DOUBLE PRECISION FUNCTION Q2SIG(XPRIME,QLAM,KAPPA,LOOPFL,NF) - Instanton-induced hard subprocess total cross section

Calculation of

$$Q^{\prime 2}\ \sigma^{(I)}_{q^\prime g}\ [{\rm nb}\ {\rm GeV}^2],$$ (1)

where $\sigma^{(I)}_{q^\prime g}$ is the instanton-induced total hard subprocess cross section taken from Ref. [1], as a function of ${\rm XPRIME}=x^\prime$, ${\rm QLAM}=Q^{\prime}/\Lambda^{(n_f)}_{\overline{\rm MS}}$ and ${\rm KAPPA}=\mu_r/Q^\prime$, ${\rm LOOPFL}=\{1,2,3\}$ and ${\rm NF}=n_f$.

Q2SIG uses the form of the cross section derived in Ref. [1],

$$Q^{\prime\,2}\,\sigma_{q^\prime g}^{(I )} = d^2_{\overline{\rm MS}}\frac{\sqrt{12}}{2^{16}}\pi^{15/2} ((\xi_\... ...} \frac{\Delta_1\beta_0}{D(\tilde{S})}\right)^{7/2} \omega (\xi_\ast )^{2n_f-1}$$ (2)

$$\times \frac{(\xi_\ast -2)^3 v^{\ast\,5} } {(v_\ast -\tilde{S})^{... ...t ) D\left( \ln \left( \frac{D(\tilde{S})}{\sqrt{\xi_\ast -2}} \right)\right)}}$$

$$\times \left( \frac{4\pi} {\alpha_{\overline{\rm MS}}\left(\mu _r... ...t(1-\ln\left(\frac{v_\ast\mu_r}{Q^\prime}\right)\right) \,\tilde{S}\right] \, ,$$

expressed entirely in terms of the collective coordinate saddle points $v_\ast\equiv Q^\prime\rho_\ast$ and $\xi_\ast$, satisfying

$$\frac{1}{2}\frac{\sqrt{\frac{1-{x^\prime}}{{x^\prime}}}} {\sqrt{\... ...\left(\rho_\ast\mu_r\right)\right) \frac{dS^{(I\overline{I})}}{d\xi}(\xi_\ast ) = 0 \,,$$ (3)

$$\left( \frac{1}{2}\,\sqrt{\frac{1-{x^\prime}}{{x^\prime}}} \sqrt{... ...ht) Q^\prime\rho_\ast+ \Delta_1\beta_0 S^{(I\overline{I})}(\xi_\ast ) -\Delta_2 = 0\, .$$ (4)

In Eq. (2) we have introduced the shorthands

$$\tilde{S}(\xi_\ast )\equiv \Delta_1 \beta_0 S^{(I\overline{I... ...(\xi_\ast ))\equiv \frac{d}{d\ln (\xi_\ast -2)}f(\xi_\ast ) ,$$ (5)

as well as the following parameters

$$d_{\overline{\rm MS}} = \frac{{\rm e}^{5/6}}{\pi^2}\,{\rm e}^{-4.534+0.292\,n_f} ,$$ (6)

$$\Delta_1 \equiv 1+\frac{\beta_1}{\beta_0} \frac{\alpha_{\overline{\rm MS}}(\mu_r)... ...ex} \Delta_2\equiv 12\,\beta_0 \frac{\alpha_{\overline{\rm MS}}(\mu_r)}{4\pi} ,$$ (7)

$$\beta_0 = 11-\frac{2}{3}n_f; \hspace{1cm} \beta_1= 102-\frac{38}{3}\,n_f.$$ (8)

The $I\overline{I}$-action, $S^{(I\overline{I})}(\xi )$, as well its $\xi$-derivatives, entering the expression (2) for the cross section, are calculated in the subroutine ACTION. For default settings of the contrôl flags in QIINIT (VALFLAG=.TRUE.), it is given by the exact valley form [2,3],

$$S^{(I\overline{I})}(\xi ) = 1-\frac{12}{f(\xi )} - \frac{96}{f(\xi )^2} +\frac{48}{f(\xi )^3}... ... 3f(\xi )+8\right] \ln\left[ \frac{1}{2\xi }\bigl( f(\xi ) +4\bigr)\right] \, ,$$ (9)

$$f(\xi ) = \xi^2+\sqrt{\xi^2-4}\xi-4 \, .$$ (10)

The fermionic overlap $\omega (\xi)$ is calculated in the subroutine OMEGA, which uses a simple but accurate approximation for the exact result from Ref. [1],

$$\omega (\xi )= \frac{4}{(\xi +1/2)^{3/2}}.$$ (11)

The system (3) and (4) of saddle point equations is treated as follows: In a first step, it is solved explicitly for $\rho_\ast$ in terms of $\xi_\ast$,

$$v_\ast \equiv Q^\prime \rho_\ast = 2\, D(\tilde{S})\, W\le... ...}}{D(\tilde{S})} \right] \right\}} {2\,D(\tilde{S})} \right) ,$$ (12)

where $W$ denotes the Lambert $W$-function, i.e. the solution of $W(x)\exp(W(x))=x$. The latter is provided by the function LAMBERTW. The saddle-point $\xi_\ast ({x^\prime}, Q^\prime/\Lambda^{(n_f)}_{\overline{\rm MS}}, \mu_r/Q^\prime )$ is then found by inserting Eq. (12) into

$${x^\prime}= \frac{(\xi_\ast -2)} {(\xi_\ast +2)+4 \tilde{S}(\tilde{S}-2 v_\ast)/v^{\ast\,2} } ,$$ (13)

which is equivalent to Eq. (4), and solving numerically for $\xi_\ast$. This is done by calling the function XI. The numerical solution $\xi_\ast ({x^\prime}, Q^\prime/\Lambda^{(n_f)}_{\overline{\rm MS}}, \mu_r/Q^\prime )$ can then be inserted into Eq. (12) to obtain $v_\ast ({x^\prime},Q^\prime/\Lambda^{(n_f)}_{\overline{\rm MS}}, \mu_r/Q^\prime)$.

The values of $4\pi/\alpha_{\overline{\rm MS}}$ are calculated in the subroutine XQS.