large logarithms
15.5 A more general form of the RGE: anomalous dimensions and running masses
The reader may have wondered why, for QCD, all the graphs of figure 15.6 are needed, whereas for QED we got away with only figure 11.3. The reason for the simplification in QED was the equality between the renormalization constants Z1 and Z2, which therefore cancelled out in the relation between the renormalized and bare chargeseande0, as briefly stated before equation (15.8) (this equality was discussed in section 11.6). We recall thatZ1 is the field strength renormalization factor for the charged fermion in QED, andZ1
is the vertex part renormalization constant; their relation to the counter terms
[ ¯
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, 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 Fourier transform of (15.71) is, of course, the Feynman propagator:
S˜F ' (q 2) = d4 xe iq·x<Ω|T (ψˆ(x)ψˆ(0))|Ω>. (15.72) Suppose we now ask: what is the large −q2 behaviour of (15.72) for space-like
2 >> m2
q2, with −q where m is the fermion mass? This sounds very similar to the question answered in 15.2.3 for the quantity S(|q2|/μ2, eμ). However,
15.5. A more general form of the RGE 133
FIGURE 15.6
Possible behaviour ofβfunctions. (a) The slope is positive near the origin (as in QED), and negative near α=α∗. (b) The slope is negative at the origin (as in QCD), and positive nearαs=α∗s.
it here:
R˜'F(|q2|/μ2), αμ),= ˜R'F(1, α(|q2|/μ2)) exp {{ t
0
dt'γ2(α(t')) }
. (15.76) The first factor is the expected one from section 15.2.3; the second results from the addition of the γ2 term in (15.73). Suppose now that β(α) has a zero at some pointα∗, in the vicinity of whichβ(α)≈ −B(α−α∗) withB >0.
Then, near this point the evolution ofαis given by (cf (15.39)) ln(|q2|/μ2) =
{ α(|q2|) αμ
= dα
−B(α−α∗), (15.77) which implies
α(|q2|) =α∗+ constant ×(μ2/|q2|)B. (15.78) Thus asymptotically for large|q2|, the coupling will evolve to the ‘fixed point’
α∗. In this case, at sufficiently large−q2, the integral in (15.76) can be eval-uated by setting α(t') =α∗, and ˜R'F will scale with an anomalous dimension γ2(α∗) determined by the fixed point value of α. The behaviour of such an αis shown in figure 15.6(a). We emphasize that there is no reason to believe that the QED β function actually does behave like this.
The point α∗ in figure 15.6(a) is called an ultraviolet-stable fixed point:
α ‘flows’ towards it at large|q2|. In the case of QCD, theβ function starts out negative, so that the corresponding behaviour (with a zero at a α∗s /= 0) would look like that shown in figure 15.6(b). In this case, the reader can check (problem 15.5) that α∗s is reached in the infrared limit q2 →0, and so α∗s is called an infrared-stable fixed point. Clearly it is the slope ofβ near the fixed point that determines whether it is u-v or i-r stable. This applies equally to a fixed point at the origin, so that QED is i-r stable at α= 0 while QCD is u-v stable atαs= 0.
132 15. QCD II: Asymptotic Freedom, the Renormalization Group the latter was dimensionless whereas (recalling that ψˆ has mass dimension 3 2 ) S˜F ' (q2) has dimension M−1 . This dimensionality is, of course, just what a propagator of the free-field form i/(/q − m) would provide.
Accordingly, we extract this (q/)−1 factor (compare σ/σpt) and consider the dimensionless ratio R˜' F(|q2|/μ2, αμ) = q/S˜F ' (q2). We might guess that, just as for S(|q2|/μ2, αμ), to get the leading large |q2| behaviour we will need to calculate R˜' F to some order in αμ, and then replace αμ by α(|q2|/μ2). But this is not quite all. The factor Z2 in (15.71) will – as noted above – depend on the renormalization scale μ, just as Z3 of (15.15) did. Thus when we change μ, the normalization of the ψˆ’s will change via the Z 2 21 factors – of course by a finite amount here – and we must include this change when writing down the analogue of (15.33) for this case (i.e. the condition that the ‘total change, on changing μ, is zero’). The required equation is
| ∂ ∂ |
μ2 + β(αμ) + γ2(αμ) R˜'F (|q 2|/μ2, αμ) = 0. (15.73)
∂μ2
||
||
αμ ∂αμ
The solution of (15.73) is somewhat more complicated than that of (15.33).
We can gain insight into the essential difference caused by the presence of γ2 by considering the special case β(αμ) = 0. In this case, we easily find
R˜F ' (|q 2|/μ2, αμ)∝ (μ2)−γ2 (αμ). (15.74) R˜'
But since F can only depend on μ via |q2|/μ2, we learn that if β = 0 then the large |q2| behaviour of R˜' F is given by (|q2|/μ2)γ2 – or, in other words, that at large |q2|
1 (
|q2|)γ2(αμ)
S˜F ' (|q 2|/μ2, αμ)∝ . (15.75) /q μ2
Thus, at a zero of the β-function, S˜F ' has an ‘anomalous’power law dependence on |q2| (i.e. in addition to the obvious /q−1 factor), which is controlled by the parameter γ2. The latter is called the ‘anomalous dimension’ of the fermion field, since its presence effectively means that the |q2| behaviour of S˜F ' is not determined by its ‘normal’ dimensionality M−1 . The behaviour (15.75) is often referred to as ‘scaling with anomalous dimension’, meaning that if we multiply |q2| by a scale factor λ, then S˜F ' is multiplied byλγ2(αμ)−1 rather than just λ−1 . Anomalous dimensions turn out to play a vital role in the theory of critical phenomena – they are, in fact, closely related to ‘critical exponents’
(see section 16.4.3, and Peskin and Schroeder 1995, chapter 13). Scaling with anomalous dimensions is also exactly what occurs in deep inelastic scattering of leptons from nucleons, as we shall see in section 15.6.
The full solution of (15.73) for β /= 0 is elegantly discussed in Coleman (1985), chapter 3; see also Peskin and Schroeder (1995) section 12.3. We quote
132 15. QCD II: Asymptotic Freedom, the Renormalization Group the latter was dimensionless whereas (recalling that ˆψhas mass dimension 32) S˜F' (q2) has dimension M−1. This dimensionality is, of course, just what a propagator of the free-field formi/(/q−m) would provide.
Accordingly, we extract this (q/)−1 factor (compare σ/σpt) and consider the dimensionless ratio ˜R'F(|q2|/μ2, αμ) =q/S˜F' (q2). We might guess that, just as forS(|q2|/μ2, αμ), to get the leading large|q2| behaviour we will need to calculate ˜R'F to some order inαμ, and then replaceαμbyα(|q2|/μ2). But this is not quite all. The factorZ2 in (15.71) will – as noted above – depend on the renormalization scaleμ, just asZ3 of (15.15) did. Thus when we change μ, the normalization of the ˆψ’s will change via theZ212 factors – of course by a finite amount here – and we must include this change when writing down the analogue of (15.33) for this case (i.e. the condition that the ‘total change, on changingμ, is zero’). The required equation is
| μ2 ∂
∂μ2
||
||
αμ
+β(αμ) ∂
∂αμ
+γ2(αμ)
|
R˜'F(|q2|/μ2, αμ) = 0. (15.73)
The solution of (15.73) is somewhat more complicated than that of (15.33).
We can gain insight into the essential difference caused by the presence ofγ2
by considering the special caseβ(αμ) = 0. In this case, we easily find R˜'F(|q2|/μ2, αμ)∝(μ2)−γ2(αμ). (15.74) But since ˜R'F can only depend onμ via|q2|/μ2, we learn that ifβ = 0 then the large|q2| behaviour of ˜R'F is given by (|q2|/μ2)γ2 – or, in other words, that at large|q2|
S˜F' (|q2|/μ2, αμ)∝ 1 /q
(|q2| μ2
)γ2(αμ)
. (15.75)
Thus,at a zero of theβ-function, ˜SF' has an ‘anomalous’power law dependence on|q2|(i.e. in addition to the obvious/q−1 factor), which is controlled by the parameterγ2. The latter is called the ‘anomalous dimension’ of the fermion field, since its presence effectively means that the|q2|behaviour of ˜SF' is not determined by its ‘normal’ dimensionality M−1. The behaviour (15.75) is often referred to as ‘scaling with anomalous dimension’, meaning that if we multiply|q2|by a scale factorλ, then ˜SF' is multiplied byλγ2(αμ)−1rather than justλ−1. Anomalous dimensions turn out to play a vital role in the theory of critical phenomena – they are, in fact, closely related to ‘critical exponents’
(see section 16.4.3, and Peskin and Schroeder 1995, chapter 13). Scaling with anomalous dimensions is also exactly what occurs in deep inelastic scattering of leptons from nucleons, as we shall see in section 15.6.
The full solution of (15.73) for β /= 0 is elegantly discussed in Coleman (1985), chapter 3; see also Peskin and Schroeder (1995) section 12.3. We quote
{{ }
15.5. A more general form of the RGE 133
FIGURE 15.6
Possible behaviour of β functions. (a) The slope is positive near the origin (as in QED), and negative near α = α∗ . (b) The slope is negative at the origin (as in QCD), and positive near αs = α∗ s .
it here:
t
R˜F ' (|q 2|/μ2), αμ), = R˜F ' (1, α(|q 2|/μ2)) exp dt'γ2(α(t')) . (15.76)
0
The first factor is the expected one from section 15.2.3; the second results from the addition of the γ2 term in (15.73). Suppose now that β(α) has a zero at some point α∗, in the vicinity of which β(α)≈ −B(α−α∗) with B >0.
Then, near this point the evolution of α is given by (cf (15.39)) { α(|q 2|) dα
ln(|q 2|/μ2) = = , (15.77)
αμ −B(α −α∗) which implies
α(|q 2|) = α ∗ + constant ×(μ2/|q 2|)B . (15.78) Thus asymptotically for large |q2|, the coupling will evolve to the ‘fixed point’
α∗ . In this case, at sufficiently large −q2, the integral in (15.76) can be eval-uated by setting α(t') = α∗, and R˜' F will scale with an anomalous dimension γ2(α∗) determined by the fixed point value of α. The behaviour of such an α is shown in figure 15.6(a). We emphasize that there is no reason to believe that the QED β function actually does behave like this.
The point α∗ in figure 15.6(a) is called an ultraviolet-stable fixed point:
α ‘flows’ towards it at large |q2|. In the case of QCD, the β function starts out negative, so that the corresponding behaviour (with a zero at a α∗ s /= 0) would look like that shown in figure 15.6(b). In this case, the reader can check (problem 15.5) that α∗ s is reached in the infrared limit q2 →0, and so α∗ s is called an infrared-stable fixed point. Clearly it is the slope of β near the fixed point that determines whether it is u-v or i-r stable. This applies equally to a fixed point at the origin, so that QED is i-r stable at α = 0 while QCD is u-v stable at αs = 0.
15.6. QCD corrections to the parton model predictions: scaling violations135 ables. Here the γi are the anomalous dimensions relevant to the quantity R, andγmis an analogous ‘anomalous mass dimension’, arising from finite shifts in the mass parameter when the scaleμ2is changed. Just as with the solution (15.76) of (15.73), the solution of (15.79) is given in terms of a ‘running mass’
m(|q2|). Formally, we can think of γm in (15.79) as analogous to β(αs) and lnmas analogous toαs. Then equation (15.41) for the runningαs,
∂αs(|q2|)
∂t =β(αs(|q2|)) (15.80) where t= ln(|q2|/μ2), becomes
∂(lnm(|q2|))
∂t =γm(αs(|q|2)). (15.81) Equation (15.81) has the solution
m(|q2|) =m(μ2) exp [ ln|q2|
lnμ2
d ln|q'2|γm(αs(|q'2|). (15.82) To one-loop order in QCD, γm(αs) turns out to be −1παs (Peskin and Schroeder 1995, section 18.1). Inserting the one-loop solution for αs in the form (15.53), we find
m(|q2|) =m(μ2)
|ln(μ2/Λ2) ln(|q2|/Λ2)
|πβ10
, (15.83)
where (πβ0)−1= 12/(33−2Nf). Thus the quark masses decrease logarithmi-cally as |q2|increases, rather like αs(|q2|). It follows that, in general, quark mass effects are suppressed both by explicit m2/|q2| factors, and by the log-arithmic decrease given by (15.83). Further discussion of the treatment of quark masses is contained in Ellis, Stirling and Webber (1996), section 2.4;
see also the review by Manohar and Sachrajda in Nakamura et al. 2010.