyons and this gauge fixing operator, we find it equals the Veneziano amplitude up to a sign factor. The sign factor is interpreted as a signed intersection number.The result implies that the mostly BRST exact operator, which is originally used to provide two-point string amplitudes, correctly fixes the $PSL(2,\mathbb{R})$ gauge symmetry for general amplitudes. We conjectur…
This paper is a short summary of already submitted papers hep-th/0410242 and hep-th/0502231. It provides a self contained description of earlier obtained results for physicists with traditional mathematical background.
Positivity of the Veneziano Amplitude in Ten Dimensions
Mani L. Bhaumik Institute for Theoretical Physics Department of Physics and Astronomy University of California, Los Angeles（加州大学洛杉矶分校曼尼·L·巴乌米克理论物理学院物理与天文学系）
In a previous paper (hep-th/0304045) it has been argued that tachyonic Dirac-Born-Infeld (DBI) actions can be obtained from open string theory in a limit, which generalizes the usual massless DBI limit. In the present note we review this construction focusing on a key property of the proposed tachyon effective actions: how they reproduce appropriate Veneziano amplitudes in a suitably defined kinematical region. Possible extensions and interesting open problems are briefly discussed.
We consider weakly-coupled theories of massive higher-spin particles. This class of models includes, for instance, tree-level String Theory and Large-N Yang-Mills theory. The S-matrix in such theories is a meromorphic function obeying unitarity and crossing symmetry. We discuss the (unphysical) regime $s,t \gg 1$, in which we expect the amplitude to be universal and exponentially large. We develop methods to study this regime and show that the amplitude necessarily coincides with the Veneziano amplitude there. In particular, this implies that the leading Regge trajectory, $j(t)$, is asymptotic…
University of Science and Technology of China（中国科学技术大学）；Interdisciplinary Center for Theoretical Study, University of Science and Technology of China（中国科学技术大学理论物理交叉中心）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论研究中心）
Although QCD can be treated perturbatively in the high energy limit, lower energies requre uses of nonperturbative methods such as ADS/CFT and/or Abelian reduction.These methods are not equivalent. In this paper we provide arguments in favor of the Abelian reduction methods. This is achieved by further developing results of our recent works reanalyzing Veneziano and Veneziano-like amplitudes and the models associated with these amplitudes. It is shown, that the obtained new partition function for these amplitudes can be mapped exactly into that for the Polychronakos-Frahm spin chain model reco…
We consider the geometric transition and compute the all-genus topological string amplitudes expressed in terms of Hopf link invariants and topological vertices of Chern-Simons gauge theory. We introduce an operator technique of 2-dimensional CFT which greatly simplifies the computations. We in particular show that in the case of local Calabi-Yau manifolds described by toric geometry basic amplitudes are written as vacuum expectation values of a product vertex operators and thus appear quite similar to the Veneziano amplitudes of the old dual resonance models. Topological string amplitudes can…
The dual resonance model, which was a precursor of string theory was based upon the idea that two-particle scattering amplitudes should be expressible equivalently as a sum of contributions of an infinite number of $s$ channel poles each corresponding to a finite number of particles with definite spin, or as a similar sum of $t$ channel poles. The famous example of Veneziano \cite{ven} satisfies all these requirements, and is additionally ghost free.We recall other trajectories which provide solutions to the duality constraints, e.g. the general Mobiüs trajectories and the logarithmic trajecto…
In this part of our four parts work (e.g see Part I, hep-th/0410242) we use the theory of polynomial invariants of finite pseudo-reflection groups in order to reconstruct both the Veneziano and Veneziano-like (tachyon-free) amplitudes and the generating function reproducing these amplitudes. We demonstrate that such generating function can be recovered with help of the finite dimensional exactly solvable N=2 supersymmetric quantum mechanical model known earlier from works by Witten, Stone and others. Using the Lefschetz isomorphisms theorem we replace traditional supersymmetric calculations by…
Bootstrap AdS Veneziano Amplitude with Arbitrary Kaluza-Klein Modes
Zhejiang Institute of Modern Physics, School of Physics, Zhejiang University（浙江大学物理学院现代物理研究所）；Joint Center for Quanta-to-Cosmos Physics, Zhejiang University（浙江大学量子与宇宙学联合中心）
String theory offers an elegant and concrete realization of how to consistently couple states of arbitrarily high spin. But how unique is this construction? In this paper we derive a novel, multi-parameter family of four-point scattering amplitudes exhibiting i) polynomially bounded high-energy behavior and ii) exchange of an infinite tower of high-spin modes, albeit with a finite number of states at each resonance. These amplitudes take an infinite-product form and, depending on parameters, exhibit mass spectra that are either unbounded or bounded, thus corresponding to generalizations of the…
In a series of published papers we reanalyzed treatments of the Veneziano amplitudes and the models associated with them. In this work we demonstrate that the already obtained partition function for these amplitudes can be exactly mapped into that for the Polychronakos-Frahm spin chain which, in turn, is obtainable from the Richardson-Gaudin XXX model. Reshetikhin and Varchenko demonstrated that such a model is recoverable from their WKB-type analysis of solutions of the Knizhnik-Zamolodchikov (K-Z) equations. The linear independence of solutions of these equations is controlled by determinant…
The bosonic string theory evolved as an attempt to find physical/quantum mechanical model capable of reproducing Euler's beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In recent publication hep-th/0212189 [IJMPA 19 (2004) 1655] an attempt was made to restore the missing links. The resul…
In this paper we use the analyticity properties of the scattering amplitude in the context of the conformal mapping techniques. The Schwarz-Christoffel and Riemann-Schwarz functions are used to map the upper half -plane onto a triangle. We use the known asymptotic and threshold behaviors of the scattering amplitude to establish a connection between the values of the Regge trajectory functions and the angles of the triangle. This geometrical interpretation allows a link between values of the Regge trajectory functions and the generators of the invariance group of Moebius transformations associa…
A d-dimensional rational polytope P is a polytope whose vertices are located at the nodes of d-dimensional Z-lattice. Consider a number of points inside the inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart polynomial of P counts the number of such lattice points (nodes) inside the inflated P and (may be) at its faces (including vertices). In Part I (hep-th/0410242) of our four parts work we noticed that the Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sence of statistical mechanics) for the …
We give a careful definition of the open string propagator in Schnabl gauge and present its worldsheet interpretation. The propagator requires two Schwinger parameters and contains the BRST operator. It builds surfaces by gluing strips of variable width to the left and to the right of off-shell states with contracted or expanded local frames. We evaluate explicitly the four-point amplitude of off-shell tachyons. The computation involves a subtle boundary term, crucial to enforce the correct exchange symmetries. Interestingly, the familiar on-shell physics emerges even though string diagrams pr…
We attack the long-standing problem of finding the AdS dual of N = 2 superconformal QCD, the N=2 super Yang Mills theory with gauge group SU(N_c) and N_f = 2 N_c fundamental hyper multiplets. The theory admits a Veneziano expansion of large N_c and large N_f, with N_f/N_c and lambda = g^2 N_c kept fixed. The topological structure of large N diagrams motivates a general conjecture: the flavor-singlet sector of a gauge theory in the Veneziano limit is dual to a closed string theory; single closed string states correspond to "generalized single-trace" operators, where adjoint letters and flavor-c…
We compute the four-point function of scalar operators in CFTs with weakly broken higher spin symmetry at arbitrary 't Hooft coupling. We use the known three-point functions in these theories, the Lorentzian OPE inversion formula and crossing to fix the result up to the addition of three functions of the cross ratios. These are given by contact Witten diagrams in AdS and manifest non-analyticity of the OPE data in spin. We use Schwinger-Dyson equations to show that such terms are absent in the large $N$ Chern-Simons matter theories. The result is that the OPE data is analytic in spin up to $J=…
We provide a brief summary of a method to calculate improvements to the Veneziano Amplitude, creating sub-leading non-linearities in the Regge trajectory of states. We formulate it as an extension of a computation by Makeenko and Olesen. We begin in a confining gauge theory coupled to matter, rewriting the meson scattering amplitude as a specific path integral over shapes and sizes of closed Wilson loops using the worldline formalism. We then prescribe how to further the computation at strong coupling by employing holography, which provides a prescription for the expectation value of these Wil…
The Veneziano amplitude was put forward as a solution to the axioms of the S-matrix bootstrap. However, unitarity, reflected in the positivity of the coefficients in the Gegenbauer expansion of the amplitude is not obvious. In this note we compute the generating function of these coefficients in terms of the Appell hypergeometric function. We use this to read off an exact form of this coefficient on the leading Regge trajectory in $D=4$. We find that it decays with the spin but always remains positive. Since for large spin these coefficients are expected to be smaller than those on the sublead…
In this paper we construct the operator connecting the three vertex in half string field theory and the dual model vertex of Sciuto, Caneshi Schwimmer and Veneziano (SCSV). This construction results in an explicit conformal transformation linking the two interactions in the matter sector at all levels. Thus establishing the equivalence in the matter sector between the two theories at least for N = 3.
We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of t…
We propose a holographic computation of the $2\rightarrow 2$ meson scattering in a curved string background, dual to a QCD-like theory. We recover the Veneziano amplitude and compute a perturbative correction due to the background curvature. The result implies a small deviation from a linear trajectory, which is a requirement of the UV regime of QCD.
We consider four scalar mesons scattering in large-Nc QCD. Using the worldline formalism we show that the scattering amplitude can be written as a formal sum over Wilson loops. The AdS/CFT correspondence maps this sum into a sum over string worldsheets in a confining background. We then argue that for well separated mesons the sum is dominated by flat space configurations. Under additional assumptions about the dual string path integral we obtain the Veneziano amplitude.
We point out some common qualitative features of the Coon amplitude$\unicode{x2014}$a family of deformations of the Veneziano amplitude with logarithmic Regge trajectories$\unicode{x2014}$and the open string scattering amplitude for strings ending on a D-brane in AdS. Both reduce to the Veneziano amplitude at relatively low energies. Both systems have an accumulation point in their spectrum, with an infinite number of states below a certain energy. The approach to this point is very similar. Both have the same high-energy fixed-angle behavior. Nevertheless, we find some differences in the spec…
A relativistic quantum model of particle scattering near the horizon of a microscopic black hole unifies gravity and the harmonic-oscillator force. The model is obtained by modifying a harmonic-oscillator nonstandard Lagrangian for a closed system of relativistic quarks to eliminate any cosmic background frame. Formulated in terms of the Planck units, the model has only one parameter, the cosmic number N2. Other results include (1) quark confinement through cluster decomposition rather than a binding potential, (2) lepton substructure without violation of known experimental results, (3) a spon…
The history of discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler's beta function. Such an analogue had in fact been known in mathematics literature at least in 1922 and was studied subsequently by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function is markedly different from that described in physics literature. This paper aims to bridge the gap between the existing treatments. Preserving all results of conformal field theories in…
Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the four-point functions at arbitrary 't Hooft coupling $λ$ in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the $λ$ expansions of the correlation…
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…
Using string scattering amplitudes of open bosonic string on a single $D$-brane, we construct a local field theoretical action for tachyon fields. Cubic local interactions between various particles, belonging to the particle spectrum of string may be directly followed from three-string scattering amplitude. These cubic local interactions may generate perturbative non-local four-particle interactions, which may contribute to four-string scattering amplitude. It was observed that tachyon field in open bosonic string theory must be represented by a complex field in order to reproduce the Venezian…
A natural SL(2,Z) invariant generalization of the Veneziano amplitude in type IIB superstring theory is investigated. It includes certain perturbative and non-perturbative (D-instanton) contributions, and it reduces to the correct expressions in different limits. The singularities are poles in the $s$-$t$-$u$ channels, corresponding to the exchange of particles with a mass spectrum coinciding with that of $(p,q)$ string states. We describe the general structure of the associated perturbative corrections to the effective action.
The relation between the spin and the mass of an infinite number of particles in a $q$-deformed dual string theory is studied. For the deformation parameter $q$ a root of unity, in addition to the relation of such values of $q$ with the rational conformal field theory, the Fock space of each oscillator mode in the Fubini-Veneziano operator formulation becomes truncated. Thus, based on general physical grounds, the resulting spin-(mass)$^2$ relation is expected to be below the usual linear trajectory. For such specific values of $q$, we find that the linear Regge trajectory turns into a square-…
We review a number of perturbative calculations describing the interactions of D-branes with massless elementary string states. The form factors for the scattering of closed strings off D-branes are closely related to the Veneziano amplitude. They show that, in interactions with strings, D-branes acquire many of their physical features: the effective size of D-branes is of order the string scale and expands with the energy of the probe, while the fixed angle scattering amplitudes fall off exponentially. We also calculate the leading process responsible for the absorption of closed strings: the…
New approach to p-adic and adelic strings, which takes into account that not only world sheet but also Minkowski space-time and string momenta can be p-adic and adelic, is formulated. p-Adic and adelic string amplitudes are considered within Feynman's path integral formalism. The adelic Veneziano amplitude is calculated. Some discreteness of string momenta is obtained. Also, adelic coupling constant is equal to unity.
Infrared Consistency and the Uniqueness of String Amplitudes
National Taiwan University（国立台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；Max Planck-IAS-NTU Center for Particle Physics, Cosmology and Geometry（马克斯·普朗克研究所-台湾大学粒子物理、宇宙学与几何中心）；University of Science and Technology of China（中国科学技术大学）
We report on new results in Witten's cubic string field theory for the off-shell factor in the 4-tachyon amplitude that was not fully obtained explicitly before. This is achieved by completing the derivation of the Veneziano formula in the Moyal star formulation of Witten's string field theory (MSFT). We also demonstrate detailed agreement of MSFT with a number of on-shell and off-shell computations in other approaches to Witten's string field theory. We extend the techniques of computation in MSFT, and show that the j=0 representation of SL(2,R) generated by the Virasoro operators $L_{0},L_{\…
