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* "Polarised Amplitudes and Soft-Virtual Cross Sections for $b\bar b \rightarrow ZH$ at NNLO in QCD" *
*  Taushif Ahmed, A.H. Ajjath, Long Chen, Prasanna K. Dhani,  Pooja Mukherjee and V. Ravindran       *
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(A) 
Linearly_Polarised_Partial_Amplitudes_Tree.m
Linearly_Polarised_Partial_Amplitudes_1Loop.m
Linearly_Polarised_Partial_Amplitudes_2Loop.m
Linearly_Polarised_Partial_Amplitudes_2Loop_ABJ.m
..................................................

Each of the above file is a list of 6 entries saving the UV renormalized form of projections obtained using linearly polarised projectors listed in eq.(3.5) in the same ordering, at the tree, one-loop and two-loop order expanded up to ep^4, ep^2 and ep^0, respectively. More specifically, we present 

\mathcal{P}^{\mu}_j \bar{v}(p_2)\mathbf{\Gamma}_{\mu} u(p_1)
for j=1,2,3,4,5,6.

The suffix "ABJ" denotes the contribution from the set of two-loop anomalous diagrams. For this set, colorfactor[CF*NF]=CF.
The full 2-loop result is the sum of expressions in Linearly_Polarised_Partial_Amplitudes_2Loop.m and Linearly_Polarised_Partial_Amplitudes_2Loop_ABJ.m.

The formula needed to compose the polarized amplitudes from these quantities is given in eq.(3.6).

In these expressions, all color factors are wrapped under the Head "colorfactor" with nf denoting the number of massless quark flavors i.e. 5 and CA, CF are the Casimirs in adjoint and fundamental representations which are given explicitly in eq.(4.4). The symbol "I" denotes the imaginary unit, i.e. square root of -1 and the symbol ep denotes the dimensional regulator, defined as (4-D)/2.

The Madelstam variables, s, t, mz2 are defined by eq.(2.2):
s = (p1+p2)^2, t=(p1-q1)^2, u=(p2-q1)^2, p1^0=p2^2=0, q1^2=m1^2=m_z^2, q2^2=m2^2=m_h^2

The dimensionless variables are defined as:
x = m^2 (1+x) (1+xy), t=-m^2 x z, q1^2=m^2, q2^2=m^2 x^2 y 

The results are expressed in terms of Logarithm, Classical polylogarithm and Li (2,2,x,y).
The letters which appear in the expressions are defined as
{l1 -> x, l2 -> 1 + x, l3 -> y, l4 -> 1 - y, l5 -> z, l6 -> 1 - z, 
    l7 -> -y + z, l8 -> 1 + y - z, l9 -> 1 + x y, l10 -> 1 + x z, 
    l11 -> x y + z, l12 -> 1 + y + x y - z, l13 -> 1 + x + x y - x z, 
    l14 -> 1 + y + 2 x y - z + x^2 y z, 
    l15 -> 2 x y + x^2 y + x^2 y^2 + z - x^2 y z, 
    l16 -> 1 + x + y + x y + x y^2 - z - x z - x y z, 
    l17 -> 1 + y + x y + y^2 + x y^2 - z - y z - x y z, 
    l18 -> -x y + z + x z + x y z, l19 -> -y + z + y z + x y z}
The Kallen function is written as root = Sqrt[ s^2 + m1^4 + m2^4 - 2 (s m1^2 + m1^2 m2^2 + m2^2 s) ]

The electroweak couplings, which are suppressed in the aforementioned files, are explained in the Appendix A with expressions given in eq.(A.2).



(B) 
Cross_Section_Soft_Virtual.m
............................

This is a list of 3 entries corresponding to cross sections at leading order, NLO and NNLO. The expressions at NLO and NNLO are in the soft-virtual approximation. 

delta=delta(1-z)
D0,D1,D2,D3 are the plus distributions defined in section 7, eq.(7.2).
