Measurements of a s as a quantitative test of QCD

1. Edinburgh, 6 February ‘03 Measurements of as as a quantitative test of QCD G. Altarelli CERN • A concise review of asymptotic freedom and of the running coupling • Measurements of as and comparison with experiment

2. The Standard Model } } Electroweak Strong SU(3) colour symmetry is exact! The EW symmetry is spont. broken down to U(1)Q Higgs sector (???) Gauge Bosons 8 gluons gA W±, Z, g Matter fields: 3 generations of quarks (coloured) and leptons + 2 more replicas

3. QCD is an unbroken SU(3)Colour gauge theory with triplet quarks: Defs: (CABC: SU(3) str. const., tA: generator repr.) (gAm is a gluon field) (D: covariant derivative) ; (es: SU(3) gauge coupling)

4. Physical QCD vertices m -iesgmtA pmA p+q+r=0 esCABC[gmn(p-q)l+perm] rlC qnB Note: es2 mB lA -ies2 [CABFCCDF (gln gmr- glr gmn) +perm] nC rD

5. QCD is a "simple" theory but with an extremely rich dynamical content: •Complex hadron spectrum •Confinement •Asymptotic freedom •Spontaneous breaking of (approx.) chiral symm. •Highly non trivial vacuum [Instantons, U(1)A symm. breaking, strong CP violation (?)] •Phase transitions [Deconfinement (q-g plasma), chiral symm. restauration,…...] • • •

6. How do we get predictions from QCD? • Non perturbative methods •Lattice simulations (great ongoing progress) •Effective lagrangians * Chiral lagrangians * Heavy quark effective theories ********* •QCD sum rules •Potential models (quarkonium) • Perturbative approach Based on asymptotic freedom. It still remains the main quantitative connection to experiment.

7. Classical gauge th. lagrangian Gauge fixing terms Ghosts Quantisation Feynman rules Pert. Theory Infinities Cut off L Regularisation Renormalisation Redef. of m, as , Zi (w.f. norm'n) Perturbative quantum gauge th.

8. Massless QCD and scale invariance In the QCD lagrangian quark masses are the only parameters with dimensions. Naively we would expect massless QCD to be scale invariant(dimensionless observables should not depend on the absolute energy scale, but only on ratios of energy-dimensional variables) The massless limit should be relevant for the asymptotic large energy limit of processes which are non singular for m->0.

9. This naïve expectation is false! For massless QCD the scale symmetry of the classical theory is destroyed by regularisation and renormalisation which introduce a dimensional parameter in the quantum version of the theory (LQCD). [When a symmetry of the classical theory is necessarily destroyed by quantisation, regularisation and renormalisation one talks of an "anomaly"] While massless QCD is finally not scale invariant, the departures from scaling are asymptotically small, logarithmic and computable (in massive QCD there are additional mass corrections suppressed by powers).

10. Hard processes At the "parton" level (q and g) we can apply the asymptotics from massless QCD to processes with the following properties: • all relevant energy variables are large xi : scaling variables Ei= xiQ Q>>m • no infrared singularities ("infrared safe") • finite for m ->0 (no mass singularities.) To satisfy these criteria processes must be sufficiently "inclusive": • add all final states with massless g emission • add all mass degenerate final states

11. Examples of important hard processes p Q • e+e- -> hadrons P' (p+p')2=s=Q2 At parton level the final state is qq + n gluons + n' qq pairs (i.e. totally inclusive). The conversion of partons into hadrons does not affect the rate (some smearing over a Q bin can be needed for probability 1) • l + N -> l' + hadrons (Deep Inelastic Scattering: DIS) k ' k q p

12. Regularisation and Renormalisation In general: •A dimensional "cut off" L is introduced (must be gauge invariant) • The dependence on the cut-off is eliminated by a redefinition of m, es and Z using suitable renormalisation conditions. Ren. mass: position of the propag. pole. Wave funct'n ren. Z: residue at the pole. The ren. coupling esis, for example, defined in terms of a ren. 3-point vertex at some momenta.

13. In particular in massless QCD: If we start with m0=0 the mass is not renorm. because it is protected by a symmetry (chiral symm.) -> m=0 The coupling es can be defined in terms of the 3-gluon coupling at a scale -m2: p2 Vbare(p2,q2,r2)=ZVren(p2,q2,r2) (Z=Zg3/2ZV) q2 r2 es = Vren(- m2, - m2, - m2) •The scale cannot be zero (infrared sing.)! • - m2<0: no absorptive parts Similarly Zg can be defined by the inverse propagator at p2= - m2

14. Computing all diagrams (with L) + …. + + p2=q2=r2 { { Vren Z Note: VBare depends on L but not on m Both Z and Vren depend on m

15. Renormalisation group equation (We write a for a or as in QED or QCD) In general: GBare(L2, a0, pi2)=Z Gren(m2, a, pi2) so that: or Finally the RGE can be written as:

16. The running coupling QCD or QED The running coupling a(t) is fixed by the beta function: or The m dependence starts at 1-loop: +…. + e e e e

17. By explicit calculation at 1-loop one finds: b(a) ~ + ba2 + ... QED: The sum is over all fermions of charge Qe QCD: b(a) ~ - ba2 + ... nf is the number of quark flavours Recall: If a(t) is small, we can compute b in pert. th. The sign in front of b decides whether: a(t) increases with t or Q2 (QED) or a(t) decreases with t or Q2 (QCD). QCD is "asymptotically free". All and only non-abelian gauge th. are asymptotically free (in 4-dim.)

18. Going back to the equation: this leads to: a(0)=a t=0->Q=m We replace b(a)~±ba2, integrate and do a small algebra. We find: In QCD we have: Note •a decreases logaritmically in Q2 •a dimensional parameter L replaces m. Do not confuse L= LQCD with L =UV cutoff!

19. b(a) ~ ±ba2(1+b'a+….) In general the perturbative coeff.s of b(a) depend on the def. of a, the renorm. scheme etc. But both b and b' are indep. Here is a sketch of the proof: QCD: Taking b' into account: this is LQCD not the cutoff!!

20. Summarising: we started from the massless classical theory and we ended up with QCD where an energy scale L appears (L= L). L depends on the definition of as (i.e. the reg. procedure, the ren. scheme…) and on the number of excited flavours nf . Definition of as We have introduced the ren. coupling as in terms of the 3-g ren. vertex at p2=-m2 (momentum subtraction). The value of as (hence L) in this scheme depends on m. But the most common def. of as is in the framework of dimensional reg. Dim. reg. is a gauge and Lorentz inv. reg. that is most simply implemented in calculations. It consists in formulating the theory in d<4 space-time dimensions.

21. Nowadays the MS definition of as (based on dimensional regularisation) is adopted, because the corresponding renormalisation technique is simplest to implement in complicated calculations. For example, it can be realised diagram by diagram. The third coefficient of the beta function b(a) is also known in MS. Translated in numbers, for nf=5 one obtains: Tarasov, Vladimirov, Zharkov,Yu which means good apparent convergence

22. Dependence of L from nf QED and QCD are theories with decoupling: quarks with mass m>Q do not contribute to the running of a up to the scale Q. So for 2mc<Q<2mb the relevant asymptotics is for nf=4, while for 2mb<Q<2mt nf=5. Going across the 2mb threshold, the b(a) coeff.s change, so the a(t) slope changes. But a(t) is continous so that L4 and L5 are different: a(t) 2mb From matching a(Q2) ---> L5~0.65 L4

23. Measurements of as(mZ) PDG’02 summary on as(mZ) MS Not the Gospel! as(mZ)=0.1172±0.002

24. The main methods for as at LEP/SLC are: • inclusive Z decay, Rl, sh, GZ • inclusive t decay •event shapes and jet rates Inclusive: dQCD is known to NNLO accuracy: dNP are power suppressed (1/Q2)n terms governed by the OPE. Here Q=mZ or mt Clearly the Z case is apriori more reliable because mZ>>mt .

25. At the Z from Rl only (assuming the standard EW theory, (mtexp)): as(mZ)=0.1224±0.0038 Better, one can use all info from Rl, GZ, sh,… and in general take as(mZ) as a parameter to be fitted from the EW precision tests. One obtains: as(mZ)=0.1187±0.0027 LEP1 only: All EW Data: as(mZ)=0.1181±0.0027 The dominant sources of error are mH and higher orders in the QCD expansion. Error from power corrections very small.

26. as from Rt Rthas a number of advantages that, at least in part, compensate the smallness of mt=1.777 GeV: • Rt is more inclusive than Re+e-(s). • one can use analiticity to go to |s|= mt2 Im s |s|= mt2 Re s • factor (1-s/mt2)2 kills sensitivity to Re s= mt2 (thresholds)

27. Still the quoted result looks a bit too precise as(mZ)=0.1181±0.0007(exp)±0.003(th) This precision is obtained by taking for granted that corrections suppressed by 1/mt are negligible. 2 Rt ~ Rt0[1+dpert+dnp] This is because in the massless theory: In fact there are no dim 2 operators (e.g. gmgm is not gauge invariant) except for light quark m2 (m~few MeV). Most people believe that. I am not sure that the gap is not filled by ambiguities of o(L2/mt2) from dpert.

28. as from scaling violations in DIS The splitting funct.s P are completely known to NLO accuracy: asP ~ asP1+as2P2 +... Floratos et al; Gonzales-Arroyo et al; Curci et al; Furmanski et al More recently the NNLO results have been derived for the first few moments (N<13,14). Larin, van Ritbergen, Vermaseren+Nogueira The full NNLO calculation is in progress and could be finished soon. The scaling violations are clearly observed and the NLO QCD fits are already excellent. These fits provide •an impressive set of QCD tests •measurements of q(x,Q2), g(x,Q2) •measurements of as(Q2)

29. Proton Structure Function F2(x,Q2)

30. Example of NLO QCD evolution fit

31. QCD predicts the Q2 dependence of F(x, Q2) not the x shape. But the Q2 dependence is related to the x shape by the QCD evolution eqs. For each x-bin approximately a straight line in dlogF(x, Q2)/dlog Q2 : the log slope. [Q2 span and precision of data not much sensitive to curvature] The scaling violations of non-singlet str. functs. would be ideal: small dep. on input parton densities But for Fp-Fn exp. errors add up in difference, and F3nN not terribly precise (and come essentially from only one experiment CCFR)

32. For xF3, using NNLO moments for N=1,3,..,13, the following results have been derived. Using Bernstein moments A combination of Mellin moments which emphasizes a value of x and a given spread in order to be sensitive to the interval where the measured points are as(mZ)=0.1153±0.0063 Santiago, Yndurain ‘01 as(mZ)=0.1174±0.0043 Maxwell, Mirjalili ‘02 Here the error from scale dep. not included (A model dep. scale fixing is chosen) Using Mellin moments as(mZ)=0.1190±0.0060 Kataev, Parente, Sidorov ‘02 Good overall agreement Not very precise ±0.006

33. When one measures as from scaling viols. in F2 from e or m beams, data are abundant errors small but:as gluon correlation Using data on p from SLAC, BCDMS, E665 and HERA, NLO kernels + NNLO for N=2,4,..,12: as(mZ)=0.1166±0.0013 (!!th error?) Santiago, Yndurain ’01 [Bernstein moments] Or using data on p from SLAC, BCDMS, NMC and HERA, NLO kernels + NNLO for N=2,4,..,12: as(mZ)=0.1143±0.0013(exp)+th error Alekhin ‘02 [Mellin moments] The difference in central value between these nominally most precise determinations makes clear that the total error ≥±0.003

34. Non singlet evolution eqs. become non diagonal as gluon partons are also relevant: The full set becomes g Recall: g The quark density with fraction y times the kernel for a gluon in a quark with fraction x/y of the parent long. mom. Gribov,Lipatov; Altarelli,Parisi

35. This estimate of TH errors is strenghtened by dispersion of results from other analyses Using data on p from BCDMS and NMC, NLO kernels, truncated moments Moments from x0 to 1 in measured range, coupled eqs. as(mZ)=0.122±0.006 Forte, Latorre, Magnea, Piccione ‘02 All lepton data, including HERA and CCFR, NLO evol. eqs. as(mZ)=0.119±0.004 Martin, Roberts, Stirling, Thorne ‘01 Compare with e+e- ->Z -> hadrons: as(mZ)=0.118±0.003 Looks great but…..

36. In my opinion the situation of as in DIS is not yet completely satisfactory (while it is for as in e+e-). Data have shown large changes in recent past: as(mZ)=0.113±0.005 BCDMS+SLAC (e,m) Milzstein, Virchaux CCFR (n) F2 and F3 combined, first gave as(mZ)=0.111±0.005 But then (energy calibration) moved to as(mZ)=0.119±0.005 •Problems from matching systematics of different experiments. •Analysis methods still not completely optimised and convergent

37. s from event shape and jet rates in e+e- QCD predicts a hierarchy of topologies: 2-jets, 3 jets, 4-jets…. 2-jets: angular distr. ~1+cos2q 3-jets: o(as) qqg Here x1,2 refer to energy fractions of (massless) quarks. 4-jets: o(as2) qqgg, qqqq •••

38. An example: multiplicities of jets at e+e- Based on a jet definition ycut is a resolution parameter ycut smaller more jets

39. LEP Measurements

40. Summarising: there is a good agreement among many different measures of as(mZ). This is a very convincing test of QCD. The value quoted by PDG 2002 is (MS): as(mZ)=0.1172±0.002 The corresponding value of L (for nf=5) is: L(5) = 216±25 MeV L is the scale of mass that finally appears in massless QCD. It is the scale where as(mZ) is of order 1.

41. In conclusion: Measurements of as(Q) at different scales clearly show the running of the QCD coupling S. Bethke, 2000