High-energy Massive String Scatterings from Orientifold Planes

We calculate bosonic massive closed string states at arbitrary mass levels scattered from Orientifold planes in the high-energy, fixed angle limit. For the case of O-particle scatterings, we obtain infinite linear relations among high-energy scattering amplitudes of different string states. We also confirm that there exist only closed string Regge poles in the form factor of the O-particle amplitudes as expected. For the case of O-domain-wall scatterings, we find that, like the well-known D-instanton scatterings, the amplitudes behave like field theory scatterings, namely UV power-law without infinite Regge poles. In addition, we discover that there exist only finite number of t-channel closed string poles in the form factor of O-domain-wall scatterings, and the masses of the poles are bounded by the masses of the external legs. We thus confirm that all massive closed string states do couple to the O-domain-wall.


I. INTRODUCTION
Being a consistent theory of quantum gravity, string theory is remarkable for its soft ultraviolet structure. This is mainly due to two closely related fundamental characteristics of high-energy string scattering amplitudes. The first is the softer exponential fall-off behavior of the form factors of high-energy string scatterings in contrast to the power-law (hard) behavior of point particle field theory scatterings. The second is the existence of infinite Regge poles in the form factor of string scattering amplitudes. Recently high-energy, fixed angle string scattering amplitudes [1,2,3] was reinvestigated for massive string states at arbitrary mass levels [4,5,6,7,8,9,10,11,12]. An infinite number of linear relations among string scattering amplitudes were obtained. The most important new ingredient of these calculations is the zero-norm states (ZNS) [13,14,15] in the old covariant first quantized (OCFQ) string spectrum. The existence of these infinite linear relations constitutes the third fundamental characteristics of high energy string scatterings. Other approaches related to this development can be found in [16], These linear relations persist [17] for string scattered from generic Dp-brane [18] except D-instanton and D-domain-wall. For the scattering of D-instanton, the form factor exhibits the well-known power-law behavior without Regge pole structure, and thus resembles a field theory amplitude. For the special case of D-domain-wall scattering [19], it was discovered [20] that its form factor behaves as power-law with infinite open Regge pole structure at high energies. This discovery makes D-domain-wall scatterings an unique example of a hybrid of string and field theory scatterings. Moreover, it was shown [20] that the linear relations break down for the D-domain-wall scattering due to this unusual power-law behavior. This result seems to imply the coexistence of linear relations and soft UV structure of string scatterings.
Recent study of high-energy scatterings of compatified closed string justified this conjecture [21]. In order to further uncover the mysterious relations among these three fundamental characteristics of string scatterings, namely, the soft UV structure, the existence of infinite Regge poles and the newly discovered linear relations stated above, it will be important to study more string scatterings, which exhibit the unusual behaviors in the high energy limit.
In this paper, we calculate massive closed string states at arbitrary mass levels scattered from Orientifold planes in the high-energy, fixed angle limit. The scatterings of massless states from Orientifold planes were calculated previously by using the boundary states for-malism [22,23], and more recently [24] on the worldsheet of real projected plane RP 2 .
Many speculations were made about the scatterings of massive string states, in particular, for the case of O-domain-wall scatterings. It is one of the purposes of this paper to clearify these speculations and to discuss their relations with the three fundamental characteristics of high-energy string scatterings stated above. For the generic Op-planes with p ≥ 0, one expects to get the infinite linear relations except O-domain-wall scatterings. For simplicity, we consider only the case of O-particle scatterings. For the case of O-particle scatterings, we obtain infinite linear relations among high-energy scattering amplitudes of different string states. We also confirm that there exist only t-channel closed string Regge poles in the form factor of the O-particle scatterings amplitudes as expected. For the case of O-domain-wall scatterings, we find that, like the well-known D-instanton scatterings, the amplitudes behave like field theory scatterings, namely UV power-law without Regge pole. In addition, we discover that there exist only finite number of t-channel closed string poles in the form factor of O-domain-wall scatterings, and the masses of the poles are bounded by the masses of the external legs. We thus confirm that all massive closed string states do couple to the O-domain-wall as was conjectured previously [19,24]. This is also consistent with the boundary state descriptions of O-planes. For both cases of O-particle and O-domain-wall scatterings, we confirm that there exist no s-channel open string Regge poles in the form factor of the amplitudes as O-planes were known to be not dynamical. However, the usual claim that there is a thinkness of order √ α ′ for the O-domain-wall is misleading as the UV behavior of its scatterings is power-law instead of exponential fall-off. This paper is organized as following. In section II, we write down a class of high-energy vertex operators at general mass levels for the scatterings of Orientifold planes. We then calculate the scattering from O-particle. In section III, we calculate the scatterings from O-domain-wall and discuss the pole structure in the form factor. A brief conclusion and discussion are given in section IV.

II. HIGH-ENERGY O-PARTICLE SCATTERINGS
We will use the real projected plane RP 2 as the worldsheet diagram for the scatterings of Orientifold planes. The standard propagators of the left and right moving fields are In addition, there are also nontrivial correlator between the right and left moving fields as well As in the usual convention [18], the matrix D reverses the sign for fields satisfying Dirichlet boundary condition. The wave functions of a tensor at general mass level can be written as The vertex operators corresponding to the above wave functions are For simplicity, we are going to calculate one tachyon and one massive closed string state scattered from the O-particle in the high-energy limit. One expects to get similar results for the generic Op-plane scatterings with p ≥ 0 except O-domain-wall scatterings, which will be discussed in section III. For this case D µν = −δ µν , and the kinematic setup are 10) where e P , e L and e T are polarization vectors of the tensor state k 2 on the high-energy scattering plane. One can easily calculate the following kinematic relations in the highenergy limit We define 22) and the Mandelstam variables can be calculated to be In the high-energy limit, we will consider an incoming tachyon state k 1 and an outgoing tensor state k 2 of the following form For simplicity, we have omited above a possible high-energy vertex (α L −1 ) r ⊗ (α L −1 ) r ′ [17,21]. For this case, with momentum conservation on the O-planes, we have

30)
(2.31) One can easily see that To fix the modulus group on RP 2 , choosing z 1 = r and z 2 = 0 and we have Similarly, for the case of A 2 , we have The scattering amplitude on RP 2 can therefore be calculated to be The integral in Eq.(2.36) can be calculated as following where we have used the following identities of the hypergeometric function F (α, β, γ, x) To further reduce the scattering amplitude into beta function, we use the momentum conservation in Eq.(2.27) and the identity (1 + α) F (−α, 1, 2 + β, −1) + (1 + β) F (−β, 1, 2 + α, −1) We finally end up with O-planes are not dynamical. This is in contrast to the D-particle scatterings [17] where both infinite s-channel open string Regge poles and t-channel closed string Regge poles exist in the form factor. We will see that the fundamental characteristics of O-domain-wall scatterings are very different from those of O-particle scatterings as we will now discuss in the next section.

III. HIGH-ENERGY O-DOMAIN-WALL SCATTERINGS
For this case the kinematic setup is In the high-energy limit, the angle of incidence φ is identical to the angle of reflection θ and Diag D µν = (−1, 1, −1). The following kinematic relations can be easily calculated e T · k 2 = e L · k 2 = 0, (3.6) We define 16) and the Mandelstam variables can be calculated to be The first term of high-energy scatterings from O-domain-wall is The second term can be similarly calculated to be The scattering amplitudes of O-domain-wall on RP 2 can therefore be calculated to be By using the similar techanique for the case of O-particle scatterings, the integral above can be calculated to be One thus ends up with .
where we have used c 0 ≡ 1 2 (M 2 1 + M 2 ) in the high-energy limit. It is easy to see that the larger the mass M of the external leg is, the more numerous the closed string poles are. We thus confirm that all massive string states do couple to the O-domain-wall as was conjectured previously [19,24]. This is also consistent with the boundary state descriptions of O-planes. However, the claim that there is a thinkness of order √ α ′ for the O-domain-wall is misleading as the UV behavior of its scatterings is power-law instead of exponential falloff. This concludes that, in contrast to the usual behavior of high-energy, fixed angle string scattering amplitudes, namely soft UV, linear relations and the existence of infinite Regge poles, O-domain-wall scatterings, like the well-known D-instanton scatterings, behave like field theory scatterings.

IV. CONCLUSIONS AND DISCUSSIONS
In this paper, we calculate bosonic massive closed string states at arbitrary mass levels scattered from Orientifold planes in the high-energy limit. We have concentrated on the discussions of three fundamental characteristics of high-energy, fixed angle string scattering amplitudes, namely soft UV, infinite Regge poles and infinite linear relations discovered recently. For the case of O-particle scatterings, we obtain infinite linear relations among highenergy scattering amplitudes of different string states at each fixed mass level. Moreover, the amplitude was found to be UV soft, namely, exponential fall-off behavior. We also confirm that there exist only infinite t-channel closed string Regge poles in the form factor of the O-particle scatterings amplitudes as expected. For the case of O-domain-wall scatterings, we find that, like the well-known D-instanton scatterings, the amplitudes behave like field theory scatterings, namely UV power-law without infinite Regge poles. In addition, we discover that there exist only finite number of t-channel closed string poles in the form factor, and the masses of the poles are bounded by the masses of the external legs. We thus confirm that all massive closed string states do couple to the O-domain-wall as was conjectured previously [19,24]. This is also consistent with the boundary state descriptions