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Chapter 10 - QCD - Coggle Diagram
Chapter 10 - QCD
SU(3)
local gauge symmetry: \(\phi(x) \rightarrow \phi'(x) = e^{ig_S\mathbf{\alpha(x)}\cdot\mathbf{T}} \phi(x)\)
\(\mathbf{T} = \frac{1}{2}\mathbf{\lambda}\):
8 generators
of SU(3), related to
Gell Mann matrices
(3x3).
Each wavefunction has
3
additional d.o.f containing "color": {r, g, b}
Local symmetry is preserved by introducing 8 new field \(G_\mu\) which transforms \(G_\mu^k \rightarrow G'_\mu = G_\mu^k - \partial_\mu\alpha_k - g_Sf_{ijk}\alpha_i G^j_\mu\)
\(f_{ijk}\) is the
structure constant
of SU(3): \([\lambda_i, \lambda_j] = if_{ijk}\lambda_k\)
And covariant derivative \(\partial_\mu \rightarrow \partial_\mu + ig_S G_\mu^\alpha T_\alpha\)
Dirac Eq becomes: \(i\gamma^\mu(\partial_\mu + ig_S G^\alpha_\mu T^\alpha)\psi - m\psi = 0\)
SU(3) is non-abelian -> gluons have
self-interaction
!
Implication:
gluons also carry color charge
! (in QED, photons are electrically neutral)
Result:
color confinement
!
QCD
Vertices
: \(qqg, ggg, gggg\)
QCD
current: \(j^\mu =\Big( -\frac{1}{2}ig_S c_j^\dagger \lambda^a c_i\Big) \times \bar{u}(p_3) \gamma^\mu u(p_1)\)
\(c_1 = r = (1,0,0)\), \(c_2 = g = (0,1,0)\), \(c_3 = b = (0,0,1)\)
\(\lambda^a\): gluon corresponding to generator \(T^a\)
\(c_j^\dagger \lambda^a c_i = \lambda^a_{ij}\)
Therefore, \(\lambda\) acts on 3x3 color wavefunction, and \(\gamma\) acts on 4x4 Dirac spinor.
Feynman rule
vertex: \( -\frac{1}{2}ig_S \lambda^a_{ij}\gamma^\mu\)
Propagator: \(-i\frac{g_{\mu\nu}}{q^2}\delta^{ab}\)
a, b
corresponds to colors of gluon
emitted/absorbed
by vertices \(\mu, \nu\); they should have
same color
.
Gluons
must carry
both color and anti-color
to preserve total color at vertex. Therefore, correspond of
off-diagonal elements
of Gell Mann matrices (changes two different colors).
8 gluons corresponding to octets; all carry non-zero color charge (no singlet).
Color confinemnet
Since no free quark has been observed, Assumption: colored objects (e.g.
hadron, meson
) are
confined
to
color-singlet states
; no objects with non-zero color charge can propagate as free particles.
Two quarks separate apart -> Flux tube with constant energy density -> potential \(\propto \kappa r\) ->
infinite energy
to separate quarks!
Gluons are also confined
-> no free gluon...
Hadron states
All hadron must be in color-singlet state. E.g. for meson: \(\psi_{c} = \frac{1}{3}(r\bar{r} + g\bar{g} + b\bar{b})\). Similar total anti-sym wavefunction exists for qqq.
Color for quark combinations:
\(q\bar{q}\): \(3\bigotimes \bar{3} = 8 \bigoplus 1\)
\(qqq\): \(3\bigotimes 3 \bigotimes 3 = 10 \bigoplus 8 \bigoplus 8 \bigoplus 1\)
Others combinations have no color singlet state! e.g. for \(qq\), \(3\bigotimes 3 = 6 \bigoplus 3\)
Running \(\alpha_s\)
At low \(q^2\), \(\alpha_s\) is large (confinement); at
large
\(q^2\), \(\alpha_s\) becomes small (
asymptotic freedom
).
Why? High-order
loop corrections
to propagator is
absorbed
to the
definition of charge
.
QED:
increase
with energy \(\alpha(q^2) = \frac{\alpha(\mu^2)}{1 - A\alpha(\mu^2)\ln(q^2/\mu^2)}\)
from q=0 to 193 GeV, \(\alpha\) changes from 1/137 to 1/127
QCD:
decrease
with energy \(\alpha_s(q^2) = \frac{\alpha_S(\mu^2)}{1 + B\alpha_s(\mu^2)\ln(q^2/\mu^2)}\)
At 1GeV, \(\alpha_s=O(1)\). At \(|q|=m_Z\), \(\alpha_s^2(m_Z^2) = 0.11\)
Why color?
Neutral pion decay cross section is 9 times larger than expected;
\(\Gamma \propto N_c^2(Q_u^2 - Q_d^2)\);
3 colors
explains!
\(e^+e^- \rightarrow \gamma \rightarrow q\bar{q}/\mu^+\mu^-\) is sensitive to hidden d.o.f.
R = \(\frac{\Gamma(ee\rightarrow hadron)}{\Gamma(ee\rightarrow \mu^2\mu^-)} = N_c\sum_qe_q^2\)
Final state \(q\bar{q}\) could be produced in \(r\bar{r}, g\bar{g}, b\bar{b}\)
Result:
R consistent with 3 color charges
.
Jets
Hadronization is fast (quark separation creates more quarks/gluons ->
jets
, mostly in original parton direction)
