# NSR Open Super-string in the proper-time gauge I: Free Field Theory

Taejin Lee  Department of Physics, Kangwon National University, Chuncheon
24341 Korea<EMAIL_ADDRESS>
###### Abstract

We study the Neveu-Schwarz-Ramond (NSR) open super-string theory in the
proper-time gauge. The string field action is obtained by evaluating the
Polyakov string path integral. In this study, we focus on the open-string
free-field action, which corresponds to the string path integral on a strip.
Depending on the periodicity of the fermion fields, the open super-string has
two sectors: The Neveu-Schwarz (NS) and Ramond (R) sectors. We can impose the
gauge conditions to fix the (super-) reparametrization invariance on the two-
dimensional metric and its super-partner on the string world sheet to secure
the covariance, in contrast to the light-cone gauge condition. Accordingly,
the proper-time emerges in the NS sector and both proper-time and its super-
partner appear in the R-sector. Integration leads to free-string field actions
in both sectors.

open super-string, covariant string field theory, NSR super-string

###### pacs:

11.25.-w, 11.25.Sq, 11.30.Pb

###### Contents

  1. I Introduction
  2. II Canonical Quantization of NSR Super-String
  3. III NSR Open super-string
  4. IV Covariant Proper-Time Gauge condition for the NSR Open Super-String
    1. IV.0.1 Neveu-Schwarz Sector
    2. IV.0.2 Ramond Sector
  5. V The Off-Shell Propagator for the NSR Open Super-String
    1. V.1 Neveu-Schwarz Sector
    2. V.2 Ramond Sector
  6. VI Free Field Action for the NSR Open Super-String
  7. VII Conclusions and Discussions

##  I Introduction

Quantum field theory describes the dynamics of nature in terms of the quantum
fields of point particles. Since its introduction by Dirac Dirac , it has been
the language in which one has attempted to understand fundamental forces.
Quantum field theory flourished as quantum electrodynamics and quantum
chromodynamics, which describes the strong interaction, and electroweak theory
in the 1970s, which combines electromagnetism and weak interaction in a
unified scheme.

Currently, we are confronting a new challenge of constructing quantum field
theories of strings and replacing point particles with strings as fundamental
objects. For the super-string, two approaches to construct quantum field
theory are available: the light-cone field theory of the NSR super-string
mandelstam1974a ; Green1983a ; Green1983b ; Mandelstam1986 ; Grossperi1987 ;
Sin1989  and Witten’s super-string field theory based on the extended BRST
symmetry Witten1986nucl . Each approach helps us to understand the super-
string dynamics and mechanism of the super-string field theory to a great
extent; however, both approaches have limitations. The light-cone super-string
theory is not manifestly covariant and is plagued with various divergences
Greensite1987a ; Greensite1987b ; Greensite1988 ; Green1988 ; Ishibashi2017 .
Witten’s super-string field theory also suffers from divergence owing to the
mid-point contact interaction Wendt1989  .

In this study, we propose a new covariant approach based on the Polyakov
string path integral Polyakov1981 . Because the Polyakov string path integral
is well-defined and finite, we expect that the divergence issues associated
with the other two approaches may be resolved using this approach. When we
evaluate the Polyakov string path integral on a strip of string worksheet, we
can obtain a covariant field theoretical propagator if we impose the gauge
condition to fix the reparametrization invariance. This approach has been
applied to the bosonic string theory Lee88 ; TLeeJKPS2017 ; Lee2017d ;
TLee2017cov ; TLeeEPJ2018 ; Lai2017S ; TLee2019PLB . An important advantage of
this approach is that it is easy to include higher interaction terms, and we
can easily evaluate three -and four-string interactions to confirm local gauge
invariance. The interacting NSR superstring in the proper-time gauge may be
free of the notorious picture changing problem of conformal field theory
formulation, if properly developed. As I put as the sub-title of the paper,
”Free Field Theory”, this paper will serve as a preliminary to a work TLeeprep
in this direction.

As a first step toward constructing a covariant interacting super-string
theory, in the present study, we will focus on the free-field action of NSR
super-strings. To obtain the free-field action, we evaluate the Polyakov
string path integral on a strip with two spatial boundaries. By applying
canonical quantization, we determine that the Hamiltonian only comprises
constraints. To secure the covariance, we impose the gauge condition on the
world-sheet metric and its super-partner. Depending on the periodicity of the
fermion fields, the NSR super-string has two sectors: periodic in the NS
sector and antiperiodic in the Ramond sector. We demonstrate that the NS
sector has a bosonic modular parameter that becomes the proper time, and the R
sector has both bosonic modular and fermionic supermodular parameters. After
integrating both the modular and super-modular parameters, we obtain a Dirac
propagator in the Ramond sector. In the NS sector, we obtain a Klein-Gordon
type propagator, integrating the modular parameter (proper time).

##  II Canonical Quantization of NSR Super-String

The reparametrization-invariant and local super-symmetric actions for the NSR
super-string are given by:

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int d^{2}\xi L,$
$\displaystyle L$ $\displaystyle=$
$\displaystyle\sqrt{-h}\Biggl{\\{}-\frac{1}{2}h^{\alpha\beta}\frac{\partial
X^{\mu}}{\partial\xi^{\alpha}}\frac{\partial
X_{\mu}}{\partial\xi^{\beta}}-\frac{i}{2}\bar{\psi}^{\mu}\gamma^{a}\frac{\partial\psi_{\mu}}{\partial\xi^{\alpha}}+\frac{1}{2}F^{\mu}F_{\mu}+\frac{1}{2}\bar{\chi}_{\alpha}\gamma^{\beta}\frac{\partial
X^{\mu}}{\partial\xi^{\beta}}\gamma^{\alpha}\psi_{\mu}$ (1)
$\displaystyle+\frac{1}{16}\bar{\psi}^{\mu}\psi_{\mu}\bar{\chi}_{\alpha}\gamma^{\beta}\gamma^{\alpha}\chi_{\beta}\Biggr{\\}}$

An auxiliary field $F^{\mu}$ is introduced to render the local super-symmetric
algebra closed off-shell. The action is invariant under the reparametrization
given by

$\displaystyle{\delta}\xi$ $\displaystyle=$
$\displaystyle\zeta^{a},~{}~{}{\delta}X^{\mu}=\zeta^{\alpha}\partial_{\alpha}X^{\mu},~{}~{}{\delta}\psi^{\mu}=\zeta^{\alpha}\partial_{\alpha}\psi^{\mu},$
$\displaystyle{\delta}e_{\alpha}{}^{a}$ $\displaystyle=$
$\displaystyle\zeta^{\beta}\partial_{\beta}e_{\alpha}{}^{a}+e_{\beta}{}^{a}\partial_{\alpha}\zeta^{\beta},~{}~{}{\delta}\xi_{\alpha}=\zeta^{\beta}\partial_{\beta}\chi_{a}+\chi_{\beta}\partial_{\alpha}\zeta^{\beta}$
(2)

and local super-symmetric transformation given by

$\displaystyle{\delta}X^{\mu}=$ $\displaystyle=$ $\displaystyle
i\epsilon\psi^{\mu},~{}~{}{\delta}\psi^{\mu}=\gamma^{\alpha}\left(\partial_{\alpha}X^{\mu}-\frac{1}{2}\bar{\chi}_{a}\psi^{\mu}\right)\epsilon,$
$\displaystyle{\delta}e_{\alpha}{}^{a}$ $\displaystyle=$
$\displaystyle\epsilon\gamma^{\alpha}\chi_{a},~{}~{}{\delta}\chi_{a}=2iD_{\alpha}\epsilon$
(3)

where the covariant derivative $D_{\alpha}$ is defined by

$\displaystyle
D_{\alpha}e=\partial_{\alpha}\epsilon-\frac{1}{2}\omega_{\alpha}\gamma^{5}\epsilon,$
(4)

with the connection

$\displaystyle\omega_{a}=-\frac{1}{\epsilon}e_{\alpha}{}^{a}\epsilon^{\beta\gamma}\partial_{\beta}e_{\gamma}{}^{b}\eta_{ab}+\frac{1}{2}\bar{\chi}_{\alpha}\gamma^{5}\gamma^{\beta}\chi_{\beta}.$
(5)

In addition to the local super-symmetric transformation and reparametrization,
the action is laso-invariant under conformal transformation.

$\displaystyle{\delta}X^{\mu}=0,~{}~{}{\delta}\psi^{\mu}=-\frac{1}{2}\epsilon\psi^{\mu},~{}~{}{\delta}h_{\alpha\beta}=2\epsilon
h_{\alpha\beta},~{}~{}{\delta}\chi_{a}=\frac{1}{2}\epsilon\chi_{\alpha}$ (6)

and the super-conformal transformation

$\displaystyle{\delta}X^{\mu}=0,~{}~{}{\delta}\psi^{\mu}=0,~{}~{}{\delta}h_{\alpha\beta}=0,~{}~{}{\delta}\chi^{\alpha}=\gamma^{\alpha}\epsilon.$
(7)

At the critical dimensions, we set the conformal factor to $e^{\phi}=1$ and
fix the super-conformal invariance, setting

$\displaystyle\gamma^{\alpha}\chi_{\alpha}=0.$ (8)

To construct the Hamiltonian describing the dynamics of the string , an
extended one-dimensional object, we express the metric $h_{\alpha\beta}$ in
terms of the lapse and shift functions

$\displaystyle
h^{\alpha\beta}=\frac{1}{N_{1}}\left(\begin{array}[]{cc}-1&N_{2}\\\
N_{2}&(N_{1})^{2}-(N_{2})^{2}\end{array}\right).$ (11)

Accordingly, zweibein $e_{\alpha}{}^{a}$ is in terms of the lapse, and the
shift functions are expressed as

$\displaystyle\left(e_{\alpha}{}^{a}\right)=\frac{1}{\sqrt{N_{1}}}\left(\begin{array}[]{cc}1&-N_{2}\\\
0&N_{1}\end{array}\right).$ (14)

Two orthogonal vectors on the world-sheet are given as

$\displaystyle{\boldsymbol{e}}_{\hat{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N_{1}}}\left({\boldsymbol{\partial}}_{\tau}-N_{2}{\boldsymbol{\partial}}_{\sigma}\right)$
$\displaystyle{\boldsymbol{e}}_{\hat{1}}$ $\displaystyle=$
$\displaystyle\sqrt{N_{1}}{\boldsymbol{\partial}}_{\sigma}$ (15)

The Hamiltonian can be obtained by taking the Legendre transformation of the
Lagrangian and defining the canonical conjugates
$(P^{\mu},\Pi^{\alpha},\zeta)$ to
$(X_{\mu},N_{\alpha},\lambda=-\sqrt{2}N^{\frac{5}{4}}_{1}\bar{\chi}_{1})$:

$\displaystyle P^{\mu}=\frac{\partial
L}{\partial\dot{X}_{\mu}},~{}~{}\Pi^{\alpha}=\frac{\partial
L}{\partial\dot{N}_{a}},~{}~{}\zeta=\frac{\partial L}{\partial\dot{\lambda}}.$
(16)

From the defining equations of the momenta, we obtain

$\displaystyle
P^{\mu}=\frac{1}{N_{1}}(\dot{X}^{\mu}-N_{2}X^{\prime\mu})+\bar{\chi}^{0}\psi^{\mu},$
(17)

and the first class primary constraints

$\displaystyle\Pi_{\alpha}=0,~{}~{}\zeta=0.$ (18)

With some algebra, we determine

$\displaystyle\bar{H}$ $\displaystyle=$ $\displaystyle P^{\mu}\dot{X}_{\mu}-L$
(19) $\displaystyle=$ $\displaystyle\frac{N_{1}}{2}(P^{2}+X^{\prime
2})+N_{2}P^{\mu}X^{\prime}_{\mu}+\frac{\lambda}{\sqrt{2}}(\gamma^{5}P^{\mu}+X^{\prime\mu})N^{-\frac{1}{4}}_{1}\psi_{\mu}+\frac{1}{2}\bar{\psi}^{\mu}\gamma^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\psi_{\mu}.$

Here, we adopt a simple two- dimensional $\gamma$ matrix algebra
$\gamma^{\alpha}\gamma^{\beta}\gamma_{\alpha}=0$, the Fierz rearrangement

$\displaystyle\bar{\psi}_{a}\psi_{2}\psi_{3}=-\frac{1}{2}\sum_{i}\bar{\psi}_{1}\Gamma_{i}\psi_{3}\Gamma^{i}\psi_{2},~{}~{}~{}\Gamma_{i}=1,\gamma_{a},\gamma_{5}.$
(20)

We need to scale $\psi\rightarrow N^{-\frac{1}{4}}_{1}\psi^{\mu}$ to express
the fermion in canonical form:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{N_{1}}{2}\left(P^{2}+X^{\prime
2}+i\psi^{\mu}\gamma^{5}\psi^{\prime}_{\mu}\right)+N_{2}\left(P^{\mu}X^{\prime}_{\mu}+\frac{i}{2}\psi^{\mu}\psi^{\prime}_{\mu}\right)$
(21)
$\displaystyle+\frac{\lambda}{\sqrt{2}}\left(\gamma^{5}P^{\mu}+X^{\prime\mu}\right)\psi_{\mu}-v_{\alpha}\Pi^{\alpha}-u\zeta$

where Lagrangian multipliers $v_{\alpha}$ and $u$ are introduced to enforce
the primary constraints. The primary constraints $\Pi_{\alpha}=0$ and
$\zeta=0$ lead to the secondary constraints given as follows:

$\displaystyle P^{2}+X^{\prime 2}+i\psi^{\mu}\gamma^{5}\psi^{\prime}_{\mu}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
P^{\mu}X^{\prime}_{\mu}+\frac{i}{2}\psi^{\mu}\psi^{\prime}_{\mu}$
$\displaystyle=$ $\displaystyle 0,$ (22)
$\displaystyle\frac{1}{\sqrt{2}}\left(\gamma^{5}P^{\mu}+X^{\prime\mu}\right)\psi_{\mu}$
$\displaystyle=$ $\displaystyle 0.$

##  III NSR Open super-string

We may extend the domain of $\sigma$, initially defined as $[0,\pi]$ for the
open super-string, to $[-\pi,\pi]$, such that the field variables for the open
string and those for the closed super-string are defined in the same domain.
Furthermore, for the open super-string, we impose the following folding
conditions on the dynamical variables:

$\displaystyle X^{\mu}(\sigma)$ $\displaystyle=$ $\displaystyle
X^{\mu}(-\sigma),~{}~{}P^{\mu}(\sigma)=p^{\mu}(-\sigma),~{}~{}N_{1}(\sigma)=N_{1}(-\sigma),~{}~{}N_{2}(\sigma)=-N_{2}(-\sigma),$
$\displaystyle\Pi_{1}(\sigma)$ $\displaystyle=$
$\displaystyle\Pi_{1}(-\sigma),~{}~{}\Pi_{2}(\sigma)=-\Pi_{2}(-\sigma),~{}~{}\psi^{\mu+}(\sigma)=\psi^{\mu-}(-\sigma),~{}~{}\lambda^{\mu+}(\sigma)=-\lambda^{\mu-}(-\sigma)$
(23)

The conditions are also read in terms of notmal modes as follows

$\displaystyle X^{\mu}_{n}$ $\displaystyle=$ $\displaystyle
X^{\mu}_{-n},~{}~{}P^{\mu}_{n}=P^{\mu}_{-n},~{}~{}N^{1}_{n}=N^{1}_{-n},~{}~{}N^{2}_{n}=-N^{2}_{-n}$
(24)

and

$\displaystyle\begin{cases}\psi^{\mu+}_{n}=\psi^{\mu-}_{n},~{}~{}\lambda^{+}_{n}=-\lambda^{-}_{-n}~{}~{}~{}~{}~{}&\text{for
the Ramond sector,}\\\
\psi^{\mu+}_{n+\frac{1}{2}}=\psi^{\mu-}_{n+\frac{1}{2}},~{}~{}\lambda^{+}_{n+\frac{1}{2}}=-\lambda^{-}_{-n-\frac{1}{2}}~{}~{}~{}~{}~{}&\text{for
the Neuveu-Schwarz sector.}\end{cases}$ (25)

With these conditions, we can write action $S$ in the Ramond sector as
follows:

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{\tau_{f}}_{\tau_{i}}d\tau\sum_{n}\Biggl{\\{}P^{\mu}_{n}\dot{X}_{\mu
n}+\frac{i}{2}\psi^{\mu}_{-n}\dot{\psi}_{\mu
n}+\bar{\Pi}_{n}\dot{\bar{N}}_{n}+\bar{\zeta}_{n}\dot{\bar{\lambda}}_{n}-\bar{N}_{n}L^{R}_{n}$
(26)
$\displaystyle-\bar{\lambda}_{n}F_{n}+v_{n}\bar{\Pi}_{n}+u_{n}\bar{\zeta}_{n}\Biggr{\\}},$

where in the Ramond sector

$\displaystyle\bar{N}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(N_{1n}+N_{2n}\right),~{}~{}\bar{N}_{0}=N_{1}0,~{}~{}\bar{\Pi}_{n}=\frac{1}{2}(\Pi_{1n}+\Pi_{2n}),~{}~{}\bar{\Pi}_{0}=\Pi_{10},n\not=0,$
$\displaystyle\bar{\lambda}_{n}$ $\displaystyle=$
$\displaystyle\lambda^{+}_{n}+\lambda^{-}_{n},~{}~{}\bar{\zeta}_{n}=\zeta^{+}_{n}+\zeta^{-}_{n},$
(27)

and in the Neveu-Schwarz sector

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{\tau_{f}}_{\tau_{i}}d\tau\sum_{n}\Biggl{\\{}P^{\mu}_{n}\dot{X}_{\mu
n}+\frac{i}{2}\psi^{\mu}_{-n-\frac{1}{2}}\dot{\psi}_{\mu
n+\frac{1}{2}}+\bar{\Pi}_{n}\dot{\bar{N}}_{n}+\bar{\zeta}_{n+\frac{1}{2}}\dot{\bar{\lambda}}1_{n+\frac{1}{2}}-\bar{N}_{n}L^{R}_{n}$
(28)
$\displaystyle-\bar{\lambda}_{n+\frac{1}{2}}G_{n+\frac{1}{2}}+v_{n}\bar{\Pi}_{n}+u_{n+\frac{1}{2}}\bar{\zeta}_{n+\frac{1}{2}}\Biggr{\\}},$

with

$\displaystyle\bar{N}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(N_{1n}+N_{2n}\right),~{}~{}\bar{N}_{0}=N_{1}0,~{}~{}\bar{\Pi}_{n}=\frac{1}{2}(\Pi_{1n}+\Pi_{2n}),~{}~{}\bar{\Pi}_{0}=\Pi_{10},n\not=0,$
$\displaystyle\bar{\lambda}_{n+\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\lambda^{+}_{n+\frac{1}{2}}+\lambda^{-}_{n+\frac{1}{2}},~{}~{}\bar{\zeta}_{n+\frac{1}{2}}=\zeta^{+}_{n+\frac{1}{2}}+\zeta^{-}_{n+\frac{1}{2}},$
(29)

##  IV Covariant Proper-Time Gauge condition for the NSR Open Super-String

####  IV.0.1 Neveu-Schwarz Sector

In the Neveu-Schwarz sector, the secondary constraints $L^{R}_{n}$,
$G_{n+\frac{1}{2}}$ form the super-Virasoro algebra with the central charge

$\displaystyle\left[L^{NS}_{n},L^{NS}_{m}\right]$ $\displaystyle=$
$\displaystyle(n-m)L^{NS}_{n+m}+\frac{d}{8}n(n^{2}-1){\delta}(n+m),$
$\displaystyle\left[L^{LS}_{n},G_{m+\frac{1}{2}}\right]$ $\displaystyle=$
$\displaystyle\left(\frac{n}{2}-n-\frac{1}{2}\right)G_{n+m+\frac{1}{2}},$ (30)
$\displaystyle\left[G_{n+\frac{1}{2}},G_{m-\frac{1}{2}}\right]$
$\displaystyle=$ $\displaystyle 2L^{NS}_{n+m}+\frac{d}{2}n(n+1){\delta}(n+m).$

The canonical generator that generates the reparametrization and local
supersymmetric transformation is constructed to be in the NS sector.

$\displaystyle\Omega_{NS}(\epsilon,\omega)$ $\displaystyle=$
$\displaystyle\dot{\epsilon}_{n}\bar{P}i_{n}+\epsilon_{n}\Biggl{\\{}L^{NS}_{n}-i(2n-j)\bar{N}_{j-n}\bar{\Pi}_{j}+i\left(\frac{3}{2}n-j-\frac{1}{2}\right)\bar{\lambda}_{j-n+\frac{1}{2}}\bar{\zeta}_{j+\frac{1}{2}}\Biggr{\\}}$
(31)
$\displaystyle+\dot{\omega}_{n+\frac{1}{2}}\bar{\zeta}_{n+\frac{1}{2}}-\omega_{n+\frac{1}{2}}\Biggl{\\{}G_{n+\frac{1}{2}}+2i\bar{\lambda}_{j-n-\frac{1}{2}}\bar{\Pi}_{j}-\frac{i}{2}(3n-j+1)\bar{N}_{j-n}\bar{\zeta}_{j+\frac{1}{2}}\Biggl{\\}}$

To secure the Lorentz covariance, it is desirable to impose gauge fixing
conditions on the Lagrangian multipliers $\bar{N}_{n}$ and
$\bar{\lambda}_{n+\frac{1}{2}}$. The Lagrangian multipliers transform the
gauge transformations by

$\displaystyle{\delta}\bar{N}_{n}$ $\displaystyle=$
$\displaystyle\dot{\epsilon}_{n}+i(n-2m)\epsilon_{m}\bar{N}_{n-m}-2i\omega_{m+\frac{1}{2}}\bar{\lambda}_{n-m-\frac{1}{2}},$
$\displaystyle{\delta}\bar{\lambda}_{n+\frac{1}{2}}$ $\displaystyle=$
$\displaystyle-\dot{\omega}_{n+\frac{1}{2}}+i\left(n-\frac{3}{2}n+\frac{1}{2}\right)\epsilon_{m}\bar{\lambda}_{n-m+\frac{1}{2}}-\frac{i}{2}(n-3m-1)\omega_{m+\frac{1}{2}}\bar{N}_{n-m}.$
(32)

We may choose the covariant gauge condition for the NSR open super-string in
the Neveu-Schwarz sector by

$\displaystyle\bar{N}_{n(\not=0)}$ $\displaystyle=$ $\displaystyle
0,~{}~{}\dot{\bar{N}}_{0}=0,$ $\displaystyle\bar{\lambda}_{n+\frac{1}{2}}$
$\displaystyle=$ $\displaystyle 0.$ (33)

We can confirm that this proper-time gauge fixes the gauge degrees of freedom
associated with the reparametrization and local super-symmetry completely and
consistently. Near the gauge-fixing hypersurface defined by Eq. (IV.0.1), the
infinitesimal gauge transformation that restores the gauge-fixing condition
along the gauge orbits is determined by

$\displaystyle\dot{\epsilon}_{n}-in\epsilon_{n}\bar{n}+\bar{N}_{n}$
$\displaystyle=$ $\displaystyle 0,~{}~{}~{}n\not=0,$
$\displaystyle\ddot{\epsilon}_{0}+\bar{N}_{0}$ $\displaystyle=$ $\displaystyle
0,$ (34)
$\displaystyle\dot{\omega}_{n+\frac{1}{2}}-\left(n+\frac{1}{2}\right)\omega_{n+\frac{1}{2}}\bar{n}-\bar{\lambda}_{n+\frac{1}{2}}$
$\displaystyle=$ $\displaystyle 0$

where $\bar{n}$ denotes constant $\bar{N}_{0}$ in the covariant gauge. This
linear differential equation for $\epsilon_{n}$ and $\omega_{n+\frac{1}{2}}$
has a unique solution:

$\displaystyle\epsilon_{n}(\tau)$ $\displaystyle=$ $\displaystyle
e^{in\bar{n}\tau}\left\\{-\int^{\tau}_{\tau_{i}}e^{-in\bar{n}\tau^{\prime}}\bar{N}_{n}dt^{\prime}+c_{n}\right\\},~{}~{}n\not=0,$
$\displaystyle\epsilon_{0}(\tau)$ $\displaystyle=$
$\displaystyle\left\\{\frac{\tau-\tau_{i})}{T}\int^{\tau_{f}}_{\tau}+\frac{\tau-\tau_{f})}{T}\int^{\tau}_{\tau_{i}}\right\\}\bar{N}_{0}d\tau^{\prime},~{}~{}T=\tau_{f}-\tau_{i}$
(35) $\displaystyle\omega_{n+\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
e^{(n+\frac{1}{2})\bar{n}\tau}\left\\{\int^{\tau}_{\tau_{i}}e^{-i(n+\frac{1}{2})\bar{n}\tau^{\prime}}\bar{\lambda}_{n+\frac{1}{2}}d\tau^{\prime}+d_{n+\frac{1}{2}}\right\\}$

where

$\displaystyle c_{n}$ $\displaystyle=$
$\displaystyle\left(1-e^{-2in\bar{n}T}\right)^{-1}\left\\{\int^{\tau_{f}}_{\tau_{i}}e^{-in\bar{n}\tau^{\prime}}\bar{N}_{n}d\tau^{\prime}+e^{-2in\bar{n}\tau_{f}}\int^{\tau_{f}}_{\tau_{i}}e^{in\bar{n}\tau^{\prime}}\bar{N}_{-n}d\tau^{\prime}\right\\}$
$\displaystyle d_{n+\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\left(1-e^{-2i(n+\frac{1}{2})\bar{n}T}\right)^{-1}\Biggl{\\{}e^{-2i(n+\frac{1}{2})\bar{n}\tau_{f}}\int^{\tau_{f}}_{\tau_{i}},e^{i(n+\frac{1}{2})\bar{n}\tau^{\prime}}\bar{\lambda}_{-n-\frac{1}{2}}d\tau^{\prime}$
(36)
$\displaystyle-\int^{\tau_{f}}_{\tau_{i}}e^{-i(n+\frac{1}{2})\bar{n}\tau^{\prime}}\bar{\lambda}_{n+\frac{1}{2}}d\tau^{\prime}\Biggr{\\}}.$

####  IV.0.2 Ramond Sector

In the Ramons sector, the constraint operators form a super-Virasoro algebra
given by

$\displaystyle\left[L^{R}_{n},L^{R}_{m}\right]$ $\displaystyle=$
$\displaystyle(n-m)L^{R}_{n+m}+\frac{d}{8}n^{3}{\delta}(n+m),$
$\displaystyle\left[L^{R}_{n},F_{m}\right]$ $\displaystyle=$
$\displaystyle\left(\frac{n}{2}-m\right)F_{n+m},$ (37)
$\displaystyle\left[F_{n},F_{m}\right]$ $\displaystyle=$ $\displaystyle
2L^{R}_{n+m}+\frac{d}{2}n^{2}{\delta}(n+m).$

The canonical generator of symmetric transformation is obtained in the Ramond
sector as

$\displaystyle\Omega_{R}(\epsilon,\omega)$ $\displaystyle=$
$\displaystyle\dot{\epsilon}_{n}\bar{\Pi}_{n}+\epsilon_{n}\left\\{L^{R}_{n}-i(2n-j)\bar{N}_{j-n}\bar{\Pi}_{j}+i\left(\frac{3n}{2}-i\right)\bar{\lambda}_{j-n}\bar{\zeta}_{j}\right\\}$
(38)
$\displaystyle+\dot{\omega}_{n}\bar{\zeta}_{n}-\omega_{n}\left\\{F_{n}+2i\bar{\lambda}_{j-n}\bar{\Pi}_{j}+i\left(\frac{3n}{2}-\frac{j}{2}\right)\bar{N}_{j-n}\bar{\zeta}_{j}\right\\}.$

We observe that the Lagrangian multipliers, $\bar{N}_{n}$ and
$\bar{\lambda}_{n}$, transform under the representation and local
supersymmetric transformation by

$\displaystyle{\delta}\bar{N}_{n}$ $\displaystyle=$
$\displaystyle\dot{\epsilon}_{n}+i(n-2m)\epsilon_{m}\bar{N}_{n-m}-2i\omega_{m}\bar{\lambda}_{n-m}$
$\displaystyle{\delta}\bar{\lambda}_{n}$ $\displaystyle=$
$\displaystyle-\dot{\omega}_{n}+i\left(n-\frac{3m}{2}\right)\epsilon_{m}\bar{\lambda}_{n-m}-\frac{i}{2}(n-3m)\omega_{m}\bar{N}_{n-m}.$
(39)

We choose the covariant gauge condition in the Ramond sector by

$\displaystyle\bar{N}_{n}=0,~{}~{}\dot{\bar{N}}_{0}=0,~{}~{}\bar{\lambda}_{n}=0,~{}~{}\dot{\bar{\lambda}}_{0}=0,~{}~{}n\not=0.$
(40)

The covariant gauge condition in Eq. (40) properly fixes gauge symmetry, we
examine the equation for infinitesimal gauge parameters that restore the
gauge-fixing condition along the gauge orbits near gauge fixing hyper-surface
by

$\displaystyle\dot{\epsilon}_{n}-in\epsilon_{n}\bar{n}-2i\omega_{n}\nu+\bar{N}_{n}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\dot{\omega}_{n}+\frac{n}{2}i\epsilon_{n}\nu-
in\omega_{n}\bar{n}-\bar{\lambda}_{n}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\ddot{\epsilon}_{0}-2i\dot{\omega}_{0}\nu+\dot{\bar{N}}_{0}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\ddot{\omega}_{0}-\dot{\bar{\lambda}}_{0}$ $\displaystyle=$
$\displaystyle 0,$

where $n\not=0$,$\bar{n}$, and $\nu$ denote $\bar{N}_{0}$ and
$\bar{\lambda}_{0}$, respectively. The existence of a unique solution to Eqs.
(IV.0.2) ensures that the chosen covariant gauge condition is complete and
consistent. With some algebra, we explicitly determine a unique solution:

$\displaystyle\epsilon_{n}(\tau)$ $\displaystyle=$ $\displaystyle
e^{in\bar{n}\tau}\left\\{-\int^{\tau}_{\tau_{i}}e^{-in\bar{n}\tau^{\prime}}(\bar{N}_{n}-2i\omega_{n}\nu)d\tau^{\prime}+c_{n}\right\\},$
$\displaystyle\epsilon_{0}(\tau)$ $\displaystyle=$
$\displaystyle\left\\{\frac{\tau-\tau_{i})}{T}\int^{\tau_{f}}_{\tau}+\frac{(\tau-\tau_{f})}{T}\int^{\tau}_{\tau_{f}}\right\\}(\bar{N}_{0}-2i\omega_{0}\nu)d\tau^{\prime},$
$\displaystyle\omega_{n}(\tau)$ $\displaystyle=$ $\displaystyle
e^{in\bar{n}\tau}\left\\{\int^{\tau}_{\tau_{i}}e^{-in\bar{n}\tau^{\prime}}(\bar{\lambda}_{n}-\frac{n}{2}i\epsilon_{n}\nu)d\tau^{\prime}+d_{n}\right\\}$
$\displaystyle\omega_{0}(\tau)$ $\displaystyle=$
$\displaystyle\left\\{\frac{(\tau_{i}-\tau)}{T}\int^{\tau_{f}}_{\tau}+\frac{(\tau_{f}-\tau)}{T}\int^{\tau}_{\tau_{i}}\right\\}\bar{\lambda}_{0}d\tau^{\prime},~{}~{}n\not=0.$
(41)

where

$\displaystyle c_{n}$ $\displaystyle=$
$\displaystyle\left(1-e^{-2in\bar{n}T}\right)^{-1}\Biggl{\\{}\int^{\tau_{f}}_{\tau_{i}}e^{-in\bar{n}\tau^{\prime}}(\bar{N}_{n}-2i\omega_{n}\nu)d\tau^{\prime}$
$\displaystyle+e^{-2in\bar{n}\tau_{f}}\int^{\tau_{f}}_{\tau_{i}}e^{in\bar{n}\tau^{\prime}}(\bar{N}_{n}-2i\omega_{-n}\nu)d\tau^{\prime}\Biggr{\\}}$
$\displaystyle d_{n}$ $\displaystyle=$
$\displaystyle\left(1-e^{-2in\bar{n}T}\right)^{-1}\Biggl{\\{}e^{-2in\bar{n}\tau_{f}}\int^{\tau_{f}}_{\tau_{i}}e^{in\bar{n}\tau^{\prime}}(\bar{\lambda}_{-n}+\frac{n}{2}i\epsilon_{-n}\nu)d\tau^{\prime}$
(42)
$\displaystyle-\int^{\tau_{f}}_{\tau_{i}}e^{-in\bar{n}\tau^{\prime}}(\bar{\lambda}_{n}-\frac{n}{2}i\omega_{n}\nu)d\tau^{\prime}\Biggr{\\}}$

In contrast to the Neveu-Schwarz sector, we cannot gauge away the fermionic
zero mode of the Lagrangian multipliers $\bar{\lambda}_{0}$ completely. This
leads to the main difference between the two sectors is that in the Ramond
sector, modular and super-modular parameters exist, while only modular
parameters exist in the Neveu-Schwarz sector.

##  V The Off-Shell Propagator for the NSR Open Super-String

In this section, we apply BRST quantization to the NSR open superstring. The
path integral adopted to represent the off-shell propagator is evaluated
explicitly.

###  V.1 Neveu-Schwarz Sector

First, we construct the BRST generator $Q_{NS}$ with the given structure
constants in Eq. (IV.0.1), introducing the fermionic ghost variables
$\eta_{n}$, $\bar{\eta}_{n}$, $\xi_{n}$, and the bosonic ghost variables
$\beta_{n+\frac{1}{2}},\bar{\beta}_{n+\frac{1}{2}},\gamma_{n+\frac{1}{2}}$

$\displaystyle Q_{NS}$ $\displaystyle=$
$\displaystyle\eta_{n}L^{NS}_{n}+\bar{\eta}_{n}\bar{\Pi}_{n}+\beta_{n+\frac{1}{2}}G_{n+\frac{1}{2}}+\bar{\beta}_{n+\frac{1}{2}}\bar{\zeta}_{n+\frac{1}{2}}-\frac{1}{2}(n-m)\eta_{n}\eta_{m}\xi_{n+m},$
(43)
$\displaystyle+\left(\frac{n}{2}-m-\frac{1}{2}\right)\eta_{n}\beta_{m+\frac{1}{2}}\gamma_{n+m+\frac{1}{2}}-\beta_{n+\frac{1}{2}}\beta_{m-\frac{1}{2}}\xi_{n+m},$

(the BRST ghost variables $\eta_{n}$ and $\xi_{n}$ may be identified in terms
of the usual $b$–$c$ ghost variables as $\eta_{n}=c_{-n},$ and
$\xi_{n}=b_{n}$.)

Second, the BRST invariant effective action is constructed as

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{\tau_{f}}_{\tau_{i}}d\tau\Bigl{\\{}P^{\mu}\dot{X}_{\mu
n}+\frac{i}{2}\psi^{\mu}_{-n-\frac{1}{2}}\dot{\psi}_{\mu(n+\frac{1}{2})}+\bar{\Pi}_{n}\dot{\bar{N}}_{n}+\bar{\zeta}_{n+\frac{1}{2}}\dot{\bar{\lambda}}_{n+\frac{1}{2}}+i\xi_{n}\dot{\eta}_{n}+i\bar{\xi}_{n}\dot{\bar{\eta}}_{n},$
(44)
$\displaystyle+i\gamma_{n+\frac{1}{2}}\dot{\beta}_{n+\frac{1}{2}}+i\bar{\gamma}_{n+\frac{1}{2}}\dot{\bar{\beta}}_{n+\frac{1}{2}}-i[Q_{NS},\Delta]\Bigr{\\}},$

Here we choose $\Delta$ by

$\displaystyle\Delta=\xi_{n}\bar{N}_{n}+\bar{\eta}_{n}\chi_{n}-\gamma_{n+\frac{1}{2}}\bar{\lambda}_{n+\frac{1}{2}}-\bar{\gamma}_{n+\frac{1}{2}}f_{n+\frac{1}{2}},$
(45)

with

$\displaystyle\chi_{n}=\frac{1}{\alpha}\bar{N}_{n},(n\not=0),\chi_{0}=0,~{}~{}f_{n+\frac{1}{2}}=\frac{1}{\alpha}\bar{\lambda}_{n+\frac{1}{2}}$
(46)

to produce covariant gauge conditions at limit $\alpha\rightarrow 0$. Scaling
of dynamic variables

$\displaystyle(\bar{\Pi}_{n},\bar{\eta}_{n})$ $\displaystyle\rightarrow$
$\displaystyle\alpha(\bar{\Pi}_{n},\bar{\eta}_{n}),~{}~{}n\not=0,$
$\displaystyle(\bar{\zeta}_{n+\frac{1}{2}},\bar{\beta}_{n+\frac{1}{2}})$
$\displaystyle\rightarrow$
$\displaystyle\alpha(\bar{\zeta}_{n+\frac{1}{2}},\bar{\beta}_{n+\frac{1}{2}}),$
(47)

we find , $\alpha\rightarrow 0$ in the limit,

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{\tau_{f}}_{\tau_{i}}d\tau\Biggl{\\{}\sum_{n}\left(P^{\mu}_{n}\dot{X}_{\mu
n}+\frac{i}{2}\psi^{\mu}_{-n-\frac{1}{2}}\dot{\psi}_{\mu
n+\frac{1}{2}}+i\xi_{n}\dot{\eta}_{n}+i\gamma_{n+\frac{1}{2}}\dot{\beta}_{n+\frac{1}{2}}\right)$
$\displaystyle+\sum_{n}\left(\bar{N}_{n}L^{NS}_{n}-\dot{\bar{\lambda}}_{n+\frac{1}{2}}G_{n+\frac{1}{2}}\right)-\sum_{n,m}\Bigl{(}(n-m)\eta_{n}\bar{N}_{m}\xi_{n+m},$
$\displaystyle+\left(\frac{n}{2}-m-\frac{1}{2}\right)\bar{N}_{n}\beta_{m+\frac{1}{2}}\gamma_{n+m+\frac{1}{2}}+\left(\frac{n}{2}-m-\frac{1}{2}\right)\eta_{n}\bar{\lambda}_{m+\frac{1}{2}}\gamma_{n+m+\frac{1}{2}}$
$\displaystyle+2\beta_{n+\frac{1}{2}}\bar{\lambda}_{m-\frac{1}{2}}\xi_{n+m}\Bigr{)}-\sum_{n}\left(\bar{\zeta}_{n+\frac{1}{2}}\bar{\lambda}_{n+\frac{1}{2}}+i\bar{\beta}_{n+\frac{1}{2}}\bar{\gamma}_{n+\frac{1}{2}}\right),$
.
$\displaystyle-\sum_{n}{}^{\prime}(\bar{\Pi}_{n}N_{n}-i\bar{\eta}_{n}\xi_{n})+\bar{\Pi}_{0}\dot{\bar{N}}_{0}+i\bar{\xi}\dot{\bar{\eta}}_{0}-i\xi_{0}\bar{\eta}_{0}\Biggr{\\}}$
(48)

We integrate the conjugates of Lagrangian multipliers $\bar{\Pi}_{n}$ and
$\bar{\zeta}_{n+\frac{1}{2}}$, and obtain the action parameterized by modular
parameter $\bar{n}$. Ghost variables $(\bar{\eta}_{n},\bar{\xi}_{n})$,
$n\not=0$, and $(\bar{\beta}_{n+\frac{1}{2}},\bar{\gamma}_{n+\frac{1}{2}})$
can be trivially integrated. By further integrating over the ghost zero modes
$(\eta_{0},\xi_{0})$ and $(\bar{\eta}_{0},\bar{\xi}_{0})$, we obtain the
factor

$\displaystyle Det[\partial^{2}_{\tau}]=T.$ (49)

Finally, we obtain the resultant action as

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{\tau_{f}}_{\tau_{i}}d\tau\Biggl{\\{}\sum_{n}\left(P^{\mu}_{n}\dot{X}_{\mu
n}+\frac{i}{2}\psi^{\mu}_{-n-\frac{1}{2}}\dot{\psi}_{\mu
n+\frac{1}{2}}+i\gamma_{n+\frac{1}{2}}\dot{\beta}_{n+\frac{1}{2}}\right)$ (50)
$\displaystyle+i\sum_{n}{}^{\prime}\xi_{n}\dot{\eta}_{n}-\bar{n}\left(L^{NS}_{0}+\sum_{n}\left(n\eta_{n}\xi_{n}-(n+\frac{1}{2})\beta_{n+\frac{1}{2}}\gamma_{n+\frac{1}{2}}\right)\right)\Biggr{\\}},$

That is the off-shell propagator for the Neveu-Schwarz sector

$\displaystyle G_{NS}=T\int d\bar{n}\int
D[X,P]D[\psi]D[\eta,\xi]D[\beta,\gamma]e^{iS},$ (51)

We observe that the path integral in Eq. (51), representing the off-shell
propagator, is a typical path integral representation for the transition
matrix element with Hamiltonian $H$:

$\displaystyle H$ $\displaystyle=$ $\displaystyle L^{NS}_{0}+L^{NS}_{gh},$
$\displaystyle L^{NS}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{n}(P^{2}_{n}+n^{2}X^{2}_{n})+\sum_{n\geq
0}\left(n+\frac{1}{2}\right)\left(\psi^{\mu}_{-n-\frac{1}{2}}\psi_{\mu
n+\frac{1}{2}}\right),$ $\displaystyle.L^{NS}_{gh}$ $\displaystyle=$
$\displaystyle\sum_{n}\left(n\eta_{n}\xi_{n}-\left(n+\frac{1}{2}\right)\beta_{n+\frac{1}{2}}\gamma_{n+\frac{1}{2}}\right).$
(52)

Defining proper time $s=T\bar{n}$, we obtain the off-shell propagator in the
NS sector as follows:

$\displaystyle G_{NS}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}ds\langle
X_{f},\psi_{f},\eta_{f},\beta_{f}|\exp\left\\{,-s(L^{NS}_{0}+L^{NS}_{gh})\right\\}|X_{i},\psi_{i},\eta_{i},\beta_{i}\rangle$
(53) $\displaystyle=$ $\displaystyle\langle
X_{f},\psi_{f},\eta_{f},\beta_{f}|\frac{1}{L^{NS}_{0}+L^{NS}_{gh}-i\epsilon},|X_{i},\psi_{i},\eta_{i},\beta_{i}\rangle$

###  V.2 Ramond Sector

We can construct the BRST generator $Q_{R}$ for the Ramond sector using the
structural constants given in Eq. (IV.0.2). Similarly,

$\displaystyle Q_{R}$ $\displaystyle=$
$\displaystyle\eta_{n}L^{R}_{n}+\bar{\eta}_{n}\bar{\Pi}_{n}+\beta_{n}F_{n}+\bar{\beta}_{n}\bar{\zeta}_{n}-\frac{1}{2}(n-m)\eta_{n}\eta_{m}\xi_{n+m}$
(54)
$\displaystyle+\left(\frac{n}{2}-m\right)\eta_{n}\beta_{m}\gamma_{n+m}-\beta_{n}\beta_{m}\xi_{n+m}$

The BRST invariant action may be written as

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{\tau_{f}}_{\tau_{i}}d\tau\Biggl{\\{}\sum_{n}\left(P^{\mu}_{n}\dot{X}_{\mu
n}+\frac{i}{2}\psi^{\mu}_{-n}\dot{\psi}_{\mu
n}+i\xi_{n}\dot{\eta}_{n}+i\gamma_{n}\dot{\beta}_{n}\right)+\bar{\Pi}_{0}\dot{\bar{N}}_{0}+\bar{\zeta}_{0}\dot{\bar{\lambda}}_{0}$
(55)
$\displaystyle+i\bar{\xi}_{0}\dot{\bar{\eta}}_{0}+i\bar{\gamma}_{0}\dot{\bar{\beta}}_{0}-\sum_{n}(\bar{N}_{n}L^{R}_{n}+\bar{\lambda}_{n}F_{n})+\sum_{n,m}\Biggl{(}(n-m)\bar{N}_{n}\eta_{m}\xi_{n+m}$
$\displaystyle-\left(\frac{n}{2}-m\right)\bar{N}_{n}\beta_{m}\gamma_{n+m}+\left(n-\frac{m}{2}\right)\bar{\lambda}_{n}\eta_{m}\gamma_{n+m}+2\bar{\lambda}_{n}\beta_{m}\xi_{n+m}\Biggr{)}$
$\displaystyle-\sum_{n}{}^{\prime}\left(\bar{\Pi}_{n}\bar{N}_{n}-\bar{\zeta}_{n}\bar{\lambda}_{n}-i\bar{\eta}_{n}\bar{\xi}_{n}-i\beta_{n}\bar{\gamma}_{n}\right)\Biggr{\\}},$

We integrate $\bar{\pi}_{n}$ and $\bar{\zeta}_{n}$, and as a result, obtain
the action parametrized by the modular and super-modular parameters in the
covariant gauge

$\displaystyle\int d\bar{n}d\nu e^{iS}.$ (56)

Integrations over the ghost variables $(\bar{\eta}_{n},\bar{\xi}_{n})$
$n\not=0$ and $(\bar{\beta}_{n},\bar{\gamma}_{n})$, $n\not=0$ are trivial. We
perform further integrations over the ghost zero modes $(\eta_{0},\xi_{0})$,
$(\beta_{0},\gamma_{0})$, $(\bar{\eta}_{0},\bar{\xi}_{0})$, and
$(\bar{\beta}_{0},\bar{\gamma}_{0})$ which do not induce a nontrivial factor
in the measure, however, in contrast to the Neveu-Schwarz sector

Furthermore, we reach the point where the off-shell propagator for the Ramond
sector is explicitly evaluated.

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{\tau_{f}}_{\tau_{i}}d\tau\Biggl{\\{}\sum_{n}\left(P^{\mu}_{n}\dot{X}_{\mu
n}+\frac{i}{2}\psi^{\mu}_{-n}\dot{\psi}_{\mu
n}\right)+i\sum_{n}{}^{\prime}(\xi_{n}\dot{\eta}_{n}+\gamma_{n}\dot{\beta}_{n})$
(57)
$\displaystyle-\bar{n}\left(L^{R}_{0}+\sum_{n}{}^{\prime}n(\eta_{n}\xi_{n}-\beta_{n}\gamma_{n})\right)-\nu\left(F_{0}-\sum_{n}{}^{\prime}\left(\frac{n}{2}\eta_{n}\gamma_{n}+2\beta_{n}\xi_{n}\right)\right)\Biggr{\\}},$

after ghost zero modes were integrated. The precise definition of the path
integral representation of the off-shell propagator in the Ramond sector is as
follows:

$\displaystyle
G_{R}(X_{f},\psi_{f},\eta_{f},\beta_{f};X_{i},\psi_{i},\eta_{i},\beta_{i})=\int
d\bar{n}d\nu\int D[X,P]D[\psi]D[\eta,\xi]D[\beta,\gamma]e^{i\bar{S}}.$ (58)

Defining the proper time $s=T\bar{n}$ and its supersymmetric counterpart
$\varrho=T\nu$, we rewrite the path integral as

$\displaystyle
G_{R}(X_{f},\psi_{f},\eta_{f},\beta_{f};X_{i},\psi_{i},\eta_{i},\beta_{i})$
$\displaystyle=$ $\displaystyle\int^{\infty}_{0}ds\int d\varrho\langle
X_{f},\psi_{f},\eta_{f},\beta_{f}|\exp\Bigl{\\{}-is(L^{R}_{0}+L^{R}_{gh})$
(59)
$\displaystyle-i\varrho(F_{0}+F_{gh})\Bigr{\\}}|X_{i},\psi_{i},\eta_{i},\beta_{i}\rangle,$

We define

$\displaystyle L^{R}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{n}(P^{2}_{n}+n^{2}X^{2}_{n})+\frac{1}{2}\sum_{n>0}n\left(\psi^{m}_{-n}\psi_{\mu
n}-\psi^{\mu}_{n}\psi_{\mu(-n)}\right),$ $\displaystyle L^{R}_{gh}$
$\displaystyle=$
$\displaystyle\sum_{n}{}^{\prime}n\left(\eta_{n}\xi_{n}-\beta_{n}\gamma_{n}\right),$
$\displaystyle F_{0}$ $\displaystyle=$ $\displaystyle\sum_{n\geq
0}\frac{1}{\sqrt{2}}\left(P^{\mu}_{n}+inX^{\mu}_{n}\right)\psi_{\mu
n}+\sum_{n>0}\frac{1}{\sqrt{2}}\left(P^{\mu}_{n}-inX^{\mu}_{n}\right)\psi_{\mu(-n)},,$
$\displaystyle F_{gh}$ $\displaystyle=$
$\displaystyle\sum_{n}{}^{\prime}\left(\frac{n}{2}\eta_{n}\gamma_{n}-2\beta_{n}\xi_{n}\right).$
(60)

Observing that

$\displaystyle\left[L^{R}_{0}+L^{R}_{gh},F_{0}+F_{gh}\right]=0,$ (61)

and

$\displaystyle\left(F_{0}+F_{gh}\right)^{2}=L^{R}_{0}+L^{R}_{gh},$ (62)

we can simply perform integration over the modular and super-modular
parameters

$\displaystyle
G_{R}(X_{f},\psi_{f},\eta_{f},\beta_{f};X_{i},\psi_{i},\eta_{i},\beta_{i})$
$\displaystyle=$ $\displaystyle\langle
X_{f},\psi_{f},\eta_{f},\beta_{f}|\frac{F_{0}+F_{gh}}{L^{R}_{0}+L^{R}_{gh}-i\epsilon}|X_{i},\psi_{i},\eta_{i},\beta_{i}\rangle$
(63) $\displaystyle=$ $\displaystyle\langle
X_{f},\psi_{f},\eta_{f},\beta_{f}|\frac{1}{F_{0}+F_{gh}}|X_{i},\psi_{i},\eta_{i},\beta_{i}\rangle.$

##  VI Free Field Action for the NSR Open Super-String

The expressions of the propagators obtained in the previous sections reveal
the structure of the free-field actions for the NSR open super-string. The
free-field action is constructed such that the off-shell propagators can be
deduced in terms of the action through the quantum field theoretical
expression. The free-string field action is given by:

$\displaystyle S=\int
D[X,\psi,\eta,\beta]_{NS}\Phi\left(L^{NS}_{0}+L^{NS}_{gh}-i\epsilon\right)\Phi+,\int
D[X,\psi,\eta,\beta]_{R}\Psi\left(F_{0}+F_{gh}\right)\Psi.$ (64)

We discuss the structure of action $S$ in detail. We may define the ”creation”
and ”annihilation” operators

$\displaystyle\psi^{\mu{\dagger}}_{j}=\psi^{\mu}_{-j},~{}~{}r^{\dagger}_{j}=\beta_{j},~{}~{}r_{j}=-\gamma_{j},~{}~{}s^{\dagger}_{j}=\gamma_{-j},~{}~{}s_{j}=\beta_{-j}$
(65)

where $j$ can appropriately be a positive integer or half-integer. They
satisfy the commutation relations of harmonic oscillators or anti-commutation
relations of Grassman harmonic oscillators:

$\displaystyle[\psi^{\mu}_{i},\psi^{{\dagger}\nu}_{j}]_{+}=\eta^{\mu\nu}{\delta}_{ij},~{}~{}[r_{i},r^{\dagger}_{j}]={\delta}_{ij},~{}~{}[s_{i},s^{\dagger}_{j}]={\delta}_{ij}$
(66)

The structure of field action becomes transparent in terms of creation and
annihilation operators. In the Neveu-Schwarz sector, the kinetic operator is
given as

$\displaystyle L^{NS}_{0}$ $\displaystyle=$ $\displaystyle
p^{2}+N^{NS}-\frac{d}{16}$ $\displaystyle=$ $\displaystyle
p^{2}+\sum_{n>0}na^{{\dagger}\mu}_{n}a_{\mu n}+\sum_{n\geq
0}\left(n+\frac{1}{2}\right)\psi^{{\dagger}\mu}_{n+\frac{1}{2}}\psi_{\mu
n+\frac{1}{2}}-\frac{d}{16}.$ $\displaystyle L^{NS}_{gh}$ $\displaystyle=$
$\displaystyle N^{NS}_{gh}+\frac{1}{8}$ (67) $\displaystyle=$
$\displaystyle\sum_{n>0}\sum_{i=1}^{2}na^{{\dagger}i}_{\rm gh}{}_{n}a^{i}_{\rm
gh}{}_{n}+\sum_{n\geq
0}\left(n+\frac{1}{2}\right)\left(r^{\dagger}_{n+\frac{1}{2}}r_{n+\frac{1}{2}}+s^{\dagger}_{n+\frac{1}{2}}s_{n+\frac{1}{2}}\right),$

where $a^{1}_{\rm gh}{}_{n},a^{2}_{\rm gh}{}_{n}$ may be written in terms of
the usual $b$–$c$ ghost variables as

$\displaystyle b_{zz}(\sigma)$ $\displaystyle=$
$\displaystyle\frac{b_{0}}{2}+\frac{1}{2}\sum_{n=1}\left(a_{1n}^{\text{gh}}e^{-in\sigma}-Ia^{{\dagger}\text{gh}}_{2n}e^{in\sigma}\right),$
(68a) $\displaystyle b_{\bar{z}\bar{z}}(\sigma)$ $\displaystyle=$
$\displaystyle\frac{b_{0}}{2}+\frac{1}{2}\sum_{n=1}\left(a_{1n}^{\text{gh}}e^{in\sigma}-ia^{{\dagger}\text{gh}}_{2n}e^{-in\sigma}\right),$
(68b) $\displaystyle c^{z}(\sigma)$ $\displaystyle=$
$\displaystyle\frac{c_{0}}{2}+\frac{1}{2}\sum_{n=1}\left(a^{{\dagger}\text{gh}}_{1n}e^{in\sigma}+ia_{2n}^{\text{gh}}e^{-in\sigma}\right),$
(68c) $\displaystyle c^{\bar{z}}(\sigma)$ $\displaystyle=$
$\displaystyle\frac{c_{0}}{2}+\frac{1}{2}\sum_{n=1}\left(a^{{\dagger}\text{gh}}_{1n}e^{-in\sigma}+ia_{2n}^{\text{gh}}e^{in\sigma}\right).$
(68d)

They satisfy

$\displaystyle[a^{{\dagger}i}_{\rm gh}{}_{n},a^{j}_{\rm
gh}{}_{m}]_{+}={\delta}_{nm}{\delta}^{ij}.$ (69)

In the Ramond sector,

$\displaystyle F_{0}$ $\displaystyle=$ $\displaystyle
p_{\mu}\psi^{\mu}_{0}+\sum_{n>0}\sqrt{n}\left(a^{{\dagger}\mu}_{n}\psi_{\mu
n}+a^{\mu}_{n}\psi^{\dagger}_{\mu n}\right),$ $\displaystyle F_{\rm gh}$
$\displaystyle=$
$\displaystyle-\sum_{n>0}\left(\frac{n}{2}(a^{1{\dagger}}_{\rm
gh}{}_{n}+a^{2}_{\rm gh}{}_{n}s^{\dagger}_{n})+2(r^{\dagger}_{n}a^{1}_{\rm
gh}{}_{n}+s_{n}a^{2{\dagger}}_{\rm gh}{}_{n})\right)$ (70)

Because $\psi^{\mu}_{0}$ satisfies the Clifford algebra

$\displaystyle[\psi^{\mu}_{0},\psi^{\nu}_{0}]_{+}=\eta^{\mu\nu},$ (71)

we may represent them by ten dimensional $G$ matrices

$\displaystyle\psi^{\mu}_{0}=\Gamma^{\mu}.$ (72)

We can define a mass operator $M$ to make the structure of the action in the
Ramond sector more apparent.

$\displaystyle M$ $\displaystyle=$
$\displaystyle\sum_{n>0}\sqrt{n}\left(a^{{\dagger}\mu}_{n}\psi_{\mu
n}+a^{\mu}_{n}\psi^{\dagger}_{\mu n}\right)~{}~{}M_{gh}=F_{gh},$
$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\sum_{n>0}n\left(a^{{\dagger}\mu}_{n}A_{\mu
n}+\psi^{{\dagger}\mu}_{n}\psi_{\mu n}\right)=N^{R}$ $\displaystyle
M^{2}_{gh}$ $\displaystyle=$
$\displaystyle\sum_{n>0}n\left(a^{{\dagger}1}_{\rm gh}{}_{n}a^{1}_{\rm
gh}{}_{n}+a^{{\dagger}2}_{\rm gh}{}_{n}a^{2}_{\rm
gh}{}_{n}+r^{\dagger}_{n}r_{n}+s^{\dagger}_{n}s_{n}\right)=N^{R}_{gh}$ (73)

Thus, the excited state of the super-string in the Neveu-Schwarz sector
describes a particle with a mass given by the eigenvalues of mass operators
$m^{2}=N^{NS}+N^{NS}_{gh}-1/2$ at the critical dimensions $d=10$. The ground
state of the super-string in the Neveu-Schwarz sector is tachyonic. This
tachyonic ground state is removed by the GSO (Gliozzi-Scherk-Olive) projection
operator GSO1977 . However, a Dirac-type particle that has a mass $m$,
$m^{2}=N^{R}+N^{R}_{gh}$, can realize an excited state of the super-string in
the Ramond sector. The ground state in the Ramond sector has ten- dimensional
spinor index with $E_{\rm ground}=0$.

The GSO-projected free-field action is defined in terms of the total fermion
number, $F$, which is BRST invariant.

$\displaystyle F=\begin{cases}\sum_{n=1}\left(\psi^{{\dagger}\mu}_{n}\psi_{\mu
n}+r^{\dagger}_{n}r_{n}+s^{\dagger}_{n}s_{n}\right),~{}~{}~{}~{}~{}~{}&\text{Ramond
sector}\\\ ,\sum_{n=0}\left(\psi^{{\dagger}\mu}_{n+\frac{1}{2}}\psi_{\mu
n+\frac{1}{2}}+r^{\dagger}_{n+\frac{1}{2}}r_{n+\frac{1}{2}}+s^{\dagger}_{n+\frac{1}{2}}s_{n+\frac{1}{2}}\right),~{}~{}~{}~{}~{}~{}&\text{Neveu-
Schwarz sector}\end{cases}$

by

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int[X,\psi,\eta,\beta]_{NS}\Phi\left(1-(-1)^{F}\right)\left(-\partial^{\mu}\partial_{\mu}+N^{NS}+N^{NS}_{gh}-\frac{1}{2}\right)\Phi,$
(74)
$\displaystyle+\frac{1}{2}\int[X,\psi,\eta,\beta]_{R}\Psi\left(1+\Gamma^{11}(-1)^{F}\right)\left(\partial_{\mu}\Gamma^{\mu}+M+M_{gh}\right)\Psi$

##  VII Conclusions and Discussions

We studied the Polyakov string path integral of an NSR open superstring on a
strip. The evaluation of the path integral was performed in a manifestly
covariant manner. This was achieved by choosing the covariant gauge conditions
that were imposed on the metric and its super-symmetric counterpart
(gravitino) on the strip. The NSR open super-string had two sectors, depending
on the periodicity of the fermion field: the NS sector with a periodic
condition and the Ramond sector with an anti-periodic sector. In the proper
time gauge, the NS sector had modular (proper-time) parameter, and the R
sector had both modular and supermodular parameters. Integration of modular
and super-modular parameters yielded string propagators in both sectors. We
demonstrated that the path integrals represent field-theoretical off-shell
string propagators, which may be obtained from string free-field actions
Terao96 ; Ohta86 ; deAlwis86 ; Date86 . Hence, this study revealed the
connection between geometric and algebraic approaches.

This study can be extended in various directions. The immediate extension may
be to evaluate the Polyakov string-path integrals of the NSR super string over
the Riemann surface, describing three and four interacting strings.
Accordingly, we will obtain full string vertex operators, just as in the case
of bosonic string theory. The (super)string theory in the proper-time gauge is
deformable to Witten’s (super)string. However, in the proper-time gauge, it is
easy to deal with string interactions of an arbitrary number of strings
because the Riemann surfaces in the proper time gauges are free of conical
singularity, which is the main obstacle in evaluating the scattering
amplitudes of higher string interactions. In the presence of conical
singularity, it is difficult to prove the local (non-Abelian) gauge invariance
of scattering amplitudes on multiple $Dp$-branes. It is also unclear whether
the extended BRST impedance is violated by higher-order string interactions.
As in the case of bosonic string theory, we expect that the scattering
amplitudes of the NSR super-string with three and four strings can be
calculated in the proper-time gauge without difficulty.

It is important to derive conventional QFT from corresponding super-string
accurately. Otherwrise, it may be difficult to systematically calculate the
stringy corrections. However, the fermionic action is introduced by hand to
compatible with the spectrum of the free superstring. In the case of the
interacting theory, things get even worse if we employ the (super)-conformal
field theory. Unlike the theory in the proper-time gauge, the conventional NSR
superstring, based on the conformal field theory possesses zero modes. Because
the conformal field theory for the NSR superstring is defined on a sphere in
contrast to the theory in the proper-time gauge, which is defined on a
cylindrical surface. To deal with these zero modes, we need to insert some
(picture changing) operators at the midpoint of the string. Yet this midpoint
insertion is difficult to be defined properly. In contrast to the conventional
approach, in the proper-time gauge approach the interacting NSR superstring,
which is free of this problem, may be derived from the Polyakov string path
integral consistently. This work is important as a preliminary work along this
direction.

###### Acknowledgements.

TL was supported by the National Research Foundation of Korea(NRF) grant,
funded by the Korean government (MSIT) (2021R1F1A106299311). Part of this
study was conducted during the author’s visit to the APCTP (Korea, Pohang).
This study has been worked also with the support of a research grant of
Kangwon National Universtiy in 2019.

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# Constraints on Possible Monopole-Dipole Interactions of WISPs from the
Transverse Relaxation Time of Polarized 3He Gas

Changbo Fua,b Thomas R. Gentilea and William M. Snowb aNational Institute of
Standards and Technology, Gaithersburg, MD 20899
bCenter for Exploration of Energy and Matter, Indiana Univ., Bloomington, IN
47408

###### Abstract

Various theories beyond the Standard Model predict new particles with masses
in the sub-eV range with very weak couplings to ordinary matter. A $P$-odd,
$T$-odd, spin-dependent interaction between polarized and unpolarized matter
is one such possibility. Such a monopole-dipole interaction can be induced by
the exchange of spin-$0$ particles. The presence of a possible monopole-dipole
interaction between fermion spins and unpolarized matter would cause an
decreased transverse spin relaxation time $T_{2}$ for a confined gas of
polarized nuclei. By reanalyzing previously existing data on the spin
relaxation times of polarized 3He in gas cells with pressure in the millibar
range and applying the well-established theory of spin relaxation for magnetic
field gradients to gradients in a possible monopole-dipole field, we present
new laboratory constraints on the strength and range of such an interaction.
These constraints represent to our knowledge the best limits on such
interactions for the neutron with ranges between $0.01$ cm and 1 cm.

##  1 Introduction

The possible existence of new particles beyond the Standard Model with masses
in the sub-eV range and very weak couplings to ordinary matter is starting to
attract increased attention[1]. These possible particles are now starting to
be referred to in the literature as WISPs (Weakly-Interacting sub-eV
Particles)[1, 2]. A $P$-odd and $T$-odd interaction between polarized and
unpolarized matter proportional to ${\vec{s}}\cdot{\vec{r}}$ (spin and
distance) is one such possibility. An interaction involving a scalar coupling
$g_{s}$ at one vertex and a pseudoscalar coupling $g_{p}$ at the other vertex
(sometimes referred to as a monopole-dipole interaction in the literature) can
be induced by the exchange of spin-$0$ particles between two vertexes. The
axion is one possible example of such a particle, and many laboratory
experiments and astrophysical observations have searched for it[3].

The monopole-dipole interaction potential between nucleon spins and
unpolarized matter is given by[4]

$V=\hbar^{2}g_{s}g_{p}\frac{\hat{\mathbf{\sigma}}\cdot\mathbf{\hat{r}}}{8\pi
m_{n}}\left(\frac{1}{r\lambda}+\frac{1}{r^{2}}\right)e^{-r/\lambda},$ (1)

where $m_{n}$ is the mass of the nucleon at the polarized vertex, $\hbar$ is
the reduced Planck constant, $\hbar\hat{\sigma}/2$ is the fermion spin,
$\lambda=\hbar/m_{a}c$ is the range of the interaction, $m_{a}$ is the mass of
axion, and $\hat{\mathbf{r}}={\mathbf{r}}/r$ is the unit vector between the
dipole and the mass.

Measurements of the longitudinal relaxation time $T_{1}$ of polarized 3He were
recently used to constrain the interaction strength and range of such a
possible monopole-dipole interaction coupling to neutrons[5, 6]. The method
takes the advantage of the fact that the motion of a polarized species through
a possible monopole-dipole field gradient will cause the polarization to
decay. In Ref. [Serebrov2009423] the author constrained the monopole-dipole
interaction by using the depolarization rate of ultracold neutrons (UCN) in a
material trap. In Ref. [Yuri-T1-3He] the author employed the longitudinal
relaxation time $T_{1}$ of polarized 3He gas at pressures of a few bar to set
a constraint on the monopole-dipole interaction strength.

In this work we constrain the strength of a possible monopole-dipole
interaction of neutrons by using previous measurements of the transverse spin
relaxation time $T_{2}$ of polarized 3He gas at pressures of a few mbar. We
find that this is the best constraint to our knowledge in the distance range
from $0.01$ cm to $1$ cm for a monopole-dipole interaction involving neutrons.
This constraint could be greatly improved in dedicated experiments.

##  2 Spin Relaxation in a Low Pressure Cell

Consider a dipole in a uniform magnetic field $\textbf{B}_{0}$ along the z
axis and a very large mass block of thickness $d$ at position $z$. If a
monopole-dipole interaction exists, the potential has the form[5]

$V(\mathbf{r})=\frac{g_{s}g_{p}N\hbar^{2}\lambda}{4m_{n}}e^{-z/\lambda}(1-e^{-d/\lambda}),$
(2)

where $N$ is the nucleon density of the matter. Consider a spherical cell with
radius $R$ containing low density polarized gas. The polarized gas in the cell
sees both the monopole-dipole field $V$ and the normal magnetic field
$\textbf{B}_{0}$. If $R\gg\lambda$, the average variation of the monopole-
dipole interaction potential over the whole cell can be written as

$\langle\Delta V\rangle\approx\frac{4\pi R^{2}\lambda}{\frac{4}{3}\pi
R^{3}}\frac{g_{s}g_{p}N\hbar^{2}\lambda}{4m_{n}}(1-e^{-d/\lambda}).$ (3)

Because both the monopole-dipole interaction and the magnetic interaction
$\vec{\mu}\cdot\vec{B}$ are of the form $\vec{s}\cdot\vec{\hat{r}}f(r)$, one
would expect an extra contribution to the spin relaxation due to the monopole-
dipole interaction.[7] In analogy with a magnetic field gradient, the
transverse relaxation rate induced by the monopole-dipole field gradient can
be written as[8]

$\frac{1}{T_{2}^{{}^{\prime}}}=(\delta\omega)^{2}\,\tau_{c},$ (4)

where $\tau_{c}$ is the correlation time for the relaxation mechanism under
consideration and $\hbar\,\delta\omega$ is the interaction energy in the
monopole-dipole interaction field. The decrease in the transverse relaxation
rate $\delta\omega$ by the factor $\delta\omega\,\tau_{c}$ due to the motion
of the dipole is known as motional narrowing. Using the correlation time for
the random motion of polarized atoms in an ideal gas
$\tau_{c}\approx\frac{R^{2}}{2D}$, where $D$ is diffusion constant, the
transverse relaxation rate from the monopole-dipole interaction becomes

$\frac{1}{T_{2}^{{}^{\prime}}}\approx\left[\frac{3\lambda^{2}\,g_{s}g_{p}N\hbar^{2}}{4R\,m_{n}}(1-e^{-d/\lambda})\right]^{2}\frac{R^{2}}{2D}.$
(5)

The relaxation rates from independent processes add, yielding a total rate
$1/T_{2}$ giving by

$\frac{1}{T_{2}}=\frac{1}{T_{2}^{(w)}}+\frac{1}{T_{2}^{(\partial
B)}}+\frac{1}{T_{2}^{(dd)}}+\frac{1}{T_{2}^{{}^{\prime}}},$ (6)

where we list the dominant sources: wall collision ($1/T_{2}^{(w)}$),
inhomogeneity of the external magnetic field ($1/T_{2}^{(\partial B)}$),
relaxation related to the dipole-dipole interaction ($1/T_{2}^{(dd)}$), and
inhomogeneity of the monopole-dipole field ($1/T_{2}^{{}^{\prime}}$). By
assuming that the difference between the measured $T_{2}$ and that calculated
from theory is due to the monopole-dipole interaction, one can set a limit on
$g_{s}g_{p}$ using

$g_{s}g_{p}\leq\frac{4\,m_{n}}{3N\hbar\lambda^{2}}\sqrt{\frac{2D}{T_{2}}}\frac{1}{(1-e^{-d/\lambda})},\
\ (\lambda\ll R).$ (7)

##  3 Monopole-dipole interaction coupling constant constraints on the neutron
from previously existing measurements of $T_{2}$ spin relaxation measurements
of 3He

We now use these relations to set limits on the monopole-dipole interaction on
the neutron spins in polarized 3He. In Ref. [long-T2-Gemmel-low-P], $T_{2}$
measurements are reported for a spherical aluminosilicate glass cell of radius
$R=3$ cm filled with 3He to a pressure of $4.5$ mbar. The transverse
relaxation time was measured to be $T_{2}=(60.2\pm 0.1)$ h. The diffusion
constant is $D=0.04$ m2/s. The nucleon density of the aluminosilicate glass is
$N=1.31\times 10^{30}$ m-3. Assuming a cell wall thickness of $1.5$ mm,
typical for glass cells used in magnetometry work of this type, from the
relations above the limit on a monopole-dipole interaction involving the
neutron is

$\displaystyle g_{s}^{(n)}g_{p}^{(n)}\leq\frac{9.8\times
10^{-23}}{\lambda^{2}(1-e^{-0.15/\lambda})},\,\ \lambda[{\rm cm}]\ll 3\ {\rm
cm}.$ (8)

In Ref. [McGregor-T2], a 13.1 mbar, 4.96-cm-diameter spherical cell was used
to study transverse spin relaxation times for polarized 3He in measured
magnetic field gradients. The diffusion constant of 3He in the cell is
$D=0.0137$ m2/s. They measured $T_{2}$ in a nonmagnetic building with low
magnetic field gradients to be 14.86 h. Assuming the cell wall nucleon density
was $1.3\times 10^{30}$ m-3 and the wall thickness is 2 mm, the constraint on
the monopole-dipole interaction strength is

$\displaystyle g_{s}^{(n)}g_{p}^{(n)}\leq\frac{1.2\times
10^{-22}}{\lambda^{2}(1-e^{-0.2/\lambda})},\,\ \lambda[{\rm cm}]\ll 5\ {\rm
cm},$ (9)

which is very close to the constraint above.

file=LowPgraph-1.eps,width=4.1in

Figure 1: Constraints on the monopole-dipole coupling strength
$g_{s}^{(n)}g_{p}^{(n)}$: 1. Solid line, this work, by reanalyzing the data in
Ref. [long-T2-Gemmel-low-P]; 2. Dot line, this work, by reanalyzing the data
in Ref. [McGregor-T2]; 3. Short dash line, from Ref. [Serebrov2009423]; 4.
Dash-dot line, from Ref. [Yuri-T1-3He]; 5. Dash-dot-dot line, from Ref.
[Youdin-HgCs]; 6\. Short-dot line, from Ref. [Bab-UCN].

Finally, we note that there are unpublished reports of even longer $T_{2}$
(140 h) for polarized 3He gas in glass cells[10]. Unfortunately the pressure
and the wall material information are not shown in the reports, hence we do
not use these results to set a limit.

From measurements and theoretical calculations[11], it is known that the
polarization of the 3He nucleus is dominated by the neutron polarization, with
only a small contribution from orbital motion and other effects. Therefore
these limits are to a good approximation interpretable simply as constraints
on a neutron monopole-dipole interaction. With the inclusion of constraints
from $T_{2}$ measurements in other nuclei, such as 129Xe, it would be possible
to place separate constraints on the monopole-dipole interactions of neutrons
and protons. This work is in progress.

##  4 Summary

In this work we present constraints on the strength and range of a possible
monopole-dipole interaction involving the neutron. These new laboratory
limits, set by reanalysis of previous measurements of the transverse spin
relaxation time $T_{2}$ in polarized 3He cells, are the best in existence for
the neutron to our knowledge in the range $0.01$ cm to 1 cm. Constraints on
the monopole-dipole interaction using this method can be significantly
improved in dedicated experiments.

This work was supported in part by the National Science Foundation under award
PHY-0116146.

## References

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Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC
BY 4.0).

In A. Martin, K. Hinkelmann, H.-G. Fill, A. Gerber, D. Lenat, R. Stolle, F.
van Harmelen (Eds.), Proceedings of the AAAI 2021 Spring Symposium on
Combining Machine Learning and Knowledge Engineering (AAAI-MAKE 2021) -
Stanford University, Palo Alto, California, USA, March 22-24, 2021.

# Deep Learning Approaches for Forecasting Strawberry Yields and Prices Using
Satellite Images and Station-Based Soil Parameters

Mohita Chaudhary<EMAIL_ADDRESS>Department of Electrical and Computer
Engineering, University of Waterloo, Ontario, Canada  Mohamed Sadok Gastli
<EMAIL_ADDRESS>Lobna Nassar<EMAIL_ADDRESS>Fakhri Karray
<EMAIL_ADDRESS>
(2021)

###### Abstract

Computational tools for forecasting yields and prices for fresh produce have
been based on traditional machine learning approaches or time series modeling.
We propose here an alternate approach based on deep learning algorithms for
forecasting strawberry yields and prices in Santa Barbara county, California.
Building the proposed forecasting model comprises three stages: first, the
station-based ensemble model (ATT-CNN-LSTM-SeriesNet_Ens) with its compound
deep learning components, SeriesNet with Gated Recurrent Unit (GRU) and
Convolutional Neural Network LSTM with Attention layer (Att- CNN-LSTM), are
trained and tested using the station-based soil temperature and moisture data
of Santa Barbara as input and the corresponding strawberry yields or prices as
output. Secondly, the remote sensing ensemble model (SIM_CNN-LSTM_Ens), which
is an ensemble model of Convolutional Neural Network LSTM (CNN-LSTM) models,
is trained and tested using satellite images of the same county as input
mapped to the same yields and prices as output. These two ensembles forecast
strawberry yields and prices with minimal forecasting errors and highest model
correlation for five weeks ahead forecasts. Finally, the forecasts of these
two models are ensembled to have a final forecasted value for yields and
prices by introducing a voting ensemble. Based on an aggregated performance
measure (AGM), it is found that this voting ensemble not only enhances the
forecasting performance by 5% compared to its best performing component model
but also outperforms the Deep Learning (DL) ensemble model found in literature
by 33% for forecasting yields and 21% for forecasting prices.

###### keywords:

Deep LearningSatellite ImagesPrice YieldForecasting Fresh
ProduceAttentionSeries-Net

##  1 Introduction

In Fresh produce Supply Chain Management (FSCM), a crucial part of the
procurement process is to find a model which helps in precisely predicting the
farmers’ prices. These prices are highly affected by the yields hence the
availability of accurate yields values to train the forecasting model is
crucial [1]. This study focuses on strawberries as fresh produce, whose yield
depends on various parameters related to weather, soil, synthetic factors,
irrigation, and others. These factors are quite uncertain therefore the
forecasting of strawberry yields and prices is a challenging task. Moreover,
from a humanitarian point of view, the United Nations World Food Programme has
reported that around 821 million people around the world suffer from hunger
[2], and that number has been growing drastically since the start of the
COVID-19 pandemic. This is part of the reason why the United Nations have
included ending hunger and the betterment of food security as part of their
main goals in their 2030 Agenda for Sustainable Development [3]. A key aspect
to overcome these issues and a significant challenge facing food security is
the ability to reliably estimate crop yields using forecasting models; the
main objective of this work. It should be noted that such forecasting models
for strawberries can be applied on numerous similar fresh produces for better
yields estimates to sustain food security.

The acquisition of data can be very expensive with limited availability.
California is chosen since the data for its strawberry yields, prices, and
soil parameters can be acquired from various publicly available online
sources. Due to the frequent absence of localized data, using remote sensing
data such as satellite images is important since they can cover larger
geographic areas. Santa Maria, which lies within Santa Barbara county, is
primarily considered in this work because it is one of the largest stations
for strawberry produce in California; it is considered the leading state
producer for strawberries [4]. The yields and prices are predicted by
manipulating historical input data to extract features that can capture as
many yields and prices trends as possible. Currently used forecasting tools
for fresh produce have limited performance since they do not consider an
extensive set of influential factors affecting yields and prices. They are
also incapable of capturing the complex patterns in big data sources of prices
transactions which provide valuable information on the underlying processes
affecting fresh produce prices and quantities; this became feasible by the
advent of the state-of-the-art machine learning and DL techniques.

The scope of this work is forecasting both strawberry yields and prices using
input features related to soil parameters. The yields and prices values are
forecasted 5 weeks ahead. It is found that the past 20 weeks values of the
soil parameters affect the yields and hence this is the lag considered to
forecast the yields and prices. An aggregated measure is used to gauge the
performance of the forecasting models, which are compared to a simple LSTM
model and a compound DL ensemble model proposed in [5]. The voting regressor
ensemble of ATT-CNN-LSTM-SeriesNet_Ens and SIM_CNN-LSTM_Ens models outperforms
its compound component models as well as simple DL models such as LSTM.
Moreover, it enhances the performance of the compound DL ensemble model
described in [5] by up to 33%.

Section 2 highlights literature review and previous work done in this area, it
also summarizes major findings and limitations. The details of the assembled
forecasting compound DL models are provided in Section 3. The datasets, data
preprocessing, and results of the conducted experiments are presented and
analyzed in Section 4. Finally, the drawn conclusion along with future work
are found in Section 5.

##  2 Literature Review

Various methods are used for yields prediction like Artificial Neural Networks
[6, 7], K-Nearest Neighbors [8, 4] and Simple Long Short Term Memory Networks
(LSTM) [9]. The weather parameters are used as input parameters to predict the
yields in [10]; while in [11, 5], the corresponding prices to strawberry
yields are predicted using various DL compound models like ConvLSTM, CNN-LSTM,
CNN-LSTM-GRU with attention along with DL ensemble models. Adding a self-
attention layer improves the yield prediction results to quite an extent.
Moreover, the DL models are recommended over nondeep ML and non-ML models for
forecasting. In [12], various machine learning and DL imputation techniques
for missing values in datasets are discussed. In [9], the weather parameters,
soil moisture dynamic data and soil quality static data across various
counties in Iowa, USA are used as input to forecast the yearly corn yields
using a basic LSTM network. The static parameters remain constant over years
for a specific region and they are useful only when different regions are
considered. The work presented here considers the dynamic parameters solely as
input and ignores the static ones since only the Santa Maria county is
considered for forecasting yields and prices. In [13], counter-propagation
artificial neural networks (CP-ANNs) and Supervised Kohonen Networks (SKNs)
are used to predict the wheat yields using a found set of influential soil
parameters.

As for the use of remote sensing in crop yields forecasting, existing
approaches are investigated. Three recent methods and applications are
presented in [14, 15, 16]. Their general framework involves using a
preprocessed collection of satellite-based data to train neural networks for
yields prediction. The first approach is conducted in [14] by J. You et al.,
where they use satellite images that are publicly available to predict annual
soybean yields for specific counties in the USA. The images used comprise 7
bands of surface reflectance and 2 bands of temperature. The authors
preprocessed these images into histograms due to their large size, then fed
them into several prediction models for comparison. The models used are
Convolutional Neural Network (CNN) and LSTM. Their results indicate that the
CNN models are better than LSTM in predicting yields. However, it should be
noted that this approach investigates yearly yields and not daily forecasts.
This is further investigated by J. Phongpreecha in [15], who considers
moisture in addition to surface reflectance and temperature satellite images
to predict annual corn yields. He also implements dimensionality reduction to
histograms as described by [14]. The investigated models are a Custom CNN-
LSTM, a Separable CNN-LSTM, a CNN-LSTM, a 3D-CNN, and a CNN-LSTM-3D CNN
network. The ConvLSTM is found to be the best performing model in forecasting
annual corn yields. A similar approach is conducted in [16] where the authors
investigate such application in southern Brazil on annual soybean yields.
Their approach considers satellite images in addition to precipitation data
obtained from weather stations. Three models are tested, namely multivariate
OLS linear regression, random forest, and LSTM where, LSTM outperforms the
other two models in forecasting.

From the literature review, it is evident that neural networks, specifically
CNN, LSTM, and combinations of both, are best suited for the forecasting
application. The limitations of the presented approaches are either
considering data collected from localized stations for forecasting, such data
is not readily available for all croplands on earth, or giving annual
predictions of yields using weekly satellite images. Thus, daily predictions
are essential with more complex deep leaning models for higher performance.
Furthermore, the satellite images should be used alongside the stationary data
to overcome the frequent problem of data scarcity.

##  3 Proposed Models and Methodology

Two models are used for forecasting which are then fed into an ensemble to
combine their outputs into a final forecasted value. The first model uses
station-based soil data as input while the other relies on soil data captured
by satellite images instead. Compound DL models are proved to perform better
than simple DL and nondeep ML models as described in [5]. The simple DL model,
LSTM, and the ensemble model described in [5] are compared to the proposed
model.

###  3.1 The Station-based Model (ATT-CNN-LSTM-SeriesNet_Ens)

This model uses the station-based soil data to forecast the strawberry yields
and prices. After preprocessing, the data is fed into two compound DL models
namely Att-CNN-LSTM and SeriesNet with GRU. The forecasted output values of
each of these two models are then fed as input into a voting ensemble which
outputs the final forecasted yields or prices.

####  3.1.1 Att-CNN-LSTM

Figure 1: Architecture of CNN-LSTM with Attention

The self-attention layer helps in focusing on the essential details in the
input data. In the proposed model, the attention layer uses additive attention
and the sigmoid activation function. Figure 1 shows the architecture of the
Att-CNN-LSTM compound model. The self-attention is a mechanism which deals
with the different positions of a single sequence and then computes the
representation of that sequence. The layer is applied atop every unit of the
sequence; additive attention is used. The attention function helps in mapping
a query and a set of key-value pairs to an output. Here, the keys, output,
values and queries are all considered as vectors [17].

The Additive Attention works by using a feed forward network to calculate the
compatibility function [18]. The equations are described in (1), (2), (3), and
(4).

$h_{t,t^{\prime}}=\tanh{(x_{t}^{T}W_{t}+x_{t^{\prime}}^{T}W_{x}+b_{t})}$ (1)
$e_{t,t^{\prime}}=\sigma(W_{a}h_{t,t^{\prime}}+b_{a})$ (2)
$a_{t}=\mathrm{softmax}(e_{t})$ (3)
$l_{t}=\sum_{t^{\prime}}a_{t,t^{\prime}}x_{t^{\prime}}$ (4)

where $\sigma$ is the element-wise sigmoid function and $W_{x}$ and $W_{t}$
are the weight matrices corresponding to $x_{t}^{T}$ and $x_{t^{\prime}}^{T}$.
$W_{a}$ is the weight matrix corresponding to the non-linear combination of
$W_{x}$ and $W_{t}$, while $b_{t}$ and $b_{a}$ are the bias vectors [17].
Equation (4) shows how the attention value $l_{t}$ is computed. The
probability distribution $a_{t}$ and compatibility score $e_{t,t^{\prime}}$
must be calculated first to find the value of attention. The compatibility
score is calculated using the hidden representation $h_{t,t^{\prime}}$ of
$x_{t}^{T}$ and $x_{t^{\prime}}^{T}$. The use of Attention layer contributes
significantly to performance by reducing the forecasting error of the DL
models. The improvement in performance is evident in the results reported by
[17, 18, 19, 20] in domains such as natural language processing, fresh produce
related predictions and healthcare questionnaires.

Stacking CNNs and LSTMs together helps in utilizing the strength of each of
these models [21, 18]. The CNNs help in extracting the spatial features
whereas the LSTMs help in extracting the temporal features and using their
combination highly improves the forecasting performance. The attention layer
is added to the CNN-LSTM compound network to improve its performance.

####  3.1.2 SeriesNet with GRU

Figure 2: Architecture of SeriesNet with GRU

In this model, the GRU is used alongside the dilated causal convolution. The
GRU network is a two-layer network with attention module in between. The
output at the end is flattened and fed into a single neuron dense layer. The
number of layers and neurons alter as per the application. The architecture of
the model is illustrated in Figure 2. Gated Recurrent Unit (GRU) [22] is a
newer version of RNNs and is quite similar to LSTMs [23]. GRUs use hidden
states rather than using the cell state or memory to transfer the information.
They have two gates, a reset gate, and an update gate. The update gate acts
similarly to the forget gate and input gates of LSTM. The reset gate, on the
contrary, is used to decide how much past information has to be forgotten.
GRUs are faster than LSTMs since they have fewer tensor operations. Both LSTMs
and GRUs are designed to overcome the short-term memory issues faced by RNNs.
Since RNNs face vanishing or exploding gradient issues, GRUs are introduced to
mitigate those issues. The structure is similar to those of LSTMs and consists
of gates that ensure that the issue of gradient is not encountered.

The traditional time series forecasting models are unable to effectively
extract essential data features; thus the authors in [24] came up with a novel
forecasting architecture called SeriesNet. The SeriesNet consists of two
networks, an LSTM network and a dilated causal convolution network. The
dilated convolution handles the loss of resolution or coverage due to the
down-sampling operation in image semantic segmentation [25]. It uses dilated
convolutions to systematically aggregate multi-scale contextual information
and improve the accuracy of image recognition. The causal convolution ensures
that the convolution kernel of CNN can perform convolution operations exactly
in time sequence [26], and that the convolution kernel only reads the current
and historical information. The LSTM network aims to learn holistic features
and to reduce dimensionality of multi-conditional data. The combined results
of the networks help the models to learn multi-range and multilevel features
from time series data, hence it has higher predictive accuracy compared to
other models. The SeriesNet with GRU model uses residual learning as well as
batch normalization to improve generalization.

###  3.2 The Remote Sensing Ensemble Model (SIM_CNN-LSTM_Ens)

This model is an ensemble of CNN-LSTM. It incorporates the capabilities of
both CNN and LSTM by treating them as layers. The architecture for the CNN-
LSTM model used for yields forecasting is presented in Figure 3.2, which
differs from the prices forecasting model. It should also be noted that the
models are fine tuned through trial and error of multiple configurations. As
for the CNN-LSTM prices forecasting model, its architecture is presented in
Figure 3.2. The preprocessing is implemented using the Geospatial Data
Abstraction Library (GDAL) [27] in Python to import the images into a
processable format. After preprocessing, the features are fed into the
designed models. These models are implemented uisng Python’s Tensoflow Keras
library [28] due to its user-friendliness.

figureCNN-LSTM architecture for yields
forecasting using satellite images

figureCNN-LSTM architecture for prices
forecasting using satellite images

###  3.3 Ensemble of ATT-CNN-LSTM-SeriesNet_Ens and SIM_CNN-LSTM_Ens

The previous approaches extract different features from their data, therefore
averaging their forecasted outputs could potentially lead to a better overall
forecast. Hence, a voting ensemble is required to achieve this averaging.
Figure 3 illustrates the overall architecture of the final proposed model. The
outputs of both compound models are combined by the averaging ensemble to
obtain the final forecast.

###  3.4 Evaluation Metrics

The metrics used for analysis are the Mean Absolute Error (MAE), the Root
Mean-Squared Error (RMSE), R-Squared coefficient ($R^{2}$), and the Aggregated
Measure (AGM). The unit in which MAE is measured is pounds/acre for yield and
US dollars for price. The AGM is a measure composed of all the previously
mentioned metrics, whose purpose is to attempt to incorporate the information
captured by all three metrics into one metric to simplify the process of
deciding the best performing model [11, 29]. The measure is negatively-
oriented, meaning lower scores indicate better performance. It is
mathematically defined in (5).

$AGM=\frac{RMSE+MAE}{2}\times(1-R^{2})$ (5)

Figure 3: Block diagram of the proposed ensemble model

##  4 Datasets, Experiments and Analysis

###  4.1 Datasets and Preprocessing

####  4.1.1 Station-based Data

There are various parameters related to the soil affecting the yields and
prices of fresh produce namely: soil moisture, soil temperature, solar
radiation, surface temperature, PDSI (Palmer Drought Severity Index) [30,
31],… etc. Choosing the most influential parameters is a major challenge due
to the high correlation amongst those parameters. Hence, the Random Forest
feature selection method in Python with scikit-learn is used and the soil
moisture and temperature are selected accordingly as the most important
parameters. Moreover, soil moisture and temperature are the two soil
parameters that can be obtained from satellite images as well. The station-
based soil data is downloaded from the National Oceanic and Atmospheric
Administration website [32]. The data for strawberry yields and prices are
downloaded from the California Strawberry Commission website [33]. A lag of
the past 20 weeks, i.e. 140 days, of soil parameters values is found to affect
the yields forecasting and prices values 5 weeks ahead. Two daily parameters
are considered, soil temperature and moisture, hence the total number of input
parameters adds up to 280 (2 parameters x 140 days). After normalizing the
data, the Principal Component Analysis (PCA) [34] is applied on the 280
parameters and it is found that the first 36 parameters gave the maximum
proportion of variance therefore are chosen to train and test the forecasting
model along with the corresponding yields or prices output. The total number
of available samples is 2812 (from year 2011 to 2019) out of which 80% are
used for training and 20% for testing.

####  4.1.2 Remote Sensing Data

The remote sensing data contains two sets of satellite images; one for surface
temperature data and the other for moisture data. The images in the
temperature dataset, obtained from [35], are taken daily, whereas those in the
moisture dataset, obtained from [36], are taken every 3 days. Therefore, each
moisture image is duplicated twice, once for the day before the original image
and once for the day after, in order to have daily moisture values along with
the daily temperature. This approach is an approximation of the missing days
based on the closest known data to those days. Samples of the original images
obtained from the datasets for Santa Barbara county are presented in Figure
4.1.2. The moisture bands are for the mainland moisture levels, which is why
the data is not available for the islands and coastal areas. In addition, a
land cover mask is applied to all images, obtained from the MODIS database
[35]. The mask maintains the pixel values of the image parts of interest,
while setting pixels not within the mask to a zero value, to ensure that the
model only trains on pixel values that correspond to crop lands hence
minimizing the number of pixels processed. The total number of samples is 2977
(from year 2011 to 2019) out of which 80% are used for training and 20% for
testing.

figure
Image samples from the temperature
bands (left) and the moisture bands (right)

figureInput sample histogram structure

As shown in [14], the images are too large to be fed directly into the model.
Thus, dimensionality reduction is applied by converting them into histograms
of pixel frequency counts. This reduction is based on the assumption of
permutation invariance, meaning that shuffling the pixels has no effect on the
information retained in the image. The histograms are then normalized based on
their pixel values. Surface reflectance bands used in [14] and [15] are
omitted due to their negligible improvement to the prediction performance for
the considered county and fresh produce. The number of bins, $b$, is set to be
32; as it is a reasonable spread of pixel counts as found in [14]. Moreover,
the range of pixel values for each band is chosen to maximize the spread of
the pixels count distribution. The structure is visualized in Figure 4.1.2;
where the three axes represent the three dimensions of the histogram. Before
being fed to the models, the bins and bands dimensions are flattened into a
single dimension to fit the models.

###  4.2 Experiments and Analysis

The block diagram in Figure 3 depicts the proposed final model, which has a
voting ensemble which averages the forecasted yields or prices resulting from
the two proposed DL compound ensembles described in Section 3.1 and Section
3.2. The main idea behind this averaging is that the final forecast should
pick the trends forecasted by both of the component models.

For yield forecasting, Figure 4.2 shows the true yields versus the yields
forecasted by ATT-CNN-LSTM-SeriesNet_Ens, SIM_CNN-LSTM_Ens, and their
averaging ensemble. The figure shows that the voting ensemble successfully
follows the general trend of the true values despite its inability to describe
the sharp fluctuations. Table 1 presents different metrics scores of the
forecasted values for the different models. Although SIM_CNN-LSTM_Ens
outperforms ATT-CNN-LSTM-SeriesNet_Ens in AGM, the averaging ensemble
outperforms both of its weaker component models with its lowest AGM. This
indicates that there is some information obtained by the latter that enhances
the performance compared to the individual models. Based on AGM, the voting
ensemble enhances the forecasting performance over that of the DL ensemble
model in [5] by 33% and that of the simple LSTM DL by 56%.

figureForecasted vs. true yields values

figureForecasted vs. true prices values

For prices forecasting, Figure 4.2 depicts the true prices versus the prices
forecasted by ATT-CNN-LSTM-SeriesNet_Ens, SIM_CNN-LSTM_Ens, and the averaging
ensemble of both. Table 2 presents different performance metrics scores for
the tested models. Although ATT-CNN-LSTM-SeriesNet_Ens outperforms SIM_CNN-
LSTM_Ens with a noticeable AGM difference, the voting ensemble outperforms
both of these compound components with its lowest AGM. The voting ensemble
persists to be the highest performing compared to the LSTM model, 53%
improvement, and to the ensemble model in [5], 21% improvement in AGM.

Table 1: Results for using LSTM, SIM_CNN-LSTM_Ens, and ATT-CNN-LSTM-
SeriesNet_Ens to forecast yield

Score | LSTM | Ensemble in [5] | SIM_CNN-LSTM_Ens | ATT-CNN-LSTM-SeriesNet_Ens | Voting Ensemble
---|---|---|---|---|---
MAE | 53.1 | 42.5 | 39.1 | 40.7 | 37.0
RMSE | 70.8 | 62.2 | 55.2 | 58.8 | 54.6
$R^{2}$ | 0.780 | 0.83 | 0.866 | 0.848 | 0.869
AGM | 13.6 | 9.0 | 6.3 | 7.5 | 6.0

Table 2: Results for using LSTM, SIM_CNN-LSTM_Ens, and ATT-CNN-LSTM-
SeriesNet_Ens to forecast price

Score | LSTM | Ensemble in [5] | SIM_CNN-LSTM_Ens | ATT-CNN-LSTM-SeriesNet_Ens | Voting Ensemble
---|---|---|---|---|---
MAE | 0.268 | 0.21 | 0.227 | 0.214 | 0.208
RMSE | 0.341 | 0.27 | 0.292 | 0.263 | 0.264
$R^{2}$ | 0.609 | 0.72 | 0.712 | 0.766 | 0.764
AGM | 0.119 | 0.07 | 0.0748 | 0.0557 | 0.0555

##  5 Conclusion and Future Work

This paper explores the application of multiple DL models in forecasting
strawberry yields and prices. Station-based data is used to train the ATT-CNN-
LSTM-SeriesNet_Ens, while remote sensing-based data is used to train the
SIM_CNN-LSTM_Ens for forecasting. It is found that the SIM_CNN-LSTM_Ens model
does better at yields forecasting while the ATT-CNN-LSTM-SeriesNet_Ens model
is better at prices forecasting. The voting ensemble of these models
outperforms its individual components since each component provides a
different forecasting behavior. Moreover, it further proves to perform better
by up to 33% compared to the most recent DL ensemble forecasting model. An
acknowledged limitation is the restricted ability of the deployed models to
capture steep fluctuations in yields and prices, making them not predictable
with the available tools. Moreover, having an ensemble of various compound DL
models is computationally expensive. Potential future work is needed to
further improve the ability of the models to capture steep fluctuations in
both yields and prices. Generalization of the models application to other
fresh produces is also required; this is achieved through transfer learning to
forecast yields and prices of FPs similar to strawberries efficiently with
minimal retraining.

###### Acknowledgements.

The authors would like to acknowledge the financial support provided by
Loblaws corporation, NSERC CRD program and Mitacs.

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