Kristen Williams
Jacksonville State University
Dr. Carl Gagliardi
Cyclotron Institute
Texas A&M University
1

What is the Standard Model?


“Standard 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.
2

How do these particles interact?
 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
3

Is the Standard Model “it?”







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).
4

Intro to Muon Decay


 The 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:
   e
   e

   e
   e

5

Driving force for muon decay
 Direct muon decay is governed by the weak interaction as described by the
SM.
 This interaction:
• is CPT invariant
• involves the W boson
6

How does this interaction look?
 The W boson facilitates muon decay according to the following Feynman diagram:
•
Feynman diagram for muon decay http://www.departments.bucknell.edu/physics/animations/
Feynman_diagrams/muon_s.html
7

Let’s simplify the picture.


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.”
8

Simplified differential decay probability d dxd
2  cos

 x
2 




3(1
 

P

  cos
2

3


(4
1 x
 
2
3

 x
0
(1
 x ) x


(4 x

3)



 x

2
E e
 m
 x
0

2 m e
 m


0.00967
9

Michel Parameters
Current measurements seem to agree with SM predictions.
ρ
δ
η
ξ
3/4
3/4
0
1
0.75080
± 0.00032(stat.)
± 0.00097(syst.)
± 0.00023
1
0.74964
± 0.00066(stat.)
± 0.00112(syst.) 2
-0.0036 ± 0.0069
3
P
μ
ξ 1
N/A
1.0003
± 0.0006(stat.)
± 0.0038(syst.) 4
10

Matrix Element
Yields 12 distinct coupling constants.
4 G
F
2




,
 
, g

  e

   e n

 m




11

Coupling Constants



90% confidence level for 10 of the 12 constants 3
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.
g
S
RR g
S
LR g
S
RL g
S
LL
0 <0.067
0 <0.088
0 <0.417
0 <0.550
g
V
RR g
V
LR g
V
RL g
V
LL
0 <0.034
0 <0.036
0 <0.104
1 >0.960
g
T
RR g
T
LR g
T
RL g
T
LL
0
0 <0.025
0 <0.104
0
≡ 0
≡ 0
12

What if an effective
coupling does occur?
tensor
 M.V. Chizhov, a theorist at CERN, proposes inclusion of a new, non-local tensor interaction when describing muon decay.


This would predict a non-zero value for g
T
RR
.
Chizhov presents this value as a new variable, κ , and calculates κ
= 0.013
.
5
13

How will this change the SM view?

Chizhov’s κ affects the both the isotropic and anisotropic terms of the decay spectrum by addition of an extra linear term.
5 d dxd
2  cos

 x
2 




3(1
 

P

  cos
2

3


(4
1 x
 
2
3

 x
0
(1
 x ) x
(4 x
 

3 x
0
 x
3
 x
0
(2
 x ) x






14

Changes to the current Michel parameters
 Furthermore, Chizhov states that a nonzero value of κ will cause the values of the Michel parameters to vary according to the following equations: 5
 
3
4
(1

2

2
)
  
3
4

1

4

2



1 2

2

3
4

1

6

2

15

Is Chizhov’s prediction realistic?


While theory assumes ≡ 0, experiment has only been successful at narrowing the value: < 0.024.
RR
2
Within this limit, Chizhov’s value, 0.013, is certainly plausible.
 Our goal: set limits on the value of and determine if the existence of κ will alter the SM view of muon decay.
16

Our approach to test Chizhov’s model





Many of the current muon decay measurements and fits have been conducted by TWIST .
TWIST performs its fits within a specific fiducial region in accordance with the capabilities of the TRIUMF detector.
For the existing TWIST measurements, this region was:
• p e
< 50 MeV/c
• p
•
|p
T
<38.5 MeV/c z
| >13.7 MeV/c
•
0.50<|cos θ |<0.84
Previous TWIST fits have not included Chizhov’s linear terms.
Thus, 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.
17

Where we began
 We assumed that each parameter in the
SM spectrum would change by some small amount, with each extra piece being a function of kappa:
•
SM +
Δ ρ ( κ ) +
Δ ξ ( κ ) +
Δ ξ δ ( κ )
18

Finding these functions


2 
To quantify
Δ ρ ( κ )
, we first performed a Χ 2 minimization of the isotropic piece of the spectrum over a given range of
κ values:


( , )

( ,
 
)

2

2
Ich
 This chart shows how the function shifts when the minimum energy of the fit range is changed from 10 MeV to 20
MeV.
drho vs. kappa
0
-0.0005
-0.001
-0.0015
-0.002
-0.0025
-0.003
-0.0035
-0.004
-0.05
-0.03
-0.01
0.01
0.03
0.05
kappa
19

Finding more functions
 We then applied this same method to the decay asymmetry in order to find functions for
ξ
κ
and
ξ
κ
:

2  

( , )

( ,

,
  
)

2

2
Ach
A

Anisotropic
Isotropic
20

How do these functions look?
dxi vs. kappa
Again, we observed small shifts in position between minimum energies of 10 and 20 MeV.
0.011
0.006
0.001
-0.004
-0.05
-0.03
-0.01
kappa
0.01
0.03
0.05
dxidelta vs. kappa
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
-0.0005
-0.0015
-0.0025
-0.0035
-0.0045
-0.0055
-0.0065
-0.0075
-0.0085
kappa What do these shifts represent?
21

What is the source of these shifts?
 We 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.
22

Focusing on the linear changes
 In order to better quantify the linear variations, we replaced the Michel parameters with their SM values: d dxd
2  cos

 x
2 




3(1
  cos

2
3

3
 
1
(4 x
 
2
3

4
3 x
0
 x

(4 x
 
3
 x
0
(2
 x ) x






23

How drastic are these changes?
dxidelta vs. kappa drho vs. kappa
0.0006
0.0004
0.0002
0
-0.0002
-0.05
-0.03
-0.01
0.01
0.03
0.05
-0.0004
-0.0006
kappa
These graphs reveal the shifts in the linear piece at minimum energies of
10 , 15 , and 20 MeV.
0.01
0.008
0.006
0.004
0.002
0
-0.004
-0.006
-0.008
-0.01
0.001
0.0008
0.0006
0.0004
0.0002
0
-0.0002
-0.05
-0.0004
-0.0006
-0.001
-0.03
-0.01
kappa
-0.030
-0.010
0.010
0.030
0.050
0.01
0.03
kappa
0.05
24

Using these results to limit κ
 From these graphs, we can see how the coefficients of the linear pieces change for different minimum energies.
 We can redefine these coefficients according to the following: c x
0

3 c x
0

4
   
3
4 c x
0

25

Finding the effective
 Solving for the change in
δ can be done in a method similar to that used by
TWIST when calculating the value of δ :




3
4
 1

1
 c x
0 c x
0

 
4


2

2
2
<0.04 κ 2
 eff

3
4


 c

 c
 
) x
0
 
2 
(
0
)
2

2


26

Examining this result for past
fit range


For past TWIST fits, the momentum range has been approximately 19< p e
<50 MeV/c.
We were interested in quantifying the effective
δ in this range for comparison to the most recent experimental value.
 eff

3
4
 1

( c

 c
 
) x
0
 
6


3
4
 1

(17.7) x
0
 
6

2
2
0.128
 
4.5

2
4
27

Limiting κ from
 Matching this equation to TWIST ’s published value of
δ yields a 90% confidence level range for κ .
 eff
0.128
 
4.5

2
4
 
TWIST

0.040
0.012
28

Limiting κ from
 We then repeated this in order to set another
90% confidence limit which is sensitive to the effective
ρ .
 eff
0.003
 
1.5

2
4
 
TWIST

0.031
0.029
29

Final result
 Combined, these two ranges set a final limit on the possible value of

0.031
0.012

30

Looking ahead




As mentioned, this value range for
κ is based on an analysis with the momentum range of past TWIST measurements:
19< p e
<50 MeV/c.
One avenue to explore would be how this limit varies within the context of a broader momentum range.
In the future, TWIST hopes to extend to 51.5 MeV/c.
Since minimum energy affects the fit coefficients
—which factor into the effective parameter calculations — other, more precise limits for
κ could be achieved.
Fit coefficients at different minimum energies
20.0
19.5
19.0
18.5
18.0
17.5
17.0
8 10
19.7
19.2
18.7
18.2
12 14 16 18 minimum energy (MeV/c)
17.7
17.2
20 22
31





Acknowledgements
Dr. Carl Gagliardi
Dr. Sherry Yennello
Cyclotron Institute at TAMU
Recent papers:
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; hepex/0605100.
5.
M.V. Chizhov, hep-ph/0405073.
32