The SL(K+3,C) Symmetry of the Bosonic String Scattering Amplitudes

We discover that the exact string scattering amplitudes (SSA) of three tachyons and one arbitrary string state, or the Lauricella SSA (LSSA), in the 26D open bosonic string theory can be expressed in terms of the basis functions in the infinite dimensional representation space of the SL(K+3,C) group. In addition, we find that the K+2 recurrence relations among the LSSA discovered by the present authors previously can be used to reproduce the Cartan subalgebra and simple root system of the SL(K+3,C) group with rank K+2. As a result, the SL(K+3,C) group can be used to solve all the LSSA and express them in terms of one amplitude. As an application in the hard scattering limit, the SL(K+3,C) group can be used to directly prove Gross conjecture [1-3], which was previously corrected and proved by the method of decoupling of zero norm states [4-10].


I. INTRODUCTION
One of the most important issue of string theory is its spacetime symmetry structure. It has been widely believed that there exist huge spacetime symmetries of string theory. One way to study string symmetry is to calculate string scattering amplitudes (SSA). Indeed, it was conjectured by Gross [1][2][3] that there exist infinite number of linear relations among high energy, fixed angle or hard SSA of different string states. This conjecture was later corrected and explicitly proved in [4][5][6][7][8][9] by using the method of decoupling of zero-norm states [10]. Moreover, these infinite linear relations are so powerful that they can be used to reduce the number of independent hard SSA from ∞ down to 1. Other approaches of stringy symmetries can be found at [11][12][13][14][15][16]. For more details, see [17] for a recent review.
On the other hand, it was found that the high energy, fixed momentum transfer or Regge SSA of three tachyons and one arbitrary string states can be expressed in terms of a sum of Kummer functions U [18][19][20], which were then shown to be the first Appell function F 1 [20]. Regge stringy recurrence relations [19,20] can then be 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 pointed out in [18,21], and for each mass level the ratios among hard SSA can be extracted from Regge SSA. It was then conjectured that the SL(5; C) dynamical symmetry of the Appell function F 1 [22] is crucial to probe high energy spacetime symmetry of string theory.
More recently, the Lauricella string scattering amplitudes (LSSA) [23] of three tachyons and one arbitrary string state in the 26D open bosonic string theory valid for arbitrary energies were calculated and expressed in terms of the D-type Lauricella functions F (K) D . Moreover, it was shown that [24] there exist K + 2 recurrence relations among F (K) D which (together with a multiplication theorem of F (1) D ) can be used to derive recurrence relations among LSSA and reduce the number of independent LSSA from ∞ down to 1.
In this paper, we will show the existence of the spacetime symmetry group structure of the LSSA. To be more specific, we will demonstrate that the LSSA can be expressed in terms of the basis functions in the infinite dimensional representation space of the SL(K + 3, C) group [25,26] which contains the SO(2, 1) spacetime Lorentz group. In addition, we find that the K + 2 recurrence relations among the LSSA discovered by the present authors [24] previously can be used to reproduce the Cartan subalgebra and simple root system of the We thus have demonstrated, for the first time, the existence of a spacetime symmetry group of the 26D open bosonic string theory. As a result, the SL(K + 3, C) group can be used to solve all the LSSA and express them in terms of one amplitude. As an application in the hard scattering limit, the SL(K + 3, C) group can be used to directly prove Gross conjecture [1][2][3], which was previously corrected and proved by the method of decoupling of zero norm states [4][5][6][7][8][9][10].

II. REVIEW OF THE LSSA
In this section, we first review the LSSA of three tachyons and one arbitrary string states of 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 There are three polarizations on the scattering plane [4,5] e T = (0, 0, 1), For later use, we define k X i ≡ e X · k i for X = (T, P, L) .
It is important to note that SSA of three tachyons and one arbitrary string state with polarizations orthogonal to the scattering plane vanish. Thus the Lorentz spacetime symmetry group is SO(2, 1). The (s, t) channel of the LSSA can be calculated to be [23] A (r T n ,r P m ,r L l ) st . (2.11) The D-type Lauricella function F (K) D is one of the four extensions of the Gauss hypergeometric function to K variables and is defined as where (α) n = α · (α + 1) · · · (α + n − 1) is the Pochhammer symbol. There was a integral representation of the Lauricella function F which was used to calculate Eq.(2.10).
As an application of Eq.(2.10), it can be shown that in the hard scattering limit e P = e L [4,5], the leading order LSSA corresponds to r T 1 = N − 2m − 2q, r L 1 = 2m and r L 2 = q, and the LSSA in the hard scattering limit can be calculated to be [23] which gives the ratios [17] A 15) and is consistent with the previous result [4][5][6][7][8][9]. The first example calculated was the ratios at mass level M 2 = 4 [4,5] T The ratios among SSA in Eq.(2.15) and Eq.(2.16) are generalization of ratios among field theory scattering amplitudes. Let's consider a simple analogy from particle physics. The ratios of the nucleon-nucleon scattering processes (a) p + p → d + π + , can be calculated to be (ignore the tiny mass difference between proton and neutron) from SU(2) isospin symmetry. Is there any symmetry group structure which can be used to calculate SSA ratios in Eq.(2.15) and Eq.(2.16)? This is the main issue we want to address in this paper and it turns out that the relevant group is the noncompact SL(K + 3, C) group as we will discuss in the rest of the paper. Since the spacetime symmetry group of the LSSA needs to include the noncompact Lorentz group SO(2, 1), the noncompact SL(K + 3, C) group seems to be a reasonable one.

III. THE SL(4,C) SYMMETRY
In this section, for illustration we first consider the simplest K = 1 case with SL(4, C) symmetry. For a given K, there can be LSSA with different mass level N. For illustration, for K = 1 as an example there are three types of LSSA To calculate the group representation of the LSSA for K = 1, we first define [26] Note that the LSSA in Eq.(2.10) for K = 1 corresponds to the case a = 1 = c, and can be written as We are now ready to introduce the 15 generators of SL(4, C) group [25,26] E α = a (x∂ x + a∂ a ) , and calculate their operations on the basis functions [25,26] Note, for example, that since β is a nonpositive integer, the operation by E −β will not be terminated as in the case of the finite dimensional representation of a compact Lie group.
Here the representation is infinite dimensional. On the other hand, a simple calculation gives which suggest the Cartan subalgebra we find out that each of the triplets [25,26] In the following, we want to further relate the SL(4, C) group to the recurrence relations of F The three recurrence relations can be used to derive recurrence relations among LSSA in Eq.(3.1).
In the following we will show that the three recurrence relations can be used to reproduce the Cartan subalgebra and simple root system of the SL(4, C) group with rank 3. With the identification in Eq.(3.2), the first recurrence relation in Eq.(3.8) can be rewritten as By using the identity the recurrence relation then becomes which means Similarly for the second recurrence relation in Eq.(3.9), we obtain After some calculations, we end up with Finally the third recurrence relation in Eq.(3.10) can be rewritten as which gives after some computation where we have used the operation of E α in Eq. (3.5). The next step is to use the definition where we have used the definition of E β in Eq. Similarly, we can check the operation of E α . Note that the first recurrence relation in Eq.(3.8) can be rewritten as where we have used the operation of E β in Eq.(3.5). The next step is to use the definition of E β in Eq.(3.4) to obtain Finally we check the operation of E γ . Note that Eq.(3.9) can be written as The next step is to use the definition and operation of E αγ to obtain After some simple computation, we get We thus have shown that the extended LSSA f b ac (α; β; γ; x) in Eq.(3.2) with arbitrary a and c form an infinite dimensional representation of the SL(4, C) group. Moreover, the 3 recurrence relations among the LSSA can be used to reproduce the Cartan subalgebra and simple root system of the SL(4, C) group with rank 3. The recurrence relations are thus equivalent to the representation of the SL(4, C) symmetry group.

IV. THE GENERAL SL(K + 3,C) SYMMETRY
To calculate the group representation of the LSSA for general K, we first define [26] Note that the LSSA in Eq.(2.10) corresponds to the case a = 1 = c, and can be written as It is possible to generalize the SL(4, C) symmetry group for the K = 1 case discussed in the previous section to the general SL(K + 3, C) group. We first introduce the (K + 3) 2 − 1 generators of SL(K + 3, C) group (k = 1, 2, ...K) [25,26] Note that we have used the upper indices to denote the "raising operators" and the lower indices to denote the "lowering operators". The number of generators can be counted by the following way. There are 1 E α , K E β 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 lowering operators. In addition, there are K (K − 1) E β k βp and K + 2 J , the Cartan subalgebra. In sum, the total number of generators are 2(3K + 3) + K(K − 1) + K + 2 = (K + 3) 2 − 1. It is straightforward to calculate the operation of these generators on the basis functions (k = 1, 2, ...K) [26] , where, for simplicity, we have omitted those arguments in f b 1 ···b K ac which remain the same after the operation. The commutation relations of the SL(K + 3) Lie algebra can be calculated in the following way. In addition to the Cartan subalgebra for the K+2 generators {J α , J β k , J γ }, satisfies the commutation relations in Eq.(3.7).
There are K +2 fundamental recurrence relations among F where m = 1, 2, ...K. The three types of recurrence relations can be used to derive recurrence relations among LSSA and reduce the number of independent LSSA from ∞ down to 1 [24].
In the following we will show that the three types of recurrence relations above imply the Cartan subalgebra of the SL(K + 3, C) group with rank K + 2. With the identification in Eq.(4.1), the first type of recurrence relation in Eq.(4.7) can be rewritten as which gives The second type of recurrence relation in Eq. (4.8) can be rewritten as which gives Eq.(4.16) can be written as The third type of recurrence relation in Eq.(4.9) can be rewritten as (m = 1, 2, ...K) which gives can be rewritten as After operation of E β j , we obtain which gives the consistent result In the previous publication, it was shown that [24] the K + 2 recurrence relations among can be used to derive recurrence relations among LSSA and reduce the number of independent LSSA from ∞ down to 1. We conclude that the SL(K + 3, C) group can be used to derive infinite number of recurrence relations among LSSA, and one can solve all the LSSA and express them in terms of one amplitude.
Finally, in addition to Eq.(4.6), there is a simple way to write down the Lie algebra commutation relations of SL(K + 3, C),namely [26] [E ij , with the identifications In addition, we find that the K + 2 recurrence relations among the LSSA can be used to reproduce the Cartan subalgebra and simple root system of the SL(K +3, C) group with rank K + 2. Thus the recurrence relations are equivalent to the representation of SL(K + 3, C) group of the LSSA. As a result, the SL(K + 3, C) group can be used to solve all the LSSA and express them in terms of one amplitude [24] . As an application in the hard scattering limit, the SL(K + 3, C) group can be used to directly prove Gross conjecture [1][2][3], which was previously corrected and proved by the method of decoupling of zero norm states [4][5][6][7][8][9][10].
There are some special properties in the SL(K + 3, C) group representation of the LSSA, which make it different from the usual symmetry group representation of a physical system.
First, the set of LSSA does not fill up the whole representation space V . For example, for states f b 1 ···b K ac (α; β 1 , · · · , β K ; γ; x 1 , · · · , x K ) in V with a = 1 or c = 1, they are not LSSA.