Question

Accelerators such as the Triangle Universities Meson Facility (TRIUMF) in British Columbia produce secondary beams of pions by having an intense primary proton beam strike a target. Such "meson factories" have been used for many years to study the interaction of pions with nuclei and, hence, the strong nuclear force. One reaction that occurs is\({{\rm{\pi }}^{\rm{ + }}}{\rm{ + p}} \to {{\rm{\Delta }}^{{\rm{ + + }}}} \to {{\rm{\pi }}^{\rm{ + }}}{\rm{ + p}}\), where the \({{\rm{\Delta }}^{{\rm{ + + }}}}\)is a very short-lived particle. The graph in Figure \({\rm{33}}{\rm{.26}}\)shows the probability of this reaction as a function of energy. The width of the bump is the uncertainty in energy due to the short lifetime of the\({{\rm{\Delta }}^{{\rm{ + + }}}}\).

(a) Find this lifetime.

(b) Verify from the quark composition of the particles that this reaction annihilates and then re-creates a d quark and a \({\rm{\bar d}}\)antiquark by writing the reaction and decay in terms of quarks.

(c) Draw a Feynman diagram of the production and decay of the \({{\rm{\Delta }}^{{\rm{ + + }}}}\)showing the individual quarks involved.

Short answer

a. The life time of \({{\rm{\Delta }}^{{\rm{ + + }}}}\) particle is \(3.28 \times {10^{ - 24}}\;{\rm{s}}\).

b. The reaction annihilates and then recreates a d quark and a \({\rm{\bar d}}\) antiquark.

c. Feynman diagram for the production and decay of \({{\rm{\Delta }}^{{\rm{ + + }}}}\)particle is shown below.

Step-by-step solution

Concept

According to quark model, there are six types of quark- up, down, strange, charm, top and bottom- whose combination form the basic structure of hadrons. Baryons are the combination of three quarks or antiquarks and mesons are a combination of a quark and an antiquark.

Given data

Uncertainty in energy \(\Delta E = 100\;{\rm{MeV}}\)

Find the life time

(a)

The uncertainty principle for energy and time is,

\(\Delta E.\Delta t = \frac{h}{{4\pi }}\)

Whereas,\(\Delta E\)is uncertainty in energy measurement and\(\Delta t\)is uncertainty in time measurement

As a result, uncertainty in the lifetime of the\({{\rm{\Delta }}^{{\rm{ + + }}}}\) particle\(\Delta t = \frac{h}{{4\pi \Delta E}}\)

\(\begin{array}{c}\Delta t = \frac{{6.6 \times {{10}^{ - 34}}\;{\rm{J}}{\rm{.s}}}}{{4 \times 3.14 \times \left( {100 \times {{10}^6}} \right)\;{\rm{MeV}} \times 1.6 \times {{10}^{ - 19}}}}\\ = 3.28 \times {10^{ - 24}}\;{\rm{s}}\end{array}\)

Therefore, the required life time of \({{\rm{\Delta }}^{{\rm{ + + }}}}\) particle created at TRIUMF is \(3.28 \times {10^{ - 24}}\;{\rm{s}}\).

Check from the quark composition of the particles that this reaction annihilates

(b)

Considering the given information:

Reaction is given by,\({\pi ^ + } + p \to {\Delta ^{ + + }} \to {\pi ^ + } + p\)

The\({{\rm{\pi }}^{\rm{ + }}}{\rm{ = u\bar d}}\)quark structure

Proton quark structure\({\rm{ = uud}}\)

The\({{\rm{\Delta }}^{{\rm{ + + }}}}\)quark structure

\({\rm{ = uuu}}\)

As a result, in terms of quark structure, the reaction is given by,

\(\begin{array}{l}{\pi ^ + } + p \to {\Delta ^{ + + }} \to {\pi ^ + } + p\\u\bar d + uud \to uuu \to u\bar d + uud\end{array}\)

According to the above reaction in terms of quarks, the reaction annihilates and then recreates a d quark and a\({\rm{\bar d}}\) antiquark.

Therefore, the required reaction annihilates and then recreates a d quark and a \({\rm{\bar d}}\) antiquark.

Draw a Feynman diagram of the production

(c)Considering the given information:

Reaction is given as, \({\pi ^ + } + p \to {\Delta ^{ + + }} \to {\pi ^ + } + p\)

Below is the Feynman diagram for the production and decay of the \({{\rm{\Delta }}^{{\rm{ + + }}}}\) particle.

Therefore, the Feynman diagram for the production and decay of \({{\rm{\Delta }}^{{\rm{ + + }}}}\)particle is drawn.