Extracting GF from τµ :
Obtaining the Central Value and Propagated Errors
Kevin Lynch
Boston University
Version 1.1: September 16, 2006
Abstract
While measuring and reporting the positive muon lifetime, τµ+ , is the primary goal of our experiment,
extracting GF is the primary theoretical motivation. In this note, I review the theory connecting GF
and τµ . In particular, I calculate numerical values of the relevant expressions connecting measurements
and Standard Model theory with GF , paying special attention to the error budget. Using the current
world average value of τµ , I obtain exactly the current world average GF , giving confidence that we can
correctly interpret our experiment.
1
Introduction to the Fermi Theory
In the early 1930s, Enrico Fermi turned his attention to the problem of nuclear beta decay. [1, 2] It was later
realized that muon decay is a manifestation of similar underlying physics. The contact interaction framework
Fermi developed, expressed in modern field theoretical Lagrangian language as
"
#
√
LFermi = −2 2Gµ e γ α PL νe µ γα PL νµ
,
(1.1)
despite being non-renormalizable, is still well suited to describing the low energy limit of the full electroweak
interaction framework. In particular, the full Lagrangian of the Fermi theory neatly separates the parameterization of the weak interaction and electromagnetic physics. The theory is finite to leading order in GF ,
and to all orders in αem . [3] A triumph of the Standard Model was the realization that the same underlying
bosonic physics described decay of muons and beta decay with the same parameter GF = Gβ = Gµ . The low
energy behavior of the full renomalizable Standard Model electroweak interaction (including, if desired, radiative corrections) can be mapped onto the Fermi form, giving a convenient parameterization for comparing
the results of different precision experiments.
For our purposes, the most interesting piece of this program is the extraction of the Fermi constant
from the muon lifetime, pioneered within the SU(2)L × U(1)Y theory by Sirlin.[4–6] The measurement of
the lifetime is remarkably clean and unambiguous (at least in particle physics terms), and the theoretical
extraction of GF from the measurement is unimpeded by theoretical limits. In particular, there are three
simplifying properties that arise in the calculation:
1. Difficult, multi-scale weak interaction corrections need not be applied. The lifetime factors into a purely
weak piece and a purely non-weak piece. All of the weak part is defined to be GF , so no calculations
need to be done!
2. QED loop corrections are small, obviating the need for high loop order corrections, even for a ppm scale
measurement. This occurs because the coupling is small (α/π ≈ 2 · 10−3), and loops are suppressed by
powers of the coupling.
3. Hadronic uncertainties are virtually non-existent. Any corrections to weak boson propagators are
folded into GF , and hence not calculated. The first hadronic correction, then, arises at two-loop order
in QED, and are thus suppressed by at least (α/π)2 ≈ 5 · 10−6 .
1
Despite the relative simplicity I’ve described, an extraction of GF at the ppm level requires calculating
through two-loops in QED, which is formidable no matter how skilled a theorist you are. The problem was
only solved in 1999 by van Ritbergen and Stuart (vRS) description [7]. In the rest of this note, I will refer
to their organization of the relations, given here
!
X
G2F m5µ
1
(1.2)
qi = Γ0 (1 + q) .
1+
=
τµ
192π 3
i
Here, Γ0 encodes the tree level, massless electron limit that most field theory students calculate, while the qi
encode the massive phase space and QED loop correction data. It is important to note that this description
differs in a small yet important way from the “standard” description originally described by Sirlin [5], and
still reported in the current PDG review [8] of electroweak physics:
!
G2F m5µ
3m2µ
1
.
(1.3)
=
(1 + q0 ) (1 + q1 ) 1 +
2
τµ
192π 3
5MW
where I have recast the expression in the vRS notation to the extent possible. The final term, from the
W boson propagator, is embedded in GF in the vRS description, because it is a purely weak interaction
phenomenon. [7] Up to now, it hasn’t mattered – it is only a 0.52 ppm correction – but does matter at the
two loop level. Additionally, it’s not clear how to include the higher loop QED correction in this framework.
We must make it clear that we use the vRS framework, and not the PDG framework in our result.
You may ask about where all those weak interaction loops end up. The answer is that we take GF as
input, and extract fundamental parameters from that, just as we extract GF from the muon lifetime. In the
standard normalization, then, we have
!
X
GF
g2
√ =
(1.4)
ri ,
1+
2
8MW
2
i
where the ri are the weak interaction loop order parameters. This can be recast in another form, where the
other observables are manifest [9]:
!
2
X
M
πα
em
2
1− W
MW
(1.5)
ri .
1+
=√
MZ20
2GF
i
Additionally, the ri are closely related to the ρ parameter, so the calculation of these corrections are simplified
by all the work done on ρ in understanding the violations of weak isospin in, for example, the t/b sector. I
won’t say any more about these issues, as they take me too far afield.
For the rest of this note, my goal is to provide the results needed to extract the Fermi constant from
our measurement of the lifetime, in the vRS framework. I have attempted to carefully survey the relevant
literature and extract the information necessary to extract GF from τµ , while properly accounting for all of
the known and estimated experimental and theoretical uncertainties. In particular, the relative error budget
of the various contributions from Equation 1.2 can be expressed as
δGF
5 δmµ
1 δτµ
1 δ(1 + q)
=−
−
−
.
GF
2 mµ
2 τµ
2 (1 + q)
(1.6)
To each of the terms on the right-hand side of this expression, I devote a section below. To follow up,
I’ll describe the tools I used, and calculate the current World Average lifetime as a cross check. Finally,
I’ll mention a potential complication (a potential opportunity, if you want to look at it that way) to our
extraction. Then I’ll leave you with references and appendices (or perhaps appendicitis, but I sure hope
not. . . )
2
The muon mass
The muon mass is accurately determined by never measuring the muon mass; instead, the determination
uses an indirect method that relies on relative and absolute measurements of magnetic moments, along with
2
indirect measurements of the electron mass. [10] The quantum mechanical definition of the magnetic moment
of a charged Dirac fermion is given by
~µ = g
Q ~
~s ,
2m 2
(2.1)
where m is the fermion mass, and ~s is the unit spin three-vector. Taking the ratio of the expressions for
both electrons and muons gives
µe gµ
µe /µp gµ
mµ
=
=
.
me
µµ ge
µµ /µp ge
(2.2)
In the last equality, I have introduced the proton magnetic moment, as the lepton-proton moment ratios are
better measured than the lepton-lepton ratio. Inserting the experimental values on the right-hand side gives
a lepton mass ratio good to a tens of parts per billion:
me
= (4.83633167 ± 0.00000013) · 10−3
mµ
δ(me /mµ )
≈ 2.7 · 10−8
(me /mµ )
=⇒
(2.3)
Obviously, we have to convert that mass ratio into a mass through multiplication; if we already knew
the muon mass, we wouldn’t be going through this exercise, so clearly we need the electron mass. As luck
would have it, measuring the electron mass is easy. Again, a mass ratio is what we’re after, with comparison
of the electron mass to something of known mass. In this case, 12 C is used, as it has the definitional mass
12 u. The actual measurement is a comagnetometer measurement of the cyclotron behavior of electrons and
12 6+
C ; an atomic physics correction is applied to account for the lower mass of the ionized carbon (relative
to the definitional neutral carbon). Then, a conversion from u to eV (determined from the Planck constant;
see the next section), must be applied.
mµ me
MeV
mµ (MeV) =
.
(2.4)
12 (1 + ε)
me m12 C 6+
u
When the dust clears, we have a very precise measurement of the muon mass:
mµ = 0.1056583692 ± 0.0000000094 GeV
=⇒
δmµ
≈ 8.9 · 10−8 .
mµ
(2.5)
The precision is so high, despite the number of intervening measurements and calculations, because most
of the steps involve high precision frequency ratio measurements, where almost all systematic uncertainties uncertainties cancel between numerator and denominator. The uncertainty on the muon mass has no
meaningful impact on the relative uncertainty of GF at the 1 ppm level.
3
The lifetime in GeV
µLan will determine the µ+ lifetime in units of master clock ticks. Much like the µ+ mass conversion, the
lifetime has to be converted to the natural units of the Fermi constant, which is GeV−1 . This extraction
requires both a tick-to-time conversion and a time-to-energy conversion. The first conversion comes from the
clock frequency. At this point, we’re still blind to the actual frequency, but we know the error: the hardware
clock has an absolute and relative frequency stability after calibration of 10−8 . The second conversion
requires Planck’s constant; the current CODATA recommendation [10] is
~ = (6.58211915 ± 0.00000056) · 10−25 GeV s
=⇒
δ~
≈ 8.5 · 10−8 .
~
(3.1)
Planck’s constant is measured by the moving-coil Watt balance technique [10, 11]:
[The value of ~] is determined by comparing electrical power known in terms of a Josephson
voltage and quantized Hall resistance to the equivalent mechanical power known in the SI unit
W = m2 kgs−3 .
The uncertainties on these conversion factors will not result in a meaningful increase in the relative
uncertainty of GF when expressed in natural (GeV) units.
3
4
Theory corrections
The non-weak theory corrections to the extraction of GF are very large on the scale of a 1 ppm measurement:
2200 ppm. Luckily, these corrections are known to better than 0.25 ppm.
4.1
Running αem
Once you begin to engage in loopology, you have to be careful of “constants”, as they no longer are; they run
with energy. The trick is to self-consistently calculate the running of αem with scale, and then figure out the
proper scale for evaluation. The former has been done by many people in many places (among them, a few
generations of field theory students). In this problem, unlike many higher loop calculations, even the latter
is simple - there is only one scale, mµ . Although the calculation is somewhat involved beyond one loop, the
result at two loops including all perturbative, contributions and some logarithmic enhancements, is
2
15αem (0)3
me
αem (0)
1
2
ln
+
.
(4.1)
−
αem (mµ ) = αem (0) + αem (0)
3π
4π 2
m2µ
16π 2
But one can do better, by “resumming” the large logarithms. [12] This step involved a knowledge of how
the logarithms at higher order will develop, and accounting for many of them by rearranging the previous
expression. Defining x = m2e /m2µ , you obtain
αem (mµ ) =
αem (0)
1+
αem (0)
3π
ln x
−
α3em (0)
ln x
4π 2
(4.2)
which includes all large logs of the form
αn lnn−1 x (n > 1) and α3 ln x .
The CODATA recommended value [10] for the fine structure constant is
−1
αem
(0) = 137.03599911 ± 0.00000046
=⇒
δα−1
em
≈ 3.4 · 10−9 .
α−1
em
This result is a combination of measurements of ge and QED theory predictions of ge in terms of αem .
the muon mass scale, the running value differs from the zero momentum transfer value by about 1%.
4.2
(4.3)
1
At
Massive phase space
When you first calculated the muon lifetime in your quantum field theory class, you probably did so in
the limit of massless final state fermions. But at the ppm scale, that’s obviously not a good idea, since
me /mµ ≈ 4800 ppm. At the very least, it’s clear that the mass of the electron should matter. Admittedly,
the three body phase space integrals are horrible, but if you are allowed to ignore the neutrino masses (see
Section 6.1), it is at least reasonable to attempt by hand.2 The massive phase space is given by
q0 = −8x − 12x2 ln x + 8x3 − x4
(4.5)
q0 = −187 ppm .
(4.6)
Given x = 2.3 · 10−5 , I find:
1 Even this result is about to be old news. The Gabrielse group at Harvard [13], along with a small gaggle of theorists,
announced a new measurement of ge at CIPANP 2006 that sets the gold standard for αem metrology. Their current result gives
α−1
em (0) = 137.035999710 ± 0.000000096
=⇒
δα−1
em
α−1
em
≈ 0.7 · 10−9 .
This change in αem makes absolutely no difference in our extraction, but is still pretty cool.
2 Some of you may have even done this part of the calculation in class. I still have nightmares about it. . .
4
(4.4)
4.3
One-loop QED
The one-loop corrections have a long history. As a first step, you might try to calculate the loops in the
massless limit. Those corrections date to 1958 [14, 15], and have a relatively simple form:
αem (x)
25
q1 =
− 3ζ(2) .
(4.7)
π
8
I’ll get back to that ζ(s) in a moment.
At this point, most of us would say, “Well, that’s neat,” and drop it. But there are always a few
masochists out there. In 1989, Nir [16] figured out how to include the mass dependencies3 :
25
3
αem (x)
(4.8)
− 3ζ(2) − (34 + 12 ln x) x + 96ζ(2)x 2 + O(x ln2 x) .
q1 =
π
8
Let’s take a brief diversion into the land of number theory. We all know ln x, but many of us haven’t
met the Riemann ζ function. On the real line, for s > 1, define [17]
1
ζ(s) =
Γ(s)
Z
∞
0
∞
X
1
us−1
.
du
=
s
eu − 1
n
n=1
(4.9)
The Riemann ζ function is related to lots of other important number theory functions, and the distribution
of prime numbers. For us, of course, all that matters are the values at a few points:
π2
6
ζ(3) = 1.2020569032 . . .
ζ(2) =
(4.10)
(4.11)
4
ζ(4) =
π
.
90
(4.12)
Numerically, the massless and massive one-loop corrections are indistinguishable as far as the muon
lifetime is concerned:
q1me 6=0 = −4238.86 ppm
q1me =0 = −4238.90 ppm
4.4
(4.13)
Two-loop QED
Following the completion of the one-loop program, estimates of the theory uncertainties from uncalculated,
higher order contributions stood at about 30 ppm on GF . Calculation of the two-loop corrections made our
experiment possible. All of the perturbative two-loop contributions, and the important non-perturbative
ones, were calculated at the end of the 1990s by van Ritbergen and Stuart. I won’t attempt a discussion
of the details, as they are quite heavy; if you are interested in the detail, brush up on your field theory,
particularly the chapter on renormalization, then read the vRS paper [7].
4.4.1
Pure gauge corrections
This includes all photon-only internal loops. [12] This class of diagrams is fully perturbative.
qγγ =
αem (mµ )
π
2 11047 1030
223
67
−
ζ(2) −
ζ(3) + ζ(4) + 53ζ(2) ln 2
2592
27
36
8
2
αem (mµ )
.
= 3.55877
π
(4.14)
(4.15)
3 To be clear, he performed the calculation in the context of heavy quark corrections to semi-leptonic b-quark decays, where
the corrections are very large, but the result translates easily to the muon case.
5
4.4.2
Electron and muon loops
These two differ, because the electron loops have divergences that cancel off against soft, final state e+ e−
pairs. This class includes loop corrections to photon propagators internal to the diagrams, and corrections
to collinear photon emission. The electron loops contribute [12]
2 1009 77
8
αem (mµ )
−
+ ζ(2) + ζ(3)
qe =
π
288
36
3
(4.16)
2
αem (mµ )
,
= 3.22034
π
while the muon loops contribute [12, 18]
2 αem (mµ )
16987 85
64
qµ =
− ζ(2) − ζ(3)
π
576
36
3
2
αem (mµ )
.
= −0.0364333
π
4.4.3
(4.17)
Hadronic and Tau Loops
The hadronic and tau corrections are small and well controlled. These are calculated by the use of dispersion
relations, as most low energy hadronic activity must be, but they are highly suppressed by (α/π)2 ; unlike
the muon g − 2 case, then, these contributions are not controlling. [18]
2
αem (mµ )
qhadronic = (−0.042 ± 0.002)
.
(4.18)
π
The tau loop correction is vanishingly small, as expected from the decoupling theorem and the muon contribution in the previous subsection. [18]
2
αem (mµ )
qτ = (−0.00058)
(4.19)
π
4.4.4
Two-loop summary
Numerically, the two-loop theory will supply a small correction, completely in line with previous estimates
of the theoretical uncertainty before their calculation:
q2me 6=0 = 36.76 ppm .
4.5
(4.20)
Error estimate for higher order terms
Confidence bounds on theory errors come from two places: uncertainties on input parameters, and estimates
of the size of uncalculated, higher order contributions to a calculation. In our case, the input parameter
uncertainties (αem (0), me /mµ , etc) have no meaningful impact on the theory error.
There are two classes of
uncalculated corrections that are estimated by vRS [7]. There exist some O α2 QED logarithmic corrections
of form4
2
α 2 m 2
me
e
em
ln2
.
(4.21)
π
mµ
mµ
Estimates of the coefficient of this term come from looking at related tree and one-loop logarithms, and lead
to an error estimate from these and smaller terms of
δGF /GF < 1.7 · 10−7 .
4 Remember,
(4.22)
there’s a difference
` ´ between loops and powers of coupling constants. The two-loop corrections are completely
known, but not all of the O α2 corrections.
6
The size of the three-loop corrections is estimated in a similar way, by looking at the size of the leading
logarithmic correction at three-loops, and using that as an estimate of the contributions:
δGF /GF < 1.4 · 10−7 .
(4.23)
Summing in quadrature, vRS estimate the total theory uncertainty on GF to be less than
δGF /GF < 2.2 · 10−7 .
5
(4.24)
The extraction
To extract GF from the lifetime measurement, you need to calculate with a tool that correctly propagates
errors. There are a small number of tools, from the very expensive and powerful commercial packages,
to nearly useless online calculators. Our needs are best served by two packages that I’ve managed to
find. The first, I didn’t evaluate, because is costs big bucks: Experimental Data Analyst, a Wolfram
produced Mathematica package. [19] Instead, I turned to an open source package called Fussy. [20] I have
implemented all the calculations discussed above in the Fussy language; I have included the code and output
in Appendix A.
Using the current world average on τµ [8]
τµ = 2.19703 ± 0.00004 µs ,
(5.1)
GF
= 1.166368 ± 0.000011 GeV−2 ,
(~c)3
(5.2)
I extract the following value of GF
in complete agreement with the current world average extraction [10]
GF
= 1.16637 ± 0.00001 GeV−2 .
(~c)3
(5.3)
I’m highly confident that I’ve got this right and, given a lifetime, I can extract the same numerical value of
GF that the Particle Data Group will derive.
6
6.1
Non-standard physics scenarios
What about massive neutrinos?
Like most people, I cheated a little bit when I displayed the massive phase space in Section 4.2. I left out the
possible effect of neutrino masses. The PDG direct limit [8], mνµ < 190 keV, is such that massive neutrinos
would have a substantial effect in our experiment.5 To see the scale of the correction, look again at the phase
space correction6 [7]:
mνµ 6=0
q0
= −8x − 12x2 ln x + 8x3 − x4 − 8y
(6.2)
where y = m2ν /m2µ . With the direct limit, this shifts the numerical value for q0 by almost 30 ppm:
mνµ =0
q0
= −187 ppm
mνµ =190 keV
q0
= −213 ppm .
(6.3)
In the cosmological limit, there is no meaningful change to q0 from the massless case.
5 The cosmological limits are much tighter: m
νµ < 2 eV. If you want to take that seriously, then there’s no problem. If you
want to be more conservative, then use the direct limit.
6 The general form is somewhat more complicated (for example [21, 22]), including elliptic integrals and the like, but it isn’t
so hard to guess what it should look like. For a three-body decay, with all final states massive, you would expect (by symmetry
arguments alone) that
q0 (x, y, z) = F (x) + F (y) + F (z) + G(xy) + G(xz) + G(yz) + H(xyz) ,
(6.1)
where the function F (x) is just q0 above, and G and H encode the tighter limits on the phase space volume caused by the
massive particle limits to the phase space integral (a modification of the contours of the Dalitz plot, if you will). But we don’t
really care here, as the muon neutrino mass is so small, and the electron neutrino mass is still completely negligible.
7
6.2
What about non-standard Michel parameters?
The Fermi theory is all well and good, but it is not the most general theory you can write down for the decay
of a muon to an electron and two neutrinoes. The most general such theory was derived by Michel in the
late 1940s. [23, 24] It is “the most general, derivative-free, lepton-number-conserving, four-lepton interaction
. . . consistent with locality and Lorentz invariance”. [25]
The Michel lagrangian contains 19 independent coupling parameters, labelled gεω . In modern language [25, 26] and in a chiral basis for the fermions,
LMichel
n={S,V,T }
i
ih
h
X
GF
= −4 √
gεω eε Γn νe ρ ν µ σ Γn µω .
2 {ε,ω}={R,L}
(6.4)
where n labels the spactime tensor structure (scalar, vector, tensor)
ΓS = 1
ΓV = γ µ
i
ΓT = √ (γ µ γ ν − γ ν γ µ ) ,
2
(6.5)
ε and ω label the left and right helicity states of the electron and muon that must be summed over, and rho
and σ must be chosen consistent with n, ε, and ω.
The gεω are not directly observable; only certain combinations are. Of these, only the one called η affects
the lifetime.7 I won’t display η here, because it isn’t necessary for my purposes. However, the effect of η on
the lifetime is approximately give by
G2F m5µ
me
1
1 + q0 (x) + 4η
=
(1 + p0 ) (1 + q1 (x) + q2 (x)) ,
(6.6)
τµ
192π 3
mµ
where, as before, x = m2e /m2µ ,
p(x) = +9x − 9x2 − x3 + 6x(1 + x) ln x ,
(6.7)
and the qi are defined as before. From this, it is clear that η = 0 in the Standard Model.
The current experimental situation concerning η is somewhat muddled. The two experiments that drive
the current PDG average are inconsistent with each other at the 2σ level. The TRIUMF measurement [27]
gives η = 0.071 ± 0.037, while the PSI measurement [28] gives η = −0.007 ± 0.013. The PDG throws
up its hands, averages, and doubles the resulting error to get agreement, resulting in a PDG average of
η = 0.001 ± 0.024. Using the PDG average for η shifts the value of GF slightly, but the error shoots through
the roof, to 232 ppm.
If it’s really the case that η 6= 0, then even the measurements in the current World Average lifetime were
a waste of time, not to mention µLan, at least as far as extracting GF . However, I wouldn’t bother putting
too much emphasis on this issue, for two reasons. First, the experimental situation on the measurement of η
is muddled. Second, by discussing non-zero η, we’ve crossed the line from discussing consistency of Standard
Model predictions to consistency of the essentially open ended class of non-Standard Model scenarios, and
we really have to discuss more than just non-zero η. To summarize, the larger issues with GF interpretation
in non-Standard Models gives impetus to experimental studies of the Michel parameters, but we needn’t
concern ourselves with considering them in our PRL.
A
Fussy code
The Fussy engine is available as an open source package [20] that I built, ran, and tested on Linux. I wrote
the following code to implement the extraction of GF from the measured τµ .
7 All others modify the distributions of decay electron and neutrino momentum and polarization relative to each other and
to the muon polarization. In general, some of these parameters are not directly observable in practice, because we can’t observe
the neutrino directions and polarizations. In practice, there are only nine experimentally accessible parameters.
8
Listing 1: My Fussy source code for extracting GF from τµ .
/∗
The paramet er x h e r e i s n ot t h e s t a n d a r d x u sed i n t h e r e s t o f t h e
n o t e ! For c o d i n g s i m p l i c i t y , I u se x = me/mmu, n ot (me/mmu) ˆ 2 !
∗/
p i = 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 ; /∗ Mathworld ∗/
i n v a l p h a = 1 3 7 . 0 3 5 9 9 9 1 1pm0 . 0 0 0 0 0 0 4 6 ; /∗ CODATA 2002 ∗/
i n v a l p h a = 1 3 7 . 0 3 5 9 9 9 7 1 0pm0. 0 0 0 0 0 0 0 9 6 /∗ G a b r i e l s e 2006 ∗/
me mmu = 4 . 8 3 6 3 3 1 6 7 e−3pm0 . 0 0 0 0 0 0 1 3 e −3; /∗ CODATA 2002 ∗/
mmu = 0 . 1 0 5 6 5 8 3 6 9 2pm0. 0 0 0 0 0 0 0 0 9 4 /∗ CODATA 2002 , GeV ∗/
mnu = 2 e−6 /∗ PDF c o s m o l o g i c a l nu mu mass l i m i t ∗/
mnu = 190 e−6 /∗ PDF d i r e c t nu mu mass l i m i t ∗/
mnu mmu = mnu/mmu
zeta two = p i ˆ 2 / 6 ;
z e t a t h r e e = 1 . 2 0 2 0 5 6 9 0 3 2 ; /∗ Mathworld ∗/
zetafour = pi ˆ4/90;
G f e r m i z e r o = 1 . 1 6 6 3 9 e−5pm0 . 0 0 0 0 1 e −5; /∗ CODATA 2002 , GeVˆ−2 ∗/
Gammazero = G f e r m i z e r o ˆ2∗mmuˆ 5 / (1 9 2 ∗ p i ˆ 3 ) ;
e t a = 0 . 0 0 1pm0 . 0 2 4 /∗ PDG 2006 ∗/
/∗ hbar = 6.5821191 5 e−16pm0.0000005 6 e−16 ∗/ /∗ CODATA 2002 , eV s ∗/
hbar = 6 . 5 8 2 1 1 9 1 5 e −25pm0 . 0 0 0 0 0 0 5 6 e −25 /∗ GeV s ∗/
/∗ Running f i n e s t r u c t u r e c o n s t a n t ∗/
alphaem ( x ) {
auto alpha , t ;
a lpha = 1/ i n v a l p h a ;
t = a lpha /(1+2∗ a lpha / (3 ∗ p i ) ∗ l o g ( x ) ) − a lpha ˆ 3 / (2 ∗ p i ˆ 2 )∗ l o g ( x ) ;
return t ;
}
/∗ Phase s p a c e c o r r e c t i o n ∗/
qzero ( x){
return −8∗xˆ2−24∗x ˆ4∗ l o g ( x)+8∗xˆ6−x ˆ 8 ;
}
/∗ e t a c o r r e c t i o n ∗/
p eta ( x){
return 9∗x ˆ2 − 9∗xˆ4−xˆ6+12∗xˆ2∗(1+x ˆ 2 )∗ l o g ( x )
}
/∗ Phase s p a c e c o r r e c t i o n , m a s s i v e n e u t r i n o ∗/
qzer o nu ( x , y ) {
return −8∗xˆ2−24∗x ˆ4∗ l o g ( x)+8∗xˆ6−xˆ8 −8∗yˆ2−24∗y ˆ4∗ l o g ( y)+8∗yˆ6−y ˆ 8 ;
}
/∗ One l o o p QED c o r r e c t i o n s , mass i n d e p e n d e n t l i m i t ∗/
qone ( x ) {
auto t ;
t = 25/8 −3∗ zeta two ;
return t / p i ∗ alphaem ( x ) ;
}
/∗ F u l l one loop , mass d e p e n d e n t QED c o r r e c t i o n ∗/
9
qonem ( x ) {
auto t ;
t = 25/8 − 3∗ zeta two − (34+24∗ l o g ( x ) ) ∗ xˆ4 + 96∗ zeta two ∗x ˆ 3 ;
return t / p i ∗ alphaem ( x ) ;
}
g q e d a = 1 1 0 4 7 / 2592 − 1030/27∗ zeta two − 223/36∗ z e t a t h r e e ;
g q e d b = 67/8∗ z e t a f o u r + 53∗ zeta two ∗ l o g ( 2 ) ;
g qed = g q e d a+g q e d b ;
g
g
g
g
e l e c = −(1009/288 − 77/36∗ zeta two − 8/3∗ z e t a t h r e e ) ;
muon = 16987/576 −85/36∗ zetatwo −64/3∗ z e t a t h r e e ;
had = −0.042pm0 . 0 0 2 ;
t a u = − 0 .0 0 0 5 8 ;
/∗
p r i n t g q e d , ” ” , g e l e c , ” ” , g muon , ” ” , g had , ” ” , g t a u , ” \ n”
p r i n t g q e d+g e l e c +g muon+g h a d+g t a u , ” \ n”
∗/
/∗ Two l o o p c o r r e c t i o n s , b u i l t from c o e f f i c i e n t s ∗/
qtwo ( x ) {
return ( g qed + g e l e c + g muon + g had + g t a u ) / p i ˆ2∗ alphaem ( x ) ˆ 2 ;
}
/∗ Two l o o p c o r r e c t i o n c r o s s check , u s i n g summed c o e f f i c i e n t s ∗/
qtwoa ( x ) {
auto ta , tb , t ;
t a =156815/5184 −1036/27∗zetatwo −895/36∗ z e t a t h r e e +67/8∗ z e t a f o u r ;
tb =53∗ zeta two ∗ l o g ( 2 ) − 0 . 0 4 2pm0 . 0 0 2 − 0 . 0 0 0 5 8 ;
t = t a+tb ;
return t / p i ˆ2∗ alphaem ( x ) ˆ 2 ;
}
/∗ E s t i m a t e o f t h e remain in g ( u n c a l c u l a t e d ) d i r e c t t h e o r y e r r o r ∗/
q e r r = 0pm1 . 4 e−7 + 0pm1 . 7 e−7
/∗ F u l l t h e o r y c o r r e c t i o n s ∗/
theory ( x){
return 1+q z e r o ( x)+qonem ( x)+qtwoa ( x)+ q e r r ;
}
q0 = q z e r o (me mmu ) ;
q0nu = qzer o nu (me mmu , mnu mmu ) ;
q1 = qone (me mmu ) ;
q1m = qonem (me mmu ) ;
q2 = qtwo (me mmu ) ;
q2a = qtwoa (me mmu ) ;
/∗
a l p h a = 1/ i n v a l p h a
p r i n t ” a l p h a : ” , alpha , ” a l p h a (mmu) : ” , alphaem (me mmu)
p r i n t ” d e l t a ( a l p h a )/ a l p h a : ” , ( alphaem (me mmu)− a l p h a )/ alphaem (me mmu) , ” \ n”
∗/
10
print
print
print
print
print
print
print
” q0 : ” , q0 . v a l ∗1 e6%f , ”ppm ” , q0 . rms/ q0 . v a l ∗1 e6%f , ”ppm\n”
” q0nu : ” , q0nu . v a l ∗1 e6%f , ”ppm d e l t a ( q0nu ) / q0 : ” , ( q0nu−q0 ) / q0 , ” \n”
” q1 : ” , q1 . v a l ∗1 e6%f , ”ppm ” , q1 . rms/ q1 . v a l ∗1 e6%f , ”ppm\n”
”q1m : ” ,q1m . v a l ∗1 e6%f , ”ppm ” ,q1m . rms/q1m . v a l ∗1 e6%f , ”ppm\n”
” q2 : ” , q2 . v a l ∗1 e6%f , ”ppm ” , q2 . rms/ q2 . v a l ∗1 e6%f , ”ppm\n”
” q2a : ” , q2a . v a l ∗1 e6%f , ”ppm ” , q2a . rms/ q2a . v a l ∗1 e6%f , ”ppm\n”
” t h e o r y : ” , t h e o r y (me mmu)%e , ” \n”
/∗ L i f e t i m e i n secon ds , w i t h e r r o r ∗/
t = 2 . 1 9 7 0 3 e−6pm0 . 0 0 0 0 4 e−6
/∗ Con vert ed t o GeVˆ−1 ∗/
t g e v = t / hbar
p r i n t ” t g e v : ” , t g e v%e , ” ” , ” ppm ” , t g e v . rms/ t g e v . v a l%e
p r i n t ” r e l . e r r . t o t ” , ( t g e v . rms/ t g e v . v a l ) / ( t . rms/ t . v a l )%e , ” \n”
/∗ E x t r a c t e d GFermi ∗/
Gfermi = s q r t (1 9 2 ∗ p i ˆ 3 / ( t g e v ∗mmuˆ5∗ t h e o r y (me mmu ) ) )
p r i n t ” Gfermi ” , Gfermi%e , ” r e l . e r r . ” , Gfermi . rms/ Gfermi . v a l%e , ”\n”
/∗ E x t r a c t e d GFermi w i t h non−z e r o e t a ∗/
temp = (1+ q z e r o (me mmu)+4∗ e t a ∗me mmu∗(1+ p e t a (me mmu ) ) )
e t a t h e o r y = temp∗(1+ qone (me mmu)+qtwo (me mmu) )
G f e r m i e t a = s q r t (1 9 2 ∗ p i ˆ 3 / ( t g e v ∗mmuˆ5∗ e t a t h e o r y ) )
p r i n t ” e t a t h e o r y ” , e t a t h e o r y%e , ” \n”
p r i n t ” G f e r m i e t a ” , G f e r m i e t a%e , ” r e l . e r r . ” , G f e r m i e t a . rms/ G f e r m i e t a . v a l%e , ” \n”
p r i n t ” S h i f t = ” , ( Gfer mieta−Gfermi ) / Gfermi%e , ” \n”
quit
In this implementation, I have used very high precision values of numerical constants, much beyond what
is really needed, from [17, 29]. World average physical constants have been taken from [8, 10], along with
their associated errors. All equations are taken from [7]. The resulting output is below.
Listing 2: Output from my Fussy script.
q0 : −187.050826ppm −0.053742ppm
q0nu : −212.918820ppm d e l t a ( q0nu ) / q0 :
0 . 1 3 8 2 9 +/−
0.00000
q1 : −4238.901059ppm −0.000708ppm
q1m : −4238.859099ppm −0.000708ppm
q2 : 3 6 . 7 5 5 7 7 4ppm 2 9 8 . 5 0 3 0 3 7ppm
q2a : 3 6 . 7 5 5 7 7 4ppm 2 9 8 . 5 0 3 0 3 7ppm
t h e o r y : 9 . 9 5 6 1 0 8 e −01 +/− 2 . 2 0 5 0 0 3 e −07
t g e v : 3 . 3 3 7 8 7 6 e+18 +/− 6 . 0 7 7 1 3 7 e+13 ppm 1 . 8 2 0 6 6 0 e −05 r e l . e r r . t o t 1 . 0 0 0 0 1 1 e+00
Gfermi 1 . 1 6 6 3 6 8 e−05 +/− 1 . 0 6 2 1 7 5 e −10 r e l . e r r . 9 . 1 0 6 6 8 8 e −06
e t a t h e o r y 9 . 9 5 6 3 0 8 e −01 +/− 4 . 6 1 7 4 2 3 e −04
G f e r m i e t a 1 . 1 6 6 3 5 7 e −05 +/− 2 . 7 0 6 6 8 2 e −09 r e l . e r r . 2 . 3 2 0 6 3 0 e −04
S h i f t = −1.003555 e −05 +/− 2 . 3 2 2 3 9 3 e −04
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11
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propagation
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URL
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