(* PY751, Assignment 5, Problem 2 *)

(* e+e- -> mu+mu- with gamma + Z *)

alphaW = 1 / 29.5; gW = Sqrt[4 Pi alphaW]; s2w = 0.23;alpha = 1 / 129; e = Sqrt[4 Pi alpha]; (* alpha at the Z pole *)

MZ = 91.2; GaZ = 2.5; (* in GeV *)

gev2topbarn = 0.389 × 109; (* converts GeV-2 to picobarn *)

In[7]:=

P[s_] :=gW2

1 - s2w

1

s - MZ2 + I GaZ MZ;

al2[s_] := Abse2

s+ P[s] -

1

2+ s2w

22;

be2[s_] := Abse2

s+ P[s] (s2w)2

2;

ga2[s_] := Abse2

s+ P[s] (s2w) -

1

2+ s2w

2;

Melement[s_, ct_] :=

s2

41 + ct2 (al2[s] + be2[s] + 2 ga2[s]) +

s2

4(2 ct) (al2[s] + be2[s] - 2 ga2[s]);

dsigma[s_, ct_] :=gev2topbarn

32 Pi sMelement[s, ct];

(* and the corresponding expressions for QED *)

MQED[s_, ct_] :=s2

41 + ct2 4

e4

s2;

dsigmaQED[s_, ct_] :=gev2topbarn

32 Pi sMQED[s, ct];

In[15]:= (* Problem 2.2. *)

sigma[s_] := NIntegrate[dsigma[s, ct], {ct, -1, 1}];LogPlotsigmaroots2, {roots, 5, 120}, AxesLabel → {GeV, sigma}

Out[16]=

20 40 60 80 100 120GeV

50

100

500

1000

5000

sigma

In[23]:= (* Problem 2.3 *)

Plot10-3 s

2 Pidsigma[s, ct] /. s → 352, 10-3 s

2 PidsigmaQED[s, ct] /. s → 352,

{ct, -1, 1}, PlotRange → {0, 12}, PlotStyle → {Dashing[None], Dashing[Tiny]},

Frame → True, Axes → False, PlotLegends → {"Standard Model", "QED"}

Out[23]=

-1.0 -0.5 0.0 0.5 1.00

2

4

6

8

10

12

Standard Model

QED

In[21]:= (* Problem 2.4 *)

Afb[s_] :=3

4

al2[s] + be2[s] - 2 ga2[s]

al2[s] + be2[s] + 2 ga2[s];

PlotAfbroots2, {roots, 20, 150}, PlotRange → {-1, 1}, AxesLabel → {GeV, Afb}

Out[21]=40 60 80 100 120 140

GeV

-1.0

-0.5

0.5

1.0Afb

2 mathhw5_2019.nb