www.elsevier.com/locate/nuclphysb
High-energy massive string scatterings
from orientifold planes
Jen-Chi Lee
∗, Yi Yang
Department of Electrophysics, National Chiao-Tung University and Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan, ROC
Received 10 January 2008; accepted 31 January 2008 Available online 8 February 2008
Abstract
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.
©2008 Elsevier B.V. All rights reserved.
1. 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[1–3]
was re-investigated for massive string states at arbitrary mass levels[4–12]. An infinite number
* Corresponding author.
E-mail addresses:[email protected](J.-C. Lee),[email protected](Y. Yang). 0550-3213/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
of linear relations among string scattering amplitudes were obtained. The most important new ingredient of these calculations is the zero-norm states (ZNS)[13–15]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 compactified 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 formalism[22,23], and more recently[24]on the worldsheet of real projected plane RP2. 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 clarify 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 scatter-ings. 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 scatter-ings, the amplitudes behave like field theory scatterscatter-ings, 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 ampli-tudes as O-planes were known to be not dynamical. However, the usual claim that there is a thinkness of order√αfor 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 Section2, 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 Section3, 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 Section4.
2. High-energy O-particle scatterings
We will use the real projected plane RP2 as the worldsheet diagram for the scatterings of
orientifold planes. The standard propagators of the left and right moving fields are
(2.1)
Xμ(z)Xν(w)= −ημνlog(z− w),
(2.2) ˜Xμ(¯z) ˜Xν(¯w)= −ημνlog(¯z − ¯w).
In addition, there are also nontrivial correlator between the right and left moving fields as well (2.3)
Xμ(z) ˜Xν(¯w)= −Dμνln(1+ z ¯w).
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
(2.4) Tμ1···μn= 1 2 εμ1···μne ik·x+ (D · ε) μ1· · · (D · ε)μne iD·k·x where (2.5) εμ1···μn≡ εμ1· · · εμn.
The vertex operators corresponding to the above wave functions are
V (ε, k, z,¯z) (2.6) =1 2 εμ1···μnV μ1···μn(k, z,¯z) + (D · ε) μ1· · · (D · ε)μnV μ1···μn(D· k, z, ¯z).
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 Section3. For this case Dμν= −δμν, and the kinematic setup are
(2.7) eP = 1 M(−E, −k2,0)= k2 M, (2.8) eL= 1 M(−k2,−E, 0), (2.9) eT = (0, 0, 1), (2.10)
k1= (E, k1cos φ,−k1sin φ),
(2.11)
k2= (−E, −k2,0)
where eP, eLand eT are polarization vectors of the tensor state k
2on the high-energy scattering
plane. One can easily calculate the following kinematic relations in the high-energy limit (2.12)
eT · k2= eL· k2= 0,
(2.13)
eT · k1= −k1sin φ∼ −E sin φ,
(2.14) eT · D · k1= k1sin φ∼ E sin φ, (2.15) eT · D · k2= 0, (2.16) eL· k1= 1 M[k2E− k1Ecos φ] ∼ E2 M(1− cos φ),
(2.17) eL· D · k1= 1 M[k2E+ k1Ecos φ] ∼ E2 M(1+ cos φ), (2.18) eL· D · k2= 1 M[−k2E− k2E] ∼ − 2E2 M . We define (2.19) a0≡ k1· D · k1= −E2− k21∼ −2E2, (2.20) a0 ≡ k2· D · k2= −E2− k22∼ −2E 2, (2.21) b0≡ k1· k2= E2− k1k2cos φ ∼ E2( 1− cos φ), (2.22) c0≡ k1· D · k2= E2+ k1k2cos φ ∼ E2(1+ cos φ),
and the Mandelstam variables can be calculated to be
(2.23) t≡ −(k1+ k2)2= M12+ M 2 2− 2k1· k2= M22− 2(1 + b0), (2.24) s≡1 2k1· D · k1= 1 2a0, (2.25) u= −2k1· D · k2= −2c0.
In the high-energy limit, we will consider an incoming tachyon state k1and an outgoing tensor
state k2of the following form
(2.26)
α−1T n−m−2qαL−2q⊗˜αT−1n−m−2q˜αL−2q|0.
For simplicity, we have omitted above a possible high-energy vertex (α−1L )r⊗ ( ˜αL−1)r [17,21]. For this case, with momentum conservation on the O-planes, we have
(2.27)
a0+ b0+ c0= M12= −2.
The high-energy scattering amplitude can then be written as
ARP2= d2z1d2z2 1 2 V (k1, z1) ˜V (k1,¯z1)+ V (D · k1, z1) ˜V (D· k1,¯z1) ×1 2εTn−2qLq,Tn−2qLqV Tn−2qLq(k 2, z2) ˜VT n−2qLq (k2,¯z2) + (D · εT)n−2q(D· εL)q(D· ˜εT)n−2q (D· ˜εL)q VTn−2qLq(D· k2, z2) × ˜VTn−2qLq (D· k2,¯z2) = A1+ A2+ A3+ A4, where A1= 1 4εTn−2qLq,Tn−2qLq d2z1d2z2 V (k1, z1) ˜V (k1,¯z1)VT n−2qLq (k2, z2) (2.28) × ˜VTn−2qLq (k2,¯z2) , A2= 1 4εTn−2qLq,Tn−2qLq d2z1d2z2 V (D· k1, z1) ˜V (D· k1,¯z1)VT n−2qLq (k2, z2) (2.29) × ˜VTn−2qLq (k2,¯z2) ,
A3= 1 4(D· εT) n−2q(D· ε L)q(D· ˜εT)n−2q (D· ˜εL)q (2.30) × d2z1d2z2 V (k1, z1) ˜V (k1,¯z1)VT n−2qLq (D· k2, z2) ˜VT n−2qLq (D· k2,¯z2) , A4= 1 4(D· εT) n−2q(D· ε L)q(D· ˜εT)n−2q (D· ˜εL)q × d2z1d2z2 V (D· k1, z1) ˜V (D· k1,¯z1)VT n−2qLq (D· k2, z2) (2.31) × ˜VTn−2qLq (D· k2,¯z2) .
One can easily see that
(2.32)
A1= A4, A2= A3.
We will choose to calculate A1and A2. For the case of A1, we have
4A1= εTn−2qLq,Tn−2qLq d2z1d2z2 eik1X(z 1)eik1˜X(¯z1) ∂XTn−2qi∂2XLq × eik2X(z 2)¯∂ ˜XT n−2q i ¯∂2˜XLqeik2˜X(¯z 2) = (−1)q+q d2z1d2z2(1+ z1¯z1)a0(1+ z2¯z2)a 0|z1− z2|2b0|1 + z 1¯z2|2c0 × ieT · k1 z1− z2 − ieT · D · k1 1+ ¯z1z2 ¯z 1− ieT · D · k2 1+ ¯z2z2 ¯z 2 n−2q × −ieT · D · k1 1+ z1¯z2 z1+ ieT · k1 ¯z1− ¯z2 − ieT · D · k2 1+ z2¯z2 z2 n−2q × eL· k1 (z1− z2)2+ eL· D · k1 (1+ ¯z1z2)2¯z 2 1+ eL· D · k2 (1+ ¯z2z2)2¯z 2 2 q (2.33) × eL· D · k1 (1+ z1¯z2)2 z21+ e L· k 1 (¯z1− ¯z2)2 + eL· D · k2 (1+ z2¯z2)2 z22 q .
To fix the modulus group on RP2, choosing z1= r and z2= 0 and we have
4A1= (−1)n 1 0 dr21+ r2a0r2b0 × eT · k 1 r − eT · D · k 1 1 r n−2q · −eT · D · k1 1 r+ eT · k 1 r n−2q × eL· k1 r2 + eL· D · k1 1 r 2 q · eL· D · k1 1 r 2+eL· k1 r2 q = (−1)n(E sin φ)2n 2 cos2 φ2 Msin2φ q+qq +q i=0 q+ q i sin2 φ2 cos2 φ 2 i (2.34) × 1 0 dr21+ r2a0+2n−2(q+q)·r2b0−n+2(q+q)−2i.
Similarly, for the case of A2, we have 4A2= (−1)n 1 0 dr21+ r2a0r2c0 × eT · D · k1 r − eT · k1 1 r n−2q · −eT · k1 1 r+ eT · D · k1 r n−2q × eL· D · k1 r2 + eL· k1 1 r 2 q · eL· k1 1 r 2+eL· D · k1 r2 q = (−1)n(E sin φ)2n 2 cos2 φ 2 Msin2φ q+qq +q i=0 q+ q i sin2 φ2 cos2 φ 2 i (2.35) × 1 0 dr21+ r2a0+2n−2(q+q)r2c0−n+2i.
The scattering amplitude on RP2can therefore be calculated to be ARP2= A 1+ A2+ A3+ A4 =1 2(−1) n(Esin φ)2n 2 cos2 φ2 Msin2φ q+qq +q i=0 q+ q i sin2 φ2 cos2 φ 2 i (2.36) × 1 0 dr21+ r2a0+2n−2(q+q)·r2b0−n+2(q+q)−2i+r2c0−n+2i.
The integral in Eq.(2.36)can be calculated as following
1 0 dr21+ r2a0+2n−2(q+q)·r2b0−n+2(q+q)−2i+r2c0−n+2i = 21+a0+2n−2(q+q) 1+ b0− n + 2(q + q)− 2i × F (2 + a0+ b0+ n − 2i, 1, 2 + b0− n + 2(q + q)− 2i, −1) + 21+a0+2n−2(q+q) 1+ c0− n + 2i F (2+ a0+ c0+ n − 2(q + q) (2.37) + 2i, 1, 2 + c0− n + 2i, −1),
where we have used the following identities of the hypergeometric function F (α, β, γ , x)
(2.38) F (α, β, γ; x) = (γ ) (β) (γ − β) 1 0 dy yβ−1(1− y)γ−β−1(1− yx)−α, (2.39) F (α, β, γ , x)= 2γ−α−βF (γ− α, γ − 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) (2.40) = 21+α+β (α+ 2) (β + 2) (α+ β + 2) to get 21+a0+2n−2(q+q) 1+ b0− n + 2(q + q)− 2i F−c0+ n − 2i, 1, 2 + b0− n + 2(q + q)− 2i, −1 + 21+a0+2n−2(q+q) 1+ c0− n + 2i F−b0+ n − 2(q + q)+ 2i, 1, 2 + c0− n + 2i, −1 = (1+ c0− n + 2i) (1 + b0− n + 2(q + q)− 2i) (2+ b0+ c0− 2n + 2(q + q)) ∼ B(1 + b0,1+ c0) (1+ c0)−n+2i(1+ b0)−n+2(q+q)−2i (2+ b0+ c0)−2n+2(q+q) (2.41) ∼ B(1 + b0,1+ c0) cos2φ 2 −n+2i sin2φ 2 −n+2(q+q)−2i .
We finally end up with
ARP2= A 1+ A2+ A3+ A4 =1 2(−1) n(E sin φ)2n 2 cos2 φ2 Msin2φ q+qq +q i=0 q+ q i sin2 φ2 cos2 φ 2 i × B(1 + b0,1+ c0) cos2φ 2 −n+2i sin2φ 2 −n+2(q+q)−2i =1 2(−1) n( 2E)2n sin2 φ2 2M q+q B(1+ b0,1+ c0) q +q i=0 q+ q i cos2 φ2 sin2 φ2 i =1 2(−1) n(2E)2n 1 2M q+q B(1+ b0,1+ c0) (2.42) ∼1 2(−1) n( 2E)2n 1 2M q+q B −t 2,− u 2 .
From Eq. (2.42) we see that the UV behavior of O-particle scatterings is exponential fall-off and one gets infinite linear relations among string scattering amplitudes of different string states at each fixed mass level. Note that both t and u correspond to the closed string channel poles, while s corresponds to the open string channel poles. It can be seen from Eq.(2.42)that an infinite closed string Regge poles exist in the form factor of O-particle scatterings. Furthermore, there are no s-channel open string Regge poles as expected since 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.
3. High-energy O-domain-wall scatterings
For this case the kinematic setup is
(3.1)
eP= 1
M(−E, k2cos θ,−k2sin θ )= k2 M,
(3.2)
eL= 1
M(−k2, Ecos θ,−E sin θ),
(3.3)
eT = (0, sin θ, cos θ),
(3.4)
k1= (E, −k1cos φ,−k1sin φ),
(3.5)
k2= (−E, k2cos θ,−k2sin θ ).
In the high-energy limit, the angle of incidence φ is identical to the angle of reflection θ and Diag Dμν= (−1, 1, −1). The following kinematic relations can be easily calculated
(3.6)
eT · k2= eL· k2= 0,
(3.7)
eT · k1= −2k1sin φ cos φ∼ −E sin 2φ,
(3.8)
eT · D · k1= 0,
(3.9)
eT · D · k2= 2k2sin φ cos φ∼ E sin 2φ,
(3.10) eL· k1= 1 M k2E− k1E cos2φ− sin2φ∼2E 2 M sin 2φ, (3.11) eL· D · k1= 0, (3.12) eL· D · k2= 1 M −k2E+ k2E cos2φ− sin2φ∼ −2E 2 M sin 2φ. We define (3.13)
a0≡ k1· D · k1∼ −2E2sin2φ− 2M12cos
2φ+ M2 1, (3.14) a0 ≡ k2· D · k2= −E2− k22∼ −2E 2, (3.15) b0≡ k1· k2∼ 2E2sin2φ+ 2M12cos2φ− 1 2 M12+ M2, (3.16) c0≡ k1· D · k2= E2− k1k2∼ 1 2 M12+ M2,
and the Mandelstam variables can be calculated to be
(3.17) t≡ −(k1+ k2)2= M12+ M 2 2− 2k1· k2= M22− 2(1 + b0), (3.18) s≡1 2k1· D · k1= 1 2a0, (3.19) u= −2k1· D · k2= −2c0.
The first term of high-energy scatterings from O-domain-wall is
4A1= (−1)n 1 0 dr21+ r2a0r2b0 × eT · k1 r − eT · D · k1 1 r n−2q · −eT · D · k1 1 r+ eT · k1 r n−2q
× eL· k1 r2 + eL· D · k1 1 r 2 q · eL· D · k1 1 r 2+eL· k1 r2 q (3.20) ∼ (−1)n(Esin 2φ)2n 1 2M cos2φ q+q1 0 dr21+ r2a0r2b0−n.
The second term can be similarly calculated to be
4A2= (−1)n 1 0 dr21+ r2a0r2c0 × eT · D · k1 r − eT · k1 1 r n−2q · −eT · k1 1 r+ eT · D · k1 r n−2q × eL· D · k1 r2 + eL· k1 1 r 2 q · eL· k1 1 r 2+eL· D · k1 r2 q (3.21) ∼ (−1)n(E sin 2φ)2n−2(q+q) 2E2 M sin 2φ q+q1 0 dr21+ r2a0r2c0+n.
The scattering amplitudes of O-domain-wall on RP2can therefore be calculated to be ARP2= A 1+ A2+ A3+ A4 =1 2(−1) n(E sin 2φ)2n 1 2M cos2φ q+q (3.22) × 1 0 dr21+ r2a0r2b0−n+r2c0+n.
By using the similar technique for the case of O-particle scatterings, the integral above can be calculated to be dr21+ r2a0r2b0−n+r2c0+n =F (−a0,1+ b0− n, 2 + b0− n, −1) 1+ b0− n + F (−a0,1+ c0+ n, 2 + c0+ n, −1) 1+ c0+ n = 22+a0+b0+c0 (1+ b0− n)(1 + c0+ n) (2+ c0+ n) (2 + b0− n) (2+ b0+ c0) (3.23) = (1+ c0+ n) (1 + b0− n) (2+ b0+ c0) .
One thus ends up with
ARP2= A 1+ A2+ A3+ A4 (3.24) =1 2(−1) n(E sin 2φ)2n 1 2M cos2φ q+q (c 0+ n + 1) (b0− n + 1) (b0+ c0+ 2) .
Some crucial points of this result are in order. First, since c0is a constant in the high-energy limit,
the UV behavior of the O-domain-wall scatterings is power-law instead of the usual exponential fall-off in other O-plane scatterings. Second, there exist only finite number of closed string poles in the form factor. Note that although we only look at the high energy kinematic regime of the scattering amplitudes, it is easy to see that there exists no infinite closed string Regge poles in the scattering amplitudes for the whole kinematic regime. This is because there is only one kinematic variable for the O-domain-wall scatterings. In fact, the structure of poles in Eq.(3.24)
can be calculated to be (1+ c0+ n) (1 + b0− n) (2+ b0+ c0) = (1+ M2) (1+ b0− n) (b0+ n) = 1+ M2 (b0− n)! (b0+ n − 1)! (3.25) = 1+ M2 n−1 k=1−n 1 b0− k
where we have used c0≡12(M12+M2)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√αfor the O-domain-wall is misleading as the UV behavior of its scatterings is power-law instead of exponential fall-off. 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.
4. Conclusions and discussions
In this paper, we calculate bosonic massive closed string states at arbitrary mass levels scat-tered 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 high-energy scatter-ing amplitudes of different strscatter-ing 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 ampli-tudes 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 bound-ary state descriptions of O-planes. For both cases of O-particle and O-domain-wall scatterings, we confirm that there exist no open string Regge poles in the form factor of the amplitudes as O-planes were known to be not dynamical.
We summarize the Regge pole structures of closed strings states scattered from various D-branes and O-planes inTable 1. The s-channel and t -channel scatterings for both D-branes and
Table 1
p= −1 1 p 23 p= 24
Dp-branes X C+ O O
Op-planes X C X
“C” and “O” represent infinite closed string Regge poles and open string Regge poles respectively. “X” means there are no infinite Regge poles.
Fig. 1. There are two possible channels for closed strings scattered from D-branes/O-planes. The diagram on the left-hand side corresponds to the s-channel scatterings, and the diagram on the right-hand side is the t -channel scatterings.
O-planes are shown inFig. 1. For O-plane scatterings, the s-channel open string Regge poles are not allowed since O-planes are not dynamical. For both cases of Domain-wall scatterings, the
t-channel closed string Regge poles are not allowed since there is only one kinematic variable instead of two as in the usual cases.
Acknowledgements
We would like to thank the hospitality of University of Tokyo at Komaba, where most of this work was done. We are indebted to Prof. Tamiaki Yoneya for many of his enlightening discussions. This work is supported in part by the National Science Council, 50 billions project of Ministry of Education and National Center for Theoretical Science, Taiwan, ROC.
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