DOUBLE PRECISION FUNCTION XI(XPR,XI_MIN,XI_MAX,XQ,XMU,NF,LOOPFL) - Saddle point calculation of conformal $I\overline{I}$-distance $\xi$

In this routine the saddle point value of the conformal $I\overline{I}$-separation ${\rm XI}=\xi_\ast$ is calculated as a function of ${\rm XPR}={x^\prime}$, ${\rm XQ}=4\pi/\alpha_{\overline{\rm MS}}(Q^\prime )$, ${\rm XMU}=4\pi/\alpha_{\overline{\rm MS}}(\mu_r )$ and ${\rm NF}=n_f$. The routine uses an interpolation in the range from ${\rm XI\_MIN}$ to ${\rm XI\_MAX}$ to invert the relation ${x^\prime}={x^\prime}(\xi_\ast,\ldots)$ (see Eq. (9) below).

The system

$$\frac{1}{2}\frac{\sqrt{\frac{1-{x^\prime}}{{x^\prime}}}} {\sqrt{\... ...\left(\rho_\ast\mu_r\right)\right) \frac{dS^{(I\overline{I})}}{d\xi}(\xi_\ast ) = 0 \,,$$ (1)

$$\left( \frac{1}{2}\,\sqrt{\frac{1-{x^\prime}}{{x^\prime}}} \sqrt{... ...ht) Q^\prime\rho_\ast+ \Delta_1\beta_0 S^{(I\overline{I})}(\xi_\ast ) -\Delta_2 = 0 \,,$$ (2)

of saddle point equations, with

$$\Delta_1 \equiv 1+\frac{\beta_1}{\beta_0} \frac{\alpha_{\overline{\rm MS}}(\mu_r)... ...ex} \Delta_2\equiv 12\,\beta_0 \frac{\alpha_{\overline{\rm MS}}(\mu_r)}{4\pi} ,$$ (3)

$$\beta_0 = 11-\frac{2}{3}n_f; \hspace{1cm} \beta_1= 102-\frac{38}{3}\,n_f,$$ (4)

is treated as follows:

In a first step, it is solved explicitly for $\rho_\ast$ in terms of $\xi_\ast$,

$$v_\ast \equiv Q^\prime \rho_\ast = 2\, D(\tilde{S})\, W\le... ...}}{D(\tilde{S})} \right] \right\}} {2\,D(\tilde{S})} \right) ,$$ (5)

where

$$\tilde{S}(\xi_\ast )\equiv \Delta_1 \beta_0 S^{(I\overline{I... ...=\frac{4\pi} {\alpha_{\overline{\rm MS}}\left(\mu _r \right)},$$ (6)

and $W$ denotes the Lambert $W$-function, i.e. the solution of $W(x)\exp(W(x))=x$. The latter is provided by the function LAMBERTW, whereas the $I\overline{I}$-action is calculated in the routine ACTION. Using the 2-loop scale transformation law,

$$X(Q^\prime )-X(\mu_r ) = 2\,\beta_0\ln \left(\frac{Q^\prime}... ...ta_1}{\beta_0}}{X(Q^\prime)+\frac{\beta_1}{\beta_0}} \right) ,$$ (7)

one may eliminate the explicit dependence of $v_\ast$ on $Q^\prime/\mu_r$ in favour of an $X(Q^\prime )$ dependence, with

$${ v_\ast (\xi_\ast, X(Q^\prime ), X(\mu_r )) =}$$

$$2\, D(\tilde{S})\, W\left( \frac{ \exp\left\{ \frac{1}{2}\frac{\t... ... +\frac{1}{2}\frac{X(Q^\prime )}{\beta_0} \right\}} {2\,D(\tilde{S})} \right) .$$

In the second step, the expression Eq. (8) for $v_\ast$ is inserted into

$${x^\prime}= \frac{(\xi_\ast -2)} {(\xi_\ast +2)+4 \tilde{S}(... ...^{\ast\,2} } = {\rm funct}\,(\xi_\ast,X(Q^\prime),X(\mu_r )),$$ (9)

which is equivalent to Eq. (2). In the present routine, the inversion of Eq. (9) to give the saddle point solution $x_\ast ({x^\prime},X(Q^\prime),X(\mu_r ))$ is done by interpolation.

In a last step, the solution $\xi_\ast ({x^\prime},X(Q^\prime ), X(\mu_r) )$ can be inserted into Eq. (8) to obtain $v_\ast ({x^\prime},X(Q^\prime ), X(\mu_r))$.