Isospin and pion-nucleon scattering

Pion-nucleon scattering depends on isospin because the strong interaction is insensitive to electric charge and distinguishes particles exclusively through their isospin quantum numbers.

The pion and the nucleon carry isospin $ I = 1 $ and $ I = \tfrac{1}{2} $, respectively. As a consequence, the interaction can occur only in states with total isospin $ \tfrac{3}{2} $ or $ \tfrac{1}{2} $.

All experimentally observed processes are combinations of these two, and only these two, isospin channels.

Explanation

Consider a generic scattering process between a pion $ \pi $ and a nucleon $ N $:

\[ \pi + N \to \pi + N \]

The pion may be $ \pi^+ , \pi^0 , \pi^- $.

The nucleon, in turn, may be either a proton (p) or a neutron (n).

Depending on the electric charges involved, the scattering process may appear different at first glance. However, the strong interaction is blind to electric charge. It depends solely on isospin.

the pion $ \pi $ has isospin $ I = 1 $
the nucleon $ N $ has isospin $ I = \tfrac{1}{2} $

When the two particles are combined, the total isospin of the system can take only two possible values:

\[ 1 \otimes \tfrac{1}{2} = \tfrac{3}{2} \oplus \tfrac{1}{2} \]

Accordingly, all possible pion-nucleon scattering processes reduce to just two physically distinct cases:

scattering with total isospin $ I = \tfrac{3}{2} $
scattering with total isospin $ I = \tfrac{1}{2} $

No other values of the total isospin are permitted.

Note. The pion has isospin $ I_1 = 1 $, while the nucleon has isospin $ I_2 = \tfrac{1}{2} $. According to the rules of isospin addition, the total isospin $ I $ may assume any value between $ |I_1 - I_2| $ and $ I_1 + I_2 $, in integer or half-integer steps. In this case: \[
|1 - \tfrac{1}{2}| = \tfrac{1}{2} \] \[ 1 + \tfrac{1}{2} = \tfrac{3}{2} \] The only allowed values of the total isospin are therefore: \[
I = \tfrac{1}{2}, \ \tfrac{3}{2} \] From this it follows that: \[1 \otimes \tfrac{1}{2} = \tfrac{3}{2} \oplus \tfrac{1}{2} \] This result shows that the pion-nucleon system can exist only in one of these two total isospin states.

In pion-nucleon scattering, since the system admits only two possible values of the total isospin ( $ I = \tfrac{3}{2} $ and $ I = \tfrac{1}{2} $ ), there are only two independent scattering amplitudes:

$ \mathcal M_{3/2} $ for the channel with $ I = \tfrac{3}{2} $
$ \mathcal M_{1/2} $ for the channel with $ I = \tfrac{1}{2} $

The scattering amplitude is a complex-valued quantity that represents the probability amplitude for a given reaction to proceed through a specific physical channel.

Note. More precisely, the scattering amplitude is the fundamental quantum-mechanical object that characterizes the strength of a scattering process. Observable probabilities are obtained by taking the squared modulus of the amplitude.

The cross section is derived directly from the scattering amplitude.

The cross section is a physical quantity that measures the likelihood that a given reaction will occur when two particles collide.

In quantum mechanics, the cross section is proportional to the squared modulus of the scattering amplitude:

\[ \sigma \propto |\mathcal M|^2 \]

The cross section is not a literal geometric area, even though it has the dimensions of an area. Rather, it represents an effective area that quantifies the probability of the scattering process.

Operationally, the cross section ( \sigma ) is defined as

\[ \sigma = \frac{\text{number of reactions per second}}{\text{incident particle flux}} \]

In practical terms, it is the ratio between the number of reactions of the specified type that occur and the number of incoming particles crossing a given area per unit time.

If $ \sigma $ is large, the reaction is likely.
If $ \sigma $ is small, the reaction is rare.

For this reason, the cross section ( \sigma ) is particularly valuable. It is directly measurable in experiments and provides a quantitative measure of the probability of a scattering reaction.

Every physical reaction can be pure or mixed

In general, depending on the initial state of the scattering process, the reaction may be

pure, if the total isospin is uniquely determined
mixed, if the initial state is a superposition of different total isospin values.

The initial scattering state is the quantum-mechanical configuration of the system prior to the collision, namely the set of incoming particles together with their physical properties.

Note. In pion-nucleon scattering, the initial state is simply the pair of particles about to collide, for example $ \pi^+ + p $ or $ \pi^- + p $, together with their associated quantum numbers such as electric charge and isospin. This initial state determines how the system decomposes into isospin channels and therefore whether the reaction is pure or mixed.

Example

Consider the scattering process

\[ \pi^+ + p \to \pi^+ + p \]

In this case, the initial state is $ \pi^+ + p $.

We begin by identifying the values of the isospin projection along the z-axis.

$ \pi^+ $ has $ I_3 = +1 $
$ p $ has $ I_3 = \tfrac{1}{2} $

The total isospin projection of the initial state is therefore

\[ I_3^{ \text{tot} } = +1 + \tfrac{1}{2} = +\tfrac{3}{2} \]

Which values of the total isospin $ I $ are compatible with $ I_3 = +\tfrac{3}{2} $?

For a pion-nucleon system, the allowed values of the total isospin are

\[ I \in \left\{ \tfrac{1}{2}, \tfrac{3}{2} \right\} \]

This follows directly from the rules of isospin addition. Combining an isospin $ I = 1 $ with an isospin $ I = \tfrac{1}{2} $ yields

\[ I = 1 + \tfrac{1}{2} = \tfrac{3}{2}, \qquad I = 1 - \tfrac{1}{2} = \tfrac{1}{2} \]

Since the $ I = \tfrac{1}{2} $ multiplet does not contain a state with $ I_3 = +\tfrac{3}{2} $, the only allowed value of the total isospin is $ I = \tfrac{3}{2} $.

It follows that the scattering process $ \pi^+ + p $ is a reaction with pure isospin.

In this case, the scattering amplitude is $ \mathcal M_{3/2} $, and the cross section is simply proportional to its squared modulus:

\[ \sigma \propto |\mathcal M_{3/2}|^2 \]

No contributions from other isospin channels arise, nor are there interference terms, because the initial state has a uniquely defined total isospin $ I = \tfrac{3}{2} $.

As a result, the probability of the reaction is determined entirely by the dynamics of the $ I = \tfrac{3}{2} $ channel.

Example 2

Now consider the scattering process

\[ \pi^- + p \to \pi^- + p \]

In this case, the initial state is $ \pi^- + p $.

Once again, we determine the values of the isospin projection along the z-axis.

$ \pi^- $ has $ I_3 = -1 $
$ p $ has $ I_3 = +\tfrac{1}{2} $

The total isospin projection of the initial state is therefore

\[ I_3^{\text{tot}} = -1 + \tfrac{1}{2} = -\tfrac{1}{2} \]

Which values of the total isospin $ I $ are compatible with $ I_3 = -\tfrac{1}{2} $?

For the pion-nucleon system, the allowed values of the total isospin are

\[ I \in \left\{ \tfrac{1}{2}, \tfrac{3}{2} \right\} \]

Again, this follows from the isospin addition rules. Combining an isospin $ I = 1 $ with an isospin $ I = \tfrac{1}{2} $ yields

\[ I = 1 + \tfrac{1}{2} = \tfrac{3}{2}, \qquad I = 1 - \tfrac{1}{2} = \tfrac{1}{2} \]

In this case, both the $ I = \tfrac{1}{2} $ and $ I = \tfrac{3}{2} $ multiplets contain a state with $ I_3 = -\tfrac{1}{2} $.

Note. A total isospin multiplet with quantum number $ I $ contains all states with third-component values ranging from $ -I $ to $ +I $ in integer steps: \[ I_3 = -I, -I+1, \dots , I-1, I \] For $ I = \tfrac{1}{2} $, the allowed values are \[ I_3 = -\tfrac{1}{2}, +\tfrac{1}{2} \] while for $ I = \tfrac{3}{2} $, the allowed values are \[ I_3 = -\tfrac{3}{2}, -\tfrac{1}{2}, +\tfrac{1}{2}, +\tfrac{3}{2} \] Since the value $ I_3 = -\tfrac{1}{2} $ belongs to both sets, states with this projection exist in both the $ I = \tfrac{1}{2} $ and $ I = \tfrac{3}{2} $ multiplets.

Consequently, the initial state $ \pi^- + p $ is not a pure isospin state, but rather a linear superposition of states with total isospin $ I = \tfrac{1}{2} $ and $ I = \tfrac{3}{2} $.

\[ \pi^- p = a (I=\tfrac{3}{2}, I_3=-\tfrac{1}{2}) + b(I=\tfrac{1}{2}, I_3=-\tfrac{1}{2}) \]

Here $ a $ and $ b $ are complex coefficients satisfying the normalization condition $ |a|^2 + |b|^2 = 1 $.

This expression makes explicit that the initial state $ \pi^- + p $ does not possess a definite total isospin. Instead, it is a superposition of two states with different total isospin values, both sharing the same projection $ I_3 = -\tfrac{1}{2} $.

What changes in a mixed reaction compared to a pure reaction?

In mixed reactions, such as $ \pi^- + p $, the initial state does not have a well-defined total isospin. Physically, this means that the scattering process does not proceed through a single isospin channel, but through a coherent superposition of multiple channels.

In the case of $ \pi^- + p $, the initial state can be expressed as a linear combination of a state with $ I = \tfrac{3}{2} $ and a state with $ I = \tfrac{1}{2} $.

As a consequence, the scattering process can occur partly through the $ I = \tfrac{3}{2} $ channel and partly through the $ I = \tfrac{1}{2} $ channel. Each total isospin value is associated with a distinct scattering amplitude:

\[ \mathcal M_{3/2} \qquad \text{and} \qquad \mathcal M_{1/2} \]

The total scattering amplitude is therefore not given by a single term, but by a weighted linear combination of the amplitudes corresponding to the two isospin channels:

\[ \mathcal M = c_{3/2} \mathcal M_{3/2} + c_{1/2} \mathcal M_{1/2} \]

The coefficients $ c_{3/2} $ and $ c_{1/2} $ are fixed by the Clebsch-Gordan coefficients appearing in the decomposition of the initial state.

Note. In pure reactions, such as $ \pi^+ + p $, the initial state has a uniquely defined total isospin $ I = \tfrac{3}{2} $. In this case, only one isospin channel contributes, and the scattering amplitude coincides directly with $ \mathcal M_{3/2} $, without additional contributions or interference effects.

The state is mixed, but the dynamics may select a single channel

In mixed reactions, the different isospin channels do not necessarily contribute with equal weight. Dynamical effects can strongly enhance one channel relative to another.

For example, although the initial state $ \pi^- + p $ contains components with total isospin $ I = \tfrac{1}{2} $ and $ I = \tfrac{3}{2} $, only the $ I = \tfrac{3}{2} $ component may contribute significantly to the reaction, while the other is effectively suppressed.

This occurs because a $ \Delta $ resonance exists with isospin $ I = \tfrac{3}{2} $. Such a resonance can be formed only in the $ I = \tfrac{3}{2} $ channel and therefore strongly enhances the corresponding scattering amplitude.

Operationally, the system behaves as if two channels were available, but only one is dynamically amplified.

For this reason, in $ \pi^- + p $ scattering the interaction proceeds almost exclusively through the channel with total isospin $ I = \tfrac{3}{2} $. The $ I = \tfrac{1}{2} $ channel gives a negligible contribution.

In this regime, the state is mixed, but the dynamics effectively selects a single channel.

\[ \mathcal M_{3/2} \gg \mathcal M_{1/2} \]

The total amplitude then reduces to

\[ \mathcal M \approx c_{3/2} \mathcal M_{3/2} \]

Consequently, the cross section is proportional to

\[ |\mathcal M|^2 \approx |c_{3/2}|^2 |\mathcal M_{3/2}|^2 \]

Note. Even in a mixed reaction, the amplitude is formally given by a sum of contributions: \[ \mathcal M = c_{3/2} \mathcal M_{3/2} + c_{1/2} \mathcal M_{1/2} \] When a single channel dominates for dynamical reasons, such as the presence of the $ \Delta $ resonance, the total amplitude effectively reduces to \[ \mathcal M \approx c_{3/2} \mathcal M_{3/2} \] As a result, the cross section becomes \[ |\mathcal M|^2 \approx |c_{3/2}|^2 |\mathcal M_{3/2}|^2 \]

Since $ c_{3/2} $ is a Clebsch-Gordan coefficient, and such coefficients are fixed rational numbers determined by the algebra of isospin, the ratios of cross sections in this regime no longer depend on the detailed dynamics of the interaction, but only on the isospin structure of the states involved.

In general, when a single isospin channel dominates, differences between reactions are governed solely by algebraic coefficients.

As a result, ratios of cross sections take simple and universal numerical values.

For example, if one considers the cross sections $ \sigma $ of the reactions $ \pi^+ + p $ and $ \pi^- + p $, their ratio is equal to 3: \[ \frac{\sigma(\pi^+ p)}{\sigma(\pi^- p)} = 3 \] These values are also confirmed experimentally.

And so on.