Patterns of high energy massive string scatterings in the Regge regime

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Patterns of high energy massive string scatterings in the Regge regime

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JHEP06(2009)028

Published by IOP Publishing for SISSA

Received: April 5, 2009 Accepted: May 25, 2009 Published: June 8, 2009

Patterns of high energy massive string scatterings in

the Regge regime

Sheng-Lan Ko,a Jen-Chi Leeb and Yi Yangb

a

Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C.

b

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

E-mail: [email protected],[email protected], [email protected]

Abstract:_{We calculate high energy massive string scattering amplitudes of open bosonic} string in the Regge regime (RR). We found that the number of high energy amplitudes for each fixed mass level in the RR is much more numerous than that of Gross regime (GR) calculated previously. Moreover, we discover that the leading order amplitudes in the RR can be expressed in terms of the Kummer function of the second kind. In particular, based on a summation algorithm for Stirling number identities developed recently, we discover that the ratios calculated previously among scattering amplitudes in the GR can be extracted from this Kummer function in the RR. We conjecture and give evidences that the existence of these GR ratios in the RR persists to subleading orders in the Regge expansion of all string scattering amplitudes. Finally, we demonstrate the universal power-law behavior for all massive string scattering amplitudes in the RR.

Keywords: _{Bosonic Strings, Discrete and Finite Symmetries}

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Contents

1 Introduction 1

2 Regge scattering for M22 = 4 3

3 General mass levels 6

4 Reproducing the GR ratios in the RR 8

5 Subleading orders 11

6 Universal power law behavior 15

7 Conclusion 16

A Kinematic variables and notations 17

1 Introduction

There are two fundamental regimes of high energy string scattering amplitudes. These are the fixed angle regime or Gross regime (GR), and the fixed momentum transfer regime or Regge regime (RR). These two regimes represent two different high energy perturbation expansions of the scattering amplitudes, and contain complementary information of the theory. The UV behavior of high energy string scatterings in the GR is well known to be very soft exponential fall-off, while that of RR is hard power-law. The high energy string scattering amplitudes in the GR [1–3] were recently intensively reinvestigated for massive string states at arbitrary mass levels [4–12]. See also the developments in [13–15]. An infi-nite number of linear relations, or stringy symmetries, among string scattering amplitudes of different string states were obtained. Moreover, these linear relations can be solved for each fixed mass level, and ratios T(N,2m,q)/T(N,0,0) among the amplitudes can be obtained. An important new ingredient of these calculations is the decoupling of zero-norm states (ZNS) [16–18] in the old covariant first quantized (OCFQ) string spectrum. It is interesting to note that the calculation in [1–3] is valid only for four-tachyon amplitude, but not for all other amplitudes of excited string states. This was pointed out and the calculation was cor-rected by two independent groups [6,14] with two different approaches. Since there does not exist any algebraic structure (or group structure) of this high energy 26D spacetime sym-metry, mathematically the meaning of these infinite number of ratios remains mysterious. Another fundamental regime of high energy string scattering amplitudes is the RR [19– 24]. See also [25–27]. Since the decoupling of ZNS applies to all kinematic regimes, one

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expects some implication of this decoupling in the RR. Moreover, it is conceivable that there exists some link between the patterns of the high energy scattering amplitudes of GR and RR. With this in mind, in this paper, we give a detail calculation of high energy string scattering amplitudes in the RR. We will find that the number of high energy scattering amplitudes for each fixed mass level in the RR is much more numerous than that of GR calculated previously. On the other hand, it seems that both the saddle-point method and the method of decoupling of high energy ZNS adopted in the calculation of GR do not apply to the case of RR. However the calculation is still manageable, and the general formula for the high energy scattering amplitudes for each fixed mass level in the RR can be written down explicitly.

In contrast to the case of scatterings in the GR, we will see that there is no linear rela-tion among scatterings in the RR. Moreover, we discover that the leading order amplitudes at each fixed mass level in the RR can be expressed in terms of the Kummer function of the second kind. More surprisingly, for those leading order high energy amplitudes A(N,2m,q)_in

the RR with the same type of (N, 2m, q) as those of GR, we can extract from them the ratios T(N,2m,q)/T(N,0,0) in the GR by using this Kummer function. Mathematically, the proof of this result turns out to be highly nontrivial and is based on a summation algorithm for Stir-ling number identity derived by Mkauers in 2007 [28]. It is very interesting to see that the identity in eq. (4.3) suggested by string theory calculation can be rigorously proved by a to-tally different but sophisticated mathematical method. The derivation of these physical ra-tios from Kummer function through Stirling number identities seems to suggest another in-terpretation of these infinite number of ratios mathematically. We then proceed to calculate Regge string scattering amplitudes to subleading orders. We conjecture and give evidences that these ratios persist to all orders in the Regge expansion of high energy string scattering amplitudes for the even mass level with (N − 1) = M

2 2

2 = even. For the odd mass levels with

(N −1) = M 2 2

2 = odd, the existence of the GR ratios shows up only in the first [N/2]+1 terms

in the Regge expansion of the amplitudes. At last, as an application of our results, we show that the well known sα(t) power-law behavior of the four tachyon string scattering ampli-tude in the RR can be extended to all high energy massive string scattering ampliampli-tudes.

This paper is organized as following. In section II, after a brief review of high energy string scatterings in the GR, we first calculate all leading high energy scattering amplitudes for the mass level M2 _{= 4 in the RR. We compare the two sets of amplitudes and discover}

a link between the two. The calculation is then generalized to general mass level in the RR in section III. We show that the leading order amplitudes can be expressed in terms of the Kummer function of the second kind. In section IV, based on a summation algorithm for Stirling number identity, we show that the ratios among scattering amplitudes in the GR can be extracted from Kummer function derived in section III. In section V, we give evidences that the existence of these ratios in the RR persists to subleading orders in the Regge expansion of all high energy string scattering amplitudes. In section VI, we demonstrate the universal power-law behavior for all massive string scattering amplitudes in the RR. Finally, an appendix is devoted to the kinematics used in the text.

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2 _{Regge scattering for M}2

2 = 4

We begin with a brief review of high energy string scatterings in the GR. That is in the kinematic regime

s, −t → ∞, t/s ≈ − sin2θ

2 = fixed (but θ 6= 0) (2.1) where s, t and u are the Mandelstam variables and θ is the CM scattering angle. It was shown [7, 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, qi ≡ (αT−1)N −2m−2q(αL−1)2m(αL−2)q|0i, (2.2)

where the polarizations of the 2nd particle with momentum k2 on the scattering plane were

defined to be eP = _M1₂(E2, k2, 0) = _Mk22 as the momentum polarization, e L₌ 1

M2(k2, E2, 0) the longitudinal polarization and eT = (0, 0, 1) the transverse polarization. Note that eP approaches to 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 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 are [7,8] T(N,2m,q) T(N,0,0) = −_M1 2 2m+q 1 2 m+q (2m − 1)!!. (2.3) The ratios in eq. (2.3) can also be obtained by using the decoupling of two types of ZNS in the spectrum

Type I : L₋₁_{|xi ,} where L1|xi = L2|xi = 0, L0|xi = 0; (2.4)

Type II : (L₋₂+ 3 2L

2

−1) |exi , where L1|exi = L2|exi = 0, (L0+ 1) |exi = 0. (2.5)

As examples, for M₂2 = 4, 6, we get [4,5]

TT T T : TLLT : T(LT ): T[LT ] = 8 : 1 : −1 : −1, (2.6)

TT T T T : TT T LL : TLLLL : TT T L : TLLL : ˜TLT,T : ˜TLP,P : TLL : ˜TLL

16 : 4₃ : 1₃ _{: −}4√₉6 _{: −}√₉6 _{: −}2√₃6 : 0 : 2₃ : 0 . . . (2.7) We now turn to the discussion on high energy string scatterings in the RR. That is in the kinematic regime

s → ∞,√−t = fixed (but√−t 6= ∞). (2.8) As in the case of GR, we only need to consider the polarizations on the scattering plane, which is defined in appendix A. Appendix A also includs the kinematic set up and some formulas we need in our calculation. However, instead of using (E, θ) as the two indepen-dent kinematic variables in the GR, we choose to use (s, t) in the RR. One of the reason

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has been, in the RR, t ∼ Eθ is fixed, and it is more convenient to use (s, t) rather than (E, θ). In the RR, to the lowest order, equations (A.13) to (A.18) reduce to

eP _{· k}1 = − 1 M2 q p2_{+ M}2 1 q p2_{+ M}2 2 + p2 ≃ −_2Ms 2 , (2.9a) eL_{· k}1 = − p M2 q p2_{+ M}2 1 + q p2_{+ M}2 2 ≃ −_2Ms 2 , (2.9b) eT _{· k}1 = 0 (2.9c) and eP _{· k}3 = 1 M2 q q2_{+ M}2 3 q p2_{+ M}2 2 − pq cos θ ≃ −_2M˜t 2 ≡ − t − M22− M32 2M2 , (2.10a) eL_{· k}3 = 1 M2 p q q2_{+ M}2 3 − q q p2_{+ M}2 2 cos θ ≃ − ˜t ′ 2M2 ≡ − t + M2 2 − M32 2M2 , (2.10b) eT _{· k}3 = −q sin φ ≃ −√−t. (2.10c)

Note that eP does not approach to eL in the RR. This is very different from the case of GR. In the following discussion, we will calculate the amplitudes for the longitudinal polarization eL. For the eP amplitudes, the results can be trivially modified. There is another important difference between the high energy scattering amplitudes in the RR and in the GR. We will find that the number of high energy scattering amplitudes for each fixed mass level in the RR is much more numerous than that of GR calculated previously. On the other hand, it seems that both the saddle-point method and the method of decoupling of high energy ZNS adopted in the calculation of GR do not apply to the case of RR. In this section, we will explicitly calculate the string scattering amplitudes on the scattering plane eL, eT for the mass level M₂2= 4. In the mass level M₂2= 4 M₁2= M₃2 = M₄2_{= −2}, it turns out that there are eight high energy amplitudes in the RR

αT₋₁αT₋₁αT₋₁_{|0i, α}L₋₁αT₋₁αT₋₁_{|0i, α}L₋₁αL₋₁αT₋₁_{|0i, α}L₋₁αL₋₁αL₋₁_|0i,

αT₋₁αT₋₂_{|0i, α}T₋₁αL₋₂_{|0i, α}L₋₁αT₋₂_{|0i, α}L₋₁αL₋₂_|0i. (2.11) The s − t channel of these amplitudes can be calculated to be

AT T T = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieT _{· k}1 x − ieT _{· k}3 1 − x 3 ≃ −i √−t3 Γ − s 2 − 1 Γ₋t₂˜_{− 1} Γ u₂ + 3 · −1₈s3+1 2s , (2.12) ALT T = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieT _{· k}1 x − ieT _{· k}3 1 − x 2 ieL_{· k}1 x − ieL_{· k}3 1 − x ≃ −i √−t2 −_2M1 2 Γ −s 2 − 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 · 3 4s 3 − ₄ts2₋ t 2 + 3 s , (2.13)

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ALLT = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieT _{· k} 1 x − ieT _{· k} 3 1 − x ieL_{· k} 1 x − ieL_{· k} 3 1 − x 2 ≃ −i √−t −_2M1 2 2 _{Γ −}s 2 − 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 · " 1 4t − 9 2 s3+ 1 4t 2₊7 2t s2+(t + 6) 2 2 s # , (2.14) ALLL = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieL_{· k}1 x − ieL_{· k}3 1 − x 3 ≃ −i −_2M1 2 3 _{Γ −}s 2 − 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 · " − 11 2 t − 27 s3_{− 6 t}2+ 6ts2₋(t + 6) 3 2 s # , (2.15) AT T = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieT _{· k} 1 x − ieT _{· k} 3 1 − x eT _{· k} 1 x2 + eT _{· k} 3 (1 − x)2 ≃ −i √−t2 Γ − s 2 − 1 Γ₋t₂˜_{− 1} Γ u₂ + 3 −1₈s3+1 2s , (2.16) AT L = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieT _{· k} 1 x − ieT _{· k} 3 1 − x eL_{· k} 1 x2 + eL_{· k} 3 (1 − x)2 ≃ i √−t −_2M1 2 Γ −s 2 − 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 · − 1 8t + 3 4 s3₋1 8 t 2 − 2ts2₋ 1 4t 2 − t − 3 s , (2.17) ALT = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieL_{· k} 1 x − ieL_{· k} 3 1 − x eT _{· k} 1 x2 + eT _{· k} 3 (1 − x)2 ≃ i √−t −_2M1 2 Γ −s 2 − 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 · 3 4s 3 −₄ts2₋ t 2 + 3 s , (2.18) and ALL = Z 1 0 dx · x k1·k2 (1 − x)k2·k3 · ieL_{· k} 1 x − ieL_{· k} 3 1 − x eL_{· k} 1 x2 + eL_{· k} 3 (1 − x)2 ≃ i −_2M1 2 2 _Γ ₋s 2 − 1 Γ₋t₂˜_{− 1} Γ u₂ + 3 · 3 4t + 9 2 s3+ t2_{− 4t}s2+ 1 4t 3₊1 2t 2_{− 9t − 18} s . (2.19) From the above calculation, one can easily see that all the amplitudes are in the same leading order ∼ s3 _{in the RR, while in the GR only A}T T T_{, A}LLT _{and A}T L _{are in the}

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the other hand, one notes that, for example, the term ∼√−tt2s2 in ALLT and AT L are in the leading order in the GR, but are in the subleading order in the RR. On the contrary, the terms√_−ts3 _{in A}LLT _{and A}T L _{are in the subleading order in the GR, but are in the}

leading order in the RR. These observations suggest that the high energy string scattering amplitudes in the GR and RR contain information complementary to each other.

One can now see that the number of high energy scattering amplitudes in the RR is much more numerous than that of GR. One important observation for high energy amplitudes in the RR is for those amplitudes with the same structure as those of the GR in eq. (2.2). For these amplitudes, the relative ratios of the coefficients of the highest power of t in the leading order amplitudes in the RR can be calculated to be

AT T T _{= −i} √_−t Γ − s 2 − 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 · 1 8ts 3 ∼ 1₈, (2.20) ALLT _{= −i} √_−t − 1 2M2 2_{Γ −}s 2 − 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 1 4ts 3 ∼ 1 64, (2.21) AT L= i √_−t −_2M1 2 Γ −s 2− 1 Γ₋₂˜t _{− 1} Γ u₂ + 3 · −1₈ts3 ∼ −₃₂1 , (2.22) which reproduces the ratios in the GR in eq. (2.6). Note that the symmetrized and anti-symmetrized amplitudes are defined as

T(T L)= 1 2 T T L_{+ T}LT_, _(2.23) T[T L]= 1 2 T T L − TLT; (2.24)

and similarly for the amplitudes A(T L)_{and A}[T L]_{in the RR. Note that T}LT _{∼ (α}L

−1)(αT−2)|0i

in the GR is of subleading order in energy, while ALT in the RR is of leading order in energy. However, the contribution of the amplitude ALT to A(T L) and A[T L] in the RR will not affect the ratios calculated above. As we will see in section IV, this interesting result can be generalized to all mass levels in the string spectrum.

3 General mass levels

In this section, we calculate high energy string scattering amplitudes in the RR for the arbitrary mass levels. Instead of states in eq. (2.2) for the GR, one can easily argue that the most general string states one needs to consider at each fixed mass level N = P

n,mnkn+ mqm for the RR are

|kn, qmi = Y n>0 (αT_−n)kn Y m>0 (αL_−m)qm_|0i. _(3.1) It seems that both the saddle-point method and the method of decoupling of high energy ZNS adopted in the calculation of GR do not apply to the case of RR. However the

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calculation is still manageable, and the general formula for the high energy scattering amplitudes in the RR can be written down explicitly. In fact, by the simple kinematics eT _{· k}

1 = 0, and the energy power counting of the string amplitudes, we end up with the

following rules to simplify the calculation for the leading order amplitudes in the RR: αT_−n: 1 term (contraction of ik3· X with εT · ∂nX), (3.2)

αL_−n: (

n > 1, 1 term

n = 1 2 terms (contraction of ik1· X and ik3· X with εL· ∂nX).

(3.3)

The s − t channel scattering amplitudes of this state with three other tachyonic states can be calculated to be A(kn,qm) ₌ Z 1 0 dx xk1·k2 (1 − x)k2·k3 ieL_{· k}1 −x + ieL_{· k}3 1 − x q1 ·Y n=1 ieT _{· k} 3(n − 1)! (1 − x)n kn Y m=2 ieL_{· k} 3(m − 1)! (1 − x)m qm = −i˜t′ 2M2 q1 q1 X j=0 q1 j s −˜t jZ 1 0 dxxk1·k2−j (1 − x)k2·k3+j−P_n,m(nkn+mqm) ·Y n=1 i√_{−t(n − 1)!}kn Y m=2 i˜t′_{(m − 1)!} −_2M1 2 qm = −i˜t′ 2M2 q1 q1 X j=0 q1 j s −˜t j B (k1· k2− j + 1 , k2· k3+ j − N + 1) ·Y n=1 i√_{−t(n − 1)!}kn Y m=2 i˜t′_{(m − 1)!} −_2M1 2 qm . (3.4)

The Beta function above can be approximated in the large s, but fixed t limit as follows B (k1· k2− j + 1, k2· k3+ j − N + 1) = B −1 − s 2+ N − j, −1 − t 2 + j = Γ(−1 − s 2 + N − j)Γ(−1 − t 2+ j) Γ(u₂ + 2) ≈ B −1 − 1₂s, −1 −₂t −1 − s₂N −ju₂ + 2−N −1 − ₂t j ≈ B −1 − 1₂s, −1 −₂t −s₂−j −1 − ₂t j . (3.5) where

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is the Pochhammer symbol. The leading order amplitude in the RR can then be written as A(kn,qm)₌ −i˜t′ 2M2 q1 B −1 − 1₂s, −1 − ₂t q1 X j=0 q1 j 2 ˜ t′ j −1 − ₂t j ·Y n=1 i√_{−t(n − 1)!}kn Y m=2 i˜t′_{(m − 1)!} −_2M1 2 qm , (3.7) which is UV power-law behaved as expected. The summation in eq. (3.7) can be represented by the Kummer function of the second kind U as follows,

p X j=0 p j 2 ˜ t′ j −1 −₂t j = 2p(˜t′)−p U −p,₂t + 2 − p,˜t ′ 2 . . . (3.8) Finally, the amplitudes can be written as

A(kn,qm) ₌ − i M2 q1 U −q1, t 2+ 2 − q1, ˜ t′ 2 B −1 − s 2, −1 − t 2 ·Y n=1 i√_{−t(n − 1)!}kn Y m=2 i˜t′_{(m − 1)!} −_2M1 2 qm . (3.9) In the above, U 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−c_{M (a + 1 − c, 2 − c, x)} (a − 1)!(1 − c)! (c 6= 2, 3, 4 . . .) (3.10) where M (a, c, x) = P∞_j=0 (a)j

(c)j xj

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.} (3.11) It is crucial to note that c = ₂t _{+ 2 − q}1, and is not a constant as in the usual case, so

U in eq. (3.9) is not a solution of the Kummer equation. This will make our analysis in the next section more complicated as we will see soon. On the contrary, since a = −q1 an

integer, the Kummer function in eq. (3.8) terminated to be a finite sum. This will simplify the manipulation of Kummer function used in this paper.

4 Reproducing the GR ratios in the RR

In section II, we have learned that the relative coefficients of the highest power t terms in the leading order amplitudes in the RR can reproduce the ratios of the amplitudes in the GR for the mass level M₂2 = 4. Now we are going to generalize the calculation to the string states of the arbitrary mass levels. The leading order amplitudes of string states in the RR, which share the same structure as eq. (2.2) in the GR can be written as

A(N,2m,q) =B −1 − s 2, −1 − t 2 _√ −tN −2m−2q 1 2M2 2m+q 22m(˜t′)qU −2m , ₂t + 2 − 2m , t˜₂′ . (4.1)

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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. (4.1). This is very different from the result in the GR. In other words, the ratios A(N,2m,q)/A(N,0,0) are t-dependent functions. As was done in section II for the mass level M₂2= 4, we can extract the coefficients of the highest power of t in A(N,2m,q)/A(N,0,0). We can use the identity of the Kummer function in eq. (3.8) to calculate

A(N,2m,q) A(n,0,0) = (−1) q 1 2M2 2m+q (−t)m 2m X j=0 (−2m)j −1 − t 2 j (−2/t)j j! + O ( 1 t m+1) . (4.2) where we have replaced ˜t′ _{by t as t is large. If the leading order coefficients in eq. (}_4.2₎

extracted from the high energy string scattering amplitudes in the RR are to be identified with the ratios calculated previously among high energy string scattering amplitudes in the GR in eq. (2.3), we need the following identity

2m X j=0 (−2m)j −1 − t 2 j (−2/t)j j! = 0(−t)0+ 0(−t)−1+ . . . + 0(−t)−m+1+(2m)! m! (−t) −m_{+ O} ( 1 t m+1) . . . (4.3) The coefficient of the term On(1/t)m+1o in eq. (4.3) is irrelevant for our discussion. The proof of eq. (4.3) suggested by string theory calculation turns out to be nontrivial math-ematically. Presumably, the difficulty of the rigorous proof of eq. (4.3) is associated with the unusual non-constant c in the argument of Kummer function in eq. (4.1) as mentioned above. We first rewrite the summation on the left hand side of eq. (4.3) as

2m X j=0 (−2m)j −1 − ₂t −₂t j−1 −2_t j 1 j! = 2m X j=0 (−2m)j −1 −₂t _Xj−1 k=0 (−1)j−1−ks(j − 1, k) −₂t k −2_t j 1 j! O(t−m₎ ∼ 2m X j=m (−2m)js(j − 1, j − m) 2m j! + 2m X j=m+1 (−2m)js(j − 1, j − m − 1) 2m j! = 2m m X j=0 (−1)j+m 2m j + m s(j + m − 1, j) +2m m X j=1 (−1)j+m 2m j + m s(j + m − 1, j − 1) (4.4) where we have used the signed Stirling number of the first kind s(n, k) to expand the Pochhammer symbol. The definition of s(n, k) is

(x)n= n

X

k=0

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Thus the leading order nontrivial identity of eq. (4.3) can be written as (m > 0) f (m) ≡ m X j=0 (−1)j 2m j + m [s(j + m − 1, j − 1) + s(j + m − 1, j)] = (2m − 1)!! (4.6) where we have used the convention that

s(m − 1, −1) (

= 0 , for m > 1

= 1 , for m = 0 , s(−1, 0) = 0. (4.7) With the help of the algorithm developed by Mkauers in 2007 [28], this identity can be proved. The point is that we can find a recurrence relation of f (m) by his algorithm. However, to utilize the algorithm, we need to introduce an auxiliary variable u and define

f (u, m) ≡ m+u_X j=0 (−1)j 2m + u j + m [s(j + m − 1, j − 1) + s(j + m − 1, j)] ≡ f1(u, m) + f2(u, m) (4.8)

where f1 and f2 are the two summations, each with one Stirling number, and f (0, m) =

f (m). By the algorithm, we can prove that both f1, f2 satisfy the following recurrence

relation [28]

− (1 + 2m + u)f(u, m) + (2m + u)f(u + 1, m) + f(u, m + 1) = 0, (4.9) hence, so is f. eq. (4.9) is the most nontrivial step to prove eq. (4.6). Now, note that

f (u, 0) = u X j=0 (−1)j u j = ( 1 , u = 0 0 , u > 0 . . . (4.10) Using the recurrence relation eq. (4.9_{) and substituting (u, m) = (1, 0), (2, 0) · · · , one can} prove that

f (u, 1) = 0, _{∀u > 0.} (4.11) Similarly, by substituting (u, m) = (1, 1), (2, 1), (3, 1) · · · , one can get f(u, 2) = 0, ∀u > 0. In general, we have

f (u, m) = 0, _{∀u > 0.} (4.12) Finally we substitute u = 0 in the eq. (4.9) to obtain

− (1 + 2m)f(0, m) + 2mf(1, m) + f(0, m + 1) = 0, (4.13) which implies

f (m + 1) = (2m + 1)f (m). (4.14) Eq. (4.6) is thus proved by mathematical induction.

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The vanishing of the coefficients of (−t)0, (−t)−1, . . . , (−t)−m+1 terms on the l.h.s. of eq. (4.3) means, for 1 6 i 6 m,

g(m, i) ≡ m+i X j=0 (−1)j−i 2m j + m − i [s(j + m − 1 − i, j) + s(j + m − 1 − i, j − 1)] = 0. (4.15)

To prove this identity, we need the recurrence relation [28]

− 2(1 + m)2(1 + 2m)g(m, i) + (2 + 7m + 4m2)g(m + 1, i)

−2m(1 + m)(1 + 2m)g(m + 1, i + 1) − mg(m + 2, i) = 0. (4.16) Putting i = 0, 1, 2 . . ., and using the fact we have just proved, i.e. g(m+1, 0) = (2m+1)g(m, 0), one can show that

g(m, i) = 0 for 1 6 i 6 m. (4.17) Eq. (4.3) is finally proved. We thus have shown that for those leading order high energy amplitudes A(N,2m,q) in the RR with the same type of (N, 2m, q) as those of GR, we can extract from them the ratios T(N,2m,q)/T(N,0,0) in the GR by using the Kummer function. Mathematically, the proof of this result turns out to be highly nontrivial and is based on a summation algorithm for Stirling number identity derived by Mkauers [28]. It is very inter-esting to see that the identity in eq. (4.3) suggested by string scattering amplitude calcula-tion can be rigorously proved by a totally different but sophisticated mathematical method. In the next section, we discuss the generalization to subleading order amplitudes in the RR.

5 Subleading orders

In this section, we calculate the next few subleading order amplitudes in the RR for the mass level M2

2 = 4, 6. We will see that the ratios in eqs. (2.6) and (2.7) persist to subleading

order amplitudes in the RR. For the even mass levels with (N − 1) = M 2 2

2 = even, we

conjecture and give evidences that the existence of these ratios in the RR persists to all orders in the Regge expansion of all high energy string scattering amplitudes . For the odd mass levels with (N − 1) = M

2 2

2 = odd, the existence of these ratios will show up only in the

first [N/2]+1 terms in the Regge expansion of the amplitudes.

We will extend the kinematic relations in the RR to the subleading orders. We first ex-press all kinematic variables in terms of s and t, and then expand all relevant quantities in s :

E1 = s − (m 2 2+ 2) 2√2 , (5.1) E2 = s + (m 2 2+ 2) 2√2 , (5.2) |k2| = q E2 1 + 2, |K3| = r s 4 + 2; (5.3)

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eP · k1 = − 1 2m2 s + −_m1 2 +m2 2 , (exact) (5.4) eL· k1 = − 1 2m2 s + −_m1 2 +m2 2 − 2m2s−1− 2m2(m22− 2)s−2 −2m2(m42− 6m22+ 4)s−3− 2m2(m62− 12m42+ 24m22− 8)s−4+ O(s−5), (5.5) eT · k1 = 0. (5.6)

A key step is to express the scattering angle θ in terms of s and t. This can be achieved by solving t = − − E2− √ s 2 2 + (|k2| − |k3| cos θ)2+ |k₃|2sin2θ ! (5.7) to obtain θ = arccos   s + 2t − m22+ 6 √ s + 8 q (s+2)2 −2(s−2)m2 2+m 4 2 s   . (exact) (5.8) One can then calculate the following expansions

eP · k3 = 1 m2 E2 √ s 2 − |k2||k3| cos θ = −t + 2 − m 2 2 2m2 , (5.9) eL· k3 = 1 m2 k2 √ 2 2 − E2k3cos θ ! = −t + 2 + m 2 2 2m2 − m2 ts−1_{− m} 2−4(t + 1) + m22(t − 2) s−2 −m2 4(4 + 3t) − 12tm22+ (t − 4)m42 s−3 −m2−16(3 + 2t) + 24(2 + 3t)m22 − 24(−1 + t)m42+ (−6 + t)m62 s−4+ O(s−5), (5.10) eT · k3 = −|k3| sin θ = −√−t −1₂√−t(2 + t + m22)s−1 − 1 8√_−t[32 + 52t + 20t 2_{+ t}3 + (32 + 20t − 6t2)m2₂_{+ (8 − 3t)m}4₂]s−2 + 1 16√_−t[320 + 456t + 188t 2_{+ 22t}3_{+ t}4 − (−224 + 36t + 132t2+ 5t3)m2₂ +(−16 − 122t + 15t2)m4₂_{+ (−24 + 5t)m}6₂]s−3 + 1 128(−t)3/2[1024 + 12032t + 16080t 2_{+ 7520t}3_{+ 1432t}4_{+ 136t}5_{+ 5t}6 −4(−512 − 896t + 2232t2+ 1844t3+ 170t4+ 7t5)m2₂ +2(768 − 2240t − 2372t2+ 1172t3 + 35t4)m4₂ −4(−128 + 288t − 450t2+ 35t3)m6₂_{+ (64 + 240t − 35t}2)m8₂]s−4+ O(s−5). (5.11) We are now ready to calculate the expansions of the four amplitudes A , A , A ,

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A(LT ) for the mass level M22= 4 to subleading orders in s in the RR. These are

AT T T _∼ 1 8 √ −tts3+ 3 16 √ −tt(t + 6)s2+3t 3_{+ 84t}2_{− 68t − 864} 64 √ −t s + O(1), (5.12) ALLT _∼ 1 64 √ −t(t − 6)s3+ 3 128 √ −t(t2− 20t − 12)s2 +3t 3_{− 342t}2_{− 92t + 5016 + 1728(−t)}−1/2 512 √ −t s + O(1), (5.13) A[LT ] _{∼ −} 1 64 √ −t(t + 2)s3−₁₂₈3 √−t(t + 2)2s2+ O(s) −(3t − 8)(t + 6) 2_{[1 − 2(−t)}−1/2_] 512 √ −t s + O(1), (5.14) A(LT ) _{∼ −} 1 64 √ −t(t + 10)s3−₁₂₈1 √−t(3t2+ 52t + 60)s2+ O(s) −3[t 3_{+ 30t}2_{+ 76t − 1080 − 960(−t)}−1/2_] 512 √ −t s + O(1). (5.15) One can now easily see that the ratios of the coefficients of the highest power of t in the leading order coefficient functions 1₈ : ₆₄1 _{: −}₆₄1 _{: −}₆₄1 agree with the ratios in the GR 8 : 1 : −1 : −1 calculated in eq. (2.6) as expected. Moreover, one further obeservation is that these ratios remain the same for the coefficients of the highest power of t in the subleading orders (s2₎ 3

16 : 1283 : −1283 : −1283 and (s) 643 : 5123 : −5123 : −5123 . We conjecture

that these ratios persist to all energy orders in the Regge expansion of the amplitudes. This is consistent with the results of GR by taking both s, −t → ∞. For the mass level M2

2 = 6 [5], the amplitudes can be calculated to be

AT T T T _∼ s2 4 − s s2 4 − 1 (eT _{· k}3)4 = t 2 16s 4₊t2(t + 6) 8 s 3₊t(t3+ 24t2− 4t − 256) 16 s 2 +t(3t 3_{− 2t}2_{− 396t − 768)} 4 s − t4 4 + 166t 3_{+ 960t}2_{− 64t − 1024} s0 +(−83t4− 1536t3+ 384t2+ 21248t + 12288)s−1+ O(s−2), (5.16) AT T LL _∼ s2 4 − s s2 4 − 1 (eT _{· k}3)2(eL· k3)2 +3st 2 s 2+ 1 t 2 + 1 (eL_{· k}1)2(eT · k3)2× 1 6 −s s2 4 − 1

(t + 2)(eL_{· k}1)(eL· k3)(eT · k3)2×

1 2 = t(t − 16) 192 s 4₊t(t2− 41t − 32) 96 s 3₊t4− 132t3− 328t2+ 1984t + 2048 192 s 2 + −11t 4 32 − 11t3 4 + 163t2 3 + 184t + 128 3 s1 + −11₈ t4+ 88t3+ 744t2_{+ 304t − 1408} s0 +4 11t4+ 280t3+ 204t2_{− 4448t − 4480}s−1+ O(s−2), (5.17)

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ALLLL _∼ s2 4 − s s2 4 − 1 (eL_{· k}3)4− t t2 4 − 1 (s + 2)(eL_{· k}1)3(eL· k3) +3st 2 s 2+ 1 t 2 + 1 (eL_{· k}1)2(eL· k3)2 −s s2 4 − 1 (t + 2)(eL_{· k}1)(eL· k3)3 + t2 4 − t t2 4 − 1 (eL_{· k}1)4 = t(t − 52) 768 s 4₊t(t2− 140t + 256) 384 s 3₊ t4− 456t3+ 2816t2− 512t − 16384 768 s 2 −19t 4 64 + 6t 3 −17t 2 3 − 176t − 256 3 s1 +(3t4_{− 10t}3_{− 528t}2_{− 672t + 1792)s}0+ O(s−1), (5.18) AT T L _{∼ −} s2 4 − s s2 4 − 1 (eT _{· k}3)2(eL· k3) −st₄ s₂+ 1 t 2 + 1 (eT _{· k}3)2(eL· k1) × 1 3 +s s2 4 − 1 t 2+ 1 (eT _{· k}3)2(eL· k1) × 1 3 = −(t + 20)t 96√6 s 4₋t(t2+ 31t + 40) 48√6 s 3₋t4+ 38t3+ 224t2− 1520t − 2560 96√6 s 2 +−3t 4_{− 72t}3_{+ 2248t}2_{+ 12000t + 5120} 48√6 s 1 +67t 3_{+ 1194t}2_{+ 1344t − 3712} 2√6 s 0_{+ O(s}−1_), _(5.19) ALLL _{∼ −} s2 4 − s s2 4 − 1 (eL_{· k}3)3+ t t2 4 − 1 s 2 + 1 (eL_{· k}1)2(eL· k3) −st₄ s₂+ 1 t 2 + 1 (eL_{· k}1)2(eL· k3) + (eL· k3)2(eL· k1) +s s2 4 − 1 t 2+ 1 (eL_{· k}3)2(eL· k1) − t2 4 − t t2 4 − 1 (eL_{· k}1)3 = −t 2_{− 8t − 128} 384√6 s 4 −t 3_{− 52t}2_{− 412t + 256} 192√6 s 3 −t 4_{− 236t}3_{− 1272t}2_{+ 4832t + 15872} 384√6 s 2 +35t 4_{+ 50t}3_{− 3008t}2_{− 23728t − 14848} 96√6 s 1 −47t 4_{+ 1432t}3_{+ 24796t}2_{+ 40640t − 101376} 48√6 s 0_{+ O(s}−1_), _(5.20)

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˜ ALT,T _{∼ −} s2 4 − s s2 4 − 1 (eT _{· k}3)2(eL· k3) × 0 −st₄ s₂+ 1 t 2 + 1 (eL_{· k}1)(eT · k3)2× 1 2 +s s2 4 − 1 t 2+ 1 (eT _{· k}3)2(eL· k1) × −1 4 = −t(t + 2) 64√6 s 4 −t(t + 2) 2 32√6 s 3 −t 4_{+ 12t}3_{+ 8t}2_{− 152t − 256} 64√6 s 2 +−3t 4_+196t2_+624t+512 32√6 s 1₊ r 3 8(5t 3_+30t2 +24t− 32)s0+ O(s−1), (5.21) ALL _∼ s2 4 − s s2 4 − 1 (eL_{· k}3)2+ st 2 s 2 + 1 t 2 + 1 (eL_{· k}1)(eL· k3) + t2 4 − t t2 4 − 1 (eL_{· k}1)2 = (t + 8) 2 384 s 4₊(t3+ 20t2+ 80t − 128) 192 s 3₊t4+ 16t3+ 96t2− 880t − 3328 384 s 2 +−t 4_{+ 8t}3_{− 110t}2_{− 1648t − 1408} 48 s 1 +t 4_{− 4t}3_{− 202t}2_{− 704t + 1728} 6 s 0_{+ O(s}−1_). _(5.22)

In the above calculations, as in the case of M2

2 = 4, we have ingored a common overall

factor which will be discussed in section VI. Note that the ratios of the coefficients in the leading order t for the energy orders s4_{, s}3_{, s}2 _{reproduced the GR ratios in eq. (}_2.7_).

How-ever, the subleading terms for orders s1, s0 contain no GR ratios. Mathematically, this is because the highest power of t in the coefficient functions of s1 is 4 rather than 5, and those of s0 _{is 4 rather than 6. This is because the power of t in the kinematic relation eq. (}_5.11₎

can be as high as one wants if one goes to subleading orders, while that of eq. (5.10) is not. The sin θ factor in eq. (5.11) contributes terms of higher order powers of t, while cos θ factor in in eq. (5.10) does not. This can be seen from the kinematic relation in eq. (5.8). In general, one can easily show that the sin θ factor will contribute only for the even mass levels with (N − 1) = M

2 2

2 = even. We thus conjecture that the existence of the GR ratios

in the RR persists to all orders in the Regge expansion of all string amplitudes for the even mass level. For the odd mass levels with (N − 1) = M

2 2

2 = odd, the existence of the

GR ratios will show up only in the first [N/2] + 1 terms in the Regge expansion of the amplitudes. An interesting question is whether this phenomena persists for the case of superstring where GSO projection needs to be imposed.

6 Universal power law behavior

In the discussion of section V, we ignored an overall common factorΓ(−1−s/2)Γ(−1−t/2)_Γ(u/2+2) of the amplitudes for mass levels M2

2 = 4, 6. We paid attention only to the ratios among scattering

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of string scattering amplitudes for string states at arbitrary mass levels in the RR. The power law behavior ∼ sα(t) of the four-tachyon amplitude in the RR is well known in the lit-erature. Here we want to generalize this result to string states at arbitrary mass levels. We can use the saddle point method to calculate the leading term of gamma functions in the RR

Γ(−1 − s/2)Γ(−1 − t/2) Γ(u/2 + 2) =

Γ(−1 − s/2)Γ(−1 − t/2) Γ(−s/2 − t/2 + N − 2) ∼ s

t/2−N+1 _{(in the RR).} _(6.1)

Thus, the overall s-dependence in the amplitudes is of the form

A(kn,qm)_{∼ s}α(t) _{(in the RR)} _(6.2) where

α(t) = α(0) + α′t, α(0) = 1 and α′ = 1/2. (6.3) This generalizes the high energy behavior of the four-tachyon amplitude in the RR to string states at arbitrary mass levels. The new result here is that the behavior is universal and is mass level independent. In fact, as a simple application, one can also derive eq. (6.2) directly from eq. (3.9) by using

B

−1 − s₂, −1 −₂t

∼ sα(t). (in the RR) (6.4) We conclude that the well known ∼ sα(t) power-law behavior of the four tachyon string scattering amplitude in the RR can be extended to high energy string scattering amplitudes of arbitrary string states.

7 Conclusion

In this paper, we calculate high energy massive string scattering amplitudes of 26D open bosonic string in the Regge regime (RR). It turns out that both the saddle-point method and the method of decoupling of high energy ZNS adopted in the calculation of GR [4–12] do not apply to the case of RR. However, the general formula for the high energy scattering amplitudes for each fixed mass level in the RR can still be written down explicitly. We have found that the number of high energy amplitudes for each fixed mass level in the RR is much more numerous than that of Gross regime (GR) calculated previously [4–12].

On the other hand, there is no linear relation among scatterings in the RR in contrast to the case of scatterings in the GR. Moreover, we discover that the leading order amplitudes in the RR can be expressed in terms of the Kummer function of the second kind. In particular, based on a summation algorithm for Stirling number identity in the combinatoric number theory, we discover that the ratios calculated previously among scattering amplitudes in the GR can be extracted from this Kummer function in the RR. We conjecture and give evidences that the existence of the GR ratios in the RR persists to all orders in the Regge expansion of all string amplitudes for the even mass level with (N −1) = M22

2 = even. For the

odd mass levels with (N − 1) = M 2 2

2 = odd, the existence of the GR ratios shows up only in

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- k₁ k₂ −k3 −k4 ? 6 @ @ @ I @ @@R eT₍₁₎ eT₍₂₎ eT₍₃₎ eT₍₄₎ θ

Figure 1. Kinematic variables in the center of mass frame.

is whether this phenomena persists for the case of superstring where GSO projection needs to be imposed. Finally, we demonstrate the universal power-law behavior for all massive string scattering amplitudes in the RR. This result generalizes the well known result for the case of high energy four-point tachyon amplitudes.

Acknowledgments

This work is supported in part by the National Science Council, 50 billions project of MOE and National Center for Theoretical Science, Taiwan. We appreciated the correspondence of Dr. Manuel Mkauers at RISC, Austria for his kind help of providing us with the rigorous proof of eq. (4.6), and for informing us reference [28].

A Kinematic variables and notations

In this appendix, we list the expressions of the kinematic variables we used in the evaluation of 4-point functions in this paper. For convenience, we take the center of momentum frame and choose the momenta of particles 1 and 2 to be along the X1_{-direction. The high energy}

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The momenta of the four particles are k1 = + q p2_{+ M}2 1, −p, 0 , (A.1) k2 = + q p2_{+ M}2 2, +p, 0 , (A.2) k3 = − q q2_{+ M}2 3, −q cos φ, −q sin θ , (A.3) k4 = − q q2_{+ M}2 4, +q cos φ, +q sin θ (A.4) where p ≡ |˜p|, q ≡ |˜q| and k2

i = −Mi2. In the calculation of the string scattering

amplitudes, we use the following formulas

−k1· k2 = q p2_{+ M}2 1 · q p2_{+ M}2 2 + p2 = 1 2 s − M 2 1 − M22 , (A.5) −k2· k3 = − q p2_{+ M}2 2 · q q2_{+ M}2 3 + pq cos θ = 1 2 t − M 2 2 − M32 , (A.6) −k1· k3 = − q p2_{+ M}2 1 · q q2_{+ M}2 3 − pq cos θ = 1 2 u − M 2 1 − M32 (A.7) where the Mandelstam variables are defined as usual with

s + t + u =X

i

M_i2 _{= 2 (N − 4) .} (A.8) The center of mass energy E is defined as

E = 1 2 q p2_{+ M}2 1 + q p2_{+ M}2 2 = 1 2 q q2_{+ M}2 3 + q q2_{+ M}2 4 . (A.9) We define the polarizations of the string state on the scattering plane as

eP = 1 M2 q p2_{+ M}2 2, p, 0 , (A.10) eL= 1 M2 p, q p2_{+ M}2 2, 0 , (A.11) eT = (0, 0, 1) . (A.12)

The projections of the momenta on the scattering plane can be calculated as (here we only list the ones we need for our calculations)

eP _{· k}1 = − 1 M2 q p2_{+ M}2 1 q p2_{+ M}2 2 + p2 , (A.13) eL_{· k}1 = − p M2 q p2_{+ M}2 1 + q p2_{+ M}2 2 , (A.14) eT _{· k}1 = 0 (A.15)

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and eP _{· k}3= 1 M2 q q2_{+ M}2 3 q p2_{+ M}2 2 − pq cos θ , (A.16) eL_{· k}3= 1 M2 p q q2_{+ M}2 3 − q q p2_{+ M}2 2cos θ , (A.17) eT _{· k}3= −q sin θ = −√−t. (A.18) References

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