# Dimensions of the pion decay constant

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• anthony2005
In summary, the pion decay constant f_{\pi} is defined by <0|\overline{d}\gamma^{0}\gamma^{5}u|\pi^{+}>=f_{\pi}m_{\pi} and the dimensions do not match due to a problem with the normalization of the single-particle state ##|\pi^+ \rangle##. The standard definition of the state is given by |\vec{p} \rangle = \hat{a}^{\dagger}(\vec{p})=|\Omega \rangle, \quad [\hat{a}(\vec{p}_1,\hat{a}^{\dagger}(\vec{p}_2)]=(2 \pi
anthony2005
The pion decay constant $f_{\pi}$ is defined by
$$<0|\overline{d}\gamma^{0}\gamma^{5}u|\pi^{+}>=f_{\pi}m_{\pi}$$
where I have set the momentum $\boldsymbol{p}=0$ (and used the temporal component of the axial vector).
Now, at the right-hand-side the dimension is two (the decay constant and the mass are measured in MeV). At the left-hand side, each quark carries dimension 3/2, so the whole axial vector has dimensions 3.
The dimensions do not match! Where is the problem?

How is the single-particle state ##|\pi^+ \rangle## normalized? The standard definition is that
$$|\vec{p} \rangle = \hat{a}^{\dagger}(\vec{p})=|\Omega \rangle, \quad [\hat{a}(\vec{p}_1,\hat{a}^{\dagger}(\vec{p}_2)]=(2 \pi)^3 2 \sqrt{m^2+\vec{p}_1^2} \delta^{(3)}(\vec{p}-\vec{p}').$$
This implies that
$$\langle \vec{p}_1|\vec{p}_2 \rangle=(2 \pi)^3 2 \sqrt{m^2+\vec{p}_1^2} \delta^{(3)}(\vec{p}_1-\vec{p}_2),$$
i.e., ##|\vec{p} \rangle## as the dimension ##1/\text{energy}##, and thus your formula is dimensionally correct. You only should take care of the value of ##f_{\pi}##, which varies in the literature. Assuming that your source uses the same normalization of states your pion-decay constant is ##f_{\pi}=\sqrt{2} F_{\pi} \simeq 130 \; \text{MeV}##.

For a very nice review on chiral symmetry in QCD see

https://arxiv.org/abs/nucl-th/9706075

Thanks, that perfectly solved my problem. I wasn't thinking about the delta function carrying a dimension, but indeed if we think of $\delta\left(\boldsymbol{p}\right)\propto\int d^{3}xe^{i\boldsymbol{p}\cdot\boldsymbol{x}}$ then the measure will lead to -3.

anthony2005 said:
Thanks, that perfectly solved my problem. I wasn't thinking about the delta function carrying a dimension, but indeed if we think of $\delta\left(\boldsymbol{p}\right)\propto\int d^{3}xe^{i\boldsymbol{p}\cdot\boldsymbol{x}}$ then the measure will lead to -3.

##\delta^{(3)}(p)## has dimension -3 in post #2. The prefactor ##\sqrt{p^2 - m^2}## has dimension one, which leads to the states having dimension -1.

vanhees71

## What is the pion decay constant?

The pion decay constant, also known as fπ, is a fundamental constant in particle physics that describes the strength of the interaction between pions and other particles.

## Why is the pion decay constant important?

The pion decay constant is important because it plays a crucial role in understanding the properties and behavior of pions, which are the lightest and most abundant particles that make up the atomic nucleus. It is also an important parameter in theoretical models of strong interactions.

## How is the pion decay constant measured?

The pion decay constant is typically measured using experiments that involve high-energy particle collisions, such as those conducted at particle accelerators. These experiments allow scientists to observe the decay of pions and measure the strength of their interactions, from which the decay constant can be calculated.

## What are the dimensions of the pion decay constant?

The pion decay constant has dimensions of energy, specifically MeV (megaelectronvolts). This is because it represents the energy scale at which pions interact with other particles.

## How does the pion decay constant relate to other fundamental constants?

The pion decay constant is related to other fundamental constants through the Standard Model of particle physics. It is connected to other parameters, such as the speed of light and the Planck constant, which together help to describe the behavior of particles at the subatomic level.

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