large logarithms
15.6 QCD corrections to the parton model predictions for deep inelastic scattering: scaling violations
i
γi(αs) +γm(αs)m ∂
∂m R(|q2|/μ2, αs, m/|q|) = 0 (15.79) where the partial derivatives are taken at fixed values of the other two
vari-E
15.6. QCD corrections to the parton model predictions: scaling violations 135 ables. Here the γi are the anomalous dimensions relevant to the quantity R, and γm is an analogous ‘anomalous mass dimension’, arising from finite shifts in the mass parameter when the scale μ2 is 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 ln m as analogous to αs. Then equation (15.41) for the running αs,
∂αs(|q2|)
= β(αs(|q 2|)) (15.80)
∂t where t = ln(|q2|/μ2), becomes
∂(ln m(|q2|))
= γm(αs(|q|2)). (15.81)
∂t Equation (15.81) has the solution
[ ln |q 2|
m(|q 2|) = m(μ2) exp d ln|q '2| γm(αs(|q '2|). (15.82)
ln μ2
To one-loop order in QCD, γm(αs) turns out to be −π 1 αs (Peskin and Schroeder 1995, section 18.1). Inserting the one-loop solution for αs in the form (15.53), we find
πβ0
|ln(μ2/Λ2)| 1
m(|q 2|) = m(μ2) , (15.83)
ln(|q2|/Λ2)
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.
15.6 QCD corrections to the parton model predictions for deep inelastic scattering: scaling violations
As we saw in section 9.2, the parton model provides a simple intuitive expla-nation for the experimental observation that the nucleon structure functions in deep inelastic scattering depend, to a good first approximation, only on the dimensionless ratio x = Q2/2M ν, rather than on Q2 and ν separately;
this behaviour is referred to as ‘scaling’. Here M is the nucleon mass, and Q2 and ν are defined in (9.7) and (9.8). In this section we shall show how QCD
15.6. QCD corrections to the parton model predictions: scaling violations137
FIGURE 15.7
Electron-quark scattering via one-photon exchange.
FIGURE 15.8
Electron-quark scattering with single-gluon emission
FIGURE 15.9
Virtual single-gluon corrections to figure 15.7.
FIGURE 15.10
Electron-gluon scattering with ¯qq production.
136 15. QCD II: Asymptotic Freedom, the Renormalization Group corrections to the simple parton model, calculated using RGE techniques, pre-dict observable violations of scaling in deep inelastic scattering. As we shall see, comparison between the theoretical predictions and experimental mea-surements provides strong evidence for the correctness of QCD as the theory of nucleonic constituents.
15.6.1 Uncancelled mass singularities at order αs.
The free parton model amplitudes we considered in chapter 9 for deep inelastic lepton-nucleon scattering were of the form shown in figure 15.7 (cf figure 9.4).
The obvious first QCD corrections will be due to real gluon emission by either the initial or final quark, as shown in figure 15.8, but to these we must add the one-loop virtual gluon processes of figure 15.9 in order (see below) to get rid of infrared divergences similar to those encountered in section 14.4.2, and also the diagram of figure 15.10, corresponding to the presence of gluons in the nucleon. To simplify matters, we shall consider what is called a ‘non-singlet structure function’ F2 NS, such as F2ep −F2 en in which the (flavour) singlet gluon contribution cancels out, leaving only the diagrams of figures 15.8 and 15.9.
We now want to perform, for these diagrams, calculations analogous to those of section 9.2, which enabled us to find the e-N structure functions νW2 and M W1 from the simple parton process of figure 15.7. There are two problems here: one is to find the parton level W’s corresponding to figure 15.8 (leaving aside figure 15.9 for the moment) – cf equations (9.29) and (9.30) in the case of the free parton diagram figure 15.7; the other is to relate these parton W’s to observed nucleon W’s via an integration over momentum fractions. In section 9.2 we solved the first problem by explicitly calculating the parton level d2σi/dQ2dν and picking off the associated νW2i, W1i . In principle, the same can be done here, starting from the five-fold differential cross section for our e− + q → e + q + g process. However, a simpler – if −
somewhat heuristic – way is available. We note from (9.46) that in general F1 = M W1 is given by the transverse virtual photon cross section
1
μ(λ)Eν (λ)Wμν W1 = σT /(4π2α/K) = E
E ∗ (15.84)
2 λ=±1
where Wμν was defined in (9.3). Further, the Callan–Gross relation is still true (the photon only interacts with the charged partons, which are quarks with spin 12 and charge ei), and so
F2/x = 2F1 = 2M W1 = σT /(4π2α/2M K). (15.85) These formulae are valid for both parton and protonW1’s and Wμν ’s, with ap-propriate changes for parton masses Mˆ . Hence the parton level 2 Fˆ1 for figure 15.8 is just the transverse photon cross section as calculated from the graphs of figure 15.11, divided by the factor 4π2α/2MˆKˆ , where as usual ‘ˆ’ denotes kinematic quantities in the corresponding parton process. This cross section,
136 15. QCD II: Asymptotic Freedom, the Renormalization Group corrections to the simple parton model, calculated using RGE techniques, pre-dict observable violations of scaling in deep inelastic scattering. As we shall see, comparison between the theoretical predictions and experimental mea-surements provides strong evidence for the correctness of QCD as the theory of nucleonic constituents.
15.6.1 Uncancelled mass singularities at order αs.
The free parton model amplitudes we considered in chapter 9 for deep inelastic lepton-nucleon scattering were of the form shown in figure 15.7 (cf figure 9.4).
The obvious first QCD corrections will be due to real gluon emission by either the initial or final quark, as shown in figure 15.8, but to these we must add the one-loop virtual gluon processes of figure 15.9 in order (see below) to get rid of infrared divergences similar to those encountered in section 14.4.2, and also the diagram of figure 15.10, corresponding to the presence of gluons in the nucleon. To simplify matters, we shall consider what is called a ‘non-singlet structure function’F2NS, such asF2ep−F2enin which the (flavour) singlet gluon contribution cancels out, leaving only the diagrams of figures 15.8 and 15.9.
We now want to perform, for these diagrams, calculations analogous to those of section 9.2, which enabled us to find the e-N structure functions νW2 andM W1 from the simple parton process of figure 15.7. There are two problems here: one is to find the parton level W’s corresponding to figure 15.8 (leaving aside figure 15.9 for the moment) – cf equations (9.29) and (9.30) in the case of the free parton diagram figure 15.7; the other is to relate these partonW’s to observed nucleonW’s via an integration over momentum fractions. In section 9.2 we solved the first problem by explicitly calculating the parton level d2σi/dQ2dν and picking off the associated νW2i, W1i. In principle, the same can be done here, starting from the five-fold differential cross section for our e−+ q→ e−+ q + g process. However, a simpler – if somewhat heuristic – way is available. We note from (9.46) that in general F1=M W1 is given by the transverse virtual photon cross section
W1=σT/(4π2α/K) =1 2
E
λ=±1
E∗μ(λ)Eν(λ)Wμν (15.84) where Wμν was defined in (9.3). Further, the Callan–Gross relation is still true (the photon only interacts with the charged partons, which are quarks with spin 12 and chargeei), and so
F2/x= 2F1= 2M W1=σT/(4π2α/2M K). (15.85) These formulae are valid for both parton and protonW1’s andWμν’s, with ap-propriate changes for parton masses ˆM. Hence the parton level 2 ˆF1 for figure 15.8 is just the transverse photon cross section as calculated from the graphs of figure 15.11, divided by the factor 4π2α/2 ˆMK, where as usual ‘ˆ’ denotesˆ kinematic quantities in the corresponding parton process. This cross section,
15.6. QCD corrections to the parton model predictions: scaling violations 137
FIGURE 15.7
Electron-quark scattering via one-photon exchange.
FIGURE 15.8
Electron-quark scattering with single-gluon emission
FIGURE 15.9
Virtual single-gluon corrections to figure 15.7.
FIGURE 15.10
Electron-gluon scattering with ¯qq production.
15.6. QCD corrections to the parton model predictions: scaling violations139
FIGURE 15.12
The first process of figure 15.11, viewed as a contribution to e−-nucleon scat-tering.
FIGURE 15.13
Kinematics for the parton process of figure 15.12.
the sum is over contributing partons. The reader may enjoy checking that (15.90) does reduce to (9.34) for free partons by showing that in that case 2 ˆF1i=e2iδ(1−z) (see Halzen and Martin 1984, section 10.3, for help), so that 2F1free=E
ie2ifi(x).
To proceed further with the calculation (i.e. of (15.87) inserted into (15.90)), we need to look at the kinematics of the γq→qg process, in the CMS. Re-ferring to figure 15.13, we let k, k' be the magnitudes of the CMS momenta k,k'. Then
ˆ
s = 4k'2= (yp+q)2=Q2(1−z)/z, z=Q2/(ˆs+Q2) ˆt = (q−p')2=−2kk'(1−cosθ) =−Q2(1−c)/2z, c= cosθ ˆ
u = (q−q')2=−2kk'(1 + cosθ) =−Q2(1 +c)/2z. (15.91) We now note that in the integral (15.87) for ˆF1, when we integrate over c= cosθ, we shall obtain an infinite result
∼ [ 1
dc
1−c (15.92)
associated with the vanishing of ˆtin the ‘forward’ direction (i.e. whenqandp' 138 15. QCD II: Asymptotic Freedom, the Renormalization Group
FIGURE 15.11
Virtual photon processes entering into figure 15.8.
however, is – apart from a colour factor – just the virtual Compton cross sec-tion calculated in secsec-tion 8.6. Also, taking the same (Hand) convensec-tion for the individual photon flux factors,
2MˆKˆ = ˆs. (15.86)
Thus for the parton processes of figure 15.9, 2Fˆ1 = σˆT /(4π2α/2MˆKˆ )
s
ˆ / 1 4 πei 2ααs(μ2)( tˆ sˆ 2ˆuQ2 )
= d cosθ · · − − + (15.87)
4π2α −1 3 sˆ sˆ tˆ sˆtˆ
where, in going from (8.181) to (15.87), we have inserted a colour factor 43
(problem 14.5 (a)), renamed the variables tˆ→ u, ˆˆ u → tˆ in accordance with figure 15.11, and replaced α2 by ei 2ααs(μ2).
Before proceeding with (15.87), it is helpful to consider the other part of the calculation – namely the relation between the nucleon F1 and the parton Fˆ1. We mimic the discussion of section 9.2, but with one significant difference:
the quark ‘taken’ from the proton still has momentum fraction y (momentum yp), but now its longitudinal momentum must be degraded in the final state due to the gluon bremsstrahlung process we are calculating. Let us call the quark momentum after gluon emission zyp (figure 15.12). Then, assuming as in section 9.2 that it stays on-shell, we have
q 2 + 2zyq · p = 0 (15.88)
or x = yz, x = Q2/2q · p, q 2 = −Q2 (15.89) and we can write (cf (9.31))
F2 / 1 / 1
= 2F1 = E
dyfi(y) dz 2Fˆ
1iδ(x − yz) (15.90)
x i 0 0
where the fi(y) are the parton distribution functions introduced in section 9.2 (we often call them q(x) or g(x) as the case may be) for parton type i, and
138 15. QCD II: Asymptotic Freedom, the Renormalization Group
FIGURE 15.11
Virtual photon processes entering into figure 15.8.
however, is – apart from a colour factor – just the virtual Compton cross sec-tion calculated in secsec-tion 8.6. Also, taking the same (Hand) convensec-tion for the individual photon flux factors,
2 ˆMKˆ = ˆs. (15.86)
Thus for the parton processes of figure 15.9, 2 ˆF1 = ˆσT/(4π2α/2 ˆMK)ˆ
= sˆ 4π2α
/ 1
−1
d cosθ·4
3· πei2ααs(μ2) ˆ s
(
−ˆt ˆ s−ˆs
ˆt +2ˆuQ2 ˆ sˆt
)
(15.87) where, in going from (8.181) to (15.87), we have inserted a colour factor 43 (problem 14.5 (a)), renamed the variables ˆt → u,ˆ uˆ → tˆin accordance with figure 15.11, and replacedα2 byei2ααs(μ2).
Before proceeding with (15.87), it is helpful to consider the other part of the calculation – namely the relation between the nucleonF1 and the parton Fˆ1. We mimic the discussion of section 9.2, but with one significant difference:
the quark ‘taken’ from the proton still has momentum fractiony(momentum yp), but now its longitudinal momentum must be degraded in the final state due to the gluon bremsstrahlung process we are calculating. Let us call the quark momentum after gluon emissionzyp(figure 15.12). Then, assuming as in section 9.2 that it stays on-shell, we have
q2+ 2zyq·p= 0 (15.88)
or
x=yz, x=Q2/2q·p, q2=−Q2 (15.89) and we can write (cf (9.31))
F2
x = 2F1=E
i
/ 1 0
dyfi(y) / 1
0
dz 2 ˆF1iδ(x−yz) (15.90) where thefi(y) are the parton distribution functions introduced in section 9.2
(we often call them q(x) or g(x) as the case may be) for parton typei, and '
15.6. QCD corrections to the parton model predictions: scaling violations 139
FIGURE 15.12
The first process of figure 15.11, viewed as a contribution to e−-nucleon scat-tering.
FIGURE 15.13
Kinematics for the parton process of figure 15.12.
the sum is over contributing partons. The reader may enjoy checking that (15.90) does reduce to (9.34) for free partons by showing that in that case 2Fˆ1 i = e2i δ(1− z) (see Halzen and Martin 1984, section 10.3, for help), so that 2Ffree = E
i e2fi(x).
1 i
To proceed further with the calculation (i.e. of (15.87) inserted into (15.90)), we need to look at the kinematics of the γq → qg process, in the CMS. Re-ferring to figure 15.13, we let k, k' be the magnitudes of the CMS momenta k, k' . Then
sˆ = 4k'2 = (yp + q)2 = Q2(1 − z)/z, z = Q2/(ˆs + Q2) tˆ = (q − p ')2 = −2kk'(1 − cos θ) = −Q2(1 − c)/2z, c = cos θ
')2
uˆ = (q − q = −2kk'(1 + cosθ) = −Q2(1 + c)/2z. (15.91) We now note that in the integral (15.87) for Fˆ1, when we integrate over c = cos θ, we shall obtain an infinite result
[ 1 dc
∼ (15.92)
1− c
associated with the vanishing of tˆ in the ‘forward’ direction (i.e. when q and p
15.6. QCD corrections to the parton model predictions: scaling violations141 yzp after gluon emission becomes equal to the quark momentum yp before emission), and we expect that it can be cured by including the virtual gluon diagrams of figure 15.9, as indicated at the start of the section (and as was done analogously in the case of e+e− annihilation). This has been verified explicitly by Kim and Schilcher (1978) and by Altarelli et al. (1978 a, b;
1979). Alternatively, we follow the procedure of Altarelli and Parisi (1977).
First we regulate the divergence as z → 1 by defining a regulated function 1/(1−z)+ such that wheref(z) is any test function sufficiently regular at the end points. Now the gluon loops which will cancel the i-r divergence only contribute atz →1, in leading log approximation. Thus the i-r finite version of ˆPqq has the form
Pqq(z) = 4 3
1 +z2
(1−z)+ +Aδ(1−z). (15.100) The coefficient A is determined by the physical requirement that the net number of quarks (i.e. the number of quarks minus the number of antiquarks) does not vary withQ2. From (15.98) this implies
[ 1 0
Pqq(z)dz= 0. (15.101)
Inserting (15.100) into (15.101), and using (15.99), we find (problem 15.6)
A= 2, (15.102)
The function Pqq is called a ‘splitting function’, and it has an impor-tant physical interpretation. The quantity αs(μ2)/(2π) Pqq(z) is, for z < 1, the probability that, to first order in αs, a quark having radiated a gluon is left with a fraction z of its original momentum. Similar functions arise in QED in connection with what is called the ‘equivalent photon approximation’
(Weizs¨acker 1934, Williams 1934, Chen and Zerwas 1975). The application of these techniques to QCD corrections to the free parton model is due to Altarelli and Parisi (1977), who thereby opened the way to this simpler and more physical way of understanding scaling violations, which had previously been discussed mainly within the rather technical operator product formalism (Wilson 1969).
We must now find some way of making sense, physically, of the uncancelled mass divergence in (15.97).
140 15. QCD II: Asymptotic Freedom, the Renormalization Group are parallel). This is a divergence of the ‘collinear’ type, in the terminology of section 14.4.2 – or, as there, a ‘mass singularity’, occurring in the zero quark violates scaling. This crucial physical result is present in the lowest-order QCD correction to the parton model, in this case. As we are learning, such loga-rithmic violations of scaling are a characteristic feature of all QCD corrections to the free (scaling) parton model.
We may calculate the coefficient of the lnQ2 term by retaining in (15.87)
due to lowest-order gluon radiation. Clearly, this corrected distribution func-tion violates scaling because of the ln Q2 term. But the result as it stands cannot represent a well-controlled approximation, since it contains divergences as z → 1 and as m2 → 0.
We postpone discussion of the mass divergence until the next section. The divergence as z → 1 is a standard infrared divergence (the quark momentum
140 15. QCD II: Asymptotic Freedom, the Renormalization Group are parallel). This is a divergence of the ‘collinear’ type, in the terminology of section 14.4.2 – or, as there, a ‘mass singularity’, occurring in the zero quark mass limit. If we simply replace the propagator factor ˆt−1= [(q−p')2]−1 by reg-ulates the divergence. We have here an uncancelled mass singularity, and it violates scaling. This crucial physical result is present in the lowest-order QCD correction to the parton model, in this case. As we are learning, such loga-rithmic violations of scaling are a characteristic feature of all QCD corrections to the free (scaling) parton model.
We may calculate the coefficient of the lnQ2 term by retaining in (15.87) only the terms proportional to ˆt−1: and so, for just one quark species, this QCD correction contributes (from (15.90)) a term
Our result so far is therefore that the ‘free’ quark distribution function q(x), which depended only on the scaling variablex, becomes modified to
q(x) +αs(μ2)
due to lowest-order gluon radiation. Clearly, this corrected distribution func-tion violates scaling because of the lnQ2 term. But the result as it stands cannot represent a well-controlled approximation, since it contains divergences asz→1 and asm2→0.
We postpone discussion of the mass divergence until the next section. The divergence asz→1 is a standard infrared divergence (the quark momentum
15.6. QCD corrections to the parton model predictions: scaling violations 141 yzp after gluon emission becomes equal to the quark momentum yp before emission), and we expect that it can be cured by including the virtual gluon diagrams of figure 15.9, as indicated at the start of the section (and as was done analogously in the case of e+e− annihilation). This has been verified explicitly by Kim and Schilcher (1978) and by Altarelli et al. (1978 a, b;
1979). Alternatively, we follow the procedure of Altarelli and Parisi (1977).
First we regulate the divergence as z → 1 by defining a regulated function number of quarks (i.e. the number of quarks minus the number of antiquarks) does not vary with Q2 . From (15.98) this implies
[ 1
Pqq(z)dz = 0. (15.101)
0
Inserting (15.100) into (15.101), and using (15.99), we find (problem 15.6)
A = 2, (15.102) Altarelli and Parisi (1977), who thereby opened the way to this simpler and more physical way of understanding scaling violations, which had previously been discussed mainly within the rather technical operator product formalism (Wilson 1969).
We must now find some way of making sense, physically, of the uncancelled mass divergence in (15.97).
15.6. QCD corrections to the parton model predictions: scaling violations143 partially with respect toμ2F, and setting the result to zero, we obtain (to order αson the right-hand side) This equation is the analogue of equation (15.35) describing the running of the coupling αs with μ2, and is a fundamental equation in the theory of perturbative applications of QCD. It is called the DGLAP equation, after Dokshitzer (1977), Gribov and Lipatov (1972), and Altarelli and Parisi (1977).
The above derivation is not rigorous: a more sophisticated treatment (Georgi and Politzer 1974, Gross and Wilczek 1974) confirms the result and extends it to higher orders.
Equation (15.108) shows that, although perturbation theory cannot be used to calculate the distribution function q(x, μ2F) at any particular value μ2F=μ20, it can be used to predict how the distributionchanges(or ‘evolves’) as μ2Fvaries. (We recall from (15.105) thatq(x, μ20) can be found experimentally viaxq(x, μ20) = 2F2(x, Q2 =μ20)/e2i.) As in the case of σ(e+e− → hadrons) and the scale μ2, the choice of factorization scale is arbitrary, and would cancel from physical quantities if all powers in the perturbation series were included. Truncating atN terms results in an ambiguity of orderα(Ns +1). In deep inelastic predictions, the standard choice for scales isμ2=μ2F=Q2.
The way the non-singlet distribution changes can be understood qualita-tively as follows. The change in the distribution for a quark with momentum fractionx, which absorbs the virtual photon, is given by the integral overyof the corresponding distribution for a quark with momentum fractiony, which radiated away (via a gluon) a fraction x/y of its momentum with probabil-ity (αs/2π)Pqq(x/y). This probability is high for large momentum fractions:
high-momentum quarks lose momentum by radiating gluons. Thus there is a predicted tendency for the distribution function q(x, μ2) to get smaller at large xas μ2 increases, and larger at small x(due to the build-up of slower partons), while maintaining the integral of the distribution over xas a con-stant. The effect is illustrated qualitatively in figure 15.14. In addition, the radiated gluons produce moreqq¯pairs at smallx. Thus the nucleon may be pictured as having more and more constituents, all contributing to its total momentum, as its structure is probed on ever smaller distance (larger μ2) scales.
In general, the right-hand side of (15.108) will have to be supplemented by terms (calculable from figure 15.10) in which quarks are generated from the gluon distribution; the equations must then be closed by a corresponding one describing the evolution of the gluon distributions (Altarelli 1982). In the now commonly used notation, this generalization of (15.108) reads
μ2F∂fi/p(x, μ2F) 142 15. QCD II: Asymptotic Freedom, the Renormalization Group
15.6.2 Factorization, and the order αs DGLAP equation
This procedure is, of course, very reminiscent of ultraviolet renormaliza-tion, in which u-v divergences are controlled by similarly importing some the scale entering into the separation in (15.107), between one (uncalculable) factor which depends on the i-r parameter m but not on Q2, and the other (calculable) factor which depends on Q2 . The scale μF can be thought of as one which separates the perturbative short-distance physics from the non-perturbative long-distance physics. Thus partons emitted at small transverse momenta < μF (i.e. approximately collinear processes) should be considered as part of the hadron structure, and are absorbed into q(x, μ2 F). Partons emit-ted at large transverse momenta contribute to the short-distance (calculable) part of the cross section. Just as for the renormalization scale, the more terms
This procedure is, of course, very reminiscent of ultraviolet renormaliza-tion, in which u-v divergences are controlled by similarly importing some the scale entering into the separation in (15.107), between one (uncalculable) factor which depends on the i-r parameter m but not on Q2, and the other (calculable) factor which depends on Q2 . The scale μF can be thought of as one which separates the perturbative short-distance physics from the non-perturbative long-distance physics. Thus partons emitted at small transverse momenta < μF (i.e. approximately collinear processes) should be considered as part of the hadron structure, and are absorbed into q(x, μ2 F). Partons emit-ted at large transverse momenta contribute to the short-distance (calculable) part of the cross section. Just as for the renormalization scale, the more terms