Solving Lauricella String Scattering Amplitudes through Recurrence Relations

We show that there exist infinite number of recurrence relations valid for all energies among the open bosonic string scattering amplitudes (SSA) of three tachyons and one arbitrary string state, or the Lauricella SSA. Moreover, these infinite number of recurrence relations can be used to solve all the Lauricella SSA and express them in terms of one single four tachyon amplitude. These results extend the solvability of SSA at the high energy, fixed angle scattering limit and those at the Regge scattering limit discovered previously.


Introduction
It was widely believed that string theory consists of huge spacetime symmetries. In particular, it was conjectured [1][2][3] that in the high energy fixed angle regime there existed infinite number of linear relations among string scattering ampitudes (SSA) of different string states. Moreover, these relations are so powerful that they can be used to solve all high energy SSA and express them in terms of one single four tachyon amplitude.
Since it is a nontrivial task to compute exactly infinite number of massive higher spin SSA and the corresponding exact symmetries [4][5][6] for states at general mass levels, only SSA at high energy limit and relations among them were worked out in details in the literature [7][8][9][10][11][12] [13,14]. In fact, in the high energy fixed angle regime, the existence of these infinite linear relations among SSA was conjectured by Gross [1][2][3] and later corrected and explicitly proved in [7][8][9][10][11][12] by using decoupling of zero-norm states (ZNS) [15], and can be used to reduce the number of independent hard SSA from ∞ down to 1.
On the other hand, it was discovered that in the high energy fixed momentum transfer regime, the Regge SSA of three tachyons and one arbitrary string states can be expressed in terms of the first Appell function F 1 [14]. Regge stringy symmetries or recurrence relations, instead of linear relations in the fixed angle regime, among Regge SSA [13,14] were constructed and can be used to reduce the number of independent Regge SSA from ∞ down to 1. Moreover, an interesting link between Regge SSA and hard SSA was identified [16,17], and for each mass level the ratios among hard SSA can be extracted from the corresponding Regge SSA. See the recent review [18] for more details.
Recently a class of stringy SSA valid for all energies were exactly calculated [19]. These are the open bosonic SSA of three tachyons and one arbitrary string state or the so-called Lauricella string scattering amplitudes (LSSA). These LSSA were soon calculated by using the deformed cubic string field theory [20][21][22][23], and exactly the same results were reproduced [24] consistently. This string field theory calculation [24] can be considered as the extention of SSA calculation of low mass string states [25,26] through second quantized string to those of infinite number of higher mass string states. One important follow-up question then is whether there exist relations among these LSSA so that one can use them to solve all the LSSA and express them in terms of one single four tachyon amplitude.
In this paper, we will show that all the LSSA calculated in [19] can be solved through various recurrence relations of Lauricella functions. Moreover, all the LSSA can be expressed in terms of one single amplitude, say four tachyon string scattering amplitude. These results strongly suggest the existence of a huge spacetime symmetry of open bosonic string theory associated with SL(K + 3, C) [27,28] as suggested in [19], and are extentions of results calculated previously for both high energy, fixed angle SSA and Regge SSA [18].
In the last section of this paper, we will study the Lauricella zero norm states (LZNS) and the corresponding stringy Ward identities. In particular, we will show that the solvability of LSSA through recurrence relations imply the validity of Ward identities. However, the LZNS or the corresponding Lauricella Ward identities are not good enough to solve all the LSSA and express them in terms of one amplitude. This is in constrast to the fact that the high energy zero norm states (HZNS) at the fixed angle regime can be used to solve all the high energy SSA [7][8][9][10][11][12] and express them in terms of one single four tachyon amplitude.

The Lauricella String Scattering Amplitude
We first briefly review the calculation of the LSSA [19] of three tachyons and one arbitrary string states sitting at the second vertex. In the center of momentum frame, the kinematics are defined as The three polarization vectors on the scattering plane are [7,8] e T = (0, 0, 1), (2.2a) Note that SSA of three tachyons and one arbitrary string state with polarizations orthogonal to the scattering plane vanish. For later use, we define For the 26D open bosonic string, the general string states at mass level M 2 2 = 2(N − 1), N = n,m,l>0 (np n + mq m + lr l ) with polarizations on the scattering plane are of the form The (s, t) channel of the LSSA were calculated to be [19] A (pn;qm;r l ) st 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 was a integral representation of the Lauricella function F (K) D discovered by Appell and Kampe de Feriet (1926) [29] F (K) (2.7) The integer K in Eq.(2.5) is defined to be For a given K, there can be LSSA with different mass level N .

Solving all Lauricella string scattering amplitudes
To solve all the LSSA, one key simplification of the calculation was the observation that all arguments b m of the Lauricella functions in the LSSA are nonpositive integers. We stress that only Lauricella functions with special arguments are used in the LSSA in Eq.(2.5). As we will see that this will be the main reason of the solvability of the LSSA. There seem to be no recurrence relations for the Lauricella functions available in the literature. We will first generalize the 2 + 2 recurrence relations of the Appell functions to the K + 2 recurrence relations of the Lauricella functions. One can then use these K + 2 recurrence relations to reduce all the Lauricella functions F (K) D in the LSSA to the Gauss hypergeometry functions 2 F 1 (a, b, c). The next step is to derive a multiplication theorem for the Gauss hypergeometry functions.
The two results can then be used to prove the solvability of all LSSA. In the two steps of the proof of the solvability of all the LSSA, the property of nonpositive integers in the arguments b m of the Lauricella functions in the LSSA plays a key role in the argument.

Recurrence Relations of the LSSA
We begin with the Appell function case with K = 2. In the Appell case, there are four fundamental recurrence relations which link the contiguous functions It is straightforward to generalize the above relations and prove the following K +2 recurrence relations for the D − type Lauricella functions where m = 1, 2, ..., K. One notes that for K = 2, Eq.  To proceed for a fixed K, we first introduce two recurrence relations from Eq.
where in each of the above two equations we have omitted those arguments of F By multiplying x j and x i in Eq.(3.10) and Eq.(3.11) respectively, we can subtract the resulting two equations to take away the F (K) D (c + 1) term and obtain the following key recurrence relation One can repeatly apply Eq.(3.12) to the Lauricella functions in the LSSA in Eq.(2.5) and end up with an expression which expresses F We can repeat the process to decrease the value of K and reduce all the Lauricella functions F  Figure 1 in the text.

Reduction by a Multiplication Theorem
In this subsection, to further reduce the Gauss hypergeometry functions in the LSSA and solve all the LSSA in terms of one single amplitude, we first derive a multiplication theorem for the Gauss hypergeometry functions.
If we replace y by (y − 1)x in the following Taylor's theorem we get the identity (3.14) One can now use the derivative relation of the Gauss hypergeometry function where (a) n = a · (a + 1) · · · (a + n − 1) is the Pochhammer symbol to derive the following multiplication theorem It is important to note that the ∞ upper bound of the summation in the above equation has been replaced by |b| since b is a nonpositive integer for the cases of LSSA. In particular if we take x = 1 in Eq.(3.16), we get the following relation The factor 2 F 1 on the right hand side of the above equation can be written as Finally one identifies that which corresponds to the LSSA with the polarization (α T −1 ) p 1 = (α T −1 ) −b in Eq.(2.4). One can even use one of the 15 Gauss contiguous relations and set x = 1 which kills the second term of Eq.

Lauricella Zero Norm States and Ward Identities
In addition to the recurrence relations among LSSA, there are on-shell stringy Ward identities among LSSA. These Ward identities can be derived from the decoupling of two type of zero norm states (ZNS) in the old covariant first quantized string spectrum. However, as we will see soon that these Lauricella zero norm states (LZNS) or the corresponding Lauricella Ward identities are not good enough to solve all the LSSA and express them in terms of one amplitude.
On the other hand, in the last section, we have shown that by using (A) Recurrence relations of the LSSA, (B) Multiplication theorem of Gauss hypergeometry function and (C) the explicit calculation of four tachyon amplitude, one can explicitly solve and calculate all LSSA. This means that the solvability of LSSA through the calculations of (A), (B) and (C) imply the validity of Ward identities. Ward identities can not be identities independent of recurrence relations we used in the last section. Otherwise there will be a contradiction with the solvabilibity of LSSA.
In this section, we will study some examples of Ward identities of LSSA from this point of view. Incidentally, high energy zero norm states (HZNS) [7][8][9][10][11][12] and the corresponding stringy Ward identities at the fixed angle regime, and Regge zero norm states (RZNS) [13,14] and the corresponding Regge Ward identities at the Regge regime have been studied previously. In particular, HZNS at the fixed angle regime can be used to solve all the high energy SSA [7][8][9][10][11][12].

The Lauricella zero norm states
We will consider a smaller set of Ward identities, namely, those among the LSSA or string scattering amplitudes with three tachyons and one arbitrary string states. So we need only consider polarizations of the tensor states on the scattering plane. There are two types of zero norm states (ZNS) in the old covariant first quantum string spectrum Type I : L −1 |x , where L 1 |x = L 2 |x = 0, L 0 |x = 0; (4.1) While type I states have zero-norm at any spacetime dimension, type II states have zero-norm only at We begin with the case of mass level M 2 = 2. There is a type II ZNS and a type I ZNS Note that for the LSSA of three tachyons and one arbitrary string state, amplitudes with polarizations orthogonal to the scattering plane vanish. We define the polarizations of the 2nd tensor state with momentum k 2 on the scattering plane to be e P = 1 M 2 (E 2 , k 2 , 0) = k 2 M 2 (or e P = 1 M 2 (−E 2 , k 2 , 0)) as the momentum polarization, e L = 1 M 2 (k 2 , E 2 , 0) the longitudinal polarization and e T = (0, 0, 1) the transverse polarization. η µν = diag (−1, 1, 1). The three vectors e P , e L and e T satisfy the completeness relation where µ, ν = 0, 1, 2 and α, β = P, L, T and  (k · α −1 ) 3 + 9 2 (k · α −1 )(α −1 · α −1 ) + 9(α −1 · α −2 ) a symmetric type I spin two ZNS where α λµν −1 ≡ α λ −1 α µ −1 α ν −1 and two vector ZNS ( Note that Eq.(4.11) and Eq.(4.12) are linear combinations of a type I and a type II ZNS. This completes the four ZNS at the second massive level M 2 = 4. The scalar ZNS in Eq.(4.9) gives the LZNS For the type I spin two ZNS in Eq.(4.10), we define θ µν = α,β e α µ e β ν u αβ , (4.14) symmetric and transverse conditions on θ µν then implies The traceless condition on θ µν implies Eq.(4.15) and Eq.(4.16) give two LZNS The vector ZNS in Eq.(4.11) gives two LZNS The vector ZNS in Eq.(4.12) gives two LZNS

The Lauricella Ward identities
In this subsection, we will explicitly verify some examples of Ward identities through processes (A),(B) and (C). Process (C) will be implicitly used through the kinematics. Ward identities can not be identities independent of recurrence relations we used in processes (A),(B) and (C) in the last section. We define the following kinematics variables (for As the first example, we calculate the Ward identity associated with the LZNS in Eq.(4.7). The calculation will be based on processes (A) and (B). By using Eq.(2.5), the Ward identity we want to prove is  where Eq.(3.20) has been used to get the last equality of the above equation.
As the second example, we calculate the Ward identity associated with the LZNS in Eq.(4.8). By using Eq.(2.5), the Ward identity is We now use Eq. (3.12) in the process (A) to the second term of Eq.(4.34) to get We then use Eq.

Conclusions
In this paper we have shown that there exist infinite number of recurrence relations valid for all energies among the LSSA of three tachyons and one arbitrary string state. Moreover, these infinite number of recurrence relations can be used to solve all the LSSA and express them in terms of one single four tachyon amplitude. It will be interesting to see the relationship between the solvability of LSSA and the associated SL(K + 3, C) [27,28] symmetry of the Lauricella functions as suggested in [19]. The results of this calculation extend the solvability of SSA at the high energy, fixed angle scattering limit [7][8][9][10][11][12] and those at the Regge scattering limit [13,14] discovered previously. We have also shown that for the first few mass levels the solvability of LSSA through the calculations of recurrence relations implies the validity of Ward identities derived from the decoupling of LZNS. However the Lauricella Ward identities are not good enough to solve all the LSSA and express them in terms of one amplitude.
The extention of results for the one tensor three tachyon scatterings calculated in this paper to multitensor scatterings is currently under investigation.