§4.9Charm

Strangeness was a surprise. Somebody found particles that were produced quickly and decayed slowly, and a quantum number was invented afterwards to describe what had already been seen.

Charm went the other way round. It was predicted — for a reason that has nothing to do with spectroscopy — and then, four years later, arrived in a way nobody expected, in two laboratories in the same month.

4.9 Why anyone wanted a fourth quark

The argument has no hadrons in it at all. It is about two decays of the same kaon that ought to be comparable and are not.

import numpy as np

br_nc = 1.14e-10          # K+ -> pi+ nu nubar   (neutral current)
br_cc = 3e-2              # K+ -> pi0 e+ nu      (charged current)
print(f"  BR(K+ -> pi+ nu nubar) = {br_nc:.2e}   flavour-changing NEUTRAL current")
print(f"  BR(K+ -> pi0 e+ nu)    = {br_cc:.2e}   flavour-changing charged current")
print(f"  ratio                  = {br_nc/br_cc:.1e}")
print(f"  the neutral-current mode is {br_cc/br_nc:.1e} times rarer"
      f" -- {np.log10(br_cc/br_nc):.1f} orders of magnitude")
print("  and no symmetry known in 1970 forbade it")

  BR(K+ -> pi+ nu nubar) = 1.14e-10   flavour-changing NEUTRAL current
BR(K+ -> pi0 e+ nu)    = 3.00e-02   flavour-changing charged current
ratio                  = 3.8e-09
the neutral-current mode is 2.6e+08 times rarer -- 8.4 orders of magnitude
and no symmetry known in 1970 forbade it

In 1970 Sheldon Glashow, John Iliopoulos and Luciano Maiani proposed a way out (the GIM mechanism gim mechanism Glashow, Iliopoulos and Maiani (1970): a fourth quark, charm, introduced to explain why flavour-changing neutral current decays such as K⁺ → π⁺νν̄ are suppressed relative to the charged-current K⁺ → π⁰e⁺ν — by 8.4 orders of magnitude in branching ratio. The c contributes a second amplitude that cancels the first; the residue is set by the mass splitting in the loop, so the measured suppression predicted m_c before any charmed hadron was seen. defined in §4.9 — open in glossary ): add a fourth quark, call it charm, give it charge +2/3, and pair it with the s the way u is paired with d. The neutral-current amplitude then receives a second contribution — the same diagram with c in place of u — which is equal and opposite. The two cancel, and what survives is proportional to the small difference between the two quarks. Chapter 7 does this properly; the whole of §4.9 is what happened when people went looking for the quark it required.

What had to be true before anything was found

The GIM model is specific, and this is what makes the next two years a test rather than a search:

The November Revolution, part one: Ting at Brookhaven

The mass of the pair follows from four measured numbers:

import numpy as np
me = 0.000511
p1 = p2 = 5.0                       # GeV, one particle per arm
th = 0.6293                         # theta1 + theta2, rad

def mass(p1, p2, th):
    E1, E2 = np.sqrt(p1**2 + me**2), np.sqrt(p2**2 + me**2)
    return np.sqrt(2*me**2 + 2*E1*E2 - 2*p1*p2*np.cos(th))

m = mass(p1, p2, th)
print(f"  p1 = {p1} GeV, p2 = {p2} GeV, theta1+theta2 = {th} rad ({np.degrees(th):.1f} deg)")
print(f"  m(e+e-) = {m:.4f} GeV")

dp_rel, dth = 1e-3, 1e-3            # 0.1 % per momentum, 1 mrad on the angle
terms = {"p1 (0.1 %)": mass(p1*(1+dp_rel), p2, th) - m,
         "p2 (0.1 %)": mass(p1, p2*(1+dp_rel), th) - m,
         "angle (1 mrad)": mass(p1, p2, th + dth) - m}
for k, v in terms.items():
    print(f"    d(m) from {k:15s} = {v*1000:6.2f} MeV")
print(f"  added in quadrature: sigma(m) = {np.sqrt(sum(v**2 for v in terms.values()))*1000:.1f} MeV")
print("  the ANGLE dominates, and 5 MeV is the whole published limit on the width")

  p1 = 5.0 GeV, p2 = 5.0 GeV, theta1+theta2 = 0.6293 rad (36.1 deg)
m(e+e-) = 3.0948 GeV
  d(m) from p1 (0.1 %)      =   1.55 MeV
  d(m) from p2 (0.1 %)      =   1.55 MeV
  d(m) from angle (1 mrad)  =   4.75 MeV
added in quadrature: sigma(m) = 5.2 MeV
the ANGLE dominates, and 5 MeV is the whole published limit on the width

Part two: Richter at SPEAR

Establishing that it really is a photon-like state

The J/ψ should be $J^{PC} = 1^{--}$: it is produced in $e^+e^-$ annihilation and decays to $e^+e^-$, and both go through a single virtual photon. But the particle was so anomalous that nothing was assumed. Three checks:

1. Interference says $J^P = 1^-$, not $1^+$

Outside the resonance, $e^+e^- \to \mu^+\mu^-$ proceeds through a photon (Eq. 4.65a). On resonance there is a second route, through the J/ψ (Eq. 4.65b). If the two intermediate states have the same quantum numbers the amplitudes add and interfere; if the J/ψ were $1^+$ they could not.

2. G-parity says the decay is strong

The J/ψ is so narrow that it might have been decaying electromagnetically, as the η does. The test uses G-parity g-parity G ≡ exp(−iπI_y)·C, a multiplicative quantum number for non-strange states of zero baryon number; G(π) = −1, so an n-pion state has G = (−1)ⁿ. Only the strong interaction conserves it. defined in §3.8-3.10 — open in glossary from §3.6: the strong interaction conserves it, so a decay into $n$ pions is allowed only for one parity of $n$. Measuring $e^+e^- \to n\pi$ across the resonance, the peak appears for n = 3 and n = 5 but not for n = 2 or n = 4 — G-parity is conserved, so the decay is strong after all, and its narrowness has to be explained some other way.

3. Isospin zero, from a branching ratio

An $I = 1$ state cannot decay to $\pi^0\pi^0$ at all — the Clebsch–Gordan coefficient for two identical isovectors coupling to $I = 1$ with $I_z = 0$ vanishes (§3.8). Measurement finds $\mathrm{BR}(\pi^0\pi^0) = \mathrm{BR}(\pi^+\pi^-)$, which is the $I = 0$ prediction. So the assignment is complete: $J^{PC} = 1^{--}$, $I = 0$.

Ten days after the ψ, the same scan found a second narrow state, the ψ′ at 3686 MeV, with the same quantum numbers. And at Frascati, ADONE — designed for a maximum of $\sqrt s = 3$ GeV, agonisingly just below — was pushed above its nominal energy to 3100 MeV as soon as Ting telephoned, and saw the resonance immediately.

Measuring a width fifty times narrower than your resolution

This is the methodological heart of the section, and it generalises far beyond charm.

Integrate Eq. (4.67) over $E$. The Lorentzian integrates to $2\pi/\Gamma$, and the result is a quantity that has nothing to do with the shape:

import numpy as np

M, G = 3.097, 93e-6                      # GeV
B = dict(ee=0.06, mumu=0.06, had=0.88)
hbarc2_nb = 0.3894e6                     # (hbar c)^2 in GeV^2 nb
Ge = B["ee"] * G

areas = {f: 6*np.pi**2 * Ge * b / M**2 * hbarc2_nb for f, b in B.items()}
print("  peak areas from Eq. (4.68), integral sigma dE = 6 pi^2 Ge Bf / M^2 :")
for f, a in areas.items():
    print(f"    {f:5s}: {a*1000:8.0f} MeV nb")

print("  now keep ONLY the areas and invert:")
tot = sum(areas.values())
Ge_meas = tot * M**2 / (6*np.pi**2) / hbarc2_nb
G_meas = Ge_meas / (areas["ee"]/tot)
print(f"    sum of areas   = {tot*1000:.0f} MeV nb")
print(f"    Gamma_e        = {Ge_meas*1e6:.2f} keV")
print(f"    Gamma (total)  = {G_meas*1e6:.1f} keV   <- the resolution never entered")
print(f"  cross-check: {areas['had']*1000:.0f} MeV nb under a peak 2000 nb high implies an apparent")
print(f"    FWHM of about {2*areas['had']*1000/(np.pi*2000):.1f} MeV -- the SPEAR beam spread,"
      f" forty times the true width")

  peak areas from Eq. (4.68), integral sigma dE = 6 pi^2 Ge Bf / M^2 :
  ee   :      805 MeV nb
  mumu :      805 MeV nb
  had  :    11805 MeV nb
now keep ONLY the areas and invert:
  sum of areas   = 13415 MeV nb
  Gamma_e        = 5.58 keV
  Gamma (total)  = 93.0 keV   <- the resolution never entered
cross-check: 11805 MeV nb under a peak 2000 nb high implies an apparent
  FWHM of about 3.8 MeV -- the SPEAR beam spread, forty times the true width

The charmed mesons themselves

Two years later, at $\sqrt s = 4.02$ GeV and with the Mark I kaon–pion separation improved, the same collider found the D mesons.

Why J/ψ and ψ′ are so narrow — the φ argument, one level up

Everything now closes. The J/ψ is a $c\bar c$ pair; to decay into ordinary hadrons it must annihilate the c against the c̄, and QCD suppresses that (Chapter 6). The alternative — falling apart into a pair of open-flavour open flavour the complement of [[hidden flavour]]: a pair of hadrons that each carry the heavy flavour explicitly (DD̄, BB̄, KK̄). A hidden-flavour state stays narrow while the open-flavour channel is below threshold and its width jumps by orders of magnitude the moment that channel opens — the φ, J/ψ and ϒ all show it, and ψ(3770) at 27 MeV is the same state after the jump. defined in §4.9 — open in glossary mesons, where the c and c̄ simply walk out — is closed, because $2m_D > m_{J/\psi}$ and $2m_D > m_{\psi'}$.

This is exactly the argument §4.5 made for the φ, and the confirmation is the third level: the ψ″(3770) sits above the threshold, and its width is 27 MeV instead of 93 keV.

mD0, mDp = 1865, 1870
for name, m in [("J/psi ", 3097), ("psi'  ", 3686), ("psi'' ", 3774)]:
    d0, dp = m - 2*mD0, m - 2*mDp
    state = "OPEN  -> falls apart" if d0 > 0 else "closed -> must annihilate ccbar"
    print(f"  {name} m = {m}: m - 2m(D0) = {d0:+5d} MeV, m - 2m(D+) = {dp:+5d} MeV   {state}")

hbar, hbarc = 6.582119569e-22, 197.3269804     # MeV s, MeV fm
print("\n  widths as times and lengths:")
for name, g in [("J/psi ", 93e-3), ("psi'  ", 294e-3), ("psi'' ", 27.0), ("rho   ", 147.0)]:
    print(f"  {name} Gamma = {g:7.3f} MeV -> tau = {hbar/g:.2e} s, c*tau = {hbarc/g:8.1f} fm")
print("  the psi'' behaves like an ordinary resonance;"
      " the J/psi lives 2600 proton-crossings")

  J/psi  m = 3097: m - 2m(D0) =  -633 MeV, m - 2m(D+) =  -643 MeV   closed -> must annihilate ccbar
psi'   m = 3686: m - 2m(D0) =   -44 MeV, m - 2m(D+) =   -54 MeV   closed -> must annihilate ccbar
psi''  m = 3774: m - 2m(D0) =   +44 MeV, m - 2m(D+) =   +34 MeV   OPEN  -> falls apart

widths as times and lengths:
J/psi  Gamma =   0.093 MeV -> tau = 7.08e-21 s, c*tau =   2121.8 fm
psi'   Gamma =   0.294 MeV -> tau = 2.24e-21 s, c*tau =    671.2 fm
psi''  Gamma =  27.000 MeV -> tau = 2.44e-23 s, c*tau =      7.3 fm
rho    Gamma = 147.000 MeV -> tau = 4.48e-24 s, c*tau =      1.3 fm
the psi'' behaves like an ordinary resonance; the J/psi lives 2600 proton-crossings

The precursor nobody read: Niu, 1971

Three years before either 1974 experiment, K. Niu and collaborators at Nagoya had already photographed a charmed particle — in an emulsion chamber exposed to cosmic rays on aircraft and balloon flights.

The charm spectrum today

The cc̄ states are called charmonium, and the name is meant literally: the system is a heavy quark and a heavy antiquark orbiting each other, so it can be treated as an atom, with the level structure and spectroscopic labels of hydrogen — $1{}^3S_1$, $2{}^3S_1$, $1{}^1S_0$. Charmed baryons exist too, with one, two or three c quarks; Appendix 3 lists them.

Where this goes next

• §4.10 does all of this a third time, for beauty — a two-arm spectrometer, a narrow state below the open-flavour threshold, and a fourth level where it opens. The pattern in the ThresholdWidth figure above is meant to be reused there.
• §6.5 is the QCD statement of “annihilation is suppressed” that this section has been leaning on since §4.5 without proof.
• §7.9 is GIM done properly, with the loop diagrams and the cancellation written out, and the c → s rule derived from the Cabibbo angle.
• Chapter 8 uses the D mesons themselves: D⁰–D̄⁰ oscillation is the fourth neutral-meson mixing system, and the hardest one to see.