The $SL(K+3,\mathbb{C})$ symmetry of string scatterings from D-branes

By using the solvability of Lauricella function $F_{D}^{(K)}\left( \alpha;\beta_{1},...,\beta_{K};\gamma;x_{1},...,x_{K}\right) $ with nonpositive integer $\beta_{J}$, we show that each scattering or decay process of string and D-brane states at \textit{arbitrary} mass levels can be expressed in terms of a single Lauricella function. This result extends the previous exact $SL(K+3,\mathbb{C})$ symmetry of tree-level open bosonic string theory to include the D-brane.


I. INTRODUCTION
Motivated by the previous calculation of high energy symmetry [1,2] of string scattering amplitudes (SSA) [3][4][5], [6][7][8][9][10], it was shown [11] recently that all SSA of four arbitrary string states of the open bosonic string theory at all kinematic regimes can be expressed in terms of the D-type Lauricella functions F (K) D (α; β 1 , ..., β K ; γ; x 1 , ..., x K ) with associated exact SL(K + 3, C) symmetry (see the definition of K in Eq. (3). On the other hand, a class of polarized fermion SSA (PFSSA) at arbitrary mass levels of the R-sector of the fermionic string theory can also be expressed in terms of the D-type Lauricella functions [12]. Indeed, it can be shown [11] that these Lauricella functions form an infinite dimensional representation of the SL(K + 3, C) symmetry group. Moreover, it was demonstrated that there existed K + 2 recursion relations among the D-type Lauricella functions. These recursion relations can be used to reproduce the Cartan subalgebra and simple root system of the SL(K + 3, C) group with rank K + 2 and vice versa.
For the cases of nonpositive integer β J (see Eq. (9)) in the D-type Lauricella functions which correspond to the cases of the SSA or the Lauricella SSA (LSSA) mentioned above, the SL(K + 3, C) group or the corresponding K + 2 stringy Ward identities among the LSSA can be used to solve [13] all the LSSA and express them in terms of one amplitude. These exact Ward identities among the exact LSSA are generalizations of the linear relations with constant coefficients among SSA in the hard scattering limit conjectured by Gross [1,2] in 1988 and later corrected and proved in [6][7][8][9][10].
More recently, by using the string theory extension [14], [15,16] of the field theory BCFW on-shell recursion relations [17,18] , one can show that [19] the residues of all n-point SSA including the Koba-Nielsen (KN) amplitudes can be expressed in terms of the Lauricella functions with nonpositive integer β J . As a result, the above SL(K + 3, C) symmetry group of the 4-point LSSA was extended to the n-point LSSA with arbitrary n [20]. It is thus believed that the SL(K + 3, C) symmetry is the fundamental symmetry of the whole bosonic string theory, at least, tree-level of the bosonic string theory.
To justify the conjecture of the fundamental SL(K + 3, C) symmetry of the bosonic string theory, one needs to collect more evidences or more SSA to support the proposed exact symmetry. In this paper, we will show that the scattering and decay processes of string and D-brane states at arbitrary mass levels can again be expressed in terms of the D-type Lauricella functions with nonpositive integer β J . This result extends the previous exact SL(K +3, C) symmetry of tree-level open bosonic string theory to include the D-brane. This is also consistent with the previous results that the linear relations with constant coefficients among SSA in the hard scattering limit persist for the processes of D-brane scatterings [21] and decays [22] as they are all related to the exact SL(K + 3, C) symmetry of bosonic string theory. We will see that the calculation will be greatly simplified by using the solvability [13] of the Lauricella functions with nonpositive integer β J .
We first review the LSSA of three tachyons and one arbitrary string states in the 26D open bosonic string theory and its associated SL(K + 3, C) symmetry. The general states are of the following form [11] where is the longitudinal polarization and e T = (0, 0, 1) is the transverse polarization on the (2 + 1)dimensional scattering plane. In addition to the mass level M 2 2 = 2(N − 1) with we define another important index K for the state in Eq.(1) where X = (T, P, L) and we have put r T n = r P m = r L l = 1 in Eq.(2) in the definition of K. Intuitively, K counts the number of variaty of the α X −j oscillators in Eq.(1). For later use, we also define k X j ≡ e X · k j for X = (T, P, L) .
Note that SSA of three tachyons and one arbitrary string states with polarizations orthogonal to the scattering plane vanish.
Note that to achieve BRST invariance or physical state conditions in the old covariant quantization scheme for the state in Eq.(1), one needs to add polarizations and put on the Virasoro constraints. As an example, let's calculate the case of symmetric spin 3 state of mass level M 2 2 = 4. We first note that the three momentum polarizations defined on the scattering plane above satisfy the completeness relation where µ, ν = 0, 1, 2 and α, β = P, L, T. Diag η µν = (−1, 1, 1). We can use Eq.(5) to transform all µ, ν coordinates to coordinates α, β on the scattering plane. One gauge choice of the symmetric spin 3 state with Virasoro constraints can be calculated to be We can then use the helicity decomposition and writing ǫ µνλ = Σ µ,ν,λ e α µ e β ν e δ λ u αβδ ; α, β, δ = P, L, T to get It is now easy to see from Eq.(7) that to achieve BRST invariance the spin 3 state can be written as a linear combination of states in Eq.(1) with coefficients u T T L and u T T T .
where B(a, b) is the Beta function with (s, t) being the usual Mandelstam variables, k X i is the momentum of the ith string state projected on the X polarization. In Eq.(8), we have defined R X l ≡ −r X 1 1 , · · · , −r X l l with {a} n = a, a, · · · , a n .
for the β J in the Lauricella function and where in Eq.(10), we have defined It is important to note that all β j of F where (α) n = α · (α + 1) · · · (α + n − 1) is the Pochhammer symbol. The result in Eq. (8) can be generalized to LSSA of four arbitrary string states, and to those of n arbitrary string states (see Eq. (25)) below [19,20].
For illustration, we calculate the Lauricella functions which correspond to the LSSA for For K = 2, there are six type of LSSA It is important to note that for a given K, there are infinite number of string states with arbitrary higher mass levels. Moreover, each string state was assigned a particular value of integer K, and its associated LSSA is a basis of the SL(K + 3, C) group representation.
To demonstrate the SL(K + 3, C) symmetry of the LSSA, one first defines the basis so that the LSSA in Eq. (8) can be rewritten as [24] A One can then introduce the (K + 3) 2 − 1 generators E ij of SL(K + 3, C) group [23,24] [ to operate on the basis functions in Eq. (22). These are 1 E α , K E β k (k = 1, 2 · · · K), 1 E γ ,1 E αγ , K E β k γ and K E αβ k γ which sum up to 3K + 3 raising generators. There are also 3K + 3 corresponding lowering operators. In addition, there are K (K − 1) E β k βp and K + 2 {J α , J β k , J γ } generators, the Cartan subalgebra. In sum, the total number of generators are For the general 4-point LSSA, it is straightforward to calculate the operation of these generators on the basis functions and show the SL(K + 3, C) symmetry [24]. For the cases of higher point (n ≥ 5) LSSA, one encounters the operation on the sum of products of the Lauricella functions [19] Residue of n-point LSSA ∼ coefficient (single tensor 4-point LSSA).
Therefore, one needs to deal with product representations of SL(K + 3, C).
Indeed, we have recently applied the string theory extension of field theory BCFW onshell recursion relations [17,18] to show that the SL(K +3, C) symmetry group of the 4-point LSSA persists for general n-point SSA with arbitrary higher point couplings in string theory [20]. We thus have shown that the SL(K + 3, C) symmetry is an exact symmetry of the whole bosonic string theory, and that all n-point SSA of the bosonic string theory form an infinite dimensional representation of the SL(K + 3, C) group. Moreover, all residues of SSA in the string theory on-shell recursion prescription can be expressed in terms of the four-point LSSA.
There is an interesting issue of the stringy on-shell Ward identities or decoupling of zero-

III. SOLVABILITY OF LSSA
There exist K + 2 recurrence relations for the D-type Lauricella functions [24]. Moreover, these recurrence relations can be used to reproduce the Cartan subalgebra and simple root system of the SL(K + 3, C) group with rank K + 2 [24]. With the Cartan subalgebra and the simple roots, one can easily write down the whole Lie algebra of the SL(K + 3, C) group.
So one can construct the SL(K + 3, C) Lie algebra from the recurrence relations and vice versa.
On the other hand, one can use the K + 2 recurrence relations to deduce the following key recurrence relation [13] x j F which, for the case of nonpositive β j , can be repeatedly used to decrease the value of K and reduce all the Lauricella functions F .., β i − |β j | (assuming i < j). For example, for say K = 2, Eq.(26) reduces to For say β 1 = 0 and β 2 = −1, we get which express the Lauricella function with K = 2 in terms of those of K = 1. We can repeat similar process to decrease the value of K.
Moreover, one can further reduce the Gauss hypergeometric functions by deriving a multiplication theorem for them, and then solve [13] all the LSSA in terms of one single amplitude.
This solvability is crucial to show that all scattering and decay processes of string and Dbrane states at arbitrary mass levels can be expressed in terms of the Lauricella function and thus its associated SL(K + 3, C) symmetry.

IV. CLOSED STRING SCATTERED OFF D-BRANE
In this paper, we will consider scattering and decay processes of string and D-brane states at arbitrary mass levels. These are three classes of processes [25], [26][27][28][29][30]: In this section, we first consider process (A). In [25], the calculation was done only for the massless string states. Here we will consider scatterings of arbitrary massive string states for the bosonic string. The standard propagators of the left and right moving fields there are nontrivial correlator as well [25] X µ (z)X ν (w) = −D µν log (z −w) as a result of the Dirichlet boundary condition at the real axis. The diagonal matrix D in Eq.(29) reverses the sign for fields satisfying Dirichlet boundary condition. That is, there are p + 1 Neumann and 25 −p Dirichlet for a Dp-brane. We will follow the standard notation and make the following replacement [25] which allows us to use the standard correlators throughout our calculations. As a warm up exercise, we first consider tachyon to tachyon scattering [21] A To fix the SL (2, R) invariance, we set z 1 = iy and z 2 = i and, for the contribution of the (0, 1) interval, we obtain [21] A (0,1) In the above calculations, we have defined so that and −k 2 For the general massive tensor to another massive tensor scattering, the calculation will be very complicated as there are many new contraction terms. We will use the solvability of the LSSA discussed above to simplify the calculation. The strategy is as follows: We can simply calculate a typical term of a given process. If the result turns out to be a Lauricella function with nonpositive β j , we can then use the solvability property to argue that the final amplitude after summing up all typical terms of the process is a LSSA. To do the calculation, we first define where n a , n b and n c are integer and then define N ′ = − (2n a + n b + n c ), so that where k 2 1 = 2(N 1 − 1) and N 1 is the mass level of k 1 . It is easy to see that a typical term in the general tensor to tensor scattering can be calculated to be [21] Similarly, for the (1, ∞) interval, one gets The sum of the two channels gives [21] where 3 F 2 is a generalized hypergeometric function. For the special arguments of 3 F 2 in Eq.(45), the hypergeometric function terminates to a finite sum and, as a result, the whole scattering amplitudes consistently reduce to the usual beta function. In calculating Eq.(45), we have used the identity which can be easily proved.
At this point, one might think that the amplitude calculated in Eq.(45) is not a LSSA, and the SL(K + 3, C) group may be just a subgroup of an unknown larger symmetry group G ⊇ SL(K + 3, C) of the bosonic string theory. However, we will see that this is not the case. To show that 3 F 2 in the amplitude Eq.(45) is a LSSA, we first do a change of variable y = x 2 to get where We can solve to get We can now do the following factorization to obtain Finally, we can use to obtain the identification where x k is defined in Eq.(50). We believe that the identity in Eq.(55) derived from string theory was not known previously in the literature [31]. In conclusion, we have shown that each amplitude of process (A) can be expressed in terms of a single Lauricella function with nonpositive integer β j and thus is a LSSA. As a result, all scattering of string at arbitrary mass levels from D-brane calculated in (A) form a part of an infinite dimensional representation of the exact SL(K + 3, C) symmetry of the bosonic string theory.

V. CLOSED STRING DECAYS INTO TWO OPEN STRING
In this section, we consider process (B), namely, closed string decays into two open strings on the brane. We will adapt the same strategy used in the last section and calculate only a typical term of a given process. We begin with the kinematics of the decay process. The momentum conservation on the D-brane reads where k c is the momentum of the closed string state. In the usual three-point amplitudes, momentum conservation completely constrains the kinematics. In the presence of D-brane, the non-conservation of momentum in the directions transverse to the D-brane gives precisely one kinematic variable which can be defined to be By using Eq.(56) and Eq.(57), one easily gets We first calculate the amplitude of a closed string tachyon to decay into two open string The next step is to use where we have used Eq.(59) and defined We now turn to the general mass level case. We will again use the solvability of the LSSA discussed above to simplify the calculation as before. A typical term of an arbitrary massive closed string state decays into two arbitrary massive open string states can be written as where n a , n b and n c are related to mass levels of k c , k 1 and k 2 . At this point, we expect after summing up all terms in the calculation, a real amplitude will be obtained. So we are going to calculate only the real part of A where N ≡ |n c − n a |, and see whether the final answer is a Lauricella function. Eq.(64) can be further reduced to where we have defined which are higher mass level generalization of Eq.(62). We can use the change of variable Finally, we obtain .
To derive a Lauricella function in Eq.(68), we note that So Eq.(68) can be written as with nonpositive β j to argue that the final amplitude after summing up all typical terms of the decay process is a LSSA.
By using the duplication formula for the gamma function the result in Eq.(72) can be further reduced to The factor Γ(−2t) Γ 2 (−t+1) can also be found in [25,27] for the massless string/D-brane decay process.

VI. CONCLUSION
In conclusion, in this paper we have shown that each amplitude of processes (A) and (B) for arbitrary massive string/D-brane states can be expressed in terms of a single Lauricella function with nonpositive integer β j and thus is a LSSA. To obtain the final results, we have used the solvability of the LSSA with nonpositive integer β j to simplify the calculation.
In addition to the scattering processes calculated in (A), all decay amplitudes of string/Dbrane states at arbitrary mass levels calculated in (B) also form a part of an infinite dimensional representation of the exact SL(K + 3, C) symmetry of the bosonic string theory. The results in this paper extends the previous exact SL(K + 3, C) symmetry of tree-level open bosonic string theory to include the D-brane.