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src/constants.jl

@@ -1,7 +1,5 @@
-# FIXME: implement the constants
-
 """
-    ELECTRON_MASS
+	ELECTRON_MASS
 
 The mass of the electron in its rest frame, given as 0.51099895000 MeV
 
@@ -12,7 +10,7 @@ This value is taken from the PDG 2020 recommended values of the fundamental phys
 - The Review of Particle Physics: P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020)
 
 """
-ELECTRON_MASS
+const ELECTRON_MASS = 0.51099895000
 
 """
    MUON_MASS 
@@ -25,7 +23,7 @@ This value is taken from the PDG 2020 recommended values of the fundamental phys
 - The Review of Particle Physics: P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020)
 
 """
-MUON_MASS
+const MUON_MASS = 105.6583755
 
 """
   ALPHA 
@@ -38,4 +36,5 @@ This value is taken from the PDG 2020 recommended values of the fundamental phys
 - The Review of Particle Physics: P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020)
 
 """
-ALPHA
+const ALPHA = 0.0072973525643
+const ALPHA_UNC = 0.0000000000011

src/cross_section.jl

@@ -1,12 +1,10 @@
 
-@inline function _rho(E,m) 
-    #
-    # FIXME: implement this!
-    #
+@inline function _rho(E, m)
+	sqrt(E^2 - m^2)
 end
 
 """
-    differential_cross_section(E_in::Real, cos_theta::Real)
+	differential_cross_section(E_in::Real, cos_theta::Real)
 
 Calculates the differential cross section for the process ``e^+ e^- \\to \\mu^+ \\mu^-`` 
 at tree level in quantum electrodynamics (QED). The calculation is performed in the center-of-momentum 
@@ -32,7 +30,7 @@
 
 # Example
 
 ```jldoctest
 julia> E_in = 1e3 # MeV
 1000.0
 
@@ -40,20 +38,22 @@
 0.5
 
 julia> differential_cross_section(E_in, cos_theta)
-4.164686491998452e-12
+4.164686491998452e-12 
+SIMONE: this is wrong, correct value is 4.188129051838138e-12
 ```
 
 # References
 - Schwartz 2014: M.D. Schwartz, "Quantum Field Theory and the Standard Model", Cambridge University Press, New York (2014)
 """
 function differential_cross_section(E_in, cos_theta)
-    #
-    # FIXME: fill me in
-    #
+	rho_e = _rho(E_in, ELECTRON_MASS)
+	rho_mu = _rho(E_in, MUON_MASS)
+	tmp = (E_in^4 + rho_e^2 * rho_mu^2 * (cos_theta)^2 + E_in^2 * (ELECTRON_MASS^2+MUON_MASS^2))
+	return tmp * ALPHA^2 * rho_mu / (16.0 * E_in^6) / rho_e
 end
 
 """
-    total_cross_section(E_in::Real)
+	total_cross_section(E_in::Real)
 
 Calculates the total cross section for the process ``e^+ e^- \\to \\mu^+ \\mu^-`` at tree level,
 as a function of the initial electron energy `E_in`.
@@ -85,9 +85,11 @@
 ```
 """
 function total_cross_section(E_in)
-    #
-    # FIXME: fill me in
-    #
+	rho_e = _rho(E_in, ELECTRON_MASS)
+	rho_mu = _rho(E_in, MUON_MASS)
+	pref = pi * (ALPHA^2) * rho_mu / (8.0 * E_in^6) / rho_e
+	tmp = (2.0 * E_in^4 + 2.0 * rho_e^2 * rho_mu^2 / 3.0 + 2.0 * E_in^2 * (ELECTRON_MASS^2+MUON_MASS^2))
+	return pref * tmp
 end
 
 

src/four_momentum.jl

@@ -21,19 +21,18 @@
 FourMomentum(en = 4.0, x = 1.0, y = 2.0, z = 3.0)
 ```
 """
-struct FourMomentum{T<:Real}
-    #
-    # FIXME: fill me in
-    #
+struct FourMomentum{T<:Real} 
+    en::T  # Energy component
+    x::T   # Spatial component in the x-direction
+    y::T   # Spatial component in the y-direction
+    z::T   # Spatial component in the z-direction
 end
 # type promotion on construction
 FourMomentum(en,x,y,z) = FourMomentum(promote(en,x,y,z)...)
 
 # return the element type
 function Base.eltype(::FourMomentum{T}) where T 
-    #
-    # FIXME:
-    #
+    return T
 end
 
 # Overload Base.show for pretty printing of FourMomentum; plain text version
@@ -72,7 +71,8 @@
 """
 function Base.:+(p1::FourMomentum, p2::FourMomentum) 
 
-    # FIXME: fill me in
+    #element-wise addition of the four-momentum components
+    return FourMomentum(p1.en + p2.en, p1.x + p2.x, p1.y + p2.y, p1.z + p2.z)
 
 end
 
@@ -98,7 +98,8 @@
 ```
 """ 
 function Base.:-(p1::FourMomentum, p2::FourMomentum) 
-    # FIXME: fill me in
+    # element-wise subtraction of the four-momentum components
+    return FourMomentum(p1.en - p2.en, p1.x - p2.x, p1.y - p2.y, p1.z - p2.z)
 end
 
 
@@ -120,7 +121,8 @@
 ```
 """
 function Base.:*(a::Real, p::FourMomentum) 
-    # FIXME: fill me in
+    # scalar multiplication of the four-momentum components
+    return FourMomentum(a * p.en, a * p.x, a * p.y, a * p.z)
 end
 
 
@@ -149,17 +151,35 @@
 """
 function minkowski_dot(p1::FourMomentum, p2::FourMomentum)
     # Minkowski metric: (+,-,-,-)
-    # FIXME: fill me in
+    return p1.en * p2.en - p1.x * p2.x - p1.y * p2.y - p1.z * p2.z
 end
 
 function _construct_moms_from_coords(E_in, cos_theta, phi)
-    #
-    # FIXME: fill me in
-    #
+    # Calculate the energy and momentum components for the electron-positron pair
+    E = E_in / 2  # Each particle has half the total energy in the center-of-mass frame
+    p = sqrt(E^2 - ELECTRON_MASS)  # Assuming electron mass is 0.511 MeV/c^2
+
+    # Calculate the spatial components based on the scattering angle and azimuthal angle
+    px = p * sqrt(1 - cos_theta^2) * cos(phi)
+    py = p * sqrt(1 - cos_theta^2) * sin(phi)
+    pz = p * cos_theta
+
+    # Construct four-momenta for e-, e+, mu-, mu+
+    e_minus = FourMomentum(E, px, py, pz)
+    e_plus = FourMomentum(E, -px, -py, -pz)
+    mu_minus = FourMomentum(E, px, py, pz)
+    mu_plus = FourMomentum(E, -px, -py, -pz)
+
+    return Dict("e-" => e_minus, "e+" => e_plus, "mu-" => mu_minus, "mu+" => mu_plus)
 end
 
 # TODO: 
 # consider using NamedTuples instead
+
+
+
+
+
 """
     coords_to_dict(E_in::Real, cos_theta::Real, phi::Real)
 
@@ -172,7 +192,7 @@
 
 # Example
 
 ```jldoctest
 julia> mom_dict = coords_to_dict(1e3,0.9,pi/4)
 Dict{String, FourMomentum{Float64}} with 4 entries:
   "mu+" => (1000.0, -306.495431, -306.495431, -894.962239)…
@@ -197,7 +217,38 @@
 ```
 """
 function coords_to_dict(E_in,cos_theta,phi)
-    #
-    # FIXME: fill me in
-    #
+
+    # Validate input types
+    if !(isa(E_in, Real) && isa(cos_theta, Real) && isa(phi, Real))
+        throw(ArgumentError("All inputs must be of type Real."))
+    end
+
+    # Validate ranges
+    if cos_theta < -1 || cos_theta > 1
+        throw(ArgumentError("cos_theta must be in the range [-1, 1]."))
+    end
+    if phi < 0 || phi >= 2 * pi
+        throw(ArgumentError("phi must be in the range [0, 2π)."))
+    end
+
+    return _construct_moms_from_coords(E_in, cos_theta, phi)
+
+
+
 end
+
+function extractThreeMomentum(p::FourMomentum)
+
+    return (p.x, p.y, p.z)
+end
+
+function extractEnergy(p::FourMomentum)
+
+    return p.en
+end
+
+function extractNegativeThreeMomentum(p::FourMomentum)
+
+    return (-p.x, -p.y, -p.z)
+end
+