higher-order QCD corrections - The Standard Model and Beyond

q_r^α qrα

e⁻ e⁺

γ G

e⁻ e⁺

Figure 5.5 e⁺e⁻ →q_rαq

¯

_r^α

. Left: the blobs represent the quark hadronization. Right:

higher-order QCD corrections.

quarks turn into jets of hadrons such as pions (low momentum [soft] gluons or quarks may be exchanged between the jets to ensure color neutrality). There are also higher-order QCD corrections, which can be calculated perturbatively. One usually expresses the theoretical and experimental result in terms of the ratio

R(s) = σ(e⁺e⁻→ hadrons)

σ(e⁺e⁻→µ⁺µ⁻) (5.10)

at CM energy√s, where the denominator is the lowest order theoretical expression in (2.232) with a running coupling,σ(e⁺e⁻ →µ⁺µ⁻) = 4πα²(s)/3s. R(s) is convenient because the largest energy dependence cancels in the ratio, and experimentally because the luminosity also cancels. R(s) counts the number of quark colors and flavors, weighted by the quark electric charge-squarede²_r, wheree_r=²₃ for [u, c, t] and−¹3 for [d, s, b]. The lowest-order prediction (i.e., ignoring α_s) is R=N_cP

re²_r, whereN_c is the number of colors and only then_q quarks lighter than√s/2 should be included in the sum. For QCD (Nc= 3) this is

R= 5

3 for [u, d], 6

3 for [uds], 10

3 for [udsc], 11

3 for [udscb]. (5.11) The higher-order QCD corrections have been computed to 4 loops (Baikov et al., 2012).

Neglecting quark masses,⁵ one predicts R=Nc

e²_r

1 + αs

π +cⁿ₂^qαs

π ²

+cⁿ₃^qαs

π ³

+cⁿ₄^qαs

π ⁴

+· · ·

(5.12) fornqquark flavors. The higher-order terms use the value of the runningαsats. Fornq = 5, for example, c⁵₂= 1.40902,c⁵₃=−12.80, andc⁵₄=−80.434. References to quark mass and Z exchange corrections are given in (Patrignani, 2016). The QCD (Nc = 3) prediction is in excellent agreement with the experimental result in Figure 5.6. R also excludes an alternative quark model involving integer electric charges (Nambu and Han, 1974), at least

5Similar to QED, there are infrared singularities associated with both virtual and real gluons as their energy approaches zero, as well as mass(or collinear)singularities that occur in the limit of a massless quark due to gluons radiated parallel to the quark. These can be shown to cancel under realistic conditions using dimensional regularization or by introducing a fictitious gluon mass. (The latter is only possible for diagrams not involving the non-abelian gauge vertices, since it would violate gauge invariance). See (Field, 1989; Salam, 2010a) for detailed discussions.

Figure 5.6

Experimental data on

R(s) compared with the lowest order (dashes) and

three-loop (solid) QCD predictions. There are steps at the

s,c, and b

thresholds, given approximately by the locations of the

(s¯

s), J/ψ

(c¯

c), and Υ (b

¯

b) resonances.

The perturbative prediction works extremely well above a few GeV, provided one includes the threshold resonances. At high energies,

boson exchange strongly dominates over one photon exchange. Plot courtesy of the Particle Data Group (Pa-trignani, 2016).

under the assumption that the quarks can be treated as pointlike (Problem 5.1). Further tests of QCD in e⁺e⁻ annihilation are reviewed in (Kluth, 2006).

Other evidence for color and QCD includes the ratio of nonleptonic and leptonic decays of theW, sinceW⁻→qq¯, whereqq¯=du, s¯ ¯ccounts the number of colors, while the leptonic decays into`⁻ν¯_`, `=e, µ, τ, do not. In particular, the branching ratio

B W⁻ →e⁻ν⁻

= Γ (W⁻→e⁻ν⁻)

Γ (W⁻→q¯q) + Γ (W⁻ →`⁻ν)¯ ' 1

3 + 2Nc −−−→_N

c=3 11% (5.13) (up to small corrections from QCD, fermion mass, quark mixing, etc.), in agreement with the experimental value∼10.7%.

Another probe is theDrell-Yanprocess in which

p⁽⁻⁾p →`⁺`⁻+ hadrons, `⁻=e⁻, µ⁻, τ⁻ (5.14) at large (p`⁺+p_`−)² 1 GeV². This is dominated by the qq¯annihilation through a γ or Z, as shown in Figure 5.7, and can be thought of as the inverse to e⁺e⁻ → qq. The¯ q_rα and ¯q_r^β can each be in one of three color states, but only the combination in which α=β can contribute, leading to a cross section 1/Nccompared to what would be expected without color, in agreement with observation. To see this, consider p¯p scattering in the approximation of considering the valence quarks only. Using the baryon wave function in (5.1),

σcolor = 3×2×2× 1

√6 2

× 1

√6 2

σno-color= 1

3σno-color, (5.15)

q γ, Z¯ q

¯ p p

−

π⁰ q

γ γ

Figure 5.7

Left: Drell-Yan process, ¯

pp→γ, Z →`⁻`⁺

. Right: The

π⁰→

2γ decay.

where the 3 represents the three colors that can annihilate, and 2² represents the color assignments of the remaining quarks, all of which add incoherently.⁶

One can useSU(2)×SU(2) chiral symmetry to show that the chiral anomaly associated with the triangle diagram in Figure 5.7 (with π⁰ coupling to the axial isospin generator T_R³ −T_L³) dominates the π⁰ → 2γ decay amplitude in the m_π → 0 limit (Adler, 1969;

Donoghue et al., 2014). This again counts the number of quark colors, so that Γ π⁰→2γ

∼ Nc

² α²m³_π0

32π³f_π² = 7.75(2) Nc

3 ²

eV, (5.16)

where fπ= 130.5(1) MeV is the pion decay constant, which is associated with the sponta-neous breaking of the chiral symmetry and is measured inπ⁺→µ⁺νdecay. The prediction forNc = 3 is increased to∼8.10 eV by chiral-breaking corrections (Bernstein and Holstein, 2013), consistent with the experimental value, 7.6(3) eV.

5.3 SIMPLE QCD PROCESSES

In this section we sketch the derivation of some simple QCD processes at tree level. Some of the calculations are similar to the QED calculations in Chapter 2, except that one has to properly take the color factors into account. Others involve the gluon self-interactions, which have no QED analog. Processes involving non-abelian vertices are extremely tedious to carry out by hand, and are best handled by specialized computer algebra programs (see the list of websites in the bibliography and the example notebooks on the book website).

However, we will illustrate one relatively simple example. Higher-order calculations involve all of the subtleties of gauges and ghost loops (the latter may even appear in tree-level calculations involving external gluons), which are treated in standard field theory texts.

Color Identities

The calculations are greatly simplified by the use of certain color identities listed in Table 5.1 for the fundamental representation matricesLⁱ₃=λⁱ/2 (denoted in this section by Lⁱ) and theSU(3) structure constantsf_ijkdefined in Tables 3.1 and 3.2. They follow easily from or are special cases of the identities given in Sections 3.1.2 and 3.1.3 and in the Problems in Chapter 3.

6An equivalent derivation is to simply average over the Nc colors of the interacting qand ¯q, so that σ∝(1/Nc)²PNc

α,β=1δ^αβ = 1/Nc.

TABLE 5.1 SU(3) color identities^aforLⁱ≡λⁱ/2 andfijk. LⁱLⁱ=⁴₃I Tr LⁱL^j

= ¹₂δ^ij f_ijkf_ijm= 3δkm f_ijkf_ijk = 24

f_ijmf_klmf_ijnf_kln= 72 f_ijmf_klmf_iknf_jln= 36 Tr LⁱL^jL^k

=¹₄(dijk+if_ijk) Tr LⁱL^jL^k

if_ijm=−³4δ^km Tr LⁱL^jL^k

if_ijk=−6 Tr LⁱL^jLⁱL^k

=−12¹δ_jk Tr LⁱL^jLⁱL^j

=−²3

Tr LⁱL^jL^jLⁱ

=¹⁶₃ Tr LⁱL^j

Tr LⁱL^j

= 2 filmdjmk−fimkdjlm+fijmdmlk= 0

f_ilmf_jmk−f_imkf_jlm+f_ijmf_mlk= 0 (Jacobi identity)

aLⁱLⁱ≡P

iLⁱLⁱ,Iis the 3×3 identity matrix, and the indices run from 1 to 8. There is no distinction between upper and lower indices.

qqandqq¯Scattering

qq and q¯q scattering via gluon exchange are very much like the simple QED processes described in Section 2.8. Here we will neglect the quark masses for simplicity, but it is straightforward to include them (as would be necessary for the production of a heavy quark, such asu¯u→tt). The process¯ qrβq¯rα →qsγq¯sδ, whereα, β, γ, andδ are color indices and r6=sare flavor indices, proceeds through anschannel gluon, as shown in the first diagram in Figure 5.8. The only differences compared with the QED processe⁻e⁺ →ff¯in (2.225) on page 46 andFigure 2.15 are that −e²Q_f →g_s² and that there are color factors on the vertices and gluon propagator, as shown inFigure 4.1, yielding

M_{f i}= (−ig_su¯3γ_µL^k_γδv4)

−ig^µρδ^ik s

(−ig_s ¯v2γ_ρLⁱ_αβu1)

= ig_s²

s Lⁱ_αβLⁱ_γδ(¯u₃γ_µv₄ v¯₂γ^µu₁).

(5.17)

The calculation of the spin-average cross section proceeds as in e⁻e⁺ → ff, Equa-¯ tion (2.227), except it is now convenient to do a color average as well, in which one averages (sums) over initial (final) quark colorsα= 1· · ·3. Similarly, in processes involving external gluons one averages (sums) over initial (final) gluon color indicesi= 1· · ·8. Neglecting the

δ^ik

qβ(p1) q_γ(p₃)

¯ qα(p2)

¯ qδ(p4)

Lⁱ L^k

δ^ik q_β(p₁)

q_γ(p₃)

¯ q_α(p₂)

¯ q_δ(p₄) L^k

− Lⁱ

Figure 5.8

Diagrams for

q_rβq

¯

_rα→q_sγq

¯

_sδ

, where

and

are flavor indices. Only the

diagram on the left contributes for

=

quark masses, the expression 2Q²_fe⁴(t²+u²)/s², derivable from (2.227), is replaced by scattering in (2.233)–(2.235). The first two terms are obtained by the replacement e⁴ → 2g⁴_s/9, just as in (5.18). However, the third (interference) term now has the color factor

The spin and color-averaged squared matrix elements for a number of 2 → 2 QCD processes, neglecting masses, are listed inTable 5.2.More extensive listings, including mass effects and extensions to supersymmetry, may be found in (Patrignani, 2016).

TABLE 5.2 Spin and color-averaged squared amplitudes|M¯|²/g_s⁴ for various QCD subprocesses,^acharacterized by kinematic subprocess invariantss,t, andu.

|M¯|²/g⁴_s 90^◦

ar6=swhen flavor indices are given. The last two processes involve the production of a hypothetical spin-0 color-tripletq0, such as one encounters in supersymmetry. Masses are neglected. The last column is the numerical value for CM scattering angle 90^◦, wheret=u=−s/2 (neglecting masses). Expanded from (Combridge et al., 1977; Barger and Phillips, 1997).

GG→q₀q¯₀

To illustrate a non-trivial non-abelian vertex, consider the processGG→q0q¯0, where q0 is a hypothetical spin-0 color triplet, such as a scalar quark in supersymmetry. There are four tree-level diagrams, as shown in Figure 5.9 Using the vertex factors from Figure 4.1, the

k, σ vectors. The straightforward way to calculate|M¯|²would be to first take the absolute square and then use (2.121) on page 28 for the gluon polarization sums. However, this would be extremely tedious. The calculation would be simplified if the second term in (2.121) did not contribute, but this requires the calculation to be done in a gauge invariant way (cf., the discussion of Compton scattering in Section2.8). In particular, for a non-abelian theory one must include the negative contribution of fictitious ghost pairs (Sterman, 1993, Section 8.5; Peskin and Schroeder, 1995, Section 17.4). Although this is straightforward, it can be avoided by introducing explicit expressions for the polarization vectors, just as we did in (2.151).

For theq₀q¯₀final state it is simpler to calculate the amplitudes rather than their absolute squares, analogous to the fermion helicity calculations in Section2.9. The four-momenta in the CM frame are

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