Further Couplings

In this section we collect information on the two routines for running $\alpha_{\mathrm{s}}$ and $\alpha_{\mathrm{em}}$, and on other couplings of standard and non-standard particles found in the PYDAT1 common block. Although originally begun for applications within the traditional particle sector, this section has rapidly expanded towards the non-standard aspects, and is thus more of interest for applications to specific processes. It could therefore equally well have been put somewhere else in this manual. Several other couplings indeed appear in the PARP array in the PYPARS common block, see section , and the choice between the two has largely been dictated by availability of space.

Purpose:

to calculate the running electromagnetic coupling constant $\alpha_{\mathrm{em}}$. Expressions used are described in ref. [Kle89]. See MSTU(101), PARU(101), PARU(103) and PARU(104).

Q2 :

the momentum transfer scale $Q^2$ at which to evaluate $\alpha_{\mathrm{em}}$.

Purpose:

to calculate the running strong coupling constant $\alpha_{\mathrm{s}}$, e.g. in matrix elements and resonance decay widths. (The function is not used in parton showers, however, where formulae rather are written in terms of the relevant $\Lambda$ values.) The first- and second-order expressions are given by eqs. () and (). See MSTU(111) - MSTU(118) and PARU(111) - PARU(118) for options.

Q2 :

the momentum transfer scale $Q^2$ at which to evaluate $\alpha_{\mathrm{s}}$.

Purpose:

to give running masses of $\d$, $\u$, $\mathrm{s}$, $\c$, $\b$ and $\t$ quarks according to eq. . For all other particles, the PYMASS function is called by PYMRUN to give the normal mass. Such running masses appear e.g. in couplings of fermions to Higgs and technipion states.

KF :

flavour code.

Q2 :

the momentum transfer scale $Q^2$ at which to evaluate $\alpha_{\mathrm{s}}$.

Note:

The nominal values, valid at a reference scale
$Q^2_{\mathrm{ref}} = \max((\mathtt{PARP(37)} m_{\mathrm{nominal}})^2 , 4\Lambda^2)$,
are stored in PARF(91)-PARF(96).

Purpose:

to give access to a number of status codes and parameters which regulate the performance of the program as a whole. Here only those related to couplings are described; the main description is found in section .

MSTU(101) :

(D=1) procedure for $\alpha_{\mathrm{em}}$ evaluation in the PYALEM function.

= 0 :

$\alpha_{\mathrm{em}}$ is taken fixed at the value PARU(101).

= 1 :

$\alpha_{\mathrm{em}}$ is running with the $Q^2$ scale, taking into account corrections from fermion loops ($\mathrm{e}$, $\mu$, $\tau$, $\d$, $\u$, $\mathrm{s}$, $\c$, $\b$).

= 2 :

$\alpha_{\mathrm{em}}$ is fixed, but with separate values at low and high $Q^2$. For $Q^2$ below (above) PARU(104) the value PARU(101) (PARU(103)) is used. The former value is then intended for real photon emission, the latter for electroweak physics, e.g. of the $\mathrm{W}/ \mathrm{Z}$ gauge bosons.

MSTU(111) :

(I, D=1) order of $\alpha_{\mathrm{s}}$ evaluation in the PYALPS function. Is overwritten in PYEEVT, PYONIA or PYINIT calls with the value desired for the process under study.

= 0 :

$\alpha_{\mathrm{s}}$ is fixed at the value PARU(111). As extra safety, $\Lambda=$PARU(117) is set in PYALPS so that the first-order running $\alpha_{\mathrm{s}}$ agrees with the desired fixed $\alpha_{\mathrm{s}}$ for the $Q^2$ value used.

= 1 :

first-order running $\alpha_{\mathrm{s}}$ is used.

= 2 :

second-order running $\alpha_{\mathrm{s}}$ is used.

MSTU(112) :

(D=5) the nominal number of flavours assumed in the $\alpha_{\mathrm{s}}$ expression, with respect to which $\Lambda$ is defined.

MSTU(113) :

(D=3) minimum number of flavours that may be assumed in $\alpha_{\mathrm{s}}$ expression, see MSTU(112).

MSTU(114) :

(D=5) maximum number of flavours that may be assumed in $\alpha_{\mathrm{s}}$ expression, see MSTU(112).

MSTU(115) :

(D=0) treatment of $\alpha_{\mathrm{s}}$ singularity for $Q^2 \to 0$ in PYALPS calls. (Relevant e.g. for QCD $2 \to 2$ matrix elements in the $p_{\perp}\to 0$ limit, but not for showers, where PYALPS is not called.)

= 0 :

allow it to diverge like $1/\ln(Q^2/\Lambda^2)$.

= 1 :

soften the divergence to $1/\ln(1 + Q^2/\Lambda^2)$.

= 2 :

freeze $Q^2$ evolution below PARU(114), i.e. the effective argument is $\max(Q^2,$PARU(114)$)$.

MSTU(118) :

(I) number of flavours $n_f$ found and used in latest PYALPS call.

PARU(101) :

(D=0.00729735=1/137.04) $\alpha_{\mathrm{em}}$, the electromagnetic fine structure constant at vanishing momentum transfer.

PARU(102) :

(D=0.232) $\sin^2 \! \theta_W$, the weak mixing angle of the standard electroweak model.

PARU(103) :

(D=0.007764=1/128.8) typical $\alpha_{\mathrm{em}}$ in electroweak processes; used for $Q^2 >$PARU(104) in the option MSTU(101)=2 of PYALEM. Although it can technically be used also at rather small $Q^2$, this $\alpha_{\mathrm{em}}$ value is mainly intended for high $Q^2$, primarily $\mathrm{Z}^0$ and $\mathrm{W}^{\pm}$ physics.

PARU(104) :

(D=1 GeV$^2$) dividing line between `low' and `high' $Q^2$ values in the option MSTU(101)=2 of PYALEM.

PARU(105) :

(D=1.16639E-5 GeV$^{-2}$) $G_{\mathrm{F}}$, the Fermi constant of weak interactions.

PARU(108) :

(I) the $\alpha_{\mathrm{em}}$ value obtained in the latest call to the PYALEM function.

PARU(111) :

(D=0.20) fix $\alpha_{\mathrm{s}}$ value assumed in PYALPS when MSTU(111)=0 (and also in parton showers when $\alpha_{\mathrm{s}}$ is assumed fix there).

PARU(112) :

(I, D=0.25 GeV) $\Lambda$ used in running $\alpha_{\mathrm{s}}$ expression in PYALPS. Like MSTU(111), this value is overwritten by the calling physics routines, and is therefore purely nominal.

PARU(113) :

(D=1.) the flavour thresholds, for the effective number of flavours $n_f$ to use in the $\alpha_{\mathrm{s}}$ expression, are assumed to sit at $Q^2 =$PARU(113) $\times m_{\mathrm{q}}^2$, where $m_{\mathrm{q}}$ is the quark mass. May be overwritten from the calling physics routine.

PARU(114) :

(D=4 GeV$^2$) $Q^2$ value below which the $\alpha_{\mathrm{s}}$ value is assumed constant for MSTU(115)=2.

PARU(115) :

(D=10.) maximum $\alpha_{\mathrm{s}}$ value that PYALPS will ever return; is used as a last resort to avoid singularities.

PARU(117) :

(I) $\Lambda$ value (associated with MSTU(118) effective flavours) obtained in latest PYALPS call.

PARU(118) :

(I) $\alpha_{\mathrm{s}}$ value obtained in latest PYALPS call.

PARU(121) - PARU(130) :

couplings of a new $\mathrm{Z}'^0$; for fermion default values are given by the Standard Model $\mathrm{Z}^0$ values, assuming $\sin^2 \! \theta_W = 0.23$. Since a generation dependence is now allowed for the $\mathrm{Z}'^0$ couplings to fermions, the variables PARU(121) - PARU(128) only refer to the first generation, with the second generation in PARJ(180) - PARJ(187) and the third in PARJ(188) - PARJ(195) following exactly the same pattern. Note that e.g. the $\mathrm{Z}'^0$ width contains squared couplings, and thus depends quadratically on the values below.

PARU(121), PARU(122) :

(D=-0.693,-1.) vector and axial couplings of down type quarks to $\mathrm{Z}'^0$.

PARU(123), PARU(124) :

(D=0.387,1.) vector and axial couplings of up type quarks to $\mathrm{Z}'^0$.

PARU(125), PARU(126) :

(D=-0.08,-1.) vector and axial couplings of leptons to $\mathrm{Z}'^0$.

PARU(127), PARU(128) :

(D=1.,1.) vector and axial couplings of neutrinos to $\mathrm{Z}'^0$.

PARU(129) :

(D=1.) the coupling $Z'^0 \to \mathrm{W}^+ \mathrm{W}^-$ is taken to be PARU(129)$\times$(the Standard Model $\mathrm{Z}^0 \to \mathrm{W}^+ \mathrm{W}^-$ coupling) $\times (m_{\mathrm{W}}/m_{\mathrm{Z}'})^2$. This gives a $\mathrm{Z}'^0 \to \mathrm{W}^+ \mathrm{W}^-$ partial width that increases proportionately to the $\mathrm{Z}'^0$ mass.

PARU(130) :

(D=0.) in the decay chain $\mathrm{Z}'^0 \to \mathrm{W}^+ \mathrm{W}^- \to 4$ fermions, the angular distribution in the $\mathrm{W}$ decays is supposed to be a mixture, with fraction 1-PARU(130) corresponding to the same angular distribution between the four final fermions as in $\mathrm{Z}^0 \to \mathrm{W}^+ \mathrm{W}^-$ (mixture of transverse and longitudinal $\mathrm{W}$'s), and fraction PARU(130) corresponding to $\mathrm{h}^0 \to \mathrm{W}^+ \mathrm{W}^-$ the same way (longitudinal $\mathrm{W}$'s).

PARU(131) - PARU(136) :

couplings of a new $\mathrm{W}'^{\pm}$; for fermions default values are given by the Standard Model $\mathrm{W}^{\pm}$ values (i.e. $V-A$). Note that e.g. the $\mathrm{W}'^{\pm}$ width contains squared couplings, and thus depends quadratically on the values below.

PARU(131), PARU(132) :

(D=1.,-1.) vector and axial couplings of a quark-antiquark pair to $\mathrm{W}'^{\pm}$; is further multiplied by the ordinary CKM factors.

PARU(133), PARU(134) :

(D=1.,-1.) vector and axial couplings of a lepton-neutrino pair to $\mathrm{W}'^{\pm}$.

PARU(135) :

(D=1.) the coupling $\mathrm{W}'^{\pm} \to \mathrm{Z}^0 \mathrm{W}^{\pm}$ is taken to be PARU(135)$\times$(the Standard Model $\mathrm{W}^{\pm} \to \mathrm{Z}^0 \mathrm{W}^{\pm}$ coupling) $\times (m_{\mathrm{W}}/m_{W'})^2$. This gives a $\mathrm{W}'^{\pm} \to \mathrm{Z}^0 \mathrm{W}^{\pm}$ partial width that increases proportionately to the $\mathrm{W}'$ mass.

PARU(136) :

(D=0.) in the decay chain $\mathrm{W}'^{\pm} \to \mathrm{Z}^0 \mathrm{W}^{\pm} \to 4$ fermions, the angular distribution in the $\mathrm{W}/ \mathrm{Z}$ decays is supposed to be a mixture, with fraction 1-PARU(136) corresponding to the same angular distribution between the four final fermions as in $\mathrm{W}^{\pm} \to \mathrm{Z}^0 \mathrm{W}^{\pm}$ (mixture of transverse and longitudinal $\mathrm{W}/ \mathrm{Z}$'s), and fraction PARU(136) corresponding to $\H ^{\pm} \to \mathrm{Z}^0 \mathrm{W}^{\pm}$ the same way (longitudinal $\mathrm{W}/ \mathrm{Z}$'s).

PARU(141) :

(D=5.) $\tan\beta$ parameter of a two Higgs doublet scenario, i.e. the ratio of vacuum expectation values. This affects mass relations and couplings in the Higgs sector. If the Supersymmetry simulation is switched on, IMSS(5) nonvanishing, PARU(141) will be overwritten by RMSS(5) at initialization, so it is the latter varaible that should be set.

PARU(142) :

(D=1.) the $\mathrm{Z}^0 \to \H ^+ \H ^-$ coupling is taken to be PARU(142)$\times$(the MSSM $\mathrm{Z}^0 \to \H ^+ \H ^-$ coupling).

PARU(143) :

(D=1.) the $\mathrm{Z}'^0 \to \H ^+ \H ^-$ coupling is taken to be PARU(143)$\times$(the MSSM $\mathrm{Z}^0 \to \H ^+ \H ^-$ coupling).

PARU(145) :

(D=1.) quadratically multiplicative factor in the $\mathrm{Z}'^0 \to \mathrm{Z}^0 \mathrm{h}^0$ partial width in left-right-symmetric models, expected to be unity (see [Coc91]).

PARU(146) :

(D=1.) $\sin(2\alpha)$ parameter, enters quadratically as multiplicative factor in the $\mathrm{W}'^{\pm} \to \mathrm{W}^{\pm} \mathrm{h}^0$ partial width in left-right-symmetric models (see [Coc91]).

PARU(151) :

(D=1.) multiplicative factor in the $\L _{\mathrm{Q}} \to \mathrm{q}\ell$ squared Yukawa coupling, and thereby in the $\L _{\mathrm{Q}}$ partial width and the $\mathrm{q}\ell \to \L _{\mathrm{Q}}$ and other cross sections. Specifically, $\lambda^2/(4\pi) =$PARU(151) $\times \alpha_{\mathrm{em}}$, i.e. it corresponds to the $k$ factor of [Hew88].

PARU(153) :

(D=0.) anomalous magnetic moment of the $\mathrm{W}^{\pm}$ in process 20; $\eta = \kappa - 1$, where $\eta = 0$ ($\kappa = 1$) is the Standard Model value.

PARU(155) :

(D=1000. GeV) compositeness scale $\Lambda$.

PARU(156) :

(D=1.) sign of interference term between standard cross section and composite term ($\eta$ parameter); should be $\pm 1$.

PARU(157) - PARU(159) :

(D=3*1.) strength of SU(2), U(1) and SU(3) couplings, respectively, in an excited fermion scenario; cf. $f$, $f'$ and $f_s$ of [Bau90].

PARU(161) - PARU(168) :

(D=5*1.,3*0.) multiplicative factors that can be used to modify the default couplings of the $\mathrm{h}^0$ particle in PYTHIA. Note that the factors enter quadratically in the partial widths. The default values correspond to the couplings given in the minimal one-Higgs-doublet Standard Model, and are therefore not realistic in a two-Higgs-doublet scenario. The default values should be changed appropriately by you. Also the last two default values should be changed; for these the expressions of the minimal supersymmetric Standard Model (MSSM) are given to show parameter normalization. Alternatively, the SUSY machinery can generate all the couplings for IMSS(1), see MSTP(4).

PARU(161) :

$\mathrm{h}^0$ coupling to down type quarks.

PARU(162) :

$\mathrm{h}^0$ coupling to up type quarks.

PARU(163) :

$\mathrm{h}^0$ coupling to leptons.

PARU(164) :

$\mathrm{h}^0$ coupling to $\mathrm{Z}^0$.

PARU(165) :

$\mathrm{h}^0$ coupling to $\mathrm{W}^{\pm}$.

PARU(168) :

$\mathrm{h}^0$ coupling to $\H ^{\pm}$ in $\gamma \gamma \to \mathrm{h}^0$ loops, in MSSM $\sin(\beta-\alpha)+\cos(2\beta)\sin(\beta+\alpha) / (2\cos^2 \! \theta_W )$.

PARU(171) - PARU(178) :

(D=7*1.,0.) multiplicative factors that can be used to modify the default couplings of the $\H ^0$ particle in PYTHIA. Note that the factors enter quadratically in partial widths. The default values for PARU(171) - PARU(175) correspond to the couplings given to $\mathrm{h}^0$ in the minimal one-Higgs-doublet Standard Model, and are therefore not realistic in a two-Higgs-doublet scenario. The default values should be changed appropriately by you. Also the last two default values should be changed; for these the expressions of the minimal supersymmetric Standard Model (MSSM) are given to show parameter normalization. Alternatively, the SUSY machinery can generate all the couplings for IMSS(1), see MSTP(4).

PARU(171) :

$\H ^0$ coupling to down type quarks.

PARU(172) :

$\H ^0$ coupling to up type quarks.

PARU(173) :

$\H ^0$ coupling to leptons.

PARU(174) :

$\H ^0$ coupling to $\mathrm{Z}^0$.

PARU(175) :

$\H ^0$ coupling to $W^{\pm}$.

PARU(176) :

$\H ^0$ coupling to $\mathrm{h}^0 \mathrm{h}^0$, in MSSM $\cos(2\alpha) \cos(\beta+\alpha) - 2 \sin(2\alpha) \sin(\beta+\alpha)$.

PARU(177) :

$\H ^0$ coupling to $\mathrm{A}^0 \mathrm{A}^0$, in MSSM $\cos(2\beta) \cos(\beta+\alpha)$.

PARU(178) :

$\H ^0$ coupling to $\H ^{\pm}$ in $\gamma \gamma \to \H ^0$ loops, in MSSM $\cos(\beta-\alpha) - \cos(2\beta)\cos(\beta+\alpha) / (2\cos^2 \! \theta_W )$.

PARU(181) - PARU(190) :

(D=3*1.,2*0.,2*1.,3*0.) multiplicative factors that can be used to modify the default couplings of the $\mathrm{A}^0$ particle in PYTHIA. Note that the factors enter quadratically in partial widths. The default values for PARU(181) - PARU(183) correspond to the couplings given to $\mathrm{h}^0$ in the minimal one-Higgs-doublet Standard Model, and are therefore not realistic in a two-Higgs-doublet scenario. The default values should be changed appropriately by you. PARU(184) and PARU(185) should be vanishing at the tree level, in the absence of CP violating phases in the Higgs sector, and are so set; normalization of these couplings agrees with what is used for $\mathrm{h}^0$ and $\H ^0$. Also the other default values should be changed; for these the expressions of the Minimal Supersymmetric Standard Model (MSSM) are given to show parameter normalization. Alternatively, the SUSY machinery can generate all the couplings for IMSS(1), see MSTP(4).

PARU(181) :

$\mathrm{A}^0$ coupling to down type quarks.

PARU(182) :

$\mathrm{A}^0$ coupling to up type quarks.

PARU(183) :

$\mathrm{A}^0$ coupling to leptons.

PARU(184) :

$\mathrm{A}^0$ coupling to $\mathrm{Z}^0$.

PARU(185) :

$\mathrm{A}^0$ coupling to $\mathrm{W}^{\pm}$.

PARU(186) :

$\mathrm{A}^0$ coupling to $\mathrm{Z}^0 \mathrm{h}^0$ (or $\mathrm{Z}^*$ to $\mathrm{A}^0 \mathrm{h}^0$), in MSSM $\cos(\beta-\alpha)$.

PARU(187) :

$\mathrm{A}^0$ coupling to $\mathrm{Z}^0 \H ^0$ (or $\mathrm{Z}^*$ to $\mathrm{A}^0 \H ^0$), in MSSM $\sin(\beta-\alpha)$.

PARU(188) :

As PARU(186), but coupling to $\mathrm{Z}'^0$ rather than $\mathrm{Z}^0$.

PARU(189) :

As PARU(187), but coupling to $\mathrm{Z}'^0$ rather than $\mathrm{Z}^0$.

PARU(190) :

$\mathrm{A}^0$ coupling to $\H ^{\pm}$ in $\gamma \gamma \to \mathrm{A}^0$ loops, 0 in MSSM.

PARU(191) - PARU(195) :

(D=4*0.,1.) multiplicative factors that can be used to modify the couplings of the $\H ^{\pm}$ particle in PYTHIA. Currently only PARU(195) is in use. See above for related comments.

PARU(195) :

$\H ^{\pm}$ coupling to $\mathrm{W}^{\pm} \mathrm{h}^0$ (or $\mathrm{W}^{* \pm}$ to $\H ^{\pm} \mathrm{h}^0$), in MSSM $\cos(\beta-\alpha)$.

PARU(197):

(D=0.) $\H ^0$ coupling to $\mathrm{W}^{\pm} \H ^{\mp}$ within a two-Higgs-doublet model.

PARU(198):

(D=0.) $\mathrm{A}^0$ coupling to $\mathrm{W}^{\pm} \H ^{\mp}$ within a two-Higgs-doublet model.

PARJ(180) - PARJ(187) :

couplings of the second generation fermions to the $Z'^0$, following the same pattern and with the same default values as the first one in PARU(121) - PARU(128).

PARJ(188) - PARJ(195) :

couplings of the third generation fermions to the $Z'^0$, following the same pattern and with the same default values as the first one in PARU(121) - PARU(128).