The simplest hard process is *Re*+*e*− that we have already started to discuss. *R* is dimensionless and in perturbation theory is given by *R* = *NC i Q*2 *i F (t , αs)*, where *F* = 1 + *o(αS)*. We have already mentioned that for this process the "anomalous dimension" function vanishes: *γ (αs)* = 0 because of electric charge non renormalisation by strong interactions. Let us review how this happens in detail. The diagrams that are relevant for charge renormalisation in QED at 1-loop are shown in Fig. 4.8. The Ward identity that follows from gauge invariance in QED imposes that the vertex (*ZV* ) and the self-energy (*Zf* ) renormalisation factors cancel and the only divergence remains in *Zγ* , the vacuum polarization of the photon. So the charge is only renormalised by the photon blob, hence it is universal (the same factor for all fermions, independent of their charge) and is not affected by QCD at 1-loop. It is true that at higher orders the photon vacuum polarization diagram is affected by QCD (for example, at 2-loops we can exchange a gluon between the quarks in the photon loop) but the renormalisation induced by the vacuum polarisation diagram remains independent of the nature of the fermion to which the photon line is attached. The gluon contributions to the vertex (*ZV* ) and to the

**Fig. 4.8** Diagrams for charge renormalisation in QED at 1-loop

**Fig. 4.9** Real and virtual diagrams relevant for the computation of R at 1-loop accuracy

self-energy (*Zf* ) cancel because they have exactly the same structure as in QED, so that *γ (αs)* = 0.

At 1-loop the diagrams relevant for the computation of R are shown in Fig. 4.9. There are virtual diagrams and real diagrams with one additional gluon in the final state. Infrared divergences cancel between the interference term of the virtual diagrams and the absolute square of the real diagrams, according to the Bloch-Nordsieck theorem. Similarly there are no mass singularities, in agreement with the Kinoshita-Lee-Nauenberg theorem, because the initial state is purely leptonic and all degenerate states that can appear at the given order are included in the final state. Given that *γ (αs)* = 0 the RGE prediction is simply given, as we have already seen, by *F (t , αs)* = *F*[0*, αs(t)*]. This means that if we do, for example, a 2-loop calculation, we must obtain a result of the form:

$$F(t, \alpha\_s) = \left[1 + c\_1 \alpha\_s (1 - b\alpha\_s t) + c\_2 \alpha\_s^2 + o(\alpha\_s^3)\right] \tag{4.57}$$

In fact we see that this form, taking into account that from Eq. (4.47) we have:

$$\alpha\_s(t) \sim \frac{\alpha\_s}{1 + b\alpha\_s t} \sim \alpha\_s(1 - b\alpha\_s t + \text{ } \dots \text{)}\tag{4.58}$$

can be rewritten as

$$F(t, \alpha\_s) = 1 + c\_1 \alpha\_s(t) + c\_2 \alpha\_s^2(t) + o(\alpha\_s^3(t)) \ = F[0, \alpha\_s(t)] \tag{4.59}$$

The content of the RGE prediction is, at this order, that there are no *αst* and *(αst)*2 terms (the leading log sequence must be absent) and the term of order *α*2 *s t* has the coefficient that allows to reabsorb it in the transformation of *αs* into *αs(t)*.

At present the first three coefficients have been computed in the *MS* scheme [15]. Clearly *c*1 = 1*/π* does not depend on the definition of *αs* but *c*2 and *c*3 do. The subleading coefficients also depend on the scale choice: if instead of expanding in *αs(Q)* we decide to choose *αs(Q/*2*)* the coefficients *c*2 and *c*3 change. In the *MS* scheme, for *γ* -exchange and *nf* = 5, which are good approximations for 2*mb << Q << mZ*, one has:

$$F[0, \alpha\_s(t)] = 1 + \frac{\alpha\_s(t)}{\pi} + 1.409\dots(\frac{\alpha\_s(t)}{\pi})^2 - 12.8\dots(\frac{\alpha\_s(t)}{\pi})^3 + \dots \quad (4.60)$$

Similar perturbative results at 3-loop accuracy also exist for *RZ* = *(Z* → *hadrons)/
(Z* → *leptons)*, *Rτ* = *(τ* → *ντ* +*hadrons)/
(τ* → *ντ* +*leptons)*, etc. We will discuss these results later when we deal with measurements of *αs*.

The perturbative expansion in powers of *αs(t)* takes into account all contributions that are suppressed by powers of logarithms of the large scale *Q*2 ("leading twist" terms). In addition there are corrections suppressed by powers of the large scale *Q*2 ("higher twist" terms). The pattern of power corrections is controlled by the light-cone Operator Product Expansion (OPE) [16] which (schematically) leads to:

$$F = \text{pert.} + r\_2 \frac{m^2}{\mathcal{Q}^2} + r\_4 \frac{<0 | Tr[\mathbf{F}\_{\mu\nu} \mathbf{F}^{\mu\nu}] | 0>}{\mathcal{Q}^4} + \dots \, + r\_6 \frac{<0 | O\_6 | 0>}{\mathcal{Q}^6} + \dots \,\tag{4.61}$$

Here *m*2 generically indicates mass corrections, notably from b quarks, for example (t quark mass corrections only arise from loops, vanish in the limit *mt* → ∞ and are included in the coefficients as those in Eq. (4.60) and the analogous ones for higher twist terms), **F***μν* = *A F A μν tA*, *O*6 is typically a 4-fermion operator, etc. For each possible gauge invariant operator the corresponding power of *Q*2 is fixed by dimensions.

We now consider the light-cone OPE in some more detail. *Re*+*e*− ∼ *(Q*2*)* where *(Q*2*)* is the scalar spectral function related to the hadronic contribution to the imaginary part of the photon vacuum polarization *Tμν* :

$$T\_{\mu\nu} = (-g\_{\mu\nu}Q^2 + q\_{\mu}q\_{\nu})\Pi(Q^2) = \int \exp iq \, x < 0 |J\_{\mu}^{\dagger}(\mathbf{x})J\_{\nu}(\mathbf{0})| 0 > dx = $$

$$= \sum\_{n} < 0 |J\_{\mu}^{\dagger}(\mathbf{0})|n > < n|J\_{\nu}(\mathbf{0})|0 > (2\pi)^4 \delta^4(q - p\_n) \tag{4.62}$$

For *Q*2 → ∞ the *x*2 → 0 region is dominant. To all orders in perturbation theory the OPE can be proven. Schematically, dropping Lorentz indices, for simplicity, near *x*2 ∼ 0 we have:

$$J^{\dagger}(\mathbf{x})J(\mathbf{0}) = I(\mathbf{x}^2) + E(\mathbf{x}^2) \sum\_{n=0}^{\infty} c\_n(\mathbf{x}^2) \mathbf{x}^{\mu\_1} \dots \mathbf{x}^{\mu\_n} \cdot O^n\_{\mu\_1 \dots \mu\_n}(\mathbf{0})$$

$$+ \text{less sing. terms} \tag{4.63}$$

Here *I (x*2*)*, *E(x*2*)*,. . . , *cn(x*2*)* are c-number singular functions, *On* is a string of local operators. *E(x*2*)* is the singularity of free field theory, *I (x*2*)* and *cn(x*2*)* contain powers of log*μ*2*x*2 in interaction. Some *On* are already present in free field theory, other ones appear when interactions are switched on. Given that *(Q*2*)* is related to the Fourier transform of the vacuum expectation value of the product of currents, less singular terms in *x*2 lead to power suppressed terms in 1*/Q*2. The perturbative terms come from *I (x*2*)* which is the leading twist term. The logarithmic scaling violations induced by the running coupling are the logs in *I (x*2*)*.

## *4.5.2 The Final State in e***+***e***−** *Annihilation*

Experiments on *e*+*e*− annihilation at high energy provide a remarkable possibility of systematically testing the distinct signatures predicted by QCD for the structure of the final state averaged over a large number of events. Typical of asymptotic freedom is the hierarchy of configurations emerging as a consequence of the smallness of *αs(Q*2*)*. When all corrections of order *αs(Q*2*)* are neglected one recovers the naive parton model prediction for the final state: almost collinear events with two backto-back jets with limited transverse momentum and an angular distribution as *(*1 + cos2 *θ )* with respect to the beam axis (typical of spin 1/2 parton quarks: scalar quarks would lead to a sin2 *θ* distribution). At order *αs(Q*2*)* a tail of events is predicted to appear with large transverse momentum *pT* ∼ *Q/*2 with respect to the thrust axis (the axis that maximizes the sum of the absolute values of the longitudinal momenta of the final state particles). This small fraction of events with large *pT* mostly consists of three-jet events with an almost planar topology. The skeleton of a three-jet event, at leading order in *αs(Q*2*)*, is formed by three hard partons *qqg*¯ , the third being a gluon emitted by a quark or antiquark line. The distribution of three-jet events is given by:

$$\frac{1}{\sigma} \frac{d\sigma}{d\mathbf{x}\_1 d\mathbf{x}\_2} = \frac{2\alpha\_s}{3\pi} \frac{\mathbf{x}\_1^2 + \mathbf{x}\_2^2}{(1 - \mathbf{x}\_1)(1 - \mathbf{x}\_2)}\tag{4.64}$$

here *x*1*,*2 refer to energy fractions of massless quarks: *xi* = 2*Ei/* √*s* with *x*1 + *x*2 + *x*3 = 2. At order *α*2 *s (Q*2*)* a hard perturbative non planar component starts to build up and a small fraction of four-jet events *qqgg* ¯ or *qqq*¯ *q*¯ appear, and so on.

A quantitatively specified definition of jet counting must be introduced for precise QCD tests and for measuring *αs*, which must be infrared safe (i.e. not altered by soft particle emission or collinear splittings of massless particles) in order to be computable at parton level and as much as possible insensitive to the transformation of partons into hadrons. One introduces a resolution parameter *ycut* and a suitable pair variable; for example [17]:

$$\mathbf{y}\_{lj} = \frac{\min(E\_l^2, E\_j^2)(1 - \cos\theta\_{lj})}{s} \tag{4.65}$$

The particles i,j belong to different jets for *yij > ycut* . Clearly the number of jets becomes a function of *ycut* : there are more jets for smaller *ycut* . Measurements of *αs(Q*2*)* have been performed starting from jet multiplicities, the largest error coming from the necessity of correcting for non-perturbative hadronisation effects.

#### *4.5.3 Deep Inelastic Scattering*

Deep Inelastic Scattering (DIS) processes have played and still play a very important role for our understanding of QCD and of nucleon structure. This set of processes actually provides us with a rich laboratory for theory and experiment. There are several structure functions that can be studied, *Fi(x, Q*2*)*, each a function of two variables. This is true separately for different beams and targets and different polarizations. Depending on the charges of l and l' (see Eq. (4.13)) we can have neutral currents (*γ* ,Z) or charged currents in the l-l' channel (Fig. 4.6). In the past DIS processes were crucial for establishing QCD as the theory of strong interactions and quarks and gluons as the QCD partons. At present DIS remains very important for quantitative studies and tests of QCD. The theory of scaling violations for totally inclusive DIS structure functions, based on operator expansion or diagrammatic techniques and renormalisation group methods, is crystal clear and the predicted *Q*2 dependence can be tested at each value of x. The measurement of quark and gluon densities in the nucleon, as functions of x at some reference value of *Q*2, which is an essential starting point for the calculation of all relevant hadronic hard processes, is performed in DIS processes. At the same time one measures *αs(Q*2*)* and the DIS values of the running coupling can be compared with those obtained from other processes. At all times new theoretical challenges arise from the study of DIS processes. Recent examples are the so-called "spin crisis" in polarized DIS and the behaviour of singlet structure functions at small x as revealed by HERA data. In the following we will review the past successes and the present open problems in the physics of DIS.

The cross-section *σ* ∼ *LμνWμν* is given in terms of the product of a leptonic (*Lμν* ) and a hadronic (*Wμν* ) tensor. While *Lμν* is simple and easily obtained from the lowest order electroweak (EW) vertex plus QED radiative corrections, the complicated strong interaction dynamics is contained in *Wμν* . The latter is proportional to the Fourier transform of the forward matrix element between the nucleon target states of the product of two EW currents:

$$W\_{\mu\nu} = \int dx \, \exp iqx \, \lnot \, p |J\_{\mu}^{\dagger}(\mathbf{x})J\_{\nu}(\mathbf{0})| \, p > \tag{4.66}$$

Structure functions are defined starting from the general form of *Wμν* given Lorentz invariance and current conservation. For example, for EW currents between unpolarized nucleons we have:

$$\begin{split} W\_{\mu\nu} &= \left( -g\_{\mu\nu} + \frac{q\_{\mu}q\_{\nu}}{q^2} \right) W\_1(\nu, \mathcal{Q}^2) + (p\_{\mu} - \frac{m\nu}{q^2}q\_{\mu})(p\_{\nu} - \frac{m\nu}{q^2}q\_{\nu}) \frac{W\_2(\nu, \mathcal{Q}^2)}{m^2} - 1 \\ &- \frac{i}{2m^2} \epsilon\_{\mu\nu\lambda\rho} p^\lambda q^\rho \ W\_3(\nu, \mathcal{Q}^2) \end{split}$$

*W*3 arises from VA interference and is absent for pure vector currents. In the limit *Q*2 *>> m*2, x fixed, the structure functions obey approximate Bjorken scaling which in reality is broken by logarithmic corrections that can be computed in QCD:

$$m\,W\_1(\upsilon,\,\mathcal{Q}^2) \to F\_1(\upsilon)$$

$$\upsilon\,W\_{2,3}(\upsilon,\,\mathcal{Q}^2) \to F\_{2,3}(\upsilon) \tag{4.67}$$

The *γ* − *N* cross-section is given by (*Wi* = *Wi(Q*2*,ν)*):

$$\frac{d\sigma^{\prime}}{d|\mathcal{Q}^{2}dv} = \frac{4\pi\alpha^{2}E^{\prime}}{\mathcal{Q}^{4}E} \cdot \left[2\sin^{2}\frac{\theta}{2}W\_{\text{l}} + \cos^{2}\frac{\theta}{2}W\_{\text{l}}\right] \tag{4.68}$$

while for the *ν* − *N* or *ν*¯ − *N* cross-section one has:

$$\frac{d\sigma^{\upsilon,\bar{\upsilon}}}{d\,Q^2d\upsilon} = \frac{G\_F^2 E'}{2\pi E} (\frac{m\_W^2}{Q^2 + m\_W^2})^2 \cdot \left[2\sin^2\frac{\theta}{2}W\_1 + \cos^2\frac{\theta}{2}W\_2 \pm \frac{E + E'}{m}\sin^2\frac{\theta}{2}W\_3\right] \tag{4.69}$$

(*Wi* for photons, *ν* and *ν*¯ are all different, as we shall see in a moment).

In the scaling limit the longitudinal and transverse cross sections are given by:

$$
\sigma\_L \sim \frac{1}{s} [\frac{F\_2(\mathbf{x})}{2\mathbf{x}} - F\_1(\mathbf{x})]
$$

$$
\sigma\_{RH,LH} \sim \frac{1}{s} [F\_1(\mathbf{x}) \pm F\_3(\mathbf{x})]
$$

$$
\sigma\_T = \sigma\_{RH} + \sigma\_{LH} \tag{4.70}
$$

where L, RH, LH refer to the helicity 0*,* 1*,* −1*,* respectively, of the exchanged gauge vector boson.

In the '60's the demise of hadrons from the status of fundamental particles to that of bound states of constituent quarks was the breakthrough that made possible the construction of a renormalisable field theory for strong interactions. The presence of an unlimited number of hadrons species, many of them with large spin values, presented an obvious dead-end for a manageable field theory. The evidence for constituent quarks emerged clearly from the systematics of hadron spectroscopy. The complications of the hadron spectrum could be explained in terms of the quantum numbers of spin 1/2, fractionally charged, u, d and s quarks. The notion of colour was introduced to reconcile the observed spectrum with Fermi statistics. But confinement that forbids the observation of free quarks was a clear obstacle towards the acceptance of quarks as real constituents and not just as fictitious entities describing some mathematical pattern (a doubt expressed even by Gell-Mann at the time). The early measurements at SLAC of DIS dissipated all doubts: the observation of Bjorken scaling and the success of the "naive" (not so much after all) parton model of Feynman imposed quarks as the basic fields for describing the nucleon structure (parton quarks).

In the language of Bjorken and Feynman the virtual *γ* (or, in general, any gauge boson) sees the quark partons inside the nucleon target as quasi-free, because their (Lorentz dilated) QCD interaction time is much longer than *τγ* ∼ 1*/Q*, the duration of the virtual photon interaction. Since the virtual photon 4-momentum is spacelike, we can go to a Lorentz frame where *Eγ* = 0 (Breit frame). In this frame *q* = *(Eγ* = 0; 0*,* 0*, Q)* and the nucleon momentum, neglecting the mass *m << Q*, is *p* = *(Q/*2*x*; 0*,* 0*,* −*Q/*2*x)* (note that this correctly gives *q*2 = −*Q*2 and *x* = *Q*2*/*2*(p*·*q)*). Consider (Fig. 4.10) the interaction of the photon with a quark carrying a fraction y of the nucleon 4-momentum: *pq* = *yp* (we are neglecting the transverse components of *pq* which are of order *m*). The incoming parton with *pq* = *yp* absorbs the photon and the final parton has 4-momentum *p q* . Since in the Breit frame the photon carries no energy but only a longitudinal momentum *Q*, the photon can only be absorbed by those partons with *y* = *x*: then the longitudinal component of *pq* = *yp* is −*yQ/*2*x* = −*Q/*2 and can be flipped into +*Q/*2 by the photon. As a result, the photon longitudinal momentum +*Q* disappears, the parton quark momentum changes of sign from −*Q/*2 into +*Q/*2 and the energy is not changed. So the structure functions are proportional to the density of partons with fraction x of the nucleon momentum, weighted with the squared charge. Also, recall that the helicity of a massless quark is conserved in a vector (or axial vector) interaction. So when the momentum is reversed also the spin must flip. Since the process is collinear there is no orbital contribution and only a photon with helicity ±1 (transverse photon) can be absorbed. Alternatively, if partons were spin zero only longitudinal photons would instead contribute.

Using these results, which are maintained in QCD at leading order, the quantum numbers of the quarks were confirmed by early experiments. The observation that *R* = *σL/σT* → 0 implies that the charged partons have spin 1/2. The quark charges

**Fig. 4.10** Schematic diagram for the interaction of the virtual photon with a parton quark in the Breit frame

were derived from the data on the electron and neutrino structure functions:

$$F\_{ep} = 4/9u(\mathbf{x}) + 1/9d(\mathbf{x}) + \dots ; \qquad F\_{en} = 4/9d(\mathbf{x}) + 1/9u(\mathbf{x}) + \dots$$

$$F\_{\bar{\nu}p} = F\_{\bar{\nu}n} = 2d(\mathbf{x}) + \dots ; \qquad F\_{\bar{\nu}n} = F\_{\bar{\nu}p} = 2u(\mathbf{x}) + \dots \quad \text{(4.71)}$$

where *F* ∼ 2*F*1 ∼ *F*2*/x* and *u(x)*, *d(x)* are the parton number densities in the proton (with fraction x of the proton longitudinal momentum), which, in the scaling limit, do not depend on *Q*2. The normalisation of the structure functions and the parton densities are such that the charge relations hold:

$$\int\_0^1 [\mu(\mathbf{x}) - \tilde{\mu}(\mathbf{x})] d\mathbf{x} = 2, \quad \int\_0^1 [d(\mathbf{x}) - \tilde{d}(\mathbf{x})] d\mathbf{x} = 1, \quad \int\_0^1 [s(\mathbf{x}) - \tilde{s}(\mathbf{x})] d\mathbf{x} = 0 \tag{4.72}$$

Also it was proven by experiment that at values of *Q*2 of a few GeV2, in the scaling region, about half of the nucleon momentum, given by the momentum sum rule:

$$\int\_{0}^{1} \mathbb{I} \sum\_{l} (q\_{l}(\mathbf{x}) + \bar{q}\_{l}(\mathbf{x})) + \text{g}(\mathbf{x}) \| \mathbf{x} \mathbf{d} \mathbf{x} = \mathbf{1} \tag{4.73}$$

is carried by neutral partons (gluons).

In QCD there are calculable log scaling violations induced by *αs(t)*. The parton rules just introduced can be summarised in the formula:

$$F(\mathbf{x}, t) = \int\_{\mathcal{X}}^{1} dy \frac{q\_0(\mathbf{y})}{\mathbf{y}} \sigma\_{point}(\frac{\mathbf{x}}{\mathbf{y}}, \alpha\_s(t)) + o(\frac{1}{\mathcal{Q}^2}) \tag{4.74}$$

Before QCD corrections *σpoint* = *e*2*δ(x/y* − 1*)* and *F* = *e*2*q*0*(x)* (here we denote by *e* the charge of the quark in units of the positron charge, i.e. *e* = 2*/*3 for the *u* quark). QCD modifies *σpoint* at order *αs* via the diagrams of Fig. 4.11. Note that the integral is from x to 1, because the energy can only be lost by radiation before interacting with the photon (which eventually wants to find a fraction *x*, as we have

**Fig. 4.11** First order QCD corrections to the virtual photon-quark cross-section: (a) leading order with (b) one-loop virtual correction; (c-d) next-to-leading order real emission

explained). From a direct computation of the diagrams one obtains a result of the following form:

$$\sigma\_{\text{polnt}}(z, \mathfrak{a}\_{\mathfrak{s}}(t)) \cong e^2 [\delta(z - \mathfrak{l}) + \frac{\mathfrak{a}\_{\mathfrak{s}}}{2\pi} (t \cdot P(z) + f(\mathfrak{z}))] \tag{4.75}$$

For *y>x* the correction arises from diagrams with real gluon emission. Only the sum of the two real diagrams in Fig. 4.11 is gauge invariant, so that the contribution of one given diagram is gauge dependent. There is a special form of axial gauge, called physical gauge, where, among real diagrams, the diagram of Fig. 4.11c gives the whole *t*-proportional term. It is obviously not essential to go to this gauge, but this diagram has a direct physical interpretation: a quark in the proton has a fraction *y>x* of the parent 4-momentum; it then radiates a gluon and looses energy down to a fraction *x* before interacting with the photon. The log arises from the virtual quark propagator, according to the discussion of collinear mass singularities in Eq. (4.12). In fact in the massless limit one has:

$$\text{propagator} = \frac{1}{r^2} = \frac{1}{(k-h)^2} = \frac{-1}{2E\_k E\_h} \cdot \frac{1}{1 - \cos\theta}$$

$$= \frac{-1}{4E\_k E\_h} \cdot \frac{1}{\sin^2 \theta / 2} \propto \frac{-1}{p\_T^2} \tag{4.76}$$

where *pT* is the transverse momentum of the virtual quark. So the square of the propagator goes like 1*/p*4 *T* . But there is a *p*2 *T* factor in the numerator, because in the collinear limit, when *θ* = 0 and the initial and final quarks and the emitted gluon are all aligned, the quark helicity cannot flip (vector interaction) so that the gluon should carry helicity zero but a real gluon can only have ±1 helicity. Thus the numerator vanishes as *p*2 *T* in the forward direction and the cross-section behaves as:

$$
\sigma \sim \int\_{}^{\mathcal{Q}^2} \frac{1}{p\_T^2} dp\_T^2 \sim \log \mathcal{Q}^2 \tag{4.77}
$$

Actually the log should be read as log *Q*2*/m*2 because in the massless limit a genuine mass singularity appears. In fact the mass singularity connected with the initial quark line is not cancelled because we do not have the sum of all degenerate initial states, but only a single quark. But in correspondence to the initial quark we have the (bare) quark density *q*0*(y)* that appear in the convolution integral. This is a non perturbative quantity that is determined by the nucleon wave function. So we can factorize the mass singularity in a redefinition of the quark density: we replace *q*0*(y)* → *q(y, t)* = *q*0*(y)* + *q(y, t)* with:

$$
\Delta q(\mathbf{x}, t) = \frac{\alpha\_s}{2\pi} t \int\_{\chi}^{1} dy \frac{q\_0(\mathbf{y})}{\mathbf{y}} \cdot P(\frac{\chi}{\mathbf{y}}) \tag{4.78}
$$

Here the factor of *t* is a bit symbolic: it stands for log *Q*2*/km*2 and what we exactly put below *Q*2 depends on the definition of the renormalised quark density, which also fixes the exact form of the finite term *f (z)* in Eq. (4.75).

The effective parton density *q(y, t)* that we have defined is now scale dependent. In terms of this scale dependent density we have the following relations, where we have also replaced the fixed coupling with the running coupling according to the prescription derived from the RGE:

$$F(\mathbf{x},t) = \int\_{\mathbf{x}}^{1} dy \frac{q(\mathbf{y},t)}{\mathbf{y}} e^2 [\delta(\frac{\mathbf{x}}{\mathbf{y}} - 1) + \frac{a\_s(t)}{2\pi} f(\frac{\mathbf{x}}{\mathbf{y}})] = e^2 q(\mathbf{x},t) + o(a\_s(t))$$

$$\frac{d}{dt}q(\mathbf{x},t) = \frac{\alpha\_s(t)}{2\pi} \int\_{\mathbf{x}}^{1} d\mathbf{y} \frac{q(\mathbf{y},t)}{\mathbf{y}} \cdot P(\overset{\mathbf{x}}{\underset{\mathbf{y}}{\rightleftharpoons}}) + o(\alpha\_s(t)^2) \tag{4.79}$$

We see that in lowest order we reproduce the naive parton model formulae for the structure functions in terms of effective parton densities that are scale dependent. The evolution equations for the parton densities are written down in terms of kernels (the "splitting functions") that can be expanded in powers of the running coupling. At leading order, we can interpret the evolution equation by saying that the variation of the quark density at *x* is given by the convolution of the quark density at *y* times the probability of emitting a gluon with fraction *x/y* of the quark momentum.

It is interesting that the integro-differential QCD evolution equation for densities can be transformed into an infinite set of ordinary differential equations for Mellin moments [2]. The moment *fn* of a density *f (x)* is defined as:

$$f\_n = \int\_0^1 d\mathbf{x} \mathbf{x}^{n-1} f(\mathbf{x}) \tag{4.80}$$

By taking moments of both sides of the second of Eqs. (4.79) one finds, with a simple interchange of the integration order, the simpler equation for the *n*-th moment:

$$\frac{d}{dt}q\_n(t) = \frac{\alpha\_s(t)}{2\pi} \cdot P\_n \cdot q\_n(t) \tag{4.81}$$

To solve this equation we observe that:

$$\log \frac{q\_n(t)}{q\_n(0)} = \frac{P\_n}{2\pi} \int\_0^t \alpha\_s(t)dt \;= \frac{P\_n}{2\pi} \int\_{\alpha\_s}^{\alpha\_s(t)} \frac{d\alpha'}{-b\alpha'} \tag{4.82}$$

where we used Eq. (4.31) to change the integration variable from *dt* to *dα(t)* (denoted as *dα* ) and *β(α)* −*bα*2 + *...*. Finally the solution is:

$$q\_n(t) = \mathbb{I}\frac{\alpha\_s}{\alpha\_s(t)}\mathbb{I}^{\frac{\rho\_0}{2\pi b}} \cdot q\_n(0) \tag{4.83}$$

#### 4 QCD: The Theory of Strong Interactions 113

The connection of these results with the RGE general formalism occurs via the light cone OPE (recall Eq. (4.66) for *Wμν* and Eq. (4.63) for the OPE of two currents). In the case of DIS the c-number term *I (x*2*)* does not contribute, because we are interested in the connected part *
* − *<* 0| *...* |0 *>*. The relevant terms are:

$$J^{\dagger}(\mathbf{x})J(\mathbf{0}) = E(\mathbf{x}^2) \sum\_{n=0}^{\infty} c\_n(\mathbf{x}^2) \mathbf{x}^{\mu\_1} \dots \mathbf{x}^{\mu\_n} \cdot O^{\mu}\_{\mu\_1 \dots \mu\_n}(\mathbf{0}) \, \mathbf{+} \text{ less sing. terms} \tag{4.84}$$

A formally intricate but conceptually simple argument (Ref. [6], page 28) based on the analiticity properties of the forward virtual Compton amplitude shows that the Mellin moments *Mn* of structure functions are related to the individual terms in the OPE, precisely to the Fourier transform *cn(Q*2*)* (we will write it as *cn(t, α)*) of the coefficient *cn(x*2*)* times a reduced matrix element *hn* from the operators *On*: *< p*|*On μ*1*...μn (*0*)*|*p >*= *hnpμ*1 *...pμn* :

$$c\_n
 } \to M\_n = \int\_0^1 d\mathbf{x} \mathbf{x}^{n-1} F(\mathbf{x}) \tag{4.85}$$

Since the matrix element of the products of currents satisfy the RGE so do the moments *Mn*. Hence the general form of the *Q*2 dependence is given by the RGE solution (see Eq. (4.33)):

$$M\_n(t, \alpha) \,=\, c\_n[0, \alpha(t)] \exp\int\_{\alpha}^{\alpha(t)} \frac{\wp\_n(\alpha')}{\beta(\alpha')} d\alpha' \cdot h\_n(\alpha) \tag{4.86}$$

In lowest order, identifying in the simplest case *Mn* with *qn*, we have:

$$\gamma\_n(\alpha) = \frac{P\_n}{2\pi}\alpha + \dots, \qquad \beta(\alpha) = \left. -b\alpha^2 + \dots \right|\_{n=1} \tag{4.87}$$

and

$$q\_n(t) = q\_n(0) \exp\int\_{\alpha}^{\alpha(t)} \frac{\chi\_n(\alpha')}{\beta(\alpha')} d\alpha' \;= [\frac{\alpha\_s}{\alpha\_s(t)}]^{\frac{p\_n}{2\pi b}} \cdot q\_n(0) \tag{4.88}$$

which exactly coincides with Eq. (4.83).

Up to this point we have implicitly restricted our attention to non-singlet (under the flavour group) structure functions. The *Q*2 evolution equations become non diagonal as soon as we take into account the presence of gluons in the target. In fact the quark which is seen by the photon can be generated by a gluon in the target (Fig. 4.12).

**Fig. 4.12** Lowest order diagram for the interaction of the virtual photon with a

The quark evolution equation becomes:

$$\frac{d}{dt}q\_l(\mathbf{x},t) = \frac{a\_s(t)}{2\pi} [q\_l \otimes P\_{qq}] + \frac{a\_s(t)}{2\pi} [\mathbf{g} \otimes P\_{q\mathbf{g}}] \tag{4.89}$$

where we introduced the shorthand notation:

$$\mathbb{E}\left[q\otimes P\right] = \mathbb{E}\left[P\otimes q\right] = \int\_{\mathcal{X}}^{\mathrm{l}} d\mathbf{y} \frac{q(\mathbf{y},t)}{\mathbf{y}} \cdot P(\frac{\mathbf{x}}{\mathbf{y}}) \tag{4.90}$$

(it is easy to check that the convolution, like an ordinary product, is commutative). At leading order, the interpretation of Eq. (4.89) is simply that the variation of the quark density is due to the convolution of the quark density at a higher energy times the probability of finding a quark in a quark (with the right energy fraction) plus the gluon density at a higher energy times the probability of finding a quark (of the given flavour i) in a gluon. The evolution equation for the gluon density, needed to close the system, can be obtained by suitably extending the same line of reasoning to a gedanken probe sensitive to colour charges, for example a virtual gluon. The resulting equation is of the form:

$$\frac{d}{dt}\mathbf{g}(\mathbf{x},t) = \frac{\alpha\_{\mathcal{S}}(t)}{2\pi} [\sum\_{l} (q\_l + \bar{q}\_l) \otimes P\_{\mathcal{S}\mathcal{G}}] + \frac{\alpha\_{\mathcal{S}}(t)}{2\pi} [\mathbf{g} \otimes P\_{\mathcal{S}\mathcal{G}}] \tag{4.91}$$

The explicit form of the splitting functions in lowest order [18, 19] can be directly derived from the QCD vertices [19]. They are a property of the theory and do not depend on the particular process the parton density is taking part in. The results are:

$$\begin{aligned} P\_{qq} &= \frac{4}{3} \mathbf{l} \frac{\mathbf{l} + \mathbf{x}^2}{(\mathbf{l} - \mathbf{x})\_+} + \frac{3}{2} \delta(\mathbf{l} - \mathbf{x}) \mathbf{l} + o(\alpha\_s), \\\ P\_{\mathcal{B}q} &= \frac{4}{3} \frac{\mathbf{l} + (\mathbf{l} - \mathbf{x})^2}{\mathbf{x}} + o(\alpha\_s) \end{aligned}$$

$$\begin{aligned} P\_{\theta\mathcal{R}} &= \frac{1}{2} [x^2 + (1 - x)^2] + o(a\_\delta) \\ P\_{\mathcal{R}\mathcal{R}} &= 6 [\frac{x}{(1 - x)\_+} + \frac{1 - x}{x} + x(1 - x)] + \frac{33 - 2n\_f}{6} \delta(1 - x) + o(a\_\delta) \end{aligned}$$

For a generic non singular weight function *f (x)*, the "+" distribution is defined as:

$$\int\_{0}^{1} \frac{f(\mathbf{x})}{(1-\mathbf{x})\_{+}} d\mathbf{x} = \int\_{0}^{1} \frac{f(\mathbf{x}) - f(\mathbf{l})}{1-\mathbf{x}} d\mathbf{x} \tag{4.93}$$

The *δ(*1 − *x)* terms arise from the virtual corrections to the lowest order tree diagrams. Their coefficient can be simply obtained by imposing the validity of charge and momentum sum rules. In fact, from the request that the charge sum rules in Eq. (4.72) are not affected by the *Q*2 dependence one derives that

$$\int\_{0}^{1} P\_{qq}(\mathbf{x})d\mathbf{x} \, \, = \, \mathbf{0} \tag{4.94}$$

which can be used to fix the coefficient of the *δ(*1 − *x)* terms of *Pqq* . Similarly, by taking the t-derivative of the momentum sum rule in Eq. (4.73) and imposing its vanishing for generic *qi* and *g*, one obtains:

$$\int\_{0}^{1} \left[ P\_{qq}(\mathbf{x}) + P\_{\mathcal{R}q}(\mathbf{x}) \right] \mathbf{x} d\mathbf{x} = 0, \qquad \int\_{0}^{1} [2n\_f P\_{q\mathbf{g}}(\mathbf{x}) + P\_{\mathcal{R}\mathbf{g}}(\mathbf{x})] \mathbf{x} d\mathbf{x} = 0. \tag{4.95}$$

At higher orders the evolution equations are easily generalised but the calculation of the splitting functions rapidly becomes very complicated. For many years the splitting functions were only completely known at NLO accuracy [20]: *αsP* ∼ *αsP*1 + *α*2 *s P*2 + *...*. Then in recent years the NNLO results *P*3 have been first derived in analytic form for the first few moments and, then the full NNLO analytic calculation, a really monumental work, was completed in 2004 by Moch, Vermaseren and Vogt [21]. Beyond leading order a precise definition of parton densities should be specified. One can take a physical definition (for example, quark densities can be defined as to keep the LO expression for the structure function *F*2 valid at all orders, the so called DIS definition [22], and the gluon density could be defined starting from *FL*, the longitudinal structure function, or a more abstract specification (for example, in terms of the *MS* prescription). Once the definition of parton densities is fixed, the coefficients that relate the different structure functions to the parton densities at each fixed order can be computed. Similarly the higher order splitting functions also depend, to some extent, from the definition of parton densities, and a consistent set of coefficients and splitting functions must be used at each order.

The scaling violations are clearly observed by experiment and their pattern is very well reproduced by QCD fits at NLO. Examples are seen in Fig. 4.13a–d [23].

**Fig. 4.13** A recent NLO fit of scaling violations from Ref. [23], for different *x* ranges, as functions of *Q*2

These fits provide an impressive confirmation of a quantitative QCD prediction, a measurement of *qi(x, Q*2 0*)* and *g(x, Q*2 0*)* at some reference value *Q*2 0 of *Q*2 and a precise measurement of *αs(m*2 *Z)*.

#### **<IP_ADDRESS> Resummation for Deep Inelastic Structure Functions**

At small or at large values of *x* (with *Q*2 large) those terms of higher order in *αs* in either the coefficients or the splitting functions which are multiplied by powers of log 1*/x* or log *(*1 − *x)* eventually become important and should be taken into account. Fortunately the sequences of leading and subleading logs can be evaluated at all orders by special techniques and resummed to all orders.

For large x resummation [24] I refer to the recent papers [25, 26] (the latter also involving higher twist corrections, which are important at large x) where a list of references to previous work can be found.

Here we will briefly summarise the small-x case for the singlet structure function which is the dominant channel at HERA, dominated by the sharp rise of the gluon and sea parton densities at small x. The small *x* data collected by HERA can be fitted reasonably well even at the smallest measured values of *x* by the NLO QCD evolution equations, so that there is no dramatic evidence in the data for departures. This is surprising also in view of the fact that the NNLO effects in the evolution have recently become available and are quite large. Resummation effects have been shown to resolve this apparent paradox. For the singlet splitting function the coefficients of all LO and NLO corrections of order [*αs(Q*2*)*log 1*/x*] *n* and *αs(Q*2*)*[*αs(Q*2*)*log 1*/x*] *n*, respectively, are explicitly known from the BFKL analysis of virtual gluon-virtual gluon scattering [27, 28]. But the simple addition of these higher order terms to the perturbative result (with subtraction of all double counting) does not lead to a converging expansion (the NLO logs completely overrule the LO logs in the relevant domain of *x* and *Q*2). A sensible expansion is only obtained by a proper treatment of momentum conservation constraints, also using the underlying symmetry of the BFKL kernel under exchange of the two external gluons, and especially, of the running coupling effects (see the recent papers [29, 30] and refs. therein). In Fig. 4.14 we present the results for the dominant singlet splitting function *xP (x, αs(Q*2*))* for *αs(Q*2*)* ∼ 0*.*2. We see that while the NNLO perturbative splitting function sharply deviates from the NLO approximation at small x, the resummed result only shows a moderate dip with respect to the NLO perturbative splitting function in the region of HERA data, and the full effect of the true small x asymptotics is only felt at much smaller values of x. The related effects are not very important for processes at the LHC but could become relevant for next generation hadron colliders.

#### **<IP_ADDRESS> Polarized Deep Inelastic Scattering**

In polarized DIS one main question is how the proton helicity is distributed among quarks, gluons and orbital angular momentum: 1*/*2 + *g* + *Lz* = 1*/*2 (for a recent review, see, for example, [31]). For a parton density *p* (either a quark or a gluon) *p* indicates the first moment of the net helicity difference *p*+ − *p*− in a polarized proton with helicity +1*/*2 or:

$$
\Delta p(\mathcal{Q}^2) = \int\_0^1 d\mathbf{x} \left[ p\_+(\mathbf{x}, \mathcal{Q}^2) - p\_-(\mathbf{x}, \mathcal{Q}^2) \right] \tag{4.96}
$$

Experiments have shown that the quark moment is small (the "spin crisis"): values from a recent fit [32] are *exp* ∼ 0*.*21 ± 0*.*14 and *gexp* ∼ 0*.*50 ± 1*.*27 at *Q*2 = 1 GeV2 (see also [33]). This is surprising because is conserved in perturbation theory at LO (i.e. it does not evolve in *Q*2). For conserved quantities we would expect that they are the same for constituent and for parton quarks. But actually the conservation of is broken by the axial anomaly. In perturbation theory the conserved density is actually = + *nf /*2*παs g* [34]. Note that also *αsg* is conserved in LO, that is *g* ∼ *logQ*2. This behaviour is not controversial but it will take long before the log growth of *g* will be confirmed by experiment! But to establish this behaviour would show that the extraction of *g* from the data is correct and that the QCD evolution works as expected. If *g* is large enough it could account for the difference between partons () and constituents ( ). From the spin sum rule it is clear that the log increase should cancel between *g* and *Lz*. This cancelation is automatic as a consequence of helicity conservation in the basic QCD vertices. From the spin sum rule one obtains that either *g*+*Lz* is large or there are contributions to at very small x outside of the measured region. *g* can be measured indirectly by scaling violations and directly from asymmetries, e.g. in *cc*¯ production. Existing measurements by Hermes, Compass, and at RHIC are still crude but show no hint of a large *g* at accessible value of *x* and *Q*2. Present data are consistent with *g* large enough to sizeably contribute to the spin sum rule but there is no indication that *αsg* can explain the difference between constituents and parton quarks. The perspectives of better measurements are good at Compass and RHIC in the near future.

#### *4.5.4 Factorisation and the QCD Improved Parton Model*

The parton densities defined and measured in DIS are instrumental to compute hard processes initiated by hadronic collisions via the Factorisation Theorem (FT). Suppose you have a hadronic process of the form *h*1 + *h*2 → *X* + *all* where *hi* are hadrons and *X* is some triggering particle or pair of particles which specify the large scale *Q*2 relevant for the process, in general somewhat, but not much, smaller than s, the total c.o.m. squared mass. For example, in *pp* or *pp*¯ collisions, *X* can be a *W* or a Z or a virtual photon with large *Q*2, or a jet at large transverse momentum *pT* , or a pair of heavy quark-antiquark of mass M. By "all" we mean a totally inclusive collection of gluons and light quark pairs. The FT states that for the total crosssection or some other sufficiently inclusive distribution we can write, apart from power suppressed corrections, the expression:

$$\sigma(\mathbf{s}, \mathbf{\tau}) = \sum\_{AB} \int d\mathbf{x}\_1 d\mathbf{x}\_2 p\_{1A}(\mathbf{x}\_1, \mathcal{Q}^2) p\_{2B}(\mathbf{x}\_2, \mathcal{Q}^2) \sigma\_{AB}(\mathbf{x}\_1 \mathbf{x}\_2 \mathbf{s}, \mathbf{\tau}) \tag{4.97}$$

Here *τ* = *Q*2*/s* is a scaling variable, *piC* are the densities for a parton of type C inside the hadron *hi*, *σAB* is the partonic cross-section for parton-A + parton-B→ *X* + *all* . This result is based on the fact that the mass singularities that are associated with the initial legs are of universal nature, so that one can reproduce the same modified parton densities, by absorbing these singularities into the bare parton densities, as in deep inelastic scattering. Once the parton densities and *αs* are known from other measurements, the prediction of the rate for a given hard process is obtained with not much ambiguity (e.g from scale dependence or hadronisation effects). The NLO calculation of the reduced partonic cross-section is needed in order to correctly specify the scale and in general the definition of the parton densities and of the running coupling in the leading term. The residual scale and scheme dependence is often the most important source of theoretical error. In the following we consider a few examples.

A comparison of data and predictions on the production of jets at large √*s* and *pT* in *pp* or *pp*¯ collisions is shown in Fig. 4.15 [9, 35].

This is a particularly significant test because the rates at different c.o.m. energies and, for each energy, at different values of *pT* span over many orders of magnitude.

This steep behaviour is determined by the sharp falling of the parton densities at large x. Also the corresponding values of √*s* and *pT* are large enough to be well inside the perturbative region. The overall agreement of the data from ISR, UA1,2 and CDF and D0 is spectacular. Similar results also hold for the production of photons at large *pT* . The collider data [36], shown in Fig. 4.16 [9], are in fair agreement with the theoretical predictions. For the same process less clear a situation is found with fixed target data. Here, first of all, the experimental results show some internal discrepancies. Also, the *pT* accessible values being smaller, the theoretical uncertainties are larger. But it is true that the agreement is poor, so that the necessity of an "intrinsic" transverse momentum of partons inside the hadron of over 1 GeV has been claimed, which theoretically is not justified (rather, given the sharp falling down at large *pT* , it could be interpreted as a correction for *pT* calibration errors).

For heavy quark production at colliders [42] the agreement is very good for the top crosssection at the Tevatron (Fig. 4.17) [43, 44]. The bottom production at the Tevatron has for some time represented a problem [45]. The total rate and the *pT* distribution of b quarks observed at CDF appeared in excess of the prediction, up to the largest measured values of *pT* . But this is a complicated case, with different scales being present at the same time: √*s*, *pT* , *mb*. Finally the problem has been solved (Fig. 4.18) by better taking into account a number of small effects from resummation of large logarithms, the difference between b hadrons and b partons, the inclusion of better fragmentation functions etc. [46].

**Fig. 4.16** Single photon production in *pp*¯ colliders as function of *pT* [9]

**Fig. 4.17** The *t* production cross-section at the Tevatron *pp*¯ collider [44]

Drell-Yan processes, including lepton pair production via virtual *γ* , W or Z exchange, offer a particularly good opportunity to test QCD. The process is quadratic in parton densities, and the final state is totally inclusive, while the large scale is specified and measured by the invariant mass squared *Q*2 of the lepton pair which itself is not strongly interacting (so there no dangerous hadronisation effects). The QCD improved parton model leads directly to a prediction for the total rate as a function of *Q*2. The value of the LO cross-section is inversely proportional to the number of colours *NC* because a quark of given colour can only annihilate with an

antiquark of the same colour to produce a colourless lepton pair. The order *αs(Q*2*)* corrections to the total rate were computed long ago and found to be particularly large [22, 38], when the quark densities are defined from the structure function *F*2 measured in DIS at *q*2 = −*Q*2. The ratio *σcorr/σLO* of the corrected and the Born cross-sections, was called K-factor, because it is almost a constant in rapidity. In recent years also the NLO full calculation of the K-factor was completed, a very remarkable calculation [37]. The QCD predictions have been best tested for W and Z production at CERN *SppS*¯ and Tevatron energies. *Q* ∼ *mW,Z* is large enough to make the prediction reliable (with a not too large K-factor) and the ratio √*τ* = *Q/*√*s* is not too small. Recall that in lowest order *x*1*x*2*s* = *Q*2 so that the parton densities are probed at x values around √*τ* . We have √*τ* = 0*.*13 − 0*.*15 (for W and Z production, respectively) at √*s* = 630 GeV (CERN *SppS*¯ Collider) and √*τ* = 0*.*04 − 0*.*05 at the Tevatron. In this respect the prediction is more delicate at the LHC, where √*τ* ∼ 5*.*7 − 6*.*5 · 10−3. One comparison of the experimental total rates at the Tevatron [48] with the QCD predictions is shown in Fig. 4.19, together with the expected rates at the LHC (based on the structure functions obtained in [23]).

The calculation of the W/Z *pT* distribution has been a classic problem in QCD. For large *pT* , for example *pT* ∼ *o(mW )*, the *pT* distribution can be reliably computed in perturbation theory, which was done up to NLO in the late '70's and early '80's. A problem arises in the intermediate range *QCD << pT << mW* , where the bulk of the data is concentrated, because terms of order *αs(p*2 *T )*log *m*2 *W /p*2 *T* become of order one and should included to all orders [39]. At order *αs* we have:

$$\frac{1}{\sigma\_0} \frac{d\sigma\_0}{dp\_T^2} = (1+A)\delta(p\_T^2) + \frac{B}{p\_T^2} \log \frac{m\_W^2}{p\_T^2} \ \ + \frac{C}{(p\_T^2)\_+} \ + D(p\_T^2) \tag{4.98}$$

**Fig. 4.19** Data vs. theory for *W* and *Z* production at the Tevatron (√*s* = 1.8 TeV) together with the corresponding predictions for the LHC (√*s*= 1.4 TeV) [48]

where A, B, C, D are coefficients of order *αs*. The "+" distribution is defined in complete analogy with Eq. (4.93):

$$\int\_{0}^{p\_{TMX}^{2}} g(z)f(z)\_{+} dz = \int\_{0}^{p\_{TMX}^{2}} [g(z) - g(0)]f(z) dz \tag{4.99}$$

The content of this, at first sight mysterious, definition is that the singular "+" terms do not contribute to the total cross-section. In fact for the cross-section the weight function *g(z)* = 1 and we obtain:

$$
\sigma = \sigma\_0 [(1+A) \, + \, \int\_0^{p\_{TMAX}^2} D(z) dz] \, \tag{4.100}
$$

The singular terms, of infrared origin, are present at the non completely inclusive level but disappear in the total cross-section. Arguments have been given that these singularities are expected to exponentiate. Explicit calculations in low order support the exponentiation which leads to the following expression:

$$\frac{1}{\sigma\_0} \frac{d\sigma\_0}{dp\_T^2} = \int \frac{d^2b}{4\pi} \exp\left(-ib \cdot p\_T\right) (1+A) \exp S(b) \tag{4.101}$$

with:

$$S(b) = \int\_0^{p\_{TMX}} \frac{d^2 k\_T}{2\pi} [\exp i k\_T \cdot b - 1] [\frac{B}{k\_T^2} \log \frac{m\_W^2}{k\_T^2} + \frac{C}{k\_T^2}] \tag{4.102}$$

At large *pT* the LO perturbative expansion is recovered. At intermediate *pT* the infrared *pT* singularities are resummed (the Sudakov log terms, which are typical of vector gluons, are related to the fact that for a charged particle in acceleration it is impossible not to radiate, so that the amplitude for no soft gluon emission is exponentially suppressed). However this formula has problems at small *pT* , for example, because of the presence of *αs* under the integral for *S(b)*:

**Fig. 4.20** QCD predictions for the *W pT* distribution compared with recent D0 data at the Tevatron (√*s* = 1.8 TeV)[49] [40]

presumably the relevant scale is of order *k*2 *T* . So it must be completed by some non perturbative ansatz or an extrapolation into the soft region. All the formalism has been extended to NLO accuracy, where one starts from the perturbative expansion at order *α*2 *s* , and generalises the resummation to also include NLO terms of order *αs(p*2 *T )* 2 log*m*2 *W /p*2 *T* (see, for example, [40]). The comparison with the data is very impressive. In Fig. 4.20 we see the *pT* distribution as predicted in QCD (with a number of variants that mainly differ in the approach to the soft region) compared with some recent data at the Tevatron [49].

A great effort is being devoted to the preparation to the LHC. Calculations for specific processes are being completed. A very important example is Higgs production via *g* + *g* → *H*. The amplitude is dominated by the top quark loop, as discussed in Chap. 3 [51]. The NLO corrections turn out to be particularly large [52], as seen in Fig. 4.21. Higher order corrections can be computed either in the effective lagrangian approach, where the heavy top is integrated away and the loop is shrunk down to a point [53] [the coefficient of the effective vertex is known to *α*4 *s* accuracy [54]], or in the full theory. At the NLO the two approaches agree very well for the rate as a function of *mH* [55]. The NNLO corrections have been computed in the effective vertex approximation [56] (see Fig. 4.21). Beyond fixed order resummation of large logs were carried out [57]. Also the NLO EW contributions have been computed [58]. Rapidity (at NNLO) [59] and *pT* distributions (at NLO) [60] have also been evaluated. At smaller *pT* the large logarithms [*log(pT /mH )*] *n* have been

**Fig. 4.21** The Higgs gluon fusion cross section in LO, NLO and NLLO

resummed in analogy with what was done long ago for W and Z production [61]. For additional recent works on Higgs physics at colliders see, for example, [62].

The activity on event simulation also received a big boost from the LHC preparation (see, for example, the review [50]). General algorithms for performing NLO calculations numerically (requiring techniques for the cancellation of singularities between real and virtual diagrams) have been developed (see, for example, [65]). The matching of matrix element calculation of rates together with the modeling of parton showers has been realised in packages, as for example in the MC@NLO [63] or POWHEG [64] based on HERWIG. The matrix element calculation, improved by resummation of large logs, provides the hard skeleton (with large *pT* branchings) while the parton shower is constructed by a sequence of factorized collinear emissions fixed by the QCD splitting functions. In addition, at low scales a model of hadronisation completes the simulation. The importance of all the components, matrix element, parton shower and hadronisation can be appreciated in simulations of hard events compared with the Tevatron data.

At different places in the previous pages we have seen examples of resummation of large logs. This is a very important chapter of modern QCD. The resummation of soft gluon logs enter in different problems and the related theory is subtle. I refer the reader here to some recent papers where additional references can be found [66]. A particularly interesting related development has to do with the so called non global logs (see, for example, [67]). If in the measurement of an observable some experimental cuts are introduced, which is a very frequent case, then a number of large logs can arise from the corresponding breaking of inclusiveness. The discussion of this problem has led to rethinking the theory of final state observables. It is also important to mention the development of software for the automated implementation of resummation (see, for example, [68]).

Before closing this section I would like to mention some very interesting developments at the interface between string theory and QCD, twistor calculus. A precursor work was the Parke-Taylor result in 1986 [69] on the amplitudes for n incoming gluons with given helicities [70]. Inspired by dual models, they derived a compact formula for the maximum non vanishing helicity violating amplitude (with n−2 plus and 2 minus helicities) in terms of spinor products. Using the relation between strings and gauge theories in twistor space Witten developed in '03 [71] a formalism in terms of effective vertices and propagators that allows to compute all helicity amplitudes. The method, alternative to other modern techniques for the evaluation of Feynman diagrams [73], leads to very compact results. Since then rapid progress followed (for reviews, see [72]): for tree level processes powerful recurrence relations were established [74], the method was extended to include massless external fermions [75] and also external EW vector bosons [76] and Higgs particles [77]. The level already attained is already important for multijet events at the LHC. And the study of loop diagrams has been started. In summary, this road looks new and promising.

# **4.6 Measurements of α***s*

Very precise and reliable measurements of *αs(m*2 *Z)* are obtained from *e*+*e*− colliders (in particular LEP) and from deep inelastic scattering.

# *4.6.1 αs from e***+***e***−** *Colliders*

The main methods at *e*+*e*− colliders are: (a) Inclusive hadronic Z decay, *Rl*, *σh*, *σl*, *Z*. (b) Inclusive hadronic *τ* decay. (c) Event shapes and jet rates.

As we have seen, for a quantity like *Rl* we can write a general expression of the form:

$$R\_l = \frac{\Gamma(Z, \mathfrak{r} \to hadrons)}{\Gamma(Z, \mathfrak{r} \to leptons)} \sim R^{EW} (1 + \delta\_{QCD} + \delta\_{NP}) + \dots \tag{4.103}$$

where *REW* is the electroweak-corrected Born approximation, *δQCD*, *δNP* are the perturbative (logarithmic) and non perturbative (power suppressed) QCD corrections. For a measurement of *αs* at the Z (in the following we always refer to the *MS* definition of *αs*) one can use all info from *Rl*, *Z* = 3*l* + *h* and (f=h or l) *σf* = 12*π
l
f /(m*2 *Z* 2 *Z)*. In the past the measurement from *Rl* was preferred (by itself it leads to *αs(mZ)* = 0*.*1226 + 0*.*0058 − 0*.*0036) but at LEP there is no reason for that. In all these quantities *αs* enters through *h*, but the measurements of, say, *Z*, *Rl* and *σl* are really independent (they are affected by entirely different systematics: *Z* is extracted from the line shape, *Rl* and *σl* are measured at the peak but *Rl* does not depend on the absolute luminosity while *σl* does).The most sensitive single quantity is *σl*. The combined value from the measurements at the Z (assuming the validity of the SM and a light Higgs mass) is [78]:

$$
\alpha\_s(m\_Z) = 0.119 \pm 0.003\tag{4.104}
$$

For a relatively light Higgs (even if not as light as from the fit to EW observables) the final error is mainly experimental with a theoretical component from our ignorance of *mH* , of higher orders in the QCD expansion [79] and also from uncertainties on the Bhabha luminometer (which affect *σh,l*) [80]. By adding all other electroweak precision electroweak tests (in particular *mW* ) one similarly finds [41]:

$$
\alpha\_s(m\_Z) = 0.1185 \pm 0.003\tag{4.105}
$$

We now consider the measurement of *αs(mZ)* from *τ* decay. *Rτ* has a number of advantages that, at least in part, tend to compensate for the smallness of *mτ* = 1*.*777 GeV. First, *Rτ* is maximally inclusive, more than *Re*+*e*− *(s)*, because one also integrates over all values of the invariant hadronic squared mass:

$$R\_{\mathbf{r}} = \frac{1}{\pi} \int\_0^{m\_{\mathbf{r}}^2} \frac{ds}{m\_{\mathbf{r}}^2} (1 - \frac{s}{m\_{\mathbf{r}}^2})^2 Im \, \Pi\_{\mathbf{r}}(\mathbf{s}) \tag{4.106}$$

The perturbative contribution is known at NNLO. Analyticity can be used to transform the integral into one on the circle at |*s*| = *m*2 *τ* :

$$R\_{\mathbb{T}} = \frac{1}{2\pi i} \oint\_{|s|=m\_{\mathbb{T}}^2} \frac{ds}{m\_{\mathbb{T}}^2} (1 - \frac{s}{m\_{\mathbb{T}}^2})^2 \Pi\_{\mathbb{T}}(s) \tag{4.107}$$

Also, the factor *(*1 − *s m*2 *τ )* 2 is important to kill the sensitivity the region *Re*[*s*] = *m*2 *τ* where the physical cut and the associated thresholds are located. Still the quoted result [81] looks a bit too precise:

$$
\alpha\_s(m\_\pi) = 0.345 \pm 0.010\tag{4.108}
$$

or

$$
\alpha\_s(m\_Z) = 0.1215 \pm 0.0012 \tag{4.109}
$$

This precision is obtained by taking for granted that corrections suppressed by 1*/m*2 *τ* are negligible. This is because, in the massless theory, the light cone expansion is given by:

$$\delta\_{NP} = \frac{ZERO}{m\_{\tau}^2} \quad + c\_4 \cdot \frac{}{m\_{\tau}^4} \quad + c\_6 \cdot \frac{}{m\_{\tau}^6} \quad + \dotsb \tag{4.110}$$

In fact there are no dim-2 Lorentz and gauge invariant operators. For example, *gμgμ* is not gauge invariant. In the massive theory, the ZERO is replaced by light quark mass-squared *m*2. This is still negligible if *m* is taken as a lagrangian mass of a few MeV. If on the other hand the mass were taken to be the constituent mass of order *QCD*, this term would not be at all negligible and would substantially affect the result (note that *αs(mτ )/π* ∼ 0*.*1 ∼ *(*0*.*6 GeV*/mτ )*2 and that *QCD* for three flavours is large). For example, the PDG value and estimate of the error is [9]:

$$
\alpha\_{\rm s}(mz) = 0.120 \pm 0.003. \tag{4.111}
$$

Most people believe the optimistic version. I am not convinced that the gap is not filled up by ambiguities of 0*(*2 *QCD/m*2 *τ )* from *δpert* [82]. In any case, one can discuss the error, but it is true and remarkable, that the central value from *τ* decay, obtained at very small *Q*2, is in good agreement with all other precise determinations of *αs* at more typical LEP values of *Q*2.

Important determinations of *αs(mZ)* are obtained from different infrared safe observables related to event rates and jet shapes in *e*+*e*− annihilation. The main problem of these measurements is the large impact of non perturbative hadronization effects on the result and therefore on the theoretical error. The perturbative part is known at NLO. One advantage is that the same measurements can be repeated at different √*s* values (e.g. with the same detectors at LEP1 or LEP2) allowing for a direct observation of the energy dependence. A typical result, from jets and event shapes at LEP, quoted in Ref. [83], is given by:

$$
\alpha\_s(m\_Z) = 0.121 \pm 0.005. \tag{4.112}
$$

Recently the rate of 4-jet events (proportional to *α*2 *s* ) at LEP as function of *ycut* has been used [84], for which a NLO theoretical calculation exists [85]. The quoted result is *αs(mZ)* = 0*.*1176 ± 0*.*0022 (the actual error could be somewhat larger because the ambiguity from hadronisation modeling is always debatable).

# *4.6.2 αs from Deep Inelastic Scattering*

QCD predicts the *Q*2 dependence of *F (x, Q*2*)* at each fixed x, not the x shape. But the *Q*2 dependence is related to the x shape by the QCD evolution equations. For each x-bin the data allow to extract the slope of an approximately straight line in *dlogF (x, Q*2*)/dlogQ*2: the log slope. The *Q*2 span and the precision of the data are not much sensitive to the curvature, for most x values. A single value of *QCD* must be fitted to reproduce the collection of the log slopes. For the determination of *αs* the scaling violations of non-singlet structure functions would be ideal, because of the minimal impact of the choice of input parton densities. We can write the non-singlet evolution equations in the form:

$$\frac{d}{dt}\log F(\mathbf{x},t) = \frac{\alpha\_s(t)}{2\pi} \int\_{\mathbf{x}}^1 \frac{d\mathbf{y}}{\mathbf{y}} \frac{F(\mathbf{y},t)}{F(\mathbf{x},t)} P\_{qq}(\frac{\mathbf{x}}{\mathbf{y}}, \alpha\_s(t)) \tag{4.113}$$

where *Pqq* is the splitting function. At present NLO and NNLO corrections are known. It is clear from this form that, for example, the normalisation error on the input density drops away, and the dependence on the input is reduced to a minimum (indeed, only a single density appears here, while in general there are quark and gluon densities). Unfortunately the data on non-singlet structure functions are not very accurate. If we take the difference of data on protons and neutrons, *Fp* − *Fn*, experimental errors add up in the difference and finally are large. The *F*3*νN* data are directly non-singlet but are not very precise. A determination of *αs* from the CCFR data on *F*3*νN* has led to [86]:

$$
\alpha\_s(m\_Z) = 0.119 \pm 0.006\tag{4.114}
$$

A recent analysis of the same data leads to *αs(mZ)* = 0*.*119 ± 0*.*002 [87], but the theoretical error associated with the method and with the choice adopted for the scale ambiguities is not considered. A fit to non singlet structure functions in electroor muon-production extracted from proton and deuterium data at the NNLO level was performed in Ref. [88] with the result:

$$
\alpha\_s(m\_Z) = 0.114 \pm 0.002\tag{4.115}
$$

When one measures *αs* from scaling violations on *F*2 from e or *μ* beams, the data are abundant, the errors small but there is an increased dependence on input parton densities and especially a strong correlation between the result on *αs* and the input on the gluon density. There are complete and accurate derivations of *αs* from scaling violations in *F*2. In a well known analysis by Santiago and Yndurain [89], the data on protons from SLAC, BCDMS, E665 and HERA are used with NLO kernels plus the NNLO first few moments. The analysis is based on an original method that uses projections on a specially selected orthogonal basis, the Bernstein polynomials. The quoted result is given by:

$$
\alpha\_s(m\_Z) = 0.1163 \pm 0.0014\tag{4.116}
$$

(these authors also quote *αs(mZ)* = 0*.*115 ± 0*.*006 for *F*3 data in *νN* scattering). A different analysis by Alekhin [90] of existing data off proton and deuterium targets with NNLO kernels and a more conventional method leads to

$$
\alpha\_s(m\_Z) = 0.114 \pm 0.002\tag{4.117}
$$

In both analyses the dominant error is theoretical and, in my opinion, should be somewhat larger than quoted. An interesting perspective on theoretical errors can be obtained by comparing analyses with different methods. We add the following examples. From truncated moments (but with a limited set of proton data and NLO kernels) [91]: *αs(mZ)* = 0*.*122±0*.*006, from Nachtmann moments (which take into account some higher twist terms) and proton data [92]: *αs(mZ)* = 0*.*1188±0*.*0017. A combination of measurements at HERA by H1 and Zeus, also including final state jet observables, leads to *αs(mZ)* = 0*.*1186 ± 0*.*0051 [93], most of the error being theoretical. Finally, to quote a number that appears to me as a good summary of the situation of *αs(mZ)* from DIS one can take the result from a NNLO analysis of available data by the MRST group [94] as quoted by Particle Data Group, W.-M. Yao et al. [9]:

$$
\alpha\_s(m\_Z) = 0.1167 \pm 0.0036\tag{4.118}
$$

If we compare these results on *αs* from DIS with the findings at the Z, given by Eq. (4.105), we see that the agreement is good, with the value of *αs* from the most precise DIS measurements a bit on the low side with respect to *e*+*e*−.

# *4.6.3 Summary on αs*

There are a number of other determinations of *αs* which are important because they arise from qualitatively different observables and methods. For example [9, 83], some are obtained from the Bjorken sum rule and the scaling violations in polarized DIS, from *ϒ* decays, from the 4-jet rate in *e*+*e*−. A special mention deserves the "measurement" of *αs* from lattice QCD [95]. A number of hadronic observables, in particular *ϒ* − *ϒ* splitting, pion and kaon decay constants, the *Bs* mass and the baryon mass are used to fix the lattice spacing and to accurately tune the QCD simulation. The value of *αs* is then obtained by computing non perturbatively a number of quantities related to Wilson loops that can also be given in perturbation theory. The result is then evolved with state of the art beta functions to obtain *αs(mZ)* = 0*.*1170 ± 0*.*0012. This result is interesting for its really special nature but it is not clear that the systematics due to the lattice technology is as small as claimed.

Summarising: there is very good agreement among many different measurements of *αs*. In Fig. 4.22 [83], a compilation of the data is reported with each measurement plotted at the scale of the experiment, which shows the consistency of the measurements and the running of *αs*. This is a very convincing, quantitative test of QCD. If I take the values of *αs(mZ)* from precision electroweak data, Eq. (4.105), from *τ* decay with the central value as in Eq. (4.109) but the larger error as in Eq. (4.111), from jets in *e*+*e*−, Eq. (4.112), and from DIS, Eq. (4.118), the average is :

$$
\alpha\_{\rm s}(m\,\mathrm{z}) = 0.119 \pm 0.002\,\tag{4.119}
$$

**Fig. 4.22** The running of *αs* as determined from present data [83]

For comparison, the average value quoted by PDG 2006 is *αs(mZ)* = 0*.*1176 ± 0*.*0020 while Ref. [83] gives *αs(mZ)* = 0*.*1189 ± 0*.*0010.

0.5

The value of (for *nf* = 5) which corresponds to Eq. (4.119) is:

$$
\Lambda\_{\mathfrak{F}} = 221 \pm 25 \text{ MeV} \tag{4.120}
$$

 is the scale of mass that finally appears in massless QCD. It is the scale where *αs()* is of order one. Hadron masses are determined by . Actually the *ρ* mass or the nucleon mass receive little contribution from the quark masses (the case of pseudoscalar mesons is special, as they are the pseudo Goldstone bosons of broken chiral invariance). Hadron masses would be almost the same in massless QCD.

## **4.7 Conclusion**

We have seen that perturbative QCD based on asymptotic freedom offers a rich variety of tests and we have described some examples in detail. QCD tests are not as precise as for the electroweak sector. But the number and diversity of such tests has established a very firm experimental foundation for QCD as a theory of strong interactions. The field of QCD appears as one of great maturity but also of robust vitality with many rich branches and plenty of new blossoms. The physics content of QCD is very large and our knowledge, especially in the non perturbative domain, is still very limited but progress both from experiment (Tevatron, RHIC, LHC......) and from theory is continuing at a healthy rate. And all the QCD predictions that we were able to formulate and to test appear to be in very good agreement with experiment.

**Acknowledgments** I am very grateful to Richard Ball, Keith Ellis, Stefano Forte, Ashutosh Kotwal, Vittorio Lubicz, Lorenzo Magnea, Michelangelo Mangano, Silvano Simula, Hartmut Wittig for their help and advise.

#### **References for 4**

Disclaimer: This list of references is by far incomplete. It is just meant to provide a guide to the vast literature on QCD. So it includes only a few old papers and, for most of the issues, some relatively recent works where to look for a more complete list of references.

#### **References**

and R. Petronzio, Phys. Lett. 97B (1980) 438; E.G. Floratos, R. Lacaze and C. Kounnas, Phys. Lett. 99B (1981) 89, 285; R. T. Herrod and S. Wada, Z. Phys. C9 (1981) 351.

JHEP **0709** (2007) 018 [arXiv:0707.2373 [hep-ph]]; P. M. Nadolsky, C. Balazs, E. L. Berger and C.-P. Yuan, Phys. Rev. D **76** (2007) 013008 [hep-ph/0702003 [HEP-PH]]; C. Balazs, E. L. Berger, P. M. Nadolsky and C.-P. Yuan, Phys. Rev. D **76** (2007) 013009 [arXiv:0704.0001 [hep-ph]]; M. Ciccolini, A. Denner and S. Dittmaier, Phys. Rev. Lett. **99** (2007) 161803 [arXiv:0707.0381 [hep-ph]].

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made.

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# **Chapter 5 QCD on the Lattice**

**Hartmut Wittig**

# **5.1 Introduction and Outline**

Since Wilson's seminal papers of the mid-1970s, the lattice approach to Quantum Chromodynamics has become increasingly important for the study of the strong interaction at low energies, and has now turned into a mature and established technique. In spite of the fact that the lattice formulation of Quantum Field Theory has been applied to virtually all fundamental interactions, it is appropriate to discuss this topic in a chapter devoted to QCD, since by far the largest part of activity is focused on the strong interaction. Lattice QCD is, in fact, the only known method which allows ab initio investigations of hadronic properties, starting from the QCD Lagrangian formulated in terms of quarks and gluons.

# *5.1.1 Historical Perspective*

In order to illustrate the wide range of applications of the lattice formulation, we give a brief historical account below.

First applications of the lattice approach in the late 1970s employed analytic techniques, predominantly the strong coupling expansion, in order to investigate colour confinement and also the spectrum of glueballs. While these attempts gave valuable insights, it soon became clear that in the case of non-Abelian gauge theories such expansions were not sufficient to produce quantitative results.

H. Wittig (-)

Johannes-Gutenberg-Universität Mainz, Mainz, Germany e-mail<EMAIL_ADDRESS>
First numerical investigations via Monte Carlo simulations, focusing in particular on the confinement mechanism in pure Yang–Mills theory, were carried out around 1980. The following years saw already several valiant attempts to study QCD numerically, yet it was realized that the available computer power was grossly inadequate to incorporate the effects of dynamical quarks. It was then that the socalled "quenched approximation" of QCD was proposed as a first step to solving full QCD numerically. This approximation rests on the *ad hoc* assumption that the dominant non-perturbative effects are mediated by the gluon field. Hadronic observables can then be computed on a pure gauge background with far less numerical effort compared to the real situation where quarks have a feedback on the gluon field. The main focus of activity during the 1980s was on bosonic theories: numerical simulations were used to compute the glueball spectrum in pure Yang– Mills theory. Another important result during this period concerned *φ*4-theory and the implications of its supposed "triviality" for the Higgs-Yukawa sector of the Standard Model. Using a combination of analytic and numerical techniques, the triviality of *φ*4 theory could be rigorously established.

Except for a brief spell of activity around the turn of the decade to simulate QCD with dynamical fermions, most projects in the 1990s were devoted to explore quenched QCD. Having recognized that the available computers and the efficiency of known algorithms were by far not sufficient to perform "realistic" simulations of QCD with controlled errors, lattice physicists resorted to exploring the quenched approximation and its limitations for a number of phenomenologically interesting quantities. Although the systematic error that arises by neglecting dynamical quarks could not be quantified reliably, many important quantities, such as quark and hadron masses, the strong coupling constant and weak hadronic matrix elements, were computed for the first time. One of the icons of that period was surely a plot of the masses of the lightest hadrons in the continuum limit of quenched QCD, produced by the CP-PACS Collaboration: their results indicated that the quenched approximation works surprisingly well (at least for these quantities), since the computed spectrum agreed with experimental determinations at the level of 10%. Simultaneously, a number of sophisticated techniques have been developed during the 1990s, thereby helping to control systematic effects, mainly pertaining to the influence of lattice artefacts, as well as the renormalization of local operators in the lattice regularized theory and their relation to continuum schemes such as MS. Perhaps the most significant development at the end of the 1990s was the clarification of the issue of chiral symmetry and lattice regularization. Following this work it is now understood under which conditions the lattice formulation is compatible with chiral symmetry. The importance of this development extends far beyond QCD and implies new prospects for the non-perturbative study of chiral gauge theories.

Since 2000 the focus has decidedly shifted from the quenched approximation to serious attempts to simulate QCD with dynamical quarks, thereby tackling the biggest remaining systematic uncertainty. Progress in this area has not just been determined by the vast increase in computer power since the very first Monte Carlo simulations, but rather by the development of new algorithmic ideas, combined with the use of alternative discretizations that are numerically more efficient. At the time of writing this contribution (2007), the whole field is actually in a state of transition: although the quenched approximation is being abandoned, the latest results from simulations with dynamical quarks have not yet reached the same level of accuracy in regard to controlling systematic errors due to lattice artefacts and effects from renormalization, as compared to earlier quenched calculations. It can thus be expected that many of the results discussed later in this chapter will soon be superseded by more accurate numbers. In turn, the quenched approximation will be completely obsolete in a few years time, except perhaps to test new ideas or for exploratory studies of more complex quantities.

#### *5.1.2 Outline*

We begin with an introduction of the basic concepts of the lattice formulation of QCD. This shall include the field theoretical foundations, discretizations of the QCD Lagrangian, as well as simulation algorithms and other technical aspects related to the actual calculation of physical observables from suitable correlation functions. The following sections deal with various applications. Lattice calculations of the hadron spectrum are described in Sect. 5.3. Section 5.4 is devoted to lattice investigations of the confinement phenomenon. Determinations of the fundamental parameters of QCD, namely the strong coupling constant and quark masses are a major focus of this article, and are presented in Sect. 5.5. Another important property of QCD, namely the spontaneously broken chiral symmetry, is discussed in some detail in Sect. 5.6, which also includes a brief introduction into analytical nonperturbative approaches to the strong interaction, based on effective field theories. Lattice calculations of weak hadronic matrix elements, which serve to pin down the elements of the Cabibbo–Kobayashi–Maskawa matrix, are covered in Sect. 5.7. We end this contribution with a few concluding remarks.

In addition to the topics listed above, lattice simulations of QCD have also made important contributions to the determination of the phase structure of QCD, including results for the critical temperature of the deconfinement phase transition. Nevertheless, in this chapter we restrict the discussion to QCD at zero temperature and refer the reader to other parts of this volume.

#### **5.2 The Lattice Approach to QCD**

The essential features of the lattice formulation can be summarized by the following statement:

Lattice QCD is the non-perturbative approach to the gauge theory of the strong interaction through regularized, Euclidean functional integrals. The regularization is based on a discretization of the QCD action which preserves gauge invariance at all stages.

This definition includes all basic ingredients: starting from the functional integral itself avoids any particular reference to perturbation theory. This is what we mean when we call lattice QCD an ab initio method. The Euclidean formulation, which is obtained by rotating to imaginary time, reveals the close relation between Quantum Field Theory and Statistical Mechanics. In particular, the Euclidean functional integral is equivalent to the partition function of the corresponding statistical system. This equivalence is particularly transparent if the field theory is formulated on a discrete space-time lattice. Via this relation, the whole toolkit of condensed matter physics, including high-temperature expansions, and, perhaps most importantly, Monte Carlo simulations, are at the disposal of the field theorist.

Many of the basic concepts introduced in this section are discussed in several common textbooks on the subject [1–4], which can be consulted for further details.

#### *5.2.1 Euclidean Quantization*

The generic steps in the Euclidean quantization procedure of a lattice field theory are the following:

We shall now illustrate this procedure for a simple example, namely the theory for a neutral scalar field.

**Step 1** Consider a real, classical field *φ(x)*, with *x* = *(x*0*, x*1*, x*2*, x*3*)*, whose time variable *x*0 is obtained by analytically continuing *t* to −i*x*0. The Euclidean action *S*E[*φ*] is defined as

$$\mathrm{SE}[\phi] = \int \mathrm{d}^4 x \left\{ \frac{1}{2} \partial\_\mu \phi(\mathbf{x}) \partial\_\mu \phi(\mathbf{x}) + V(\phi) \right\}, \qquad \partial\_\mu \equiv \frac{\partial}{\partial x\_\mu}, \tag{5.1}$$

where

$$V(\phi) = \frac{1}{2}m^2\phi(\mathbf{x})^2 + \frac{\lambda}{4!}\phi(\mathbf{x})^4. \tag{5.2}$$

**Step 2** In order to discretize the theory, a hyper-cubic lattice, E, is introduced as the set of discrete space-time points, i.e.

$$\Lambda\_{\mathcal{E}} = \left\{ \mathbf{x} \in \mathbb{R}^4 \, \middle| \, \mathbf{x}^0/a = 1, \dots, N\_{\mathcal{l}}; \, \mathbf{x}^j/a = 1, \dots, N\_{\mathcal{S}} \, \middle| \, \, j = 1, 2, 3 \right\},$$

$$T = N\_{\mathcal{l}} a, \ L = N\_{\mathcal{S}} a. \tag{5.3}$$

Thus, any space-time point is an integer multiple of the lattice spacing *a*. The total number of lattice sites is *N*t × *N*3 s , while the physical space-time volume is *T* × *L*3. The discretized action is then given by

$$\mathcal{S}\_{\rm E}[\phi] = a^4 \sum\_{\mathbf{x} \in \Lambda\_{\rm E}} \left\{ \frac{1}{2} d\_{\mu} \phi(\mathbf{x}) d\_{\mu} \phi(\mathbf{x}) + \frac{1}{2} m^2 \phi(\mathbf{x})^2 + \frac{\lambda}{4!} \phi(\mathbf{x})^4 \right\},\tag{5.4}$$

where the lattice derivatives can be defined as

$$d\_{\mu}\phi(\mathbf{x}) := \frac{1}{a} \left( \phi(\mathbf{x} + a\hat{\mu}) - \phi(\mathbf{x}) \right) \qquad \text{``forward'' derivative}, \qquad (5.5)$$

$$d^\*\_{\mu}\phi(\mathbf{x}) := \frac{1}{a} \left(\phi(\mathbf{x}) - \phi(\mathbf{x} - a\hat{\mu})\right) \qquad \text{``backward'' derivative.} \qquad (5.6)$$

Here and below *μ*ˆ denotes a unit vector in direction of *μ*. Via a Fourier transform, the Euclidean lattice E is related to the dual lattice, ∗ E, defined by

$$\begin{aligned} \Lambda\_{\rm E}^{\*} &= \left\{ p \in \mathbb{R}^{4} \, \middle| \, p\_{0} = \frac{2\pi}{T} n^{0}, \, p\_{j} = \frac{2\pi}{L} n^{j} \right\} \\\ n^{0} &= -\frac{N\_{\rm t}}{2}, -\frac{N\_{\rm t}}{2} + 1, \dots, \, \frac{N\_{\rm t}}{2} - 1, \quad n^{j} = -\frac{N\_{\rm s}}{2}, -\frac{N\_{\rm s}}{2} + 1, \dots, \, \frac{N\_{\rm s}}{2} - 1. \end{aligned} \tag{5.7}$$

This not only implies that the momenta *p*0 and *pj* are quantized in units of 2*π/T* and 2*π/L*, respectively, but also that a momentum cutoff has been introduced, since

$$-\frac{\pi}{a} \le p\_{\mu} \le \frac{\pi}{a}.\tag{5.8}$$

As we shall see below, this way of introducing a momentum cutoff can be extended to gauge theories in such a way that gauge invariance is respected. An important point to realize is that the lattice action is not unique: it is only required that the discretized expression for *S*E reproduces the continuum result as the lattice spacing *a* is taken to zero.

**Step 3** The theory is quantized via the Euclidean functional integral

$$Z\_{\mathbb{E}} := \int D[\phi] \, \mathrm{e}^{-S\_{\mathbb{E}}[\phi]}, \qquad D[\phi] = \prod\_{\chi \in \Lambda\_{\mathbb{E}}} \mathrm{d}\phi(\chi). \tag{5.9}$$

Here one sees explicitly that the discretization procedure has given a mathematical meaning to the integration measure, which reduces to that of an ordinary, multipledimensional integration.

One can now define Euclidean correlation functions of local fields through

$$\langle \phi(\mathbf{x}\_{1}) \cdots \phi(\mathbf{x}\_{n}) \rangle = \frac{1}{Z\_{\mathrm{E}}} \int D[\phi] \phi(\mathbf{x}\_{1}) \cdots \phi(\mathbf{x}\_{n}) \mathbf{e}^{-S\_{\mathrm{E}}[\phi]}.\tag{5.10}$$

In the continuum limit, these correlation functions approach the Schwinger functions, which encode the physical information about the spectrum within the Euclidean formulation. Osterwalder and Schrader [5] have laid down the general criteria which must be satisfied such that the information in Minkowskian spacetime can be reconstructed from the Schwinger functions.

**Step 4** The particle spectrum is extracted from the exponential fall-off of the Euclidean two-point correlation function. To this end, one must define the Euclidean time evolution operator. The *transfer matrix* T describes time propagation by a finite Euclidean time interval *a*. The functional integral can be expressed in terms of the transfer matrix as

$$Z\_{\rm E} = \text{Tr}\, \mathsf{T}^{N\_t},\tag{5.11}$$

where the trace is taken over the basis |*α* of the Hilbert space of physical states. In order to obtain expressions which are more reminiscent of those in Minkowski space-time, one can define a Hamiltonian HE by

$$\mathbf{T} = \mathbf{\bar{e}}^{-a\mathsf{H}\_{\mathsf{E}}}.\tag{5.12}$$

If |*α* denotes an eigenstate of the transfer matrix with eigenvalue *λα*, i.e.

$$\mathsf{T}|\alpha\rangle = \lambda\_{\alpha}|\alpha\rangle = \mathsf{e}^{-aE\_{a}}|\alpha\rangle,\tag{5.13}$$

then one can work out the spectral decomposition of the two-point correlation function, viz.

$$
\langle \phi(\mathbf{x}) \phi(\mathbf{y}) \rangle = \frac{1}{Z\_{\mathrm{E}}} \int D[\phi] \phi(\mathbf{x}) \phi(\mathbf{y}) \mathbf{e}^{-S \to [\phi]} \tag{5.14}
$$

$$=\sum\_{\alpha} \mathbf{e}^{-(E\_{\alpha}-E\_{0})(\chi\_{0}-\chi\_{0})} \langle \alpha | \hat{\phi}(0,\vec{\chi}) | 0 \rangle \langle 0 | \hat{\phi}(0,\vec{\chi}) | \alpha \rangle. \tag{5.15}$$

Here, the quantity *(Eα* − *E*0*)* is the so-called mass gap, i.e. the energy of the state |*α* above the vacuum. For large Euclidean time separations *(x*0 − *y*0*)* the lowest state dominates the two-point function, i.e. all higher states die out exponentially. The spectral decomposition of the two-point function forms the basis for numerical simulations of lattice field theories, as the mass (or energy) of a given state is given by the dominant exponential fall-off at large Euclidean times (see Sect. 5.2.3).

## *5.2.2 Lattice Actions for QCD*

Our goal now is to find a lattice transcription of the Euclidean QCD action in the continuum, i.e.

$$\mathbf{S\_{QCD}} = \int \mathbf{d}^4 \mathbf{x} \left\{ -\frac{1}{2g\_0^2} \text{Tr} \left( F\_{\mu\nu} F\_{\mu\nu} \right) + \sum\_{f=u,d,s\dots} \bar{\psi}\_f \left( \chi\_\mu D\_\mu + m\_f \right) \psi\_f \right\}, \quad (5.16)$$

where *g*0 denotes the gauge coupling, and our conventions are chosen such that the covariant derivative is defined through

$$D\_{\mu} = \partial\_{\mu} + A\_{\mu},\tag{5.17}$$

while the field tensor reads

$$F\_{\mu\nu} = \partial\_{\mu}A\_{\nu} - \partial\_{\nu}A\_{\mu} + [A\_{\mu}, A\_{\nu}], \qquad A\_{\mu}^{\dagger} = -A\_{\mu}. \tag{5.18}$$

Before attempting to write down a discretized version, we must first elucidate the notion of a lattice gauge field in a non-Abelian theory. In fact, in this case it turns out that the gauge potential *Aμ* must be abandoned when the theory is discretized. The reason is that the familiar non-Abelian transformation law, i.e.

$$A\_{\mu}(\mathbf{x}) \rightarrow \mathbf{g}(\mathbf{x}) A\_{\mu}(\mathbf{x}) \mathbf{g}(\mathbf{x})^{-1} + \mathbf{g}(\mathbf{x}) \partial\_{\mu}(\mathbf{x}) \mathbf{g}(\mathbf{x})^{-1}, \quad \mathbf{g}(\mathbf{x}) \in \text{SU}(\mathfrak{H}), \tag{5.19}$$

no longer holds exactly when *∂μ* is replaced by its discrete counterpart *dμ* of Eq. (5.5). Strict gauge invariance at the level of the regularized theory cannot be maintained in this fashion.

The definition of a lattice gauge field relies on the concept of the parallel transporter. If a quark moves in the presence of a background gauge field from *y* to *x*, it picks up a non-Abelian phase factor, given by

$$U(\mathbf{x}, \mathbf{y}) = \text{P.O.exp}\left\{-\int\_{\mathbf{y}}^{\mathbf{x}} \text{d}z\_{\mu} \, A\_{\mu}(z)\right\},\tag{5.20}$$

where "P.O." denotes path ordering, as a consequence of the non-Abelian nature of the gauge field. By contrast to the gauge potential *Aμ*, which is an element of the Lie algebra of SU(3), the parallel transporter *U (x, y)* is an element of the gauge group itself. On the lattice, the parallel transporter between neighbouring lattice sites *x* and *x* + *aμ*ˆ is called *link variable:*

$$U(\mathbf{x}, \mathbf{x} + a\hat{\mu}) \equiv U\_{\mu}(\mathbf{x}), \qquad U(\mathbf{x} + a\hat{\mu}, \mathbf{x}) = U(\mathbf{x}, \mathbf{x} + a\hat{\mu})^{-1} = U\_{\mu}(\mathbf{x})^{-1}. \tag{5.21}$$

A consistent and manifestly gauge invariant discretization of QCD is obtained by identifying the gauge degrees of freedom with the link variables *Uμ(x)*, which transform under the gauge group as

$$U\_{\mu}(\mathbf{x}) \to \mathbf{g}(\mathbf{x}) \, U\_{\mu}(\mathbf{x}) \, \mathbf{g}(\mathbf{x} + a\hat{\mu})^{-1}, \quad \mathbf{g}(\mathbf{x}), \, \mathbf{g}(\mathbf{x} + a\hat{\mu}) \in \mathbf{SU}(\mathfrak{H}).\tag{5.22}$$

The connection with the gauge potential *Aμ(x)* is somewhat subtle: if *Uμ(x)* denotes a *given* link variable in the discretized theory, it can be used to *define* a vector field *Aμ(x)* as an element of the Lie algebra of SU(3) via

$$\mathfrak{e}^{aA\_{\mu}(\chi)} \equiv U\_{\mu}(\mathfrak{x}).\tag{5.23}$$

In turn, if *A*c *μ* is a *given* gauge potential in the continuum theory, one can always find a link variable which approximates *A*c *μ* up to cutoff effects.

Now we turn to the problem of defining a discretized version of the Yang–Mills action. To this end we define the *plaquette Pμν (x)* as the product of link variables around an elementary square of the lattice:

$$P\_{\mu\nu}(\mathbf{x}) \equiv U\_{\mu}(\mathbf{x}) U\_{\nu}(\mathbf{x} + a\hat{\mu}) U\_{\mu}(\mathbf{x} + a\hat{\nu})^{-1} U\_{\nu}(\mathbf{x}) \,^{-1}. \tag{5.24}$$

A graphical representation is shown in Fig. 5.1. Using the transformation property in Eq. (5.22), it is easy to convince oneself that this object is manifestly gauge invariant. Moreover, it serves to define the simplest discretization of the Yang–Mills action, the Wilson plaquette action [6]

$$S\_{\rm G}[U] = \beta \sum\_{\chi \in \Lambda\_{\rm E}} \sum\_{\mu < \nu} \left( 1 - \frac{1}{3} \text{Re } \text{Tr } P\_{\mu \nu}(\chi) \right). \tag{5.25}$$

It has become a standard textbook exercise to verify that for small lattice spacings

$$\mathcal{S}\_{\mathcal{G}}[U] \longrightarrow -\frac{1}{2\mathbf{g}\_0^2} \int \mathbf{d}^4 \mathbf{x} \, \text{Tr} \,(F\_{\mu\nu} F\_{\mu\nu}) + \mathcal{O}(a),\tag{5.26}$$

provided that one relates the parameter *β* to the bare gauge coupling via *β* = 6*/g*2 0 in Eq. (5.25). We have remarked already that the discretization of a field theory is not

**Fig. 5.1** Graphical representation of the plaquette *Pμν (x)* in the *(μ, ν)*-plane. The arrow between sites *x* + *aμ*ˆ and *x* denotes the link variable *Uμ(x)*

unique, and hence one is free to add further gauge invariant terms to the plaquette action which formally vanish as *a* → 0, but which produce a discretization with an accelerated rate of convergence to the continuum limit. The most widely chosen alternatives are the Symanzik [7] and Iwasaki [8] actions.

Quark and antiquark fields, *ψ(x)* and *ψ(x)* ¯ , are associated with the lattice sites and transform under the gauge group as

$$
\psi(\mathbf{x}) \to \mathbf{g}(\mathbf{x})\psi(\mathbf{x}), \qquad \bar{\psi}(\mathbf{x}) \to \bar{\psi}(\mathbf{x})\mathbf{g}(\mathbf{x})^{-1}.\tag{5.27}
$$

Using the transformation property of the link variables, it is straightforward to write down a discretized version of the covariant derivative, i.e.

$$\nabla\_{\mu}\psi(\mathbf{x}) := \frac{1}{a}\left(U\_{\mu}(\mathbf{x})\psi(\mathbf{x} + a\hat{\mu}) - \psi(\mathbf{x})\right)$$

$$\nabla\_{\mu}^{\*}\psi(\mathbf{x}) := \frac{1}{a}\left(\psi(\mathbf{x}) - U\_{\mu}(\mathbf{x} - a\hat{\mu})^{-1}\psi(\mathbf{x} - a\hat{\mu})\right),\tag{5.28}$$

where ∇*μ* and ∇∗ *μ* denote the "forward" and "backward" derivatives, respectively. Finally, we note that in Euclidean space-time, the Dirac matrices can be defined to satisfy + *γμ, γν* , = 2*δμν*.

Before we attempt to construct the fermionic part of the action of lattice QCD, it is useful to identify the basic properties that the discretized, massless Dirac operator, *D*, should satisfy:

Locality, i.e. the absence of long-ranged interactions, is a basic property of any quantum field theory describing elementary particles. Property (b) implies that the correct continuum behaviour of the quark-gluon interaction is reproduced. Furthermore, condition (c) ensures that the correct fermion spectrum is obtained: fermion masses are associated with poles of {*D(p)* - }−1, which, in the continuum theory, only occur at vanishing four-momentum. Finally, property (d) ensures that the massless theory respects chiral symmetry.

Using the definition of the covariant derivative and the conventions for the Dirac matrices in Euclidean space-time, we can now write down the simplest discretized version of the massless lattice Dirac operator:

$$D\_{\rm disc} = \frac{1}{2} \gamma\_{\mu} (\nabla\_{\mu} + \nabla\_{\mu}^{\*}). \tag{5.29}$$

It turns out, however, that this "naïve" discretization violates condition (c) and therefore produces spurious fermionic degrees of freedom. This is the so-called fermion doubling problem, which is most easily explained by considering *D*disc in momentum space for the free theory. The Fourier transform yields

$$\tilde{D}\_{\text{disc}}(p) = \text{i}\gamma\_{\mu}\frac{1}{a}\sin(ap\_{\mu}) = \text{i}\gamma\_{\mu}p\_{\mu} + \text{O}(a^{2}).\tag{5.30}$$

The discretization procedure has thus replaced *pμ* by a sine function. While the Taylor expansion guarantees that condition (b) is satisfied, the occurrence of sin*(apμ)* implies that *D*-disc*(p)* vanishes not only at *pμ* = 0, but also at *π/a* for *μ* = 0*,...,* 3 in the permitted range of momenta, thereby violating condition (c). The massless propagator {*D*-disc*(p)*}−1 therefore has 24 = 16 poles, and thus there is a 16-fold degeneracy of the fermion spectrum.

As we shall see below, the fermion doubling problem is closely linked with the issue of chiral symmetry on the lattice. For now we simply list the various methods that have been devised to address fermion doubling. Historically the first was due to Wilson ("Wilson fermions") [6]. Here, the degeneracy is lifted completely, but the price to pay is the explicit breaking of chiral symmetry at the level of the regularized theory. Another method, due to Kogut and Susskind ("staggered fermions") [9], is based on the idea of spreading individual spinor components over the corners of an elementary hypercube of the lattice. Although the degeneracy is only lifted partially (from 16 to 4), this formulation has the advantage of leaving a subgroup of chiral symmetry unbroken. More recent developments include the use of socalled "domain wall" [10, 11] or "overlap" [12] fermions. These formulations leave chiral symmetry unbroken in principle, and also succeed in lifting the degeneracy completely. Finally, there are the so-called "perfect" actions [13], which are based on a renormalization group approach and which are in principle completely free of lattice artefacts. An exact realization of the perfect action which can be used in simulations is, however, difficult to obtain. In practice, one typically uses a so-called truncated fixed point action. Domain wall and overlap fermions, as well as perfect actions are particular realizations of a class of discretizations dubbed "Ginsparg-Wilson fermions". They have the remarkable feature that chiral symmetry is preserved, while the fermion doubling problem is completely avoided. We shall come back to this issue in more detail below.

For now we turn specifically to Wilson's treatment of the fermion doubling problem. It exploits the fact that the discretization is not unique. Thus, one can add a term to *D*disc, which formally vanishes as *a* → 0, but which pushes the masses of the unwanted doubler states to the cutoff scale at any non-zero value of the lattice spacing. Explicitly, the massless Wilson-Dirac operator *D*w reads

$$D\_{\mathbb{W}} = \frac{1}{2} \chi\_{\mu} (\nabla\_{\mu} + \nabla\_{\mu}^{\*}) + ar \nabla\_{\mu}^{\*} \nabla\_{\mu},\tag{5.31}$$

where *r* is the so-called Wilson parameter, which is usually set to one. The Fourier transform of *D*w for a trivial gauge field reads

$$\tilde{D}\_{\mathbb{W}}(p) = \mathrm{i}\gamma\_{\mu}\frac{1}{a}\sin(ap\_{\mu}) + \frac{2r}{a}\sin^{2}\left(\frac{ap\_{\mu}}{2}\right),\tag{5.32}$$

which explicitly demonstrates (for the free theory, at least) that the poles at *pμ* = *π/a* receive additional contributions proportional to *r/a*, which is of order of the cutoff for *r* = O*(*1*)*. Although this procedure leads to a complete lifting of the degeneracy,1 it has a number of unwanted features: first, it should be noted that the Wilson fermion action differs from the classical action in the continuum by terms of order *a*, as a result of adding the counterterm proportional to *r*. By contrast, the leading discretization effects of the Wilson plaquette action for Yang–Mills theory are only O*(a*2*)*. The Wilson fermion formulation will thus have a reduced rate of convergence towards the continuum limit. Secondly, the addition of the Wilson term results in an explicit breaking of chiral symmetry, since the massless theory is no longer invariant under global axial rotations, such as

$$
\psi(\mathbf{x}) \to \mathbf{e}^{\mathrm{i}a\gamma\varsigma} \psi(\mathbf{x}), \qquad \bar{\psi}(\mathbf{x}) \to \bar{\psi}(\mathbf{x}) \mathbf{e}^{\mathrm{i}a\gamma\varsigma}, \tag{5.33}
$$

which implies that property (d) is violated. While the rate of convergence to the continuum limit can be accelerated by employing what is known as "O*(a)* improvement" (see below), the explicit breaking of chiral symmetry cannot be cured within the Wilson theory. Thus, quantities like the quark condensate, which arises from the spontaneous breaking of chiral symmetry, cannot be studied in a conceptually "clean" manner using Wilson fermions. A detailed discussion how this can be achieved with the help of a more sophisticated fermionic discretization ("Ginsparg-Wilson fermions") is presented in Sect. 5.6. However, for most applications of lattice QCD, explicit chiral symmetry breaking is merely an inconvenience, but no serious obstacle.

We have already remarked when discussing the discretized Yang–Mills part of the QCD action that the non-uniqueness of the discretization opens the possibility to construct lattice actions with an accelerated rate of convergence towards the continuum limit. A systematic way how to do this is the so-called Symanzik improvement programme [14], in which lattice artefacts can be removed order by order in the lattice spacing. In a nutshell, the improvement programme amounts to extending the renormalization procedure of a field theory to the level of irrelevant operators, i.e. operators that formally vanish as *a* → 0. In this sense one adds suitable counterterms, which for any non-zero value of *a* produce a cancellation of the cutoff effects at a given order, provided that their coefficients are tuned appropriately. For QCD with Wilson fermions, Sheikholeslami and Wohlert [15] have shown that the Symanzik improvement programme to lowest order is realized by adding one O*(a)* counterterm to the Wilson-Dirac operator *D*w. The resulting expression in the massless case reads

$$D\_{\rm sw} = D\_{\rm w} + \frac{ia}{4} c\_{\rm sw} \sigma\_{\mu\nu} \widehat{F}\_{\mu\nu},\tag{5.34}$$

1That the degeneracy is indeed completely lifted in the presence of a non-trivial gauge field can be verified in numerical simulations.

**Fig. 5.2** Four plaquettes that must be summed over to yield the quantity *Qμν (x)* in the lattice definition of the field strength tensor. The site *x* is at the center of the "clover" leaf

where *σμν* = i 2 [*γμ, γν* ], and *F* .*μν* is a lattice transcription of the gluon field strength tensor *Fμν* . A suitable representation of *F* .*μν* in terms of plaquette variables is given by

$$
\widehat{F}\_{\mu\nu}(\mathbf{x}) = \frac{1}{8a^2} \left( \mathcal{Q}\_{\mu\nu}(\mathbf{x}) - \mathcal{Q}\_{\nu\mu}(\mathbf{x}) \right), \tag{5.35}
$$

where *Qμν (x)* is the sum of the four plaquettes emanating from the site *x*, as depicted in Fig. 5.2. The object *Qμν (x)* is aptly called "clover" leaf. In order to remove all lattice artefacts of order *a* in hadron masses, the improvement coefficient *c*sw must be fixed by imposing a suitable improvement condition. Without going into details here, we note that it is possible to find such a condition, which can also be evaluated at the non-perturbative level [16, 17]. The resulting, non-perturbatively O*(a)* improved Wilson action can then be used to compute, say, hadron masses whose values differ from the continuum result by terms of only O*(a*2*)*.

The Wilson-Dirac operator for a quark with bare mass *m*0 is simply *(D*w + *m*0*)*. However, the form of the Wilson fermion action, *S*W F [*U, ψ,ψ* ¯ ], which is found in the literature is usually expressed in terms of the "hopping parameter" *κ* rather than *m*0. By rescaling the fermion fields according to

$$
\psi(\mathbf{x}) \to \sqrt{2\kappa} \,\psi(\mathbf{x}), \qquad \bar{\psi}(\mathbf{x}) \to \bar{\psi}(\mathbf{x}) \sqrt{2\kappa}, \tag{5.36}
$$

one obtains

$$\mathcal{S}\_{\mathbb{F}}^{\mathbb{W}}[U,\bar{\psi},\psi] \equiv a^4 \sum\_{\boldsymbol{\chi} \in \Lambda\_{\mathbb{E}}} \bar{\psi}(\boldsymbol{\chi})(D\_{\mathbb{W}} + m\boldsymbol{\omicron})\psi(\boldsymbol{x})$$

$$= a^4 \sum\_{\boldsymbol{\chi} \in \Lambda\_{\mathbb{E}}} \left\{-\kappa \sum\_{\mu=0}^3 \frac{1}{a} \Big[\bar{\psi}(\boldsymbol{x})(\boldsymbol{r} - \boldsymbol{\gamma}\_{\mu})U\_{\mu}(\boldsymbol{x})\psi(\boldsymbol{x} + a\hat{\mu})\Big] \right.$$

$$+ \bar{\psi}(\boldsymbol{x} + a\hat{\mu})(\boldsymbol{r} + \boldsymbol{\gamma}\_{\mu})U\_{\mu}(\boldsymbol{x})^{-1}\psi(\boldsymbol{x})\Big\}$$

$$+ \bar{\psi}(\boldsymbol{x})\psi(\boldsymbol{x})\Big\}. \tag{5.37}$$

#### 5 QCD on the Lattice 149

The hopping parameter *κ* is related to the bare mass *m*0 via

$$\kappa = \frac{1}{2am\_0 + 8r},\tag{5.38}$$

while the dimensionless parameter *r* is usually set to one. Taken together with the plaquette action of Eq. (5.25), the Wilson action for QCD is thus conveniently parameterized in terms of the bare parameters*(β, κ)*, with *β* = 6*/g*2 0 and *κ* as above, instead of the bare gauge coupling and quark mass *(g*0*, m*0*)*.

Another consequence of adding the Wilson term to the naïve lattice action is the resulting additive renormalization of the quark mass. In other words, the point where the quark mass vanishes is a priori unknown. The value that must be subtracted is called the critical quark mass, which corresponds to the critical value of the hopping parameter, *κc*. The bare subtracted quark mass is then given by

$$m = \frac{1}{2a} \left(\frac{1}{\kappa} - \frac{1}{\kappa\_c}\right). \tag{5.39}$$

From Eq. (5.38) one easily infers that the critical value of *κ* in the free theory occurs at

$$\kappa\_c = \frac{1}{8}, \qquad r = 1,\tag{5.40}$$

while for non-zero *g*0 the value of *κc* must be determined, for instance, by adjusting *κ* to the point where the pion mass vanishes.

We now turn to discussing one alternative to using Wilson's solution to the fermion doubling problem, namely the so-called "staggered" (or Kogut-Susskind) fermions. One might think that the doubling problem arises since there are too many fermion degrees of freedom in the discretized theory, if one associates a four-component Dirac spinor with each individual lattice site. Pictorially, the main idea of Kogut and Susskind was to "thin out" the degrees of freedom by distributing single spinor components over different lattice sites. In their particular formulation, the 16 corners of a four-dimensional hypercube serve to accommodate the individual components of four Dirac spinors. Therefore, if these hypercubes are regarded as the main building blocks for the fermionic discretization, rather than the lattice sites themselves, this procedure will result in a partial lifting of the degeneracy from 16 fermion species down to four. It is clear, though, that a simple distribution of spinor components is not sufficient to define the action, since the Dirac matrices mix different spinor components. Thus, the staggered fermion action is only obtained after performing a diagonalization in spinor space, which then decouples the individual components.

Rather than describing the details of this procedure, which can be found in most textbooks, we simply state the result. Starting from the usual four-component spinor and performing a spin-diagonalization, the lattice action for staggered fermions with bare mass *m*0 coupled to the gauge field is derived as

$$\begin{split} S\_{\rm F}^{\rm stag}[U,\bar{\chi},\chi] &= a^4 \sum\_{x \in \Lambda\_{\rm E}} \sum\_{a=1}^4 \left\{ m\_0 \bar{\chi}\_a(x) \chi\_a(x) \\ &+ \frac{1}{2a} \sum\_{\mu=0}^3 \eta\_{\mu}(x) \Big[ \bar{\chi}\_a(x) U\_{\mu}(x) \chi\_a(x+a\hat{\mu}) - \bar{\chi}\_a(x+a\hat{\mu}) U\_{\mu}(x)^{-1} \chi\_a(x) \Big] \right\}, \quad (5.41) \end{split} \tag{5.41}$$

where *χα* denotes a one-component Grassmann variables. The spin-diagonalization has thus replaced the Dirac matrices *γμ* by real, position-dependent phase factors *ημ(x)*, which are given by

$$\eta\_0(\mathbf{x}) = 1, \qquad \eta\_j(\mathbf{x}) = (-1)^{n\_0 + \ldots + n\_{j-1}}, \quad n\_j = \mathbf{x}\_j/a. \tag{5.42}$$

At the level of the classical action, the spinor components are completely decoupled, and the action is decomposed into four identical pieces. In order to occupy all 16 corners of a four-dimensional hypercube with one-component Grassmann variables, one needs four Dirac spinors, each of which contributes a term like Eq. (5.41) to the overall action. This produces the fourfold degeneracy of staggered fermions, with the remnant doubler states being referred to as "tastes", in order to distinguish them from physical flavours. The formulation using the one-component fields within a hypercube can be re-expressed in terms of the spin-taste basis [18], from which one can infer directly that the taste symmetry is broken. However, one axial generator of the taste symmetry remains unbroken. The fermion mass in the staggered approach is therefore protected against any additive renormalization through the associated global axial U(1) symmetry, unlike the case of the Wilson action. While the various tastes decouple in the continuum limit, non-vanishing interactions between the tastes at O*(a*2*)* in the lattice spacing are induced, leading to large lattice artefacts. The Symanzik improvement programme can be employed to reduce these taste-changing interactions [19], and the resulting "improved staggered fermions" (the so-called "Asqtad"-action being one particular example [20]) have been widely used in a series of simulations.

For a long time lattice physicists have struggled to find a fermionic discretization which would both solve the doubling problem and be compatible with chiral symmetry. In fact, physicists grew increasingly doubtful that this could be achieved, following the proof of a "No-Go theorem" by Nielsen and Ninomiya [21], which stated that the conditions (a)–(d) mentioned above could not be satisfied simultaneously. Since one does not want to give up locality and property (b), this would imply that either (c) or (d) must be violated. Indeed, the Wilson and staggered discretizations seem to confirm this expectation: while the Wilson fermion action removes all doublers, it breaks chiral symmetry, leading to an additive renormalization of the quark mass, as well as several other consequences. By contrast, the staggered formulation preserves a U(1) subgroup of chiral symmetry at the price of only partially removing the spurious degrees of freedom.

#### 5 QCD on the Lattice 151

A way to circumvent the Nielsen–Ninomiya theorem was already pointed out by Ginsparg and Wilson in 1982 [22], when they suggested to relax condition (d) in favour of

$$
\gamma\_{\mathfrak{F}} D + D \gamma\_{\mathfrak{F}} = a D \gamma\_{\mathfrak{F}} D. \tag{5.43}
$$

However, it was not before 1997 that this condition—now commonly referred to as the Ginsparg-Wilson relation—was confronted with a non-trivial solution. It was shown [23] that the so-called "perfect action" constructed from a renormalization group approach satisfied equation (5.43). It was also realized that any lattice Dirac operator, which is a solution to the Ginsparg-Wilson relation, also satisfies the Atiyah–Singer index theorem, i.e.

$$\{\gamma\xi, D\} = aD\gamma\xi D \quad \Leftrightarrow \quad \text{index}(D) = a^{\xi} \sum\_{\chi \in \Lambda\_{\mathbb{E}}} \frac{1}{2} \text{Tr} \ (\gamma\xi D) = n \text{--} - n\_{+}, \quad (5.44)$$

such that the operator *D* exhibits |*n*− − *n*+| exact chiral zero modes. Finally, it was shown [24] that the Ginsparg-Wilson relation implies an exact symmetry of the associated action, with infinitesimal variations proportional to

$$
\delta\psi = \chi\_{\mathbb{S}}(1 - aD)\psi, \qquad \delta\bar{\psi} = \bar{\psi}\gamma\mathfrak{s}. \tag{5.45}
$$

Moreover, this symmetry reproduces the correct chiral anomaly in the flavour singlet case, and therefore all the hallmarks of the correct chiral behaviour are present in the lattice theory: chiral zero modes, an exact index theorem and the chiral anomaly derived from the Ward identities associated with the exact symmetry.

Another line in the development of lattice fermion actions that preserve chiral symmetry goes back to Kaplan's domain wall fermion approach [10], which was subsequently applied to QCD by Furman and Shamir [11]. Without going into detail, we state that the basic idea is to introduce an extra, fifth dimension and to couple the fermions to a mass defect (the so-called "domain wall height") in that extra dimension. To make this more explicit, let *x,y* denote the coordinates in the fourdimensional bulk, and *s, t* the coordinates in the 5th dimension, which has finite length *N*5. The gauge fields are trivial in the 5th direction, and the Dirac operator then has the general structure

$$D\_{\rm dwf}(\mathbf{x}, \mathbf{s}; \mathbf{y}, t) = D^{\parallel}(\mathbf{x}, \mathbf{y}) \delta\_{\rm sI} + \delta(\mathbf{x} - \mathbf{y}) D\_{\rm sI}^{\perp} \tag{5.46}$$

where *D(x, y)* is the usual Wilson-Dirac operator with a *negative* mass term, −*M*, which represents the domain wall height. The operator *D*⊥ *st* couples fermions in the 5th dimension and contains the physical bare quark mass *m*0. It can then be shown that for *m*0 = 0 and in the limit *N*5 → ∞ there are no fermion doublers and, more importantly, chiral modes of opposite chirality are trapped in the four-dimensional domain walls at *s* = 1*, N*5.

However, in a real lattice simulation of domain wall fermions, one has to work with a finite value of *N*5, so that the decoupling of chiral modes is not exact. One expects, though, an exponential suppression of the remnant chiral symmetry breaking effects, and this has been confirmed in several simulations. Furthermore, the rate of suppression may be accelerated by optimizing the choice of lattice action for the gauge fields. Hence, the domain wall formulation of QCD offers a method to realize almost exact chiral symmetry at non-zero lattice spacing at the expense of simulating a five-dimensional theory.

Another operator which correctly reproduces the chiral properties of QCD at non-zero lattice spacing was constructed by Neuberger [12]. Its definition is

$$D\_N = \frac{1}{\overline{a}} \left( 1 - \frac{A}{\sqrt{A^\dagger A}} \right), \quad A = 1 + s - aD\_\mathbf{W}, \quad \overline{a} = \frac{a}{1+s}, \tag{5.47}$$

where *D*w is the massless Wilson-Dirac operator, and |*s*| *<* 1 is a tunable parameter. By defining *Q* = −*γ*5*A*, one can rewrite Eq. (5.47) as

$$D\_{\rm N} = \frac{1}{\overline{a}} \left( 1 + \chi \text{sign}(\mathcal{Q}) \right). \tag{5.48}$$

The Neuberger-Dirac operator *D*N removes all doublers from the spectrum, and can easily be shown to satisfy the Ginsparg-Wilson relation [12]. The occurrence of an inverse square root in *D*N raises two issues. First, it is a priori not clear whether or not *D*N is local. Second, the application of *D*N in a computer program is potentially very costly, since the sign-function of the matrix *Q* must be implemented using, for instance, a polynomial approximation.

In order to qualify as a viable discretization of the quark action, "strict" locality, meaning that only fields in a local neighbourhood of a given lattice site are coupled, is not actually required. If *D(x, y)* denotes a generic lattice Dirac operator which couples fields at sites *x* and *y*, then a sufficient condition for locality of *D* is the exponential suppression of non-local interactions, i.e.

$$\|D(\mathbf{x}, \mathbf{y})\| \le \mathbf{e}^{-\mathbf{y}\|\mathbf{x}-\mathbf{y}\|/a},\tag{5.49}$$

where |*x* − *y*| is the distance between sites and · denotes a suitably defined matrix norm. In Ref. [25] it was shown that the Neuberger-Dirac operator *D*N is local in the sense of Eq. (5.49), provided that the lattice spacing in physical units2 is not larger than about 0*.*13 fm. As far as the issue of numerical efficiency is concerned, we note that the most widely used approximations of sign*(Q)* with good convergence properties include Chebysheff or Zolotarev polynomials, as well as rational fractions.

2So far we have not discussed how to assign physical units to the lattice spacing *a*. This is described in Sect. 5.2.4.

#### 5 QCD on the Lattice 153

The last fermionic discretization we wish to mention here was originally constructed to address another problem of Wilson's discretization, namely the fact that they are not protected against the occurrence of zero modes for *any* non-zero value of the bare quark mass. These unphysical zero modes manifest themselves as "exceptional" configurations, which occur with a certain frequency in numerical simulations with Wilson quarks and which can lead to strong statistical fluctuations. The problem can be cured by introducing a so-called "chirally twisted" mass term, after which the fermionic part of the QCD action in the continuum assumes the form

$$\mathcal{S}\_{\rm F}^{\rm tm;cont} = \int \mathrm{d}^4 x \,\bar{\psi}(\mathbf{x}) (\chi\_{\mu} D\_{\mu} + m + \mathrm{i}\mu\_{\mathrm{q}} \chi\_{\mathrm{S}} \tau^3) \psi(\mathbf{x}). \tag{5.50}$$

Here, *μ*q is the twisted mass parameter, and *τ* 3 is a Pauli matrix. The standard action in the continuum can be recovered via a global chiral field rotation:

$$
\psi'(\mathbf{x}) = \mathbf{e}^{\mathrm{i}\alpha \gamma\_3 \mathbf{r}^3/2} \psi(\mathbf{x}), \qquad \bar{\psi}'(\mathbf{x}) = \bar{\psi}(\mathbf{x}) \mathbf{e}^{\mathrm{i}\alpha \gamma\_3 \mathbf{r}^3/2}. \tag{5.51}
$$

Fixing the twist angle *α* by requiring that tan *α* = *μ*q*/m* one finds

$$\mathbf{d}S\_{\mathbf{F}}' = \int \mathbf{d}^4 \mathbf{x} \,\, \bar{\psi}'(\mathbf{x}) (\chi\_{\mu} D\_{\mu} + M) \psi'(\mathbf{x}), \qquad M = \sqrt{m^2 + \mu\_{\mathbf{q}}^2}, \tag{5.52}$$

which demonstrates the complete equivalence of the twisted formulation with "ordinary" QCD. The lattice action of twisted mass QCD for *N*f = 2 flavours is defined as [26]

$$\mathcal{S}\_{\rm F}^{\rm tm}[U,\bar{\psi},\psi] = a^4 \sum\_{\mathbf{x} \in \Lambda \underline{\mathbf{e}}} \bar{\psi}(\mathbf{x}) (D\_{\mathbf{w}} + m\_0 + \mathrm{i}\mu\_{\mathbf{q}}\gamma\_{\mathbf{5}}\tau^3) \psi(\mathbf{x}). \tag{5.53}$$

Although this formulation breaks physical parity and flavour symmetries, is has a number of advantages over standard Wilson fermions. In particular, the presence of the twisted mass parameter *μ*q protects the discretized theory against unphysical zero modes. Another attractive feature of twisted mass lattice QCD is the fact that the leading lattice artefacts are of order *a*2 without the need to add the Sheikholeslami-Wohlert term [27], even though the Wilson-Dirac operator is used in Eq. (5.53). Although the problem of explicit chiral symmetry breaking remains, the twisted formulation is particularly useful to circumvent some of the problems that are encountered in connection with the renormalization of local operators on the lattice. Recent review of twisted mass lattice QCD can be found in [28, 29].

We wish to end this part with a few general remarks. Although we have discussed discretizations of the QCD action in some detail, including the most recent developments, many more variants of the basic types of action—including several different combinations of fermionic and pure gauge parts—can be found in the literature. This reflects the fact that the discretization is not unique. The actual choice of lattice action in a particular simulation will influence the convergence rate to the continuum limit, the algorithmic efficiency, the renormalization properties of local operators, or—in the case of domain wall fermions—the extent to which chiral symmetry is realized. Depending on the properties of a particular discretization, the choice of lattice action can be optimized for the physics one wishes to study.

#### *5.2.3 Functional Integral and Observables*

The lattice formulation provides a regularization of non-Abelian gauge theories whilst preserving the gauge invariance at all stages of the calculation. This comes at a price, since all continuous space-time symmetries are broken explicitly and must be recovered in the continuum limit. Nevertheless, the lattice regularized theory inherits all consequences of gauge invariance, including renormalizability. Moreover, the lattice regularizes the theory without any reference to perturbation theory. By contrast, in continuum schemes like the MS scheme of dimensional regularization the cutoff is only defined after fixing the order of the perturbative expansion. As we shall see below, observables in lattice QCD are directly given in terms of functional integrals, which can be evaluated stochastically using Monte Carlo integration. In this way, any use of perturbation theory is completely avoided.

For concreteness, let us assume that we have made a particular choice for the Yang–Mills part *S*G[*U*] and the fermionic part *S*F[*U, ψ,ψ* ¯ ], for instance, the Wilson plaquette action and Wilson fermions. Let denote an observable, which is represented by a polynomial in the quark and antiquark fields and the link variables. The expectation value, , is defined through the Euclidean functional integral3

$$
\langle \Omega \rangle = \frac{1}{Z} \int D[U] D[\bar{\psi}, \psi] \,\Omega \,\mathrm{e}^{-\mathrm{Sg}[U] - \mathrm{S\bar{r}}[U, \bar{\psi}, \psi]},\tag{5.54}
$$

where *Z* is fixed by the condition1= 1. The functional integral involves an integration over the gauge group and over all fermionic degrees of freedom, the latter being represented by anti-commuting (Grassmann) variables. Since the fermionic action, *S*F[*U, ψ,ψ* ¯ ] is bilinear in the quark and antiquark fields, the integration over the Grassmann variables is Gaussian and can be performed analytically. This yields

$$\langle \Omega \rangle = \frac{1}{Z} \int \prod\_{\chi \in \Lambda \boxtimes} \prod\_{\mu=0}^{3} \mathrm{d}U\_{\mu}(\mathbf{x}) \,\widetilde{\Omega} \, \{ \det D\_{\mathrm{lat}} \}^{N\_{\mathrm{f}}} \, \mathrm{e}^{-\mathrm{Sg}[U]}.\tag{5.55}$$

3Here and in the following we drop the subscript "E" on the partition function *Z*.

5 QCD on the Lattice 155

Equation (5.55) requires some further explanation:

The numerical evaluation of via Monte Carlo integration proceeds as follows. One starts by generating a set of gauge configurations using a computer program. One configuration in the set represents the collection of all link variables on a given lattice, i.e.

$$\left\{ U\_{\mu}(\mathbf{x}) \,|\, \mathbf{x} \in \Lambda\_{\mathbf{E}}, \,\mu = 0, \ldots, 3 \right\},\tag{5.56}$$

for which we shall use the shorthand {*Uμ(x)*} below. A collection of an infinite number of configurations is called an *ensemble*. The statistical weight, *W*, of an individual configuration is given by

$$W = \{ \det D\_{\rm lat} \}^{N\_{\rm f}} \text{ e }^{-S\_{\rm G}[U]}. \tag{5.57}$$

In other words, the composition of the ensemble is determined by a probability distribution, which is given by the negative exponentiated classical action in the integrand of the Euclidean functional integral. Owing to the weight factor, the integrand of the functional integral will be strongly peaked around those configurations for which *W* is large. This particular feature makes the expectation value amenable to a Monte Carlo treatment. The key idea is to replace the ensemble by a *finite* sample of *N*cfg gauge configurations, which is dominated by those configurations for which *W* is large. Provided that one can construct a suitable algorithm, the sample will then consist predominantly of those configurations which give a large contribution to the Euclidean functional integral and thus . Such a procedure is called *importance sampling*.

Technically, the sample is produced by generating a sequence of configurations via a *Markov process:*

$$\left\{U\_{\mu}(\mathbf{x})\right\}\_{1} \longrightarrow \left\{U\_{\mu}(\mathbf{x})\right\}\_{2} \longrightarrow \dots \longrightarrow \left\{U\_{\mu}(\mathbf{x})\right\}\_{N\_{\text{cfg}}}.\tag{5.58}$$

One assigns a probability for the transition from + *Uμ(x)*, *i* to + *Uμ(x)*, *i*+1, which is usually a function of the statistical weights of the two configurations, *Wi* and *Wi*+1, respectively. For each individual configuration in the sequence one then evaluates the observable, which yields the estimates *i, i* = 1*,...,N*cfg. The expectation value is related to the mean value via

$$
\langle \Omega \rangle = \lim\_{N\_{\rm cfg} \to \infty} \overline{\Omega}, \qquad \overline{\Omega} = \frac{1}{N\_{\rm cfg}} \sum\_{l=1}^{N\_{\rm cfg}} \Omega\_l. \tag{5.59}
$$

In other words, in the limit of infinite statistics the mean value converges to the ensemble average which is identical to the expectation value. An important consequence of approximating the ensemble average by the sample average is a non-zero value of the *variance*. Hence, in order to specify the results from a Monte Carlo integration completely, one must also quote the statistical error which is given by the square root of the variance.

In the standard algorithms that implement Markov processes (such as the Metropolis algorithm [30]), the transition probabilities for going from one configuration to another are determined by comparing the statistical weights for *local* variations in the field variables. This guarantees computational efficiency, since the variation of individual link variables does not involve global information from the entire lattice. In Eq. (5.55) the dynamical effects of the quark fields are incorporated via the determinant of the lattice Dirac operator. The determinant, however, is a non-local object, which is expensive to compute. When the first efforts were made to compute observables in QCD in the 1980s, the available computer power did not allow for the inclusion of the quark determinant. Instead, lattice physicists resorted to what is known as the "quenched approximation", which is based on the assumption that the bulk of non-perturbative contributions is carried by the gauge field, so that the determinant is set to a constant:

$$\text{Quenched approximation:} \qquad \det D\_{\text{lat}} = 1 \quad \Leftrightarrow \quad N\_{\text{f}} = 0. \tag{5.60}$$

The resulting gain in computer time amounts to several orders of magnitude. In the quenched approximation the effects of virtual quark loops are entirely suppressed. As a consequence, results for observables are afflicted with an unknown systematic error. As we shall see later, there are several quantities (for instance, the masses of the lightest hadrons) for which the quenching error amounts to just 10–15%. Although this justifies the use of the quenched approximation to some extent, it is clear that dynamical quark effects must be taken into account, in order to arrive at reliable, non-perturbative predictions with a total accuracy at the percent level.

Modern algorithms for dynamical quarks, such as the Hybrid Monte Carlo algorithm [31], do not evaluate the quark determinant directly. Rather, one exploits the property that the determinant can be rewritten as a functional integral over bosonic fields, which is then evaluated stochastically. Thereby one avoids computing a global object, but the computational effort involved in the stochastic estimation of the quark determinant is still large compared with the quenched approximation. More details can be found in Sect. 5.2.6 below.

Correlation functions, i.e. the expectation values of polynomials in the quark and gluon fields, are the most important quantities, since they determine implicitly the particle spectrum of the theory. As was discussed already in Sect. 5.2.1, the link between correlation functions and the particle spectrum is provided by the transfer matrix T. For lattice QCD with Wilson fermions, the existence of a positive transfer matrix was rigorously established [32].

As a concrete example we shall discuss the two-point correlation function of a charged kaon. A polynomial of quark fields with the quantum numbers of the kaon is given by

$$\phi \mathbf{k}(\mathbf{x}) = (\bar{\mu}\gamma\mathbf{s}\mathbf{s})\,(\mathbf{x}),\tag{5.61}$$

where the parentheses indicate summation over spinor and colour components of the fields. Mostly one is interested in correlation functions in which all spatial points have been summed over and which therefore only depend on the Euclidean time separation. We define

$$C\_{\mathbf{K}}(\mathbf{x}\_{0};\vec{p}) = \sum\_{\vec{\chi}} \mathbf{e}^{\dagger \vec{p} \cdot \vec{\chi}} \left\langle \phi\_{\mathbf{K}}(\mathbf{x}) \phi\_{\mathbf{K}}^{\dagger}(\mathbf{0}) \right\rangle. \tag{5.62}$$

The inclusion of the phase factor in conjunction with the summation over *x* amounts to a projection onto spatial momentum *p*. On a finite lattice with periodic boundary conditions*C*K*(x*0; *p)* must be symmetric under *x*0 ↔ *T* −*x*0. Therefore, the spectral decomposition of *C*K*(x*0; *p)* reads

$$C\_{\mathbf{K}}(\mathbf{x}\_{0};\vec{p}) = \sum\_{a} \frac{\left| \langle 0 | \phi\_{\mathbf{K}}(0) | a \rangle \right|^{2}}{2\epsilon\_{a}(\vec{p})} \left\{ \mathbf{e}^{-\epsilon\_{a}(\vec{p}) \chi\_{0}} + \mathbf{e}^{-\epsilon\_{a}(\vec{p}) (T-\chi\_{0})} \right\},\tag{5.63}$$

where the sum runs over all states in the kaon channel with fixed momentum *p*, and *α(p)* is the mass gap (see Sect. 5.2.1).4 For large Euclidean times *x*0 the ground state dominates. If we further set *p* = 0, then the asymptotic form of the two-point function reads

$$\lim\_{m\_0 \to \infty} \mathcal{C}\_{\mathbf{K}}(\mathbf{x}\_0; \vec{p}) = \frac{\left| \langle 0 | \phi \mathbf{k} \,(\mathbf{0} \, \middle|\, K \rangle \right|^2}{m\_{\mathbf{K}}} \mathbf{e}^{-m\_{\mathbf{K}}T/2} \cosh \left( m\_{\mathbf{K}} (T/2 - \mathbf{x}\_0) \right), \qquad (5.64)$$

where *m*K = 0*(p)* |*p*=0 is the mass of the kaon, and the sum of the two exponentials has been re-expressed using the cosh function. Owing to the ordering 0*(p) <* 1*(p) < . . .* , the higher excited states are exponentially suppressed. The functional

4In the commonly normalization of hadron states one includes a factor 2*α(p)* in the denominator.