The critical parameters and the nature of the phase transition depend on the number of quark flavors and their masses (see fig.2.6). But actually this charge is not conserved due to the topological structure of the QCD vacuum (instantons) as discussed in the following (for an introduction, see [308]). For the vector potential A.1/, which is a pure gauge and thus part of the "vacuum", the winding number is generally defined by

## Massless QCD and Scale Invariance

The associated Goldstone boson, the axion, is not actually massless due to the chiral anomaly. When a symmetry of the classical theory is necessarily destroyed by quantization, regularization and renormalization, one speaks of an "anomaly". The point-like cross-section in the denominator is given by the point D4˛2=3s, where DQ2D4E2 is the squared total energy center of mass and Q is the mass of the exchanged virtual gauge boson.

## The Renormalization Group and Asymptotic Freedom

An important point is the appearance of the fluid coupling, which determines the asymptotic deviations from scaling. It is clear that there is no dependence of the basic 3-gluon node on the lowest order. In short, in QED and QCD quarks with m Q do not contribute to nf in the coefficients of the corresponding ˇ function.

## More on the Running Coupling

It is sufficient to extend the usual algebra in a straightforward way such as gD2g;I, where I is the D-dimensional identity matrix, D.D2/ or Tr. In the original MS recipe, only 1= was subtracted (and this clearly plays the role of a cutoff), while log4 and E were not. Later, since these constants always appear in the expansion of functions, it was decided to change MS to MS.

Note that the MS definition of ˛ is different from that in the momentum subtraction scheme, because the finite terms (those beyond logs) are different. It is interesting to note that the expansion coefficients are of order 1 or 10 (only for the last one), so the MS expansion seems quite well behaved.

On the Non-convergence of Perturbative Expansions

OneCeannihilation experiments at high energy provide a unique opportunity to systematically test the various signatures predicted by QCD for the final state structure averaged over a large number of events. Asymptotic freedom is characterized by a hierarchy of configurations that arise as a result of the small size of ˛s.Q2/. Neglecting all corrections of order ˛s.Q2/, we recover the prediction of the naïve parton model for the final state: nearly collinear events with two parallel jets with limited transverse momentum and angular distribution 1Ccos2 with respect to the beam axis (typical of parton quarks with spin 1/2, while when scalar quarks would lead to a sin2 distribution).

According to order˛s.Q2/, a tail of events is predicted to occur with large transverse momentum pT Q=2 with respect to a suitably defined jet axis (e.g. the thrust axis, see below). To order˛s2.Q2/, a hard, disruptive, nonplanar component begins to build up, and a small fraction of four-ray eventsqqggN orqqqNN q appears, and so on. Event shape variables, defined based on the series of 4-momenta of end-state particles, are introduced to describe the topological structure of the end-state energy flow in a quantitative way [154].

A quantitatively specified definition of jets and of the number of jets in one event (jet count) should be introduced for accurate QCD testing and measurement of beams, which should be infrared safe (i.e. not modified by soft particle emission or collinear splitting of massless particles ) to be computable at the parton level and as insensitive as possible to the transformation of partons into hadrons (see for example [294]). The inclusive clustering is done by identifying the smallest of the distances and, if it is adij, by recombining particle iandj, while if it is, ia jet is called and removed from the list. Ris is the radius of the radius, i.e. the radius of a cone that by definition contains the radius.

In general, it can be shown that the forp 0 behavior of the jet algorithm with respect to soft radiation is quite similar to that observed for the kT algorithm.

## Deep Inelastic Scattering

### The Longitudinal Structure Function

After SLAC established cross-section dominance, it took about 40 years before meaningful data on the longitudinal structure function FL were obtained [see (2.85)].

### Large and Small x Resummations for Structure Functions

More important is the small resummation of x because the single structure functions are large in this field ofx (while all the structure functions vanish near x D 1). Here we briefly summarize the small-x case for the single structure function, which is the dominant channel in HERA, dominated by the sharp increase in the gluon density and sea partition in smallx. The smallx data collected by HERA can be fit well enough, even at the smallest measured values of x, by the QCD NLO evolution equations, so that there is no dramatic evidence in the data for departures.

This is surprising also considering that the NNLO effects in development have recently become available and are quite large [292]. For the singlet splitting function, the coefficients for all LO and NLO corrections of order Œ˛s.Q2/log1=xnand˛s.Q2/Œ˛s.Q2/log1=xn, respectively, are explicitly known from Balitski, Fadin, Kuraev, Lipatov ( BFKL) analysis of virtual gluon-virtual gluon scattering [191,284]. But the simple addition of these higher-order terms to the perturbative result (subtracting all double-counting) does not lead to a convergent expansion (the NLO logs completely override the LO logs in the relevant domain ofxandQ2).

A sensible expansion is obtained only by a proper treatment of momentum conservation constraints, also by using the underlying symmetry of the BFKL core during exchange of the two external gluons, and especially by the ongoing coupling effects (see the analysis in [49,141] and references therein). We see that while the perturbative NNLO splitting function strongly deviates from the NLO approximation at smallx, the resummed result shows only a moderate decrease compared to the NLO perturbative splitting function in the region. The related effects are not very important for most processes at the LHC, but may become relevant for the next generation of hadron colliders.

### Polarized Deep Inelastic Scattering

In any case, it is a less pronounced crisis than it used to be in the past. Based on the spin sum rule, one finds that either gCLzi is relatively large, or there is a contribution to ˙ at very small x outside the measured range. Denoting the first moment of net helicity carried by sumqC Nqbyq, we have the relations [104,159].

The X distribution of g1 is known down to tox104 on proton and deuterium, and the first moment of g1 does not appear to come much from the unmeasured interval at smallx (also theoretically g1 should be smooth at smallx [190]). The value of 0:11 from total inclusive data and SU.3/ appears to be consistent with the value extracted from single-particle inclusive DIS (SIDIS), where one obtains an almost vanishing result for a fit to all data [159,326] leading to confusing results. Indeed, there is an apparent tension between the first moments as determined using the approximate SU.3/symmetry and from fitting data on SIDIS (x 0:001) (especially for the odd density).

But the adequacy of the SIDIS data is questionable (especially the kaon data that fix) and so is their theoretical treatment (for example, the use of partons results at too low energy and the ambiguities in the kaon fragmentation function). In terms of conserved quantities, we would expect them to be the same for constituents and for parton quarks. From the spin sum rule it is clear that the log increment should cancel between g and Lz.

The present data, affected by large errors (see, in particular, [303] for a discussion of this point) are consistent with a significant contribution of the rotation sum rule in (2.112), but there is no indication that effects can explain the difference between component and parton quarks.

## Hadron Collider Processes and Factorization

*Vector Boson Production**Jets at Large Transverse Momentum**Heavy Quark Production**Higgs Boson Production*

A great deal of effort has been and is being devoted to the theoretical preparation and interpretation of LHC experiments. In the following we will detail a number of more important and simpler examples, without any claim to completeness. A comparison of the experimental total rates for W and Z with the QCD predictions at hadron colliders [327] is shown in Fig.2.23.

Velocity distributions of produced W and Z have also been measured with fair accuracy at the Tevatron and LHC and predicted at UFOs [55]. It is very important to determine its mass and couplings for various accurate predictions of SM. Scale dependence of the total cross section at LO (blue), NLO (red) and NNLO (black) as a function of mtop (left) orp.

The peak mass (and e˛s value) can be determined from the cross section, assuming QCD is correct, and compared to the most accurate value from the final decay state. The upper pole mass value is derived in [27] from the cross section data, using the best available parton density with the correlated value of ˛s, ismpole D173:3˙2:8GeV. Recently, CDF has obtained [7] the first measurement of the differential cross section of the top quark pair production as a function of cos, with the top quark production angle.

We now turn to the discussion of the inclusive production cross section of the SM Higgs (for a review and list of references see [165]).

## Measurements of ˛ s

As we have seen in Section 2.7.1, for a quantity siRl we can write a general expression of the form The combined value from measurements at Z (assuming the validity of SM and the. However, sensitivity to hadronic effects in the vicinity of the cutoff is still a non-negligible source of theoretical error that the formulation of duality violation models proves to reduce.

Over time, awareness of the problem of higher-order control and non-perturbative effects has increased. A significant part of the progress comes from experimental measurements of the moments of the £ decay mass distributions defined by varying the weighting function in the integral in (2.129). The range of Q2 and the precision of the data are not very sensitive to curvature for most values of x.

For determining of˛s, the scale violations of non-singlet structure functions would be ideal, due to the minimal impact of the choice of input part densities. In the second line, taken from [254], the large error also includes an ambiguity estimate based on the gluon density parametrization. Via the sum rule for momentum conservation, this also limits the small values of the same density.

As a result, one obtains a striking direct confirmation of the unfolding of the coupling according to the prediction of the renormalization group.

Conclusion