Effects of D-instantons in string amplitudes

We investigate the different energy regimes in the conjectured SL(2,Z) invariant four graviton scattering amplitude that incorporates D-instanton contributions in 10d type IIB superstring theory. We show that the infinite product over SL(2,Z) rotations is convergent in the whole complex plane s,t. For high energies s>>1, fixed scattering angle, and very weak coupling g<<1/s, the four-graviton amplitude exhibits the usual exponential suppression. As the energy approaches 1/g, the suppression gradually diminishes until there appears a strong amplification near a new pole coming from the exchange of a (p,q) string. At energies s<<1/\sqrt{g}, the pure D instanton contribution to the scattering amplitude is found to produce a factor $A_4^{Dinst}\cong \exp (c g^{3/2}e^{-{2\pi\over g}} s^3)$. At energies $1/\sqrt{g}<<s<<1/g $, the D-instanton factor becomes $A_4^{Dinst}\cong \exp (2 e^{-{2\pi\over g_s}+\pi g_s s^2})$. At higher energies s>>1/g the D-instanton contribution becomes very important, and one finds an oscillatory behavior which alternates suppression and amplification. This suggests that non-perturbative effects can lead to a high-energy behavior which is significantly different from the perturbative string behavior.


Introduction
A problem of interest is understanding what are the concrete effects that nonperturbative corrections can have in superstring theory, in particular, how they affect the high-energy behavior of string amplitudes. In ten-dimensional type IIB superstring theory, the source of non-perturbative corrections are the D instantons.
Computing the contribution of multiply-charged D instantons directly is complicated. However, combining different pieces of information, Green and Gutperle [1] conjectured the exact modular function that multiplies the R 4 term in the type IIB effective action, which exactly incorporates the infinite set of multiply-charged D instanton corrections.
One of the constraints on the effective action used by Green and Gutperle is precisely SL(2, Z) invariance. The SL(2, Z) symmetry of type IIB superstring theory requires that the effective action must be invariant under SL(2, Z) transformations to all orders in the α ′ expansion. In particular, this implies that graviton scattering amplitudes must be SL(2, Z) invariant, since there is a direct correspondence between the terms in the effective action and the momentum expansion of the scattering amplitude.
In [2,3] an SL(2, Z) invariant four graviton amplitude was constructed by applying a simple SL(2, Z) symmetrization of the tree-level string theory four graviton amplitude. The construction follows essentially the same rule used by Green and Gutperle to symmetrize the R 4 term. It was conjectured that this scattering amplitude incorporates the full series of D instanton corrections with the different D-instanton numbers. This symmetric amplitude satisfies a number of consistency conditions. In particular, corrections of perturbative origin appear with an integer power of g 2 s . This is non-trivial and does not hold for any symmetrization. It is also consistent with the conjecture that high derivative terms in the type II effective action of the form H 4k−4 R 4 should not receive perturbative contributions beyond genus k [4]. By construction, it reproduces the exact R 4 term proposed in [1], and it can be viewed as a tree-level amplitude that accounts for the exchange of (p, q) string states [5].
These (p, q) string states have a simple eleven-dimensional origin [6]. Type IIB superstring theory is obtained from M theory by compactification on a 2-torus and taking the zero area limit at fixed torus moduli. In this limit, most membrane states get an infinite mass, except a certain set of states that represent the (p, q) strings of uncompactified 10d type IIB string theory. These states are precisely the states that contribute as simple poles in the SL(2, Z) invariant amplitude of [2,3].
In this work we investigate the properties of the SL(2, Z) invariant amplitude. In particular, we factorize the pure D instanton contribution and study the high energy limit.

This gives
The scattering amplitude A sl(2) 4 adds to the Virasoro amplitude perturbative and nonperturbative contributions. They are seen explicitly by expanding the Eisenstein functions at large Using the asymptotic expansion for the Bessel function K r−1/2 , we see that the E k+1/2 (τ ) terms in the amplitude are of the form Note that the non-perturbative contributions are O(e − 2πm g s ), where m = wn is an integer number. The coefficient 2πm is crucial in order to have a one-to-one correspondence between these terms and instanton contributions. It is a remarkable fact that the product over SL(2, Z) rotations automatically generates the full series of D-instanton contributions.
We summarize the main properties of A sl(2) 4 : 1) It is SL(2, Z) invariant. This is explicit in the Einstein frame, 2) It adds perturbative g 2k s and non-perturbative O(e − 2πm g s ) corrections to the Virasoro amplitude.
3) It has simple poles in the s-t-u channels at s pq = n, t pq = n, u pq = n, n = 0, 1, 2, ... corresponding to a tree-level exchange of particles with masses (2.13) 4) It reproduces the exact (proportional to E 3/2 (τ )) R 4 term conjectured in [1], containing a one-loop correction and the full D-instanton contributions. It also reproduces the exact ζ(5)E 5/2 (τ )∇ 4 R 4 term conjectured in [18] (moreover, in [17] there was a calculation of a genus one term in ∇ 6 R 4 which was found proportional to 2ζ(3)ζ(2), which differs from the prediction of the A sl (2)  4 amplitude only by a factor of 2).
The spectrum (2.13) is the spectrum of (p, q) string states [5]: This spectrum corresponds to the zero winding sector of the spectrum studied in [6,26] for the nine-dimensional type IIB string theory. Setting τ 1 = C (0) = 0, the full spectrum in D = 9 is given by where R 10 is the radius of the compact tenth dimension. In the limit R 10 → ∞, one must set, as usual, the winding number w 10 to zero to have finite mass. The term proportional to and one gets eq. (2.14). It is important to note that a charged D string has mass M = O(1/g s ), as seen from (2.15). The neutral (p, q) strings of ten dimensions have masses given by (2.14) of order M = O(1/ √ g s ) for q = 0. This is why the product over SL(2, Z) rotations produces poles at α ′ s = O(1/g s ).
The collection of states (2.14) are the only quantum states of M-theory compactified on a 2-torus that remain of finite mass after taking the zero-area limit of the torus that leads to ten-dimensional type IIB string theory [3]. The scattering amplitude A sl(2) 4 can thus be viewed as a tree-level scattering amplitude where all these states are exchanged.
The scattering amplitude A sl(2) 4 does not describe loop effects such as discontinuity cuts (see [3,17] for discussions). In particular, it should not be a good approximation of the full scattering amplitude at g s = O(1) and αs large. It represents an improvement of the Virasoro amplitude at g s ≪ 1 (or at the S-dual situation, g s ≫ 1), where D-instanton (and some perturbative) contributions have been incorporated.
The exchange of (p, q) string states is clear from the pole structure of (2.2). It becomes manifest by writing where α ′ M 2 mn = 4|m + nτ | . This generalizes the analogous formula for the Virasoro amplitude, with δ 0 written in the form Now the sum in (2.16) contains not only the terms α ′ M 2 m = 4m, but all the terms M 2 mn , representing all (p, q) string states.

Convergence properties
The scattering amplitude A sl(2) 4 is defined through an infinite product (2.2) over a pair (p, q) of relatively prime integers, i.e. integers (p, q) having greatest common divisor equal to one. An important issue is what are the convergence properties of this product.
To study the convergence, we write We have to look at the behavior of terms with large p, q. For any given s, t, u, there are positive integers (p 0 , q 0 ) such that all terms with p > p 0 , q > q 0 have |p + qτ | ≫ s, t, u and s pq , t pq , u pq small. These terms have the behavior where we have used s +t +u = 0. The sum over p, q of |p+qτ | −3 is known to be convergent [25] (in particular, the full sum over p, q of |p + qτ | −3 defines E 3 2 (τ ), see (2.6)). Therefore the sum in (3.1) is convergent. Since we have made no assumption about s, t, the series

Approach to the different scales
Let us assume g s ≪ 1, and examine the different scales that appear as the center-ofmass energy is increased from zero. We set C (0) = τ 1 = 0, so that τ = iτ 2 = ig −1 s .

Region α
(s, t) → 1 stu , and one recovers the supergravity tree-level four graviton amplitude.
In general, for any α ′ s ≪ 1 g s , one has A sl(2) 4 (s, t) → A 0 4 (s, t), one recovers the Virasoro amplitude (2.9) with its usual properties: simple poles ats = n,t = n,ū = n, with n positive (recall that in the physical region of elastic scattering s is positive, while t, u are negative). The high energy behavior is as follows: i) High α ′ s, fixed scattering angle ϕ: ii) High α ′ s, fixed t: .

(4.2)
It is useful to split the scattering amplitude A represents the pure D-instanton contribution, terms proportional to e −2πwn/g s coming K r−1/2 in (2.10). They are given by These series converge fors < 1 g s . In the regions ≪ 1 g s , the leading behavior of δ pert is just given by the first term in the series (4.4), δ pert ∼ = 2π 2 9 (s 3 +t 3 +ū 3 ).  Fig. 1: δ(s, t) as a function ofs = α ′ s 4 at fixed scattering angle ϕ = π 2 for g s = 0.01. The straight line is δ 0 (s, t).
Assuming g s ≪ 1, one can use the asymptotic form of the Bessel function (2.11). Then one obtains that in this region α ′ s ≪ 1 g s , the D-instanton contribution is given by = 2sign(s)e − 2π g s e πg ss 2 Erf( πg ss 2 ) + (s → t) + (s → u) , where Erf is the error function. There are two regimes,s ≪ 1/ √ g s , so that πg ss 2 ≪ 1, and 1/ √ g s ≪s ≪ 1/g s . In the first case, we get This is a positive contribution in the physical region of the Mandelstam parameters, but it is negligible compared to δ 0 and δ pert . In the second case, we get This is still tiny, since in this region πg ss 2 ≪ 2π/g s .

Region
In this case one begins to see simple poles at α ′ s = m 2 + n 2 /g 2 s with n = 0. Since g s is small, there is an accumulation of poles with m = 0, 1, 2, ... near the pole at n = 1, at n = 2, etc. These poles are not seen in a coarse grain plot of the amplitude, since they appear at special points. for g s = 0.01: a) The tree-level Virasoro part δ 0 is a straight line. b) The perturbative part δ pert has cusps and is positive, giving an amplification effect. c) The D instanton part δ Dinst is still negligible ats < 320. Figure 1 shows δ(s, t) (the logarithm of the amplitude, see (2.4)) as a function of s for large s, t, u and fixed scattering angle ϕ = π 2 and for g s = 0.01. One can see that at the beginning there is the straight line with negative slope as in (4.1), reproducing the usual suppression of the Virasoro amplitude. Then δ(s, t) becomes positive, producing an amplification of the amplitude nears = 1/g s (see figure 2). As s is further increased, the amplitude diminishes; then it is amplified again nears = 3/g s .
The behavior can be understood as a combination of the effects of δ 0 and δ pert , since the D-instanton contribution δ Dinst is still negligible in this region for g s = 0.01. Figure 2 shows the two contributions separately.

Region α
To examine the behavior in this region, we again consider the three contributions δ 0 , δ pert , δ Dinst separately.
The perturbative part (4.4) can be resummed explicitly, with the result [3]: Its high energy behavior is shown in Figure 3, which indicates a behavior δ pert ∼ = const. s. More precisely, it is bounded between two straight lines: 1.22s < δ pert < 1.69s. On the other hand, we have from (4.1): δ 0 ∼ = −α ′ a 0 s . The pure D instanton part δ Dinst can be computed from δ Dinst = δ − δ 0 − δ pert , where δ(s, t) and δ pert are computed from the convergent sums (3.1) and (4.9). Numerically, one finds that δ Dinst oscillates between negative and positive values, which are of the same order of magnitude as δ 0 , δ pert . This gives rise to a behavior which alternates strong suppression and amplification of the amplitude as s is increased.
The asymptotic behavior at very large s is unclear since numerical precision is worst at high s. It is plausible that at α ′ s ≫ 1/g s there are higher genus corrections not contained in A sl(2) 4 which become important. Among the different types of corrections, there are gravitational corrections corresponding to multiple exchange of gravitons [23,27]. In the present case of high energy and fixed scattering angle, the dominant genus h contribution is known [22], though it is unclear how to resum the full series [28].
We find remarkable that the product over SL(2, Z) rotations produces a convergent, mathematically well defined amplitude, and that the infinite D-instanton sum produces significant changes in the high energy behavior. It would be interesting to understand how to incorporate higher genus corrections to A sl(2) 4 in an SL(2, Z) invariant way.