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Zero-norm states and High-energy Symmetries of String Theory

We derive stringy Ward identities from the decoupling of two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. These Ward identities are valid to all energy and all loop orders in string perturbation theory. The high-energy limit of these stringy Ward identities can then be used to fix the proportionality constants between scattering amplitudes of different string states algebraically without referring to Gross and Mende's saddle point calculation of high-energy string-loop amplitudes. As examples, all Ward identities for the mass level 4 and 6 are derived, their high-energy limits are calculated and the proportionality constants between scattering amplitudes of different string states are determined. In addition to those identified before, we discover some new nonzero components of high-energy amplitudes not found previously by Gross and Manes. These components are essential to preserve massive gauge invariances or decouple massive zero-norm states of string theory. A set of massive scattering amplitudes and their high energy limits are calculated explicitly for each mass level to justify our results.


I. INTRODUCTION
In this paper, all Ward identities for the mass level M 2 = 4, 6 will be derived, their high-energy limits are calculated and the the proportionality constants between scattering amplitudes of different string states are determined directly from these Ward identities.
General formula of high-energy amplitudes for arbitrary mass levels will be given in terms of those of tachyons. In addition to those identified before, we discover some new nonzero components of high-energy amplitudes at each mass level not found previously by Gross and Manes [10]. These components are essential to preserve massive gauge invariances or decouple massive zero-norm states of string theory. A set of massive scattering amplitudes and their high energy limits are calculated explicitly for each mass level M 2 = 4, 6 to justify our results. This paper is organized as following. In section II, we derive stringy Ward identities for the mass level M 2 = 4 [11], and then take high-energy limits of them to determine the proportionality constants between scattering amplitudes of different string states algebraically. At the subleading order energy, one finds 6 unknown amplitudes and 4 equations. Presumably, they are not proportional to each other or the proportional coefficients do depend on the scattering angle φ CM . This result will be confirmed at section III. In section III, the high energy limits of a set of string-tree level amplitudes with one tensor at mass level M 2 = 4 and three tachyons are explicitly calculated to justify the results of section II. The whole program is then generalized to mass level M 2 = 6 in section IV. We make a comparision of our results with those of Gross and Manes [10] in section V. Finally a brief conclusion is given in section VI.

II. HIGH-ENERGY STRINGY WARD IDENTITIES OF MASS LEVEL M 2 = 4
In the OCFQ spectrum of open bosonic string theory, the solutions of physical states conditions include positive-norm propagating states and two types of zero-norm states which were neglected in the most literature. They are [12] Type I : L −1 |x , where L 1 |x = L 2 |x = 0, L 0 |x = 0; (2.1) Type II : (L −2 + 3 2 Equations (2.1) and (2.2) can be derived from Kac determinant in conformal field theory.
While type I states have zero-norm at any spacetime dimension, type II states have zero-norm only at D=26. The existence of type II zero-norm states signals the importance of zero-norm states in the structure of the theory of string. In the first quantized approach of string theory, the stringy on-shell Ward identities are proposed to be (for our purpose we choose four-point amplitudes in this paper) states was given in [5]. A more thorough understanding of the solution of these equations and their relation to space-time ω ∞ symmetry of toy D=2 string was discussed in [8]. For our purpose here, there are four zero-norm states at mass level M 2 = 4, the corresponding Ward identities were calculated to be [11] k µ θ νλ T (µνλ) where θ µν is transverse and traceless, and θ ′ λ and θ λ are transverse vectors. In each equation, we have chosen, say, v 2 (k 2 ) to be the vertex operators constructed from zero-norm states and k µ ≡ k 2µ . Note that eq.(2.6) is the inter-particle Ward identity corresponding to D 2 vector zero-norm state obtained by antisymmetrizing those terms which contain α µ −1 α ν −2 in the original type I and type II vector zero-norm states.   where µ, ν = 0, 1, 2 and α, β = P, L, T. Diag η µν = (−1, 1, 1).One can now transform all µ, ν coordinates in eqs.(2.4)-(2.7) to coordinates α, β. For eq(2.4), we have θ µν = e µ L e ν L − e µ T e ν T or θ µν = e µ L e ν T + e µ T e ν L . In the high energy E → ∞, fixed angle φ CM limit, one identifies e P = e L and eq. (2.4) gives ( we drop loop order χ here to simplify the notation) In eqs (2.12) and (2.13), we have assigned a relative energy power for each amplitude. For each longitudinal L component, the order is E 2 and for each transverse T component, the order is E. This is due to the definitions of e L and e T in eqs (2.9) and (2.10), where e L got one energy power more than e T .By eq. (2.12), the E 6 term of the energy expansion for T LLL is forced to be zero. As a result, the possible leading order term is E 4 . Similar rule applies to T LLT in eq(2.13). For eq(2.5), we have θ ′µ = e µ L or θ ′µ = e µ T and one gets, in the high energy limit, For the D 2 Ward identity, eq.(2.6), we have θ µ = e µ L or θ µ = e µ T and one gets, in the high energy limit, It is important to note that T [LL] in eq.(2.16) originate from the high energy limit of T [P L] , and the antisymmetric property of the tensor forces the leading E 4 term to be zero. Finally the singlet zero norm state Ward identity, eq.(2.7), imply, in the high energy limit, Note that these proportionality constants are, as conjectured by Gross [4], independent of the scattering angle φ CM and the loop order χ of string perturbation theory. They are also independent of particles chosen for vertex v 1,3,4 . Most importantly, we now understand that they originate from zero-norm states in the OCFQ spectrum of the string! The subleading order amplitudes corresponding to E 2 appear in eqs.(2.12), (2.14), (2.16) and (2.18). One has 6 unknown amplitudes and 4 equations. Presumably, they are not proportional to each other or the proportional coefficients do depend on the scattering angle φ CM . We will justify this point later in our sample calculation in section III. Our calculation here is purely algebraic without any integration and is independent of saddle point calculation in [3,4,10]. It is important to note that our result in eq.(2.19) is gauge invariant as it should be since we derive it from Ward identities (2.4)-(2.7). On the other hand, the result obtained in [10] with T 3 T T T ∝ T 3 [LT ] , and T 3 LLT = 0 in the leading order energy at this mass level is, on the contrary, not gauge invariant. In fact, with T 3 LLT = 0, an inconsistency arises, for example, between eqs. (2.13) and (2.15). We give one example here to illustrate the meaning of the massive gauge invariant amplitude. To be more specific, we will use two different gauge choices to calculate the high-energy scattering amplitude of symmetric spin three state. The first gauge choice is In the high-energy limit, using the helixity 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, we get The second gauge choice is In the high-energy limit, similar calculation gives It is now easy to see that the first and second terms of eq.(2.21) will not contribute to the high-energy scattering amplitude of the symmetric spin three state due to the spin term of eq.(2.23), which corresponds to the amplitude T 3 LLT , was missing in the calculation of Ref [10]. We will discuss this issue in section V.
To further justify our result, we give a sample calculation in section III.

III. A SAMPLE CALCULATION OF MASS LEVEL M 2 = 4
In this section, we give a detailed calculation of a set of sample scattering amplitudes to explicitly justify our results presented in section II. Since the proportionality constants in eq.(2.19) are independent of particles chosen for vertex v 1,3,4 . For simplicity, we will choose them to be tachyons. For the string-tree level χ = 1, with one tensor v 2 and three tachyons v 1,3,4 , all scattering amplitudes of mass level M 2 = 4 were calculated in [11]. They are ( In deriving eqs. (3.1) to (3.4), we have made the SL(2, R) gauge fixing by choosing To calculate the high energy expansions (s, t → ∞, s t = fixed ) of these scattering amplitudes, one needs the followong energy expansion formulas where ξ = sin φ CM 2 and η = cos φ CM 2 . The high-energy expansions of Mandelstam variables are given by We can now explicitly calculate all amplitudes in eq.(2.19). After some algebra, we get 14) ) is the high-energy limit of with s + t + u = 2n − 8, and we have calculated it up to the next leading order in E. We thus have justified eq.(2.19) with T 3 T T T = −8E 9 T (3) sin 3 φ CM and T 5 LLT = 0. We have also checked that T 6 LLL = T 4 LLL = T 4 LT T = T 4 (LL) = 0 as claimed in section II. Note that, unlike the leading E 9 order, the angular dependences of E 7 order are different for each amplitudes. The subleading order amplitudes corresponding to T 2 (E 8 order) appear in eqs.(2.12), (2.14), (2.16) and (2.18).
One has 6 unknown amplitudes. An explicit sample calculation gives In this section we generalize the calculation of sections II and III to mass level M 2 = 6.
There are four positive-norm physical propagating states at this mass level [13], a totally symmetric spin four state, a mixed symmetric spin three state, a symmetric spin two state and a scalar state. There are nine zero-norm states at this mass level. One can use the simplied method [5] to calculate all of them. The spin three and spin two zero-norm states are (from now on, unless otherwise stated, each spin polarization is assumed to be transverse, traceless and is symmetric with respect to each group of indices) where α µν −1 = α µ −1 α ν −1 etc. There are two type I degenerate vector zero-norm states which can be calculated as following (4.5) The L 1 and L 2 constraints of equation (2.1) give The type II vector zero-norm state is The type I singlet zero-norm state was calculated to be [5] Ansatz The L 1 and L 2 constraints of equation ( Finally the type II singlet zero-norm state can be calculated to be where a : b : c = 75 : 39 : 19. We are now ready to calculate the high-energy Ward identities. The high-energy limit of stringy Ward identity corresponding to eq.(4.1) are where T µνλ is the amplitude corresponding to α (µν Eqs.(4.12) and (4.13) correspond to θ µνλ = −e µ L e ν L e λ L +3e µ (L e ν T e λ T ) and θ µνλ = −3e µ (L e ν L e λ T ) +e µ T e ν T e λ T respectively. Similarly eq.(4.2) gives where T µν is the amplitude corresponding to α µ −1 α ν −3 and T µν,λ is the amplitude corresponding to mixed symmetric part of α µν −1 α λ −2 , that is, first symmetrizing w.r.t. µν and then antisymmetrizing w.r.t. µλ. This is exactly the amplitude for the positive-norm mixed symmetric spn three state. The type I symmetric spin two zero-norm state eq.(4.3) gives, in the high-energy limit, where T µν is the amplitude corresponding to α µν −2 . The E 6 order of T 6→4 P L,L in eq. (4.15) is forced to be zero in the high-energy limit (e P = e L ) due to the antisymmetric property of the tensor T µν,λ w.r.t.µλ. It is important to note that in deriving eqs.(4.15) and (4.16), we have made the following irreducible decomposition of the term in eq.(4.3). The first term with totally symmetric spin three index corresponds to the gauge artifact of the positive-norm spin four state, and the mixed symmetric tensor structure of the second term is exactly the same as that of the positive-norm spin three state. In general, there are three other possible mixed symmetric spin three terms, which do not appear in eq. (4.17). This is a nontrivial consistent check of zero-norm states spectrum in the OCFQ string. We will see similar mechanism happens in our later calculation. The type II symmetric spin two zero-norm state eq.(4.4) gives, in the high-energy limit, Two type I vector zero-norm states eq.(4.7) give, in the high-energy limit, where T µ is the amplitude corresponding to α µ −4 . Note that T 6→4 LP,P in eq. (4.20) is identical to T 6→4 LL,P in eqs.(4.15) and (4.18) in the high-energy limit. However, T 2 LP,P and T 2 LL,P can be different. Also T 5 P P,T in eq.(4.21) is zero since it equals to T 5 LL,T in eq.(4.14), which is zero, in the high-energy limit. However, T 3 P P,T and T 3 LL,T can be different. In deriving eqs.(4.20) and (4.21), in addition to (4.17), one needs another projection formula Again, the first term of eq.(4.22) with totally symmetric spin three index corresponds to the gauge artifact of the positive-norm spin four state, and the mixed symmetric tensor structure of the second term is exactly the same as that of the positive-norm spin three state. This is another consistent check of zero-norm states spectrum in the OCFQ string.
In the following, we will use eqs.(4.17) and (4.22) whenever they are needed. Type II vector zero-norm state eq.(4.6) gives, in the high-energy limit, (4.24) Note that T 5 T L,L in eq.(4.23) is identical to T 5 T P,P in eq. (4.21) in the high-energy limit. Finally, type I and type II singlet zero-norm states give, in the high-energy limit,  It is very important to note that in the E 4 order, one gets one more amplitude T 4 LP,P , and in the E 3 order, one gets another amplitude T 3 P P,T described after eq.(4.21). It can be checked by eqs.(4.12)-(4.26) that all the amplitudes of orders E 8 E 7 E 6 and E 5 are zero. So the real leading order amplitudes correspond to E 4 , which appear in eqs. Note that there are two equations for (4.21). One has 7 equations with 9 amplitudes, . Presumably, they are not proportional to each other or the proportional coefficients do depend on the scattering angle φ CM . Our calculation here is again purely algebraic without any integration and is independent of saddle point calculation in [3,4,10]. It is important to note that our result in eq.(4.27) is gauge invariant. On the other hand, the result obtained in [10] with increases much faster than that of positive-norm states [5]. To further justify our result, we give a sample calculation in the following.
Since the proportionality constants in eq.(4.27) are independent of particles chosen for vertex v 1, 3,4 . For simplicity, we will choose them to be tachyons. For the string-tree level χ=1, with one tensor v 2 and three tachyons v 1,3,4 , all scattering amplitudes for mass level M 2 = 6 were explicitly calculated in [14]. They are We can now explicitly calculate all amplitudes in eq.(4.27). After a lengthy algebra, we have justified eq.(4.27) with T 4 T T T T = 16E 12 sin 4 φ CM T (4) in the high-energy limit. We have also checked that T 8 LLLL = T 6 LLLL = T 6 LLL = T 6 T T LL = T 6 LP,P = T 6 LL,P = 0 and T 4 LP,P = T 4 LL,P as claimed above. The calculation of T LLLL ,for example, gives where n is the number of T and T (0) is the high energy four tachyons amplitude. As a result, all high-energy string scattering amplitudes can be expressed in terms of those of tachyons.

V. A COMPARISION WITH SADDLE POINT CALCULATION
To compare our results with Ref [10], we briefly review the works in [3,4,10]. In Ref [10], it was shown that the high-energy, fixed angle scattering amplitudes of oriented open strings can be obtained from those of closed strings calculated by Gross and Mende [3] by using the reflection principle. First, from eq. (2.3), one notes that the high-energy limit α ′ → ∞ is equivalent to the semi-classical limit of first-quantized string theory. In this limit, the closed string G-loop scattering amplitudes is dominated by a saddle point in the moduli space − → m.
For the oriented open string amplitudes, the saddle point configuration can be constructed from an associated configuration of the closed string via reflection principle. It was also found that the Euler number χ of the oriented open string saddle is always where G is the genus of the associated closed string saddle. Thus the integral in eq. (2.3) is dominated in the α ′ → ∞ limit by an associated G-loop closed string saddle point in X µ , − → m i and ξ i . The closed string classical trajectory at G-loop order was found to behave at the saddle point as [3] X µ c1 (z) = which leads to the χ-th order open string four-tachyon amplitude ).
to extremize E G if we neglect the mass of the tachyons in the high-energy limit. For excited string states, it was found that only polarizations in the plane of scattering will contribute to the amplitude at high energy. To leading order in the energy E, the products of e T and e L with ∂ n X are given by [10] e T · ∂ n X ∼ i(−) n (n − 1)! λ n E sin φ CM , n > 0; (5.3) e L · ∂ n X ∼ i(−) (n−1) (n − 1)! λ n E 2 sin 2 φ CM 2m 2 n−2 l=0 λ l , n > 1; (5.4) e L · ∂ n X ∼ 0, n = 1, (5.5) where m 2 is the mass of the particle. Now, we would like to point out that naive uses of eqs.(5.3) to (5.5) will miss some high-energy amplitudes and will give, for example, a wrong result T 3 LLT = 0 [10] since e L · ∂X ∼ 0.This is inconsistent with our result eq.(2.19) or eq.(3.14). The missing terms can be seen as following. We will use the M 2 = 4 string-tree χ = 1 amplitude T LLT to illustrate our point. Let's first use the path integral calculation dx i < e ik 1 X e L · ∂Xe L · ∂Xe T · ∂Xe ik 2 X e ik 3 X e ik 4 X >, (5.6) which is similar to the calculation of moments of the Gaussian integral 2a term that was missing in the argument of section 6 of Ref [10]. Similar situations happen for n 3 and even more terms were missed. The argument can be easily generalized to − → b ∈ R 3 in the space of helixity decomposition. Eq.(5.6) corresponds to the case of n = 3. It can be checked that some terms with the same energy order as T T T T survive in the calculation of eq.(5.6). They will be missing if one misuses eqs.(5.3) to (5.5). Similar wrong calculations will suppress many other should be non-zero high-energy amplitudes at mass level M 2 = 6 stated after eq.(4.27). Another way to calculate eq.(5.6) is to use Wick theorem. Again, naive uses of eqs.(5.3) to (5.5) will miss some high-energy amplitudes which correspond to, for example, the contraction of e ik 1 X with (e L · ∂X)(e L · ∂X). We stress here that eqs.(5.1) to (5.5) are still valid as they stand.

VI. CONCLUSION
We have shown that the physical origin of high-energy symmetries and the proportionality constants in eqs. (2.19) and (4.27) are from zero-norm states in the OCFQ spectrum.
Other related approaches of high-energy stringy symmetries can be found in [15]. The most challenging problem remained is the calculation of algebraic structure of these stringy symmetries derived from the complete zero-norm state solutions of eqs. (2.1) and (2.2) with arbitrarily high spins. Presumably, it is a complicated 26D generalization of ω ∞ of the simpler toy 2D string model [8]. Our calculation in eqs. (2.19) and (4.27) are, similar to the toy 2D string, purely algebraic without any integration which signal the powerfulness of zero-norm states and symmetries they imply. The results presented in this paper can be