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QCD: a Gauge Theory for Strong Interactions

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QCD: a Gauge Theory for Strong Interactions Corfu Summer Institute on Elementary Particle Physics, 1998 PROCEEDINGS QCD: a gauge theory for strong interactions. Roberto Petronzio Address: University of Rome \Tor Vergata", via della ricerca scientifica, 00133, rome, italy E-mail: [email protected] Abstract: I briefly review some of the perturbative methods, standard perturbation theory, heavy quark effective theory, chiral perturbation theory, large Nc QCD, lattice QCD, used to tackle the theory at different regimes. I also discuss some of the non-perturbative results obtained within lattice QCD 1. Apologies is fulfilled if Ψ is minimally coupled to the gauge field: These proceedings have major defects. First, as for the references, I only quote the ones I used ∂µΨ(x) → (∂µ − ie Aµ)Ψ(x) (2.2) for preparing them. These are mostly review and if the gauge field Lagrangian is invariant articles and the reader will find there more ex- under tensive citations. Second, I present a survey of perturbative methods for QCD, including at the Aµ → Aµ +1/e ∂µα(x) (2.3) end a part concerning non-perturbative numeri- cal results. The details on each of the methods i.e. is of the form: discussed are necessarily limited: they should be intended as a guide through some of the keywords L(A )=−1/4F F µν of each subject, written in italic. Anyway, I could µ µν − not escape for a second time from writing them Fµν = ∂µAν ∂ν Aµ (2.4) without getting into troubles with my friends The Lagrangian is quadratic in the gauge Nikos Trachas and George Zoupanos, whom I field. thank again for the pleasant athmosphere of the In the case of an SU(3) gauge symmetry, the school. phase transformation affecting the color degrees of freedom (ic): 2. Introduction Ψ(x)ic → (eiΩ(x))icjc Ψ(x)jc (2.5) The theory is characterized at microscopic level where by the presence of a gluon self interaction with a X icjc icjc k strength related by symmetry to the one of the Ω(x) = Tk α(x) (2.6) quark gluon coupling. This is due to the non k icjc abelian property of the gauge group. with Tk the 8 traceless and Hermitean gen- In the well known case of QED, the gauge erators of the SU(3) group. symmetry is abelian and the invariance of the The covariant derivative acts non-trivially in action under a local phase rotation of the Fermi colour space: fields, icjc icjc − icjc Dµ = ∂µδ ie Aµ (2.7) Ψ(x) → eiα(x)Ψ(x) (2.1) where Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio X hadrons, where only a single external invariant icjc icjc k Aµ = Tk Aµ (2.8) exist, i.e. the total center of mass energy. In this k case, the scaling law is very simple and states There are as many gauge fields as group gen- that the “adimensional cross section” is a con- erators. In this case, to preserve the invariance stant of the fermion action, the gauge field must sat- isfy a tranformation that consists of a shift of the σ(e+e− → hadrons) × E2 = constant (3.1) field ( like in the abelian case) AND of a color cm rotation: The next case, is the cross section for the hadron inclusive deep inelastic scattering of a lep- ton off a hadron. Here there are two external → Aµ Aµ +1/g ∂µΩ+i[Aµ,Ω] (2.9) momenta, the momentum transfer of the scat- The gauge action invariant under such a trans- tered electron, q , and the momentum of the tar- formation is non-quadratic and contains cubic get hadron, P . One can the form two indepen- and quartic self-interactions. det kinematic invariants, the momentum trans- fer squared, q2, and the scalar product Pq .The dimensionless cross section ( the structure func- µν L(Aµ)=−1/4Gµν G tion) can only depend, in the scaling limit, upon the dimensionless ratio −q2/2pq = x Gµν = ∂µAν − ∂ν Aµ + ig[Aν,Aµ] (2.10) Bjorken In the two examples above, the validity of The gluon self interactions lead to two ma- scaling laws suggests pointlike interactions of the jor consequences: i) the property of asymptotic photon with the hadron constituents, the charged freedom i.e. a weakly interacting theory at small quarks in this case, only affected by small correc- distances, ii) the confinement of colour charges tions governed by the small value of the strong at large distances. coupling constant experienced at large transferred An analytic solution at all distances is still momenta. The existence of a coupling “running” missing and various perturbative methods have with the scale is typical of renormalizable theo- been devised, with a lowest order supported in ries. For these theories, predictions for physical each case by specific experimental facts. quantities are free of ultraviolet divergences only In Table 1, shown at the end, I give a sum- when expressed in terms of renormalised coupling mary of the perturbative methods that I will be constant and masses, i.e. of parameters fixed briefly reviewing in these lectures, by recalling by some independent experimental input at some 2 the corresponding expansion parameter, the ba- reference physical scale q0. sic degrees of freedom, the experimental fact sup- Different choices of the reference scale bring porting the approximation and the name. to different values of the renormalized coupling in such a way that the final physical predictions 3. Perturbative QCD to all orders of perturbation theory are identical. The renormalized coupling as a function of the The experimental evidence that a perturbative renormalization scale moves on a curve of “con- expansion in the coupling constant ( very much stant physics” fixed by the integration constant like in QED) might be a good approximation un- of the renormalization group equation: der specific conditions is provided by the validity of scaling laws for processes with large momen- dα(Q2) = β(α(Q2)) = tum transfers: adimensional physical quantities, dlog(Q2) like cross sections multiplied by suitable powers −b α(Q2)2 + b α(Q2)3 + ... (3.2) of the center of mass energy, depend upon di- 0 1 mensionless invariants made from EXTERNAL The first term in the expansion has a nega- momenta only. A prototype of these processes is tive sign, opposite to the one of QED and leads the total cross section of electron positron into to a solution of the form 2 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio the collinear logaritms is then replaced by the 2 2 2 2 α(Q )=1/(b0log(Q /Λ )) (3.3) factorisation scale q0. Again, a judicious choice of the factorisation scale close to the “running where Λ defines uniquely the theory (apart scale” avoids the presence large logarithms in the from quark masses) and is renormalisation group perturbative series. The “running” with the fac- invariant. With increasing Q2, the coupling con- torisation scale of the collinear logarithms is gov- stant decreases and opens the way to a perturba- erned by the DGLAP equations that need input tive treatment of strong interactions. The iden- parton densities as boundary values. For the sim- tification of the large momentum transfer of the plest case of the “non singlet” quark densities ( process with the renormalisation scale of the “run- i.e. valence quark distributions”) the equations ning” coupling resums the potentially large loga- are of the form: rithms of the ratio of the scale of the process over the renormalization scale (“renormalization scale dqNS(x;q2) logs”) that could make the effective expansion 2 R dlog(Q ) parameter of order one and makes the pertur- 1 = dy P qq(x/y)qNS(y,Q2) (3.4) bative approximation truncated at a fixed order x y more reliable. Figures 1 and 2 show the “experi- More complicated coupled equations hold for mental” running of αS and the determination of the singlet case. The “evolution probabilities” its value at the Z0 mass reference scale. P qq(z) can be perturbatively expanded: The smallness of the coupling constant at small distances does not imply necessarily a good qq qq 2 qq convergence of the perturbative expansion. As P (Z)=αP1 (z)+α P2 (z)+... (3.5) we have seen in the case of a bad choice of the renormalization scale, each power of the coupling iii) processes with EXCLUSIVE kinematics, in the series may appear multiplied by a loga- sensitive also to “infrared logarithms” arising from rithm of the “running scale”. The “renormal- incomplete cancellation among real and virtual isation scale logs” are not the only potentially emissions of soft gluons. As an example, when dangerous for the convergence of the perturba- the Bjorken variable x tends to 1, the evolution tive expansion [2]. Indeed, hard processes can be probabilities contain large corrections due to the divided into the two following classes: presence of double logs of the type i) totally inclusive processes , like e + e−→ − 2 N hadrons, are “long distance insensitive”, i.e. do (αlog(1 x) ) (3.6) not show in perturbation theory a singular de- These are resummed by suitable techniques pendence upon quantities related to the non per- generalizing the exponentiation of infared singu- turbative aspects of the “long distance” physics larities of QED. of QCD, Figure 3 shows the agreement of the pertur- ii) inclusive, but “long distance sensitive”, bative predictions incuding the resummation of like the deep inelastic scattering, where the per- next-to-leading logarithms, i.e. those appearing turbative corrections to the lowest order contains raised to a power equal to the one of the pertur- logarithms of the running scale over the quark bative expansion minus one, for the hadronic jet mass or over any long distance regulator that inclusive cross section at the Tevatron.
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