Model and Definitions

$$S=S_{e}+S_{u}+S_{f}+S_{b}$$

$$\begin{eqnarray*} S_{e} & = & \sum _{\alpha =c,s}\frac{1}{2\pi K_{\alpha }}\int ... ..._{c}(x,\tau ')\right)D_{\xi }\left(u(x,\tau )-u(x,\tau ')\right) \end{eqnarray*}$$

String models:

1. Rigid String $\sigma =0\quad \int dx\rho =m$
2. Flactuating (Gaussian) String $n=1\quad \omega _{0}=0$
3. Very Floppy String $n>1\quad \omega _{0}=0$
4. Stationary (G&S) String $\rho =\sigma =\omega _{0}=0$

Correlation functions

$$\begin{eqnarray*} G_{u}(x,\tau ) & = & \left\langle u(x,\tau )u(0,0)\right\rangl... ...lpha }(x,\tau )}e^{-i\sqrt{2}\phi _{\alpha }(0,0)}\right\rangle \end{eqnarray*}$$

Dimensionless parameters

$$\begin{eqnarray*} D_{b}(\lambda ) & = & \frac{2D_{\xi }\alpha }{\pi v_{s}^{2}}\l... ...sqrt{\pi \epsilon }}\\ \mathcal{L} & = & \frac{L}{2\pi \alpha } \end{eqnarray*}$$

$\epsilon$ and $\eta$ are dimensionless parameters associated with the string

realization	$\epsilon$	$\eta$	$D_{b}$	$D_{f}$
Rigid String	$\frac{1}{\mathcal{M}}=\frac{1}{2m\omega _{0}}$	$\varpi =\frac{\alpha \omega _{0}}{v_{s}}$	$\frac{2D_{\xi }\alpha }{\pi v_{s}^{2}}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\sqrt{\frac{m\omega _{0}}{2\pi }}$	$\frac{2D_{\eta }\alpha }{\pi v_{c}^{2}}\sqrt{\frac{m\omega _{0}}{2\pi }}$
Flactuating String	$\epsilon =\frac{1}{2\pi \sqrt{\rho \sigma }}=\frac{v_{u}}{2\pi \sigma }$	$\eta =\frac{v_{u}}{v_{s}}$	$\frac{2D_{\xi }\alpha }{\pi v_{s}^{2}}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\sqrt{\frac{\sigma }{2v_{u}}}$	$\frac{2D_{\eta }\alpha }{\pi v_{c}^{2}}\sqrt{\frac{\sigma }{2v_{u}}}$
Very Floppy String	$\epsilon =\frac{1}{2\pi \sqrt{\rho \sigma }}=\frac{v_{u}}{2\pi \sigma }$	$\eta =\frac{v_{u}}{v_{s}}$	$\frac{2D_{\xi }\alpha }{\pi v_{s}^{2}}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\frac{1}{2}\sqrt{\frac{n-1}{\pi \epsilon \eta ^{1-\frac{1}{n}}}}$	$\frac{2D_{\eta }\alpha }{\pi v_{c}^{2}}\frac{1}{2}\sqrt{\frac{n-1}{\pi \epsilon \eta ^{1-\frac{1}{n}}}}$
Stationary String	$1$	$1$	$\frac{2D_{\xi }\alpha }{\pi v_{s}^{2}}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}$	$\frac{2D_{\eta }\alpha }{\pi v_{c}^{2}}$

Free Correlators

$$\begin{eqnarray*} G_{\alpha o}(x,\tau ) & = & -\frac{K_{\alpha }}{4}\ln \frac{x^... ...}-\left(\frac{v_{u}\tau }{\alpha }\right)^{1-\frac{1}{n}}\right] \end{eqnarray*}$$

1st order perturbation:

$$\begin{eqnarray*} \delta R_{c} & = & \frac{1}{8\left(\pi \alpha \right)^{2}}\sum... ...e \partial _{x}\phi _{3}\partial _{x}\phi _{4}\right\rangle _{c} \end{eqnarray*}$$

where $d[3,4]=dx_{3}d\tau _{3}dx_{4}d\tau _{4}\delta (x_{3}-x_{4})$ and $\phi _{i}=\phi (x_{i},\tau _{i})$

Average results

$$\begin{eqnarray*} \left\langle e^{i\epsilon _{3}\sqrt{2}\phi _{3}}e^{-i\epsilon ... ...\frac{x_{1}-x_{4}}{\left\vert r_{1}-r_{4}\right\vert^{2}}\right) \end{eqnarray*}$$

$$\begin{eqnarray*} \left\langle D_{\xi ,\eta }(u_{3}-u_{4})\right\rangle _{u} & =... ...{0}(2,3)\right)e^{-\lambda ^{2}\left(G_{0}(0)-G_{0}(3,4)\right)} \end{eqnarray*}$$

Electrons

$$\begin{eqnarray*} \frac{\delta R_{c}}{R_{c0}} & = & 1+\frac{K_{c}^{2}\epsilon ^{... ...da ^{2}\left(G_{0}(0)-G_{0}(3,4)\right)}\left(I_{+}-I_{-}\right) \end{eqnarray*}$$

$$\begin{eqnarray*} I_{\pm }(r_{1}-r_{2}) & = & \int d^{2}R\ln \left\vert r_{1}-R\... ..._{+}(r) & = & 2\pi \ln r\\ I_{-}(r) & = & \pi \cos 2\theta _{r} \end{eqnarray*}$$

Rigid String Correlations

$$\begin{eqnarray*} \delta G_{u}(r) & = & 4\sqrt{\pi }\frac{\mathcal{L}}{\mathcal{... ...{\varpi }{\alpha }\left(\left\vert r\right\vert-x\right)}\right] \end{eqnarray*}$$

with $f_{b}\left(\left\vert x\right\vert,\lambda \right)=\left\vert\frac{x}{\alpha }\... ...}{\mathcal{M}}\left(1-e^{-\varpi \left\vert x\right\vert/\alpha }\right)\right]$

Fluctuating String Correlations

$$\begin{eqnarray*} \delta G_{u} & = & \frac{1}{2\sqrt{\pi }}\epsilon ^{\frac{5}{2... ...\alpha }\right)^{-\lambda ^{2}\epsilon }\left(I_{+}-I_{-}\right) \end{eqnarray*}$$

Flow Equations

$$\begin{eqnarray*} G_{\alpha } & = & -K_{\alpha eff}\ln \frac{r}{\alpha }-d_{eff}... ...& = & -\epsilon _{eff}\ln \frac{r}{\alpha }-b_{eff}\cos 2\theta \end{eqnarray*}$$

Electrons flow

$$\begin{eqnarray*} \frac{dK_{c}}{dl} & = & -\sqrt{\pi }K_{c}^{2}\frac{v_{c}}{v_{s... ...D_{b}(\lambda )e^{-\lambda ^{2}\left(G_{0}(0)-G_{0}(3,4)\right)} \end{eqnarray*}$$

Rigid String flow

$$\begin{eqnarray*} \frac{d\varpi }{dl} & = & \varpi +4\sqrt{\pi }\frac{\mathcal{L... ...}(\lambda )\\ \frac{dD_{f}(\lambda )}{dl} & = & D_{f}(\lambda ) \end{eqnarray*}$$

when $D(\lambda )$is constant

$$\begin{eqnarray*} \frac{d\varpi }{dl} & = & \varpi +\mathcal{L}\left(D_{b}+K_{c}... ...} & = & -\frac{1}{2}K_{s}^{2}\frac{D_{b}}{\sqrt{1-e^{-\varpi }}} \end{eqnarray*}$$

Fluctuatin String flow

Define

$$\begin{eqnarray*} \mathcal{D}_{b,f}(u) & = & \epsilon \int \frac{d\lambda }{2\pi }e^{i\lambda u-\lambda ^{2}\epsilon \ln \eta }D_{b,f}(\lambda ) \end{eqnarray*}$$

$$\begin{eqnarray*} \frac{d\epsilon }{dl} & = & \sqrt{\pi }\eta \epsilon ^{\frac{3... ... \frac{dK_{s}}{dl} & = & -\frac{1}{2}K_{s}^{2}\mathcal{D}_{b}(0) \end{eqnarray*}$$

Very Floppy String flow

$$\begin{eqnarray*} \frac{dD_{b}(\lambda )}{dl} & = & \left(\frac{3}{2}+\frac{1}{2... ...{f}\right)\\ \frac{dK_{s}}{dl} & = & -\frac{1}{2}K_{s}^{2}D_{b} \end{eqnarray*}$$

Non interaction point

$$\begin{eqnarray*} S_{I} & = & -\int dxd\tau d\tau '\left[\psi _{R}^{\dagger }\ps... ...dagger }\psi _{L}(x,\tau ')D_{\eta }(u(\tau )-u(\tau '))\right]= \end{eqnarray*}$$

averaging over the string degreed of freedom, $\left\langle D(u-u')\right\rangle =f(\tau -\tau ')$, and seperating elastic and inelastic terms

$$\begin{eqnarray*} \left\langle S_{I}\right\rangle _{u} & = & -\oint dxd\tau d\ta... ... _{R}^{\dagger }\psi _{R}(x,T)\psi _{L}^{\dagger }\psi _{L}(x,T) \end{eqnarray*}$$

Define

$$F_{b,f}(l)=\int _{0}^{1}drf(r)=\frac{v_{s}}{\alpha }\int _{0}^{\alpha /v_{s}}d\tau f(\tau )$$

Then the $g_{1}$ process is modifed by

$$\begin{eqnarray*} g_{1\perp } & = & g_{1\perp }^{in}-\frac{2\alpha }{v_{s}}F_{b}... ...rt\vert}-g_{2\perp } & = & g_{c}^{in}-\frac{\alpha }{v_{s}}F_{f} \end{eqnarray*}$$

at the noninteracting point $v_{s}=v_{F}\quad D=\frac{D}{\pi v_{F}^{2}\sqrt{\pi \epsilon }}\quad \dot{y}=-Df(1)$, therefore

$$\frac{1}{\pi v_{F}}\frac{dg_{1\perp }^{in}}{dl}=\frac{dy}{dl}+\frac{d}{dl}(DF)=-Df(1)+\frac{d}{dl}(DF)$$

The string flauctuaions is of the form $f(r)=g(\eta r)$, therefore

$$\begin{eqnarray*} \frac{d\eta }{dl} & = & \eta \\ \frac{df}{d\eta } & = & \frac{r}{\eta }\frac{df}{dr} \end{eqnarray*}$$

suppose also that r $f(r)\rightarrow 0$ as $r\rightarrow 0$ then

$$\begin{eqnarray*} \frac{dF}{dl} & = & \int _{0}^{1}\frac{df}{d\eta }\frac{d\eta ... ...DF) & = & \frac{dD}{dl}F+D\frac{dF}{dl}=D(F+\frac{dF}{dl})=Df(1) \end{eqnarray*}$$

thus we get

$$\frac{dg_{1\perp }^{in}}{dl}=0$$

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