# Decay of Z boson

#### taper100

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#### fzero

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It looks like:

RunNo = run number (collider schedule),

EvNo = event number (in terms of recorded events for current run)

E1 ($E_1$) = energy of 1st electron. Units are GeV.

px1 ($p_{x,1}$)= x-component of momentum of 1st e. Units are GeV/c.

py1 = y-component "

pz1 = z-component "
The coordinates are such that the beam axis is identified with the z-axis

pt1 ($p_{T,1}$) = transverse (with respect to the beam direction) momentum of the 1st e. From the choice of axes, $p_T = \sqrt{p_x^2 + p_y^2}$.

eta1 ($\eta_1$) = pseudorapidity of 1st e. See http://en.wikipedia.org/wiki/Pseudorapidity for a detailed description. It is related to the angle between the electron path and the beam axis.

phi1 ($\phi_1$) = azimuthal angle of 1st e. This is measured in the x-y plane and appears to be in degrees.

Q1 ($q_1$) = charge of the particle, -1 for the electron, +1 for the positron.

The other set of variables are the same for the other particle.

Another link with a pictorial description of the transverse momentum, pseudorapidity, and azimuthal angle is http://www-cdf.fnal.gov/physics/new/qcd/ue_escan/etaphi.html.

#### taper100

Excellent, that is very helpful. Thanks. In addition, for my statistics project I need to construct a confidence interval in one part of the project. Using the given data, do you have any ideas of a physical quantity I can construct the confidence interval from (such as mass). For example, is there any relation between the given quantities that would yield the mass of the z boson? I'm a math major and I don't know a whole lot about high energy physics although I have some familiarity with quantum mechanics. So any help would be appreciated.

#### fzero

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Excellent, that is very helpful. Thanks. In addition, for my statistics project I need to construct a confidence interval in one part of the project. Using the given data, do you have any ideas of a physical quantity I can construct the confidence interval from (such as mass). For example, is there any relation between the given quantities that would yield the mass of the z boson? I'm a math major and I don't know a whole lot about high energy physics although I have some familiarity with quantum mechanics. So any help would be appreciated.
This is a great question and the answer is yes. In the center-of-mass frame, the Z-boson is at rest, so its energy is just given by its mass:

$$E_\mathrm{com} = m_Z c^2.$$

After the decay, the center of mass energy is given by

$$E_\mathrm{com}^2 = (E_1+E_2)^2 - |\vec{p}_1 + \vec{p}_2|^2 c^2.$$

Since this decay conserves momentum, we can equate these values to derive a formula for the mass of the Z-boson:

$$m_Z = \sqrt{ \left( \frac{E_1+E_2}{c^2}\right)^2 - \left|\frac{\vec{p}_1 + \vec{p}_2}{c}\right|^2 }.$$

In practice, the distribution values computed from the collected energy and momentum data is assumed to be a Gaussian distribution centered on the true Z mass.

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#### mfb

Mentor
In practice, the distribution values computed from the collected energy and momentum data is assumed to be a Gaussian distribution centered on the true Z mass.
That is a good assumption for most particles, where the width of the distribution is dominated by the resolution of the detector. The Z boson is different, you can see its natural line shape - a Breit-Wigner distribution, together with some details you probably don't want to care about. This article gives an introduction.

Usual particle physics results here would be the central mass and the width of the distribution, together with confidence intervals. The cross-section is interesting, too, but it cannot be determined with this dataset.

By the way: The energy values for the electrons in the .csv are not measured, they are calculated from the momentum and the known electron mass.

#### fzero

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Homework Helper
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That is a good assumption for most particles, where the width of the distribution is dominated by the resolution of the detector. The Z boson is different, you can see its natural line shape - a Breit-Wigner distribution, together with some details you probably don't want to care about. This article gives an introduction.

Usual particle physics results here would be the central mass and the width of the distribution, together with confidence intervals. The cross-section is interesting, too, but it cannot be determined with this dataset.
Yes, I should not have forgotten that, thanks!. It would be good for taper100 to actually use the Breit-Wigner distribution, since he'd have a chance to discuss the statistical properties of something other than a boring Gaussian.

#### taper100

Thanks to everyone for the help. I tried to use the equation for mz to calculate the mass. However, I observed a lot of variation and I got a mean around 143 GeV/c^2. I know that the actual mass of the Z boson is around 91 Gev/c^2, so I believe that I must have done something wrong. I used the following equation
sqrt((E1+E2)^2 + (px1+px2)^2+(py1+py2)^2+(pz1+pz2)^2).
Did I do anything wrong which would cause me to not obtain mass in units of Gev/c^2?

#### fzero

Science Advisor
Homework Helper
Gold Member
Thanks to everyone for the help. I tried to use the equation for mz to calculate the mass. However, I observed a lot of variation and I got a mean around 143 GeV/c^2. I know that the actual mass of the Z boson is around 91 Gev/c^2, so I believe that I must have done something wrong. I used the following equation
sqrt((E1+E2)^2 + (px1+px2)^2+(py1+py2)^2+(pz1+pz2)^2).
Did I do anything wrong which would cause me to not obtain mass in units of Gev/c^2?
Sorry, I had a very important minus sign missing:

$$m_Z = \sqrt{ \left( \frac{E_1+E_2}{c^2}\right)^2 - \left|\frac{\vec{p}_1 + \vec{p}_2}{c}\right|^2 },$$

so sqrt((E1+E2)^2 - (px1+px2)^2-(py1+py2)^2-(pz1+pz2)^2) is the right equation to use.

I just noticed that there is a final column that looks very much like the $m_Z$ as computed from the data. If I could get my spreadsheet to work I would check.

#### taper100

ok, I adjusted for the minus sign and the results do indeed give the mass in the final column. Thanks for the help. The mean of all of the masses is 74.67 Gev/c^2 with a standard deviation of 27.14. Could you explain why the mean isn't closer to the actual boson mass. I figured the mean would be closer to the actual mass due to the large sample size (n=663). Is there a physical phenomenon that I am not taking into account?

#### mfb

Mentor
Not all events in the dataset are coming from Z bosons. You have some background events - pairs of electrons from other sources. You have more background events in the lower mass region. Therefore, the average of all events is not useful. You need some fit to the Z-peak, plus (probably optional) some model for the background. I would neglect all events with a very low calculated mass (<40 GeV). If the Breit-Wigner shape does not fit to the remaining events, you can add an exponential function* to account for those background events.

*in a real analysis for a publication, you would have to study the background shape. An exponential distribution should be a good approximation.

#### taper100

ok that makes sense, thanks again. What I wanted to do with this statistics project was to construct a confidence interval for the mean of the z boson mass. Is there any method to determine which of the events are background events? If i neglect the low masses as you said, will the rest of the masses be majority those of z bosons?

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