an introduction to qcd - welcome to scippscipp.ucsc.edu/~dine/ph222/qcd_lecture.pdf · an...

An Introduction to QCD

Physics 222 2011, Advanced Quantum Field Theory

Michael DineDepartment of Physics

University of California, Santa Cruz

May 2011

Physics 222 2011, Advanced Quantum Field Theory An Introduction to QCD

e+e− Annihilation

The total cross section in e+e− annihilation is a particularlysimple object to study. It is a function of only one dimensionalquantity, s. So, provided it is infrared finite, it can only involveαs(µ), and log(s/µ2), where µ is a renormalization scale.Simply choosing s = µ2, gives that

R =σ(e+e− → hadrons)

σ(e+e− → µ+µ−)=∑

f

Q2f (1 +

∞∑n=1

cn

(αs(s)

4π

)n

). (1)


Establishing the finiteness of the total inclusive cross section iseasy once one establishes the connection of this quantity to thevacuum polarization tensor. This is a consequence of unitarity,as expressed in Peskin and Schroeder’s equation 7.49,

2ImM(a→ a) =∑

f

∫dΠfM∗(a→ f )M(a→ f ). (2)

In this case, the process a→ f corresponds to the amplitude tostart with the vacuum state and produce quarks, antiquarks,etc. by the action of the electromagnetic current.

σ(e+e− → hadrons) =16π3α2

sImΠ(s). (3)


The first term can be read off from QED calculations; thesecond was calculated by yours truly and others a long timeago[?]. Specifying this term requires precisely defining what ismeant by αs; redefinitions of the form

αs = α′s + aα2s + . . . (4)

change the expression at higher orders. Such redefinitionscorrespond to different choices of “renormalization scheme."


Finiteness of Π(q2)

The infrared finiteness of Π(q2) follows from general principlesof quantum field theory. Indeed, we are familiar from ourstudies of Feynman graphs, that if all momenta are Euclidean,one can Wick rotate so as to obtain well-behaved Euclideanintegrals. By simple power counting arguments one can seethat in four dimensions there are no infrared divergences. Onecan then analytically continue in q2 to timelike momenta(Feynman parameters are analytic functions of the momenta,essentially because they are well-behaved for all Euclideanmomenta, including complex; study some simple graphs tounderstand how this works).

So the discontinuity in Π is also well behaved (it just arrivesfrom the discontinuity of terms like (log(−q2))n uponcontinuation to timelike q2.


It is interesting to understand how discontinuities come about ina quantity like Π(q2). Consider, for simplicity, a scalar fieldtheory with φ3 interactions, and study the analog of the vacuumpolarization. Take the scalar to have mass m. Then ifq2 > 4m2, the momenta are such that the two internal lines canbe on shell; this is precisely when a discontinuity arises. (Thisis the generalization of the statement that for q2 < 0 one canperform a Wick rotation, avoiding consideration of iε terms inpropagators, and thus any imaginary part in the diagram).


Consider how the propagators behave near their singularities.Writing

1k2 −m2 + iε

=1

(k0 − Ek + iε)(k2 + Ek − iε), (5)

consider the second factor. This is something real plussomething imaginary times a δ-function:∫

dk0

k0 − Ek + iε=

12

(∫dk0

k0 − Ek + iε+

∫dk0

k0 − Ek − iε

)(6)

‘ +12

(∫dk0

k0 − Ek + iε−∫

dk0

k0 − Ek − iε

)The second integral can be done as a contour integral; the firstis real, and is called the “Principle Value".


So we have:

1k0 − Ek ± iε

= P1

k0 − Ek∓ iδ(k0 − Ek ) (7)

leading to Peskin and Schroeder’s equation 7.55:

1k2 −m2 + iε

= real− 2πiδ(k2 −m2) (8)

This, in turn, leads to the "Cutkosky rules" (p. 236), relating theimaginary parts of individual diagrams to particularcontributions to cross sections.


The Infrared Problem in QCD

Let’s look at the low momentum behavior of the Feynmandiagram for the vertex.


As k → 0, one has

Γµ ≈ −e3∫

d4k(2π)4

γρ(6p′ + m)γµ(6p + m)γρ

k2(−2p · k + iε)(−2p′ · k + iε)(9)

≈ −4e3∫

d4k(2π)4

pρ′pργµ

k2(2p · k)(2p′ · k)

(Dirac eqn. in last step).


Do k0 integral; close below, so only pick up pole from firstpropagator.

Γµ = −iγµ

(4e2

∫d3k

(2π)3p · p′

2|~k |(2p · k)(2p′ · k)

). (10)

This diverges as k → 0. No excuses! What cancels?


M(1) = e2 u(p′)γµ( 6p− 6k + m) 6ε(k)u(p)

−2p · k× [stuff] (11)

= e2 2p · εp · k

u(p′)γµu(p)× [stuff]

= e2 p · εp · k

×M(0).

HereM(0) is the zeroth order amplitude for the process withoutthe photon. Similarly

M(2) = e2 p′ · εp′ · k

u(p′)γµu(p)× [stuff] (12)

= e2 p′ · εp′ · k

×M(0).


Squaring (taking interference term) summing over photonpolarizations

dσdΩ

=

(dσdΩ

)0

e2∫

d3k(2π)3

2p · p′

(p · k)(p′ · d)

1

2|~k |. (13)


Jets in e+e− Annihilation

The cancelation of infrared divergences in e+e− annihilation is readilyunderstood. Cross sections for particular final states are divergent,but there are cancelations between vertex corrections and softphoton emissions when one considers the inclusive cross section.

Exclusive processes are more subtle. If we consider production of afixed number of quarks and gluons, this will diverge. But the lesson ofthe QED infrared cancelation is that we need to be careful whatquestions we ask. Indeed, this is presumably a good thing, since wedon’t see final state quarks or gluons, so if such computations weresensible, we would clearly contradict experiment. Instead, we can askquestions like how much energy flows into a cone in a particular solidangle, for example. This leads to notion of jets.

In e+e− annihilation, want to include soft emissions, and (nearly)colinear emission in the definition of jets. For suitable choice, nosmall invariants, and still have an expansion in αs(s).


The Parton Model

The parton model originated with Feynman. (The term “parton"seems to have been intended to avoid speaking of quarks andgluons; this may have resulted from his rivalry with Gell-Mann.)Consider two protons colliding at extremely high energy, so thattheir mass can be neglected (Feynman and others introducedthe notion of an “infinite momentum frame"). Idea is that in ahadron, there is a probability of finding a quark or gluoncarrying a fraction x of the total longitudinal momentum. At thesame time, the transverse momentum, pt , was assumed to belimited to some characteristic QCD number (consistent withdata, which shows rapid fall off of production with pt . We willdiscuss these assumptions later. One of the reasons thatpeople were so interested in the possibility of asymptoticfreedom is that these phenomena do not occur innon-asymptotically free theories.


Deep Inelastic ScatteringThe prototype for the parton model is deep inelastic scattering.Here one considers scattering of an electron or neutrino off of anucleon at large momentum transfer. One treats the scatteringas scattering off of individual quarks and gluons within theproton, assuming that they carry longitudinal momentumfraction x and essentially zero transverse momentum. Theprobability for a parton of type i to carry momentum x is givenby the “Parton distribution function", fi(x) (PDF).


We need, first, the cross section for the scattering of a (virtual)photon of momentum q2 = −Q2 off a quark. Calling theinvariants of the “parton level" scattering s, t , and u, this isgiven by:

dσdt

(e−q → e−q) =2πα2Q2

is2

[s2 + u2

t2

]. (14)

We need to express the invariants in terms of x andmeasurable quantities. The actual cross section is thenobtained by convoluting the parton cross section with the PDF(sometimes also called “distribution function").

t = q2 = −Q2; s = 2p · k = 2ξP · k = ξs, (15)

ands + t + u = 0. (16)


So the actual cross section is given by

σ(e−(k) + p(P)→ e−(k ′) + X ) =

∫ 1

0dξ∑

i

fi(ξ) (17)

×σ(e−(k)qf (ξP)→ e−(k ′) + qf (p′)).

This forms stresses the inclusive nature of the process. Indeed,at the parton level, the cross section would seem to be elastic,but in fact any sort of arrangement of the quarks and gluonsinto hadrons is considered in the final state.


Finally, the momentum fraction, ξ, is fixed in terms ofmeasurable quantities, given the assumption that the initial andfinal quarks are on shell (with negligible mass), and notingp′ = ξP + q:

ξ = x ; x =Q2

2P · q. (18)

The cross section can then be expressed in terms of variousmeasurable quantites. A convenient variable is

y =2P · q

s(19)


In terms of parton variables,

y =s + u

s(20)

andd2σ

dxdy= (∑

i

xfi(x)Q2i )

2πα2sQ4 [1 + (1− y)2]. (21)

Similar analysis can be performed for deep inelastic neutrinoscattering.


Such processes were studied at SLAC in the late 1960’s; onecould extract the f ’s, and they were seen to be roughlyindependent of Q2. These provided strong evidence for thequark model.But in ordinary field theories, this sort of scaling did not seem tohold. If one considered interactions among the partons, oneobtained corrections involving log(Q2), associated with the factthat the transverse momenta were typically of order Q2. It isasymptotic freedom of QCD which accounts for the observedapproximate scaling.


Other Applications of the Parton Model

Lepton Pair Production (Drell-Yan)

σ(qi qi → `+`−) =13

Q2f

4πα2

3s. (22)

Peskin and Schroeder discuss convenient ways to analyze thekinematics. In the end, a convolution of the parton crosssection over distribution functions associated with the twoincoming hadrons:

d4σ

dy3dy4d2p⊥= x1f (x1)x2f (x2)

1π

dσdt

(1 + 2→ 3 + 4). (23)

See Peskin and Schroeder for the definitions of the kinematicquantities appearing here.


Jet Pair Production

The formula above, eqn. 23, is applicable to other processes,most notably jet pair production.


Limitations of the Parton Model: Evolution of theStructure Functions with Q2

Actually field theories don’t behave like the parton model.Asymptotically free theories come close. The problem comesfrom diagrams with emission of quarks and gluons, in additionto the basic parton model process. This leads to two effects:large corrections (behaving as α(Q2) log(Q2/λ2), and also toan apparent breakdown of the notion of probability for findingpartons.


In fact, in QCD, amplitudes do appear to factorize∗, and aprobability interpretation goes through. As in the infraredproblem in QED, it is possible to sum the large logarithmiccorrections. This is achieved by writing differential equations,with Q2, for the distribution functions, which are now functionsof momentum. These are known as the Altareli-Parisiequations. They relate derivatives with respect to log(Q) of f tointegrals over distribution functions (over x), weighted withfunctions, known as splitting functions, which describe theamplitude to produce some other parton from the initial one.These are given in PS, eqns. 17.128-17.130.


Everything is in your book or in the PDG!


Parton Luminosities

A helpful way to think about collider physics is to think aboutparton luminosities as function of (parton and beam) energy.Here are a couple of examples.


an introduction to qcd - welcome to scippscipp.ucsc.edu/~dine/ph222/qcd_lecture.pdf · an...

Documents