Question

On July \(4,2012,\) the discovery of the Higgs boson at the Large Hadron Collider was announced. During the data-taking run, the LHC reached a peak luminosity of \(4.00 \cdot 10^{33} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}\) (this means that in an area of 1 square centimeter, \(4.00 \cdot 10^{33}\) protons collided every second). Assume that the cross section for the production of the Higgs boson in these proton-proton collisions is \(1.00 \mathrm{pb}\) (picobarn). If the LHC accelerator ran without interruption for \(1.00 \mathrm{yr}\) at this luminosity, how many Higgs bosons would be produced?

Short answer

Answer: Approximately \(1.26144 \cdot 10^5\) Higgs bosons would be produced.

Step-by-step solution

Convert the cross section into \(\mathrm{cm}^2\)

The cross section given is in picobarns (pb). We will need to convert it into square centimeters (cm^2) for our calculations. Since 1 pb = \(10^{-36} \mathrm{~cm}^2\), we have: Cross section (cm^2) = \(1.00 \mathrm{~pb} \times 10^{-36} \mathrm{~cm}^2/\mathrm{pb} = 1.00 \cdot 10^{-36} \mathrm{~cm}^2\).

Calculate the number of collisions per year

Given the luminosity is \(4.00 \cdot 10^{33} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}\), to find the number of collisions happening in 1 year, we need to multiply the luminosity by the total number of seconds in a year. Number of seconds in a year = 365 days/year * 24 hours/day * 60 minutes/hour * 60 seconds/minute = 31536000 seconds/year. Number of collisions per year = Luminosity \(\times\) Seconds in a year Number of collisions per year = \(4.00 \cdot 10^{33} \mathrm{~cm}^{-2} \mathrm{~s}^{-1} \times 3.1536 \cdot 10^7 \mathrm{~s/yr} = 1.26144 \cdot 10^{41} \mathrm{~cm}^{-2} \mathrm{~yr}^{-1}\).

Calculate the number of Higgs bosons produced

Now we can find the number of Higgs bosons produced by multiplying the cross section by the number of collisions per year. Number of Higgs bosons produced = Cross section (cm^2) \(\times\) Number of collisions per year Number of Higgs bosons produced = \((1.00 \cdot 10^{-36} \mathrm{~cm}^2) \times (1.26144 \cdot 10^{41} \mathrm{~cm}^{-2} \mathrm{~yr}^{-1}) = 1.26144 \cdot 10^5 \mathrm{~yr}^{-1}\). So, if the LHC accelerator ran without interruption for 1 year at the given luminosity, approximately \(1.26144 \cdot 10^5\) Higgs bosons would be produced.

Key concepts

Large Hadron Collider

The Large Hadron Collider (LHC) is the world's largest and most powerful particle accelerator. It was designed to allow physicists to test various predictions of high-energy physics, including the existence of the Higgs boson. Nestled beneath the border between Switzerland and France, the LHC is an engineering marvel, a circular tunnel with a circumference of 27 kilometers.

In simple terms, the LHC boosts beams of protons to high energies before they are made to collide at several points around the machine. These high-energy proton-proton collisions are the experiments that allow scientists to probe deeper into the fundamental components of matter, offering insights into the basic building blocks of the universe.

Proton-Proton Collisions

Proton-proton collisions are at the heart of what happens inside the Large Hadron Collider. But what does this mean? Essentially, this process involves accelerating two beams of protons in opposite directions and then allowing them to collide. These high-energy collisions produce a variety of particles, some of which may be completely new to science.

The intense energy of the collision 'materializes' into a multitude of particles, a phenomenon elegantly explained by Einstein's equation E=mc^2. This equation shows that energy can be converted into mass, which is precisely what happens when protons collide; the kinetic energy of the protons is converted into the mass of new particles.

Cross Section in Particle Physics

In particle physics, the term 'cross section' is used to express the likelihood of a particular interaction occurring between particles. It's a measurement of the probability that a collision will happen under certain conditions. It's important to know that a larger cross section means a higher probability of the interaction occurring.

For instance, when we say the cross section for Higgs boson production is 1 picobarn, we are quantifying the chance of this particle being produced during a collision. This doesn't mean the Higgs boson will definitely be produced in every collision, but rather gives us the expected rate of production over a very large number of collisions. Cross sections in particle physics are typically very small because the particles being collided are so tiny.

Luminosity in Accelerators

Luminosity in accelerators like the LHC is a measure of collision intensity. When physicists talk about luminosity, they're describing the number of particles that can collide in a given area over a specific period of time. High luminosity equals more collisions, which means a greater chance of observing rare events, like the creation of a Higgs boson.

When we say the LHC achieved a luminosity of 4.00 x 10^33 cm^-2 s^-1, it means that, over the course of a year with continuous operation, we could expect a vast number of collisions. It is this high rate of collisions that affords scientists enough opportunities to spot a particle as elusive as the Higgs boson.