The Lauricella Functions and Exact String Scattering Amplitudes
aa r X i v : . [ h e p - t h ] S e p The Lauricella Functions and Exact String Scattering Amplitudes
Sheng-Hong Lai, ∗ Jen-Chi Lee, † and Yi Yang ‡ Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C. (Dated: November 5, 2018)
Abstract
We discover that the 26 D open bosonic string scattering amplitudes (SSA) of three tachyonsand one arbitrary string state can be expressed in terms of the D-type Lauricella functions withassociated SL ( K + 3 , C ) symmetry. As a result, SSA and symmetries or relations among SSA ofdifferent string states at various limits calculated previously can be rederived. These include thelinear relations first conjectured by Gross [1–3] and later corrected and proved in [4–9] in the hardscattering limit, the recurrence relations in the Regge scattering limit with associated SL (5 , C )symmetry [19–21] and the extended recurrence relations in the nonrelativistic scattering limit withassociated SL (4 , C ) symmetry [24] discovered recently. Finally, as an application, we calculate anew recurrence relation of SSA which is valid for all energies. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION It has long been believed that there exist huge hidden spacetime symmetries of stringtheory. As a consistent theory of quantum gravity, string theory contains no free parameterand an infinite number of higher spin string states. On the other hand, the very softexponential fall-off behavior of string scattering amplitudes (SSA) in the hard scatteringlimit, in contrast to the power law behavior of hard field theory scattering amplitudes,strongly suggests the existence of infinite number of relations among SSA of different stringstates. These relations or symmetries soften the UV structure of quantum string theory.Indeed, this kind of infinite relations were first conjectured by Gross [1–3] and later correctedand explicitly proved in [4–9] by using decoupling of zero-norm states (ZNS) [10], and canbe used to reduce the number of independent hard SSA from ∞ down to 1.It was important to note that the linear relations obtained by decoupling of ZNS in thehard scattering limit corrected [4–6] the saddle point calculations of Gross [2], Gross andMende [1] and Gross and Manes [3]. The results of the former authors were consistent withthe decoupling of high energy ZNS or unitarity of the theory while those of the latter werenot. See one simple example to be presented in Eq.(41) in section IV. Independently, theinconsistency of the saddle point calculations of the above authors was also pointed out bythe authors of [11] using the group theoretic approach of string amplitudes [12].On the other hand, inspired by Witten’s seminal paper [13], there have been tremendousdevelopments on calculations of higher point and higher loop Yang-Mills and gravity fieldtheory amplitudes [14]. Many interesting relations among these field theory amplitudes havealso been proposed and suggested. In addition, connections between field theory and stringtheory amplitudes are currently under many investigations.Historically, there were at least three approaches to probe stringy symmetries or relationsamong scattering amplitudes of higher spin string states. These include the gauge symmetryof Witten string field theory, the conjecture of Gross [2] on symmetries or linear relationsamong SSA of different string states in the hard scattering limit by the saddle point method[1–3] and Moore’s bracket algebra approach [15–17] of stringy symmetries. See a recentreview [18] for some connections of these three approaches.Recently, it was found that the Regge SSA of three tachyons and one arbitrary stringstates can be expressed in terms of a sum of Kummer functions U [19–21], which soon later2ere shown to be the first Appell function F [21]. Regge stringy symmetries or recurrencerelations [20, 21] were then constructed and used to reduce the number of independent ReggeSSA from ∞ down to 1. Moreover, an interesting link between Regge SSA and hard SSAwas found [19, 22], and for each mass level the ratios among hard SSA can be extracted fromRegge SSA. This result enables us to argue that the known SL (5; C ) dynamical symmetryof the Appell function F [23] is crucial to probe high energy spacetime symmetry of stringtheory.More recently, the extended recurrence relations [24] among nonrelativistic low energySSA of a class of string states with different spins and different channels were constructedby using the recurrence relations of the Gauss hypergeometric functions with associated SL (4 , C ) symmetry [25]. These extended recurrence relations generalize and extend thefield theory BCJ [26] relations to higher mass and higher spin string states.To further uncover the structure of stringy symmetries, in section II of this paper wecalculate the 26 D open bosonic SSA of three tachyons and one arbitrary string states at arbitrary energies. We discover that these SSA can be expressed in terms of the D-typeLauricella functions with associated SL ( K + 3 , C ) symmetry [25]. As a result, all theseSSA and symmetries or relations among SSA of different string states at various limitscalculated previously can be rederived. These will be presented in sections III, IV and Vwhich include the recurrence relations in the Regge scattering limit [20, 21] with associated SL (5; C ) symmetry, the linear relations conjectured by Gross [2] and corrected and proved in[4–9] in the hard scattering limit and the extended recurrence relations in the nonrelativisticscattering limit [24] with associated SL (4; C ) symmetry discovered very recently. However,since not all Lauricella functions F ( K ) D with arbitrary independent arguments can be used torepresent SSA, it remained to be studied how the basis states of each SL ( K + 3 , C ) grouprepresentation for a given K relates to SSA [27].As a byproduct from the calculation of rederiving linear relations in the hard scatteringlimit directly from Lauricella functions, we propose an identity Eq.(50) which generalizes theStirling number identity Eq.(51) [19, 22] used previously to extract ratios among hard SSAfrom the Appell functions in Regge SSA. Finally, as an example, in section VI we calculatea new recurrence relation of SSA which is valid for all energies.3 I. FOUR-POINT STRING AMPLITUDES
We will consider SSA of three tachyons and one arbitrary string states put at the secondvertex. For the 26D open bosonic string, the general states at mass level M = 2( N − N = P n,m,l> (cid:0) nr Tn + mr Pm + lr Ll (cid:1) with polarizations on the scattering plane are of the form (cid:12)(cid:12) r Tn , r Pm , r Ll (cid:11) = Y n> (cid:0) α T − n (cid:1) r Tn Y m> (cid:0) α P − m (cid:1) r Pm Y l> (cid:0) α L − l (cid:1) r Ll | , k i . (1)In the CM frame, the kinematics are defined as k = (cid:18)q M + | ~k | , −| ~k | , (cid:19) , (2) k = (cid:18)q M + | ~k | , + | ~k | , (cid:19) , (3) k = (cid:18) − q M + | ~k | , −| ~k | cos φ, −| ~k | sin φ (cid:19) , (4) k = (cid:18) − q M + | ~k | , + | ~k | cos φ, + | ~k | sin φ (cid:19) (5)with M = M = M = − φ is the scattering angle. The Mandelstam variables are s = − ( k + k ) , t = − ( k + k ) and u = − ( k + k ) . There are three polarizations onthe scattering plane e T = (0 , , , (6) e L = 1 M (cid:18) | ~k | , q M + | ~k | , (cid:19) , (7) e P = 1 M (cid:18)q M + | ~k | , | ~k | , (cid:19) . (8)For later use, we define k Xi ≡ e X · k i for X = ( T, P, L ) . (9)Note that SSA of three tachyons and one arbitrary string state with polarizations orthogonalto the scattering plane vanish.For illustration, we begin with a simple case, namely, four-point function with the threetachyons and the highest spin state at mass level M = 2( N − N = p + q + r of thefollowing form | p, q, r i = (cid:0) α T − (cid:1) p (cid:0) α P − (cid:1) q (cid:0) α L − (cid:1) r | , k i . (10)4he four-point scattering amplitude can be calculated as A ( p,q,r ) st = sin( πk · k )sin( πk · k ) A ( p,q,r ) tu = sin( u + 2 − N ) π sin( s + 2 − N ) π A ( p,q,r ) tu = ( − N Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − A ( p,q,r ) tu = ( − N Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − × Z ∞ dx x k · k ( x − k · k · (cid:20) k T x + k T x − (cid:21) p · (cid:20) k P x + k P x − (cid:21) q · (cid:20) k L x + k L x − (cid:21) r = Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − (cid:0) − k T (cid:1) p (cid:0) − k P (cid:1) q (cid:0) − k L (cid:1) r × Z ∞ dx x k · k ( x − k · k · (cid:18) − ( − k T k T )) x − x (cid:19) p · (cid:18) − ( − k P k P ) x − x (cid:19) q · (cid:18) − ( − k L k L ) x − x (cid:19) r . (11)In the above calculation, we have used the string BCJ relation A ( p,q,r ) st = sin( πk · k )sin( πk · k ) A ( p,q,r ) tu , (12)which was proved by monodromy of integration of string amplitudes [28, 29] and explicitlyproved recently in [24]. We can now do a change of variable x − x = x ′ to get A ( p,q,r ) st = Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − (cid:0) − k T (cid:1) p (cid:0) − k P (cid:1) q (cid:0) − k L (cid:1) r × Z dx ′ x ′ − t − (1 − x ′ ) − u − (cid:18) − ( − k T k T ) x ′ (cid:19) p · (cid:18) − ( − k P k P ) x ′ (cid:19) q · (cid:18) − ( − k L k L ) x ′ (cid:19) r = Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − · (cid:0) − k T (cid:1) p (cid:0) − k P (cid:1) q (cid:0) − k L (cid:1) r Γ( − t − − u − s + 2 − N ) × F (3) D ( − t − , − p, − q, − r, s − N ; − k T k T , − k P k P , − k L k L ) , (13)which can be written as 5 ( p,q,r ) st = (cid:0) − k T (cid:1) p (cid:0) − k P (cid:1) q (cid:0) − k L (cid:1) r Γ( − s − N )Γ( − t − u + 2) × F (3) D ( − t − , − p, − q, − r, s − N ; − C T , − C P , − C L ) (14)if we define k Xi = e X · k i , k X k X = C X . (15)In Eq.(14), the D-type Lauricella function F ( K ) D is one of the four extensions of the Gausshypergeometric function to K variables and is defined as F ( K ) D ( a ; b , ..., b K ; c ; x , ..., x K )= X n , ··· ,n K ( a ) n + ··· + n K ( c ) n + ··· + n K ( b ) n · · · ( b K ) n K n ! · · · n K ! x n · · · x n K K (16)where ( a ) n = a · ( a + 1) · · · ( a + n −
1) is the Pochhammer symbol. There is a integralrepresentation of the Lauricella function F ( K ) D discovered by Appell and Kampe de Feriet(1926) [30] F ( K ) D ( a ; b , ..., b K ; c ; x , ..., x K )= Γ( c )Γ( a )Γ( c − a ) Z dt t a − (1 − t ) c − a − · (1 − x t ) − b (1 − x t ) − b ... (1 − x K t ) − b K , (17)which can be used to directly calculate the amplitude in Eq.(14). The relevance of theLauricella function in Eq.(17) for string scattering amplitudes was first suggested in [21].We now calculate the string four-point scattering amplitude with three tachyons and one6eneral higher spin state in Eq.(1) as following A ( p n ; q m ; r l ) st = sin( πk · k )sin( πk · k ) A ( p n ; q m ; r l ) tu = sin( u + 2 − N ) π sin( s + 2 − N ) π A ( p n ; q m ; r l ) tu = ( − N Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − · Z ∞ dx x k · k (1 − x ) k · k · Y n =1 " ( − n − ( n − k T x n + ( − n − ( n − k T ( x − n p n · Y m =1 " ( − m − ( m − k P x m + ( − m − ( m − k P ( x − m q m · Y l =1 " ( − l − ( l − k L x l + ( − l − ( l − k L ( x − l r l = ( − N Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − Z ∞ dx x k · k (1 − x ) k · k − N · Y n =1 (cid:18) k T ( − n − ( n − − ( − k T k T )( x − x ) n ] (cid:19) p n · Y m =1 (cid:18) k P ( − m − ( m − − ( − k P k P )( x − x ) m ] (cid:19) q m · Y l =1 (cid:18) k L ( − l − ( l − − ( − k L k L )( x − x ) l ] (cid:19) r l . (18)We can now do a change of variable x − x = y to get7 ( p n ; q m ; r l ) st = ( − N Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − Z dy y k · k − N (1 − y ) − k · k − k · k + N − · Y n =1 (cid:18) k T ( − n − ( n − − ( − k T k T ) y n ] (cid:19) p n · Y m =1 (cid:18) k P ( − m − ( m − − ( − k P k P ) y m ] (cid:19) q m · Y l =1 (cid:18) k L ( − l − ( l − − ( − k L k L ) y l ] (cid:19) r l = ( − N Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − · Y n =1 (cid:2) ( − n − ( n − k T (cid:3) p n Y m =1 (cid:2) ( − m − ( m − k P (cid:3) q m Y l =1 h ( − l − ( l − k L i r l · Z dy y k · k − N (1 − y ) − k · k − k · k + N − · (cid:0) − ( z Tn y ) n (cid:1) p n (cid:0) − ( z Pm y ) m (cid:1) q m (cid:0) − ( z Ll y ) l (cid:1) r l . (19)Finally the amplitude can be written in the following form A ( p n ; q m ; r l ) st = Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u − Y n =1 (cid:2) − ( n − k T (cid:3) p n · Y m =1 (cid:2) − ( m − k P (cid:3) q m Y l =1 (cid:2) − ( l − k L (cid:3) r l · Z dy y − t − (1 − y ) − u − [(1 − z Tn y )(1 − z Tn ω n y ) ... (1 − z Tn ω n − n y )] p n · [(1 − z Pm y )(1 − z Pm ω m y ) ... (1 − z Pm ω m − m y )] q m · [(1 − z Ll y )(1 − z Ll ω l y ) ... (1 − w Ll ω l − l y )] p n , (20)8hich can then be written in terms of the D-type Lauricella function F ( K ) D as following A ( p n ; q m ; r l ) st = Γ( s + 2 − N )Γ( − s − N )Γ( u + 2)Γ( − u −
1) Γ( − t − − u − s + 2 − N ) · Y n =1 (cid:2) − ( n − k T (cid:3) p n Y m =1 (cid:2) − ( m − k P (cid:3) q m Y l =1 (cid:2) − ( l − k L (cid:3) r l · F ( K ) D − t − {− p } , ..., {− p n } n , {− q } , ..., {− q m } m , {− r } , ..., {− r l } l ; s + 2 − N ; (cid:2) z T (cid:3) , ..., (cid:2) z Tn (cid:3) , (cid:2) z P (cid:3) , ..., (cid:2) z Pm (cid:3) , (cid:2) z L (cid:3) , ..., (cid:2) z Ll (cid:3) , = Γ( − s − N )Γ( − t − u + 2) Y n =1 (cid:2) − ( n − k T (cid:3) p n Y m =1 (cid:2) − ( m − k P (cid:3) q m Y l =1 (cid:2) − ( l − k L (cid:3) r l · F ( K ) D − t − {− p } , ..., {− p n } n , {− q } , ..., {− q m } m , {− r } , ..., {− r l } l ; s + 2 − N ; (cid:2) z T (cid:3) , ..., (cid:2) z Tn (cid:3) , (cid:2) z P (cid:3) , ..., (cid:2) z Pm (cid:3) , (cid:2) z L (cid:3) , ..., (cid:2) z Ll (cid:3) (21)where we have defined k Xi ≡ e X · k i , ω k = e πik , z Xk = ( − k X k X ) k (22)and { a } n = a, a, · · · , a | {z } n , (cid:2) z Xk (cid:3) = z Xk , z Xk e πik , · · · , z Xk e πi ( k − k or z Xk , z Xk ω k , ..., z Xk ω k − k . (23)The integer K in Eq.(21) is defined to be K = n X j =1 j { for all r Tj =0 } + m X j =1 j { for all r Pj =0 } + l X j =1 j { for all r Lj =0 } . (24)For a given K , there can be SSA with different mass level N .Alternatively, by using the identity of Lauricella function for b i ∈ Z − F ( K ) D ( a ; b , ..., b K ; c ; x , ..., x K ) = Γ ( c ) Γ ( c − a − P b i )Γ ( c − a ) Γ ( c − P b i ) · F ( K ) D (cid:16) a ; b , ..., b K ; 1 + a + X b i − c ; 1 − x , ..., − x K (cid:17) , (25)we can rederive the string BCJ relation [24, 28, 29] A ( r Tn ,r Pm ,r Ll ) st A ( r Tn ,r Pm ,r Ll ) tu = ( − ) N Γ (cid:0) − s − (cid:1) Γ (cid:0) s + 2 (cid:1) Γ (cid:0) u + 2 − N (cid:1) Γ (cid:0) − u − N (cid:1) = sin (cid:0) πu (cid:1) sin (cid:0) πs (cid:1) = sin ( πk · k )sin ( πk · k ) , (26)9hich gives another form of the ( s, t ) channel amplitude A ( r Tn ,r Pm ,r Ll ) st = B (cid:18) − t − , − s − (cid:19) Y n =1 (cid:2) − ( n − k T (cid:3) r Tn · Y m =1 (cid:2) − ( m − k P (cid:3) r Pm Y l =1 (cid:2) − ( l − k L (cid:3) r Ll · F ( K ) D (cid:18) − t − R Tn , R Pm , R Ll ; u − N ; ˜ Z Tn , ˜ Z Pm , ˜ Z Ll (cid:19) (27)and similarly the ( t, u ) channel amplitude A ( r Tn ,r Pm ,r Ll ) tu = B (cid:18) − t − , − u − (cid:19) Y n =1 (cid:2) − ( n − k T (cid:3) r Tn · Y m =1 (cid:2) − ( m − k P (cid:3) r Pm Y l =1 (cid:2) − ( l − k L (cid:3) r Ll · F ( K ) D (cid:18) − t − R Tn , R Pm , R Ll ; s − N ; Z Tn , Z Pm , Z Ll (cid:19) . (28)In Eq.(27) and Eq.(28), we have defined R Xk ≡ (cid:8) − r X (cid:9) , · · · , (cid:8) − r Xk (cid:9) k with { a } n = a, a, · · · , a | {z } n , (29)and Z Xk ≡ (cid:2) z X (cid:3) , · · · , (cid:2) z Xk (cid:3) with (cid:2) z Xk (cid:3) = z Xk , · · · , z Xk ( k − (30)where z Xk = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − k X k X (cid:19) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , z Xkk ′ = z Xk e πik ′ k , ˜ z Xkk ′ ≡ − z Xkk ′ (31)for k ′ = 0 , · · · , k − . With the notation introduced above, the ( s, t ) channel amplitude in Eq.(21) can berewritten as A ( r Tn ,r Pm ,r Ll ) st = B (cid:18) − t − , − s − N (cid:19) Y n =1 (cid:2) − ( n − k T (cid:3) r Tn · Y m =1 (cid:2) − ( m − k P (cid:3) r Pm Y l =1 (cid:2) − ( l − k L (cid:3) r Ll · F ( K ) D (cid:18) − t − R Tn , R Pm , R Ll ; s − N ; Z Tn , Z Pm , Z Ll (cid:19) . (32)10 II. REGGE SCATTERING LIMIT
With the exact SSA calculated in Eq.(32), Eq.(27) and Eq.(28) which are valid for allkinematic regimes, we can rederive SSA and symmetries or relations among SSA of differ-ent string states at various limits calculated previously. These include the linear relationsconjectured by Gross [1–3] and proved in [4–9] in the hard scattering limit, the recurrencerelations in the Regge scattering limit [19–21] and the extended recurrence relations in thenonrelativistic scattering limit [24] discovered recently. In this section, we first calculate theRegge scattering limit. The relevant kinematics in Regge limit are k T = 0, k T ≃ −√− t, (33) k P ≃ − s M , k P ≃ − ˜ t M = − t − M − M M , (34) k L ≃ − s M , k L ≃ − ˜ t ′ M = − t + M − M M . (35)One can easily calculate ˜ z Tkk ′ = 1 , ˜ z Pkk ′ = 1 − (cid:16) − s ˜ t (cid:17) /k e i πk ′ k ∼ s /k (36)and ˜ z Lkk ′ = 1 − (cid:16) − s ˜ t ′ (cid:17) /k e i πk ′ k ∼ s /k . (37)In the Regge limit, the SSA in Eq.(27) reduces to A ( r Tn ,r Pm ,r Ll ) st ≃ B (cid:18) − t − , − s − (cid:19) Y n =1 (cid:2) ( n − √− t (cid:3) r Tn · Y m =1 (cid:20) ( m − t M (cid:21) r Pm Y l =1 (cid:20) ( l − t ′ M (cid:21) r Ll · F (cid:18) − t − − q , − r ; − s s ˜ t , s ˜ t ′ (cid:19) . (38)where F is the Appell function. Eq.(38) agrees with the result obtained in [21] previously. IV. HARD SCATTERING LIMIT
In this section, we rederive the linear relations conjectured by Gross [1–3] and correctedand proved in [4–9] in the hard scattering limit. As we will see that the calculation will be11ore subtle than that of the Regge scattering limit. In the hard scattering limit e P = e L [4, 5], and we can consider only the polarization e L case. We first briefly review the results[18] for linear relations among hard SSA. One first observes that for each fixed mass level N only states of the following form [7, 8] | N, m, q i ≡ ( α T − ) N − m − q ( α L − ) m ( α L − ) q | , k i (39)are of leading order in energy in the HSS limit. The choice of only even power 2 m in α L − is the result of the observation [4, 5] that the naive energy order of the amplitudes willin general drop by even number of energy powers. Scattering amplitudes corresponding tostates with ( α L − ) m +1 turn out to be of subleading order in energy. Many simplificationsoccur if we apply Ward identities or decoupling of ZNS only on high energy states in Eq.(39)in the HSS limit. One important result was the discovery of the linear relations among hardSSA of different string states at each fixed mass level N [7, 8] A ( N, m,q ) st A ( N, , st = (cid:18) − M (cid:19) m + q (cid:18) (cid:19) m + q (2 m − . (40)Exactly the same results can also be obtained by two other calculations, the Virasoro con-straint calculation and the corrected saddle-point calculation [7, 8]. In the decoupling ofZNS calculations at the mass level M = 4, for example, there are four leading order SSA[4, 5] A T T T : A LLT : A ( LT ) : A [ LT ] = 8 : 1 : − − A T T T ∝ A [ LT ] , and A LLT = 0 which are inconsistent with the decoupling of ZNS or unitarity of thetheory. Indeed, a sample calculation was done [4, 5] to explicitly verify the ratios in Eq.(41).One interesting application of Eq.(40) was the derivation of relation of A ( N, m,q ) st and A ( N, m,q ) tu in the hard scattering limit [31] A ( N, m,q ) st ≃ ( − ) N sin( πk · k )sin( πk · k ) A ( N, m,q ) tu (42)where A ( N, m,q ) tu ≃ √ π ( − N − − N E − − N (cid:18) sin φ (cid:19) − (cid:18) cos φ (cid:19) − N · exp (cid:20) − t ln t + u ln u − ( t + u ) ln( t + u )2 (cid:21) . (43)12q.(42) was shown to be valid for scatterings of four arbitrary string states and was obtainedin 2006 [32], and thus was earlier than the discovery of four point field theory BCJ relations[26] and ”string BCJ relations” in Eq.(26) [24, 28, 29]. In contrast to the calculation ofstring BCJ relations [28, 29] which was motivated by the field theory BCJ relations [26],the derivation of Eq.(42) was motivated by the calculation of hard closed SSA [31] by usingKLT relation [33]. See a more detailed discussion in a recent publication [24].We are now ready to rederive Eq.(39) and Eq.(40) from Eq.(27). The relevant kinematicsare k T = 0, k T ≃ − E sin φ, (44) k L ≃ − p M ≃ − E M , (45) k L ≃ E M sin φ . (46)One can calculate ˜ z Tkk ′ = 1 , ˜ z Lkk ′ = 1 − (cid:16) − st (cid:17) /k e i πk ′ k ∼ O (1) . (47)The SSA in Eq.(27) reduces to A ( r Tn ,r Ll ) st = B (cid:18) − t − , − s − (cid:19) · Y n =1 [( n − E sin φ ] r Tn Y l =1 (cid:20) − ( l − E M sin φ (cid:21) r Ll · F ( K ) D (cid:18) − t − R Tn , R Ll ; u − N ; (1) n , ˜ Z Ll (cid:19) . (48)As was mentioned above that, in the hard scattering limit, there was a difference betweenthe naive energy order and the real energy order corresponding to the (cid:0) α L − (cid:1) r L operator inEq.(1). So let’s pay attention to the corresponding summation and write A ( r Tn ,r Ll ) st = B (cid:18) − t − , − s − (cid:19) · Y n =1 [( n − E sin φ ] r Tn Y l =1 (cid:20) − ( l − E M sin φ (cid:21) r Ll · X k r (cid:0) − t − (cid:1) k r (cid:0) u + 2 − N (cid:1) k r (cid:0) − r L (cid:1) k r k r ! (cid:16) st (cid:17) k r · ( · · · ) (49)13here we have used ( a ) n + m = ( a ) n ( a + n ) m and ( · · · ) are terms which are not relevant tothe following discussion. We then propose the following formula r L X k r =0 (cid:0) − t − (cid:1) k r (cid:0) u + 2 − N (cid:1) k r (cid:0) − r L (cid:1) k r k r ! (cid:16) st (cid:17) k r =0 · (cid:18) tus (cid:19) + 0 · (cid:18) tus (cid:19) − + · · · + 0 · (cid:18) tus (cid:19) − (cid:20) rL (cid:21) − + C r L (cid:18) tus (cid:19) − (cid:20) rL (cid:21) + O (cid:18) tus (cid:19) − (cid:20) rL (cid:21) +1 . (50)where C r L is independent of energy E and depends on r L and possibly scattering angle φ . For r L = 2 m being an even number, we further propose that C r L = (2 m )! m ! and is φ independent. We have verified Eq.(50) for r L = 0 , , , · · · , s → ∞ with t fixed) and setting r L = 2 m ,Eq.(50) reduces to the Stirling number identity, m X k r =0 (cid:0) − t − (cid:1) k r (cid:0) − s (cid:1) k r ( − m ) k r k r ! (cid:16) st (cid:17) k r ≃ m X k r =0 ( − m ) k r (cid:18) − t − (cid:19) k r ( − /t ) k r k r != 0 · ( − t ) + 0 · ( − t ) − + · · · + 0 · ( − t ) − m +1 + (2 m )! m ! ( − t ) − m + O ((cid:18) t (cid:19) m +1 ) , (51)which was proposed in [19] and proved in [22].It was demonstrated in [19] that the ratios in the hard scattering limit in Eq.(40) can bereproduced from a class of Regge string scattering amplitudes presented in Eq.(38). The keyof the mathematical proof [22] was the new Stirling number identity proposed in Eq.(51).In Eq.(50), the 0 terms correspond to the naive leading energy orders in the hard SSAcalculation. The true leading order SSA in the hard scattering limit can then be identified A ( r Tn ,r Ll ) st ≃ B (cid:18) − t − , − s − (cid:19) · Y n =1 [( n − E sin φ ] r Tn Y l =1 (cid:20) − ( l − E M sin φ (cid:21) r Ll · C r L ( E sin φ ) − (cid:20) rL (cid:21) · ( · · · ) ∼ E N − P n ≥ nr Tn − (cid:18) (cid:20) rL (cid:21) − r L (cid:19) − P l ≥ lr Ll , (52)14hich means that SSA reaches its highest energy when r Tn ≥ = r Ll ≥ = 0 and r L = 2 m beingan even number. This is consistent with the previous result presented in Eq.(39) [4–9].Finally, the leading order SSA in the hard scattering limit, i.e. r T = N − m − r L = 2 m and r L = q , can be calculated to be A ( N − m − q, m,q ) st ≃ B (cid:18) − t − , − s − (cid:19) ( E sin φ ) N (2 m )! m ! (cid:18) − M (cid:19) m + q = (2 m − (cid:18) − M (cid:19) m + q (cid:18) (cid:19) m + q A ( N, , st (53)which reproduces the ratios in Eq.(40), and is consistent with the previous result [4–9]. V. NONRELATIVISTIC SCATTERING LIMIT
In a recent paper [24] both s − t and t − u channel nonrelativistic low energy string scat-tering amplitudes of three tachyons and one leading trajectory string state at arbitrary masslevels were calculated. It was discovered that the mass and spin dependent nonrelativisticstring BCJ relations [28, 29] can be expressed in terms of Gauss hypergeometric functions.As an application, for each fixed mass level N, the extended recurrence relations amongnonrelativistic low energy string scattering amplitudes of string states with different spinsand different channels can be derived.In this section, we intend to rederive the results stated above from the Lauricella functions.In the nonrelativistic limit | ~k | ≪ M , we have k T = 0 , k T = − (cid:20) ǫ M + M ) M M ǫ | ~k | (cid:21) sin φ, (54) k L = − M + M M | ~k | + O (cid:16) | ~k | (cid:17) , (55) k L = − ǫ φ + M + M M | ~k | + O (cid:16) | ~k | (cid:17) , (56) k P = − M + O (cid:16) | ~k | (cid:17) , (57) k P = M + M − ǫ M cos φ | ~k | + O (cid:16) | ~k | (cid:17) (58)where ǫ = p ( M + M ) − M . One can easily calculate z Tk = z Lk = 0 , z Pk ≃ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) M M + M (cid:19) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (59)15he SSA in Eq.(32) reduces to A ( r Tn ,r Pm ,r Ll ) st ≃ Y n =1 h ( n − ǫ φ i r Tn Y m =1 (cid:20) − ( m − M + M (cid:21) r Pm · Y l =1 h ( l − ǫ φ i r Ll B (cid:18) M M , − M M (cid:19) · F ( K ) D (cid:18) M M R Pm ; M M ; (cid:18) M M + M (cid:19) m (cid:19) (60)where K = m X j =1 j { for all r Pj =0 } . (61)Note that for string states with r Pk = 0 for all k ≥
2, one has K = 1 and the Lauricellafunctions in the low energy nonrelativistic SSA reduce to the Gauss hypergeometric functions F (1) D = F . In particular, for the case of r T = N , r P = N , r L = N , and r Xk = 0 for all k ≥
2, the SSA reduces to A ( N ,N ,N ) st = (cid:16) ǫ φ (cid:17) N (cid:16) ǫ φ (cid:17) N · (cid:18) − M + M (cid:19) N B (cid:18) M M , − M M (cid:19) · F (cid:18) M M − N ; M M ; 2 M M + M (cid:19) , (62)which agrees with the result obtained in [24] previously. Similarly, one can calculate thecorresponding nonrelativistic t − u channel amplitude as A ( N ,N ,N ) tu = ( − N (cid:16) ǫ φ (cid:17) N (cid:16) ǫ φ (cid:17) N · (cid:18) − M + M (cid:19) N B (cid:18) M M , M M (cid:19) · F (cid:18) M M − N ; M M ; 2 M M + M (cid:19) . (63)Finally the ratio of s − t and t − u channel amplitudes is [24] A ( p,r,q ) st A ( p,r,q ) tu = ( − N B (cid:0) − M M + 1 , M M (cid:1) B (cid:0) M M , M M (cid:1) = ( − N Γ ( M M ) Γ ( − M M + 1)Γ (cid:0) M M (cid:1) Γ (cid:0) − M M + 1 (cid:1) ≃ sin π ( k · k )sin π ( k · k ) (64)16here, in the nonrelativistic limit, we have k · k ≃ − M M , (65a) k · k ≃ ( M + M ) M . (65b)We thus have ended up with a consistent nonrelativistic string BCJ relations. We stressthat the above relation is the stringy generalization of the massless field theory BCJ relation[26] to the higher spin stringy particles. VI. THE ASSOCIATE SYMMETRY GROUP OF STRING SCATTERING AM-PLITUDES
In the Lie group approach of special functions, the associate Lie group for the Lauricellafunction F ( K ) D in the SSA at each fixed K is the SL ( K + 3 , C ) group [25] which containsthe SL (2 , C ) fundamental representation of the 3 + 1 dimensional spacetime Lorentz group SO (3 , sl ( K + 3 , C ) contains the 2 + 1 dimensional so (2 ,
1) Lorentz spacetime sym-metry on the scattering plane in our case as well. In the Regge limit, the Lauricella functionin the SSA reduces to the Appell function F with associate group SL (5 , C ) [23], which is K independent. In the low energy nonrelativistic limit, the Lauricella function in the SSAreduces to the Gauss hypergeometric function F with associate group SL (4 , C ) [25], whichis also K independent.In sum, we have identified the associate exact SL ( K + 3 , C ) symmetry of string scat-tering amplitudes with three tachyons and one arbitrary string states of 26 D bosonic openstring theory. However, since not all Lauricella functions F ( K ) D with arbitrary independent arguments can be used to represent SSA, it remained to be studied how the basis states ofeach SL ( K + 3 , C ) group representation for a given K relates to SSA. This important issueis currently under investigation.Finally, with the SL ( K + 3 , C ) group and the recurrence relations of the Lauricella func-tions F ( K ) D , one can derive infinite number of recurrence relations of SSA of different stringstates which are valid for all energies, as long as all the Lauricella functions F ( K ) D in therecurrence relation representing the SSA. For a simple example, the following recurrence17elation of F ( K ) D can be verified cF ( K ) D ( b j ; c ) + c ( x j − F ( K ) D ( b j + 1; c )+( a − c ) x j F ( K ) D ( b j + 1; c + 1) = 0 , (66)which leads to the recurrence relation of SSA (cid:16) u − N (cid:17) A ( r Tn ,r Pm ,r Ll ) st − (cid:16) s (cid:17) k T A ( r ′ Tn ,r Pm ,r Ll ) st = 0 (67)where ( r ′ Tn , r Pm , r Ll ) means the group (cid:16) −{ r T − } , (cid:8) − r T (cid:9) , · · · , (cid:8) − r Tn (cid:9) n ; R Pm , R Ll (cid:17) of polar-izations. In Eq.(66), we have omitted those arguments of F ( K ) D which remain the same forall three Lauricella functions. Acknowledgments
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