Notes on high-energy limit of bosonic closed string scattering amplitudes

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Notes on high-energy limit of bosonic closed string

scattering amplitudes

Chuan-Tsung Chan

a

_{, Jen-Chi Lee}

b,∗

_{, Yi Yang}

b a_{Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan} b_{Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan}

Received 2 May 2006; accepted 24 May 2006 Available online 12 June 2006

Abstract

We study bosonic closed string scattering amplitudes in the high-energy limit. We find that the meth-ods of decoupling of high-energy zero-norm states and the high-energy Virasoro constraints, which were adopted in the previous works to calculate the ratios among high-energy open string scattering amplitudes of different string states, persist for the case of closed string. However, we clarify the previous saddle-point calculation for high-energy open string scattering amplitudes and claim that only (t, u) channel of the amplitudes is suitable for point calculation. We then discuss three evidences to show that saddle-point calculation for high-energy closed string scattering amplitudes is not reliable. By using the relation of tree-level closed and open string scattering amplitudes of Kawai, Lewellen and Tye (KLT), we calculate the high-energy closed string scattering amplitudes for arbitrary mass levels. For the case of high-energy closed string four-tachyon amplitude, our result differs from the previous one of Gross and Mende, which is NOT consistent with KLT formula, by an oscillating factor.

1. Introduction

Recently high-energy, fixed-angle behavior of string scattering amplitudes[1–3] was inten-sively reinvestigated [4–10]. The motivation was to uncover the long-sought hidden stringy space–time symmetry. An important new ingredient of this approach is the zero-norm states (ZNS)[11–13]in the old covariant first quantized (OCFQ) string spectrum. One utilizes the

de-* _{Corresponding author.}

E-mail addresses:[email protected](C.-T. Chan),[email protected](J.-C. Lee), [email protected](Y. Yang).

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coupling of zero-norm states to obtain relations among scattering amplitudes. An infinite number of linear relations among high-energy scattering amplitudes of different string states were de-rived. Moreover, these linear relations can be used to fix the proportionality constants among high-energy scattering amplitudes of different string states at each fixed mass level algebraically. Thus there is only one independent component of high-energy scattering amplitude at each fixed mass level. On the other hand, a saddle-point method was also developed to calculate the general formula of tree-level high-energy scattering amplitudes of four arbitrary string states to verify the ratios among the high-energy scattering amplitudes of different string states calculated by the above algebraic methods. Moreover, these high-energy scattering amplitudes can be expressed in terms of high-energy four tachyon scattering amplitude as conjectured by Gross in 1988[2]. However, all the above calculations were focused only on the case of open string theory.

In this paper, we generalize the calculations to high-energy closed string scattering ampli-tudes. We find that the methods of decoupling of energy zero-norm states and the high-energy Virasoro constraints, which were adopted in the previous works to calculate the ratios among high-energy open string scattering amplitudes of different string states, persist for the case of closed string. The result is simply the tensor product of two pieces of open string ratios of high-energy scattering amplitudes. However, we clarify the previous saddle-point calculation for high-energy open string scattering amplitudes and claim that only (t, u) channel of the am-plitudes is suitable for saddle-point calculation. We then discuss three evidences to show that saddle-point calculation for high-energy closed string scattering amplitudes is not reliable. By using the relation of tree-level closed and open string scattering amplitudes of Kawai, Lewellen and Tye (KLT)[14], we calculate the tree-level high-energy closed string scattering amplitudes for arbitrary mass levels. For the case of high-energy closed string four-tachyon amplitude, our result differs from the previous one of Gross and Mende[1], which is NOT consistent with KLT formula, by an oscillating factor. This means that the high-energy closed string amplitudes do not

factorize into product of two high-energy open string amplitudes in contrast to the conventional

wisdom[1,15].

2. Decoupling of zero norm states

In this section, we calculate the ratios among high-energy closed string scattering amplitudes of different string states by the decoupling of high-energy closed string ZNS. Since the calcu-lation is similar to that of open string, we will, for simplicity, work on the first massive level

M2= 8(n − 1) = 8 (n = 2) only. At this mass level, the corresponding open string Ward

identi-ties are (M2= 2 for open string, α_closed = 4αopen= 2)[16]

(1) kμθνTμν+ θμTμ= 0, (2) 3 2kμkν+ 1 2ημν Tμν₊5 2kμT μ_{= 0,}

where θν is a transverse vector. In Eqs. (1) and (2), we have chosen, say, the second vertex V2(k2)to be the vertex operators constructed from zero-norm states and kμ≡ k2μ. The other

three vertices can be any string states. Note that Eq.(1)is the type I Ward identity while Eq.(2)

is the type II Ward identity which is valid only at D= 26. The high-energy limits of Eqs.(1)

and (2)were calculated to be

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MT_{T P}3→1+ T_T1= 0,

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(5) 3M2T_LL4→2+ T_{T T}2 + 5MT_L2= 0.

In the above equations, we have defined the following orthonormal polarization vectors for the second string vertex V2(k2)

(6) eP = 1 M(E2, k2,0)= k2 M, (7) eL= 1 M(k2, E2,0), (8) eT = (0, 0, 1)

in the center-of-mass (c.m.) frame contained in the plane of scattering. We have also denoted the naive power counting for orders in energy[4,5]in the superscript of each amplitude according to the following rules, eL· k ∼ E2, eT · k ∼ E1. Note that sinceT_{T P}1 is of subleading order in

energy, in generalT_{T P}1 = T_{T L}1 . A simple calculation of Eqs.(3)–(5)shows that[16]

(9) T1 T P:T 1 T = 1 : − √ 2= 1 : −M, (10) T2 T T:T 2 LL:T 2 L = 4 : 1 : − √ 2= 2M2: 1 :−M.

It is interesting to see that, in addition to the leading order amplitudes in Eq.(10), the subleading order amplitudes in Eq.(9)are also proportional to each other. This does not seem to happen at higher mass level.

We are now back to the closed string calculation. The OCFQ closed string spectrum at this mass level are ( + + •) ⊗ ( + + •)_{. In addition to the spin-four positive-norm state}

⊗_{, one has 8 ZNS, each of which gives a Ward identity. In the high-energy limit, we have} θμν= eμ_Le_Lν − eμ_Teν_T or θμν_{= e}μ Le ν T + e μ Te ν L, θμ= e μ Lor e μ

T and one replace ημνby e μ Te

ν T. In the

following, we list only high-energy Ward identities which relate amplitudes with even-energy power in the high-energy expansion:

(1) ⊗ : (11) M(TLL,LL− TT T ,LL)+ TLL,L− TT T ,L= 0, (12) MTLT ,P T + TLT ,T = 0. (2) ⊗ •: (13) 3M2(TLL,LL− TT T ,LL)+ (TLL,T T − TT T ,T T)+ 5M(TLL,L− TT T ,L)= 0. (3) ⊗ : (14) M(TLL,LL− TLL,T T)+ TL,LL− TL,T T = 0, (15) MTP T ,LT + TT ,LT = 0. (4) ⊗ : (16) M2TLL,LL+ MTLL,L+ MTL,LL+ TL,L= 0, (17) M2TP T ,P T + MTP T ,T + MTT ,P T + TT ,T = 0. (5) ⊗ •: (18) 3M3TLL,LL+ MTLL,T T + 5M2TLL,L+ 3M2TL,LL+ TL,T T + 5M2TL,L= 0.

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(6) • ⊗ : (19) 3M2(TLL,LL− TLL,T T)+ (TT T ,LL− TT T ,T T)+ 5M(TL,LL− TL,T T)= 0. (7) • ⊗ : (20) 3M3TLL,LL+ MTT T ,LL+ 5M2TL,LL+ 3M2TLL,L+ TT T ,L+ 5M2TL,L= 0. (8) • ⊗ •: 9M4TLL,LL+ 3M2TLL,T T + 3M2TT T ,LL+ 15M3TLL,L (21) + 15M3_T L,LL+ 5MTT T ,L+ 5MTL,T T + 25M2TL,L+ TT T ,T T = 0.

Those Ward identities which relate amplitudes with odd-energy power in the high-energy expan-sion are omitted as they are subleading order in energy. The mass M in Eqs.(11) to (21)should now be interpreted as the closed string mass M2= 8. Eqs.(12), (15) and (17)are subleading order amplitudes, and one can then solve the other 8 equations to give the ratios

TT T ,T T:TT T ,LL:TLL,T T:TLL,LL:TT T ,L:TL,T T:TLL,L:TL,LL:TL,L (22) = 1 : 1 2M2: 1 2M2: 1 4M4:− 1 2M :− 1 2M:− 1 4M3:− 1 4M3: 1 4M2.

Eq.(22)is exactly the tensor product of two pieces of open string ratios calculated in Eq.(10).

3. Virasoro constraints

We consider the mass level M2= 8 (n = 2). The most general state is |2 = 1 2! μ11 μ 1 2 α μ1₁ −1α μ1₂ −1+ 1 2 μ 2 1 α μ2₁ −2 ⊗ 1 2! ˜μ11 ˜μ 1 2 ˜α ˜μ1 1 −1˜α˜μ 1 2 −1+ 1 2 ˜μ 2 1 ˜α ˜μ2 1 −2 |0, k (23) =1 4 μ1₁ μ1₂ αμ 1 1 −1α μ1₂ −1+ μ21 α μ2₁ −2 ⊗ ˜μ1 1 ˜μ12 ˜α ˜μ1 1 −1˜α˜μ 1 2 −1+ ˜μ21 ˜α ˜μ2 1 −2 |0, k. The Virasoro constraints are

(24a) L1|2 ∼ kμ11_μ1 1 μ 1 2 α μ1 2 −1+ μ21 α μ2 1 −1 ⊗ ˜μ1 1 ˜μ 1 2 ˜α ˜μ1 1 −1˜α˜μ 1 2 −1+ ˜μ21 ˜α ˜μ2 1 −2 = 0, (24b) ˜L1|2 ∼ μ1₁ μ1₂ αμ 1 1 −1α μ1₂ −1+ μ21 α μ2₁ −2 ⊗kμ11 ˜μ1 1 ˜μ12 ˜α ˜μ1 2 −1+ ˜μ21 ˜α ˜μ2 1 −1 = 0, (24c) L2|2 ∼ μ1₁ μ1₂ ημ11μ12+ 2kμ21_μ2 1 ⊗ ˜μ1 1 ˜μ 1 2 ˜α ˜μ1 1 −1˜α˜μ 1 2 −1+ ˜μ21 ˜α ˜μ2 1 −2 = 0, (24d) ˜L2|2 ∼ μ1₁ μ1₂ αμ 1 1 −1α μ1₂ −1+ μ21 α μ2₁ −2 ⊗ ˜μ1 1 ˜μ12 η˜μ 1˜μ1 12 + 2k˜μ 2 1 ˜μ2 1 = 0. Taking the high-energy limit in the above equations by letting (μi, νi)→ (L, T ), and

(25) kμi→ MeL_, _ημ1μ2→ eT_eT_, we obtain (26a) M L μ + μ αμ₋₁⊗ ˜μ1 1 ˜μ 1 2 ˜α ˜μ1 1 −1˜α˜μ 1 2 −1+ ˜μ21 ˜α ˜μ2 1 −2 = 0, (26b) μ1₁ μ1₂ αμ 1 1 −1α μ1₂ −1+ μ21 α μ2₁ −2 ⊗M L ˜μ + ˜μ ˜α₋₁˜μ = 0,

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(26c) T T + 2M L ⊗ ˜μ1 1 ˜μ 1 2 ˜α ˜μ1 1 −1˜α˜μ 1 2 −1+ ˜μ21 ˜α ˜μ2 1 −2 = 0, (26d) μ1₁ μ1₂ αμ 1 1 −1α μ1₂ −1+ μ21 α μ2₁ −2 ⊗T T + 2M L = 0, which lead to the following equations

(27a) M L μ + μ ⊗ ˜μ1 1 ˜μ 1 2 = 0, (27b) M L μ + μ ⊗ ˜μ2 1 = 0, (27c) μ1₁ μ1₂ ⊗ M L ˜μ + ˜μ = 0, (27d) μ2₁ ⊗ M L ˜μ + ˜μ = 0, (27e) T T + 2M L ⊗ ˜μ1 1 ˜μ12 = 0, (27f) T T + 2M L ⊗ ˜μ2 1 = 0, (27g) μ1₁ μ1₂ ⊗ T T + 2M L = 0, (27h) μ2₁ ⊗ T T + 2M L = 0.

The remaining indices μ,˜μ in the above equations can be set to be T or L, and we obtain (28a) M L L ⊗ L L + L ⊗ L L = 0, (28b) M L L ⊗ T T + L ⊗ T T = 0, (28c) M T L ⊗ T L + T ⊗ T L = 0, (29a) M L L ⊗ L + L ⊗ L = 0, (29b) M T L ⊗ T + T ⊗ T = 0, (30a) M L L ⊗ L L + L L ⊗ L = 0, (30b) M T T ⊗ L L + T T ⊗ L = 0, (30c) M T L ⊗ T L + T L ⊗ T = 0, (31a) M L ⊗ L L + L ⊗ L = 0, (31b) M T ⊗ T L + T ⊗ T = 0, (32a) T T ⊗ L L + 2M L ⊗ L L = 0, (32b) T T ⊗ T T + 2M L ⊗ T T = 0,

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(33) T T ⊗ L + 2M L ⊗ L = 0, (34a) L L ⊗ T T + 2M L L ⊗ L = 0, (34b) T T ⊗ T T + 2M T T ⊗ L = 0, (35) L ⊗ T T + 2M L ⊗ L = 0.

Since the transverse component of the highest spin state α₋₁T · · · αT₋₁⊗ ˜α₋₁T · · · ˜α₋₁T at each fixed mass level gives the leading order scattering amplitude, there should have even number of T at each fixed mass level. Thus Eqs.(28c), (29b), (30c) and (31b)are subleading order in energy and are therefore irrelevant. Set T T ⊗ T T = 1, we can solve the ratios from the remaining equations. The final result is

T T ,T T 1 T T ,LL= LL,T T 1/(2M2) LL,LL 1/(4M4) T T ,L= L,T T −1/(2M) LL,L= L,LL −1/(4M3) L,L 1/(4M2)

which is exactly the tensor product of two pieces of open string ratios. This result is consistent with Eq.(22)from the decoupling of high-energy zero-norm state in Section2.

4. Saddle point calculation

In this section, we calculate the tree-level high-energy closed string scattering amplitudes for arbitrary mass levels. We first review the calculation of high-energy open string scattering amplitude. The (s, t) channel scattering amplitude with V2= αμ₋₁1αμ₋₁2· · · α₋₁μn|0, k, the highest

spin state at mass level M2= 2(n − 1), and three tachyons V1,3,4is[6]

(36) Tμ1μ2...μn n;st = n l=0 (−)l n l B −s 2− 1 + l, − t 2− 1 + n − l k(μ1 1 · · · k μn−l 1 k μn−l+1 3 · · · k μn) 3 ,

where B(u, v)= ₀1dx xu−1₍₁_{− x)}v−1_{is the Euler beta function. It is now easy to calculate the}

general high-energy scattering amplitude at the M2= 2(n − 1) level

(37) TT T T ... n (s, t) −2E3 sin φc.m. n Tn(s, t)

whereTn(s, t)is the high-energy limit of

(−₂s−1)(−₂t−1)

(u₂₊₂₎ with s+ t + u = 2n − 8, and was

previously[4,6]miscalculated to be ˜Tn;st √ π (−1)n−12−nE−1−2n sinφc.m. 2 ₋₃ cosφc.m. 2 5−2n (38) × exp −sln s+ t ln t − (s + t) ln(s + t) 2 .

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One can now generalize this result to multi-tensors. The (s, t) channel of open string high-energy scattering amplitude at mass level (n1, n2, n3, n4)was calculated to be[4,6]

(39) TT1_...T2_...T3_...T4_... n1n2n3n4;st = −2E3 sin φc.m. ni_T ni(s, t).

In the above calculations, the scattering angle φc.m. in the center of mass frame is defined to be

the angle between k1 and k3. s= −(k1+ k2)2, t= −(k2+ k3)2 and u= −(k1+ k3)2 are the

Mandelstam variables. M_i2= 2(ni− 1) with ni the mass level of the ith vertex. Ti in Eq.(39)

is the transverse polarization of the ith vertex defined in Eq.(8). All other 4-point functions at mass level (n1, n2, n3, n4)were shown to be proportional to Eq.(39).

The corresponding (t, u) channel scattering amplitudes of Eqs.(37) and (39)can be obtained by replacing (s, t) in Eq.(38)by (t, u) Tn(t, u) √ π (−1)n−12−nE−1−2n sinφc.m. 2 ₋₃ cosφc.m. 2 5−2n (40) × exp −tln t+ u ln u − (t + u) ln(t + u) 2 .

We now claim that only (t, u) channel of the amplitude, Eq.(40), is suitable for saddle-point calculation. The previous saddle-point calculation for the (s, t) channel amplitude, Eq. (38), in the high-energy expansion is misleading. The corrected high-energy calculation of the (s, t) channel amplitude will be given in Eq. (57). The reason is as following. When calculating Eq.(37)from Eq.(36), one calculates the high-energy limit of

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(−s₂− 1)(−₂t − 1)

(u₂+ 2) , s+ t + u = 2n − 8,

in Eq.(36)by expanding the function with the Stirling formula

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(x)∼√2π xx−1/2e−x.

However, the above expansion is not suitable for negative real x as there are poles for (x) at

x= −n, negative integers. Unfortunately, our high-energy limit

(43a) s∼ 4E2 0, (43b) t∼ −4E2sin2 φc.m. 2  0, (43c) u∼ −4E2cos2 φc.m. 2  0,

contains this dangerous situation in the (s, t) channel calculation of Eq.(38). On the other hand, the corresponding high-energy expansion of (t, u) channel scattering amplitude in Eq.(40)is well defined. Another evidence for this point is the following. When one uses the saddle point method to calculate the high-energy open string scattering amplitudes in the (s, t) channel, the saddle-point we identified was[6–8]

(44) x0= s s+ t = 1 1− sin2(φ/2)>1,

which is out of the integration range (0, 1). Therefore, we cannot trust the saddle point calculation for the (s, t) channel scattering amplitude. On the other hand, the corresponding saddle-point

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calculation for the (t, u) channel scattering amplitude is safe since the saddle-point x0is within

the integration range (1,∞). This subtle situation becomes crucial and relevant when one tries to calculate the high-energy closed string scatterings amplitude and compare them with the open string ones.

We now discuss the high-energy closed string scattering amplitudes. There exists a celebrated formula by Kawai, Lewellen and Tye (KLT), which expresses the relation between tree ampli-tudes of closed and open string (α_closed= 4α_open = 2)

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A(4)_closed(s, t, u)= sin(πk2· k3)A(4)open(s, t) ¯A(4)open(t, u).

To calculate the high-energy closed string scattering amplitudes, one encounters the difficulty of calculation of high-energy open string amplitude in the (s, t) channel discussed above. To avoid this difficulty, we can use the well-known formula

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(x)= π

sin(π x)(1− x)

to calculate the large negative x expansion of the function. We first discuss the high-energy four-tachyon scattering amplitude which already existed in the literature. We can express the open string (s, t) channel amplitude in terms of the (t, u) channel amplitude,

A(4open-tachyon)(s, t)= (−s₂− 1)(−t₂− 1) (u₂+ 2) =sin(π u/2) sin(π s/2) (−₂t − 1)(−u₂− 1) (s₂+ 2) (47) ≡sin(π u/2) sin(π s/2)A (4-tachyon) open (t, u),

which we know how to calculate the high-energy limit. Note that for the four-tachyon case, ¯

A(4)open(t, u)= A(4)open(t, u)in Eq.(45). The KLT formula, Eq.(45), can then be used to express

the closed string four-tachyon scattering amplitude in terms of that of open string in the (t, u) channel

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A(4_closed-tachyon)(s, t, u)=sin(π t/2) sin(π u/2)

sin(π s/2) A

(4-tachyon) open (t, u)A

(4-tachyon) open (t, u).

The high-energy limit of open string four-tachyon amplitude in the (t, u) channel can be easily calculated to be

(49)

A(4open-tachyon)(t, u) (stu)−3/2exp

−sln s+ t ln t + u ln u 2

,

which gives the corresponding amplitude in the (s, t) channel

(50) A(4open-tachyon)(s, t) sin(π u/2) sin(π s/2)(stu) −3/2_exp₋sln s+ t ln t + u ln u 2 .

The high-energy limit of closed string four-tachyon scattering amplitude can then be calculated, through the KLT formula, to be

(51)

A(4_closed-tachyon)(s, t, u)sin(π t/2) sin(π u/2)

sin(π s/2) (stu)

−3_exp₋sln s+ t ln t + u ln u

4

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The exponential factor in Eq.(49)was first discussed by Veneziano[17]. Our result for the high-energy closed string four-tachyon amplitude in Eq.(51)differs from the one calculated in the literature[1]by an oscillating factor sin(π t/2) sin(π u/2)_{sin(π s/2)} [18]. We stress here that our results for

Eqs.(49), (50) and (51) are consistent with the KLT formula, while the previous calculation

in[1]is NOT.

One might try to use the saddle-point method to calculate the high-energy closed string scat-tering amplitude. The closed string four-tachyon scatscat-tering amplitude is

A(4_closed-tachyon)(s, t, u)= dx dyexp k1· k2 2 ln|z| + k2· k3 2 ln|1 − z| = dx dyx2+ y2−2(1− x)2+ y2−2 × exp −s 8ln x2+ y2−t 8ln (1− x)2+ y2 (52) ≡ dx dyx2+ y2−2(1− x)2+ y2−2exp−Kf (x, y),

where K= s₈ and f (x, y)= ln(x2+ y2)− τ ln[(1 − x)2+ y2] with τ = −_st. One can then calculate the “saddle-point” of f (x, y) to be

(53) ∇f (x, y)|_x₀₌ 1

1−τ,y0=0= 0.

The high-energy limit of the closed string four-tachyon scattering amplitude is then calculated to be A(4_closed-tachyon)(s, t, u) 2π K det∂2f (x0,y0) ∂x∂y exp−Kf (x0, y0) (54) (stu)−3exp −sln s+ t ln t + u ln u 4 ,

which is consistent with the previous one calculated in the literature[1], but is different from our result in Eq.(51). However, one notes that

(55) ∂2f (x0, y0) ∂x2 = 2(1− τ)3 τ = − ∂2f (x0, y0) ∂y2 , ∂2f (x0, y0) ∂x∂y = 0,

which means that (x0, y0)is NOT the local minimum of f (x, y), and one should not trust this

saddle-point calculation. This is the third evidence to see that there is no clear definition of saddle-point in the calculation of the high-energy open string scattering amplitude in the (s, t) channel, and thus the invalid saddle-point calculation of high-energy closed string scattering amplitude.

Finally we calculate the high-energy closed string scattering amplitudes for arbitrary mass levels. The (t, u) channel open string scattering amplitude with V2= α₋₁μ1α₋₁μ2· · · αμ₋₁n|0, k, the

highest spin state at mass level M2= 2(n − 1), and three tachyons V1,3,4can be calculated to be

(56) Tμ1μ2...μn n;tu = n l=0 n l B −t 2+ n − l − 1, − u 2 − 1 k(μ1 1 · · · k μn−l 1 k μn−l+1 3 · · · k μn) 3 .

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In calculating Eq.(56), we have used the Mobius transformation y=x−1_x to change the inte-gration region from (1,∞) to (0, 1). One notes that Eq.(56)is NOT the same as Eq.(36)with

(s, t)replaced by (t, u), as one would have expected from the four-tachyon case discussed in the paragraph after Eq.(45). In the high-energy limit, one easily sees that

(57)

Tn(s, t) (−)n

sin(π u/2)

sin(π s/2)Tn(t, u),

which is the generalization of Eq. (47)to arbitrary mass levels. Eq.(57) is the correction of Eqs.(37) and (38)as claimed in the paragraph after Eq.(40). The (s, t) channel of high-energy open string scattering amplitudes at mass level (n1, n2, n3, n4)can then be written as, apart from

an overall constant,

A(4)_open(s, t) (−)nisin(π u/2)

sin(π s/2) −2E3 sin φc.m. ni_T ni(t, u) (58) (−)nisin(π u/2) sin(π s/2)(stu) ni −3 2 _exp −sln s+ t ln t + u ln u 2 .

Finally the total high-energy open string scattering amplitude is the sum of (s, t), (t, u) and (u, s) channel amplitudes, and can be calculated to be

A(4)_open (−)nisin(π s/2)+ sin(πt/2) + sin(πu/2)

sin(π s/2) (59) × (stu) ni −3 2 exp −sln s+ t ln t + u ln u 2 .

By using Eqs. (45) and (57), the high-energy closed string scattering amplitude at mass level

(n1, n2, n3, n4)is calculated to be, apart from an overall constant, A(4)_closed(s, t, u) (−)nisin(π t/2) sin(π u/2)

sin(π s/2) −2E3 sin φc.m. 2ni_T ni(t, u) 2

(−)nisin(π t/2) sin(π u/2)

sin(π s/2) (60) × (stu)ni−3_exp −sln s+ t ln t + u ln u 4 ,

whereTni(t, u)is given by Eq.(40). For the case of four-tachyon scattering amplitude at mass

level (0, 0, 0, 0), Eq.(60)reduces to Eq.(51). All other high-energy closed string scattering am-plitudes at mass level (n1, n2, n3, n4)are proportional to Eq.(60). The proportionality constants

are the tensor product of two pieces of open string ratios.

5. Conclusion

In conclusion, we have used the methods of decoupling of high-energy zero-norm states and the high-energy Virasoro constraints to calculate the ratios among high-energy closed string scat-tering amplitudes of different string states. The result is exactly the tensor product of two pieces of open string ratios calculated before. However, we clarify the previous saddle-point calculation for high-energy open string scattering amplitudes and show that only (t, u) channel of the am-plitudes is suitable for saddle-point calculation. We also discuss three evidences, Eqs.(43),(44)

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and (55), to show that saddle-point calculation for high-energy closed string scattering ampli-tudes is not reliable. Instead of using saddle-point calculation adopted before, we then propose to use the formula of Kawai, Lewellen and Tye (KLT) to calculate the high-energy closed string scattering amplitudes for arbitrary mass levels. For the case of high-energy closed string four-tachyon amplitude, our result differs from the previous one of Gross and Mende, which is NOT consistent with KLT formula, by an oscillating factor. The oscillating prefactors in Eqs. (59)

and (60)imply the existence of infinitely many zeros and poles in the string scattering

ampli-tudes even in the high-energy limit. Physically, the presence of poles simply reflects the fact that there are infinite number of resonances in the string spectrum[18], and the presence of zeros reflects the coherence of string scattering.

Acknowledgements

This work is supported in part by the National Science Council and National Center for The-oretical Science, Taiwan, ROC.

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