String Tree Amplitudes Research Articles

On primary relations at tree-level in string theory and field theory

By the use of cyclic symmetry, KK relations and BCJ relations, one can reduce the number of independent $N$-point color-ordered tree amplitudes in gauge theory and string theory from $N!$ to $(N-3)!$. In this paper, we investigate these relations at tree-level in both string theory and field theory. We will show that there are two primary relations. All other relations can be generated by the primary relations. In string theory, the primary relations can be chosen as cyclic symmetry as well as either the fundamental KK relation or the fundamental BCJ relation. In field theory, the primary relations can only be chosen as cyclic symmetry and the fundamental BCJ relation. We will further show a kind of more general relation which can also be generated by the primary relations. The general formula of the explicit minimal-basis expansions for color-ordered open string tree amplitudes will be given and proven in this paper.

Stringy KLT relations, global symmetries, and E 7(7)-violation

We study consequences of the Kawai-Lewellen-Tye (KLT) relations applied to tree amplitudes in toroidal compactifications of string theory to four dimensions. The closed string tree amplitudes with massless external states respect a global SU(4)xSU(4) symmetry, which is enhanced to the SU(8) R-symmetry of N=8 supergravity in the field theory limit. Our analysis focuses on two aspects: (i) We provide a detailed account of the simplest SU(8)-violating amplitudes. We classify these processes and derive explicit superamplitudes for all local 5- and 6-point operators with SU(4)xSU(4) symmetry at order alpha'^3. Their origin is the dilatonic operator exp(-6 phi) R^4 in the closed-string effective action. (ii) We expand the 6-point closed string tree amplitudes to order alpha'^3 and use two different methods to isolate the SU(8)-singlet contribution from exp(-6 phi) R^4. This allows us to extract the matrix elements of the unique SU(8)-invariant supersymmetrization of R^4. Their single-soft scalar limits are non-vanishing. This demonstrates that the N=8 supergravity candidate counterterm R^4 is incompatible with continuous E_7(7) symmetry. From the soft scalar limits, we reconstruct to quadratic order the SU(8)-invariant function of scalars that multiplies R^4, and show that it satisfies the Laplace eigenvalue equation derived recently from supersymmetry and duality constraints.

Disk relations for tree amplitudes in minimal coupling theory of gauge field and gravity

KLT relations on S 2 factorize closed string amplitudes into product of open string tree amplitudes. The field theory limits of KLT factorization relations hold in minimal coupling theory of gauge field and gravity. In this paper, we consider the field theory limits of relations on D 2 . Though the relations on D 2 and KLT factorization relations hold on worldsheets with different topologies, we find the field theory limits of D 2 relations also hold in minimal coupling theory of gauge field and gravity. We use the D 2 relations to give three- and four-point tree amplitudes where gluons are minimally coupled to gravitons. We also give a further discussion on general tree amplitudes in minimal coupling theory of gauge field and gravity. In general, any tree amplitude with M gravitons in addition to N gluons can be given by pure-gluon tree amplitudes with N + 2 M legs.

Relations between closed string amplitudes at higher-order tree level and open string amplitudes

KLT relations almost factorize closed string amplitudes on S 2 by two open string tree amplitudes which correspond to the left- and the right-moving sectors. In this paper, we investigate string amplitudes on D 2 and RP 2 . We find that KLT factorization relations do not hold in these two cases. The relations between closed and open string amplitudes have new forms. On D 2 and RP 2 , the left- and the right-moving sectors are connected into a single sector. Then an amplitude with closed strings on D 2 or RP 2 can be given by one open string tree amplitude except for a phase factor. The relations depends on the topologies of the world-sheets. Under T-duality, the relations on D 2 and RP 2 give the amplitudes between closed strings scattering from D-brane and O-plane respectively by open string partial amplitudes. In the low energy limits of these two cases, the factorization relations for graviton amplitudes do not hold. The amplitudes for gravitons must be given by the new relations instead.

HOLOMORPHIC FACTORIZATION AND RENORMALIZATION GROUP IN CLOSED STRING THEORY

The prescription of Kawai, Lewellen and Tye for writing closed string tree amplitudes as sums of products of open string tree amplitudes, is applied to the worldsheet renormalization group equation. The main point is that regularization of the Minkowski (rather than Euclidean) worldsheet theory allows factorization into left-moving and right-moving sectors to be maintained. Explicit calculations are done for the tachyon and the (gauge fixed) graviton.

Gauge-covariantS-matrices for field theory and strings

S-matrices can be written Lorentz covariantly in terms of free field strengths for vector states, allowing arbitrary gauge choices. In string theory the vertex operators can be chosen so this gauge invariance is automatic. As examples we give four-vector (super)string tree amplitudes in this form, and find the field theory actions that give the first three orders in the slope.

Alternative String Theory in Twistor Space forN=4Super-Yang-Mills Theory

In this Letter, an alternative string theory in twistor space is proposed for describing perturbative $N=4$ super-Yang-Mills theory. Like the recent proposal of Witten, this string theory uses twistor worldsheet variables and has manifest spacetime superconformal invariance. However, in this proposal, tree-level super-Yang-Mills amplitudes come from open string tree amplitudes as opposed to coming from $D$-instanton contributions.

Cubic Twistorial String Field Theory

Witten has recently proposed a string theory in twistor space whose D-instanton contributions are conjectured to compute N=4 super-Yang-Mills scattering amplitudes. An alternative string theory in twistor space was then proposed whose open string tree amplitudes reproduce the D-instanton computations of maximal degree in Witten's model. In this paper, a cubic open string field theory action is constructed for this alternative string in twistor space, and is shown to be invariant under parity transformations which exchange MHV and googly amplitudes. Since the string field theory action is gauge-invariant and reproduces the correct cubic super-Yang-Mills interactions, it provides strong support for the conjecture that the string theory correctly computes N-point super-Yang-Mills tree amplitudes.

Path integral evaluation of D-brane amplitudes

See hep-th/9907179.

OFF-SHELL TREE AMPLITUDES IN HIKKO’S CLOSED STRING FIELD THEORY

Off-shell closed string tree amplitudes are obtained for the HIKKO’s theory, using the method of off-shell conformal field theory.

Selection rules in low energy string effective lagrangians

Selection rules which restrict elementary particle interactions are derived as the low-energy limit of superstring theory. A general mechanism is demonstrated, in four-dimensional string tree-amplitudes, whereby the supersymmetric trilinear Yukawa-like couplings in the effective lagrangian which violate lepton number vanish, terms which otherwise occur naturally in supersymmetric versions of the standard model. Explicit expressions for the vertex operators, and all cubic bosonic couplings involving the Yang-Mills gauge bosons and the graviton, together with the scalar and anti-symmetric tensor components of gravity are given. There are no √2 α' corrections to the three-point string tree-amplitudes, a result which eliminates any higher derivative trilinear couplings among these fields in the effective action. In this four-dimensional string theory, expressions are derived for the gravitational coupling constant κ and the Yang-Mills coupling constant g YM in terms of the two string parameters: the universal Regge slope parameter which is the origin of the length scale √2 α', and the dimensionless string coupling constant g. We find κ = 1 2 g√2α' and g YM = g .

Non-polynomial closed string field theory: loops and conformal maps

Recently, we proposed the complete classical action for the non-polynomial closed string field theory, which succesfully reproduced all closed string tree amplitudes. (The action was simultaneously proposed by the Kyoto group). In this paper, we analyze the structure of the theory. We (a) compute the explicit conformal map for all g-loop, p-puncture diagrams, (b) compute all one-loop, two-puncture maps in terms of hyper-elliptic functions, and (c) analyze their modular structure. We analyze, but do not resolve, the question of modular invariance.

Möbius infinity subtraction and effective action in the σ-model approach to closed string theory

We show how to express the closed string tree amplitudes and effective action in terms of the correlators in the 2D σ-model. The subtraction of Möbius infinities corresponds to taking ∂ ∂ ln ϵ ( ϵ is a 2D UV cutoff) of the regularized correlators. The effective action is represented as the integral of the “central charge” coefficient thus demonstrating the equivalence between the S-matrix and σ-model approaches to string (tree) equations of motion. The “ ∂ ∂ ln ϵ ” prescription provides a consistent way of computing tree tadpoles on non-trivial backgrounds and hence of studying string loop corrections to the equations of motion by equating the full tadpoles to zero.

Effective actions from the conformal invariance conditions of bosonic σ-models with graviton and dilaton backgrounds

We discuss generalised functional relations between the β-functions of bosonic σ-models propagating in graviton and dilaton backgrounds and field variations of the associated effective actions based on these backgrounds. We analyse the constraints on the possible form of the proportionality (non-degenerate) tensors, imposed by classical symmetries of the associated point-like action. As an application we derive the second order [O(α′)] effective action and we find that it is consistent with a ghost free combination for both fields. Moreover we show explicitly that this action is related to the one computed within the framework of string tree-amplitudes by field redefinitions.

A relation between tree amplitudes of closed and open strings

We derive a formula which expresses any closed string tree amplitude in terms of a sum of the products of appropriate open string tree amplitudes. This formula is applicable to the heterotic string as well as to the closed bosonic string and type II superstrings. In particular, we demonstrate its use by showing how to write down, without any direct calculation, all four-point heterotic string tree amplitudes with massless external particles.