CITE THIS NOTEBOOK: A curious mass ratio by Paul Abbott. Wolfram Community FEB 2 2023.
School of Physics, Mathematics and Computing
The University of Western Australia
​paul.c.abbott@uwa.edu.au

m
e
+
m
μ
+
m
τ
2

m
e
+
m
μ
+
m
τ


The following approximation appeared on Fermat's Library:
In[]:=
Here
m
e
is the electron mass,
m
μ
is the muon mass, and
m
τ
is the tau mass.
A nice problem is to check this computation.

Direct Computation

Extract the masses:
In[]:=
m[e]=
electron
PARTICLE

mass

Out[]=
510.9989461
keV/
2
c
In[]:=
m[μ]=
muon
PARTICLE

mass

Out[]=
105.6583745
MeV/
2
c
In[]:=
m[τ]=
tau
PARTICLE

mass

Out[]=
1.77686
GeV/
2
c
Compute the expression:
In[]:=
m[e]+m[μ]+m[τ]
2

m[e]
+
m[μ]
+
m[τ]

Out[]=
0.66666

Uncertainty

To extract the standard uncertainty for these masses, one can find the entity property
mass
Around
via
, which that leads to a very simple direct solution:
In[]:=
{m[e],m[μ],m[τ]}=Through
electron
PARTICLE
,
muon
PARTICLE
,
tau
PARTICLE

mass
Around

Out[]=

0.51099894
(
+33
61
-33
)

MeV/
2
c
,
105.65837
(
+25
45
-25
)

MeV/
2
c
,

+0.13
1776.86
-0.13

MeV/
2
c

Use
AroundReplace
to obtain the uncertainty in the result:
In[]:=
AroundReplace
m
e
+
m
μ
+
m
τ
2

m
e
+
m
μ
+
m
τ

,{
m
e
m[e],
m
μ
m[μ],
m
τ
m[τ]}
Out[]=
0.66666
(
+7
1
-7
)
It looks like 2/3 is not the most probable value for this expression (although it is a possible one).

Distribution

Michael Trott suggested looking at the distribution, which is also quite instructive:
In[]:=
{m[e],m[μ],m[τ]}=QuantityMagnitude[{m[e],m[μ],m[τ]}]
Out[]=
0.51099894
(
+33
61
-33
)
,105.65837
(
+25
45
-25
)
,
+0.13
1776.86
-0.13

In[]:=
td=TransformedDistribution
m
e
+
m
μ
+
m
τ
2

m
e
+
m
μ
+
m
τ

,{
m
e
NormalDistribution[m[e]["Value"],m[e]["Uncertainty"]〚1〛],
m
μ
NormalDistribution[m[μ]["Value"],m[μ]["Uncertainty"]〚1〛],
m
τ
NormalDistribution[m[τ]["Value"],m[τ]["Uncertainty"]〚1〛]};
In[]:=
HistogramRandomVariate[td,
6
10
],100,GridLines
2
3
,None,GridLinesStyleRed
Out[]=
So, clearly, 2/3 is not the most likely value for this mass ratio.