Measurementsof s

as a quantitativetest of QCD

Edinburgh, 6 February ‘03

CERN

G. Altarelli

• A concise review of asymptotic freedomand of the running coupling

• Measurements of s and comparison with experiment

G. Altarelli

The Standard Model

}

Strong}

Electroweak

SU(3) colour symmetry isexact!

The EW symmetryis spont. brokendown to U(1)Q

Gauge Bosons

8 gluons gAW±, Z,

Matter fields:3 generations of quarks (coloured) and leptons

Higgs sector (???)

+ 2 more replicas

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QCD is an unbroken SU(3)Colour

gauge theory with triplet quarks:

Defs:

(CABC: SU(3) str. const.,tA: generator repr.)

(gA is a gluon field)

; (D: covariant derivative)

(es: SU(3) gauge coupling)

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Physical QCD vertices

-iestA

pA

qBrC

p+q+r=0

esCABC[g(p-q)perm]

A B

DC

-ies2 [ CABFCCDF (gggg

perm]

Note: es2

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but with an extremely rich dynamicalcontent:

QCD is a "simple" theory

•Complex hadron spectrum•Confinement•Asymptotic freedom•Spontaneous breaking of

(approx.) chiral symm. •Highly non trivial vacuum

[Instantons, U(1)A symm. breaking, strongCP violation (?)]

•Phase transitions[Deconfinement (q-g plasma), chiral symm.restauration,…...]

• • •

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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 quantitativeconnection to experiment.

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Classical gauge th. lagrangian

Quantisation Gauge fixing termsGhosts

Feynman rules

Infinities

Pert. Theory

RegularisationRenormalisation

Perturbative quantum gauge th.

Cut off

Redef. of m, s ,Zi (w.f. norm'n)

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Massless QCD and scale invariance

In the QCD lagrangian

quark masses are the only parameterswith dimensions.

Naively we would expect massless QCDto be scale invariant (dimensionless observables should not depend on theabsolute energy scale, but only on ratios of energy-dimensional variables)

The massless limit should be relevant forthe asymptotic large energy limit of processes which are non singular for m->0.

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This naïve expectation is false!

For massless QCD the scale symmetry of theclassical theory is destroyed by regularisationand renormalisation which introduce adimensional parameter in the quantum version of the theory (QCD).

[When a symmetry of the classical theory isnecessarily destroyed by quantisation,regularisation and renormalisationone talks of an "anomaly"]

While massless QCD is finally not scaleinvariant, the departures from scaling areasymptotically small, logarithmic andcomputable (in massive QCD there areadditional mass corrections suppressedby powers).

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Hard processes

At the "parton" level (q and g) we can applythe asymptotics from massless QCD toprocesses with the following properties:

• finite for m ->0 (no mass singularities.)

• no infrared singularities ("infrared safe")

• all relevant energy variables are large

Ei= xiQ Q>>m xi : scaling variables

To satisfy these criteria processes must besufficiently "inclusive":

• add all final states with massless g emission

• add all mass degenerate final states

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Examples of important hard processes

• e+e- -> hadrons

At parton level the final state is

qq + n gluons + n' qq pairs

(i.e. totally inclusive). The conversion ofpartons into hadrons does not affect therate (some smearing over a Q bin can beneeded for probability 1)

• l + N -> l' + hadrons (Deep Inelastic Scattering: DIS)

Qp

P'(p+p')2=s=Q2

k

k '

q

p

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Regularisation and Renormalisation

•A dimensional "cut off" is introduced(must be gauge invariant)

• The dependence on the cut-off is eliminated by a redefinition of m, es and Zusing suitable renormalisation conditions.

In general:

Ren. mass: position of the propag. pole.Wave funct'n ren. Z: residue at the pole.

The ren. coupling es is, for example, defined in terms of a ren. 3-point vertex at some momenta.

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In particular in massless QCD:

If we start with m0=0 the mass is notrenorm. because it is protected by asymmetry (chiral symm.) -> m=0

The coupling es can be defined in terms of the 3-gluon coupling at a scale -2:

p2

q2r2 es = Vren(- 2, - 2, - 2)

•The scale cannot be zero (infrared sing.)!

• - 2<0: no absorptive parts

(Z=Zg3/2ZV)

Vbare(p2,q2,r2)=Z Vren(p2,q2,r2)

Similarly Zg can be defined by the inverse propagator at p2= - 2

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Computing all diagrams (with)

+ + + ….

{

Z

{Vren

Note: VBare depends on but not on

Both Z and Vren depend on

p2=q2=r2

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In general:

Renormalisation group equation

GBare(2, 0, pi2)=Z Gren(2, , pi

2)

so that:

or

Finally the RGE can be written as:

(We write for or s in QED or QCD)

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The running coupling

The running coupling (t) is fixed by thebeta function:

The dependence starts at 1-loop:

e + +….e

ee

or

QCD or QED

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By explicit calculation at 1-loop one finds:

QED: () ~ + b2 + ...

QCD: () ~ - b2 + ...

The sum is over all fermions of charge Qe

nf is the number of quark flavours

Recall:

If (t) is small, we can compute b in pert.th. The sign in front of b decides whether:(t) increases with t or Q2 (QED) or(t) decreases with t or Q2 (QCD).

QCD is "asymptotically free". All and only non-abelian gauge th.are asymptotically free (in 4-dim.)

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Going back to the equation:

We replace ()~±b2, integrate and do a small algebra. We find:

In QCD we have:

Note

• decreases logaritmically in Q2

•a dimensional parameter replaces .

Do not confuse = QCD with =UV cutoff!

this leads to: (0)=t=0->Q=

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() ~ ±b2(1+b'+….)

In general the perturbative coeff.s of ()depend on the def. of , 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 QCD not the cutoff!!

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Summarising: we started from the massless classical theory and we ended up with QCDwhere an energy scale appears (=).

depends on the definition of s (i.e. the reg. procedure, the ren. scheme…) and on the number of excited flavours nf .

Definition of s

We have introduced the ren. coupling s

in terms of the 3-g ren. vertex at p2=-2

(momentum subtraction). The value of s

(hence ) in this scheme depends on

But the most common def. of s 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.

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Nowadays the MS definition of s (based on dimensional regularisation)is adopted, because the correspondingrenormalisation technique is simplest to implement in complicated calculations.For example, it can be realised diagram bydiagram.

The third coefficient of the beta function() is also known in MS. Translated innumbers, for nf=5 one obtains:

Tarasov, Vladimirov, Zharkov,Yu

which means good apparent convergence

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Dependence of from nf

QED and QCD are theories with decoupling:quarks with mass m>Q do not contributeto the running of up to the scale Q.

So for 2mc<Q<2mb the relevant asymptoticsis for nf=4, while for 2mb<Q<2mt nf=5.

Going across the 2mb threshold, the ()coeff.s change, so the (t) slope changes.But (t) is continous so that 4 and 5

are different:

(t)

2mb

From matching (Q2)5~0.65 4

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PDG’02 summary on s(mZ) MS

s(mZ)=0.1172±0.002Not the Gospel!

Measurements of s(mZ)

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The main methods for s at LEP/SLC are:

• inclusive Z decay, Rl, h, Z

• inclusive decay• event shapes and jet rates

Inclusive:

QCD is known to NNLO accuracy:

Here Q=mZ or m

Clearly the Z case is apriori more reliablebecause mZ>>m.

NP are power suppressed (1/Q2)n termsgoverned by the OPE.

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At the Z from Rl only (assuming the standard EW theory, (mtexp)):

s(mZ)=0.1224±0.0038

Better, one can use all info from Rl, Z, h,…and in general take s(mZ) as a parameterto be fitted from the EW precision tests.

s(mZ)=0.1187±0.0027

One obtains:

LEP1 only:

All EW Data: s(mZ)=0.1181±0.0027

The dominant sources of error are mH andhigher orders in the QCD expansion.Error from power corrections very small.

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s from R

Rhas a number of advantages that,at least in part, compensate the smallness of m=1.777 GeV:

• R is more inclusive than Re+e-(s).

• one can use analiticity to go to |s|= m

Im s

Re s

|s|= m

• factor (1-s/m2)2 kills

sensitivity to Re s= m2 (thresholds)

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Still the quoted result looks a bit too precise

s(mZ)=0.1181±0.0007(exp)±0.003(th)

This precision is obtained by taking for granted that corrections suppressed by 1/m

are negligible.

2

R ~ R0[1+pert+np]

This is because in the massless theory:

In fact there are no dim 2 operators (e.g. gg is not gauge invariant)except for light quark m2 (m~few MeV) .

Most people believe that. I am not surethat the gap is not filled by ambiguitiesof o(2/m

2) from pert.

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The splitting funct.s P are completely knownto NLO accuracy: sP ~ sP1+s

2P2 +...

Floratos et al; Gonzales-Arroyo et al; Curci et al; Furmanski et al

More recently the NNLO results have beenderived for the first few moments (N<13,14).

Larin, van Ritbergen, Vermaseren+Nogueira

The full NNLO calculation is in progress andcould be finished soon.

The scaling violations are clearly observedand 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 s(Q2)

s from scaling violations in DIS

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Proton Structure Function F2(x,Q2)

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Example of NLO QCD evolution fit

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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 linein 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 inputparton densities

But for Fp-Fn exp. errors add up in difference,and F3N not terribly precise (and comeessentially from only one experiment CCFR)

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For xF3, using NNLO moments for N=1,3,..,13,the following results have been derived.

Using Bernstein momentsA combination of Mellin moments whichemphasizes a value of x and a given spreadin order to be sensitive to the interval wherethe measured points are

s(mZ)=0.1153±0.0063Santiago, Yndurain ‘01

s(mZ)=0.1174±0.0043

Maxwell, Mirjalili ‘02

Here the error from scale dep. not included(A model dep. scale fixing is chosen)

s(mZ)=0.1190±0.0060Kataev, Parente, Sidorov ‘02

Using Mellin moments

Good overall agreementNot very precise ±0.006

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When one measures s from scaling viols.in F2 from e or beams, data are abundanterrors small but: s gluon correlation

Using data on p from SLAC, BCDMS, E665 andHERA, NLO kernels + NNLO for N=2,4,..,12:

s(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:

s(mZ)=0.1143±0.0013(exp)+th error

Alekhin ‘02 [Mellin moments]

The difference in central value betweenthese nominally most precise determinationsmakes clear that the total error ≥±0.003

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Non singlet evolution eqs. become non diagonal as gluon partons are also relevant:

g

The full set becomes

Recall:

Gribov,Lipatov; Altarelli,Parisi

The quark density with fraction y timesthe kernel for a gluon in a quark withfraction x/y of the parent long. mom.

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Compare with e+e- ->Z -> hadrons:

s(mZ)=0.118±0.003

Looks great but…..

Moments from x0 to 1 in measured range, coupled eqs. s(mZ)=0.122±0.00

6

Using data on p from BCDMS and NMC,NLO kernels, truncated moments

Forte, Latorre, Magnea, Piccione ‘02

This estimate of TH errors is strenghtenedby dispersion of results from other analyses

All lepton data, including HERA and CCFR,NLO evol. eqs.

s(mZ)=0.119±0.004Martin, Roberts, Stirling, Thorne ‘01

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In my opinion the situation of s in DIS isnot yet completely satisfactory (while it isfor s in e+e-).

Data have shown large changes in recent past:

s(mZ)=0.113±0.005 BCDMS+SLAC (e,)

Milzstein, Virchaux

CCFR () F2 and F3 combined, first gave

s(mZ)=0.111±0.005

But then (energy calibration) moved to

s(mZ)=0.119±0.005

•Problems from matching systematics of different experiments.•Analysis methods still not completelyoptimised and convergent

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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+cos2

3-jets: o(s) qqg

Here x1,2 refer to energy fractions of (massless) quarks.

4-jets: o(s2) qqgg, qqqq

•••

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An example: multiplicities of jets at e+e-

Based on ajet definitionycut is a resolutionparameter

ycut smaller

more jets

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LEP Measurements

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s(mZ)=0.1172±0.002

Summarising: there is a good agreementamong many different measures of s(mZ).This is a very convincing test of QCD.

The value quoted by PDG 2002 is (MS):

The corresponding value of (for nf=5) is:

= 216±25 MeV

is the scale of mass that finally appears inmassless QCD.It is the scale where s(mZ) is of order 1.

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Measurements of s(Q) at different scales clearly show the running of the QCD coupling

S. Bethke, 2000

In conclusion: