The Lauricella Functions and Exact String Scattering Amplitudes

We discover that the 26D open bosonic string scattering amplitudes (SSA) of three tachyons and one arbitrary string state can be expressed in terms of the D-type Lauricella functions with associated SL(K+3,C) symmetry. As a result, SSA and symmetries or relations among SSA of different string states at various limits calculated previously can be rederived. These include the linear relations first conjectured by Gross [1-3] and later corrected and proved in [4-9] in the hard scattering 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 with associated SL(4,C) symmetry [24] discovered recently. Finally, as an application, we calculate a new recurrence relation of SSA which is valid for all energies.


I. INTRODUCTION
It has long been believed that there exist huge hidden spacetime symmetries of string theory. As a consistent theory of quantum gravity, string theory contains no free parameter and an infinite number of higher spin string states. On the other hand, the very soft exponential fall-off behavior of string scattering amplitudes (SSA) in the hard scattering limit, 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 string states. These relations or symmetries soften the UV structure of quantum string theory.
It was important to note that the linear relations obtained by decoupling of ZNS in the hard scattering limit corrected [4][5][6] the saddle point calculations of Gross [2], Gross and Mende [1] and Gross and Manes [3]. The results of the former authors were consistent with the decoupling of high energy ZNS or unitarity of the theory while those of the latter were not. See one simple example to be presented in Eq.(41) in section IV. Independently, the inconsistency of the saddle point calculations of the above authors was also pointed out by the 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 tremendous developments on calculations of higher point and higher loop Yang-Mills and gravity field theory amplitudes [14]. Many interesting relations among these field theory amplitudes have also been proposed and suggested. In addition, connections between field theory and string theory amplitudes are currently under many investigations.
Historically, there were at least three approaches to probe stringy symmetries or relations among scattering amplitudes of higher spin string states. These include the gauge symmetry of Witten string field theory, the conjecture of Gross [2] on symmetries or linear relations among SSA of different string states in the hard scattering limit by the saddle point method [1][2][3] and Moore's bracket algebra approach [15][16][17] of stringy symmetries. See a recent review [18] for some connections of these three approaches.
Recently, it was found that the Regge SSA of three tachyons and one arbitrary string states can be expressed in terms of a sum of Kummer functions U [19][20][21], which soon later were shown to be the first Appell function F 1 [21]. Regge stringy symmetries or recurrence relations [20,21] were then constructed and used to reduce the number of independent Regge SSA from ∞ down to 1. Moreover, an interesting link between Regge SSA and hard SSA was found [19,22], and for each mass level the ratios among hard SSA can be extracted from Regge SSA. This result enables us to argue that the known SL(5; C) dynamical symmetry of the Appell function F 1 [23] is crucial to probe high energy spacetime symmetry of string theory.
More recently, the extended recurrence relations [24] among nonrelativistic low energy SSA of a class of string states with different spins and different channels were constructed by using the recurrence relations of the Gauss hypergeometric functions with associated SL (4, C) symmetry [25]. These extended recurrence relations generalize and extend the field 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 we calculate the 26D 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-type Lauricella functions with associated SL(K + 3, C) symmetry [25]. As a result, all these SSA and symmetries or relations among SSA of different string states at various limits calculated previously can be rederived. These will be presented in sections III, IV and V which 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][5][6][7][8][9] in the hard scattering limit and the extended recurrence relations in the nonrelativistic scattering 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 to represent SSA, it remained to be studied how the basis states of each SL(K + 3, C) group representation for a given K relates to SSA [27].
As a byproduct from the calculation of rederiving linear relations in the hard scattering limit directly from Lauricella functions, we propose an identity Eq.(50) which generalizes the Stirling number identity Eq.(51) [19,22] used previously to extract ratios among hard SSA from the Appell functions in Regge SSA. Finally, as an example, in section VI we calculate a new recurrence relation of SSA which is valid for all energies.

II. FOUR-POINT STRING AMPLITUDES
We will consider SSA of three tachyons and one arbitrary string states put at the second vertex. For the 26D open bosonic string, the general states at mass level M 2 2 = 2(N − 1), N = n,m,l>0 nr T n + mr P m + lr L l with polarizations on the scattering plane are of the form In the CM frame, the kinematics are defined as For later use, we define k X i ≡ e X · k i for X = (T, P, L) .
Note that SSA of three tachyons and one arbitrary string state with polarizations orthogonal to the scattering plane vanish.
For illustration, we begin with a simple case, namely, four-point function with the three tachyons and the highest spin state at mass level M 2 2 = 2(N − 1), N = p + q + r of the following form |p, q, r = α T −1 The four-point scattering amplitude can be calculated as In the above calculation, we have used the string BCJ relation which was proved by monodromy of integration of string amplitudes [28,29] and explicitly proved recently in [24]. We can now do a change of variable x−1 which can be written as if we define In Eq. (14), the D-type Lauricella function F is one of the four extensions of the Gauss hypergeometric function to K variables and is defined as where (a) n = a · (a + 1) · · · (a + n − 1) is the Pochhammer symbol. There is a integral representation of the Lauricella function F (K) D discovered by Appell and Kampe de Feriet which can be used to directly calculate the amplitude in Eq. (14). The relevance of the Lauricella 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 one general higher spin state in Eq.(1) as following We can now do a change of variable x−1 x = y to get Finally the amplitude can be written in the following form which can then be written in terms of the D-type Lauricella function F (K) D as following where we have defined and {a} n = a, a, · · · , a n , The integer K in Eq.(21) is defined to be For a given K, there can be SSA with different mass level N.
Alternatively, by using the identity of Lauricella function for we can rederive the string BCJ relation [24,28,29] A (r T n ,r P m ,r L l ) st A (r T n ,r P m ,r L l ) tu which gives another form of the (s, t) channel amplitude and similarly the (t, u) channel amplitude In Eq. (27) and Eq.(28), we have defined R X k ≡ −r X 1 1 , · · · , −r X k k with {a} n = a, a, · · · , a n , and where for k ′ = 0, · · · , k − 1.
With the notation introduced above, the (s, t) channel amplitude in Eq.(21) can be rewritten as With the exact SSA calculated in Eq.(32), Eq. (27) and Eq.(28) which are valid for all kinematic regimes, we can rederive SSA and symmetries or relations among SSA of different string states at various limits calculated previously. These include the linear relations conjectured by Gross [1][2][3] and proved in [4][5][6][7][8][9] in the hard scattering limit, the recurrence relations in the Regge scattering limit [19][20][21] and the extended recurrence relations in the nonrelativistic scattering limit [24] discovered recently. In this section, we first calculate the Regge scattering limit. The relevant kinematics in Regge limit are One can easily calculatez In the Regge limit, the SSA in Eq. (27) reduces to where F 1 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][2][3] and corrected and proved in [4][5][6][7][8][9] in the hard scattering limit. As we will see that the calculation will be more 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] Exactly the same results can also be obtained by two other calculations, the Virasoro constraint calculation and the corrected saddle-point calculation [7,8]. in the hard scattering limit [31] A (N,2m,q) st where Eq.(42) was shown to be valid for scatterings of four arbitrary string states and was obtained in 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 of string 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 using KLT 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 kinematics One can calculatez The SSA in Eq. (27) reduces to As was mentioned above that, in the hard scattering limit, there was a difference between the naive energy order and the real energy order corresponding to the α L where we have used (a) n+m = (a) n (a + n) m and (· · · ) are terms which are not relevant to the following discussion. We then propose the following formula where C r L 1 is independent of energy E and depends on r L 1 and possibly scattering angle φ. For r L 1 = 2m being an even number, we further propose that C r L 1 = (2m)! m! and is φ independent. We have verified Eq.(50) for r L 1 = 0, 1, 2, · · · , 10. It should be noted that, taking Regge limit (s → ∞ with t fixed) and setting r L 1 = 2m, Eq.(50) reduces to the Stirling number identity, 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 be reproduced from a class of Regge string scattering amplitudes presented in Eq.(38). The key of 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 SSA calculation. The true leading order SSA in the hard scattering limit can then be identified which means that SSA reaches its highest energy when r T n≥2 = r L l≥3 = 0 and r L 1 = 2m being an even number. This is consistent with the previous result presented in Eq.(39) [4][5][6][7][8][9].

V. NONRELATIVISTIC SCATTERING LIMIT
In a recent paper [24] both s − t and t − u channel nonrelativistic low energy string scattering amplitudes of three tachyons and one leading trajectory string state at arbitrary mass levels were calculated. It was discovered that the mass and spin dependent nonrelativistic string 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 among nonrelativistic low energy string scattering amplitudes of string states with different spins and 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 1 | ≪ M 2 , we have where ǫ = (M 1 + M 2 ) 2 − 4M 2 3 . One can easily calculate where, in the nonrelativistic limit, we have We thus have ended up with a consistent nonrelativistic string BCJ relations. We stress that the above relation is the stringy generalization of the massless field theory BCJ relation [26] to the higher spin stringy particles.

PLITUDES
In the Lie group approach of special functions, the associate Lie group for the Lauricella function F (K) D in the SSA at each fixed K is the SL (K + 3, C) group [25] which contains the SL (2, C) fundamental representation of the 3 + 1 dimensional spacetime Lorentz group SO(3, 1). So sl (K + 3, C) contains the 2 + 1 dimensional so(2, 1) Lorentz spacetime symmetry on the scattering plane in our case as well. In the Regge limit, the Lauricella function in the SSA reduces to the Appell function F 1 with associate group SL (5, C) [23], which is K independent. In the low energy nonrelativistic limit, the Lauricella function in the SSA reduces to the Gauss hypergeometric function 2 F 1 with associate group SL (4, C) [25], which is also K independent.
In sum, we have identified the associate exact SL (K + 3, C) symmetry of string scat-