3-Jet Fraction in the Cross-Section for
+
e
-
e
Hadrons

A demonstration for how to get this cross-section from the Lagrangian via different jet algorithms.
June 20, 2017—Andrey Grabovskiy

Notation and Rules

Use vec for vectors and g for the metric tensor, and distinguish upper and lower indices:
In[]:=
Format[g[a_,b_],TraditionalForm]:=Subscript[g,SequenceForm[a,b]]​​Format[g[up[a_],b_],TraditionalForm]:=Subsuperscript[g,b,a]​​Format[g[b_,up[a_]],TraditionalForm]:=Subsuperscript[g,b,a]​​Format[g[up[a_],up[b_]],TraditionalForm]:=Superscript[g,SequenceForm[a,b]]​​Attributes[g]=Orderless;​​​​Format[γ[a_],TraditionalForm]:=
γ
a
​​Format[γ[up[a_]],TraditionalForm]:=
a
γ
​​​​Format[vec[a_,b_],TraditionalForm]:=
a
b
​​Format[vec[a_,up[b_]],TraditionalForm]:=
b
a
​​vec[0,_]=0;​​
Use SP for scalar product:
In[]:=
SP[0]=0;​​Attributes[SP]=Orderless;​​SP[aa_,aa_]:=SP[aa]​​SP[aa_,-bb_]:=-SP[aa,bb]​​SP[-aa_]:=SP[aa]​​Format[SP[a_,b_],TraditionalForm]:=SequenceForm[a,b]​​Format[SP[a_],TraditionalForm]:=
2
a
​​SP[0,_]=0;
Convolution rules:
In[]:=
gammasim={a___·γ[up[c_]]·γ[up[d_]]·b___vec[e_,c_]vec[e_,d_](-SP[e])a·b,g[a_,b_]vec[c_,up[b_]]vec[c,a],g[a_,up[b_]]vec[c_,b_]vec[c,a],g[a_,up[b_]]g[c_,b_]g[c,a],g[dd_,up[b_]]c___·γ[b_]·a___c·γ[dd]·a,g[dd_,b_]c___·γ[up[b_]]·a___c·γ[dd]·a,vec[ka_,b_]vec[c_,up[b_]]-SP[ka,c],g[ao_,up[ao_]]->4}
Out[]=
{a___·γ[up[c_]]·γ[up[d_]]·b___vec[e_,c_]vec[e_,d_]-(a·b)SP[e],g[a_,b_]vec[c_,up[b_]]vec[c,a],g[a_,up[b_]]vec[c_,b_]vec[c,a],g[a_,up[b_]]g[b_,c_]g[a,c],c___·γ[b_]·a___g[dd_,up[b_]]c·γ[dd]·a,c___·γ[up[b_]]·a___g[b_,dd_]c·γ[dd]·a,vec[c_,up[b_]]vec[ka_,b_]-SP[c,ka],g[ao_,up[ao_]]4}

γ Matrix Trace (Spur) Calculation

Cross-Section for q
OverLine(q)-BarGluon
Production

Cross-Section for 3-Jet Production

Questions

AUTHORSHIP INFORMATION
Andrey Grabovskiy
6/20/17