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QCD-2004 Lesson 1 : Field Theory and Perturbative QCD I 1)Preliminaries: Basic quantities in field theory 2)Preliminaries: COLOUR 3) The QCD Lagrangian and Feynman rules 4) Asymptotic freedom from e + e - -> hadrons 5) Deep Inelastic Scattering Guido Martinelli Bejing 2004

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G(x 1, x 2, x 3, x 4 )= ‹ 0 | T [ (x 1 ) (x 2 ) (x 3 ) (x 4 )] |0 › c L ( )= 1/2 ( (x) ) 2 - 1/2 m 0 2 2 (x) - 1/4! 0 4 (x) S ( )= ∫ d 4 x L ( ) The basic quantities of field theory: the Green Functions p1p1 p2p2 p3p3 p4p4 ( Z 1/2 ) 4 M( p 1, p 2, p 3, p 4, m, ) (p 1 2 -m 2 ) (p 2 2 -m 2 ) (p 3 2 -m 2 ) (p 4 2 -m 2 ) + ….

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Lim (p 1 2 -m 2 ) (p 2 2 -m 2 ) G(p 1, p 2, p 3, p 4, m, ) (p 3 2 -m 2 ) (p 4 2 -m 2 ) p 2 1,2,3,4 m 2 ( Z 1/2 ) 4 S(p 1 + p 2 p 3 + p 4 ) S-matrix element p1p1 p2p2 p3p3 p4p4 - At lowest order in perturbation theory: (p 1 2 -m 0 2 ) (p 2 2 -m 0 2 ) (p 3 2 -m 0 2 ) (p 4 2 -m 0 2 ) + +

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It can be shown that the Green Functions can be written in terms of Functional Integrals over classical fields G(x 1, x 2, x 3, x 4 )= ‹ (x 1 ) (x 2 ) (x 3 ) (x 4 ) › where Z = ∫ [d ] e i S ( ) The Functional Integral Z -1 ∫ [d ] (x 1 ) (x 2 ) (x 3 ) (x 4 ) e i S ( ) In perturbation theory e i S ( ) = e i S o ( )+ i S i ( ) ~ e i S o ( ) (1+i S i ( )- S 2 i ( )/2 + …) S i ( ) ~ O( )

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‹ x N › = Z -1 ∫ dx x N e - S (x) where Z( ) = ∫ dx e - S (x) and S (x)= x 2 + x 4 Z ( ) = ∑ n (- ) n (1/2+2 n)/n! ≈ ∑ n (- ) n (2n)!/n! For small values of the series expansion is accurate The series is however asymptotic (although Borel summable) with zero radius of convergence Starting from a certain term, which depends on the value of, an increase of the number of terms can only worsen the accuracy Field theories of interest for physics are not even Borel summable, a nonperturbative method to evaluate the FUNCTIONAL INTEGRAL is thus needed

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Z -1 ∫ [d ] (x 1 ) (x 2 ) (x 3 ) (x 4 ) e i S ( ) This integral is only a formal definition because of the infrared and ultraviolet divergences. These problems can be cured by introducing an infrared and an ultraviolet cutoff. 1) We introduce an ultraviolet cutoff by defining the fields on a (hypercubic) four dimensional lattice (x) -> ( a n) where n=( n x, n y, n z, n t ) and a is the lattice spacing; (x) -> (x) = ( (x+ a n ) - (x)) / a ; The momentum p is cutoff at the first Brioullin zone, |p| π / a The cutoff can be in conflict with important symmetries of the theory, as for example Lorentz invariance or chiral invariance This problem is common to all regularizations like for example Pauli- Villars, dimensional regularization etc.

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2) We introduce an infrared cutoff by working in a finite volume, that is n i = 1, 2, …, L and p i a = 2π k i / L with k i = 0, 1, …, L - 1 At finite volume the Green functions are subject to finite size effects The physical theory is obtained in the limit a 0 renormalizability L thermodinamic limit Non physical quantities like Green Functions may develop divergencies in this limits; S matrix elements are however finite. Z (a) = 1 + log(pa ) +... = g 2 log(p 2 a 2 ) p pp

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Z -1 ∫ [d ] (x 1 ) (x 2 ) (x 3 ) (x 4 ) e i S ( ) On a finite volume (L) and with a finite lattice spacing ( a ) this is now an integral on L 4 real variables which can be performed with Important sampling techniques Z = ∑ { = 1} e J ij i j Ising Model 2 N = 2 L 3 ≈ 10 301 for L = 10 !!!

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Wick Rotation Z -1 ∫ [d ] (x 1 ) (x 2 ) (x 3 ) (x 4 ) e i S ( ) -> Z -1 ∫ [d ] (x 1 ) (x 2 ) (x 3 ) (x 4 ) e - S ( ) This is like a statistical Boltzmann system with H = S Several important sampling methods can be used, for example the Metropolis technique, to extract the fields with weight e - S ( ) = Z -1 ∑ { (x)} n n (x 1 ) n (x 2 ) n (x 3 ) n (x 4 ) Z = ∑ { (x) } n 1 = N t -> i t E

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QCD

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The COLOUR SYMMETRIC IN CONTRAST WITH FERMI STATISTICS LET US ASSUME: 1) QUARK MODEL flavour spin colour Also: Then antisymmetric only colour singlets exist as asymptotic states

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2) The COLOUR quarks, summed over colour number of quarks in the loop assuming Q u = 2/3 and Q d = - 1/3 measured from N c =3.06 ± 0.10

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3) gauge anomaly The COLOUR A , V coupled to gauge bosons Renormalizability requires V VV AA

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4) The COLOUR A , V coupled to gauge bosons V VV octect mesons In the quark model AA

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K0K0 K+K+ K-K- K0K0 -- 00 ++ -1/2+1/2+1 33 Y +1 0 ‘‘ 88

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In the chiral limit u d u,d,s gluons AA AA AA AA anomaly absent in the case of ,K and 8

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THE QCD LAGRANGIAN Let us start with a simple example: This Lagrangian is invariant under the global symmetry transformation where are the generators of the tranformation in the representation to which belongs For a global transformation This is not true for a local transformation = (x)

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THE QCD LAGRANGIAN We look for a ``covariant” derivative such that The trasformation rules are By defining () By writing For an infinitesimal transf. we have Invariant under a gauge transf.

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THE QCD LAGRANGIAN

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Gluons and quarks (vector fields and spinors) The QCD Lagrangian : L ( ) = - 1/4 G A G A GLUONS + ∑ f=flavour q f (i D - m f ) q f QUARKS ( & GLUONS) G A = G A - G A - g 0 f ABC G B G C q f q f a (x) ( ) D I + i g 0 t A ab G A

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a b A, SU(2) t A ab = 1/2 A ab SU(3) t A ab = 1/2 A ab [t A, t B ] = i f ABC t C B, , q A,, pC,, r Feynman Rules ghost-gluon quark-gluon 4-gluons 3-gluons

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R e+e- (S) hadrons

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The COLOUR

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Large Momenta and Energies The QCD Lagrangian depends only on the quark masses and on the dimensionless couplig g 0 At large scales the theory can be studied as in the massless case -> Any dimensionless quantity as which depends only on is naively expected to become a constant at the quantum level this expectation is wrong

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Asymptotic Freedom from R e+e- (S) Let us compute Re+e- in perturbation theory: hadrons

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Asymptotic Freedom from R e+e- (S) + 2 + + d = d2d2

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Asymptotic Freedom from R e+e- (S) + 2 + + The sum of the diagrams in the dashed boxes is ultraviolet finite because of the electromagnetic current conservation

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Asymptotic Freedom from R e+e- (S) 2 + d3d3 The sum of the virtual and real emission diagrams is infrared finite because it corresponds to the physical inclusive cross-section VIRTUAL + REAL CONTRIBUTIONS

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Asymptotic Freedom from R e+e- (S) … + 2 + + The sum of these diagrams is ultraviolet divergent and renormalizes the strong coupling constant, similarly for the real emission diagrams

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Asymptotic Freedom from R e+e- (S) WE HAVE TO INTRODUCE A CUTOFF OVER THE MOMENTA IN THE LOOPS IN ORDER TO REGULARIZE THE ULTRAVIOLET DIVERGENCES WE THEN DEFINE THE ``RENORMALIZED COUPLING” S AT THE SCALE S We choose S (S) in such a way that at the scale S 0 We cannot predict Re+e-(S 0 ), which is used to define S (S 0 ), as much as in QED em is fixed from g-2 or Thomson scattering

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Asymptotic Freedom from R e+e- (S) With the previous definition of S (S 0 ) is also independent of the ultraviolet cutoff is only expressed in terms of measurable quantities or: where

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Asymptotic Freedom from R e+e- (S) is a non-perturbative constant, independent of S, like the proton mass (it depends however on the renormalization scheme and on the order computed in perturbation theory)

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Renormalization Group Equations IN GENERAL NOTE THAT 0 > 0 UNLIKE IN QED (also 1 > 0 )

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depends on the scale only because of Renormalization Group Equations scheme dependence

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SMALL SCALES = LARGE NON PERTURBATIVE EFFECTS (MASS TERMS, HIGHER TWISTS, ETC.) LARGE SCALES= SMALL SENSITIVITY TO QCD

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