Recent developments of the Lauricella string scattering amplitudes and their exact SL(K+3,C) Symmetry

In this review we propose a new perspective to demonstrate Gross conjecture on high energy symmetry of string theory. We review the construction of 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. These LSSA form an infinite dimensional representation of the SL(K+3,C) group. Moreover, we show that 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 LSSA can be used to directly prove Gross conjecture which was previously corrected and proved by the method of decoupling of zero norm states (ZNS). Finally, the exact LSSA can be used to rederive the recurrence relations of SSA in the Regge scattering limit with associated SL(5,C) symmetry and the extended recurrence relations (including the mass and spin dependent string BCJ relations) in the nonrelativistic scattering limit with associated SL(4,C) symmetry discovered recently.

In contrast to low energy string theory, many issues of high energy behavior of string theory have not been well understood yet so far.Historically, it was first conjectured by Gross [1][2][3][4][5] that there exist infinite linear relations among hard string scattering amplitudes (HSSA) of different string states.Moreover, these linear relations are so powerful that they can be used to solve all HSSA and express them in terms of one amplitude.This conjecture was later (slightly) corrected and proved by using the decoupling of zero norm states [6][7][8][9] in [10][11][12][13][14][15][16].For more details, see the recent review articles [17,18].
In this paper, we review another perspective to understand high energy behavior of string and demonstrate Gross conjecture on high energy symmetry of string theory.Since the theory of string as a quantum theory consists of infinite number of particles with arbitrary high spins and masses, one first crucial step to uncover its high energy behavior is to exactly calculate a class of SSA which contain the whole spectrum and are valid for all energies.
Recently the present authors constructed a class of such exact SSA which contain three tachyons and one arbitrary string state in the spectrum, or the Lauricella SSA (LSSA), in the 26D open bosonic string theory.
In chaper II of this review, we calculate the LSSA and express them in terms of the D-type Lauricella functions.As an application, we easily reproduce the string BCJ relation [19][20][21][22].For illustration of LSSA, we give two simple examples to demonstrate the complicated notations.We then proceed to show that the LSSA form an infinite dimensional representation of the SL(K + 3, C) group.For simplicity and as an warm up exercise, we will begin with the case of K = 1 or the SL(4, C) group.
In chapter III, we first show that there exist K + 2 recurrence relations among the D-type Lauricella functions.We then show that the corresponding 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. 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.We stress that these exact nonlinear relations among the exact LSSA are generalization of the linear relations among HSSA in the hard scattering limit conjectured by Gross.Finally we show that, for the first few mass levels, the Lauricella recurrence relations imply the validity of Ward identities derived from the decoupling of Lauricella ZNS.However these Lauricella Ward identities are not good enough to solve all the LSSA and express them in terms of one amplitude.
In chapter IV of this review, we calculate symmetries or relations among the LSSA of different string states at various scattering limits.These include the linear relations first conjectured by Gross [1][2][3][4][5] and later corrected and proved in [10,[12][13][14][15][16] in the hard scattering limit, the recurrence relations in the Regge scattering limit with associated SL(5, C) symmetry [23][24][25] and the extended recurrence relations (including the mass and spin dependent string BCJ relations) in the nonrelativistic scattering limit with associated SL(4, C) symmetry [26] discovered recently.

II. THE EXACT LSSA AND THEIR SL(K + 3, C) SYMMETRY
A. The exact LSSA One important observation of calculating the LSSA is to first note that SSA of three tachyons and one arbitrary string state with polarizations orthogonal to the scattering plane vanish.This observation will greatly simplify the calculation of the LSSA.In the CM frame, we define the kinematics as ) with M 2 1 = M 2 3 = M 2 4 = −2 and φ is the scattering angle.The Mandelstam variables are There are three polarizations on the scattering plane and they are defined to be [10,12] e T = (0, 0, 1), (2.5) ) ) ) the longitudinal polarization and e T = (0, 0, 1) the transverse polarization.For later use, we also define We can now proceed to calculate the LSSA of three tachyons and one arbitrary string states in the 26D open bosonic string theory.The general states at mass level with polarizations on the scattering plane are of the following form The (s, t) channel of the LSSA can be calculated to be [27] A where we have defined and (2.12) In Eq.(2.12), we have defined (2.14) The integer K in Eq.(2.10) is defined to be K = j {for all r T j =0} + j {for all r P j =0} + j {for all r L j =0} . (2.15) The D-type Lauricella function in Eq.(2.10) is one of the four extensions of the Gauss hypergeometric function to K variables and is defined to be where (α   which was used to calculate Eq.(2.10).

B. String BCJ relation as a by-product
Alternatively, by using the identity of the Lauricella function for b one can rederive the string BCJ relations [19][20][21][22] A This gives another form of the (s, t) channel amplitude Similarly, the (t, u) channel amplitude can be calculated to be To illustrate the complicated notations used in Eq.(2.10), we give two explicit examples of the LSSA in the following subsection.

Example one
We take the tensor state of the second vertex to be The LSSA in Eq.(2.10) can then be calculated to be where the arguments in F D are calculated to be and the order K in Eq.(2.15) is + j {for all r P j =0} + j {for all r L j =0} = 1 + 1 + 1 = 3. (2.26)

Example two
We take the tensor state to be The LSSA in Eq.(2.10) can be calculated to be (2.29) + j {for all r P j =0} + j {for all r L j =0} = (1 + 2 + 5 + 6) + 0 + 0 = 14. (2.31) In the following subsections, we discuss the exact SL(K + 3, C) symmetry of the LSSA.For simplicity, we will begin with the simple SL(4, C) symmetry with K = 1.

D. 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 levels N. As an example, for the case of K = 1 there are three types of LSSA To calculate the group representation of the LSSA for K = 1, we define [29] We see that the LSSA in Eq.(2.10) for the case of K = 1 corresponds to the case a = 1 = c, and can be written as We can now introduce the (K + 3) 2 − 1 = (1 + 3) 2 − 1 = 15 generators of SL(4, C) group [29,30] E α = a (x∂ x + a∂ a ) , and calculate their operations on the basis functions [29,30]  (2.36) It is important to 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 Indeed, if we redefine we discover that each of the following six triplets [29,30] E. The General SL(K + 3,C) Symmetry We are now ready to generalize the calculation of the previous section and calculate the group representation of the LSSA for general K.We first define [29] Note that the LSSA in Eq.(2.10) corresponds to the case a = 1 = c, and can be written as It is possible to extend the calculation of 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 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 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) [29] where, for simplicity, we have omitted those arguments in 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 }, let's redefine We discover that each of the following seven triplets [29] satisfies the commutation relations in Eq.(2.38).
Finally, in addition to Eq.(2.44), there is another compact way to write down the Lie algebra commutation relations of SL(K + 3, C).Indeed, one can check that the Lie algebra commutation relations of SL(K + 3, C) can be written as [29] [ with the following identifications

F. Discussion
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, states Indeed, there are more states in V with K ≥ 2 which are not LSSA either.We give one example in the following.For K = 2 there are six types of LSSA (ω = −1) (2.52) One can show that those states obtained from the operation by E β on either states in Eq.(2.50) to Eq.(2.52) are not LSSA.However, it can be shown in chap III that all states in V including those "auxiliary states" which are not LSSA stated above can be exactly solved by recurrence relations or the SL(K + 3, C) group and express them in terms of one amplitude.These "auxiliary states" and states with a = 1 or c = 1 in V may represent other SSA, e.g.SSA of two tachyon and two arbitrary string states etc. which will be considered in the near future.

III. SOLVING LSSA THROUGH RECURRENCE RELATIONS
In the previous section, the string scattering amplitudes of three tachyons and one arbitrary string states in the 26D open bosonic string theory.hasbeen obtained in term of the D-type Lauricella functions, i.e.LSSA in Eq.(2.10).The symmetry of the LSSA was also discussed by constructing the SL(K + 3, C) group for the D-type Lauricella functions It is natural to suspect that the LSSA are dependent each other due to the symmetry among them.In fact, we are able to show that all the LSSA are related to a single LSSA by the recurrence relations of the D-type Lauricella functions.
To solve all the LSSA, a key observation is that all arguments β m in the Lauricella functions F (K) D (α; β 1 , ..., β K ; γ; x 1 , ..., x K ) in the LSSA (2.10) are nonpositive integers.We will see that this plays a key role to prove the solvability of all the LSSA.The generalization of the 2 + 2 recurrence relations of the Appell functions to the K + 2 recurrence relations of the Lauricella functions was given in [31].One can use these K + 2 recurrence relations to reduce all the Lauricella functions F (K) D in the LSSA (2.10) to the Gauss hypergeometry functions 2 F 1 (α, β, γ).Then all the LSSA can be solved by deriving a multiplication theorem for the Gauss hypergeometry functions.
In this section, we will review these steps constructed in [31].
To simplify the notation, we will omit those arguments of which remain the same in the rest of the paper.Then the above K + 2 recurrence relations can be expressed as To proceed, we first consider the two recurrence relations from Eq.(3.10) for m = i, j with By shifting β i,j to β i,j −1 and combining the above two equations to eliminate the F (K) D (c + 1) term, we obtain the following key recurrence relation [31] x j F (K) FIG. 1: The neighborhood points in the figures are related by the recurrence relations.
One can repeatly apply Eq.(3.13) to the Lauricella functions in the LSSA in Eq.(2.10) and end up with an expression which expresses We can repeat the above process to decrease the value of K and reduce all the Lauricella functions in the LSSA to the Gauss hypergeometry functions F

B. Solving all the LSSA
In the last subsection, we have expressed all the LSSA in terms of the Gauss hypergeometry functions F (1) In this subsection, we further reduce the Gauss hypergeometry functions by deriving a multiplication theorem for them, and solve all the LSSA in terms of one single amplitude.
We begin with the Taylor's theorem By replacing y by (y − 1)x, we get the identity One can now use the derivative relation of the Gauss hypergeometry function where (α is the Pochhammer symbol, to obtain the following multiplication theorem It is important to note that the summation in the above equation is up to a finite integer |β| given β is a nonpositive integer for the cases of LSSA. In particular if we take x = 1 in Eq.(3.17), we get the following relation By using the following one of the 15 Gauss contiguous relations and set x = 1 which kills the second term of Eq.(3.19), we can reduce the argument β in 2 F 1 (α, β, c, 1) to β = −1 or 0 which corresponds to vector or tachyon amplitudes in the LSSA.This completes the proof that all the LSSA calculated in Eq.(2.10) can be solved through various recurrence relations of Lauricella functions.Moreover, all the LSSA can be expressed in terms of one single four tachyon amplitude.

C. Examples of solving LSSA
For illustration, in this subsection, we calculate the Lauricella functions which correspond to the LSSA for levels K = 1, 2, 3.
For K = 1 there are three type of LSSA (α For K = 2 there are six type of LSSA (ω = −1) (3.28) For K = 3, there are ten type of LSSA (ω All the LSSA for K = 2, 3 can be reduced through the recurrence relations in Eq. (3.13) and expressed in terms of those of K = 1.Furthermore, all resulting LSSA for K = 1 can be further reduced by applying Eq.(3.18) to Eq.(3.19) and finally expressed in terms of one single LSSA.

D. SL(K + 3,C) Symmetry and Recurrence Relations
In this subsection, we are going to use the recurrence relations of the D-type D (α; β 1 , ..., β K ; γ; x 1 , ..., x K ) to reproduce the Cartan subalgebra and simple root system of SL(K + 3, C) with rank K + 2. We will first review the case of SL(4, C) symmetry group, and then extend it to the general case of SL(K + 3, C) Symmetry.

SL(4, C) Symmetry
We first relate the SL(4, C) group to the recurrence relations of F which 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.(2.33), the first recurrence relation in Eq.(3.39) can be rewritten as By using the identity Finally we check the operation of E γ .Note that Eq.(3.40) can be written as which gives After some simple computation, we get which is consistent with the operation of E γ in Eq.(2.36).
We thus have shown that the extended LSSA f b ac (α; β; γ; x) in Eq.(2.33) 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.

SL(K + 3, C) Symmetry
The K + 2 fundamental recurrence relations among F (K) D (α; β; γ; x) or the Lauricella functions.havebeen listed in Eqs.(3.8-3.10).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.(2.39), the first type of recurrence relation in Eq.(3.8) can be rewritten as which means The second type of recurrence relation in Eq.(3.9) can be rewritten as Eq.(3.68) can be written as The third type of recurrence relation in Eq.(3.10) can be rewritten as (m = 1, 2, ...K) which gives In the above calculation, we have used the definition and operation of E βmγ in Eq. (2.41) and Eq.(2.42), respectively.
Eq.(3.72) can be written as can be rewritten as In the above calculation, we have used the definitions and operations of E β k and E α in Eq.(2.41) and Eq.(2.42), respectively.
The K + 2 equations in Eq.(3.65),Eq.(3.69) and Eq.(3.73) together with K + 2 equations for the operations {E α , E β k , E γ } are equivalent to the Cartan subalgebra and the simple root system of SL(K + 3, C) with rank K + 2. 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 Lie algebra from the recurrence relations and vice versa.
In the previous subsections, it was shown that [32] 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.

E. 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) [10,[12][13][14][15][16] and the corresponding stringy Ward identities at the fixed angle regime, and Regge zero norm states (RZNS) [24,25] 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 [10,[12][13][14][15][16].

The Lauricella zero norm states
We will consider the set of Ward identities of the LSSA with three tachyons and one arbitrary string states.Thus we only need to consider polarizations of the tensor states on the scattering plane since the amplitudes with polarizations orthogonal to the scattering plane vanish.
There are two types of zero norm states (ZNS) in the old covariant first quantum string spectrum, Type I : We begin with the case of mass level M 2 = 2.There is a type II ZNS and a type I ZNS The three polarizations defined in Eq.(2.5) to Eq.(2.7) of the 2nd tensor state with momentum k 2 on the scattering plane satisfy the completeness relation where µ, ν = 0, 1, 2 and α, β = P, L, T .and The type II ZNS in Eq.(3.78) gives the LZNS Type I ZNS in Eq.(3.79) gives two LZNS . LZNS in Eq.(3.82) and Eq.(3.83) correspond to choose θ µ = e T and θ µ = e L respectively.In conclusion, there are 3 LZNS at the mass level M 2 = 2.
At the second massive level M 2 = 4, there is a type I scalar ZNS 17 4 (k a symmetric type I spin two ZNS where For the type I spin two ZNS in Eq.(3.85), we define e α µ e β ν u αβ . (3.89) The transverse and traceless conditions on θ µν then implies u P P = u P L = u P T = 0 and u P P − u LL − u T T = 0, (3.90) which gives two LZNS (α As the examples, we calculate the Ward identities associated with the LZNS in Eq. (3.82) and Eq.(3.83).The calculation is based on processes (A) and (B).By using Eq.(2.10), the Ward identities we want to prove are or, using the kinematics variables we just defined, F (3.104) The Eq.(3.103) and Eq.(3.104) can be explicitly proved as and D (α; −2; α − γ − 1; 1) where we used of Eq.(3.13) in the process (A) to get Eq.(3.105) and Eq.(3.107), and Eq.(3.18) in the process (B) to get Eq.(3.108).The last last lines of the above equations are obtained by using Eq.(3.19).

F. Summary
In this section 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.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 [32].
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.

IV. RELATIONS AMONG LSSA IN VARIOUS SCATTERING LIMITS
In this section, we will show that there exist relations or symmetries among SSA of different string states at various scattering limits.In the first subsection, we will show that the linear relations [1][2][3][4][5] conjectured by Gross among the hard SSA (HSSA) at each fixed mass level in the hard scattering limit can be rederived from the LSSA.These relations reduce the number of independent HSSA from ∞ down to 1.
In the second subsection, we will show that the Regge SSA (RSSA) in the Regge scattering limit can be rederived from the LSSA.All the RSSA can be expressed in terms of the Appell functions with associated SL(5, C) symmetry [23][24][25].Moreover, the recurrence relations of the Appell functions can be used to reduce the number of independent RSSA from ∞ down to 1.
Finally, in the nonrelativistic scattering limit, we show that the nonrelativistic SSA (NSSA) and various extended recurrence relations among them an be rederived from the LSSA.In addition, we will also derive the nonrelativistic level M 2 dependent string BCJ relations which are the stringy generalization of the massless field theory BCJ relation [33] to the higher spin stringy particles.These NSSA can be expressed in terms of the Gauss hypergeometry functions with associated SL(4, C) symmetry [23][24][25].
A. Hard scattering limit-Proving Gross conjecture from LSSA In this subsection, we will show that the linear relations conjectured by Gross [1][2][3][4][5] in the hard scattering limit can be rederived from the LSSA.First, we briefly review the results discussed in [17,18] for the linear relations among HSSA.It was first observed that for each fixed mass level N with M 2 = 2(N − 1) the following states are of leading order in energy at the hard scattering limit [14,15] |N, 2m, q ≡ (α T Note that in Eq.(4.1)only even powers 2m in α L −1 [10][11][12] survive and the naive energy order of the amplitudes will drop by an even number of energy powers in general.The HSSA with vertices corresponding to states with an odd power in (α L −1 ) 2m+1 turn out to be of subleading order in energy and can be ignored.By using the stringy Ward identities or decoupling of two types of zero norm states (ZNS) in the hard scattering limit, the linear relations among HSSA of different string states at each fixed mass level N were calculated to be [14,15] Exactly the same result can be obtained by using two other techniques, the Virasoro constraint calculation and the corrected saddle-point calculation [14,15].The calculation of of Eq.(4.2) was first done for one high energy vertex in Eq.(4.1) and can then be easily generalized to four high energy vertices.In the decoupling of ZNS calculations at the mass level M 2 = 4, for example, there are four leading order HSSA [10,12] A T T T : which are proportional to each other.While the saddle point calculation of [5] gave , and A LLT = 0 which are inconsistent with the decoupling of ZNS or unitarity of the theory.Indeed, a sample calculation was done [10,12] to explicitly verify the ratios in Eq.(4.3).
One interesting application of Eq.(4.2) was the derivation of the ratio between A (N,2m,q) st and A (N,2m,q) tu in the hard scattering limit [19] A where A (N,2m,q) tu is the corresponding (t, u) channel HSSA.
Eq.(4.4) was shown to be valid for scatterings of four arbitrary string states in the hard scattering limit and was obtained in 2006.This result was obtained earlier than the discovery of four-point field theory BCJ relations in [33] and "string BCJ relations" in Eq.(2.19) [20][21][22].In contrast to the the calculation of string BCJ relations in [21,22] which was motivated by the field theory BCJ relations in [33], the result of Eq.(4.4) was inspired by the calculation of hard closed SSA [19] by using KLT relation [34].More detailed discussion can be found in [18,19].Now we are ready to rederive Eq.(4.1) and Eq.(4.2) from the LSSA in Eq.(2.10).The relevant kinematics are where E and φ are CM frame energy and scattering angle respectively.One can calculate The LSSA in Eq.(2.10) reduces to As was mentioned earlier 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 −1 r L 1 operator in Eq.(2.9).So let's pay attention to the corresponding summation and write which reproduces the ratios in Eq.(4.2), and is consistent with the previous result [10][11][12][13][14][15][16].

B. Regge scattering limit
There is another important high energy limit of SSA, the RSSA in the Regge scattering limit.The relevant kinematics in the Regge limit are ) where F 1 is the Appell function.Eq.(4.20) agrees with the result obtained in [25] previously.
The recurrence relations of the Appell functions can be used to reduce the number of independent RSSA from ∞ down to 1.One can also calculate the string BCJ relation in the Regge scattering limit, and study the extended recurrence relation in the Regge limit [26].

C. Nonrelativistic Scattering Limit and Extended Recurrence Relations
In this section, we discuss nonrelativistic string scattering amplitudes (NSSA) and the extended recurrence relations among them.In addition, we will also derive the nonrelativistic level M 2 dependent string BCJ relations which are the stringy generalization of the massless field theory BCJ relation [33] to the higher spin stringy particles.
We will take the nonrelativistic string scattering limit or | k 2 | << M 2 limit to calculate the mass level and spin dependent low energy SSA.In constrast to the zero slope α ′ limit used in the literature to calculate the massless Yang-Mills couplings [37,38] for superstring and the three point ϕ 3 scalar field coupling [39][40][41] for the bosonic string, we found it appropriate to take the nonrelativistic limit in calculating low energy SSA for string states with both higher spins and finite mass gaps.

Nonrelavistic LSSA
In this subsection, we first calculate the NSSA from the LSSA.In the nonrelativistic limit where ǫ = (M 1 + M 2 ) 2 − 4M 2 3 and M 1 = M 3 = M 4 = M tachyon .One can easily calculate where p 1 is an arbitrary integer.More extended recurrence relations can be similarly derived.
The existence of these low energy stringy symmetries comes as a surprise from Gross's high energy symmetries [1,3,5] point of view.Finally, in contrast to the Regge string spacetime symmetry which was shown to be related to SL(5, C) of the Appell function F 1 , here we found that the low energy stringy symmetry is related to SL(4, C) [30] of the Gauss hypergeometry functions 2 F 1 .

D. Summary
In this section, we rederive from the LSSA the relations or symmetries among SSA of different string states at three different scattering limits.We first reproduce the linear relations [14,15] of the HSSA from the LSSA in the hard scattering limit.We also obtain Appell functions F 1 and Gauss hypergeometric functions 2 F 1 with SL(5, C) and SL(4, C) symmetry in the Regge and the nonrelativistic limits respectively.In contrast to the linear relations in the hard scattering limit, we obtain extended recurrence relations for the cases of RSSA and NSSA.These two classes of recurrence relations are closely related to those of the LSSA with K = 2 and K = 1 respectively.In the end, we also show that with the nonrelativistic string BCJ relations [20], the extended recurrence relations we obtained can be used to connect SSA with different spin states and different channels.
exact LSSA and their SL(K + 3, C) Symmetry A. The exact LSSA B. String BCJ relation as a by-product C. Two simple examples of the LSSA 1. Example one 2. Example two D. The SL(4, C) Symmetry E. The General SL(K + 3,C) Symmetry F. Discussion III.Solving LSSA through Recurrence relations A. Recurrence Relations of the LSSA B. Solving all the LSSA C. Examples of solving LSSA D. SL(K + 3,C) Symmetry and Recurrence Relations 1. SL(4, C) Symmetry 2. SL(K + 3, C) Symmetry E. Lauricella Zero Norm States and Ward Identities 1.The Lauricella zero norm states 2. The Lauricella Ward identities F. Summary IV.Relations among LSSA in various scattering limits A. Hard scattering limit-Proving Gross conjecture from LSSA B. Regge scattering limit C. Nonrelativistic Scattering Limit and Extended Recurrence Relations 1. Nonrelavistic LSSA 2. Nonrelativistic string BCJ relations 3. Extended recurrence relations in the nonrelativistic scattering limit D. Summary I. INTRODUCTION
.60) Using the definition and operation of E αγ in Eq.(2.35), we obtain f b ac (α; β; γ; x) − 1 c f b ac (α; β; γ + 1; x) − E αγ ac (β − γ) f b ac (α; β; γ; x) = 0, which gives .73) It is important to see that Eq.(3.65),Eq.(3.69) and Eq.(3.73) imply the last three equations of Eq.(2.42) or the Cartan subalgebra of SL(K + 3, C) as expected.In addition to the Cartan subalgebra, we need to derive the operations of the {E α , E β k , E γ } from the recurrence relations.With the operations of Cartan subalgebra and {E α , E β k , E γ }, one can reproduce the whole SL(K + 3, C) algebra.The calculations of E α and E γ are straightforward and are similar to the case of SL(4, C) in the previous section.Here we present only the calculation of E β k .The recurrence relation in Eq.(3.8) .77) While type I ZNS exists at any spacetime dimension, type II ZNS only exists at D = 26.
.87) Note that Eq.(3.86) and Eq.(3.87) 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.(3.84) gives the LZNS 25(α P −1 ) 3

. 10 ) 1 =
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 2m and r L 2 = q, can be calculated to be A (N −2m−2q,2m,q) st