Genus one super-Green function revisited and superstring amplitudes with non-maximal supersymmetry

We reexamine genus one super-Green functions with general boundary conditions twisted by $(\alpha, \beta)$ for $(\sigma, \tau)$ directions in the eigenmode expansion and derive expressions as infinite series of hypergeometric functions. Using these, we compute one-loop superstring amplitudes with non-maximal supersymmetry, taking an example of massless vector emissions of open string type I ${\cal Z}_2$ orbifold.


Introduction
The study of one-loop superstring amplitudes [1,2] having their bosonic predecessors [3,4,5] and that of the attendant genus one Green functions have a long history. Major results and useful formulas have already been incorporated in standard textbooks [6,7] and we think that the study of this subject has in that sense been trivialized. The number of articles devoted to computations and (phenomenological) applications of superstring amplitudes in recent years are, however, relatively small and the generalized Green functions that do not satisfy ordinary periodicity or anti-periodicity on the genus one Riemann surfaces in σ or τ directions do not seem to have been systematically studied, according to our search [8], despite that they are after all two point functions of QFT free fields. These Green functions are needed in order to study scattering properties of particles in superstring compactifications [9,10,11] which carry non-maximal supersymmetry and which are soluble by free fields.
In the first half of this paper, we study these bosonic and fermionic Green functions as the inverse of Laplacian and Dirac operators respectively, exploiting the elementary method of eigen-mode expansion. In the known special cases, our computation boils down to the formula expressible in terms of the theta functions. In general, our final expression is given by an infinite series consisting of a hypergeometric function (with its argument successively shifted), which is relevant to the genus zero Green function 1 . This is in accord with the picture that the genus one Green functions can be obtained from those of genus zero by putting an infinite number of image charges. Our result can be represented as super-Green functions with worldsheet supersymmetry broken by the boundary condition. In the latter half of this paper, we demonstrate the use of these Green functions by a simple yet nontrivial example that has non-maximal supersymmetry.
In the next section, we compute the genus one Green functions with the twist angles (α, β) in (σ, τ ) directions respectively, using the eigen-mode expansion. We make exploit of partial fractions. In section 3, we recall superstring one-loop vacuum amplitudes in the worldsheet covariant path integrals and recast them with those of the light-cone operator formulation in order to circumvent the nuisance of the overall normalization. We review the case of T 4 /Z 2 orbifold .
In section 4, we demonstrate the use of the super-Green functions derived in section 2 by computing the superannulus contributions to massless vector emission amplitudes, taking an example from the case of non-maximal supersymmetry. Explicit results for 1, 2, 3 point amplitudes are compared with the vanishing amplitudes for the case of maximal supersymmetry. 1 Such Green function in fact appears in string theory under constant B field [12].
In appendix A-H, we give some details of computation and background materials quoted in the text.
2 Genus one Green functions with (α, β) boundary condition In this section, we compute the genus one Green functions with general boundary condition to be designated by (α, β), using the eigenmode expansion. We mainly consider the case of torus here. The other one-loop geometries, Klein bottle, annulus and Möbius band, can be constructed by the involution (or, the image method) as seen, for example, in [13,14].

case of α = 0
Now we would like to compute the Green function By translational invariance, we have chosen 0 in the second set of arguments.
Here we consider (2.14) Now we divide this sum into n 1 = 0 part and n 1 = 0 part to use the partial fraction decomposition in eq. (2.6).
As the result of the calculation in appendix D.2, we obtain

fermionic part
The eigen-equations are The eigenvalues can be written as Here we calculate the Green function . (2.19) Similarly, using wherez,z ′ ,θ andθ denote respectively the conjugate points of z, z ′ , θ and θ ′ .
3 Path integral of an NSR fermionic string and genus one vacuum amplitudes In this section, quoting the formula (F.8) in appendix F for the path integral of an NSR fermionic string valid for any genus, we briefly review genus one vacuum amplitudes. Introducing notation to represent the contributions of the path integrals from a worldsheet chiral boson and a fermion obeying the general boundary condition specified by α f β f , we formulate our discussion to cover a large class of cases with toroidal compactification and its orbifolding. Here in this notation, we have labelled the bosonic case by b and the fermionic case by f.

path integral formula for an NSR fermionic string
Let the bosonic coordinates, the fermionic ones, zweibein and the Rarita -Schwinger field be where the action is and For the notation in eqs. (3.1), (3.2), (3.3), please look at appendix F. Below, at one-loop, we will recast the expressions eq. (3.1) into that from the light-cone operator formalism.
3.2 superstring genus one vacuum amplitudes in flat ten dimensions

torus
In the case of torus, the boundary conditions for fermions in flat ten dimensions are specified by + (periodic) or − (antiperiodic) for both σ 1 and σ 2 directions: Similar expressions hold forψ. In the notation of section 2, r = e 2πiα , s = e 2πiβ , so that Except for the (+, +) spin structure, there is neither conformal killing spinor, nor supermoduli.
For the (+, +) spin structure, its contribution to the vacuum amplitude vanishes due to the integrations of ψ,ψ fermionic zero modes. The torus vacuum amplitude for IIB/IIA in flat ten dimensions is, therefore, simply written as whereĝ ab = 1 τ 1 , and we have chosen [16] of the IIB superstring that implements the modular invariance. We have denoted by F the fundamental region of the torus and the factor 1 2 is accounted for by SL(2, Z)/P SL(2, Z) = {{±1 2 }}. The Euclidean volume is denoted by V E . Omitting the calculations of the Weil Petersen measure factor and those of the functional determinants, we obtain where In this paper, we take a short cut to proceed further and to determine the normalization factor K by comparing the last expression eq. (3.8) with the vacuum amplitude evaluated in the light cone gauge operator formalism, written in terms of the so(8) characters. (The overall normalization can also be seen by the one-loop free energy in local field theory): Identifying eq. (3.8) with eq. (3.10), we obtain (3.13) Note that, from the point of view of one-loop free energy in local field theory, comes from a gaussian integration over one momentum, and − 1 2 d 2 τ τ 2 · · · comes from a proper time representation of log Det.

Klein bottle, annulus, möbius strip
Let us write eq. (3.8) as where L To construct an unoriented string, namely the type I superstring , one first makes the closed string sector by the Ω (twist) projection: . We obtain Here, cpf denotes the Chan Paton factor, ǫ = ±1 andχ i 2 τ 2 + 1 2 indicates that the replacement by τ → i 2 τ 2 + 1 2 in the argument is to be made only for the oscillator part. These replacements Finally, the infrared stability seen as the cancellation of the massless poles in Z I closed, one−loop + Z I open, one−loop in the transverse channel (or equivalently the cancellation of dilaton tadpoles [15,17,18,19,20,21] or infinity cancellation [22,15]) selects cpf = 2 5 = 32, ǫ = −1 and the gauge group SO(32) [23].

generalization to cases of toroidal compactification and their orbifolding
In order to proceed even further and to prepare for calculation of string scattering amplitudes in section 4, we will introduce notation for the integrand of the string one-loop partition function.
Let us, in particular, write (χXχ) IIB/IIA flat as Here we have introduced in order to represent the contribution from a single chiral boson and fermion obeying the boundary conditions (α b , β b ) and (α f , β f ) respectively: The power 8 = 10 − 2 seen in eq. (3.21) permits covariant interpretation as the 2d metric and 2d gravitino fields obey the same boundary condition as the worldsheet bosons and fermions do respectively. As a simple prototypical example, let us consider IIB string on T 4 (= (S 1 ) 4 )/Z 2 with radii of S 1 being R I , I = 5, 6, 7, 8.
and a I ≡ √ α ′ R I . The first line represents the contribution from the T 4 compactification without Z 2 insertion, the second, the third and the fourth lines represent the contributions from the untwisted sector with Z 2 insertion, the Z 2 twisted sector and the Z 2 twisted sector with the Z 2 insertion respectively. In each term inside the bracket, the first bin represents the spacetime part and the second bin the internal part.
Referring to the character of c = 1, Z 2 orbifold, we are able to see Here, the arguments of the theta constants are modulo 1 and the non-integer parts are understood to be taken. Note that, in this notation, we have included the contribution from the 2 4 = 16 fixed points in the twisted sector in eq. (3.25).

Open superstring on T 4 /Z 2
Another prototypical example which we will consider in the next section is the open string sector in the type I superstring on T 4 (= (S 1 ) 4 )/Z 2 . The partition function is (3.28) Among the many possibilities discussed in [24,21,20], where the dilaton tadpoles cancel, we will consider the simplest case where the gauge group is U(n = 16) (9) × U(d = 16) (5) with all of the D5 branes at the same fixed point and the first and the second subscripts indicate D9 and D5 brane respectively 2 . [33,34,35,36,37].

Neumann functions with arguments on the boundary
In order to proceed to the computation, we need the Neumann function for the superannulus under a variety of boundary conditions for a worldsheet boson and a worldsheet fermion specified and with the arguments set on the same boundary. The Neumann function for the Möbius strip case can be read off from the annulus case by the change of the arguments in the theta functions and will not be discussed explicitly here. For the case of ++ ν f , which is always needed, The last line of eq. (4.1) can be dropped in the calculation of amplitudes as the source J satisfies d 2 zdθdθJ = 0. We need the case +− ν f as well: Note that, for closed string models with Z 2 insertion, G supertorus −+ Here we have introduced the grassmann source η J , J = 1, 2, 3, ..., N, for this representation.
Following appendix F, we carry out the gaussian integration 3 and the sum S over the boundary conditions.
Let S = S ′ ⊕ S (++) , where S (++) is the part of the sum which contains (++) to some power in ν f . For these parity-violating cases [44], it is well-known that the amplitudes for lower N vanish. Ignoring these cases in this paper, let us denote the remaining part of the N point amplitude for the case labelled by • by Here, we have denoted by (V E δ) a product of the momentum conserving delta functions I k I (that appear from the integrations of the zero modes of the bosonic coordinates) and the volume of the compactification V c . The annulus and the Möbius strip contributions are denoted by A ′• N and M ′• N respectively, and (4.10) See appendix H for these properties.

Analysis and evaluation of N = 1, 2, 3 cases
We will now analyze a few simplest cases. Let us first obtain a few generic features of the amplitudes from the integral representation. First, in order to obtain a non-vanishing amplitude, all grassmann integrations must be saturated. Also, under the assumption made in the last subsection,

case of maximal supersymmetry
In this case, namely, in the case of flat 10d and its toroidal compactifications, it is well-known that the vanishing of these two types after the summation over ν f is established, (see, for example, [45]) by the Riemann identity eq. (C.11). In fact according to eqs. (3.23) and (3.29) and where (2n) 2 where

case of non-maximal supersymmetry
Finally, let us consider the case of type I superstring on T 4 /Z 2 . Among the summation over Coming back to eq. (4.6), we obtain the expression for A ′ 3 T 4 /Z 2 : where J ′ ν (Z 2 ) has been introduced in eq. (3.29) and ν 6 f refers to the spacetime part (bin) of ν f . Let us recall from eqs. (3.23), (3.25) , and therefore as well as from eq. (4.9) (4.20) We obtain where we have introduced shorthand notation ϑ α β (I − J) ≡ ϑ α β z I 2 − z J 2 . The second, third and fourth terms in eq. (4.21) can be further converted by using eq. (C.12): where in eq. (C.12), Finally, we obtain (4.24) Unlike the case of maximal supersymmetry, after nationalizing, each term consists of the product of different ϑ functions and we do not find the use of the Riemann identity.

A Some of the notation
The complex coordinates on the worldsheet torus are denoted by The Laplacian is defined by We introduce a real superfield by where θ andθ are Grassmann numbers, X M (z,z) and ψ M (z,z) are bosonic and fermionic fields, and F M (z,z) is a auxiliary field. The super-derivatives are defined by Hence we take C Some formulae

C.1 Gauss hypergeometric function
Gauss hypergeometric function is In order to obtain the second line in eq. (C.1), we must have • z can not be the real number which is greater than 1, • (1 − tz) −α takes the branch which goes to 1 as t → 0.

(C.6)
We also use the following notation: This function has following properties: The theta function satisfies the heat equation: For α, β = 0, 1 2 , this function satisfies the Riemann identity [46] α,β=0, 1 and also where

C.5 Zeta function
The zeta function is defined by For example, In addition, the generalized zeta function is defined by (D.1) the first term in the last line of eq. (D.1) can be computed as Next, turning to the second term, we obtain the terms in the box brackets of eq. (D.9) can be recast as follows: (D.11) Hence we obtain (z) The terms in the brackets on the last line vanish when acting ∆ = 4∂ z ∂z. (D.13) The first term in the last line of eq. (D.13) is where we have used The second term in eq. (D.13) is Using the formula (z|τ ) and where ν f = (−−), (−+), (+−). Using eqs. (E.2) and (E.3),

F Path integral of a fermionic string at one-loop
In the appendix, the path integral of a fermionic string [49,50,51] is briefly recalled. We present formulas in the flat ten dimensional case but they can be easily adapted to the other cases such as T 4 /Z 2 orbifold given in the text. The basic variables are the bosonic coordinates (2d massless scalar fields) X µ , the fermionic ones (2d massless two-component Majorana spinor fields) ψ Maj µ α , zwiebein e a m (or e m a such that g mn = e m a e n b δ ab ) and the Rarita -Schwinger field χ α m , we denoted by µ, ν, ..., α, β, ..., m, n, ... and a, b, ... ten dimensional vector indices, two dimensional spinor indices, two dimensional worldsheet indices, and two dimensional local Lorentz indices respectively. The action [52] is where Ω(CKS) is the volume of the conformal killing spinor and D ′ ζ indicates that kerP 1/2 has been excluded. The final formula is

G Superannulus
In this appendix, we apply the method of images in superspace to superannulus [15,14].
Let the conjugate point of (z, θ) be (z,θ). The involution acting on f (z, θ) associated with  H.3 eq. (4.8) at z I ∼ z J in case of maximal supersymmetry Let us check that eq. (4.8) reduces to that of [15] at z I ∼ z J in case of maximal supersymmetry.