Exponential Fall-Off Behavior of Regge Scatterings in Compactified Open String Theory

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Progress of Theoretical Physics, Vol. 128, No. 5, November 2012

Exponential Fall-Oﬀ Behavior of Regge Scatterings in Compactiﬁed Open String Theory

Song He,1,3,∗) Jen-Chi Lee2,3,∗∗) and Yi Yang2,3,∗∗∗)

1_{Institute of High Energy Physics, Chinese Academy of Sciences,} Beijing 100039, China

2_{Department of Electrophysics, National Chiao-Tung University and Physics} Division, National Center for Theoretical Sciences, Hsinchu, Taiwan, R.O.C.

3_{Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China} (Received July 31, 2012)

We calculate massive string scattering amplitudes of compactified open string in the Regge regime. We extract the complete infinite ratios among high-energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes. The complete ratios calculated by this indirect method include and extend the subset of ratios calculated previously [J. C. Lee and Y. Yang, Nucl. Phys. B784 (2007), 22; J. C. Lee, T. Takimi and Y. Yang, Nucl. Phys. B 804 (2008), 250] by the more difficult direct fixed angle calculation. In this calculation ofcompactified open string scattering, we discover a realization of arbitrary real valuesL in the identity Eq. (4·18), rather than integer value only in all previous high-energy string scattering amplitude calculations. The identity in Eq. (4·18) was explicitly proved recently in [J. C. Lee, C. H. Yan and Y. Yang, SIGMA 8 (2012), 045, arXiv:1012.5225] to link fixed angle and Regge string scattering amplitudes. In addition, we discover a kinematic regime with stringy highly winding modes, which shows the unusual exponential fall-off behavior in the Regge string scattering. This is complimentary with a kinematic regime discovered previously [J. C. Lee, T. Takimi and Y. Yang, Nucl. Phys. B804 (2008), 250] which shows the unusual power-law behavior in the high-energy fixed angle compactified string scatterings.

Subject Index: 129

§1. Introduction

There are three fundamental characteristics of high-energy fixed angle string scattering amplitudes,1)–3)which are not shared by the field theory scattering. These are the softer exponential fall-off behavior (in contrast to the hard power-law behav-ior of field theory scatterings), the infinite Regge-pole structure of the form factor and the existence of infinite number of linear relations,4)–13) or stringy symmetries, discovered recently among high-energy string scattering amplitudes of different string states. An important new ingredient to derive these linear relations is the zero-norm states (ZNS)14)–16) in the old covariant first quantized (OCFQ) string spectrum, in particular, the identification of particle symmetries induced by the inter-particle ZNS14) in the spectrum. Other approaches related to this development can be found in 17).

∗)_{E-mail: [email protected]} ∗∗)_{E-mail: [email protected]} ∗∗∗)_{E-mail: [email protected]}

at National Chiao Tung University Library on May 1, 2014

http://ptp.oxfordjournals.org/

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Recently, following an old suggestion of Mende,18)two of the present authors19) calculated high-energy fixed angle massive scattering amplitudes of closed bosonic string with some coordinates compactified on the torus. The calculation was ex-tended to the compactified open string scatterings.20) An infinite number of linear relations among high-energy scattering amplitudes of different string states were ob-tained in the fixed angle or Gross kinematic regime (GR). The UV behavior in the GR shows the usual soft exponential fall-off behavior. These results are reminiscent of the existence of an infinite number of massive ZNS in the compactified closed21) and open22) string spectrums constructed previously. In addition, it was discovered that, for some kinematic regime with super-highly winding modes at fixed angle, the so-called Mende kinematic regime (MR), these infinite linear relations break down and, simultaneously, the string amplitudes enhance to hard power-law behavior at high energies instead of the usual soft exponential fall-off behavior.

In this paper, we calculate high-energy small angle or Regge string scattering amplitudes23)–30) of open bosonic string with one coordinate compactified on the torus. The results can be generalized to more compactified coordinates. It is shown that there is no linear relations among Regge scattering amplitudes as expected. However, as in the case of noncompactified Regge string scattering amplitude cal-culation,31)–33) we can deduce the infinite GR ratios in the fixed angle from these compactified Regge string scattering amplitudes. We stress that the GR ratios cal-culated in the present paper by this indirect method from the Regge calculation are for the most general high-energy vertex rather than only a subset of GR ra-tios obtained directly from the fixed angle calculation.19), 20) In this calculation, we have used a set of master identities Eq. (4.18) to extract the GR ratios from Regge scattering amplitudes. Mathematically, the complete proof of these identities for

ar-bitrary real values L was recently worked out in 36) by using an identity of signless

Stirling number of the first kind in combinatorial theory. The proof of the identity for L = 0, 1, was previously given in 31)–33) based on a set of identities of signed Stirling number of the first kind.35) It is interesting to see that, physically, the iden-tities for arbitrary real values L can only be realized in high-energy compactified string scatterings considered in this paper. All other high-energy string scatterings calculated previously31)–33)correspond to integer values of L only. A recent work on string D-particle scatterings34)also gave integer values L.

More importantly, we discover an exponential fall-off behavior of high-energy compactified open string scatterings in a kinematic regime with highly winding modes at small angle. The existence of this regime was conjectured in 20). However, no Regge scatterings were calculated there and thus the results for the small angle scatterings extracted from the fixed angle calculation were not completed and fully reliable.31), 32) The discovery of the soft exponential fall-off behavior in this kinematic regime with small angle in compactified string scatterings is complimentary with a kinematic regime discovered previously,19), 20) which shows the unusual power-law behavior in the high-energy fixed angle compactified string scatterings. This paper is organized as the following. In§2, we set up the kinematics. In §3, we review the fixed angle compactified string scatterings. Section 4 is devoted to the compactified Regge string scatterings. We first calculate the Regge string scattering amplitudes

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and extract the most general ﬁxed angle ratios from these Regge amplitudes. We then derive a Regge regime which shows an unusual exponential fall-oﬀ behavior. A brief conclusion is made in§5.

§2. Kinematics set-up

We consider 26D open bosonic string with one coordinate compactified on S1 with radius R. It is straightforward to generalize our calculation to more compacti-fied coordinates. The mode expansion of the compacticompacti-fied coordinate is

X25(σ, τ ) = x25+ K25τ + i k=0

α25_k k e

−ikτ_{cos nσ,} _(2.1)

where K25 is the canonical momentum in the X25 direction

K25= 2πJ − θl+ θi

2πR . (2.2)

Note that J is the quantized momentum and we have included a nontrivial Wilson line with U (n) Chan-Paton factors, i, l = 1, 2...n. , which will be important in the later discussion. The mass spectrum of the theory is

M2 =K252+ 2 (N − 1) ≡ 2πJ − θl+ θi 2πR ₂ + ˆM2, (2.3) where we have deﬁned level mass as ˆM2 = 2 (N − 1) and N = _k=0α25_−kα25_k +

αμ_−kαμ_k, μ = 0, 1, 2...24. We are going to consider 4-point correlation function in this

paper. In the center of momentum frame, the kinematic can be set up to be19), 20)

k1= + p2+ M₁2, −p, 0, −K₁25 , (2.4) k2= + p2+ M₂2, +p, 0, +K₂25 , (2.5) k3= −q2+ M₃2, −q cos φ, −q sin φ, −K₃25 , (2.6) k4= −q2+ M₄2, +q cos φ, +q sin φ, +K₄25 , (2.7)

where p is the incoming momentum, q is the outgoing momentum and φ is the center of momentum scattering angle. In the high-energy limit, one includes only momenta on the scattering plane, and we have included the fourth component for the compactiﬁed direction as the internal momentum. The conservation of the fourth component of the momenta implies

m K_m25= m _2πJ m− θl,m+ θi,m 2πR = 0. (2.8)

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Note that k_i2= (K_i25)2− M_i2=− ˆM_i2. (2.9) We have −k1· k2 = p2+ M₁2· p2+ M₂2+ p2+ K₁25K₂25 = 1 2 s + k₁2+ k2₂= 1 2s − 1 2 ˆ M₁2+ ˆM₂2 , (2.10) −k2· k3 =− p2+ M₂2· q2+ M₃2+ pq cos φ + K₂25K₃25 = 1 2 t + k2₂+ k2₃= 1 2t − 1 2 ˆ M₂2+ ˆM₃2 , (2.11) −k1· k3 =− p2+ M₁2· q2+ M₃2− pq cos φ − K₁25K₃25 = 1 2 u + k₁2+ k₃2= 1 2u − 1 2 ˆ M₁2+ ˆM₃2 , (2.12)

where s, t and u are the Mandelstam variables with

s + t + u = i

ˆ

M_i2 = 2 (N − 4) . (2.13) Note that the Mandelstam variables defined above are not the usual 25-dimensional Mandelstam variables in the scattering process since we have included the inter-nal momentum K_i25 in the definition of ki. In order to define the Regge or fixed

momentum transfer regime, we deﬁne the momenta

k1 = + p2+ ˆM₁2, −p, 0, 0 , (2.14) k2 = + p2+ ˆM₂2, +p, 0, 0 , (2.15) k3 = −q2+ ˆM₃2, −q cos φ, −q sin φ, 0 , (2.16) k4 = −q2+ ˆM₄2, +q cos φ, +q sin φ, 0 (2.17) and the corresponding 25-dimensional Mandelstam variables

−k1· k2 = p2+ ˆM₁2· p2+ ˆM₂2+ p2 = 1 2 s25− ˆM₁2− ˆM₂2 , (2.18) −k2· k3 =− p2+ ˆM₂2· q2+ ˆM₃2+ pq cos φ = 1 2 t25− ˆM22− ˆM32 , (2.19) −k1· k3 =− p2+ ˆM₁2· q2+ ˆM₃2− pq cos φ = 1 2 u25− ˆM₁2− ˆM₃2 , (2.20) where

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s25+ t25+ u25=

i

ˆ

M_i2 = 2 (N − 4) . (2.21) In the high-energy limit, we deﬁne the polarizations on the scattering plane to be

eP = 1 M2 p2+ M₂2, p, 0, 0 , (2.22) eL= 1 M2 p, p2+ M₂2, 0, 0 , (2.23) eT = (0, 0, 1, 0) , (2.24)

where the fourth component refers to the compactiﬁed direction. The center of mass energy E is deﬁned as (for large p, q)

E = 1 2 p2+ M₁2+ p2+ M₂2 = 1 2 q2+ M₃2+ q2+ M₄2 . (2.25) The projections of the momenta on the scattering plane can be calculated to be (here we only list the ones we will need for our calculation)

eP · k1 =− 1 M2 p2+ M₁2 p2+ M₂2+ p2 , (2.26) eL· k1 =− p M2 p2+ M₁2+ p2+ M₂2 , (2.27) eT · k1 = 0 (2.28) and eP · k3= 1 M2 q2+ M₃2 p2+ M₂2− pq cos φ , (2.29) eL· k3= _M1 2 p q2+ M₃2− q p2+ M₂2cos φ , (2.30) eT · k3=−q sin φ. (2.31)

§3. Fixed angle regime

We begin with a brief review of high energy string scatterings for the non-compactiﬁed 26D open bosonic string in the GR. That is in the kinematic regime

s, −t → ∞, t/s ≈ − sin2 θ₂= ﬁxed (but θ = 0) where s, t and u are the Mandelstam variables for the noncompactiﬁed momenta and θ is the 26D CM scattering angle. It was shown7), 8) that for the 26D open bosonic string the only states that will survive the high-energy limit at mass level M₂2 = 2(N − 1) are of the form

|N, 2m, r ≡ (αT₋₁)N−2m−2r(αL₋₁)2m(αL₋₂)r|0, k₂ . (3.1)

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It can be shown that the high-energy vertex in Eq. (3.1) is conformal invariants up to a subleading term in the high-energy expansion. Note that eP approaches eL in the GR, and the scattering plane is deﬁned by the spatial components of eL and

eT. Polarizations perpendicular to the scattering plane are ignored because they are kinematically suppressed for four point scatterings in the high-energy limit. One can then use the saddle-point method to calculate the high energy scattering amplitudes. For simplicity, we choose k1, k3 and k4 to be tachyons and the ﬁnal result of the

ratios of high energy, ﬁxed angle string scattering amplitude are7), 8)

T(N,2m,r) T(N,0,0) = − 1 M2 _2m+r 1 2 _m+r (2m − 1)!!. (3.2)

We now review the results obtained previously for the compactified open string scatterings at fixed angle φ = finite.20) For simplicity, the second vertex was chosen to be

|N, 0, r, i, l =αT₋₁N−2rαL₋₂r|k2, l2, i, l (3.3)

at mass level ˆM₂2 = 2 (N − 1) , which was scattered with three “tachyon” states (with ˆM₁2 = ˆM₃2 = ˆM₄2 = −2). The high-energy ﬁxed angle open string scattering amplitudes with one compactiﬁed coordinate were calculated to be (the trace factor due to Chan-Paton was ignored)20)

T(N,0,r,i,l) (−iq sin φ)N

⎛ ⎝− pq2+ M₃2− qp2+ M₂2cos φ M2q2sin2φ ⎞ ⎠ r ·r j=0 r j ⎡ ⎣− p p2+ M₁2+p2+ M₂2 pq2+ M₃2− qp2+ M₂2cos φ ⎤ ⎦ j · B −1 −1 2s, −1 − 1 2t −1 −1 2s N−2j −1 − 1 2t 2j 2 +1 2u ₋₁ N , (3.4) where (a)j = a(a + 1)(a + 2)...(a + j − 1) is the Pochhammer symbol, and (a)j = aj

for large a and ﬁxed j. 3.1. Fixed winding modes

In the Gross regime, p2 K_i2 and p2 N, Eq. (3.4) reduces to

T(N,0,r,i,l) −iEsinφ2 cosφ₂ _N − 1 2M2 _r · B −1 − 1 2s, −1 − 1 2t . (3.5)

For each ﬁxed mass level N , we have the linear relation for the scattering amplitudes

T(N,0,r,i,l) T(N,0,0,i,l) = − 1 2M2 _r (3.6)

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with ratios consistent with our previous result in Eq. (3.2). Note that in Eq. (3.5) there is an exponential fall-oﬀ factor in the high-energy expansion of the beta func-tion. The inﬁnite linear relations in Eq. (3.6) “soften” the high-energy behavior of string scatterings in the GR.

3.2. Super-highly winding modes

We next consider a more interesting regime, the Mende kinematic regime (MR).20) For the case of φ = ﬁnite, the only choice to achieve UV power-law behavior is to require (we choose (K₁25)2  (K₂25)2  (K₃25)2  (K₄25)2)

K_i252 p2 N. (3.7)

In order to explicitly show that this choice of kinematic regime does lead to UV power-law behavior, it was shown that in this regime

s = constant (3.8)

in the open string scattering amplitudes. This in turn gives the desired power-law behavior of high-energy compactiﬁed open string scattering in Eq. (3.4). On the other hand, it can be shown that the linear relations break down as expected in this regime. For the choice of kinematic regime in Eq. (3.7), Eqs. (2.10) and (3.8) imply

lim p→∞ p2+ M₁2·p2+ M₂2+ p2 K₁25K₂25 = limp→∞ p2+ M₁2·p2+ M₂2+ p2 _2πl 1−θj,1+θi,1 2πR _2πl 2−θj,2+θi,2 2πR = −1. (3.9) For ﬁnite momenta J1and J2, the power-law behavior can be achieved by scattering

of string states with “super-highly” winding nontrivial Wilson lines

(θi,1− θl,1)→ ∞, (θi,2− θl,2)→ −∞. (3.10)

Note that the directions of momenta K₁25 and K₂25 are opposite. SinceK_i252 p2

and by Eq. (2.9), we can do the expansion of Eq. (3.9) to get

−K25 1 K225(1 + p 2 2(K25 1 )2)(1 + p2 2(K25 2 )2) + p 2 K₁25K₂25 =−1, (3.11)

which in turn, to the ﬁrst order of the expansion, gives

−K₁25K₂25 1 + p 2 2(K₁25)2 + p2 2(K₂25)2 + p2=−K₁25K₂25. (3.12) A simple calculation then gives

(λ1+ λ2)2 = 0, (3.13)

where signs of λ1 = _Kp25

1 and λ2 =−

p

K25

2 are chosen to be the same. It can be seen now

that the kinematic regime in Eq. (3.7) does solve Eq. (3.13). In conclusion, there is a φ = f inite regime with UV power-law behavior for the high-energy compactiﬁed open string scatterings. This new phenomenon never happens in the 26D string scatterings. The linear relations break down as expected in this regime.

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§4. Regge scatterings

We now begin to consider the compactiﬁed Regge string scatterings. It is im-portant at this point to note that in the high-energy, t25 = f inite approximation,

all ˆM_i2 can be neglected and we have cos φ 1 +2t25

s25, p sin φ √

−t25, (4.1)

where we have used Eqs. (2.18) to (2.21) to do the calculation. It is easy to see that high-energy ﬁxed t25= f inite (instead of ﬁxed t) approximation corresponds to the small angle φ or Regge regime (RR). In the high-energy limit, p2 = q2 = s25/4. In

this paper, we are going to consider two diﬀerent Regge regimes (RR) correspond-ing to ﬁxed windcorrespond-ing modes (K_i25)2 p2 and highly winding modes (K_i25)2  p2 respectively.

4.1. Fixed winding modes

We ﬁrst consider the following RR

t25= f inite, (Ki25)2 p2 N. (4.2)

In this regime

t25 t + (K225− K325)2. (4.3)

A class of high-energy vertex at ﬁxed mass level N =_n,mnpn+ mqm is31), 33)

|pn, qm, i, l = l>0

(αT_−n)pn

m>0

(αL_−m)qm|k₂, l2, i, l . (4.4)

The conformal invariant property of the above vertex was discussed in 33). Note that states containing operators α25_−n are of sub-leading order in energy and are neglected. For simplicity, we will only consider the states

|N, 2m, r, i, l =αT₋₁N−2m−2rαL₋₁2mαL₋₂r|k2, l2, i, l (4.5) at mass level ˆM₂2 = 2 (N − 1) scattered with three “tachyon” states (with ˆM₁2 =

ˆ

M₃2 = ˆM₄2 = −2). Equation (4.5) is the most general high-energy vertex in the ﬁxed angle regime. The vertex considered previously at ﬁxed angle in Eq. (3.3) corresponds to m = 0 only and thus was not completed. The relevant kinematics can be calculated to be eP · k1  − s25 2M2, e P _{· k} 3  −t25− M 2 2 − M32 2M2 =− ˜ t25 2M2; (4.6) eL· k1  − s25 2M2, e L_{· k} 3 −t25+ M 2 2 − M32 2M2 =− ˜ t₂₅ 2M2; (4.7) and eT · k1 = 0, eT · k3  −√−t25. (4.8)

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We are now ready to calculate the Regge scattering amplitudes. Note that eP =

eLin the RR.31)–33) We will calculate eLamplitudes in this paper. The corresponding

eP amplitudes can be similarly calculated. The s − t channel of the compactified Regge string scattering amplitudes in the regime Eq. (4.2) can be calculated to be (We will ignore the trace factor due to Chan-Paton in the scattering amplitude calculation. This does not affect our final results in this paper.)

A(N,2m,r,i,l)= ₁ 0 dx x k1·k2₍₁_{− x)}k2·k3 eT · k3 1− x N−2m−2r · eL· k1 −x + eL· k3 1− x 2m eL· k1 x2 + eL· k3 (1− x)2 r  (√−t25)N−2m−2r _˜ t₂₅ 2M2 r ₁ 0 dx x k1·k2₍₁_{− x)}k2·k3−N+2m ·2m j=0 2m j s25 2M2x _j −˜t 25 2M2(1− x) 2m−j = (√−t₂₅)N−2m−2r _˜ t₂₅ 2M2 r _˜ t₂₅ 2M2 2m ·2m j=0 2m j (−1)j s25 ˜ t₂₅ _j B (k1· k2− j + 1, k2· k3− N + j + 1) . (4.9)

Note that the term eL_x·k21 in the bracket is subleading in energy and can be neglected.

In the high-energy limit, the beta function in Eq. (4.9) can be approximated by

B (k1· k2− j + 1, k2· k3− N + j + 1) B −1 − 1 2s, −1 − t 2 −s 2 _−j −1 − t 2 j. (4.10) Finally, the leading order amplitude in the RR can be written as

A(N,2m,r,i,l)= B −1 − s 2, −1 − t 2  √ −t25N−2m−2r 1 2M2 _2m+r 22m(˜t₂₅)rU −2m , t 2 + 2− 2m , ˜ t₂₅ 2 , (4.11)

which is UV power-law behaved as t = f inite in the beta function by Eq. (4.3). U in Eq. (4.11) is the Kummer function of the second kind and is deﬁned to be

U (a, c, x) = π sin πc M (a, c, x) (a − c)!(c − 1)!− x1−cM (a + 1 − c, 2 − c, x) (a − 1)!(1 − c)! , (c = 2, 3, 4...) (4.12) where M (a, c, x) =∞_j=0(a)_(c)_jjx_j!j is the Kummer function of the ﬁrst kind. U and M are the two solutions of the Kummer equation

xy(x) + (c − x)y(x) − ay(x) = 0. (4.13)

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It is crucial to note that, in our case of Eq. (4.11), c = ₂t + 2− 2m and is not a constant as in the usual deﬁnition, so U in Eq. (4.11) is not a solution of the Kummer equation.

It is important to note that there is no linear relation among high-energy string scattering amplitudes of different string states for each fixed mass level in the RR as can be seen from Eq. (4.11). This is very different from the result in the GR in Eq. (3.2). In other words, the ratios A(N,2m,r,i,l)/A(N,0,0,i,l) are ˜t₂₅-dependent functions. In particular, we can extract the coefficients of the highest power of ˜t₂₅

in A(N,2m,r,i,l)/A(N,0,0,i,l). We can use the identity of the Kummer function 22m(˜t₂₅)−2m U −2m,t 2 + 2− 2m, ˜ t₂₅ 2 = ₂F0 −2m, −1 − t 2, − 2 ˜ t₂₅ ≡ 2m j=0 (−2m)_j −1 − t 2 j − 2 ˜t 25 _j j! = 2m j=0 2m j −L − ˜t25 2 j 2 ˜ t₂₅ _j (4.14) to get A(N,2m,r,i,l) A(N,0,0,i,l) = (−1) m₋ 1 2M2 _2m+r (˜t₂₅− M₂2+ M₃2)−m−r(˜t₂₅)2m+r ·2m j=0 (−2m)_j −L − ˜t25 2 j (−2/˜t₂₅)j j! + O 1 ˜ t₂₅ _m+1 , (4.15) where L = 1 − N − (K₂25)2+ K₂25K₃25. (4.16) If the leading order coeﬃcients in Eq. (4.15) extracted from the high energy string scattering amplitudes in the RR are to be identiﬁed with the complete ratios in Eq. (3.2) calculated previously among high energy string scattering amplitudes in the GR31), 32) lim ˜t 25→∞ A(N,2m,r,i,l) A(N,0,0,i,l) = − 1 M2 _2m+r 1 2 _m+r (2m − 1)!! = T (N,2m,r,i,l) T(N,0,0,i,l) , (4.17)

we need the following identity:

2m j=0 (−2m)_j −L −t˜25 2 j (−2/˜t₂₅)j j! = 0(−˜t₂₅)0+ 0(−˜t₂₅ )−1+ ... + 0(−˜t₂₅)−m+1+(2m)! m! (−˜t 25)−m+ O 1 ˜ t₂₅ _m+1 . (4.18)

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Note that the ratios calculated previously at ﬁxed angle in Eq. (3.6) corresponds to m = 0 only in Eq. (4.17) and thus was not completed. The ratios in Eq. (4.17) calculated in this paper by the indirect method through the RR amplitudes are the most general ones. The coeﬃcient of the term O1/˜t₂₅m+1

in Eq. (4.18) is irrelevant for our discussion. The proof of Eq. (4.18) turns out to be nontrivial. The standard approach by using integral representation of the Kummer function seems not applicable here. Presumably, the diﬃculty of the rigorous proof of Eq. (4.18) is associated with the nonconstant c mentioned previously.

Mathematically, the complete proof of Eq. (4.18) for arbitrary real values L was recently worked out in 36) by using an identity of signless Stirling number of the ﬁrst kind in combinatorial theory. The proof of the identity for L = 0, 1, was previously given in 31)–33) based on a set of identities of signed Stirling number of the ﬁrst kind.35) It is interesting to see that, physically, the identities for arbitrary real values

L can only be realized in high-energy compactiﬁed string scatterings considered in

this paper. This is due to the dependence of the value L on winding momenta

K_i25. All other high-energy string scattering amplitudes calculated previously31)–33)

correspond to integer value of L only. 4.2. Highly winding modes

In this subsection, we consider the more interesting RR

t25= f inite, (K_i25)2  p2 N. (4.19) In this regime, Eqs. (2.11) and (2.19) imply

t25 t − 2 p2+ ˆM₂2· q2+ ˆM₃2+ 2 p2+ M₂2· q2+ M₃2− 2K₂25K₃25. (4.20)

It is easy to see that in general∗) t is as large as p2 in this regime. The most general high-energy vertex at each ﬁxed mass level N is

|N, 2m, r, i, l =αT₋₁N−2m−2rαL₋₁2mαL₋₂r|k2, l2, i, l . (4.21)

Note that states containing operatorsα25_−nare again of sub-leading order in energy. For simplicity, we will only consider the states

|N, 0, r, i, l =αT₋₁N−2rαL₋₂r|k2, l2, i, l (4.22)

at mass level ˆM₂2 = 2 (N − 1) scattered with three “tachyon” states (with ˆM₁2 = ˆ

M₃2 = ˆM₄2=−2). The s − t channel of the high-energy scattering amplitude can be calculated to be A(N,0,r,i,l)= d4x · i<j (xi− xj)ki·kj · ieT · k1 x1− x2 + ieT · k3 x3− x2 + ieT · k4 x4− x2 N−2r · eL· k1 (x1− x2)2 + e L_{· k}₃ (x3− x2)2 + e L_{· k}₄ (x4− x2)2 r . (4.23) ∗)_{For some regime,}_{t can be ﬁnite. For example, for K}25

2 =K325 p2 N, t t25=finite by

Eq. (4·20).

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After ﬁxing the SL(2, R) gauge and using the kinematic relations Eqs. (2.26) to (2.31) and Eq. (4.1) derived previously, we have

A(N,0,r,i,l)=−i√−t₂₅N pq2+ M₃2− qp2+ M₂2 M2t25 _r · ₁ 0 dx · x k1·k2₍₁_{− x)}k2·k3−N+2r · ⎡ ⎣ p p2+ M₁2+ pp2+ M₂2 pq2+ M₃2− qp2+ M₂2 x2 − 1 (1− x)2 ⎤ ⎦ r =−i√−t₂₅N pq2+ M₃2− qp2+ M₂2 M2t25 _r ·r j=0 r j −p p2+ M₁2+ pp2+ M₂2 pq2+ M₃2− qp2+ M₂2 j · ₁ 0 dx · x k1·k2−2j₍₁_{− x)}k2·k3−N+2j =−i√−t₂₅N pq2+ M₃2− qp2+ M₂2 M2t25 _r ·r j=0 r j −p p2+ M₁2+ pp2+ M₂2 pq2+ M₃2− qp2+ M₂2 j · B −1 2s + N − 2j − 1, − 1 2t + 2j − 1 , (4.24)

where B(u, v) is the Euler beta function. We can do the high-energy approximation of the gamma function Γ (x) and end up with

A(N,0,r,i,l)=−i√−t₂₅N pq2+ M₃2− qp2+ M₂2 M2t25 _r · r j=0 r j −p p2+ M₁2+ pp2+ M₂2 pq2+ M₃2− qp2+ M₂2 j ·Γ −1 −1 2s + N − 2j Γ−1 − 1₂t + 2j Γ2 +1₂u −i√−t25N pq2+ M₃2− qp2+ M₂2 M2t25 _r · r j=0 r j −p p2+ M₁2+ pp2+ M₂2 pq2+ M₃2− qp2+ M₂2 j · B −1 − 1 2s, −1 − 1 2t −1 − 1 2t 2j −1 − 1 2s _N−2j 2 +1 2u _−N

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= i √ −t25 (u_s) _N pq2+ M₃2− qp2+ M₂2 M2t25 _r · B −1 − 1 2s, −1 − 1 2t · r j=0 r j −p p2+ M₁2+ pp2+ M₂2 pq2+ M₃2− qp2+ M₂2 4 s2 j −1 − 1 2t 2j. (4.25) Finally, since t is as large as p2 in the regime Eq. (2.26), we can easily do the summation and end up with

A(N,0,r,i,l)= i √ −t25 (u_s) _N − 1 M2 _r · B −1 − 1 2s, −1 − 1 2t −p q2+ M₃2− qp2+ M₂2 t25 + pp2+ M₁2+ pp2+ M₂2 t25 t s ₂ r , (4.26) where (_st) and (u_s) are fixed numbers. Since t is as large as s in this regime, the beta function B(−1 − s₂, −1 − t₂) in Eq. (4.26) implies that the UV behavior of the amplitude is exponential fall-off. On the other hand, it is clear that there is no linear relation in this regime. In conclusion, we have discovered a small angle φ 0 regime with UV exponential fall-off behavior for the high-energy compactified open string scatterings. This new phenomenon never happens in the 26D string scatterings.

§5. Conclusion

In this paper, we have mainly achieved three new results for high-energy string scattering amplitudes. First, we calculate massive string scattering amplitudes of compactified open string in the Regge regime. We can then extract the complete infinite ratios among high-energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes. The complete ratios calculated by this indirect method include and extend the subset of ratios calculated previously19), 20)by the more difficult direct fixed angle calculation.

Second, by studying the high-energy string scattering for the compactiﬁed open string, we discover in this paper a realization of arbitrary real values L in the identity Eq. (4.18) which was proposed recently to link ﬁxed angle and Regge string scattering amplitudes. All other high-energy string scatterings calculated previously31), 33), 34) correspond to integer value of L only. Physically, the parameter L is related to the mass level of an excited string state and can take non-integer values for Kaluza-Klein modes. Mathematically, the identity in Eq. (4.18) was explicitly proved recently for arbitrary real values L in 36) by using the signless Stirling number in combinatorial theory.

Finally, we discover a kinematic regime which shows the unusual exponential fall-off behavior in the small angle scattering. This is complimentary with a fixed angle regime discovered previously,20)which shows the unusual power-law behavior in the compactified string scatterings.

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Acknowledgements

This work is supported in part by the National Science Council, 50 Billions Project of MOE and National Center for Theoretical Science, Taiwan, R.O.C. We would like to thank the hospitality of KITPC where part of this work was completed during our visits in the summer of 2010. J. C. Lee and Yi Yang would like to thank discussions on this subject with Prof. C. I. Tan of Brown University, Prof. B. Feng of Zhejiang University and Dr. Y. Mitsuka. S. He would like to thank Prof. Mei Huang and Prof. Sang Jin Sin’s warm support. He is grateful to APCTP and CQUeST in Korea for their hospitalities at various stages of this work.

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