Back to QCD: asymptotic freedom .1 One loop calculation

The reader will of course have realized, some time back, that the quantityβ0

introduced in (15.3) must be precisely the coefficient of α²_s in the one-loop contribution to theβ-function of QCD defined by

βs= μ²∂αs

∂μ²

||

||_{fixed bareα}

s

; (15.47)

that is to say,

βs(one loop) =−β0α²_s (15.48) with

β0= 33−2Nf

12π . (15.49)

ForNf≤16 the quantityβ0ispositive, so that the sign of (15.48)) is opposite to that of the QED analogue, equation (15.36). Correspondingly, (15.44) is replaced by

αs(|q²|) = αs(μ²)

[1 +αs(μ²)β0ln(Q²/μ²)], (15.50) whereQ²=|q²|.² Then replacingαsin (15.1) by (15.50) leads to (15.7).

Thus in QCD the strong coupling runs in the opposite way to QED, be-coming smaller at large values ofQ² (or small distances) – the property of asymptotic freedom. The justly famous result (15.49) was first obtained by Politzer (1973), Gross and Wilczek (1973), and ’t Hooft. ’t Hooft’s result, announced at a conference in Marseilles in 1972, was not published. The published calculation of Politzer and of Gross and Wilczek quickly attracted enormous interest, because it immediately offered a way to understand how the successful parton model could be reconciled with the undoubtedly very strong binding forces between quarks. The resolution, we now understand, lies in quite subtle properties of renormalized quantum field theory, involving first the exposure of ‘large logarithms’, then their re-summation in terms of the running coupling, and of course the crucial sign of theβ-function. Not only did the result (15.49) explain the success of the parton model: it also, we repeat, opened the prospect of performing reliable perturbative calcula-tions in a strongly interacting theory, at least at high Q². For example, at sufficiently high Q², we can reliably compute theβ function in perturbation theory. The result of Politzer and of Gross and Wilczek, when combined with

2Except that in (15.50)αs is evaluated at largespacelikevalues ofq², whereas in (15.7) it is wanted at large timelikevalues. Readers troubled by this may consult Peskin and Schroeder (1995) section 18.5. The difficulty is evaded in the approach of section 15.6 below.

15.3. Back to QCD: asymptotic freedom 125

FIGURE 15.3

q¯q vacuum polarization correction to the gluon propagator.

the motivations for a colour SU(3) group discussed in the previous chapter, led rapidly to the general acceptance of QCD as the theory of strong interactions, a conclusion reinforced by the demonstration by Coleman and Gross (1973) that no theory without Yang-Mills fields possessed the property of asymptotic freedom.

In section 11.5.3 we gave the conventional physical interpretation of the way in which the running of the QED coupling tends to increase its value at distances short enough to probe inside the screening provided by e⁺e⁻ pairs

−1

(|q|⁻¹ << m_e ). This vacuum polarization screening effect is also present in (15.49) via the term−^2N_12π ^f , the value of which can be quite easily understood.

It arises from the ‘q¯q’ vacuum polarization diagram of figure 15.3, which is

− [2]

precisely analogous to the e⁺e diagram used to calculate Π¯_γ (q²) in QED.

The only new feature in figure 15.3 is the presence of the ^λ ₂ -matrices at each vertex. If ‘a’ and ‘b’ are the colour labels of the ingoing and outgoing gluons, the λ₂ -matrix factors must be

3 (λ_a ) (λ_b )

E (15.51)

2 _αβ 2 _βα

α,β=1

since there are no free quark indices (of type α, β) on the external legs of the diagram. It is simple to check that (15.51) has the value ¹ ₂ δ_ab (this is, in fact, the way the λ’s are conventionally normalized). Hence for one quark flavour we expect ‘α/3π’ to be replaced by ‘αs/6π’, in agreement with the second term in (15.49).

The all-important, positive, first term must therefore be due to the gluons.

The one-loop graphs contributing to the calculation of β0 are shown in figure 15.4. They include figure 15.3, of course, but there are also, characteristically, graphs involving the gluon self-coupling which is absent in QED, and also (in covariant gauges) ghost loops. We do not want to enter into the details of the calculation of β(αs) here (they are given in Peskin and Schroeder 1995, chapter 16, for example), but it would be nice to have a simple intuitive picture of the

‘antiscreening’ result in terms of the gluon interactions, say. Unfortunately no fully satisfactory simple explanation exists, though the reader may be

inter-15.3. Back to QCD: asymptotic freedom 127 is a measure of the scale at whichαsreally does become ‘strong’. The extrac-tion of a value of ΛQCDis a somewhat complicated matter, as we shall briefly indicate in the following section, but a typical value is in the region of 200 MeV. Note that this is a distance scale of order (200 MeV)⁻¹ ∼ 1 fm, just about the size of a hadron – a satisfactory connection.

15.3.2 Higher-order calculations, and experimental compar-ison

So far we have discussed only the ‘one-loop’ calculation ofβ(αs). The general perturbative expansion for βs can be written as

βs(αs) =−β0α²_s−β1α³_s−β2α⁴_s+. . . (15.55) whereβ0is the one-loop coefficient given in (15.49),β1 is the two-loop coeffi-cient, and so on. β1 was calculated by Caswell (1974) and Jones (1974), and has the value

β1= 153−19Nf

24π² . (15.56)

The three-loop coefficientβ2, obtained by Tarasovet al. (1980) and by Larin and Vermaseren (1993), is

β2= 77139−15099Nf+ 325N_f²

3456π² . (15.57)

The four-loop coefficientβ3was calculated by van Ritbergenet al. (1997) and by Czakon (2005); we shall not give it here. A technical point to note is that while β0 and β1 are independent of the scheme adopted for renormalization (see appendix O), the higher-order coefficients do depend on it; the value (15.57) is in the widely used MS scheme. Likewise, ΛQCD will be scheme-dependent (see appendix O), and the value Λ_MS will be used here (the ‘QCD’

now being understood).

Only in the one-loop approximation for βs can an analytic solution of (15.47) be obtained. However, a useful approximate solution can be found iteratively, as follows. Consider the two-loop version of (15.54), namely

ln(Q²/Λ²_MS) =−

[ dαs

β0α²_s+β1α³_s. (15.58) Expanding the denominator and integrating gives

ln(Q²/Λ²_MS) = 1 β0αs

+ b1

β0

lnαs+C, (15.59) where b1 = β1/β0 and C is a constant. In the MS scheme, C is given by C= (b1/β0) lnβ0. Then the equation forαsis

L= 1 β0αs +b1

β0lnβ0αs, (15.60) 126 15. QCD II: Asymptotic Freedom, the Renormalization Group

FIGURE 15.4

Graphs contributing to the one-loop β function in QCD. The curly line rep-resents a gluon, a dotted line a ghost (see section 13.3.3 ) and a straight line a quark.

ested to consult Hughes (1980, 1981) and Nielsen (1981) for a ‘paramagnetic’

type of explanation, rather than a ‘dielectric’ one.

Returning to (15.50), we note that the equation effectively provides a pre-diction of α_s at any scale Q², given its value at a particular scale Q² = μ² , which must be taken from experiment. The reference scale is now normally taken to be the Z⁰ mass; the value αs(m2_Z) then plays the role in QCD that α ∼ 1/137 does in QED.

Despite appearances, equation (15.50) does not really involve two param-eters – after all, (15.47) is only a first-order differential equation. By intro-ducing

ln Λ² _QCD = ln μ² − 1/(β0αs(μ²)), (15.52) equation (15.50) can be rewritten (problem 15.3) as

αs(Q²) = 1

. (15.53)

β₀ ln(Q²/Λ² _QCD) Equation (15.53) is equivalent to (cf (15.30))

[ ∞ dαs

ln (

Q²/Λ² )

= (15.54)

QCD αs(Q²) βs(one loop)

with βs(one loop) = −β0α²_s . Λ_QCD is therefore an integration constant, rep-resenting the scale at which α_s would diverge to infinity (if we extended our calculation beyond its perturbative domain of validity). More usefully, Λ_QCD

126 15. QCD II: Asymptotic Freedom, the Renormalization Group

FIGURE 15.4

Graphs contributing to the one-loopβ function in QCD. The curly line rep-resents a gluon, a dotted line a ghost (see section 13.3.3 ) and a straight line a quark.

ested to consult Hughes (1980, 1981) and Nielsen (1981) for a ‘paramagnetic’

type of explanation, rather than a ‘dielectric’ one.

Returning to (15.50), we note that the equation effectively provides a pre-diction ofαs at any scale Q², given its value at a particular scale Q² =μ², which must be taken from experiment. The reference scale is now normally taken to be the Z⁰ mass; the value αs(m²_Z) then plays the role in QCD that α∼1/137 does in QED.

Despite appearances, equation (15.50) does not really involve two param-eters – after all, (15.47) is only a first-order differential equation. By intro-ducing

ln Λ²_QCD= lnμ²−1/(β0αs(μ²)), (15.52) equation (15.50) can be rewritten (problem 15.3) as

αs(Q²) = 1

β0ln(Q²/Λ²_QCD). (15.53) Equation (15.53) is equivalent to (cf (15.30))

ln(

Q²/Λ²_QCD)

= [ ∞

αs(Q²)

dαs

βs(one loop) (15.54) withβs(one loop) =−β0α²_s. ΛQCD is therefore an integration constant, rep-resenting the scale at whichαs would diverge to infinity (if we extended our calculation beyond its perturbative domain of validity). More usefully, ΛQCD

15.3. Back to QCD: asymptotic freedom 127

is a measure of the scale at which αs really does become ‘strong’. The extrac-tion of a value of ΛQCD is a somewhat complicated matter, as we shall briefly indicate in the following section, but a typical value is in the region of 200 MeV. Note that this is a distance scale of order (200 MeV)⁻¹ ∼1 fm, just about the size of a hadron – a satisfactory connection.

15.3.2 Higher-order calculations, and experimental compar-ison

So far we have discussed only the ‘one-loop’ calculation of β(αs). The general perturbative expansion for βs can be written as

βs(αs) = −β0α² _s− β1α³ _s− β2α⁴ _s + . . . (15.55) where β0 is the one-loop coefficient given in (15.49), β1 is the two-loop coeffi-cient, and so on. β1 was calculated by Caswell (1974) and Jones (1974), and has the value

153 − 19Nf

β₁ = . (15.56)

24π²

The three-loop coefficient β2, obtained by Tarasov et al. (1980) and by Larin and Vermaseren (1993), is

77139 − 15099Nf + 325N_f²

β2 = . (15.57)

3456π²

The four-loop coefficient β3 was calculated by van Ritbergen et al. (1997) and by Czakon (2005); we shall not give it here. A technical point to note is that while β0 and β1 are independent of the scheme adopted for renormalization (see appendix O), the higher-order coefficients do depend on it; the value (15.57) is in the widely used MS scheme. Likewise, Λ_QCD will be scheme-dependent (see appendix O), and the value Λ_MS will be used here (the ‘QCD’

now being understood).

Only in the one-loop approximation for β_s can an analytic solution of (15.47) be obtained. However, a useful approximate solution can be found iteratively, as follows. Consider the two-loop version of (15.54), namely

[ dαs

ln(Q²/Λ² _MS) = − . (15.58)

β0α² _s + β1α³ _s Expanding the denominator and integrating gives

b1

ln(Q²/Λ² ) = 1

+ ln α_s + C, (15.59)

MS β0α_s β₀

where b1 = β1/β0 and C is a constant. In the MS scheme, C is given by C = (b1/β0) lnβ0. Then the equation for αs is

1 b1

L = + ln β0αs, (15.60)

β0αs β0

15.4. σ(e⁺e⁻ →hadrons) revisited 129

FIGURE 15.5

Comparison between measurements ofαsand the theoretical prediction, as a function of the energy scaleQ(Bethke 2009). (See color plate I.)

power series inαs,

σ(e⁺e⁻ →hadrons) =σpt(Q²)

| 1 +

E∞ n=1

cn(Q²/μ²)

(αs(μ²) π

)ⁿ|

, (15.63) whereμis the renormalization scale. (A similar expansion can be written for many other physical quantities too.) The coefficients fromc2onwards depend on the renormalization scheme (see appendix O), and are usually quoted in the MS scheme. c1 is the leading order (LO) coefficient, and we already know that c1 = 1 from (15.1). c2 is the next-to-leading (NLO) coefficient; c2(1) was calculated by Dine and Sapirstein (1979), Chetyrkinet al. (1979) and by Celmaster and Gonsalves (1980), and has the value 1.9857−0.1152Nf. The next-to-next-to-leading (NNLO) coefficient c3(1) was calculated by Samuel and Surguladze (1991) and by Gorishnii et al. (1991), and is equal to -12.8 for five flavours. The N3LO coefficientc4(1) (which requires the evaluation of some twenty thousand diagrams) may be found in Baikovet al. (2008) and Baikovet al. (2009).

The physical cross sectionσ(e⁺e⁻→hadrons) must be independent of the renormalization scaleμ², and this would also be true of the series in (15.63) if an infinite number of terms were kept: the μ²-dependence of the coefficients 128 15. QCD II: Asymptotic Freedom, the Renormalization Group

where we have defined L = ln(Q²/Λ² MS). In first approximation, one sets b1

to zero and finds αs = (1/β0L) as before. To obtain the next approximation, we set α_s = (1/β0L) in the b₁ term of (15.60), and solve for α_s to first order in b1. This gives (problem 15.4 (a))

1 1

α_s = − β₁ ln L. (15.61)

β³L² β0L ₀

Problem 15.4 (b) carries the calculation to the three-loop stage.

The current world average value of αs(m2_Z) is (Bethke 2009)

αs(m2_Z) = 0.1184 ± 0.0007. (15.62) The remarkable precision of this number represents extraordinary consistency among the many methods used to determine it³, which include deep inelastic scattering, electroweak fits, e⁺e⁻ → jets, and lattice calculations (see chapter 16). If (15.62) is used to determine Λ_MS from (15.61), one finds Λ_MS = 231 MeV; using the 3-loop formula of problem 15.4 (b) gives Λ_MS = 213 MeV (Bethke 2009).

These values of Λ_MS are for Nf = 5, appropriate for the Z⁰ mass region, well above the b threshold. As Q² runs to smaller values, and a quark mass threshold is crossed, N_f changes by one unit, and so correspondingly do the coefficients β0, β1, . . .. Physical quantities must however be continuous across a quark threshold. This requires that the values of α_s above and below that threshold satisfy certain matching conditions (Rodrigo and Santamaria 1993, Bernreuther and Wetzel 1982, Chetyrkin et al. 1997). These are satisfied by allowing Λ_MS to depend on N_f . At one and two loop order, the matching

con-(N_f −1) (N_f )

dition is simply αs = αs , which can be straightforwardly implemented

(N_f −1) (N_f −1)

in terms of Λ_MS and Λ_MS . In higher orders the matching conditions contain additional terms, which are required at (n−1)-loop order for an n-loop calculation of αs.

Figure 15.5 shows a summary (Bethke 2009) of measurements of α_s as a function of the energy scale Q, compared with the QCD prediction. The latter is evaluated in 4-loop approximation, using 3-loop threshold matching conditions at the masses m_c = 1.5 GeV and m_b = 4.7 GeV. The agreement is perfect, a triumph for both experiment and theory.

15.4 σ(e

⁺

e

⁻

→ hadrons) revisited

We may now return to the physical process which originally motivated this extensive detour. The perturbative corrections to σpt(Q²) are expressed as a

3With the exception of a long-standing systematic difference: results from structure functions prefer a smaller value of αs(m2_Z) than most of the others.

128 15. QCD II: Asymptotic Freedom, the Renormalization Group where we have definedL = ln(Q²/Λ²_MS). In first approximation, one setsb1

to zero and findsαs= (1/β0L) as before. To obtain the next approximation, we setαs= (1/β0L) in theb1 term of (15.60), and solve forαs to first order inb1. This gives (problem 15.4 (a))

αs= 1

β0L − 1

β₀³L²β1lnL. (15.61) Problem 15.4 (b) carries the calculation to the three-loop stage.

The current world average value ofαs(m²_Z) is (Bethke 2009)

αs(m²_Z) = 0.1184±0.0007. (15.62) The remarkable precision of this number represents extraordinary consistency among the many methods used to determine it³, which include deep inelastic scattering, electroweak fits, e⁺e⁻→jets, and lattice calculations (see chapter 16). If (15.62) is used to determine Λ_MS from (15.61), one finds Λ_MS = 231 MeV; using the 3-loop formula of problem 15.4 (b) gives Λ_MS = 213 MeV (Bethke 2009).

These values of Λ_MS are forNf = 5, appropriate for the Z⁰ mass region, well above the b threshold. AsQ² runs to smaller values, and a quark mass threshold is crossed,Nf changes by one unit, and so correspondingly do the coefficientsβ0, β1, . . .. Physical quantities must however be continuous across a quark threshold. This requires that the values ofαs above and below that threshold satisfy certain matching conditions (Rodrigo and Santamaria 1993, Bernreuther and Wetzel 1982, Chetyrkinet al. 1997). These are satisfied by allowing Λ_MSto depend onNf. At one and two loop order, the matching con-dition is simplyα^(Ns ^f−1)=α^(Ns ^f), which can be straightforwardly implemented in terms of Λ^(N_MS^f−1) and Λ^(N_MS^f−1). In higher orders the matching conditions contain additional terms, which are required at (n−1)-loop order for ann-loop calculation ofαs.

Figure 15.5 shows a summary (Bethke 2009) of measurements of αs as a function of the energy scale Q, compared with the QCD prediction. The latter is evaluated in 4-loop approximation, using 3-loop threshold matching conditions at the massesmc= 1.5 GeV andmb= 4.7 GeV. The agreement is perfect, a triumph for both experiment and theory.

15.4 σ(e

⁺

e

⁻

→ hadrons) revisited

We may now return to the physical process which originally motivated this extensive detour. The perturbative corrections toσpt(Q²) are expressed as a

3With the exception of a long-standing systematic difference: results from structure functions prefer a smaller value ofαs(m²_Z) than most of the others.

15.4. σ(e⁺e⁻ → hadrons)revisited 129

FIGURE 15.5

Comparison between measurements of αs and the theoretical prediction, as a function of the energy scale Q (Bethke 2009). (See color plate I.)

power series in αs,

| ∞ (αs(μ²))ⁿ| σ(e⁺ e ⁻ → hadrons) = σpt(Q²) 1 + E

cn(Q²/μ²) , (15.63)

n=1 π

where μ is the renormalization scale. (A similar expansion can be written for many other physical quantities too.) The coefficients from c₂ onwards depend on the renormalization scheme (see appendix O), and are usually quoted in the MS scheme. c1 is the leading order (LO) coefficient, and we already know that c1 = 1 from (15.1). c2 is the next-to-leading (NLO) coefficient; c2(1) was calculated by Dine and Sapirstein (1979), Chetyrkin et al. (1979) and by Celmaster and Gonsalves (1980), and has the value 1.9857 − 0.1152Nf. The next-to-next-to-leading (NNLO) coefficient c3(1) was calculated by Samuel and Surguladze (1991) and by Gorishnii et al. (1991), and is equal to -12.8 for five flavours. The N3LO coefficient c4(1) (which requires the evaluation of some twenty thousand diagrams) may be found in Baikov et al. (2008) and Baikov et al. (2009).

The physical cross section σ(e⁺e⁻ → hadrons) must be independent of the renormalization scale μ², and this would also be true of the series in (15.63) if an infinite number of terms were kept: the μ²-dependence of the coefficients

15.5. A more general form of the RGE 131 was given in equation (11.7). For QCD, although gauge invariance does imply generalizations of the Ward identity used to prove Z1 = Z2 (Taylor 1971, Slavnov 1972), the consequence is no longer the simple relation ‘Z1 = Z2’ in this case, due essentially to the ghost contributions. In order to see what changeZ1/=Z2would make, let us return to the one-loop calculation ofβfor QED, pretending thatZ1/=Z2. We have

e0= Z1

Z2

Z₃⁻¹²eμ (15.67)

where, because we are renormalizing at scale μ, all the Zi’s depend onμ(as in (15.15)), but we shall now not indicate this explicitly. Taking logs and differentiating with respect toμat constante0, we obtain

μ d To leading order ineμ, theγ3term in (15.70) reproduces (15.26) when (15.15) is used for Z3, the other two terms in (15.68) cancelling viaZ1 =Z2. So if, as in the case of QCD, Z1 is not equal to Z2, we need to introduce the con-tributions from loops determining the fermion field strength renormalization factor, as well as those related to the vertex parts (together with appropriate ghost loops), in addition to the vacuum polarization loop associated in the Z3.

Quantities such as γ2 and γ3 have an interesting and important signifi-cance, which we shall illustrate in the case ofγ2 for QED.Z2 enters into the relation between the propagator of the bare fermion<Ω|T( ˆψ0(x) ˆψ0(0))|Ω>and the renormalized one, via (cf (11.2))

<Ω|T(ψ(x) ˆ¯ˆ ψ(0)|Ω>= 1

Z2<Ω|T(ψ¯ˆ₀(x) ˆψ0(0))|Ω>, (15.71) where (cf section 10.1.3) |Ω> is the vacuum of the interacting theory. The Fourier transform of (15.71) is, of course, the Feynman propagator:

S˜_F^' (q²) = [

d⁴xe^iq·x<Ω|T(ψ(x) ˆ¯ˆ ψ(0))|Ω>. (15.72) Suppose we now ask: what is the large−q²behaviour of (15.72) for space-like q², with −q²>>m² wherem is the fermion mass? This sounds very similar to the question answered in 15.2.3 for the quantityS(|q²|/μ², eμ). However, 130 15. QCD II: Asymptotic Freedom, the Renormalization Group

cn(Q²/μ²) would cancel that of αs(μ²). This requirement can be imposed The second term on the right-hand side of (15.66) gives the contribution iden-tified in (15.2).

In practice only a finite number of terms n = N will be available, and a μ² -dependence will remain, which implies an uncertainty in the prediction of the cross section (and similar physical observables), due to the arbitrariness of the scale choice. This uncertainty will be of the same order as the neglected terms, i.e. of order α^N+1 _s . Thus the scale dependence of a QCD prediction gives a

Im Dokument Non-Abelian Gauge Theories (Seite 140-145)