High-energy Massive String Scatterings from Orientifold Planes

0.16 MB·English

by  Jen-Chi Lee

High-energy Massive String Scatterings from Orientifold Planes
Jen-Chi Lee∗ and Yi Yang†
Department of Electrophysics, National Chiao-Tung University and Physics Division,
National Center for Theoretical Sciences, Hsinchu, Taiwan, R.O.C.
(Dated: February 2, 2008)
Abstract
8
0
0 We calculate bosonic massive closed string states at arbitrary mass levels scattered from Orien-
2
tifold planes in the high-energy, fixed angle limit. For the case of O-particle scatterings, we obtain
n
a
J infinite linear relations among high-energy scattering amplitudes of different string states. We also

1
confirm that there exist only closed string Regge poles in the form factor of the O-particle ampli-
1

] tudes as expected. For the case of O-domain-wall scatterings, we find that, like the well-known
h
-t D-instanton scatterings, the amplitudes behave like field theory scatterings, namely UV power-
p
e law without infinite Regge poles. In addition, we discover that there exist only finite number of
h
[ t-channel closed string poles in the form factor of O-domain-wall scatterings, and the masses of

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v the poles are bounded by the masses of the external legs. We thus confirm that all massive closed
5
4 string states do couple to the O-domain-wall.
2
4
.
2
1
7
0
:
v
i
X
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
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I. INTRODUCTION
Being a consistent theory of quantum gravity, string theory is remarkable for its soft
ultraviolet structure. Thisis mainlydueto two closely relatedfundamental 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
arbitrarymass levels [4, 5, 6, 7, 8, 9, 10, 11, 12]. Aninfinitenumber oflinear 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 thethird
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 formfactorbehaves as power-law with infinite open Regge polestructure at high
energies. This discovery makes D-domain-wall scatterings an unique example of a hybrid of
stringandfieldtheoryscatterings. Moreover, itwasshown[20]thatthelinearrelationsbreak
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-
2
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 be-
have 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 orga-
nized 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.
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II. HIGH-ENERGY O-PARTICLE SCATTERINGS
We will use the real projected plane RP as the worldsheet diagram for the scatterings
2
of Orientifold planes. The standard propagators of the left and right moving fields are
Xµ(z)Xν (w) = ηµν log(z w), (2.1)
h i − −
X˜µ(z¯)X˜ν (w¯) = ηµν log(z¯ w¯). (2.2)
− −
D E
In addition, there are also nontrivial correlator between the right and left moving fields as
well
Xµ(z)X˜ν (w¯) = Dµν ln(1+zw¯). (2.3)
−
D E
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
1
T = ε eik·x +(D ε) (D ε) eiD·k·x (2.4)
µ1···µn 2 µ1···µn · µ1··· · µn
h i
where
ε ε ε . (2.5)
µ1···µn ≡ µ1 ··· µn
The vertex operators corresponding to the above wave functions are
1
V (ε,k,z,z¯) = ε Vµ1···µn (k,z,z¯)+(D ε) (D ε) Vµ1···µn (D k,z,z¯) . (2.6)
2 µ1···µn · µ1 ··· · µn ·
h i
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 = δ , and the kinematic setup are
µν µν
−
1 k
eP = ( E, k ,0) = 2, (2.7)
M − − 2 M
1
eL = ( k , E,0), (2.8)
M − 2 −
eT = (0,0,1), (2.9)
k = (E,k cosφ, k sinφ), (2.10)
1 1 1
−
k = ( E, k ,0) (2.11)
2 2
− −
4
where eP, eL and eT are polarization vectors of the tensor state k on the high-energy
2
scattering plane. One can easily calculate the following kinematic relations in the high-
energy limit
eT k = eL k = 0, (2.12)
2 2
· ·
eT k = k sinφ Esinφ, (2.13)
1 1
· − ∼ −
eT D k = k sinφ Esinφ, (2.14)
1 1
· · ∼
eT D k = 0, (2.15)
2
· ·
1 E2
eL k = [k E k Ecosφ] (1 cosφ), (2.16)
· 1 M 2 − 1 ∼ M −
1 E2
eL D k = [k E +k Ecosφ] (1+cosφ), (2.17)
· · 1 M 2 1 ∼ M
1 2E2
eL D k = [ k E k E] . (2.18)
· · 2 M − 2 − 2 ∼ − M
We define
a k D k = E2 k2 2E2, (2.19)
0 ≡ 1 · · 1 − − 1 ∼ −
a′ k D k = E2 k2 2E2, (2.20)
0 ≡ 2 · · 2 − − 2 ∼ −
b k k = E2 k k cosφ E2(1 cosφ), (2.21)
0 1 2 1 2
≡ · − ∼ −
(cid:0) (cid:1)
c k D k = E2 +k k cosφ E2(1+cosφ), (2.22)
0 1 2 1 2
≡ · · ∼
(cid:0) (cid:1)
and the Mandelstam variables can be calculated to be
t (k +k )2 = M2 +M2 2k k = M2 2(1+b ), (2.23)
≡ − 1 2 1 2 − 1 · 2 2 − 0
1 1
s k D k = a , (2.24)
1 1 0
≡ 2 · · 2
u = 2k D k = 2c . (2.25)
1 2 0
− · · −
In the high-energy limit, we will consider an incoming tachyon state k and an outgoing
1
tensor state k of the following form
2
αT n−m−2q αL q α˜T n−m′−2q′ α˜L q′ 0 . (2.26)
−1 −2 ⊗ −1 −2 | i
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
For simplicity, we have omited above a possible high-energy vertex (αL )r (α˜L )r′ [17, 21].
−1 ⊗ −1
For this case, with momentum conservation on the O-planes, we have
a +b +c = M2 = 2. (2.27)
0 0 0 1 −
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The high-energy scattering amplitude can then be written as
1
ARP2 = d2z d2z V (k ,z )V˜ (k ,z¯ )+V (D k ,z )V˜ (D k ,z¯ )
1 2 1 1 1 1 1 1 1 1
Z 2 · ·
h i
· 21εTn−2qLq,Tn−2q′Lq′VTn−2qLq (k2,z2)V˜Tn−2q′Lq′ (k2,z¯2)
+(D ε )n−2q(D ε )q(D ε˜ )n−2q′(D ε˜ )q′VTn−2qLq (D k ,z )
T L T L 2 2
· · · · ·
V˜Tn−2q′Lq′ (D k ,z¯ )
2 2
· ·
= A +A +A +A
1 2 3 4
where
1
A1 = 4εTn−2qLq,Tn−2q′Lq′ Z d2z1d2z2
V (k ,z )V˜ (k ,z¯ )VTn−2qLq (k ,z )V˜Tn−2q′Lq′ (k ,z¯ ) , (2.28)
1 1 1 1 2 2 2 2
·
D E
1
A2 = 4εTn−2qLq,Tn−2q′Lq′ Z d2z1d2z2
V (D k ,z )V˜ (D k ,z¯ )VTn−2qLq (k ,z )V˜Tn−2q′Lq′ (k ,z¯ ) , (2.29)
1 1 1 1 2 2 2 2
· · ·
D E
A = 1 (D ε )n−2q(D ε )q(D ε˜ )n−2q′(D ε˜ )q′
3 T L T L
4 · · · ·
d2z d2z V (k ,z )V˜ (k ,z¯ )VTn−2qLq (D k ,z )V˜Tn−2q′Lq′ (D k ,z¯ ) , (2.30)
1 2 1 1 1 1 2 2 2 2
·Z D · · E
A = 1 (D ε )n−2q(D ε )q(D ε˜ )n−2q′(D ε˜ )q′
4 T L T L
4 · · · ·
d2z d2z V (D k ,z )V˜ (D k ,z¯ )VTn−2qLq (D k ,z )V˜Tn−2q′Lq′ (D k ,z¯ ) .
1 2 1 1 1 1 2 2 2 2
·Z D · · · · E
(2.31)
One can easily see that
A = A ,A = A . (2.32)
1 4 2 3
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We will choose to calculate A and A . For the case of A , we have
1 2 1
4A1 = εTn−2qLq,Tn−2q′Lq′ Z d2z1d2z2·
n−2q′ q′
eik1X (z )eik1X˜ (z¯ ) ∂XT n−2q i∂2XL qeik2X(z ) ∂¯X˜T i∂¯2X˜L eik2X˜ (z¯ )
1 1 2 2
(cid:28) (cid:29)
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
= ( 1)q+q′ d2z d2z (1+z z¯ )a0 (1+z z¯ )a′0 z z 2b0 1+z z¯ 2c0
1 2 1 1 2 2 1 2 1 2
− Z | − | | |
ieT k ieT D k ieT D k n−2q
· 1 · · 1z¯ · · 2z¯
1 2
·(cid:20)z z − 1+z¯ z − 1+z¯ z (cid:21)
1 2 1 2 2 2
−
ieT D k ieT k ieT D k n−2q′
· · 1z + · 1 · · 2z
·(cid:20)− 1+z z¯ 1 z¯ z¯ − 1+z z¯ 2(cid:21)
1 2 1 2 2 2
−
eL k eL D k eL D k q
· 1 + · · 1 z¯2 + · · 2 z¯2
·(cid:20)(z z )2 (1+z¯ z )2 1 (1+z¯ z )2 2(cid:21)
1 2 1 2 2 2
−
eL D k eL k eL D k q′
· · 1 z2 + · 1 + · · 2 z2 . (2.33)
·(cid:20)(1+z z¯ )2 1 (z¯ z¯ )2 (1+z z¯ )2 2(cid:21)
1 2 1 2 2 2
−
To fix the modulus group on RP , choosing z = r and z = 0 and we have
2 1 2
1
4A = ( 1)n dr2 1+r2 a0r2b0
1
− Z
0 (cid:0) (cid:1)
eT k eT D k n−2q eT D k eT k n−2q′
· 1 · · 1r · · 1r + · 1
·(cid:20) r − 1 (cid:21) ·(cid:20)− 1 r (cid:21)
eL k eL D k q eL D k eL k q′
· 1 + · · 1r2 · · 1r2 + · 1
·(cid:20) r2 1 (cid:21) ·(cid:20) 1 r2 (cid:21)
φ q+q′ φ i
2cos2 q+q′ q +q′ sin2
= ( 1)n(Esinφ)2n 2   2 
− M sin2φ (cid:18) i (cid:19) φ
  Xi=0 cos2 
2
   
1
dr2 1+r2 a0+2n−2(q+q′) r2 b0−n+2(q+q′)−2i. (2.34)
·Z ·
0 (cid:0) (cid:1) (cid:0) (cid:1)
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Similarly, for the case of A , we have
2
1
4A = ( 1)n dr2 1+r2 a0r2c0
2
− Z
0 (cid:0) (cid:1)
eT D k eT k n−2q eT k eT D k n−2q′
· · 1 · 1r · 1r + · · 1
·(cid:20) r − 1 (cid:21) ·(cid:20)− 1 r (cid:21)
eL D k eL k q eL k eL D k q′
· · 1 + · 1r2 · 1r2 + · · 1
·(cid:20) r2 1 (cid:21) ·(cid:20) 1 r2 (cid:21)
φ q+q′ φ i
2cos2 q+q′ q +q′ sin2
= ( 1)n(Esinφ)2n 2   2 
− M sin2φ (cid:18) i (cid:19) φ
  Xi=0 cos2 
2
   
1
dr2 1+r2 a0+2n−2(q+q′) r2 c0−n+2i. (2.35)
·Z
0 (cid:0) (cid:1) (cid:0) (cid:1)
The scattering amplitude on RP can therefore be calculated to be
2
ARP2 = A +A +A +A
1 2 3 4
φ q+q′ φ i
1 2cos2 q+q′ q +q′ sin2
= ( 1)n(Esinφ)2n 2   2 
2 − M sin2φ (cid:18) i (cid:19) φ
  Xi=0 cos2 
2
   
1
dr2 1+r2 a0+2n−2(q+q′) r2 b0−n+2(q+q′)−2i + r2 c0−n+2i . (2.36)
·Z ·
0 (cid:0) (cid:1) h(cid:0) (cid:1) (cid:0) (cid:1) i
The integral in Eq.(2.36) can be calculated as following
1
dr2 1+r2 a0+2n−2(q+q′) r2 b0−n+2(q+q′)−2i + r2 c0−n+2i
Z ·
0 (cid:0) (cid:1) h(cid:0) (cid:1) (cid:0) (cid:1) i
21+a0+2n−2(q+q′)
= [ ]
1+b n+2(q+q′) 2i
0
− −
F (2+a +b +n 2i,1,2+b n+2(q +q′) 2i, 1)
0 0 0
· − − − −
21+a0+2n−2(q+q′)
+[ ]F (2+a +c +n 2(q +q′)+2i,1,2+c n+2i, 1) (2.37)
1+c n+2i 0 0 − 0 − −
0
−
where we have used the following identities of the hypergeometric function F (α,β,γ,x)
Γ(γ) 1
F(α,β,γ;x) = dy yβ−1(1 y)γ−β−1(1 yx)−α, (2.38)
Γ(β)Γ(γ β) Z − −
− 0
F (α,β,γ,x) = 2γ−α−βF (γ α,γ β,γ,x). (2.39)
− −
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To further reduce the scattering amplitude into beta function, we use the momentum con-
servation in Eq.(2.27) and the identity
(1+α)F ( α,1,2+β, 1)+(1+β)F ( β,1,2+α, 1)
− − − −
Γ(α+2)Γ(β +2)
= 21+α+β (2.40)
Γ(α+β +2)
to get
21+a0+2n−2(q+q′)
[ ]F ( c +n 2i,1,2+b n+2(q +q′) 2i, 1)
1+b n+2(q +q′) 2i − 0 − 0 − − −
0
− −
21+a0+2n−2(q+q′)
+[ ]F ( b +n 2(q +q′)+2i,1,2+c n+2i, 1)
0 0
1+c n+2i − − − −
0
−
Γ(1+c n+2i)Γ(1+b n+2(q +q′) 2i)
0 0
= − − −
Γ(2+b +c 2n+2(q +q′))
0 0
−
(1+c )−n+2i(1+b )−n+2(q+q′)−2i
B(1+b ,1+c ) 0 0
∼ 0 0 (2+b +c )−2n+2(q+q′)
0 0
φ −n+2i φ −n+2(q+q′)−2i
B(1+b ,1+c ) cos2 sin2 . (2.41)
0 0
∼ (cid:18) 2(cid:19) (cid:18) 2(cid:19)
We finally end up with
ARP2 = A +A +A +A
1 2 3 4
φ q+q′ φ i
1 2cos2 q+q′ q +q′ sin2
= ( 1)n(Esinφ)2n 2   2 
2 − M sin2φ (cid:18) i (cid:19) φ
  Xi=0 cos2 
2
   
φ −n+2i φ −n+2(q+q′)−2i
B(1+b ,1+c ) cos2 sin2
0 0
· (cid:18) 2(cid:19) (cid:18) 2(cid:19)
φ q+q′ φ i
1 sin2 q+q′ q +q′ cos2
= ( 1)n(2E)2n 2  B(1+b ,1+c )  2 
2 − 2M 0 0 (cid:18) i (cid:19) φ
  Xi=0 sin2 
2
   
q+q′
1 1
= ( 1)n(2E)2n B(1+b ,1+c )
0 0
2 − (cid:18)2M(cid:19)
1 1 q+q′ t u
( 1)n(2E)2n B , . (2.42)
∼ 2 − (cid:18)2M(cid:19) (cid:18)−2 −2(cid:19)
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
9
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
infinites-channelopenstringReggepolesandt-channelclosedstring Reggepolesexist inthe
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
1 k
eP = ( E,k cosθ, k sinθ) = 2, (3.1)
M − 2 − 2 M
1
eL = ( k ,Ecosθ, Esinθ), (3.2)
2
M − −
eT = (0,sinθ,cosθ), (3.3)
k = (E, k cosφ, k sinφ), (3.4)
1 1 1
− −
k = ( E,k cosθ, k sinθ). (3.5)
2 2 2
− −
In the high-energy limit, the angle of incidence φ is identical to the angle of reflection θ and
Diag D = ( 1,1, 1). The following kinematic relations can be easily calculated
µν
− −
eT k = eL k = 0, (3.6)
2 2
· ·
eT k = 2k sinφcosφ Esin2φ, (3.7)
1 1
· − ∼ −
eT D k = 0, (3.8)
1
· ·
eT D k = 2k sinφcosφ Esin2φ, (3.9)
2 2
· · ∼
1 2E2
eL k = k E k E cos2φ sin2φ sin2φ, (3.10)
1 2 1
· M − − ∼ M
(cid:2) (cid:0) (cid:1)(cid:3)
eL D k = 0, (3.11)
1
· ·
1 2E2
eL D k = k E +k E cos2φ sin2φ sin2φ. (3.12)
· · 2 M − 2 2 − ∼ − M
(cid:2) (cid:0) (cid:1)(cid:3)
10