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
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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. tames the singularities of collinear emission of iv) processes affected by “low x “ logarithms massless quanta from massless quanta. These in the gluon evolution probability, affecting the “collinear singularities” are universally factoris- behaviour of singlet structure functions at very able and can be “renormalized” like the usual ul- small values ( 10−4 ) of the Bjorken variables x: traviolet singularities, i.e. reabsorbed into FUNC- TIONS ( the parton distributions) normalized at 2 gg 2 some factorisation scale Q0. The “long distance Plowx = α(1/x + αlog(x)/x +(αlog(x)) /x + ...) regulator’ in the ratio over the running scale in (3.7)
3 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
The resummation implied by the DGLAP is expected to diverge FACTORIALLY for equation, leading log in the factorisation scale, large n: leads to a gluon distibution behaving at low x’s n like: fn ' n!b (3.13) √ The terms of the series first decrease and ' log 1/x g(x) e (3.8) have a minimum at n = n∗,withn∗'1/αb. The explicit resummation of the “low x lead- Nevertheless, the series can be “resummed” ing logs” is done via the BFKL equation and by making a Borel transform: leads to a behaviour like: X n fnt − B(f(t)) = f(0)δ(t)+ (3.14) g(x) ' x λg (3.9) n! n However, next-to-leading-log (NLL) correc- For the terms of the new series the factorial tions to the BFKL resummation are very large growth has been cancelled out and the original and render a bit questionable the convergence of quantity can be recovered by the inverse trans- the approach: form:
NLL LL − Z λg = λg (1 3.5αS), (3.10) ∞ −t/αS f(αS)= dte B(f(t)) (3.15) with 0 This is not possible if there are poles for λLL =2.65α (3.11) g S B(f(t)) along the integration path: the possibil- The best fits to the Hera data are today ob- ity of going complex and circumventing the pole tained by using NLL DGLAP equations together leads to an ambiguity in the choice of the path, with a a steep distribution of the gluon distribu- above or below the pole. The difference of the tion. and are shown in figure 4. two paths, i.e. the ambiguity, just amounts to The choice of the renormalisation and of the a closed circle around the pole that can be es- factorisation scales is irrelevant to all orders of timated by the Cauchy theorem. For example, perturbation theory, but it may affect a finite if: order prediction in a significant way. Altough 1 on general grounds it should be chosen “close” B(t) ' , (3.16) t − t0 to the “running scale”, its actual choice is ar- the uncertainty will be proportional to: bitrary and only the availability of higher order corrections will reduce the sensitivity of the pre- 2 dictions to such a choice [1]. Three loops calcu- et0/αS (Q ) ' lations start to be available, for example, for the 2 2 e−t0b0log(Q /Λ ) = hadonic width of the Z0 thatcanbeusedtoex- 2 2 t0b0 tract the value of the strong coupling constant at (Λ /Q ) (3.17) that scale,reaching an absolute error of 0.002. i.e. the ambiguity of the perturbative series The ultimate limitation of perturbative cal- is a power correction and is removed only by the culations comes from power corrections; i.e. from full inclusion of all power corrections. One then contributions of the type (ΛQCD/Q)P . EXPECTS a power correction to be present to The power P can be related to the conver- compensate the ambiguities of the perturbative gence of the perturbative expansion. Indeed, the series. By looking at the location of the poles generic perturbative expansion: of the Borel transform, one can predict the exis- tence of corresponding power corrections in the X n+1 cases where alternative ways to determine their f(αS)=f(0) + fnαS (3.12) n=0 presence are lacking.
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In QCD, there are poles of the Borel trans- QCD where the dominant term in the beta func- form on the real and positive 00t00 axis related to tion comes from the gluon vacuum polarization, the behaviour of the coupling constant at low val- the coefficient b0 is positive and from eq. 3.22, ues of the virtual momentum running in loop in- one obtains that the pole of the Borel transform tegrals of hard processe: they are called “renormalons”lies on the positive real axis, leading to an ambi- [3]. As an example, I report the case of the cor- guity of the form: relation of two vector currents: (Λ2/q2)a (3.23) R 4 iqx d xe < 0 | T (Jµ(x)Jν (0) | 0 > with a = 2 in this particular case. 2 2 From the Operator Product Expansion (OPE) =(qµqν −q gµν )Π(q ) (3.18) of the two currents we know indeed that the lead- A sketch of the derivation in the limit of large ing power corrections are of the type found from number of flavours goes through the following the localization of the first( the closest to the real steps: axis) renormalon pole. 1) define: The information coming from the presence of renormalons in the perturbative series is then “redundant” when there is an OPE, but may be 2 2 2 2 D(q )=4π dΠ(q )/dlog(q ) (3.19) used as an indication for the leading power cor- rections affecting processes that do not admit an The diagrams leading in the above mentioned operator product expansion analysis. limit are the iteration of fermion bubbles in the vacuum polarisation and can be resummed through the renormalisation group running of the cou- 4. Heavy Quark Effective Theory pling constant, with the result: A second useful approximation is the expansion Z around the limiting case of an infinitely massive dk2 2 ' 2 2 quark. The effective Lagrangian describing the D(Q ) 2 α(k )F (k ) Z k leading term in the approximation exhibits ad- X ∞ dη = α F (η)(−b(Nf ) ditional symmetries with respect to the ordinary S η 0 n=0 0 QCD Lagrangian [4]. The basic step is the sepa- n αSlog(η/C)) (3.20) ration of the heavy quark four momentum q into a dominant term proportional to the quark mass 2 2 2 (Nf) where η = k /q , αS = αS(q )andb0 and into a residual one, according to:, is the term of the leading order beta function coefficient proportional to the number of flavours qµ = Mvµ +kµ (4.1) ( negative with the convention of eq. 3.2). In the low η region F (η) ' A × (k2)a and the integral Correspondingly, the Dirac spinor ψ(x)can over η can be performed with the result: be factorized as follows:
−iMvx ˜ X∞ ψ(x)=e ψ(x) (4.2) D ' αn+1n!(b(Nf )/a)n (3.21) S 0 where, in turn: n=0 The Borel transform is easily calculated: 1+/v 1−/v ψ˜(x)= Q + χ− (4.3) 2 v 2 v X 1 The decomposition up to now is fully gen- n (Nf) n ' B(t)= t (b0 /a) (3.22) eral, with Q and χ− eigenstates of the velocity a − (b(Nf )t) v v n 0 operatorv / with eigenvalues, respectively. The derivation strictly relies on the large flavour By inserting the decomposition above in the number limit. If extended to the real case of QCD Lagrangian:
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of total angular momentum 0 and 1 respectively. ψ¯(i/D − M)ψ (4.4) In the angular momentum 1 sector, a degener- acy is expected for the states D∗ − D ,where we get: 2 1 the light spin combines with the orbital angular momentum to give a partial angular momentum
(Q¯ v +¯χ−v)(M(/v − 1) + i/D)(Qv + χ−v) 3/2 and the total angular momentum is 2 and 1 respectively, and for the states where the orbital- = Q¯vivDQv − χ¯−v(ivD +2M)χ−v T T light partial angular momentum is 1/2andthe +Q¯ iD/ ,χ− +¯χ− iD/ Q (4.5) v v v v total is 1 and 0 respectively.The latter have not T µ where Dµ v = 0. The field χ−v has a mass been observed experimentally because, according of 2M, a quadratic action and can be integrated to M.Wise, they are too wide to be found, The out to derive the effective Lagrangian for the Qv splitting among members of a doublet is induced field: by the spin dependent part of the order 1/M la- grangian , i.e. by the second term in the equation 1 cited. L = Q¯ (ivD + iD/ T iD/ T )Q (4.6) eff v 2M + ivD v The relevant hadron matrix element respon- sible for the splitting is: The Heavy Quark Effective Lagrangian is ob- tained by expanding in inverse powers of M the <σH ·Bl > (4.9) expression above. In particular, to the first or- der, one gets: where the Bl is the chromomagnetic field generated by the light quark. The matrix el- 1 L = L0 + L1 (4.7) ement above, taken between the members of a 2M doublet ( denoted by +/−), is given by: where,
H l 0 · − L = Q¯vivDQv < (σ B ) >= 1 T T ∗ L = −Q¯vD/ D/ Qv n /4 constant (4.10) = −Q¯ D2 Q − gQ¯ σ Gµν Q /2 (4.8) v T v v µν v where n =(2S+1)withS is the total The leading term (L0) is diagonal in Flavour spin of the state. The mass formula for an ele- and Spin space ( Spin-Flavour symmetry). The ment of the doublet can be finally parametrised coefficient of the first term in the next-to-leading as: Lagrangian L1 can be shown to be “protected” to all orders by the invariance under redefinition ¯ λ1 n λ2 of the “residual” momentum in eq. 4.1, while MH = M + Λ − (4.11) 2M 2M the one of the second term, proportional to the strong coupling constant g,isnot. The splitting inside a doublet depends upon The effective Lagrangian leads to many phe- the parameter λ2, that only depends upon the nomenological predictions: I will only quote some light system partial angular momentum. From ∗ 2 of them. B − B splitting, one gets: λ2 =0.12GeV and ∗ 2 The first refers to the mass formulae for heavy- from D − D splitting, λ2 =0.10 GeV , rather light mesons: in particular, the independence of consistently. − ∗ the leading Lagrangian from the heavy quark From the D1 D2 “doublet” one expects in 0 2 spin, leads to the prediction of a degeneracy among general a different value and gets: λ2 =0.013GeV . doublets of mesons with a given spin for the light The second application of HQET is repre- quark and different values of the heavy quark sented by the “Isgur-Wise” function, a universal spin. In the zero orbital angular momentum sec- function for semileptonic heavy → heavy0 de- tor, this applies to D − D∗ and B − B∗ states cays, crucial for the determination of the weak
6 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio interaction CKM mixing matrix. To leading or- der in 1/M , the momentum trasfer to the lep-
h+(w)=hv(w)=hA1(w)=hA3(w)=ξ(w) Hab = Q¯aqb (4.14) h−(w)=hA2(w) = 0 (4.20) with the following properties: The reduction of the form factors for the vec- tor matrix element can also be obtained from the 1+/v conservation of the vector current between the H = H 2 different flavours in the infinite quark mass limit: ∗ ' Tr(Hγ5) pseudoscalar It follows that: ∗ Tr(Hγ5) ' pseudoscalar
Tr(Hγµ) ' vector (4.15) qµ
µ 5 that leads to: h−(w) = 0. Fermion charge where Pv and Pv are the vector and pseu- doscalar meson fields, respectively. conservation also leads to the normalisation con- As an example, we derive the constraint on dition for the Isgur-Wise function at zero recoil, the form factors describing the matrix element i.e. at w =1, in the infinite quark mass limit: of a vector current among pseudoscalar heavy mesons: ξ(1) = 1 (4.23)
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5. Chiral perturbation theory The “potential” is a function of ρ2 = σ2 +π2: the symmetry of the model can also be seen as The expansion parameter of this approximation 2 an O(4) invariance in the for the vector (σ, π). Eπ is 2 where E is the energy of a light pseu- (4πfπ) π There is a total analogy with the Higgs sector doscalar. Indeed, the method relies on the con- of the standard model, and, as in that case, the struction of an effective theory among psudoscalars spontaneous breaking of the symmetry depends that embeds the spontaneous breaking of chiral upon the location of the minimum of the poten- symmetry of QCD. In the QCD Lagrangian, at tial. A spontaneously broken solution is obtained zero quark mass, both vector and axial phase when the mass term is negative and the location transformations for the fermion field: of the minimum is at: p ρ = −12µ2/λ (5.6) δψ = iαψ The “unbroken” solution has a minimum at δψ = iαγ ψ (5.1) 5 the origin that is obviously invariant : are exact symmetries, at classical level. In a → + 0 VR0VL =0 (5.7) theory with Nf flavours, the symmetry extends to SU(Nf)left × SU(Nf)right × U(1)vector,(the the residual invariance of the broken solution U(1)axial is broken by the anomaly at quantum depends upon the direction of the breaking: by level) and it is realised “a’ la Goldstone”, i.e. is choosing it proportinal to the identity in the no- spontaneously broken to U(Nf )vector. tation of equation 5.2, i.e. in the direction of the The effective theory of light pseudoscalars for σ field, one gets: Nf = 2 can be constructed starting from a linear sigma model for a set of scalar fields organised in ρ · 1 → ρ · V · 1 · V + a matrix as follows: R L · · + = ρ VR VL (5.8) Φ=σ+i~τ · ~π (5.2) IF, VR = VL, the minimum is invariant. There with the following most general renormalis- is a residual symmetry left unbroken represented able Lagrangian: by the SU(2)vector trasnformations, like in the QCD case. The physical set of states is found by expand- ∗ L = Tr(∂µΦ ∂µΦ) ing around the new minimum, i.e. by shifting the −µ2/2Tr(Φ∗Φ) − λ/48Tr((Φ∗Φ)2) (5.3) σ field:
The Lagrangian is invariant under the trans- σ ≡ S + ρ (5.9) formation: The Lagrangian in the shifted fields has now a positive mass term for the S field and a zero → + φ VRΦVL (5.4) mass term for the ~π fields: a Lagrangian for the ~π fields alone is obtained by sending the S where VL,R are unitary 2 × 2 matrices. foot- field mass to infinity while keeping the position note ?? The model exhibits an SU(2)left×SU(2)right symmetry. Note that the φ field transforms like of the minimum constant. This limit transforms an interpolating field for a “meson” psudoscalar the “double well” section of the potential into a “double delta function” potential, with a fixed made of qleft − qright¯ pair. The Lagrangian writ- tenintermsofσand π fields, reads: modulus for the O(4) vector:
σ2 + ~π2 = ρ2 (5.10) 2 2 L =((∂µσ) +(∂µπ) )/2 and the model becomes the non-linear sigma −µ2(σ2 +π2)−λ(σ2 +π2)2/4! (5.5) model.
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By measuring the fields in “ρ units, the con- predictions finite. However, these terms are or- dition above simply states that the original Φ ganized in incresing powers of the expansion pa- matrix is an SU(2) matrix. The field S is now a rameter and can be neglected if the energy of function of the ~π fields and the matrix Φ is more the pions much smaller than 4πfπ, under the as- conveniently reparametrised as: sumption that the finite part of the counterterms is not anomalously large. Φ=ei~τ ·~π/ρ (5.11) The next order (O(E4)), where there are 12 new operators entering the Lagrangian with co- The expression for Φ, expanded to the second efficients related to experimental quantities, still order in powers of ~π, reduces to the original one retains some degree of predictivity. with S expressed, to the same order, as a function Interesting extensions of the low effective La- of ~π. grangian approach are beingdeveloped for vector The non linear Lagrangian contains only a mesons and for baryons. kinetic part: the potential enforces the constraint between S and ~π fields already taken into ac- count: 6. The large Nc expansion
The approximation consists in considering the L = ρ2Tr(∂ Φ+∂ Φ)/4 (5.12) µ µ theory in the limit of large number of colours One can extract the Noether axial current: where only a subset of diagrams survive, preserv- ing the main non perturbative aspects of the the- δS J~5 = ~δ5Φ ' ρ∂ ~π (5.13) ory at finite colour number ( three...) [5]. The µ δ(∂ Φ) µ µ diagrammatic simplification was first noticed by in the one pion sector. G.’t Hooft and it is best seen by adopting his Its matrix element between a pion and the notation for describing fermions and gluons. vacuum is The first possess only one colour index and their propagation in a diagram is indicated by a line with an arrow distinguishing the fermion < 0 | J~5 | π>= µ from the antifermion case. The second belong | | ρ<0 ∂µ~π π(q)>=ρqµ (5.14) to the adjoint representation, have an upper and a lower index and are represented by two paral- By comparing with the well known PCAC lel lines with opposite arrows. To each ordinary relation, we identify ρ with f and fix the only π Feynman diagram corresponds a “color” diagram unknown constant of the effective Lagrangian for constructed according to the notation described. massless pseudoscalar interactions. Indeed, by The color lines can be open or closed. In the lat- expanding to higher orders in the “pion” fields, ter case, a color index is summed and a factor one get multifield interacting terms. The first is N is gained for the diagram. As an example, a four pion interaction with strength: c the lowest order correction due to a quark loop to the photon vacuum polarisation contains a sin- ∂ ~π2∂µ~π2/ρ2 (5.15) µ gle closed colour line ( the fermion loop) and is 2 2 Proportional to Eπ/fπ, i.e.to the expansion of order Nc. parameter of the χpT approach. To the next order, the first non trivial in The theory is NOT renormalizable ( the price the strong coupling constant, one can exchange a of decoupling the S field...), very much like the gluon inside the quark loop generating a color di- 2 2 standard model without the Higgs field. Loop agrams with two closed loops, i.e. of order Nc g , corrections need the presence of higher dimen- where I have also included the order in the cou- sional operators with proper coefficients to cancel pling constant for future use. the ultraviolet divergences. At order g4 there are three possible diagrams: In principle one should include all countert- two with the exchange of two gluons, one planar erms ( an infinite number) needed to keep the and another non-planar where the two gluon lines
9 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio cannot be drawn in a plane without crossing them, determinations of the finite parts of the coun- an a third diagram with two quark loops con- terterms of the order E4 Lagrangian compared to nected by two gluons. The corresponding “colour” their relative relevance in the large Nc limit. The diagrams have three colour loops, one colour loop average value of the group of O(1) is about 0.4 and two colour loops, respectively. Their rele- and the one of the group of O(Nc)isabout3.3, vance is then very different in the large Nc limit. showing again the qualitative agreement with large Indeed, in this limit, the expansion is governed by Nc predictions. 2 ≡ 2 the “effective coupling” gN Ncg and the limit Unfortunately, although the approximation should be taken while such a coupling is kept seems a very promising one, no real quantita- fixed. The three diagrams in the new coupling tive progress has been achieved along this line 4 4 2 4 are of order: NcgN , NcgN /Nc and NcgN /Nc re- for QCD, while it has been successfully exploited spectively. Notice that the extra fermion loop, in various 2-dimensional models. the third diagram, costs an extra inverse power of Nc. This appears also in the one-loop expres- 2 L value order sion for the beta function of the gN coupling, i where the fermion contribution to the vacuum polarisation is suppressed by an inverse power of 2L1 − L2 -0.60.5 1 Nc. L4 -0.3 0.5 1 Qualitative important consequences of the L6 -0.2 0.3 1 fermion loops suppression in the large Nc limit L7 -0.4 0.2 1 are the prediction of infinitely narrow qq¯ reso- L2 1.4 0.3 Nc nances: their decay needs the formation of an L3 -3.5 1.1 Nc extra qq¯ pair. Experimentally, such a suppression L5 1.4 0.5 Nc seems at work in the Φ decay that occurs pref- L8 0.9 0.3 Nc erentially in the KK channel, unfavoured by the L9 6.9 0.7 Nc restricted phase space, instead of the ππ channel L10 -5.5 0.7 Nc ( the Zweig rule). The latter would require an Table 2: Large N order of finite counterterms of the extra fermion loop. O(p4) χpT Lagrangian An interesting recent application of large Nc suppression is the estimate of the relevance of finite parts of the counterterms of the effective Chiral Lagrangian. The leading lagrangian con- tains only a single flavour trace ( remember that 7. Lattice QCD the group is a flavour group) and among all dia- grams, there is a leading subset in the large Nc The basic approximation of lattice QCD is the limit with a single fermion loop. The situation is discretisation of the space time: quarks and glu- different for the next-to-leading lagrangian: there ons live on a 4-d crystal. In particular, fermions are terms with a single flavour trace, like: live on lattice sites. A fermion field ψA(i, j, k, l) is a function of four discrete coordinates [6] [7]. + + The theory on the lattice is the Euclidean ex- L3 × Tr(∂µΦ ∂µΦ∂νΦ ∂νΦ) (6.1) tension of the origina Minkowski theory, where and terms with two flavour traces, like: the time has been rotated to the imaginary axis, but the original continous O(4) symmetry of the theory has been replaced by a discrete hypercu- + + L2 × Tr(∂µΦ ∂µΦ)Tr(∂νΦ ∂νΦ) (6.2) bic symmetry. The main appeal of the lattice formulation of the theory, originally proposed by In the second case, the two traces require K.Wilson , is that gauge invariance can be main- a second fermion loop and a suppression is ex- tained EXACT. The explicit construction can be pected. In Table 2, I report the experimental found by “undoing” the limits of ordinary dif-
10 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio ferential calculus, i.e. by introducing the “finite The simplest circuit that can be formed is the difference derivatives: elementary plaquette PL, i.e. a square around four neighbouring corners. The Wilson action → ∇ ∇∗ ∂µψ 1/2( µ + µ)ψ (7.1) for the gauge fields is just the sum of all possible plaquettes. where The “naive continuum limit” of the action is obtained by expanding in powers of the lattice 00 00 ∇µψ(x) ≡ ψ(x + µ) − ψ(x) spacing, traditionally called a . For this, we ∗ need the expression relating the gauge link to the ∇ ψ(x) ≡ ψ(x) − ψ(x − µ) (7.2) µ ordinary gluon fields of the continuum theory: The request of LOCAL gauge invariance reads: Z x+µ µ Uµ = exp(i dxµA ) (7.8) ψA(x) → GAB(x)ψB (x) (7.3) x where GAB (x) is a unitary SU(3) matrix, The plaquette is easily handled in the abelian is violated( as in the continuum) by the finite case: the results is the exponential of the curl of difference derivative: ψ(x)andψ(x+µ) trans- the gauge field around the elementary square: form differently under a LOCAL gauge rotation, they are at different sites, and cannot be com- R P laquette ' exp(i dx Aµ) bined together in the finite difference derivative. R perimeter µ µν Again, the solution is the introduction of a field = exp(i surface dσµν F ) 4 µν Uµ to match the gauge phase rotation of neigh- ' 1 − a F Fµν ... (7.9) bouring fermions. If: up to terms of higher order in the lattice
Uµ(x) → G(x)Uµ(x)G(x + µ) (7.4) spacing or that vanish by the antisymmetry of the gauge field tensor Fµν . then : The Dirac term is also reduced to the usual continuum expression, by neglecting higher or- Uµ(x)ψ(x + µ) → G(x)Uµ(x)ψ(x + µ) (7.5) ders in the lattice spacing: A kinetic gauge invariant lattice action for the fermions can be constructed: Uµψ(x + µ) ∼ (1 + iaAµ)ψ(x)+a∂µψ → 1 ψ(x)+a(∂µ +iAµ)ψ(x)+...(7.10) L = ψ(¯x)( ∇(U)+∇∗(U))ψ(x) (7.6) fermions 2 Notice that the coupling constant has disap- where peared in the covariant derivative. This comes from a redefinition of the gauge fields that now incorporate the coupling constant. As a conse- ∇ (U)ψ(x) ≡ U ψ(x + µ) − ψ(x) µ µ quence, the pure gauge action appears DIVIDED ∇∗ ≡ − + − µ(U)ψ(x) ψ(x) Uµ ψ(x µ) (7.7) by the coupling constant square. The properly normalized lattice action is then: The gauge fields are represented by the group elements, instead of the group generators, they X L = β PL (7.11) “live” on the links connecting neighbouring lat- gauge plaquettes tice sites ( they are also named “links”). A gauge 2 invariant interaction among them is the colour where β ≡ 2Ncolours/g . trace of any closed circuit, the “Wilson loops”. It The above limit is called “naive” because it is straightforward to check that the cyclic prop- refers to a classical continuum limit. In the quan- erty of the trace ensures the invariance, under the tum theory, the lattice spacing acts as an ultra- transformation of eq. 7.5, of the Wilson loops. violet cutoff ( the fields cannot vary at distances
11 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio smaller that the lattice spacing) and the con- The plaquette correlation provides a test of tinuum limit is affected by divergences if taken the mass of the states surviving at large dis- naively without a proper renormalization proce- tances. dure. Quantum averages are obtained from func- P < 0 | PL PL | 0 >= tional integrals of the type: P ~xP (~0,0) (~x,T ) 2 ipnx (|< 0 | PL ~ | n>|) e n ~x P(00) − → | |2 mnT (Euclidean) n Cn e (7.13) R <φ1φ2...φn >= (−S (φ)) DφeR E φ1φ2...φn For large time separation, only the lowest (− ( )) (7.12) Dφe SE φ energy state with the quantum numbers of the source survives. A free multigluon state has zero where the real exponent instead of a complex lowest energy, therefore the existence of a “mass one is a consequence of the Euclidean rotation. gap”, i.e. of a non zero mass state provides evi- Quantum averages appear like statistical av- dence for colour confinement. erages on a 4-d system in the canonical formu- In the strong coupling approximation, lattice lation. The usual sum over states of statistical QCD confines. Technically, this is a consequence mechanics is replaced by the functional integral of exact gauge invariance of the lattice functional that on a discretized space-time is a functional measure and of the action. This implies that the sum over possible gauge and fermion fields “con- link integral: figurations”. Analytic estimates for the quantum averages R DU U → (on the lattice) can be obtained in the language of stat-mech, in R Qµ µ the “high temperature approximation”, i.e. in i,µ dUµ(i)Uµ(i) = 0 (7.14) the limit when βBoltzmann → 0. From the ex- pression for the gauge action of eq. 7.11, this should be zero. Indeed, its matrix elements limit corresponds to the “strong coupling expan- can be arbitrarily changed with a gauge transfor- sion”, i.e. g2 →∞In normal perturbation the- mation and the the only invaraiant value is zero. ory, one solves exactly the free part of quantum This implies that after expanding the whole correlations and expands the rest in powers of action, the non zero integrals always imply that the coupling constant, in this case the whole ex- a link must appear with its hermitian conjugate ponential is expanded in a power series in β. (or powers of U − U + pairs). This approximation reproduces an important The connected correlation of the two plaque- feature of QCD, colour confinement, and is taken ttes at large time separation: as a good starting point of the lattice QCD ap- proach, with some caveats to be seen. A confinement test can be constructed from Q R < PLP1PL2 >= i dU exp(−β plaquettes)PL1PL2 the connected correlation of two plaqettes, when iµQ µR P dUi exp(−β plaquettes) the relative distance becomes very large. Each i,µ µ plaquette is a “gluon source” that can be decom- → Q R P( strongP coupling) i n posed into a sum of terms with a given number dU (−β plaquettes) /n!PL1PL2 iµQ µR P P (7.15) of gluons by expanding to an arbitrary order the dUi (−β plaquettes)n /n! i,µ µ expression for Uµ as a function of Aµ, starting from a 2 gluon state. If confinement is ABSENT, Remembering that links must overlap pair- massless gluons are the good degrees of freedom wise, the leading term is the one with a “tube” also at large distances, if it is PRESENT, only with the two plaquettes in the correlation as end- colour singlets survive at large distances, with a caps. Up to irrelevant constants , one gets for the non zero mass. time dependence of the correlation:
12 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
knowledge of the Λ parameter of QCD, trans- lated into the particular renormalization scheme < PL1PL2 > represented by the lattice regularization, provides ' (4β)T ' e−T log(4/β)) (7.16) an estimate of the cutoff in physical units, given i.e. a “mass gap “ M ∼ log(4/β). the value of the bare coupling. The strong cou- In the same strong coupling limit, “meson” pling approximation holds at most when g2(a) ' correlations, i.e. the correlations to leading order 10 that corresponds to a lattice spacing of a Fermi of two point-like currents made of a q − q¯ pair, or slightly below. The QCD dynamics is not “re- are obtained from a sum of paths where the q −q¯ solved” by such an approximation and the issue pair do not separate at all: a non zero “internal” is wether the important property of confinement area between the path of the quark and the one persists to higher values of β,i.e. for relatively of the antiquark would be penalized by a factor: small values of the bare coupling where: βarea. −1 Overall, the strong coupling expansion pro- a ΛQCD (7.17) vides a good starting picure of confined quarks Entering the “weak (bare) coupling region” and gluons. However, such a limit of the theory in a non perturbative way can only be achieved is very unphysical and might be very misleading: numerically. The discretization on the lattice also “compact QED” ( a version of QED on the and therefore the reduction of the theory to a lattice) exhibits confinement in this limit, while finite number of degrees of freedom is a prereq- electric charges are well propagating to large dis- uisite for a numerical estimate. tances in reality! Numerical simulations are done through the The point is that the strong coupling limit is following steps. very far from the “continuum limit” of the the- 1) one generates a set of “equilibrium” con- ory, where the lattice cutoff is small enough to figurations, distributed according the Boltzmann produce negligible effects. functional measure. As an example, one a) starts The continuum limit can be seen from a stat- from a random configuration, b) updates the field mech point of view as the approach to a second values LOCALLY until the equilibrium distribu- order ( or higher) phase transition. Indeed, in tion is reached by applying an iterative algorithm LATTICE units, the correlation length increases that leads from a field value to the next one after toward the continuum limit ( the basic length comparing the action in the two cases, as follows: measure is shrinking) like in a phase transition. The transition point is obtained at the “critical 0 φ → φ and Sold → Snew (7.18) temperature” of the system, i.e. at a critical cou- ≡ − pling. From the point of view of field theory, if ∆S Snew Sold < 0 the new value that PHYSICAL units are kept fixed and the lattice decreases the action is accepted, otherwise it is spacing goes to zero or equivalently the UV cut- accepted with a conditional probabilty off diverges. In order to approach a finite the- ≡ −∆S ory, the renormalized one, the (bare) coupling p e (7.19) must be properly tuned to keep physical quanti- The first choice correspond to the search of ties constant. the absolute minimum, i.e. to the classical limit, In both views, the continuum limit occurs the second option takes into account quantrum at a critical value of the coupling. Such a value fluctuations around the minimum. is known from the asymptotic freedom property By iterating this algorithm, one finally ob- of the theory, where the behaviour of the bare tains, after some “thermalization time”, a set of coupling upon the cutoff, at small distances, i.e. equilibrium configurations. large values of the momentum cutoff, can be ex- 2) One makes the functional integral (sum) tracted from a perturbative calculation. The in- by averaging the observables, functions of the finite cutoff limit ( the continuum limit) is ob- gauge fields, over the set of equilibrium config- tained at zero bare coupling, i.e. at β →∞.The urations collected.
13 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
Fermions are a special case. 1) they cannot be associated with C numbers ( they are non- T (Ulocal,Ublock) (7.20) commuting Grassman variables), 2) their corre- The integration over the old , local variables, lations can be worked out analytically, thanks to produces an effective action among the block ones: the quadratic form of the action. Fermion correlations, according to the Wick − e Seffective(Ublock ) = theorem, are sums of products of two point cor- R −S(U )+T (U ,U ) relations given by the “propagators”, i.e. the dUlocale local local block (7.21) inverse of the quadratic operator in the action. Besides, their integration produces a “fermion If the transformation kernel T satisfies: determinant”, i.e. the determinant of the same Z quadratic operator raised to a power equal to the T (Ulocal,Ublock ) number of flavours . The quadratic fermion op- dUblocke = 1 (7.22) erator is a function of the gauge configuration and its determinant represents a NON-LOCAL the partition function in the blocked vari- function of the links, i.e. contains Wilson loops ables is identical to the one in the local vari- of arbirtary length and complexity and not just ables. The integration of the local variables has the elementary plaquette of the pure gauge ac- tinned the degrees of freedom of the theory, but tion. They arise from the wondering on the lat- the “block” theory has the same cutoff effects of tice of virtual fermions. The updating of a single the original one. In the new variables, the lat- link in the lattice action including the fermion tice spacing is larger ( twice the old one) and the determinant requires a calculation that involves physical correlation length in units of the NEW most of the links on the lattice and not just the lattice spacing is SMALLER than the old one: neighbouring ones of the plaquettes containing The expected effective action “moves” in the the updated link. Numerically, this represents coupling constant space away from the critical a formidable task and has delayed for a long point ( small g) and toward the strong coupling time simulations including the effects of virtual region. fermion loops. The effective action itself cannot be expected to have the simple form of the original one. In The last part of these lectures is devoted to general many couplings are generated beside the a summary of some of the important results in “elementary plaquette” coupling and one is forced lattice QCD along its history. to truncate the effective action to a finite set of Strong coupling lattice QCD was invented by coupling, reducing the precision of the blocking K.Wilson in ’74. He wanted to connect the un- transformation. physical strong coupling region to the physical The strategy of the project was to integrate weak coupling region through a renormalisation numerically and iteratively local variables along group approach in configuration space, succes- the “renormalised trajectory” to move from large fully exploited in the analysis of phase transitions β ( small g) to small β ( large g)until one would in statistical mechanics. The idea is to define reach the region of large couplings where an an- an effectiv action for the theory by integrating alytic strong coupling expansion would converge the high frequecy modes of the fields. In figure in a reliable way. 5 there is a sketch of the “blocking” procedure, The project failed mainly because of the lim- leading from the original variables, the small cir- ited computer resources available at that time cles, to the blocked ones, the big circles, defined that required wild approximations for the form on a coarser lattice with a lattice spacing larger, of the effective action with a consequent loss of in the figure, by a factor two than the original precision of the method. one. The basic ingredient of the transformation In ’79, M.Creutz showed numerically that is the coupling of the new block variables to the the confinement property valid in the strong cou- site ones, defined by: pling region indeed does extend to the weak cou-
14 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio pling region by calculating the potential between of calculations made at sizeable quark masses, for two static quarks parametrised as follows: example of the order of a 100 MeV, when the to- tal lattice size does not axceed a couple of Fermi. V (R) ' α(R)/R + kR (7.23) The onset of lattice artifacts, with a given number of lattice points, depends upon the form i.e. as the sum of a Coulomb part and of a of the action that is not unique. They are of or- part linearly rising with the separation, the con- der a2 for the pure gauge part of the Wilson ac- fining potential, governed by the “string tension” tion, but of order a for the Wilson fermion action, k. where a problem not mentioned in this rapid sur- He made simulations showing the persistence vey, the fermion doubling, is solved by explicitely of the string tension up to values of β ∼ 5.4 − breaking the chirality on the lattice. 5.7 corresponding to a lattice resolution of about Indeed, there are two approaches to use the .25Fm. The latter estimate was obtained by frredom in the choice of the action to deal with comparing the value of the string tension in lat- acceptable lattice artefacts. The first is the one tice units extracted from the simulations with the already presented and due to Wilson. The ac- phenomenological value ( extracted from a sim- tion along the renormalised trajectory is “perfec- ple model for resonaces) of about half a GeV. t”, i.e. embodies the CONTINUUM properties, From the 80’s have started the simulations but is very complicated to determine. The sec- aiming to calculate the hadron spectrum. The ond approach, originally suggested by Symanzik, first, done by Hamber and Parisi, used a 53 × 10 consists in a systematic “improvement” of the lattice ! action. The lattice action can be expanded in a Since, there has been an increase in the com- power series in the lattice spacing: puting power by about a factor 106, i.e. from ma- chines with a peak speed in the Megaflop range 0 1 ( a Million of Floating Point Operations per sec- Llattice = Lcontinuum + aLcontinuum+ ond), to present machines close to the Teraflop 2 2 a Lcontinuum + ... (7.24) range. This increase in computing power is much less dramatic when translated into an improve- and the approach aims to eliminate lattice ment of the lattice resolution. The point is that artefacts up to a finite order in the lattice spac- 1 the CPU power scales with the number of points ing. For example, the Lcontinuum part can be per side N like N 4+ where is of order one in eliminated by a single “counterterms” added to the most favourable calculations like those in- the lattice action with a coupling non-perturbatively volving gluons only, but bigger for those involv- tuned. However, physical observables involving ing fermions. This leads to an improvement in composite operators in general also require an the resolution by about a factor 20 at the best. additional operator improvement. A better resolution is not the only obstacle I will present a selection of results for the to precise numerical estimates. As already men- spectrum, quenched and unquenched, for the η0, tioned, the simulations involving virtual fermions, for some quantities “running” with the scale, like 2 i.e. including in the gauge action the effective αS (Q ) orR the first moment of valence quark dis- part due to integration over fermion loops, are tribution dxxq(x, Q2). There many other sub- more demanding by about a factor 100 than those ject where there is a lot of activity in the lat- where fermion loops have been “manually” sup- tice community, and in particular on the calcu- pressed ( the “quenching” approximation). Fur- lation of hadronic matrix elements of the weak thermore, the light quark mass values cannot be Hamiltonian, on the study of the high temper- set equal to the physical ones of a few MeV with- ature properties of the theory, on bound states out stepping into severe finite volume effects that with exotic quantum numbers and on the defin- increase exponentially, for a fixed physical vol- tion of an exact chiral symmetry on the lattice ume, when the pion mass goes to zero. Most of that would allow a non-perturbative study of the the results are obtained by chiral extrapolations standard model. Figure 6 shows the octet baryon
15 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio quenched spectrum as a function of the pseu- mass with respect to the mass of the pseudoscalar doscalar mass squared both normalized by the octet, due to the anomaly. This is a fermion K∗ mass at a fixed lattice spacing slightly be- loop effect and is proportional to the number of low 0.1Fermi,atβ=6.2, and for the standard flavours running in the loop. One of the first es- and the improved cases. The same plots for the timates of the effect was done with a trick that decuplet is in figure 7 where the partial cancella- replaces fermion statistics with boson statistics. tion of lattice artifacts brings into a better agree- In this case the η0 becomes LIGHTER for larger ment with experimental data reprsented by the flavour number. The effect of virtual BOSON crosses. The results for unquenched simulations loops, with a Dirac operator (the “bermions) is are given in figure 8 and 9. In the first, show- easy to simulate by MonteCarlo ( boson fields ing the spectrum, the lowest value of the pion ARE C numbers) and has been used succesfully mass reached is higher than in quenched simula- to find the sign and the order of magnitude of tions given the much higher computing demand unquenching effects. of these calculations. Figure 9 shows the consis- tency of the lattice spacing extracted from differ- Running quantities are difficult to calculate, ent physical quantities, when extrapolated to the because one needs to follow their running over chiral limit. Indeed, in the unquenched case, the many orders of energy scale. This cannot be quark mass enters in the effective gauge action done on a single lattice that can accomodate only through the fermion determinant, changes the scales differing at most by the number of points action and therefore the value of the lattice cut- per side. A finite volume recursive method allows off. Notice that the β values of unquenched sim- to “match” simulations done at different values of ulations are much lower than those of quenched. the lattice spacing and of the bare coupling and If one takes the lattice spacing as function of the to explore a large range of energy scales. These bare coupling from perturbation theory: calculations have only been done in the quenched 2 approximation so far. Figure 11 shows the non- a =1/Λ e−1/(b0g (a)) (7.25) QCD perturbative calculation of the running coupling to be the same in the quenched and unquenched constant. The final precision attainable is of the case, one gets: order of 2 per cent at the reference Z0 mass scale.
2 unquenched gquenched b ∼ 0 (7.26) Recently some effort has been devoted to the g2 bquenched unquenched 0 calculation of the non perturbative running of the unquenched quenched and, since b0
16 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
8. Conclusions
We have been rapidly surveying different analytic perturbative approaches and we have landed on a numerical non-perturbative method. Solving QCD by computer might not be easthetically ap- pealing, but 1) it may clarify the validity of per- turbative approaches, 2) it may “join” the ana- lytic method of the strong coupling expansion at some point 3) as a byproduct pushes the develop- ment of fast algorithms and machines, 4) it is the only method we have to investigate, without fur- ther assumptions, the non perturbative aspects of the theory.
References
[1] R.Bonciani, S.Catani, M.Mangano and P.Nason, hep-ph/9801375. [2] S.Catani, talk given at the XVIII symposium on Lepton-Photon Interactions LP97, Hamburh, aug. 97. [3] M.Beneke, “Renormalons”, Phys. Reports 317 (99) 1. [4] M.B.Wise, “Heavy Quark Physics”, hep-ph/9805468 [5] A.V.Manohar, “Large N QCD”, in “Probing the standard model of particle interactions”, Ed. El- sevier, hep-ph/9802419. [6] H.J.Rothe,”Lattice gauge theories – An Intro- duction”, 1992 World Scientific Publishing [7] I.Montvay and G.Muenster, ”Quantum Fields on a Lattice”, Cambridge Univ. Press, 1994
17 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
perturbation in basic degrees of freedom supported by name
αs quarks and gluons scaling laws perturbative QCD
1 infinitely heavy mesons HQET Mq heavy quarks splittings
( Eπ )2 light pseudoscalars PCAC, χ pT 4πfπ low energy theorems
1 narrow resonances, Zweig rule, Nc expantion Ncolors planar theory sea/valence ratio
1 β ≡ 2 quarks and gluons confinement lattice QCD gbare on a crystal
Table 1: The summary of perturbative approaches
Figure 1: The running of αS
18 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
Figure 2: Determinations of αS at the Z0 mass
Figure 3: Jet inclusive cross section at Tevatron
19 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
Figure 4: Theoretical predictions are compared with observed scaling violations at Hera( DESY-Hamburg)
a
2a
Figure 5: Schematic view of a blocking procedure from local variables ( small circles) to block variables (big circles)
20 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
Figure 6: The spectrum of the baryon octet in the quenched approximation
Figure 7: The spectrum of the baryon decuplet in the quenched approximation
21 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
Figure 8: The state of the art for the spectrum in full QCD
Figure 9: Chiral extrapolation of the lattice spacing in physical units in full QCD
22 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
Figure 10: The static quark potential in full QCD
Figure 11: Non-perturbative running of αS in quenched QCD
23 Corfu Summer Institute on Elementary Particle Physics, 1998 Roberto Petronzio
Figure 12: Continuum limit of step scaling functions for the running of the the average valence momentum in quenched QCD
24