Sample Numerical Evaluation of e^+e^- -> Four Parton Loop Amplitudes

This file contains numerical evaluation of the amplitudes for e^+e^- -> q qbar Q Qbar

[eventually we should add e^+e^- -> q g g qbar]

at a couple of kinematic points (one, so far).

The amplitudes are from:

Paper I: hep-ph/9610370 , Nucl. Phys. B489 (1997), 3, for e^+e^- -> q qbar Q Qbar only

Paper II: hep-ph/9708239 , Nucl. Phys. B513 (1998), 3, for e^+e^- -> q qbar Q Qbar and e^+e^- -> q g g qbar.

The corresponding Maple and Mathematica expressions are also available.

The first kinematic point, in maple notation, is:

K[1]:= vector([0.1114840966233978, 1.7414106410531399E-02, -7.4129753043655378E-02, 8.1426239085103519E-02 ]):

K[2]:= vector([0.4007105288041996, -0.1188785829684985, 0.3809442819803379, 3.6307911430263483E-02 ]):

K[3]:= vector([0.4067364031558585, 8.5137970338974467E-02, -0.3537515560803549, -0.1817852145512735 ]):

K[4]:= vector([8.1068971416544025E-02, 1.6326506218992662E-02, 4.6937027143672323E-02, 6.4051064035906491E-02 ]):

K[5]:= vector([ - 0.5, 0 , 0, 0.5 ]):

K[6]:= vector([ - 0.5, 0 , 0, - 0.5 ]):

NOTE:

* The first component of each momentum vector is the time component (energy); the last three are the spatial components.

* An "all-outgoing" momentum convention is used, in which the signs of all components of incoming momenta (in this case K[5] and K[6]) are reversed with respect to standard, positive-energy conventions, so that \sum_i K[i] = 0.

First we give values of some of the spinor products for these kinematics.

There are different possible phase conventions for the spinor products.

Maple procedures that contain our conventions are given here.

<12> = .095848237508053706 + .36723960069159301 * I

<13> = .25212083943590259 - .036280871181179222 * I

<14> = -.0029052554710315017 + .11843447342895163 * I

<15> = -.14600042635487092 + .41423931635311632 * I

<16> = -.14600042635487092 - .093497235480468331 * I

<23> = .18556855003770264 - .77103394358663994 * I

<24> = -.087612241372900612 - .14409828206690534 * I

<25> = .50740103848885969 + .42374830545370885 * I

<26> = .50740103848885969 - .32702722136599390 * I

[12] = -.095848237508053743 + .36723960069159315 * I

[13] = -.25212083943590271 - .036280871181179240 * I

[14] = .0029052554710315036 + .11843447342895170 * I

[15] = -.14600042635487096 - .41423931635311643 * I

[16] = -.14600042635487115 + .093497235480468477 * I

s_{12} = .14405180894952101

s_{13} = .064881219291529528

s_{14} = .014035165005745017

s_{15} = -.19291033570850132

s_{16} = -.030057857538294280

s_{23} = .62892902892586126

s_{24} = .028440219733016828

s_{25} = -.43701844023446308

s_{26} = -.36440261737393612

s_{56} = 1.00

q qbar Q Qbar, helicity configuration "++":
A_6^{tree,++}(1,2,3,4) of eq. (3.10) of I, and of eq. (12.3) of II,

Atree[`q+Qb+Q-qb-`] of Maple, Atree["q+Qb+Q-qb-"] of Mathematica

-> -.12605426632500299 - .11816940321985430 * I

V^{++}(1,2,3,4) of eq. (3.11) of I, and of eq. (12.4) of II,

V[`q+Qb+Q-qb-`] of Maple, V["q+Qb+Q-qb-"] of Mathematica

-> - 2/e^2

+ (-3.3939951062996455 - 2*ln(mu^2) - 6.2831853071795864 * I)/e

- ln(mu^2)^2 - 3.3939951062996455 * ln(mu^2) + 7.8546328930259723

+ I * ( -6.2831853071795864 * ln(mu^2) - 10.662550092270676 )

F^{++}(1,2,3,4) of eq. (3.12) of I, and of eq. (12.5) of II,

F[`q+Qb+Q-qb-`] of Maple, F["q+Qb+Q-qb-"] of Mathematica

-> -.72276578786469562 + .7834890945989297 * I

[Note that it has only a dispersive part, no absorptive part, for these kinematics.]

q qbar Q Qbar, helicity configuration "+-":
A_6^{tree,+-}(1,2,3,4) of eq. (3.13) of I, and of eq. (12.6) of II,

Atree[`q+Qb-Q+qb-`] of Maple, Atree["q+Qb-Q+qb-"] of Mathematica

-> .080524399880310078 + .036637772656647830 * I

V^{+-}(1,2,3,4) of eq. (3.14) of I, and of eq. (12.7) of II,

V[`q+Qb-Q+qb-`] of Maple, V["q+Qb-Q+qb-"] of Mathematica

-> - 2/e^2

+ (-3.3939951062996455 - 2*ln(mu^2) - 6.2831853071795864 * I)/e

- ln(mu^2)^2 - 3.3939951062996455 * ln(mu^2) + 7.8546328930259723

+ I * ( -6.2831853071795864 ln(mu^2) - 10.662550092270676 )

[same as V^{++}(1,2,3,4) above]

F^{+-}(1,2,3,4) of eq. (3.15) of I, and of eq. (12.8) of II,

F[`q+Qb-Q+qb-`] of Maple, F["q+Qb-Q+qb-"] of Mathematica

-> -.021379727948831 + .1265611606653839 * I

[Only a dispersive part, for these kinematics.]

q qbar Q Qbar, subleading-color helicity configuration "sl":
A_6^{tree,sl}(1,2,3,4) of eq. (3.16) of I, and of eq. (12.9) of II,

Atreesl of Maple and Mathematica

-> -1.2932296324459240 - 1.4381552961978792 * I

V^{sl}(1,2,3,4) of eq. (3.19) of I, and of eq. (12.10) of II,

Vsl of Maple and Mathematica

-> - 2/e^2

+ (-7.0606617729663122 - 2*ln(mu^2) - 6.2831853071795864 * I)/e

- ln(mu^2)^2 - 7.0606617729663122 * ln(mu^2) - 7.8522340747300819

+ I * ( -6.2831853071795864 ln(mu^2) - 22.181723155433251 )

F^{sl}(1,2,3,4) of eq. (3.20) of I, and of eq. (12.11) of II,

Fsl of Maple and Mathematica

-> 5.825295630461210 - 4.6105373785423904 * I

[Only a dispersive part, for these kinematics.]

q qbar Q Qbar, axial-vector fermion-loop contribution "ax":
A_6^{ax}(1,2,3,4) of eq. (3.22) of I, and of eq. (12.13) of II

[not given in Maple or Mathematica files]

-> .0012706742355937123 - .00013713139735817644 * I

+ ( .000091115338712839406 - .0000036838035841907216 * I )/m_top^2

+ O(1/m_top^4)

[Only a dispersive part, for these kinematics.]