Researching | D-brane interaction, the open string pair production and its enhancement plus its possible detection

Journals >Acta Physica Sinica >Volume 69 >Issue 10 >Page 101101-1 > Article

Acta Physica Sinica
Vol. 69, Issue 10, 101101-1 (2020)

D-brane interaction, the open string pair production and its enhancement plus its possible detection

Jian-Xin Lu^* and Nan Zhang

Author Affiliations

Peng Huanwu Center for Fundamental Theory, The Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei 230026, China

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DOI: 10.7498/aps.69.20200037 Cite this Article

Jian-Xin Lu, Nan Zhang. D-brane interaction, the open string pair production and its enhancement plus its possible detection[J]. Acta Physica Sinica, 2020, 69(10): 101101-1 Copy Citation Text

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Abstract

This review article reports the recent studies, based on a series of publications by one of the present authors along with his collaborators, regarding the interaction between two D-branes, the open string pair production and its possible enhancement in Type II superstring theories. Specifically, computed is the interaction amplitude between two D-branes, placed parallel at a separation, with each carrying a general worldvolume constant flux, and discussed are the amplitude properties, say, the repulsive or attractive nature of the interaction. When at least one of the D-branes carries an electric flux, the interaction amplitude can have an imaginary part, reflecting the instability of the underlying system via the open string pair production. The decay rate and the pair production rate are both computed. In addition, the enhancement of the latter is found when the added electric and magnetic fluxes are correlated in both magnitude and direction in a certain manner. In particular, when one of the branes is D3 and the other is D1, the corresponding pair production rate becomes large enough to be tested in an earthbound laboratory. Note that the pair production rate is related to the brane separation along the direction transverse to both branes, therefore, to the extra-dimensions with respect to the brane observer. So if the underlying string theory is relevant and the D3 can be taken as our own 4-dimensional world, measuring, say, the electric current due to the pair production and comparing it against the added electric and magnetic fields to see if the measurements agree with the prediction of the computations. This can be used to verify the existence of extra-dimensions. Further, this provides also a potential new means to test the underlying string theory without the need of compactifying it to four dimensions.

Keywords

D-branes strings and branes

$ T_{p} = {1}/\Big[{(2{\text{π}})^{p} \alpha'^{\frac{p + 1}{2}} g_{\rm{s}}}\Big], $

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$ {\rm{d}} s^2 = \left(1 +\frac{k_p}{r^{7 - p}}\right)^{-\frac{7 - p}{8}} {\rm{d}} x_{\|}^2 + \left(1+ \frac{k_p}{r^{7 - p}}\right)^{\frac{p + 1}{8}} {\rm{d}} x_\bot^2, $

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$\begin{split} & |B \rangle_{\rm{NS}} = \frac{1}{2} \left[|B, + \rangle_{\rm{NS}} - |B, - \rangle_{\rm{NS}}\right],\\ & |B \rangle_{\rm{R}} = \frac{1}{2} \left[|B, + \rangle_{\rm{R}} + |B, - \rangle_{\rm{R}}\right], \end{split} $

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$ |B, \eta\rangle = \frac{c_p}{2} |B_{\rm{mat}}, \eta\rangle |B_{\rm{g}}, \eta\rangle, $

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$ |B_{\rm{mat}}, \eta\rangle = |B_X \rangle|B_\psi, \eta\rangle, \quad |B_{\rm{g}},\eta\rangle = |B_{\rm{gh}}\rangle |B_{\rm{sgh}}, \eta\rangle, $

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$ |B_X\rangle = {\rm{exp}} \left(- \sum\limits_{n = 1}^\infty \frac{1}{n} \alpha_{- n} \cdot {{M}} \cdot {\tilde \alpha}_{ - n}\right) |B_X\rangle_0, $

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$ |B_\psi, \eta\rangle_{\rm{NS}} = - {\rm{i}}\; {\rm{exp}} \left({\rm{i}} \eta \sum\limits_{m = 1/2}^\infty \psi_{- m} \cdot {{M}} \cdot {\tilde \psi}_{- m}\right)|0\rangle, $

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$ |B_\psi,\eta\rangle_{\rm{R}} = - {\rm{exp}} \bigg({\rm{i}} \eta \sum\limits_{m = 1}^\infty \psi_{- m} \cdot {{M}} \cdot {\tilde \psi}_{- m}\bigg) |B_{\psi}, \eta\rangle_{0{\rm{R}}}. $

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$ {{M}} = ([(\eta - \hat{F})(\eta + \hat{F})^{-1}]_{\alpha\beta}, -\delta_{ij}), $

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$ |B_X\rangle_0 = [- \det (\eta + \hat F)]^{1/2} \,\delta^{(9 - p)} (q^i - y^i) \prod\limits_{\mu = 0}^9 |k^\mu = 0\rangle, $

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$ |B_\psi, \eta\rangle_{0{\rm{R}}} = \bigg({{C}} { {\varGamma}}^0 { {\varGamma}}^1 \cdots { {\varGamma}}^p \frac{1 + {i} \eta \varGamma_{11}}{1 + {i} \eta } {{U}} \bigg)_{AB} |A\rangle |\tilde B\rangle. $

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$ {{U}} (\hat F) \!=\! \frac{1}{\sqrt{- \det (\eta \!+\! \hat F)}} ; {\rm{exp}} \left(\!- \frac{1}{2} {\hat F}_{\alpha\beta}{ {\varGamma}}^\alpha{ {\varGamma}}^\beta\!\right); ,$

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$ { {\varGamma}} = \langle B_{p'} (\hat F') | D |B_{p} (\hat F) \rangle, $

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$ D = \frac{\alpha'}{4 {\text{π}}} \int_{|z| \leqslant 1} \frac{{\rm{d}}^2 z}{|z|^2} z^{L_0} {\bar z}^{{\tilde L}_0}, $

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$ \begin{split} & { {\varGamma}}_{\rm{NSNS}} = \frac{1}{2} \left[{ {\varGamma}}_{\rm{NSNS}} (+) - { {\varGamma}}_{\rm{NSNS}} (-)\right], \\ & { {\varGamma}}_{{\rm{RR}}} = \frac{1}{2}\left[{ {\varGamma}}_{{\rm{RR}}} (+) + { {\varGamma}}_{{\rm{RR}}} (-) \right]. \end{split} $

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$ { {\varGamma}} (\eta' \eta) = \frac{ c_{p'} c_{p}}{4} \frac{\alpha'}{4 {\text{π}}} \int_{|z| \leqslant 1} \frac{{\rm{d}}^2 z}{|z|^2} A^X \, A^{\rm{bc}}\, A^\psi (\eta'\eta)\, A^{\beta\gamma} (\eta' \eta). $

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$ \begin{split} &A^X = \langle B'_X | |z|^{2 L^X_0} |B_X \rangle,\\ &A^\psi (\eta' \eta) = \langle B'_\psi, \eta'| |z|^{2 L_0^\psi} |B_\psi, \eta \rangle, \\ &A^{\rm{bc}} = \langle B_{\rm{gh}} | |z|^{2 L_0^{\rm{gh}}} | B_{\rm{gh}}\rangle,\\ &A^{\beta\gamma} (\eta' \eta) = \langle B_{\rm{sgh}}, \eta'| |z|^{2 L_0^{\rm{sgh}}} |B_{\rm{sgh}}, \eta\rangle. \end{split} $

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$ A^{\rm{bc}} = |z|^{- 2} \prod\limits_{n = 1}^\infty \left(1 - |z|^{2n}\right)^2, $

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$ A_{\rm{NSNS}}^{\beta\gamma} (\eta' \eta) = |z| \prod\limits_{n = 1}^\infty \left(1 + \eta'\eta |z|^{2n - 1}\right)^{-2}, $

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$ A_{{\rm{RR}}}^{\beta\gamma} (\eta' \eta) \!={}_{{\rm{R}}0}\langle B_{\rm{sgh}}, \eta'|B_{\rm{sgh}},\eta\rangle_{\rm{0R}}\, |z|^{\frac{3}{4}} \!\!\!\prod\limits_{n = 1}^\infty \!\! \left(1 \!+\! \eta'\eta |z|^{2n}\right)^{\!-2}, $

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$ { {M}}_\mu\,^\rho ({ {M}}^{\rm{T}})_\rho\,^\nu = ({ {M}}^{\rm{T}})_\mu\,^\rho { {M}} \rho\,^\nu = \delta_\mu\,^\nu, $

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$ \begin{split} A^X =\; & V_{p + 1} \frac{\left[\det(\eta + \hat F') \det(\eta + \hat F)\right]^{\frac{1}{2}}} { \left(2 {\text{π}}^2 \alpha' t\right)^{ \textstyle {\frac{9 - p}{2}}}}\, {\rm{e}}^{\textstyle {- \frac{y^2}{2{\text{π}} \alpha' t}}} \\ &\times \prod\limits_{n = 1}^\infty \left(\frac{1}{1 - |z|^{2n}}\right)^{9 - p} \\ &\times \prod\limits_{\alpha = 0}^p \frac{1}{1 - \lambda_\alpha |z|^{2n}}, \end{split} $

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$ \begin{split} A^\psi_{\rm{NSNS}} (\eta'\eta) = & \prod\limits_{n = 1}^\infty \left(1 + \eta'\eta |z|^{2n - 1}\right)^{9 - p}\\ & \times \prod\limits_{\alpha = 0}^p \left(1 + \eta'\eta \lambda_\alpha |z|^{2n - 1}\right), \end{split} $

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$ \begin{split} A^\psi_{{\rm{RR}}} (\eta'\eta) =\; &{}_{R0}\langle B'_\psi, \eta' |B_\psi, \eta\rangle_{\rm{0R}}\, |z|^{\frac{5}{4}} \\ &\times\prod\limits_{n = 1}^\infty \left(1 + \eta'\eta |z|^{2n}\right)^{9 - p}\\ & \times \prod\limits_{\alpha = 0}^p \left(1 + \eta' \eta \lambda_\alpha |z|^{2n}\right), \end{split} $

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$ { {W}} = { {M}}{{ {M}}'^{\rm{T}}} = \left( {\begin{array}{*{20}{c}} {{{ {w}}_{(1 + p) \times (1 + p)}}}&0\\ 0&{{{ {\mathbb{I}}}_{(9 - p) \times (9 - p)}}} \end{array}} \right),$

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$ \begin{split} & { {w}}_{\alpha}\,^{\beta} = ({ {s s}}'^{\rm T})_{\alpha}\,^{\beta} \\ =\; & \left[({ {\mathbb{I}}} - \hat F) ({ {\mathbb{I}}}+ \hat F)^{-1} ({ {\mathbb{I}}}+ \hat F')({ {\mathbb{I}}}- \hat F')^{-1}\right]_{\alpha}\,^{\beta}, \end{split} $

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$ { {s}}_{\alpha}\,^{\beta} = [({ {\mathbb{I}}}- \hat F) ( { {\mathbb{I}}} + \hat F)^{-1}]_{\alpha}\,^{\beta}, $

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$ W W^{\rm{T}} = W^{\rm{T}} W = \mathbb{I}_{10 \times 10}. $

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$ \begin{split} & \langle 0| {\rm{e}}^{-\tfrac{1}{n} \alpha_n^\mu ({ {M}}')_\mu\,^\nu \tilde\alpha_{n \nu}} |z|^{ 2\alpha_{- n}^\tau \alpha_{n \tau}} \\ & \times {\rm{e}}^{ - \tfrac{1}{n} \alpha_{-n}^\rho ({ {M}})_\rho\,^\sigma \tilde\alpha_{-n\sigma} }| 0\rangle \\ =& \langle 0| {\rm{e}}^{-\tfrac{1}{n} \alpha_n^\mu ({ {M}}')_\mu\,^\nu \tilde\alpha_{n\nu}} {\rm{e}}^{ - \tfrac{|z|^{2n}}{\, \,n} \alpha_{-n}^\rho ({ {M}})_\rho\,^\sigma \tilde\alpha_{-n\sigma} } | 0\rangle, \end{split} $

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$ \langle 0| {\rm{e}}^{\textstyle - \frac{1}{n} \tilde \alpha_{n}^{\mu} \alpha_{n \mu} }{\rm{e}}^{\textstyle - \frac{|z|^{2n}}{\,\,n} \alpha_{-n}^\rho { {W}}_\rho\,^\sigma \tilde\alpha_{-n\sigma}} | 0\rangle, $

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$ \langle 0| {\rm{e}}^{\textstyle - \frac{1}{n} \tilde \alpha^{\mu}_{n} \alpha_{n \mu} }{\rm{e}}^{\textstyle - \frac{|z|^{2n}}{\,\,n} \lambda_{\rho}\, \alpha^{\rho}_{-n} \tilde\alpha_{-n\rho}} | 0\rangle. $

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$ \begin{split} & \left(1 - \lambda_{0} |z|^{2n}\right) \left(1 - \lambda^{-1}_{0} |z|^{2n}\right) \\ & \times\left(1 - \lambda_{1} |z|^{2n}\right)\left(1 - \lambda^{-1}_{1} |z|^{2n}\right) \\ =\; & 1 - \sum\limits_{\alpha = 0}^{1} \left(\lambda_{\alpha} + \lambda^{-1}_{\alpha}\right) |z|^{2n} \\ & + \left[2 + \left(\lambda_{0} + \lambda^{-1}_{0}\right)\left(\lambda_{1} + \lambda^{-1}_{1}\right)\right] |z|^{4n} \\ & - \sum\limits_{\alpha = 0}^{1} \left(\lambda_{\alpha} + \lambda^{-1}_{\alpha}\right) |z|^{6n} + |z|^{8n}\\ = \; & 1 - {\rm{tr}} { {w}}\, |z|^{2n} + \frac{1}{2} \left[\left({\rm{tr}} { {w}}\right)^{2} - {\rm{tr}} { {w}}^{2}\right] |z|^{4n} \\ & - {\rm{tr}} { {w}} \, |z|^{6n} + |z|^{8n} . \end{split} $

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$\begin{split} \varGamma_{\rm{NSNS}} (\eta'\eta) =\;& \frac{ V_{p + 1} \sqrt{\det (\eta + \hat F')\det(\eta + \hat F)}}{(8 {\text{π}}^2 \alpha')^{\frac{1 + p}{2}}} \int_0^\infty \frac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} {\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}\, |z|^{-1} \prod\limits_{n = 1}^\infty \left(\frac{1 + \eta'\eta |z|^{2n - 1}}{1 - |z|^{2n}}\right)^{7 + \delta_{p, {\rm{even}}} - p} \\ & \times\prod\limits_{\alpha = 0}^{[\frac{p - 1}{2}]} \, \frac{\left(1 + \eta'\eta \lambda_\alpha |z|^{2 n - 1}\right)\left(1 +\eta'\eta \lambda^{-1}_\alpha |z|^{2 n - 1}\right)}{\left(1 - \lambda_\alpha |z|^{2n}\right) \left(1 - \lambda^{-1}_\alpha |z|^{2n}\right)},\end{split}$

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$\begin{split} \varGamma_{{\rm{RR}}} (\eta'\eta) =\; &\frac{V_{p + 1} \sqrt{\det(\eta + \hat F')\det(\eta + \hat F)}}{(8{\text{π}}^2\alpha')^{\frac{1 + p}{2}}} {}_{\rm{0R}}\langle B', \eta'| B, \eta\rangle_{\rm{0R}} \int_0^\infty \frac{{\rm{d}}t}{t^{\frac{9 - p}{2}}} \, {\rm{e}}^{- \frac{y^2}{2{\text{π}} \alpha' t}}\prod\limits_{n = 1}^\infty \left(\frac{1 + \eta' \eta |z|^{2n}}{1 - |z|^{2n}}\right)^{7 + \delta_{p, {\rm{even}}} - p} \\& \times\prod\limits_{\alpha = 0}^{[\frac{p - 1}{2}]} \, \frac{\left(1 + \eta'\eta \lambda_\alpha |z|^{2 n}\right)\left(1 +\eta'\eta \lambda^{-1}_\alpha |z|^{2 n}\right)}{\left(1 - \lambda_\alpha |z|^{2n}\right) \left(1 - \lambda^{-1}_\alpha |z|^{2n}\right)}. \end{split}$

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$\frac{c^2_p}{16 {\text{π}} (2 {\text{π}}^2 \alpha')^{\frac{7 - p}{2}} } = \frac{1}{(8{\text{π}}^2\alpha')^{\frac{1 + p}{2}}}, \;\; \int_{|z|\leqslant 1}\frac{ {\rm{d}}^2 z}{|z|^2} =2{\text{π}}^2 \int_0^\infty {\rm{d}}t, $

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$ {}_{\rm{0R}}\langle B', \eta'| B, \eta\rangle_{\rm{0R}} \; \equiv {}_{\rm{0R}}\langle B_{\rm{sgh}}, \eta'| B_{\rm{sgh}}, \eta\rangle_{\rm{0R}}{}_{\rm{0R}}\langle B'_\psi, \eta'| B_\psi, \eta\rangle_{\rm{0R}}. $

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${}_{\rm{0R}}\langle B', \eta'| B, \eta\rangle_{\rm{0R}} =- \frac{ 2^{4} \,\delta_{\eta\eta', +}}{\sqrt{\det(\eta + \hat F)\det(\eta + \hat F')}} \sum\limits_{n = 0} ^{\left[\frac{p + 1}{2}\right]}\left(\frac{1}{2^{n} n!}\right)^{2} \left(2 n\right)! \hat F_{\left[\alpha_{1}\beta_{1}\right. \cdots} \hat F_{\left.\alpha_{n}\beta_{n}\right]} \hat F'^{\left[\alpha_{1}\beta_{1}\right. \cdots} \hat F'^{\left.\alpha_{n}\beta_{n}\right]}. $

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$ \begin{split} \varGamma_{\rm{NSNS}} = \; & \frac{1}{2} \left[ \varGamma_{\rm{NSNS}} (+) - \varGamma_{\rm{NSNS}} (-)\right] =\frac{ V_{p + 1} \sqrt{\det (\eta + \hat F')\det(\eta + \hat F)}}{2\,(8 {\text{π}}^2 \alpha')^{\frac{1 + p}{2}}}\\ &\times \int_0^\infty \frac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} {\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}\, |z|^{-1} \left[\prod\limits_{n = 1}^{\infty} A_{n} (+) - \prod\limits_{n = 1}^{\infty} A_{n} (-) \right], \end{split} $

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$ \begin{split} \varGamma_{{\rm{RR}}} \; = \; & \frac{1}{2} \left[ \varGamma_{{\rm{RR}}} (+) + \varGamma_{{\rm{RR}}} (-)\right] = \frac{V_{p + 1} \sqrt{\det(\eta + \hat F')\det(\eta + \hat F)}}{2\,(8{\text{π}}^2\alpha')^{\frac{1 + p}{2}}} \\ &\times {}_{\rm{0R}}\langle B', \eta'| B, \eta\rangle_{\rm{0R}} \int_0^\infty \frac{{\rm{d}}t}{t^{\frac{9 - p}{2}}} \, {\rm{e}}^{- \frac{y^2}{2{\text{π}} \alpha' t}} \prod\limits_{n = 1}^{\infty} B_{n} (+), \end{split} $

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$\begin{split} & A_{n} (\pm) = \left(\frac{1 \pm |z|^{2n - 1}}{1 - |z|^{2n}}\right)^{7 + \delta_{p, {\rm{even}}} - p} \prod\limits_{\alpha = 0}^{[\frac{p - 1}{2}]} \, \frac{\left(1 \pm \lambda_\alpha |z|^{2 n - 1}\right)\left(1 \pm \lambda^{-1}_\alpha |z|^{2 n - 1}\right)}{\left(1 - \lambda_\alpha |z|^{2n}\right) \left(1 - \lambda^{-1}_\alpha |z|^{2n}\right)}, \\ & B_{n} (+) = \left(\frac{1 + |z|^{2n}}{1 - |z|^{2n}}\right)^{7 + \delta_{p, {\rm{even}}} - p} \prod\limits_{\alpha = 0}^{[\frac{p - 1}{2}]} \, \frac{\left(1 + \lambda_\alpha |z|^{2 n}\right)\left(1 + \lambda^{-1}_\alpha |z|^{2 n}\right)}{\left(1 - \lambda_\alpha |z|^{2n}\right) \left(1 - \lambda^{-1}_\alpha |z|^{2n}\right)}. \end{split}$

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$ \begin{split} \varGamma_{\rm{NSNS}} = \; & \dfrac{ 2^{\left[\frac{p + 1}{2}\right]} \, V_{p + 1} \sqrt{\det (\eta + \hat F')\det(\eta + \hat F)}}{2\,(8 {\text{π}}^2 \alpha')^{\frac{1 + p}{2}}}\int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} \dfrac{{\rm{d}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}}{\left[\eta ({i} t)\right]^{12 - 3 \left[\frac{p + 1}{2}\right]}} \, \prod\limits_{\alpha = 0}^{\left[\frac{p-1}{2}\right]} \dfrac{\sin ({\text{π}} \nu_{\alpha})}{\theta_{1} (\nu_{\alpha} | it) } \\ & \times \left[\left[\theta_{3} (0 | it) \right]^{4 - \left[\frac{p + 1}{2}\right]} \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \theta_{3} (\nu_{\alpha} | it) - \left[\theta_{4} (0 | it)\right]^{4 - \left[\frac{p + 1}{2}\right]} \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \theta_{4} (\nu_{\alpha} | it)\right]. \end{split} $

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$ \begin{split} \varGamma_{{\rm{R}}R} \; &= \frac{2^{\left[\frac{p + 1}{2}\right] - 4} \,V_{p + 1} \sqrt{\det(\eta + \hat F')\det(\eta + \hat F)}}{2\,(8{\text{π}}^2\alpha')^{\frac{1 + p}{2}}} {}_{\rm{0R}}\langle B', \eta'| B, \eta\rangle_{\rm{0R}} \int_0^\infty \frac{{\rm{d}}t}{t^{\frac{9 - p}{2}}} \, \frac{{\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}}{\left[\eta (it)\right]^{12 - 3 \left[\frac{p + 1}{2}\right]}} \\ & \!\!\!\!\!\! \times \left[\theta_{2} (0 | it) \right]^{4 - \left[\frac{p + 1}{2}\right]} \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \frac{ \theta_{2} (\nu_{\alpha} | it)}{\theta_{1} (\nu_{\alpha} | it) } \, \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \frac{\sin ({\text{π}} \nu_{\alpha})}{\cos({\text{π}} \nu_{\alpha})}, \end{split} $

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$ {}_{\rm{0R}}\langle B', \eta'| B, \eta\rangle_{\rm{0R}} = - 2^{4}\,\delta_{\eta\eta', +}\, \prod\limits_{\alpha = 0}^{\left[\frac{p -1}{2}\right]} \cos({\text{π}} \nu_{\alpha}). $

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$ \begin{split} \; & \varGamma_{p, p} = \varGamma_{\rm{NSNS}} + \varGamma_{{\rm{RR}}}, = \frac{ 2^{\left[\frac{p + 1}{2}\right]} \, V_{p + 1} \sqrt{\det (\eta + \hat F')\det(\eta + \hat F)}}{2\,(8 {\text{π}}^2 \alpha')^{\frac{1 + p}{2}}} \int_0^\infty \frac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} \frac{{\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}}{\left[\eta (it)\right]^{12 - 3 \left[\frac{p + 1}{2}\right]}} \, \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \frac{\sin ({\text{π}} \nu_{\alpha})}{\theta_{1} (\nu_{\alpha} | it) } \\ & ~~ \times \left[\left[\theta_{3} (0 | it) \right]^{4 - \left[\frac{p + 1}{2}\right]} \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \theta_{3} (\nu_{\alpha} | it) - \left[\theta_{4} (0 | it)\right]^{4 - \left[\frac{p + 1}{2}\right]} \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \theta_{4} (\nu_{\alpha} | it) - \left[\theta_{2} (0 | it) \right]^{4 - \left[\frac{p + 1}{2}\right]} \prod\limits_{\alpha = 0}^{\left[\frac{p - 1}{2}\right]} \theta_{2} (\nu_{\alpha} | it)\right]. \end{split}$

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$ \begin{split} 2 \,\theta_{1} (w |\tau) \theta_{1} (x |\tau) \theta_{1} (y |\tau) \theta_{1} (z |\tau) = \;& \theta_{1} (w' |\tau) \theta_{1} (x' |\tau) \theta_{1} (y' |\tau) \theta_{1} (z' |\tau) + \theta_{2} (w' |\tau) \theta_{2} (x' |\tau) \theta_{2} (y' |\tau) \theta_{2} (z' |\tau) \\ & - \theta_{3} (w' |\tau) \theta_{3} (x' |\tau) \theta_{3} (y' |\tau) \theta_{3} (z' |\tau) + \theta_{4} (w' |\tau) \theta_{4} (x' |\tau) \theta_{4} (y' |\tau) \theta_{4} (z' |\tau), \end{split} $

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$ \begin{split} & 2 w' \;=\; - w + x + y + z, ~~ 2 x' \;=\; w - x + y + z,~~ 2 y' \;=\; w + x - y + z,~~ 2 z' \;=\; w + x + y - z. \end{split}$

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$ \begin{split} \varGamma_{p,p} = & \dfrac{ 2^3 \, V_{p + 1}\, \left[\det (\eta + \hat F')\det(\eta + \hat F)\right]^{\frac{1}{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{(8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} \dfrac{{\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}}{\eta^{3} (it)} \\ & \times \dfrac{ \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2}\right| it \right)\theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2}\right| it \right)}{\theta_{1} (\nu_{0} | it)\theta_{1} (\nu_{1} | it)\theta_{1} (\nu_{2} | it)}\\ & \!\!\!\!\!\!\!\!\!\!\!= \dfrac{ 2^2 \, V_{p+ 1} \left[\det (\eta + \hat F')\det(\eta + \hat F)\right]^{\frac{1}{2}} \left[ \sum\limits_{\alpha = 0}^{2}\cos^{2}({\text{π}} \nu_{\alpha}) - 2\prod\limits_{\alpha = 0}^{2} \cos({\text{π}}\nu_{\alpha}) - 1\right]}{(8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t}{t^{\frac{9 - p}{2}}} {\rm{e}}^{- \dfrac{y^2}{2{\text{π}}\alpha' t}} \prod\limits_{n = 1}^{\infty} C_{n}, \end{split}$

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$ C_{n} = \frac{\tilde C_{n} }{(1 - |z|^{2n})^{2} \prod\limits_{\alpha = 0}^{2} \left[1 - 2 |z|^{2n} \cos(2{\text{π}} \nu_{\alpha}) + |z|^{4n}\right]}, $

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$ \begin{split} \tilde C_{n} = \; & \left[1 - 2 |z|^{2n} \cos[{\text{π}}(\nu_{0} + \nu_{1} + \nu_{2})] + |z|^{4n}\right] \left[1 - 2 |z|^{2n} \cos[{\text{π}}(\nu_{0} - \nu_{1} + \nu_{2})] + |z|^{4n}\right] \\ & \times \left[1 - 2 |z|^{2n} \cos[{\text{π}}(\nu_{0} + \nu_{1} - \nu_{2})] + |z|^{4n}\right] \left[1 - 2 |z|^{2n} \cos[{\text{π}}(\nu_{0} - \nu_{1} - \nu_{2})] + |z|^{4n}\right] \qquad\quad \end{split} $

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$ \begin{split} \stackrel{{\text{或}}}{ = } \; & \left[1 - 2 |z|^{2n} {\rm{e}}^{{\rm{i}}{\text{π}}\nu_{0}} \cos[{\text{π}}(\nu_{1} + \nu_{2})] + {\rm{e}}^{2{\text{π}} {\rm{i}} \nu_{0}} |z|^{4n}\right]\left[1 - 2 |z|^{2n} {\rm{e}}^{{\rm{i}}{\text{π}}\nu_{0}} \cos[{\text{π}}(\nu_{1} - \nu_{2})] + {\rm{e}}^{2{\text{π}} {\rm{i}} \nu_{0}} |z|^{4n}\right] \\ & \times \left[1 - 2 |z|^{2n} {\rm{e}}^{- {\rm{i}}{\text{π}}\nu_{0}} \cos[{\text{π}}(\nu_{1} + \nu_{2})] + {\rm{e}}^{- 2{\text{π}} {\rm{i}} \nu_{0}} |z|^{4n}\right]\left[1 - 2 |z|^{2n} {\rm{e}}^{- {\rm{i}}{\text{π}}\nu_{0}} \cos[{\text{π}}(\nu_{1} - \nu_{2})] + {\rm{e}}^{-2{\text{π}} {\rm{i}} \nu_{0}} |z|^{4n}\right]. \end{split} $

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$ \begin{split} & \varGamma_{p,p} = - \dfrac{ 2^3 \, i \, V_{p + 1}\, \left[\det (\eta + \hat F')\det(\eta + \hat F)\right]^{\frac{1}{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{(8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t'} {t'^{\frac{p - 3}{2}}} \dfrac{{\rm{e}}^{- \frac{y^2 t'}{2{\text{π}}\alpha'}}}{\eta^{3} (it')} \\ &\qquad \times \dfrac{ \theta_{1} \left(\left. \dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2} i t' \right| it' \right) \theta_{1} \left(\left. \dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2} it' \right| it' \right) \theta_{1} \left(\left. \dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2} it' \right| it' \right)\theta_{1} \left(\left. \dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2} it' \right| it' \right)}{\theta_{1} (\nu_{0} it' | it')\theta_{1} ( \nu_{1} it' | it')\theta_{1} ( \nu_{2} it' | it')}\\ = & \dfrac{2^2 \, V_{p+ 1} \big[\det (\eta + \hat F')\det(\eta + \hat F)\big]^{\frac{1}{2}} }{(8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \int_0^\infty \!\! \dfrac{{\rm{d}} t}{t^{\frac{p - 3}{2}}} {\rm{e}}^{- \frac{y^2 t}{2{\text{π}}\alpha'}} \prod\limits_{\alpha = 0}^{2} \dfrac{\sin ({\text{π}} \nu_{\alpha})}{\sinh ({\text{π}} \nu_{\alpha} t)} \left[ \sum\limits_{\alpha = 0}^{2}\cosh^{2}({\text{π}} \nu_{\alpha} t) \!-\! 2 \prod\limits_{\alpha = 0}^{2} \cosh({\text{π}} \nu_{\alpha} t) \!-\! 1\right] \!\!\prod\limits_{n = 1}^{\infty} Z_{n}, \end{split}$

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$ Z_{n} = \frac{\tilde Z_{n} }{(1 - |z|^{2n})^{2} \prod\limits_{\alpha = 0}^{2} \left[1 - 2 |z|^{2n} \cosh(2{\text{π}} \nu_{\alpha} t) + |z|^{4n}\right]}, $

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$ \begin{split} \tilde Z_{n} \; = \;& \left[1 - 2 |z|^{2n} \cosh[{\text{π}}(\nu_{0} + \nu_{1} + \nu_{2}) t] + |z|^{4n}\right]\left[1 - 2 |z|^{2n} \cosh[{\text{π}}(\nu_{0} - \nu_{1} + \nu_{2})t] + |z|^{4n}\right] \\ & \times \left[1 - 2 |z|^{2n} \cosh[{\text{π}}(\nu_{0} + \nu_{1} - \nu_{2})t] + |z|^{4n}\right]\left[1 - 2 |z|^{2n} \cosh[{\text{π}}(\nu_{0} - \nu_{1} - \nu_{2})t] + |z|^{4n}\right] \qquad \qquad\qquad \end{split} $

(53)

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$ \begin{split} \stackrel{{\text{或}}}{ = } \; & \left[1 - 2 |z|^{2n} {\rm{e}}^{-{\text{π}}\nu_{0}t} \cosh[{\text{π}}(\nu_{1} + \nu_{2})t] + {\rm{e}}^{-2{\text{π}} \nu_{0} t} |z|^{4n}\right] \left[1 - 2 |z|^{2n} {\rm{e}}^{- {\text{π}}\nu_{0} t} \cosh[{\text{π}}(\nu_{1} - \nu_{2})t] + {\rm{e}}^{- 2{\text{π}} \nu_{0} t} |z|^{4n}\right] \\&\times \left[1 - 2 |z|^{2n} {\rm{e}}^{{\text{π}}\nu_{0} t} \cosh[{\text{π}}(\nu_{1} + \nu_{2})t] + {\rm{e}}^{ 2{\text{π}} \nu_{0}t} |z|^{4n}\right] \left[1 - 2 |z|^{2n} {\rm{e}}^{{\text{π}}\nu_{0}t} \cosh[{\text{π}}(\nu_{1} - \nu_{2}) t] + {\rm{e}}^{2{\text{π}} \nu_{0}t} |z|^{4n}\right]. \end{split}$

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$\eta (\tau) = \frac{1}{(- {\rm{i}} \tau)^{1/2}} \eta \left(- \frac{1}{\tau}\right), \;\; \theta_1 (\nu | \tau) = {\rm{i}} \frac{{\rm{e}}^{- {\rm{i}} {\text{π}} \nu^2/\tau}}{(- {\rm{i}} \tau)^{1/2}} \theta_1 \left(\left.\frac{\nu}{\tau} \right| - \frac{1}{\tau}\right),$

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$ \begin{split} \varGamma_{p,p} = \; & \dfrac{ 2^2 \, V_{p+ 1} \left[\det (\eta + \hat F')\det(\eta + \hat F)\right]^{\frac{1}{2}} }{(8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t}{t^{\frac{p - 3}{2}}} {\rm{e}}^{- \frac{y^2 t}{2{\text{π}}\alpha'}} \dfrac{\sinh({\text{π}}\bar\nu_{0})}{\sin({\text{π}}\bar\nu_{0} t)} \prod\limits_{\alpha = 1}^{2} \frac{\sin ({\text{π}} \nu_{\alpha})}{\sinh ({\text{π}} \nu_{\alpha} t)} \\&\times \left[\left(\cosh({\text{π}}\nu_{1} t) - \cosh({\text{π}}\nu_{2} t) \right)^{2} + 4 \sin^{2}\frac{{\text{π}}\bar\nu_{0} t}{2} \left(\cosh({\text{π}} \nu_{1} t) \cosh({\text{π}}\nu_{2} t) - \cos^{2} \frac{{\text{π}}\bar\nu_{0}t}{2}\right)\right]\prod\limits_{n = 1}^{\infty} Z_{n},\end{split} $

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$ \begin{split} {\cal{W}}_{p, p} = \; & - \dfrac{2 \,{\rm{Im}} \varGamma}{V_{p + 1}} = \dfrac{ 2^3 \, \left[\det (\eta + \hat F')\det(\eta + \hat F) \right]^{\frac{1}{2}} \sinh ({\text{π}} \bar\nu_{0}) \sin({\text{π}}\nu_{1}) \sin({\text{π}}\nu_{2})}{\bar \nu_{0} (8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}}\\ & \times \sum\limits_{k = 1}^{\infty} (-)^{k + 1} \left(\frac{\bar\nu_{0}}{k}\right)^{\frac{p - 3}{2}} {\rm{e}}^{- \frac{k y^2 }{2{\text{π}}\bar\nu_{0} \alpha'}} \frac{\left(\cosh \frac{k{\text{π}} \nu_{1}}{\bar\nu_{0}} - (-)^{k} \cosh \frac{k {\text{π}} \nu_{2}}{\bar\nu_{0}} \right)^{2}}{\sinh\frac{k {\text{π}}\nu_{1}}{\bar\nu_{0}} \sinh\frac{k{\text{π}}\nu_{2}}{\bar\nu_{0}}} Z_{k} (\bar\nu_{0}, \nu_{1},\nu_{2}), \end{split} $

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$Z_{k} (\bar\nu_{0}, \nu_{1}, \nu_{2}) = \prod\limits_{n = 1}^{\infty}\dfrac{[1 - 2 (-)^{k} |z_{k}|^{2n} \cosh\dfrac{k {\text{π}}(\nu_{1} + \nu_{2})}{\bar\nu_{0}} + |z_{k}|^{4n}]^{2} [1 - 2 (-)^{k}|z_{k}|^{2n} \cosh\dfrac{k{\text{π}}(\nu_{1} - \nu_{2})}{\bar\nu_{0}} + |z_{k}|^{4n}]^{2} }{(1 - |z_{k}|^{2n})^{4} [1 - 2 |z_{k}|^{2n} \cosh \dfrac{2k{\text{π}}\nu_{1}}{\bar\nu_{0}} + |z_{k}|^{4n}] [1 - 2 |z_{k}|^{2n} \cosh \dfrac{2k{\text{π}}\nu_{2}}{\bar\nu_{0}} + |z_{k}|^{4n}]},$

(58)

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${\cal{W}}^{(1)}_{p, p} = \dfrac{ 2^3 \, \left[\det (\eta + \hat F')\det(\eta + \hat F) \right]^{\frac{1}{2}} \sinh ({\text{π}} \bar\nu_{0}) \sin({\text{π}}\nu_{1}) \sin({\text{π}}\nu_{2})}{\bar \nu_{0} (8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \bar\nu_{0}^{\frac{p - 3}{2}} {\rm{e}}^{- \frac{ y^2 }{2{\text{π}}\bar\nu_{0} \alpha'}} \dfrac{\left(\cosh \frac{{\text{π}} \nu_{1}}{\bar\nu_{0}} + \cosh \frac{ {\text{π}} \nu_{2}}{\bar\nu_{0}} \right)^{2}}{\sinh\dfrac{{\text{π}}\nu_{1}}{\bar\nu_{0}} \sinh\dfrac{{\text{π}}\nu_{2}}{\bar\nu_{0}}} Z_{1} (\bar\nu_{0}, \nu_{1},\nu_{2}).$

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$ { M} = ([(\eta - \hat{F})(\eta + \hat{F})^{-1}]_{\alpha\beta}, - \delta_{ij}), $

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$ { M}_{p} = ([(\eta_{p} - \hat{F}_{p})(\eta_{p} + \hat{F}_{p})^{-1}]_{\alpha'\beta'}, - \delta_{i'j'}). $

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$\begin{split}& {({ F_6})_{\alpha \beta }} \\=\; & {\left( {\begin{array}{*{20}{c}} {{{({{ F}_p})}_{\alpha '\beta '}}}&{}&{}&{}&{}&{}\\ {}&0&{{{\hat g}_1}}&{}&{}&{}\\ {}&{ - {{\hat g}_1}}&0&{}&{}&{}\\ {}&{}&{}& \ddots &{}&{}\\ {}&{}&{}&{}&0&{{{\hat g}_{\frac{{6 - p}}{2}}}}\\ {}&{}&{}&{}&{ - {{\hat g}_{\frac{{6 - p}}{2}}}}&0 \end{array}} \right)_{7 \times 7}}.\end{split} $

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$ {({{ M}_6})_{\alpha \beta }} = \left( {\begin{array}{*{20}{c}} {{{({{ M}_p})}_{\alpha '\beta '}}}&{}&{}&{}&{}&{}\\ {}&{\dfrac{{1 - \hat g_1^2}}{{1 + \hat g_1^2}}}&{\dfrac{{2{\mkern 1mu} {{\hat g}_1}}}{{1 + \hat g_1^2}}}&{}&{}&{}\\ {}&{\dfrac{{2{\mkern 1mu} {{\hat g}_1}}}{{1 + \hat g_1^2}}}&{\dfrac{{1 - \hat g_1^2}}{{1 + \hat g_1^2}}}&{}&{}&{}\\ {}&{}&{}& \ddots &{}&{}\\ {}&{}&{}&{}&{\dfrac{{1 - \hat g_{\frac{{6 - p}}{2}}^2}}{{1 + \hat g_{\frac{{6 - p}}{2}}^2}}}&{\frac{{2{\mkern 1mu} {{\hat g}_{\frac{{6 - p}}{2}}}}}{{1 + \hat g_{\frac{{6 - p}}{2}}^2}}}\\ {}&{}&{}&{}&{\dfrac{{2{\mkern 1mu} {{\hat g}_{\frac{{6 - p}}{2}}}}}{{1 + \hat g_{\frac{{6 - p}}{2}}^2}}}&{\dfrac{{1 - \hat g_{\frac{{6 - p}}{2}}^2}}{{1 + \hat g_{\frac{{6 - p}}{2}}^2}}} \end{array}} \right).$

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$ \dfrac{ \theta_{1} \left(\left.\frac{\nu_{0} + \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2}\right| it \right)\theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2}\right| it \right)}{\theta_{1} (\nu_{0} | it)\theta_{1} (\nu_{1} | it)\theta_{1} (\nu_{2} | it) \eta^{3} (it)} \prod\limits_{\alpha = 0}^{2} \sin {\text{π}} \nu_{\alpha}. $

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$ \sqrt{\det(\eta_{p} + \hat F'_{p})\det(\eta_{p} + \hat F_{p})} \,V_{\rm{NN}} \left(2 {\text{π}}^{2} \alpha' t\right)^{- \frac{{\rm{DD}}}{2}} {\rm{e}}^{- \frac{y^{2}}{2{\text{π}}\alpha' t}}, $

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$ \begin{split} & F' = \left( {\begin{array}{*{20}{c}} 0&{ - \hat f'}&0&0&0& \cdots \\ {\hat f'}&0&0&0&0& \cdots \\ 0&0&0&{ - \hat g'}&0& \cdots \\ 0&0&{\hat g'}&0&0& \cdots \\ 0&0&0&0&0& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots &{} \end{array}} \right),\\ & F = \left( {\begin{array}{*{20}{c}} 0&{ - \hat f}&0&0&0& \cdots \\ {\hat f}&0&0&0&0& \cdots \\ 0&0&0&{ - \hat g}&0& \cdots \\ 0&0&{\hat g}&0&0& \ldots \\ 0&0&0&0&0& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}} \right). \end{split} $

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$ \begin{split} {\cal{W}}^{(1)} \;= \; &\dfrac{8\, |\hat f - \hat f'||\hat g - \hat g'|}{(8{\text{π}}^2 \alpha')^{\frac{1 + p}{2}}} \bar\nu_0^{\frac{p - 3}{2}} \dfrac{\left[\cosh\dfrac{{\text{π}} \nu_{1}}{\bar\nu_0} + 1\right]^2}{\sinh \dfrac{{\text{π}} \nu_1}{\bar\nu_0}} \\ & \times {\rm{e}}^{- \tfrac{ y^2}{2{\text{π}} \alpha'\bar \nu_0}} Z (\bar \nu_{0}, \nu_{1}),\\[-10pt] \end{split}$

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$ Z (\bar\nu_{0}, \nu_{1}) = \prod\limits_{n = 1}^\infty \frac{\Big [1 + {\rm{e}}^{- \tfrac{2 n {\text{π}}}{\bar\nu_0} \left(1 - \tfrac{\nu_1}{2 n}\right)}\Big]^4 \Big[1 + {\rm{e}}^{- \tfrac{2 n {\text{π}}}{\bar\nu_0}\left(1 + \tfrac{\nu_1}{2 n}\right)}\Big]^4}{\Big(1 - {\rm{e}}^{- \tfrac{2 n {\text{π}}}{\bar\nu_0}}\Big)^6 \Big[1 - \, {\rm{e}}^{- \tfrac{2 n {\text{π}}}{\bar\nu_0} \left(1 - \tfrac{\nu_1}{ n}\right)}\Big] \Big[1 - \, {\rm{e}}^{- \tfrac{2 n {\text{π}}}{\bar\nu_0}\left(1 + \tfrac{\nu_1}{ n}\right)}\Big]}. $

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$ \tanh({\text{π}}\bar\nu_0) = \dfrac{|\hat f - \hat f'|}{1- \hat f \hat f'},\quad \tan ({\text{π}}\nu_1) = \dfrac{|\hat g - \hat g'|}{1 + \hat g \hat g'}. $

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$ \begin{split} & \left(2{\text{π}} \alpha'\right)^{\frac{1 + p}{2}} {\cal{W}}^{(1)} \\ \approx \; & \dfrac{ \bar\nu_{0} \nu_{1}}{2}\left(\frac{\bar\nu_0}{4{\text{π}}}\right)^{\frac{p - 3}{2}} \dfrac{\left[\cosh\dfrac{{\text{π}} \nu_{1}}{\bar\nu_0} + 1\right]^2}{\sinh \dfrac{{\text{π}} \nu_1}{\bar\nu_0}} \, {\rm{e}}^{- \tfrac{ y^2}{2{\text{π}} \alpha'\bar\nu_0}}. \end{split} $

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$ {\cal{W}}^{(1)} = \dfrac{2\,(e E) (e B)}{(2{\text{π}})^{2}} \dfrac{\left[\cosh\dfrac{{\text{π}} B}{E} + 1\right]^2}{\sinh \dfrac{{\text{π}} B}{E}} \, {\rm{e}}^{- \tfrac{{\text{π}} m^{2}}{e E}}, $

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$ m = T_{f} y = \dfrac{y}{2{\text{π}}\alpha'}. $

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$ \alpha '{M^2} = - \alpha '{p^2} = \begin{cases}{\dfrac{y^{2}}{4{\text{π}}^{2} \alpha'} + N_{\rm{R}}}, & { ({\rm{R}}{\text{}}{\text{分部}}),}\\ {\dfrac{y^{2}}{4{\text{π}}^{2} \alpha'} + N_{\rm{NS}} - \dfrac{1}{2}}, & {({\rm{NS}}{\text{}}{\text{分部}}),} \end{cases}$

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$ \begin{split} &N_{\rm{R}} = \sum\limits_{n = 1}^{\infty} (\alpha_{- n} \cdot \alpha_{n} + n d_{-n} \cdot d_{n}),\\ &N_{\rm{NS}} = \sum\limits_{n = 1}^{\infty} \alpha_{- n} \cdot \alpha_{n} + \sum\limits_{r = 1/2}^{\infty} r d_{- r} \cdot d_{r}. \end{split} $

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$ {\cal{W}}_{\rm{scalar}} = \dfrac{(e E) (e B)}{2 (2{\text{π}})^{2}} {\rm{csch}} \left(\dfrac{{\text{π}} B}{E}\right) \, {\rm{e}}^{- \tfrac{{\text{π}} m^{2}_{0}}{e E}}, $

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$ {\cal{W}}_{\rm{spinor}} = \dfrac{(e E) (e B)}{(2{\text{π}})^{2}} \coth\left(\dfrac{{\text{π}} B}{E}\right) \, {\rm{e}}^{- \tfrac{{\text{π}} m_{1/2}^{2}}{e E}}, $

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$ {\cal{W}}_{\rm{vector}} = \dfrac{(e E) (e B)}{ 2 (2 {\text{π}})^{2}} \frac{ 2 \cosh \dfrac{2 {\text{π}} B}{E} + 1}{ \sinh \dfrac{{\text{π}} B}{E}} \, {\rm{e}}^{- \tfrac{{\text{π}} m^{2}_{1}}{e E}}. $

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$\begin{split} {\cal{W}}^{(1)} \;=\;& 16\, {\cal{W}}_{\rm{scalar}} = 8\, {\cal{W}}_{\rm{spinor}}\\ = \;&\frac{16}{3} \, {\cal{W}}_{\rm{vector}} = \frac{ 8 (e E)^{2}}{(2 {\text{π}})^2 } \, {\rm{e}}^{- \tfrac{ {\text{π}} m^{2}}{e E }}. \end{split}$

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$ \begin{split} &{\cal{W}}^{(1)} \approx \frac{ (e E) (e B)}{(2 {\text{π}})^2 } \, {\rm{e}}^{- \tfrac{ {\text{π}} (m^{2} - e B)}{e E }}, \\ & {\cal{W}}_{\rm{vector}} \approx \frac{(e E) (e B)}{ (2 {\text{π}})^{2}} \, {\rm{e}}^{- \tfrac{{\text{π}} (m^{2}_{1} - e B)}{e E}}, \\ &{\cal{W}}_{\rm{scalar}} \approx \frac{(e E) (e B)}{ (2{\text{π}})^{2}} \, {\rm{e}}^{- \tfrac{{\text{π}} (m^{2}_{0} + e B)}{e E}},\\ &{\cal{W}}_{\rm{spinor}} \approx \frac{(e E) (e B)}{(2{\text{π}})^{2}} \, {\rm{e}}^{- \tfrac{{\text{π}} m_{1/2}^{2}}{e E}}. \end{split}$

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$ {{E}}^{2}_{\rm{S}} = (2 N + 1) e B - g_{\rm{S}}\; e\; B \cdot S + m^{2}_{\rm{s}}, $

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$ \begin{split} & {{E}}^{2}_{S = 0} = e B + m^{2}_{0}, \\ &{{E}}^{2}_{S = 1/2} = m^{2}_{1/2}, \\ & {{E}}^{2}_{S = 1} = - e B + m^{2}_{1}. \end{split} $

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$ {\cal{W}}_{\rm{s}} = \frac{ (e E) (e B)}{(2 {\text{π}})^2 } {\rm{e}}^{- \tfrac{{\text{π}} {\rm{E}}^{2}_{\rm{S}}}{e E}}, $

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$ {\cal{W}}^{(1)} = 5 {\cal{W}}_{\rm{scalar}} + 4 {\cal{W}}_{\rm{spinor}} + {\cal{W}}_{\rm{vector}} . $

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$ \begin{split} &{ F'_{\alpha \beta }}= \\ &{\left( {\begin{array}{*{20}{c}} {{{({{ F'}_{p'}})}_{\alpha '\beta '}}}\! & \!{}\! & \!{}\! & \!{}\! & \!{}\! & \!{}\\ {}\! & \!0\! & \!{{{\hat g'}_1}}\! & \!{}\! & \!{}\! & \!{}\\ {}\! & \!{ - {{\hat g'}_1}}\! & \!0\! & \!{}\! & \!{}\! & \!{}\\ {}\! & \!{}\! & \!{}\! & \! \ddots \! & \!{}\! & \!{}\\ {}\! & \!{}\! & \!{}\! & \!{}\! & \!0\! & \!{{{\hat g'}_{\frac{\kappa }{2}}}}\\ {}\! & \!{}\! & \!{}\! & \!{}\! & \!{ - {{\hat g'}_{\frac{\kappa }{2}}}}\! & \!0 \end{array}} \right)_{(p + 1) \times (p + 1)}},\end{split}$

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$ \sqrt{\det(\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})} \,V_{\rm{NN}} \left(2 {\text{π}}^{2} \alpha' t\right)^{- \frac{{\rm{DD}}}{2}} {\rm{e}}^{- \frac{y^{2}}{2{\text{π}}\alpha' t}}. $

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$ \sqrt{\det(\eta_{p} + \hat F'_{p})\det(\eta_{p} + \hat F_{p})} \,V_{{\rm{p}} + 1} \left(2 {\text{π}}^{2} \alpha' t\right)^{- \frac{9 - p}{2}} {\rm{e}}^{- \frac{y^{2}}{2{\text{π}}\alpha' t}}. $

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$\begin{split} &\sqrt{\det(\eta_{p} + \hat F'_{p})\det(\eta_{p} + \hat F_{p})} \\= \;& \hat g'_{1} \cdots \hat g'_{\kappa/2} \sqrt{\det(\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})}. \end{split}$

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$ T_{p} \int (C_{p' + 1} \wedge \hat F'\wedge \hat F'\cdots \wedge \hat F')_{p + 1}, $

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$ T_{p} V_{\kappa} \hat g'_{1} \cdots \hat g'_{\kappa/2} \int C_{p' + 1}, $

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$ N T_{p'} = T_{p} V_{\kappa} \hat g'_{1} \cdots \hat g'_{\kappa/2}, $

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$ V_{\kappa} = \frac{N}{\hat g'_{1} \cdots \hat g'_{\kappa/2}} \frac{T_{p'}}{T_{p}}. $

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$ \begin{split} &\sqrt{\det(\eta_{p} + \hat F'_{p})\det(\eta_{p} + \hat F_{p})} \\ &\times V_{{{p}} + 1} \left(2 {\text{π}}^{2} \alpha' t\right)^{- \tfrac{9 - p}{2}} {\rm{e}}^{- \tfrac{y^{2}}{2{\text{π}}\alpha' t}}\\ = \;&\hat g'_{1} \cdots \hat g'_{\kappa/2} \sqrt{\det(\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})} \\ & \times V_{p' + 1} \frac{N}{\hat g'_{1} \cdots \hat g'_{\kappa/2}} \frac{T_{p'}}{T_{p}} \left(2 {\text{π}}^{2} \alpha' t\right)^{- \tfrac{9 - p}{2}} {\rm{e}}^{- \tfrac{y^{2}}{2{\text{π}}\alpha' t}}\\ =\;& N \frac{c_{p'}}{c_{p}} \sqrt{\det(\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})}\\ & \times V_{p' + 1} \left(2 {\text{π}}^{2} \alpha' t\right)^{- \tfrac{9 - p}{2}} {\rm{e}}^{- \tfrac{y^{2}}{2{\text{π}}\alpha' t}}, \end{split}$

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$ \begin{split} \varGamma_{p,p'}\; = \; &{\varGamma_{p, p}}/{N}\\ =\;& \dfrac{ 2^3 \, V_{p + 1}\, \left[\det (\eta_{p} + \hat F'_{p})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{N (8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} \dfrac{{\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}}{\eta^{3} (it)} \\ & \times \dfrac{ \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2}\right| it \right)\theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2}\right| it \right)}{\theta_{1} (\nu_{0} | it)\theta_{1} (\nu_{1} | it)\theta_{1} (\nu_{2} | it)}\\ =\;& \dfrac{c_{p'}}{c_{p}} \dfrac{ 2^3 \, V_{p' + 1}\, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{ (8 {\text{π}}^2 \alpha')^{\frac{p + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} \dfrac{{\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}}{\eta^{3} (it)} \\ & \times \dfrac{ \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2}\right| it \right)\theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2}\right| it \right)}{\theta_{1} (\nu_{0} | it)\theta_{1} (\nu_{1} | it)\theta_{1} (\nu_{2} | it)} \end{split}, $

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$ \begin{split} \varGamma_{p,p'}\; = \; &{\varGamma_{p, p}}/{N}\\ =\;& \dfrac{ 2^3 \, V_{p' + 1}\, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{ 2^{\frac{\kappa}{2}} (8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} \dfrac{{\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}}}{\eta^{3} (it)} \\ & \times \dfrac{ \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2}\right| it \right)\theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2}\right| it \right)}{\theta_{1} (\nu_{0} | it)\theta_{1} (\nu_{1} | it)\theta_{1} (\nu_{2} | it)}\\ = \;& \dfrac{ 2^2 \, V_{p'+ 1} \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} \left[ \sum\limits_{\alpha = 0}^{2}\cos^{2}({\text{π}} \nu_{\alpha}) - 2\prod\limits_{\alpha = 0}^{2} \cos({\text{π}}\nu_{\alpha}) - 1\right]}{ 2^{\frac{\kappa}{2}}(8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \\ & \times \int_0^\infty \dfrac{{\rm{d}} t}{t^{\frac{9 - p}{2}}} {\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}} \prod\limits_{n = 1}^{\infty} C_{n}, \end{split} $

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$ \begin{split} \varGamma_{p,p'} \; =\; & - \dfrac{ 2^3 \, i \, V_{p' + 1}\, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{{1}/{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{2^{\frac{\kappa}{2}}(8 {\text{π}}^2 \alpha')^{\tfrac{p' + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{p - 3}{2}}} \dfrac{{\rm{e}}^{- \tfrac{y^2 t}{2{\text{π}}\alpha' }}}{\eta^{3} (it)} \\ & \times \dfrac{ \theta_{1} \left(\left. \dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2} i t \right| it \right) \theta_{1} \left(\left. \dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2} it \right| it \right) \theta_{1} \left(\left. \dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2} it \right| it \right)\theta_{1} \left(\left. \dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2} it \right| it \right)}{\theta_{1} (\nu_{0} it | it)\theta_{1} ( \nu_{1} it | it)\theta_{1} ( \nu_{2} it | it)}\\ =\; & \dfrac{ 2^2 \, V_{p'+ 1} \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} }{2^{\frac{\kappa}{2}} (8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t}{t^{\frac{p - 3}{2}}} {\rm{e}}^{- \tfrac{y^2 t}{2{\text{π}}\alpha'}} \prod\limits_{\alpha = 0}^{2} \dfrac{\sin ({\text{π}} \nu_{\alpha})}{\sinh ({\text{π}} \nu_{\alpha} t)} \\ & \times \left[ \sum\limits_{\alpha = 0}^{2}\cosh^{2}({\text{π}} \nu_{\alpha} t)- 2 \prod\limits_{\alpha = 0}^{2} \cosh({\text{π}} \nu_{\alpha} t) - 1\right] \prod\limits_{n = 1}^{\infty} Z_{n}, \\[-15pt] \end{split}$

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$ \sim {\rm{e}}^{- \tfrac{y^2 t}{2{\text{π}}\alpha'}} {\rm{e}}^{{\text{π}} (\nu_{2} - \nu_{1} - \nu_{0}) t} = {\rm{e}}^{- 2{\text{π}} t\left(\tfrac{y^{2}}{(2{\text{π}})^{2} \alpha'} - \tfrac{\nu_{2} - \nu_{1} - \nu_{0}}{2}\right)}, $

(96)

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$ \begin{split} {\cal{W}}_{p, p'} = \;& - \frac{2\, {\rm{Im}} \varGamma}{V_{p' + 1}}\; =\; \dfrac{ 2^{3- {\kappa}/{2}} \, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} \sinh({\text{π}}\bar\nu_{0}) \sin({\text{π}}\nu_{1}) \sin({\text{π}}\nu_{2})} {\bar\nu_{0} (8 {\text{π}}^2 \alpha')^{\tfrac{p' + 1}{2}}} \sum\limits_{k = 1}^{\infty} (-)^{k + 1} \left(\dfrac{\bar\nu_{0}}{k}\right)^{\tfrac{p - 3}{2}} \\ &\times \dfrac{\left(\cosh\dfrac{k {\text{π}} \nu_{1}}{\bar\nu_{0}} - (-)^{k} \cosh\dfrac{k{\text{π}}\nu_{2}}{\bar\nu_{0}}\right)^{2}}{\sinh\dfrac{k{\text{π}}\nu_{1}}{\bar\nu_{0}} \sinh\dfrac{k{\text{π}}\nu_{2}}{\bar\nu_{0}}} {\rm{e}}^{- \tfrac{k y^2 }{2{\text{π}}\alpha' \bar\nu_{0}}}\, Z_{k} (\bar\nu_{0}, \nu_{1}, \nu_{2}), \\[-25pt] \end{split} $

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$ Z_{k} (\bar\nu_{0}, \nu_{1},\nu_{2}) \! = \! \prod\limits_{n = 1}^{\infty} \dfrac{\left[ \left(1 \!+\! |z_{k}|^{4n } - 2 (-)^{k} |z_{k}|^{2n} \cosh\dfrac{k {\text{π}} \nu_{1}}{\bar\nu_{0}}\cosh\dfrac{k{\text{π}} \nu_{2}}{\bar\nu_{0}}\right)^{2} - 4 |z_{k}|^{4n} \sinh^{2}\dfrac{k{\text{π}}\nu_{1}}{\bar\nu_{0}} \sinh^{2}\dfrac{k {\text{π}} \nu_{2}}{\bar\nu_{0}}\right]^{2}} { (1 - |z_{k}|^{2n})^{2}\left(1 - 2 |z_{k}|^{2n} \cosh\dfrac{2 {\text{π}} k \nu_{1}}{\bar\nu_{0}} + |z_{k}|^{4n}\right)\left(1 - 2 |z_{k}|^{2n} \cosh\dfrac{2 {\text{π}} k \nu_{2}}{\bar\nu_{0}} + |z_{k}|^{4n}\right)}, $

(98)

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$ \begin{split} {\cal{W}}^{(1)}_{p, p'} = \;& \dfrac{ 2^{3- \frac{\kappa}{2}} \, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} \sinh({\text{π}}\bar\nu_{0}) \sin({\text{π}}\nu_{1}) \sin({\text{π}}\nu_{2})} { (8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \bar\nu_{0}^{\frac{p - 5}{2}} \, {\rm{e}}^{- \tfrac{y^2 }{2{\text{π}}\alpha' \bar\nu_{0}}} \dfrac{\left(\cosh\dfrac{ {\text{π}} \nu_{1}}{\bar\nu_{0}} + \cosh\dfrac{{\text{π}}\nu_{2}}{\bar\nu_{0}}\right)^{2}} {\sinh\dfrac{{\text{π}}\nu_{1}}{\bar\nu_{0}}\sinh\dfrac{{\text{π}}\nu_{2}}{\bar\nu_{0}}} \\ &\times Z_{1} (\bar\nu_{0}, \nu_{1}, \nu_{2}). \\[-10pt] \end{split}$

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$ F = \left( {\begin{array}{*{20}{c}} 0&{\hat f}&0&0\\ { - \hat f}&0&0&0\\ 0&0&0&{\hat g}\\ 0&0&{ - \hat g}&0 \end{array}} \right),\quad F' = \left( {\begin{array}{*{20}{c}} 0&{\hat f'}\\ { - \hat f'}&0 \end{array}} \right).$

(100)

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$ \begin{split} \varGamma_{3, 1}\; = \;&\dfrac{2 V_{2} \sqrt{(1 - \hat f'^{2})(1 - \hat f^{2})(1 + \hat g^{2})} (\cosh ({\text{π}} \bar\nu_{0}) - \cos({\text{π}} \nu_{1}))^{2}}{8 {\text{π}}^{2} \alpha'} \int_{0}^{\infty} \dfrac{{\rm{d}} t}{t^{3}} \; {\rm{e}}^{- \frac{y^{2}}{2 {\text{π}} \alpha' t}} \\ &\times \prod\limits_{n = 1}^{\infty} \dfrac{[1 - 2 |z|^{2n} {\rm{e}}^{-{\text{π}} \bar\nu_{0}} \cos({\text{π}}\nu_{1}) + {\rm{e}}^{- 2{\text{π}} \bar\nu_{0}} |z|^{4n}]^{2} [1 - 2 |z|^{2n} {\rm{e}}^{{\text{π}} \bar\nu_{0}} \cos({\text{π}}\nu_{1}) + {\rm{e}}^{ 2{\text{π}} \bar\nu_{0}} |z|^{4n}]^{2} }{(1 - |z|^{2n})^{4} [1 - 2 |z|^{2n} \cosh(2{\text{π}} \bar\nu_{0}) + |z|^{4n}] [1 - 2 |z|^{2n} \cos(2{\text{π}}\nu_{1}) + |z|^{4n}]}, \end{split} $

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$ \tanh {\text{π}} \bar\nu_{0} = \frac{|\hat f - \hat f'|}{1 - \hat f \hat f'}, \quad \tan {\text{π}} \nu_{1} = \frac{1}{\hat g}, $

(102)

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$ \begin{split} \varGamma_{3, 1} \; =\;& \dfrac{2 V_{2} |\hat f - \hat f'| }{8 {\text{π}}^{2} \alpha'} \int_{0}^{\infty} \dfrac{{\rm{d}} t}{t} \; {\rm{e}}^{- \frac{y^{2} t}{2 {\text{π}} \alpha' }} \dfrac{[\cosh ({\text{π}}\nu_{1} t) - \cos({\text{π}}\bar\nu_{0} t)]^{2}}{\sin({\text{π}}\bar\nu_{0} t) \sinh({\text{π}}\nu_{1} t)}\\ &\times\prod\limits_{n = 1}^{\infty} \dfrac{|1 - 2 |z|^{2n} {\rm{e}}^{- i {\text{π}} \bar\nu_{0} t} \cosh ({\text{π}} \nu_{1} t) + |z|^{4n} {\rm{e}}^{- 2 i {\text{π}} \bar\nu_{0} t}|^{4}}{(1 - |z|^{2n})^{4} [1 - 2 |z|^{2n} \cosh (2{\text{π}} \nu_{1} t) + |z|^{4n}][1 - 2 |z|^{2n} \cos(2{\text{π}}\bar\nu_{0} t) + |z|^{4n}]}. \end{split}$

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${\cal{W}}= \frac{4 |\hat f - \hat f'|}{8 {\text{π}}^{2} \alpha'} \sum\limits_{k = 1}^{\infty} \frac{(- )^{k - 1}}{k} {\rm{e}}^{- \frac{k y^{2}}{2{\text{π}} \alpha' \bar\nu_{0}}} \frac{\left[\cosh \dfrac{{\text{π}} k \nu_{1} }{\bar\nu_{0}} - (-)^{k}\right]^{2}}{\sinh \dfrac{{\text{π}} k \nu_{1} }{\bar\nu_{0}}} Z_{k} (\bar\nu_{0}, \nu_{1}), $

(104)

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$Z_{k} (\bar\nu_{0}, \nu_{1})= \prod\limits_{n = 1}^{\infty} \frac{\left[1 - 2 (-)^{k} |z_{k}|^{2n} \cosh \dfrac{{\text{π}} k \nu_{1} }{\bar\nu_{0}} + |z_{k}|^{4n} \right]^{4}}{(1 - |z_{k}|^{2n})^{6} \left[1 - 2 |z_{k}|^{2n} \cosh \dfrac{2{\text{π}} k \nu_{1} }{\bar\nu_{0}} + |z_{k}|^{4n}\right]}, $

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${\cal{W}}^{(1)}= \dfrac{4 |\hat f - \hat f'|}{8 {\text{π}}^{2} \alpha'} {\rm{e}}^{- \frac{y^{2}}{2{\text{π}} \alpha' \bar\nu_{0}}} \dfrac{\left[\cosh \dfrac{{\text{π}} \nu_{1} }{\bar\nu_{0}} + 1\right]^{2}}{\sinh \dfrac{{\text{π}} \nu_{1} }{\bar\nu_{0}}} \prod\limits_{n = 1}^{\infty} \dfrac{\left[1 + 2 |z_{1}|^{2n} \cosh \dfrac{{\text{π}} \nu_{1} }{\bar\nu_{0}} + |z_{1}|^{4n} \right]^{4}}{(1 - |z_{1}|^{2n})^{6} \left[1 - 2 |z_{1}|^{2n} \cosh \dfrac{2{\text{π}} \nu_{1} }{\bar\nu_{0}} + |z_{1}|^{4n}\right]}.$

(106)

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$ {\cal{W}}^{(1)}_{{\rm{D3}}/{\rm{D1}}} = \frac{e E}{2 {\text{π}} }\, {\rm{exp}}\bigg[{- \frac{{\text{π}} \left(m^{2} - \frac{\nu_{1}}{2 \alpha'}\right)}{e E}}\bigg], $

(107)

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$ { F'_3} = \left( {\begin{array}{*{20}{c}} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&{\hat g'}\\ 0&0&{ - \hat g'}&0 \end{array}} \right), $

(108)

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$ \tan({\text{π}}\nu_{1}) = \frac{\left|\hat g - \hat g'\right|}{1 + \hat g \hat g'}. $

(109)

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$ \begin{split} &\alpha '{M^2} = - \alpha '{p^2} \\ ={}& \begin{cases}{\dfrac{y^{2}}{4{\text{π}}^{2} \alpha'} + N_{\rm{R}},}&{({\rm{R}}{\text{}}{\text{分部}}),}\\ {\dfrac{y^{2}}{4{\text{π}}^{2} \alpha'} + N_{\rm{NS}} - \dfrac{1}{2},}&{({\rm{NS}}{\text{}}{\text{分部}}),} \end{cases} \end{split} $

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$\begin{split} &N_{\rm{R}} = \sum\limits_{n = 1}^{\infty} (\alpha_{- n} \cdot \alpha_{n} + n d_{-n} \cdot d_{n}),\\ &N_{\rm{NS}} = \sum\limits_{n = 1}^{\infty} \alpha_{- n} \cdot \alpha_{n} + \sum\limits_{r = 1/2}^{\infty} r d_{- r} \cdot d_{r}. \end{split} $

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$ \alpha' {\rm{E}}^{2}_{\rm{NS}} = (2 N + 1) \frac{\nu_{1}}{2} - \nu_{1} \, S + \alpha' M^{2}_{\rm{NS}}, $

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$ S = \sum\limits_{n = 1}^{\infty} (a^{+}_{n} a_{n} - b^{+}_{n} b_{n}) + \sum\limits_{r = 1/2}^{\infty} (d^{+}_{r} d_{r} - \tilde d^{+}_{r} \tilde d_{r}), $

(113)

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$ \alpha' M^{2}_{\rm{NS}} = \frac{y^{2}}{4 {\text{π}}^{2} \alpha'} + N_{\rm{NS}} - \frac{1}{2}, $

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$\begin{split} N_{\rm{NS}} ={}& \sum\limits_{n = 1}^{\infty} n (a^{+}_{n} a_{n} + b^{+}_{n} b_{n}) \\ & + \sum\limits_{r = 1/2}^{\infty} r (d^{+}_{r} d_{r} + \tilde d^{+}_{r} \tilde d_{r}) + N^{\perp}_{\rm{NS}}, \end{split} $

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$ (a^{+}_{1})^{\tilde n} d^{+}_{1/2} |0 \rangle_{\rm{NS}}, $

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$ \alpha' {\rm{E}}^{2}_{\rm{NS}} = - \frac{\nu_{1}}{2} + (1 - \nu_{1}) (S - 1) + \dfrac{y^{2}}{4 {\text{π}}^{2} \alpha'}, $

(117)

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$ \alpha' M^{2}_{\rm{NS}} = \frac{y^{2}}{4 {\text{π}}^{2} \alpha'} + S - 1, $

(118)

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$ \alpha' {\rm{E}}^{2}_{\rm{NS}} = - \frac{\nu_{1}}{2} + \frac{y^{2}}{4 {\text{π}}^{2} \alpha'}, $

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$ \begin{split} &{\cal{W}}^{(1)} = \frac{2\,(e E) (e B)}{(2{\text{π}})^{2}} \dfrac{\left[\cosh\dfrac{{\text{π}} B}{E} + 1\right]^2}{\sinh \dfrac{{\text{π}} B}{E}} \, {\rm{e}}^{- \tfrac{{\text{π}} m^{2}}{e E}}, \\ & {\cal{W}}_{\rm{vector}} = \dfrac{(e E) (e B)}{ 2 (2 {\text{π}})^{2}} \frac{ 2 \cosh \dfrac{2 {\text{π}} B}{E} + 1}{ \sinh \dfrac{{\text{π}} B}{E}} \, {\rm{e}}^{- \tfrac{{\text{π}} m^{2}}{e E}}, \\ &{\cal{W}}^{(1)}_{{\rm{D3}}/{\rm{D1}}} = \dfrac{e E}{2 {\text{π}} }\, {\rm{e}}^{- \tfrac{{\text{π}} \left(m^{2} - \frac{\nu_{1}}{2 \alpha'}\right) }{e E}}, \end{split}$

(120)

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$ {\cal{W}}^{(1)} = {\cal{W}}_{\rm{vector}} = \dfrac{ (e E) (e B)}{(2 {\text{π}})^2 } \, {\rm{e}}^{- \tfrac{ {\text{π}} (m^{2} - e B)}{e E }}, $

(121)

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$ \begin{split} \varGamma_{p, p'} \; = \;& \dfrac{ 2^3 \, V_{p' + 1}\, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{ 2^{\frac{\kappa}{2}} (8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{9 - p}{2}}} \dfrac{{\rm{e}}^{- \dfrac{y^2}{2{\text{π}}\alpha' t}}}{\eta^{3} (it)} \\ & \times \dfrac{ \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2}\right| it \right) \theta_{1} \left(\left.\dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2}\right| it \right)\theta_{1} \left(\left.\dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2}\right| it \right)}{\theta_{1} (\nu_{0} | it)\theta_{1} (\nu_{1} | it)\theta_{1} (\nu_{2} | it)}\\ = \;& \dfrac{ 2^2 \, V_{p'+ 1} \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} \left[ \sum_{\alpha = 0}^{2}\cos^{2}({\text{π}} \nu_{\alpha}) - 2\prod\limits_{\alpha = 0}^{2} \cos({\text{π}}\nu_{\alpha}) - 1\right]}{ 2^{\frac{\kappa}{2}}(8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \\ & \times \int_0^\infty \dfrac{{\rm{d}} t}{t^{\frac{9 - p}{2}}} {\rm{e}}^{- \frac{y^2}{2{\text{π}}\alpha' t}} \prod\limits_{n = 1}^{\infty} C_{n}, \end{split} $

(122)

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$ \begin{split} \varGamma_{p,p'} \; = \;& - \dfrac{ 2^3 \, i \, V_{p' + 1}\, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}}\prod\limits_{\alpha = 0}^{2} \sin ({\text{π}} \nu_{\alpha}) }{2^{\frac{\kappa}{2}}(8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t} {t^{\frac{p - 3}{2}}} \dfrac{{\rm{e}}^{- \frac{y^2 t}{2{\text{π}}\alpha' }}}{\eta^{3} (it)} \\ & \times \dfrac{ \theta_{1} \left(\left. \dfrac{\nu_{0} + \nu_{1} + \nu_{2}}{2} i t \right| it \right) \theta_{1} \left(\left. \dfrac{\nu_{0} - \nu_{1} + \nu_{2}}{2} it \right| it \right) \theta_{1} \left(\left. \dfrac{\nu_{0} + \nu_{1} - \nu_{2}}{2} it \right| it \right)\theta_{1} \left(\left. \dfrac{\nu_{0} - \nu_{1} - \nu_{2}}{2} it \right| it \right)}{\theta_{1} (\nu_{0} it | it)\theta_{1} ( \nu_{1} it | it)\theta_{1} ( \nu_{2} it | it)}\\ = \;& \dfrac{ 2^2 \, V_{p'+ 1} \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} }{2^{\frac{\kappa}{2}} (8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \int_0^\infty \dfrac{{\rm{d}} t}{t^{\frac{p - 3}{2}}} {\rm{e}}^{- \frac{y^2 t}{2{\text{π}}\alpha'}} \prod\limits_{\alpha = 0}^{2} \dfrac{\sin ({\text{π}} \nu_{\alpha})}{\sinh ({\text{π}} \nu_{\alpha} t)} \\ & \times \left[ \sum\limits_{\alpha = 0}^{2}\cosh^{2}({\text{π}} \nu_{\alpha} t)- 2 \prod\limits_{\alpha = 0}^{2} \cosh({\text{π}} \nu_{\alpha} t) - 1\right] \prod\limits_{n = 1}^{\infty} Z_{n}, \end{split} $

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$\begin{split} {\cal{W}}_{p, p'} \; = \;& \dfrac{ 2^{3- \frac{\kappa}{2}} \, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} \sinh({\text{π}}\bar\nu_{0}) \sin({\text{π}}\nu_{1}) \sin({\text{π}}\nu_{2}} {\bar\nu_{0}) (8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \sum\limits_{k = 1}^{\infty} (-1)^{k + 1} \left(\dfrac{\bar\nu_{0}}{k}\right)^{\frac{p - 3}{2}} \\ &\times \dfrac{\left(\cosh\dfrac{k {\text{π}} \nu_{1}}{\bar\nu_{0}} - (-1)^{k} \cosh\dfrac{k{\text{π}}\nu_{2}}{\bar\nu_{0}}\right)^{2}}{\sinh\dfrac{k{\text{π}}\nu_{1}}{\bar\nu_{0}} \sinh\dfrac{k{\text{π}}\nu_{2}}{\bar\nu_{0}}} {\rm{e}}^{- \frac{k y^2 }{2{\text{π}}\alpha' \bar\nu_{0}}}\, Z_{k} (\bar\nu_{0}, \nu_{1}, \nu_{2}), \end{split} $

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$ \begin{split} {\cal{W}}^{(1)}_{p, p'} \; = \;& \dfrac{ 2^{3- \frac{\kappa}{2}} \, \left[\det (\eta_{p'} + \hat F'_{p'})\det(\eta_{p} + \hat F_{p})\right]^{\frac{1}{2}} \sinh({\text{π}}\bar\nu_{0}) \sin({\text{π}}\nu_{1}) \sin({\text{π}}\nu_{2})} { (8 {\text{π}}^2 \alpha')^{\frac{p' + 1}{2}}} \bar\nu_{0}^{\frac{p - 5}{2}} \, {\rm{e}}^{- \frac{y^2 }{2{\text{π}}\alpha' \bar\nu_{0}}}\\ & \times \dfrac{\left(\cosh\dfrac{ {\text{π}} \nu_{1}}{\bar\nu_{0}} + \cosh\dfrac{{\text{π}}\nu_{2}}{\bar\nu_{0}}\right)^{2}}{\sinh\dfrac{{\text{π}}\nu_{1}}{\bar\nu_{0}}\sinh \dfrac{{\text{π}}\nu_{2}}{\bar\nu_{0}}} \, Z_{1} (\bar\nu_{0}, \nu_{1}, \nu_{2}). \end{split} $

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Jian-Xin Lu, Nan Zhang. D-brane interaction, the open string pair production and its enhancement plus its possible detection[J]. Acta Physica Sinica, 2020, 69(10): 101101-1

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