1 Setting limits on a new parameter outside of Standard Model muon decay. Kristen Williams Jacksonville State University Dr. Carl Gagliardi Cyclotron Institute.

1

Setting limits on a new parameter outside of Standard

Model muon decay.

Kristen Williams Jacksonville State University

Dr. Carl Gagliardi Cyclotron InstituteTexas A&M University

2

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.

3

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

4

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).

5

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:

ee

ee

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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

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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

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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.”

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Simplified differential decay probability

022

2 (1 )3(1 ) (4 3) 3

32cos

cos 1 (4 3)3

xx x x

d xx

dxdP x x

2 eE

mx

0 2 0.00967em

xm

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Michel ParametersCurrent measurements seem to agree with SM predictions.

ρ 3/4 0.75080 ± 0.00032(stat.) ± 0.00097(syst.) ± 0.000231

δ 3/4 0.74964± 0.00066(stat.)± 0.00112(syst.)2

η 0 -0.0036 ± 0.00693

ξ 1 N/A

Pμξ 11.0003± 0.0006(stat.)± 0.0038(syst.)4

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Matrix ElementYields 12 distinct coupling constants.

, ,, ,

4( ) ( )

2F

e n mS V TR L

Gg e

12

Coupling Constants

90% confidence level for 10 of the 12 constants3

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. SgRRSgLRSgRLSgLL

0 <0.067

0 <0.088

0 <0.417

0 <0.550

VgRRVgLRVgRLVgLL

0 <0.034

0 <0.036

0 <0.104

1 >0.960

TgRRTgLRTgRLTgLL

0 ≡ 0

0 <0.025

0 <0.104

0 ≡ 0

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What if an effective RR tensor coupling does occur?

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 .Chizhov presents this value as a new

variable, κ, and calculates κ = 0.013.5

TgRR

TgRR

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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

002

2

0

32 (1 )3(1 ) (4 3) 3

33 (2 )2cos

cos 1 (4 3)3

xxx x x

d x xx

x xdxdP x x

x

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Changes to the current Michel parameters

Furthermore, Chizhov states that a non-zero value of κ will cause the values of the Michel parameters to vary according to the following equations:5

23

(1 2 )4

21 2

231 6

4 23

1 44

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Is Chizhov’s prediction realistic?

While theory assumes ≡ 0, experiment has only been successful at narrowing the value: < 0.024.2

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.

TgRR

TgRR

TgRR

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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:

• pe<50 MeV/c

• pT<38.5 MeV/c

• |pz|>13.7 MeV/c

• 0.50<|cosθ|<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.

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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 + Δρ(κ) + Δξ(κ) + Δξδ(κ)

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Finding these functions To quantify Δρ(κ), we first

performed a Χ2 minimization of the isotropic piece of the spectrum over a given range of κ values:

This chart shows how the function shifts when the minimum energy of the fit range is changed from 10 MeV to 20 MeV.

222

( , ) ( , )

Ich

ISOch x k ISOfit x

drho vs. kappa

-0.004-0.0035

-0.003-0.0025

-0.002-0.0015

-0.001-0.0005

0

-0.05 -0.03 -0.01 0.01 0.03 0.05

kappa

drho

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Finding more functions

We then applied this same method to the decay asymmetry in order to find functions for Δξ(κ) and Δξδ(κ) :

222

( , ) ( , , , )

Ach

Ach x k Afit x

AnisotropicA

Isotropic

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How do these functions look?Again, we observed small shifts in positionbetween minimum energies of 10 and 20 MeV.

What do these shifts represent?

dxi vs. kappa

-0.004

0.001

0.006

0.011

-0.05 -0.03 -0.01 0.01 0.03 0.05

kappa

dxi

dxidelta vs. kappa

-0.0085

-0.0075

-0.0065

-0.0055

-0.0045

-0.0035

-0.0025

-0.0015

-0.0005

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

kappa

dxid

elta

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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.

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Focusing on the linear changes

In order to better quantify the linear variations, we replaced the Michel parameters with their SM values:

02

2

0

323(1 ) (4 3)

3

cos 3 (2 )2cos 1(1 (4 3)

33

4

4

3

)

xx x

xdx

dxd x xx x

x

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How drastic are these changes?

These graphs reveal theshifts in the linear piece at minimum energies of 10, 15, and 20 MeV.

dxidelta vs. kappa

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

-0.05 -0.03 -0.01 0.01 0.03 0.05

kappa

dxid

elta

dxi vs. kappa

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

-0.050 -0.030 -0.010 0.010 0.030 0.050

kappa

dx

i

drho vs. kappa

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

-0.05 -0.03 -0.01 0.01 0.03 0.05

kappa

drho

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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:

0

34c x

0c x

0

34c x

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Finding the effective δ

Solving for the change in δ can be done in a method similar to that used by TWIST when calculating the value of δ: 2

0

20

31 4

41 2

c x

c x

2 2 20 0

31 ( ) 6 ( )

4eff c c x c x

<0.04 κ2

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Examining this result for past TWIST fit range

For past TWIST fits, the momentum range has been approximately 19<pe<50 MeV/c.

We were interested in quantifying the effective δ in this range for comparison to the most recent experimental value.

20

31 ( ) 6

4eff c c x 2

0

31 (17.7) 6

4x

230.128 4.5

4

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Limiting κ from δ

Matching this equation to TWIST’s published value of δ yields a 90% confidence level range for κ.

230.128 4.5

4eff

0.74964 0.00130TWIST

0.040 0.012

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Limiting κ from ρ

We then repeated this in order to set another 90% confidence limit which is sensitive to the effective ρ.

230.003 1.5

4eff

0.75080 0.00105TWIST

0.031 0.029

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Final result

Combined, these two ranges set a final limit on the possible value of κ.

Note the comparison to Chizhov’s value: κ ≈ 0.013.

0.031 0.012

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Looking ahead As mentioned, this value range for

κ is based on an analysis with the momentum range of past TWIST measurements: 19<pe<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

19.7

19.2

18.7

18.2

17.7

17.217.0

17.5

18.0

18.5

19.0

19.5

20.0

8 10 12 14 16 18 20 22

minimum energy (MeV/c)

Cxi

-Cxi

del

ta

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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;

hep-ex/0605100.5. M.V. Chizhov, hep-ph/0405073.