Setting limits on a new parameter outside of Standard Model muon
decay.
Kristen Williams – Jacksonville State University
Dr. Carl Gagliardi – Cyclotron Institute
Texas A&M University
WHAT IS THE STANDARD MODEL?
CURRENT KNOWLEDGE AND LIMITS
EXAMINING THIS THEORY
“Standard
The
We
4GF

2  S ,V ,T
 ,  R ,L
gS
RR
gS
LR
gS
RL
gS
LL
0
0
0
0
<0.067
<0.088
<0.417
<0.550
g e  ( e )n (  )m  



gV
RR
gV
LR
gV
RL
gV
LL
0
0
0
1
<0.034
<0.036
<0.104
>0.960
gT
RR
gT
LR
gT
RL
gT
LL
0
≡0
muon decay is governed by the weak interaction as described by the
SM.
This interaction:
•is CPT invariant
•involves the W boson
Due to its large mass, the W+ boson will propagate a finite, statistically
insignificant distance ~ 0.0025 fm.
Thus, the decay can be localized “at a point.”
[Feynman diagrams for muon decay]
References:
1.TWIST Collaboration, J.R. Musser et al., Phys. Rev. Lett. PRL 94, 101805 (2005).
2.TWIST Collaboration, A. Gaponenko et al., Phys. Rev. D 71, 071101(R) (2005).
3.C. A. Gagliardi, R.E. Tribble, and N.J. Williams, Phys. Rev. D 72, 073002 (2005).
4.TWIST Collaboration, B. Jamieson et al., submitted to Phys. Rev. D; hep-ex/0605100.
5.M.V. Chizhov, hep-ph/0405073.
0
-0.0036 ± 0.00693
ξ
1
N/A
Pμξ
1
drho vs. kappa
-0.05
0
-0.0005
-0.001
-0.0015
-0.002
-0.0025
-0.003
-0.0035
-0.004
-0.03
-0.01
0.01
0.03
0.05
We
0
0
<0.025
dxi vs. kappa
0.011
kappa
0.006
0.001
1.0003
± 0.0006(stat.)
± 0.0038(syst.)4
<0.104
≡0
These graphs
show how the functions
shift when the minimum
energy of the fit range is
changed from 10 to 20
MeV.
fit each graph to a polynomial
trendline and found that the quadratic
pieces are unaffected by the minimum
energy.
Only the linear pieces change when
the energy range is adjusted.
This confirms our hypothesis that the
linear contribution from κ is sensitive to
the energy range of the fit.
dxidelta vs. kappa
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
-0.05
-0.004
0.02
0.03
-0.03
0.04
-0.01
0.01
0.03
0.05
order to better quantify the linear
variations, we replaced the Michel
parameters with their SM values.
From these graphs, we can see how the coefficients
of the linear pieces change for different minimum
energies. We can then redefine the coefficients
(below) and use the TWIST experimental values to set
limits on κ.
0.05
In
kappa
-0.0015
-0.0025
-0.0035
-0.0045
-0.0055
-0.0065
-0.0075
-0.0085
kappa
dxidelta vs. kappa
drho vs. kappa
0.001
0.0008
NEW THEORY TO TEST THE SM
0.0006
0.0006
M.V.
Chizhov, a theorist at CERN, proposes inclusion of a new, non-local
tensor interaction when describing muon decay.
T
This would predict a non-zero value for g
RR .
Chizhov presents this value as a new variable, κ, and calculates
T
κ = g RR ≈ 0.013.5
0.0004
0.0004
dxidelta
These graphs show
the shifts in the linear
pieces at 10, 15, and
20 MeV.
0.0002
0
-0.0002-0.05
0.0002
0
-0.0002-0.05
-0.03
-0.01
0.01
0.03
0.05
-0.0004
-0.03
-0.01
0.01
0.03
0.05
-0.0004
0.01
-0.0006
0.008
dxi vs.
-0.0006
kappa
-0.0008
-0.001
kappa
0.006
kappa
0.004
0.002
dxi
Direct


 e  
e 
η
2
 Ich
Ach( x, k )  Afit ( x,  ,  ,  ) 

2
 
2
 Ach
-0.0005
0
The
   e 
e 
3/4
0.74964
± 0.00066(stat.)
± 0.00112(syst.)2
[Matrix element]
[Current limits set in 2005.3]
muon (a lepton) has a mass over 200 times the electron
~105.7 MeV.
Thus, it will decay after a mean life of only ~ 2.2 μs.
While the muon can decay via 3 different modes, the primary mode (~100%)
produces an electron and two neutrinos:
δ
3/4
 ISOch( x, k )  ISOfit ( x,  ) 2
2
dxi
[Simplified differential probability spectrum]
ρ
0.75080
± 0.00032(stat.)
± 0.00097(syst.)
± 0.000231
 
2
drho
3(1  x)  2  (4 x  3)  3 x (1  x) 
0

d 2
3
x 
2
x 

2

dxd cos


 P  cos 1  x  (4 x  3)


 

3
began the fits by assuming that each parameter in the SM spectrum would
change by some small amount, with each extra piece being a function of kappa:
SM + Δρ(κ) + Δξ(κ) + Δξδ(κ)
To quantify Δρ(κ), Δξ(κ), and Δξδ(κ) we performed χ2 minimizations of the
isotropic piece and the decay asymmetry over a given range of κ values:
drho
INTRO TO MUON DECAY
simplified differential decay probability (shown below) is parameterized by
the four Michel parameters: ρ, η, ξ, δ.
New measurements of all four parameters have been published in the past
year. As shown in the table, the current values seem to agree nicely with the
SM.
90% confidence levels have also been set for 10 of the 12 coupling constants
yielded from the decay matrix element.
RR and LL tensor couplings do not occur when the decay is localized “at a
point.”
Thus, these two constants are assumed to be identically zero.
dxidelta
Model” (SM) is the name given to the current theory of elementary
particles and how they interact.
These particles are classified as fermions (leptons and quarks) or bosons.
The SM describes nature on atomic and subatomic scales where interactions
are governed not by gravity, but by the other 3 forces:
•Electromagnetic force - acts on charged particles;
force carrier - photon
•Strong force - binds the components of the nucleus;
force carrier - gluon
•Weak force - describes particle decay;
force carriers - Z and W bosons
When the theory was developed in the 1970’s, it incorporated all knowledge of
particle physics at that time.
Since then, it has continued to successfully predict the outcomes of a number
of experiments.
Thus, the goal of much of current particle physics research is to test the SM’s
limits.
In each realm of particle physics, we ask, “How adequate is the SM?”
One test of the SM is a rigorous study of one well-known weak interaction muon decay.
Since the SM specifies exactly how this decay should occur, any unexpected
observations would be of great interest.
Searching for such deviations is the goal of TWIST (TRIUMF Weak Interaction
Symmetry Test).
0
-0.050
-0.002
-0.030
-0.010
0.010
0.030
0.050
3
  c x0   c x0
4
3
1  c x0  4 2 
 4
Combined, these two ranges set a

final, 90% confidence level limit on the

1  c x0  2 2
possible value of κ.
3
 eff  1  (c  c ) x0  6 2  (c x0 )2  2 
[Chizhov’s simplified differential probability spectrum]
4

0.031


0.012
T
3
Setting limits on κ from
2
While theory assumes g
has only been successful at
RR ≡ 0,Texperiment

 This value range for κ is based on an

1

(
c

c
)
x


6

2


0
TWIST values.

g
narrowing the value:
<
0.024.
RR
4
analysis with the momentum range of
 Within this limit, Chizhov’s value, 0.013, is certainly plausible.
3
past TWIST measurements:
2
T and determine if the existence of κ will
3
Our goal: set limits on the value of g
 1  (17.7) x0  6 
  c x0
RR
19<pe<50 MeV/c.
alter the SM view of muon decay.
4
4
In the future, TWIST hopes to extend
Many of the current muon decay measurements and fits have been conducted
3
2
3
2 
to 51.5 MeV/c.
by TWIST.

0.128


4.5

eff   0.003  1.5
4
4
TWIST performs its fits within a specific fiducial region in accordance with the
Since minimum energy affects the fit
capabilities of the TRIUMF detector.
coefficients—which factor into the


0.74964

0.00130


0.75080

0.00105
TWIST
TWIST
Previous TWIST fits have not included Chizhov’s linear terms.
effective parameter calculations—
Chizhov’s
κ affects both the isotropic and anisotropic terms of the decay
spectrum by addition of an extra linear term and predicts new values for each of
the Michel parameters.5
3
3
2
2
3
x

2

(1

x
)


(1

2

)


1

6

3(1  x) 



0
(4
x

3)

3

x

4
4
0


d 2
3
x
x
2
x 

3
2
3

x
(2

x
)
2

dxd cos
2


0


1

4

 P  cos 1  x  (4 x  3) 




 1  2


4

3
x

 
our approach was to perform a similar fit for κ and set a limit on how
sensitive the linear pieces are to the chosen energy range.
Thus,
0.031    0.029
-0.004
-0.006
-0.008
-0.01
kappa

0.040    0.012
other, more precise limits for κ could
be achieved.