Introduction#
Warning
These pages are under development.
While the Standard Model provides an incredibly accurate description of the most fundamental constituents of matter, it remains difficult to describe interactions at the level of bound states (hadrons). This is because the fundamental force that dominates at this range―the strong force―is characterized by an asymptotic freedom: it has a strong coupling constant at low momentum transfer.
One the one hand, this asymptotic behavior of the strong force forces quarks to bound together in colorless groups (color confinement), giving particle physicists an amazingly varied spectrum of quark combinations to study. On the other, the asymptotic running of the coupling constant makes it almost impossible to predict from first principles how bound states of quarks interact at lower energies.
Theoreticians have developed several descriptive models and numerical tools (lattice QCD most importantly), but these models often rest on several assumptions or fail to describe larger systems. It therefore remains important to study particle interactions at the lower energies.
Partial Wave Analysis is a collection of techniques that allows us to perform these studies.
Todo
Importance of decomposing interactions
Strong QCD coupling constant at low momentum transfer
Cannot compute intermediate states from theory
Need to collect properties: mass, width, spin, parity
Scattering theory#
Tip
An extensive, chronological lists of publications on scattering theory can be found here.
Transition amplitude#
Todo
Amplitudes and intensities
Scattering operator \(S\) (S-matrix)
Lead-up to \(T\)-matrix (transition operator)
Distinction dynamical part and angular part
Partial waves#
Todo
Decomposition into partial waves [Peters, 2004], p.3:
Consider an incident wave \(\left|i\right> = Psi_i\). In an experimental setting, we can assume a vanishing potential at \(t\rightarrow\infty\), which allows us to expand the incoming wave in terms of Legendre polynomials \(P_l\), with \(l\) the angular orbital momentum:
\[\Psi_i(r,\theta) = \sum_{l=0}^\infty U_l(r) P_l(\theta) \]The \(U_l\) factor can be parametrized further. At this stage, it is important to point that the angular orbital momentum of the incoming state is used to characterize specific systems. Here, it is common to use the labels used for electron orbitals:
\(l\)
Label
Origin
0
s
“sharp”
1
p
“principal”
2
d
“diffuse”
3
f
“fundamental”
4
g
rest is alphabetical
…
…
…
In PWA, it is therefore common to distinguish these wave contributions as \(S\)-wave, \(P\)-wave, etc.
Todo
Separating out angular and radial wave functions using Legendre polynomials
What we can see from this and how it relates to analysis techniques
(?) Difference with amplitude analysis
Possible initial states: 1, 2, and multi-body
Suitable for propagation, not re-scattering
Isobar model#
Mandelstam variables#
Show code cell source
import matplotlib.pyplot as plt
from feynman import Diagram
fig, ax = plt.subplots(1, 3, figsize=(9, 3), frameon=False, tight_layout=True)
fig.patch.set_visible(False)
for a in ax:
a.axis("off")
a.set_xlim(0, 1)
a.set_ylim(0, 0.8)
s_channel = Diagram(ax[0])
# Vertices
v_p1 = s_channel.vertex(xy=(0.1, 0.7), marker="")
v_p2 = s_channel.vertex(tuple(v_p1.xy), dy=-0.6, marker="")
v_p3 = s_channel.vertex(tuple(v_p1.xy), dx=0.8, marker="")
v_p4 = s_channel.vertex(tuple(v_p3.xy), dy=-0.6, marker="")
v1 = s_channel.vertex(tuple(v_p1.xy), dx=0.2, dy=-0.3, marker="")
v2 = s_channel.vertex(tuple(v_p3.xy), dx=-0.2, dy=-0.3, marker="")
# Lines
p1 = s_channel.line(v_p1, v1, arrow=False)
p2 = s_channel.line(v1, v_p2, arrow=False)
interaction = s_channel.line(v1, v2, style="dotted", arrow=False)
e4 = s_channel.line(v2, v_p3, arrow=False)
e5 = s_channel.line(v_p4, v2, arrow=False)
# Labels
v_p1.text("$p_1$", x=-0.06, y=0)
v_p2.text("$p_2$", x=-0.06, y=0)
v_p3.text("$p_3$", x=+0.06, y=0)
v_p4.text("$p_4$", x=+0.06, y=0)
s_channel.text(0.5, 0.08, "$s$-channel", fontsize=15)
s_channel.text(0.5, 0, "$s = (p_1 + p_2)^2 = (p_3 + p_4)^2$", fontsize=12)
s_channel.plot()
t_channel = Diagram(ax[1])
# Vertices
v_p1 = t_channel.vertex(xy=(0.1, 0.7), marker="")
v_p2 = t_channel.vertex(tuple(v_p1.xy), dy=-0.6, marker="")
v_p3 = t_channel.vertex(tuple(v_p1.xy), dx=0.8, marker="")
v_p4 = t_channel.vertex(tuple(v_p3.xy), dy=-0.6, marker="")
v1 = t_channel.vertex(tuple(v_p1.xy), dx=0.4, dy=-0.15, marker="")
v2 = t_channel.vertex(tuple(v_p2.xy), dx=0.4, dy=+0.15, marker="")
# Lines
p1 = t_channel.line(v_p1, v1, arrow=False)
p2 = t_channel.line(v1, v_p3, arrow=False)
interaction = t_channel.line(v1, v2, style="dotted", arrow=False)
e4 = t_channel.line(v2, v_p2, arrow=False)
e5 = t_channel.line(v_p4, v2, arrow=False)
# Labels
v_p1.text("$p_1$", x=-0.06, y=0)
v_p2.text("$p_2$", x=-0.06, y=0)
v_p3.text("$p_3$", x=+0.06, y=0)
v_p4.text("$p_4$", x=+0.06, y=0)
t_channel.text(0.5, 0.08, "$t$-channel", fontsize=15)
t_channel.text(0.5, 0, "$t = (p_1 - p_3)^2 = (p_4 - p_2)^2$", fontsize=12)
t_channel.plot()
u_channel = Diagram(ax[2])
# Vertices
v_p1 = u_channel.vertex(xy=(0.1, 0.7), marker="")
v_p2 = u_channel.vertex(tuple(v_p1.xy), dy=-0.6, marker="")
v_p3 = u_channel.vertex(tuple(v_p1.xy), dx=0.8, marker="")
v_p4 = u_channel.vertex(tuple(v_p3.xy), dy=-0.6, marker="")
v1 = u_channel.vertex(tuple(v_p1.xy), dx=0.4, dy=-0.15, marker="")
v2 = u_channel.vertex(tuple(v_p2.xy), dx=0.4, dy=+0.15, marker="")
# Lines
p1 = u_channel.line(v_p1, v1, arrow=False)
p2 = u_channel.line(v1, v_p4, arrow=False)
interaction = u_channel.line(v1, v2, style="dotted", arrow=False)
e4 = u_channel.line(v2, v_p2, arrow=False)
e5 = u_channel.line(v_p3, v2, arrow=False)
# Labels
v_p1.text("$p_1$", x=-0.06, y=0)
v_p2.text("$p_2$", x=-0.06, y=0)
v_p3.text("$p_3$", x=+0.06, y=0)
v_p4.text("$p_4$", x=+0.06, y=0)
u_channel.text(0.5, 0.08, "$u$-channel", fontsize=15)
u_channel.text(0.5, 0, "$u = (p_1 - p_4)^2 = (p_3 - p_2)^2$", fontsize=12)
u_channel.plot();
