The Short Distance Regime
The short distance regime,|Q| 1 GeV, has weak coupling,gs1 (asymptotic freedom).
It can therefore be described in terms of point-like quarks and gluons with interactions that can be calculated in perturbation theory. For example, in adeep inelastic scatteringprocess, e−p→e−X, the final e− is observed but the final hadronic states, represented byX, are not observed and are summed over. (Such processes are known as inclusive, as opposed to exclusive ones such as e−p→ e−p+ 3π or e−p→ e−∆+ involving a definite final state.) For Q2 ≡ −q2 1 GeV2 the process is described to first approximation by the photon interacting with a point-like quark, as shown in Figure 5.4. Other parts of the diagram, involving the distribution of quarks within the proton, and the process of the remaining quarks in the proton and the scattered quark turning into hadrons (hadronization), are non-perturbative effects. The concept of describing inclusive sums over final states of hadrons in terms of perturbative calculations involving quarks and gluons is known asquark-hadron duality(e.g., Melnitchouk et al., 2005).
5.1 THE QCD LAGRANGIAN
The quark fields are denoted qrα, whereα= 1,2,3 or R,G,B is the gauged color andr= u, d, s, c, b, tis the (ungauged) flavor index. An alternate notation isuα=quα, etc. The Dirac indices are suppressed. The quarks transform under the fundamental (3) representation of SU(3), and are non-chiral,LiL =LiR = λ2i. The Dirac adjoint field transforms as a 3∗ and is written ¯qαr. There are 8 Hermitian gauge fields (gluons),Gi=Gi†, i= 1· · ·8. The QCD
γ(q)
p e−(k)
X1
X2
e−(k)
Figure 5.4
Deep inelastic scattering. The interaction between the lepton
e−and the quark can be described perturbatively by one photon exchange. However, the distri-bution of quarks in the proton, and the hadronization of the remaining and scattered quarks (the blobs) into unobserved hadrons
X=
X1+
X2are non-perturbative.
Lagrangian density is LQCD =−1
4GiµνGiµν +X
r
¯
qrαiD6 αβqrβ−X
r
mrq¯αrqrα+θQCD
32π2gs2GiµνG˜iµν, (5.2) where the field strength tensor is
Giµν =∂µGiν−∂νGiµ−gsfijkGjµGkν. (5.3) The quark gauge covariant derivative is
Dµβα ≡(Dµ)αβ=∂µδαβ+igs
√2Gµβα , (5.4)
where
Gαβ = Gβα†
=
8
X
i=1
Giλiαβ
√2, (5.5)
with Gαα = 0, represents the gluon field in tensor or matrix notation. Themr in (5.2) are thecurrentquark masses. They are actually generated by spontaneous symmetry breaking in the full standard model (including the chiral electroweak part), but can be considered as bare masses when considering QCD alone. Without loss of generality (for QCD) they can be taken to be real, nonnegative, and diagonal in flavor.
The interaction terms in (5.3) and (5.4) are the same for all six flavors, so they are invariant under a global chiralU(6)×U(6) flavor symmetry. However, mc, mb,andmtare large compared to the typical scale of the strong interactions, Λ ∼ (200−300) MeV, so the symmetry is badly broken except at very high energy. More useful is an approximate SU(3) flavor symmetry in the limit mu ∼md ∼ms, which is valid at the 25% level, and the even better (1%) SU(2) isospin symmetry, which holds in the limitmd ∼mu.2 These symmetries are enhanced to become chiral form= 0; e.g., formu=md= 0 the continuous symmetries ofLQCD become
SU(3)color
| {z }
gauge
×SU(2)×SU(2)
| {z }
global
×U(1)×U(1)
| {z }
globalB,BA
, (5.6)
2In fact,mu,md, andmsare not really degenerate compared to each other. However,muandmdare both very small compared to Λ, so they are “degenerate” in the sense thatmu/Λ∼md/Λ∼0. Similarly, SU(3) holds approximately becausems∼100 MeV (mu,d) is smaller than Λ though non-negligible.
while theSU(2)×SU(2) becomesSU(3)×SU(3) ifms→0 as well. TheU(1) factors are separateLandRchiral baryon numbers. The sum (difference) of the generators corresponds to baryon (“axial baryon”) number B(BA). However, BA is not a good symmetry of the strong interactions, and its unsuccessful prediction in the quark model is known as theaxial U(1)A problem. Its resolution by non-perturbative effects in QCD will be commented on in Section 5.8.3.
Another difficulty is the strongCP problem, which refers to the last term in (5.2). The strong interactions are observed to be reflection invariant (i.e., parity P is conserved), as well as invariant under charge conjugation (C), time reversal (T), and the products CP and CP T. The first three terms in (5.2) respect these symmetries. However, it is possible to add the finalstrongCP term toLQCD, where θQCD is a dimensionless constant, and
G˜iµν ≡1
2µνρσGiρσ (5.7)
is the dual field strength tensor. (G and ˜G are related by exchanging the analogs of the electric and magnetic fields.) The strongCP term is gauge invariant and does not spoil the renormalizability of QCD. However, for θQCD 6= 0 it violates P, T, and CP symmetries, and stringent experimental limits on the electric dipole moment of the neutron require
|θQCD|.10−11−10−10(e.g., Kim and Carosi, 2010). For pure QCD it is possible to simply impose these symmetries, i.e., to takeθQCD = 0. However, as will be discussed inChapter 10, this becomes problematic in the context of the full standard model, which has other sources ofCP violation.
5.2 EVIDENCE FOR QCD
QCD is the unique renormalizable field theory consistent with the observations that ex-isted by ca. 1970, and since that time there has not been any serious competing theory.3 Nevertheless, it is interesting to review some of the evidence for the ingredients of QCD.
Spin-12 Quarks and Spin-1 Gluons
The first evidence for spin-12 quarks was the success of the constituent quark model, which successfully classified a large number of hadrons in terms of three flavors of quarks. The spin-12 nature is essential for this classification, as is evident from the construction of the nucleons out of three quarks or of the spin-1 ρ from a qq¯ pair in an S-wave. The very simple description of the approximate flavor symmetries of the strong interactions and their breaking in the quark model, i.e., (3.113) on page 110, also provided evidence.
By 1970, dynamical evidence emerged from the deep-inelastic scattering processe−p→ e−X, withQ2 =−q2 1 GeV2, shown inFigure 5.4 and to be described in Section 5.5.
The rate observed at the Stanford Linear Accelerator Center (SLAC) at largeQ2was much larger than would be expected for a “big fuzzy” proton, calling for the existence of point-like partonconstituents of the proton, reminiscent of the discovery of the atomic nucleus in the Rutherford experiment. In principle, the partons could be spin-0, spin-12, or higher.
However, the angular distribution of the scattered electron established that they are
spin-1
2 (the Callan-Gross relation), consistent with being quarks. This was later confirmed in
3In fact, prior to the development of QCD many physicists seriously considered abandoning the ideas of field theory or of any fundamental dynamical equations for the strong interactions, in favor of thebootstrap, which postulated that there was a unique S-matrix consistent with the ideas of unitarity, analyticity, crossing, etc. (See, e.g., Eden et al., 1966; Collins, 1977).
the distributions observed in deep inelasticµ±N and (−)ν N scattering and in their relative strengths.
Additional evidence emerged a few years later in the study of e+e− → hadrons. The observed cross section falls as 1/s, wheresis the square of the CM energy. This is consistent with the behavior expected for the production of point-like quarks (up to higher-order QCD corrections), as in (2.232) on page 46, and not with the more rapid falloff expected without such constituents. These later hadronize into collimated clusters of hadrons known asjets. The spin-12 nature was established by the observed 1 + cos2θangular distribution of (2.231) for the jets, as opposed to the sin2θ distribution predicted for spin-0 (Problem 2.23). On the other hand, the positive evidence for quarks was apparently contradicted by the non-observation of isolated quarks. The resolution of that conflict had to await the development of QCD and the notion of infrared slavery.
The first direct evidence for spin-1 gluons came from the observation of distinct 3-jet events frome+e−→q¯qGand of the planar broadening of events in which the third jet could not be resolved, by the TASSO and other collaborations at PETRA (DESY) in 1979 (see, e.g., Wu, 1984; Bethke, 2007). Another type of compelling evidence is indirect, i.e., the observed asymptotic freedom of the strong interactions requires a non-abelian gauge theory.
Other advantages of color octet gluons were especially emphasized in (Fritzsch et al., 1973).
Evidence for Color
The observed hadrons are all color singlets. Nevertheless, with the benefit of hindsight, the color quantum number was already needed in the original quark model. That is because the quark assignments for the baryons and hyperons (baryons involving an s quark) were totally symmetric in the flavor, spin, and space indices. In particular, the Ω−, which was successfully predicted by theSU(3) model (Section 3.2.3), was interpreted as
|Ω−i=|s↑s↑s↑i, (5.8)
where the arrows all represent spin-up with respect to a reference axis. The Ω− is therefore symmetric in flavor (all s quarks), spin (all spins in the same direction), and in space indices (the orbital angular momenta are zero). However, this violates the spin and statistics theorem, which follows from the union of relativity and quantum mechanics (see, e.g., Streater and Wightman, 2000), and which requires that all physical spin-12 states should be antisymmetric. This fundamental difficulty with the quark model is easily resolved by the introduction of the color quantum number, under the assumption that the Ω− and other baryon/hyperon states are color singlets, since the projection of 3×3×3 onto the singlet is totally antisymmetric, as in (5.1),
|Ω−i ∝αβγ|s↑αs↑βs↑γi. (5.9) Thus, the color quantum number was needed even before the development of QCD,4, where it played the additional role of a gauge quantum number.
There are other tests based on counting the number of colors that contribute to an amplitude or rate. The leading diagrams fore+e− →hadrons are shown inFigure 5.5.At short enough distance, one may regard the process as first producing quark qrα and its antiquark, which may be computed perturbatively, followed by hadronization, in which the
4An apparently alternative resolution of the statistics problem, parastatistics (Greenberg, 2008), is in fact equivalent to the existence of the color quantum number.
γ
¯ qrα qrα
e− e+
γ G
e− e+
G
γ
e− e+
Figure 5.5 e+e− →qrαq