Spin polarization independence of hard polarized fermion string scattering amplitudes

We calculate a class of polarized fermion string scattering amplitudes (PFSSA) at arbitrary mass levels. We discover that, in the hard scattering limit, the functional forms of the non-vanishing PFSSA at each fixed mass level are independent of the choices of spin polarizations. This result justifies and extends Gross conjecture on high energy string scattering amplitudes to the fermionic sector. In addition, this peculiar property of hard PFSSA is to be compared with the usual spin polarization dependence of the hard polarized fermion field theory scatterings.


I. INTRODUCTION
One important characteristic of string scattering amplitudes (SSA) is its very soft exponential fall-off behavior in the hard scattering limit. This behavior is closely related to the existence of infinite linear relations among hard SSA of different string states at each fixed mass level. Moreover, these linear relations are so powerful that they can be used to solve all hard SSA and express them in terms of one amplitude. This means that there is only one hard SSA f (E, θ) at each fixed mass level which is very different from the usual spin dependence of hard fermion field theory scatterings. This important high energy symmetry of string theory was first conjectured by Gross [1][2][3] and later corrected and proved by using the decoupling of zero norm states [4] in [5][6][7][8][9][10]. For more details, see the recent review [11].
However, all calculations that have been done so far are only for boson SSA of either the bosonic string theory [5][6][7][8][9] or the NS sector (both GSO even and odd) of the fermionic string theory [10]. So it will be important and of interest to see whether one can extend Gross conjecture to the R sector of the fermionic string theory.
Since it is a nontrivial task to construct the general massive fermion string vertex operators, as the first step in this letter, we choose to calculate polarized fermion string scattering amplitudes (PFSSA) at arbitrary mass levels which involve the leading Regge trajectory fermion string state of the R sector (α ′ ≡ 1 2 ) [12] χ α (m 1 ...m n−1 mn) i∂X m 1 · · · i∂X m n−1 (i∂X mn δ γ α − in which the tensor-spinor wavefunction χ α (m 1 ...m n−1 mn) satisfies the on-shell conditions which include a γ traceless condition. One of the reason for choosing this leading Regge trajectory state is that the corresponding vertex operator has been constructed in the literature [12]. The construction was mainly based on the complete construction of the first massive level states for both NS and R sectors [13].
On the other hand, since Gross conjecture was shown to be valid for both GSO even and odd states in the NS sector [10], for simplicity in this paper, we are going to ignore the GSO projection, and the other three string states in the SSA will be chosen to be one massless fermion and two tachyon states (GSO odd).
The state in Eq.(1.1) is a combination of (α i −1 ) n |α R and (α i −1 ) n−1 (d j −1 ) |ᾱ R (in the lightcone gauge language). For the case of n = 1 [13], for example, the vector-spinor χ α µ is a 10D Majorana spinor that forms an irreducible massive representation of the Lorentz group. In the corresponding four dimensional case, the vector-spinor χ α µ tranforms as the product of a four-vector and a Dirac spinor, and satisfies the Rarita-Schwinger equations which is the case of spin s = 3 2 field equation of the more general Bargmann-Wigner equation with spin s ≥ 1 2 . Note that Eq.(1.4) is similar to the γ traceless condition in Eq.(1.2). It was shown for the bosonic SSA that at each fixed mass level M 2 = 2(N − 1) only tensor states of the following form [8,9] are of leading order in energy in the hard scattering limit. In Eq.(1.5), e P = 1 M (E, k, 0) = k 2 M 2 the momentum polarization, e L = 1 M (k, E, 0) the longitudinal polarization and e T = (0, 0, 1) the transverse polarization are the three polarizations on the scattering plane [5,6]. In the hard scattering limit, one can identify e P = e L [5,6]. It was remarkable to discover that all the hard bosonic SSA at each fixed mass level share the same functional forms with the following ratios [8,9] T (N,2m,q) Thus there is only one hard SSA T (N,0,0) at each fixed mass level. For the leading Regge trajectory states we are considering in this paper, we set q = 0.
In this paper we will be mainly concerning with the spinor polarizations in the SSA calculation. So, for simplicity, we will be writing where u α satisfies the 10D Dirac type equation in Eq.(1.3). For the leading hard SSA of the Regge trajectory states, one can choose to put all the tensor polarizations ǫ m 1 = · · · = ǫ mn = ǫ T .

II. POLARIZED FERMION STRING SCATTERING AMPLITUDES (PFSSA)
In this section, we will calculate the PFSSA with the following four vertex operators in the 10D open superstring theory: the massless spinor the massive spinor (α ′ = 1 2 ) and two tachyons In the above, we have chosen the total ghost charges sum up to −2. The correlators of the worldsheet boson X µ , worldsheet fermion ψ µ , spin field S A and ghost field φ are The PFSSA we want to calculate can be written as (2.10) Let's calculate A 1 first, whose correlator can be written as The first correlators in Eq.(2.12) was calculated in Eq.(2.6) [12], and the other two can be calculated to be and (2.14) For the s − t channel amplitude, we take z 1 = 0, z 3 = 1, z 4 → ∞ (0 ≤ z 2 ≤ 1) and, for simplicity, set all ǫ 1 = · · · = ǫ n = ǫ, we get Finally the integration in A 1 can be performed [14] and we obtain Similar techanique can be used to calculate A 2 whose correlator can be written as The second and the third correlators in Eq.(2.19) were calculated in Eq.(2.13) and Eq.(2.14) respectively. The first correlator can be written as where the composite operators Kβ λ was defined to be [12] Kβ λ = ψ 2λ γβ γ ρ ψ ρ 2 S 2γ . (2.21) The correlation functions containing spin fields S α and the composite operators Kβ λ can be found in [12].

III. HARD SCATTERING LIMIT
In this section, we will calculate the hard scattering limit of the PFSSA we obtained in the previous section. We will concentrate on the spinor polarizations and ignore the parts of the tensor polarizations. To do so we need to solve 10D Dirac equation and calculate explicitly the two factors in Eq.(2.16) and Eq.(2.25) We will follow the definition in [16] to calculate the 10D Dirac matrices. The ground states of the R sector are degenerate and can be labeled by where each of the s a is ± 1 2 in the s basis. To simplify the notation, we will ignore the factor  Then in d = 2k + 2, where γ µ is the 2 k × 2 k Dirac matrices in d − 2 dimensions and I is the 2 k × 2 k identity matrix. We list all the 10D Dirac matrices calculated in the following We begin with the calculation of C matrix in Eq.(3.27) and Eq.(3.28), which is defined to be So we have and

39)
The next step is to solve 10D Dirac equation and calculate explicitly the spinors u 1 and u 2 in Eq.(3.27) and Eq.(3.28). In the CM frame, we have the kinematics , +q cos θ, +q sin θ .
For our case here, u 1 is a massless spinor, so we have The 10D Dirac equation can be calculated to be which can be solved to be For the massive u 2 we have The 10D Dirac equation can be calculated to be Let's first assume where U 2 is a 2-spinor. If we put Eq.(3.50) into Eq.(3.49), we get which can be solved to be So the first class of solutions of u 2 is Alternatively, we can assume For this case, Dirac equation reduces to and we get the second class of solutions We are now ready to calculate the vector components of u 1 Γ µ Cu 2 in Eq.(3.27) and Eq.(3.28) which are to be contracted with k 4 and k 2 . One needs only calculate the first three components of the vector.
On the other hand, it is crucial to note that the last three components of Γ 0 C, Γ 1 C and Γ 2 C in Eq.(3.37), Eq.(3.38) and Eq.(3.39) are all off-diagonal matrices. In order to get non-vanishing amplitudes, one is forced to choose different spin sign factors for each of the last three spin components of u 1 and u 2 . We will see that the choice of u and u 2 as The first three component of 2 can be calculated to be and in Eq.(3.28) For the second case, as an example, we choose u 1 as in Eq.(3.57) and u 2 as The first three component of u 1 Γ µ Cu (2) 2 can be calculated to be and .
and in Eq.(3.28) so one gets in the hard scattering limit Finally the only leading order amplitude in the hard scattering limit is 2 k 4µ = E sin θ. and they are not all proportional to each other. Note that the usual unpolarized cross section obtained by summing over final spins and averaging over the initial spins in the hard scattering limit is 1 4 spins |M| 2 ∼ (1 + cos 2 θ) . (4.76) The second example is the lowest order process of the elastic scattering of a spin-one-half particle by a spin-zero particle such as e − π + −→ e − π + [18]. The non-vanishing amplitudes were shown to be [18] M(e − R π + −→ e − R π + ) = M(e − L π + −→ e − L π + ) ∼ cos θ 2 , (4.77) M(e − R π + −→ e − L π + ) = M(e − L π + −→ e − R π + ) ∼ sin θ 2 . (4.78) They are again not all proportional to each other.
This paper is the first attack by the present authors to probe high energy, higher spin fermion string scatterings. There are many interesting related issues which remained to be studied. To name a few examples, are there linear relations among hard fermion SSA so that all the fermion SSA can be solved and expressed in terms of one amplitude? can these relations be extended to connect hard SSA of string states of NS sector and R sector ? We will come back to these interesting topics in the near future.