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QCD Phenomenology at High Energy

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QCD Phenomenology at High Energy QCD Phenomenology at High Energy Bryan Webber CERN Academic Training Lectures 2008 Lecture 1: Basics of QCD ● QCD Lagrangian ❖ Gauge invariance ❖ Feynman rules ● Running Coupling ❖ Beta function ❖ Charge screening ❖ Lambda parameter ● Renormalization Schemes ● History of Asymptotic Freedom ● Non-perturbative QCD ❖ Infrared divergences Lagrangian of QCD ● Feynman rules for perturbative QCD follow from Lagrangian 1 A αβ L = − F F + q¯a(i6D − m)abqb + Lgauge−fixing 4 αβ A flavoursX A A Fαβ is field strength tensor for spin-1 gluon field Aα , A A A ABC B C Fαβ = ∂αAβ − ∂βAα − gf Aα Aβ Capital indices A,B,C run over 8 colour degrees of freedom of the gluon field. Third ‘non-Abelian’ term distinguishes QCD from QED, giving rise to triplet and quartic gluon self-interactions and ultimately to asymptotic freedom. 2 ABC ● QCD coupling strength is αS ≡ g /4π. Numbers f (A,B,C = 1, ..., 8) are structure constants of the SU(3) colour group. Quark fields qa (a = 1, 2, 3) are in triplet colour representation. D is covariant derivative: C C (Dα)ab = ∂αδab + ig t Aα ab “ C C” (Dα)AB = ∂αδAB + ig(T Aα )AB ● t and T are matrices in the fundamental and adjoint representations of SU(3), respectively: tA, tB = if ABCtC, T A, T B = if ABCT C ˆ ˜ ˆ ˜ 1 A ABC αβ where (T )BC = −if . We use the metric g = diag(1,–1,–1,–1) and set h¯ = c = α 1. 6D is symbolic notation for γ Dα. Normalisation of the t matrices is A B AB 1 Tr t t = TR δ , TR = . 2 ● Colour matrices obey the relations: 2 A A N − 1 t t = CF δac , CF = ab bc 2N XA C D ABC ABD CD Tr T T = f f = CA δ , CA = N XA,B 4 Thus CF = 3 and CA = 3 for SU(3). 2 Gauge Invariance ● QCD Lagrangian is invariant under local gauge transformations. That is, one can redefine quark fields independently at every point in space-time, ′ qa(x) → qa(x) = exp(it · θ(x))abqb(x) ≡ Ω(x)abqb(x) without changing physical content. ● Covariant derivative is so called because it transforms in same way as field itself: ′ ′ Dαq(x) → Dαq (x) ≡ Ω(x)Dαq(x) . (omitting the colour labels of quark fields from now on). Use this to derive transformation property of gluon field A ′ ′ ′ Dαq (x) = ∂α + igt · Aα Ω(x)q(x) ′ ≡ (∂αΩ(x))`q(x)+Ω(x)∂α´q(x)+ igt · AαΩ(x)q(x) A A where t · Aα ≡ A t Aα . Hence P ′ −1 i −1 t · A = Ω(x)t · AαΩ (x)+ ∂αΩ(x) Ω (x) . α g ` ´ 3 ● Transformation property of gluon field strength Fαβ is ′ −1 t · Fαβ(x) → t · Fαβ(x) = Ω(x)Fαβ(x)Ω (x) . Contrast this with gauge-invariance of QED field strength. QCD field strength is not gauge invariant because of self-interaction of gluons. Carriers of the colour force are themselves coloured, unlike the electrically neutral photon. ● Note there is no gauge-invariant way of including a gluon mass. A term such as 2 α m A Aα is not gauge invariant. This is similar to QED result for mass of the photon. On the other hand quark mass term is gauge invariant. 4 Feynman Rules ● Use free piece of QCD Lagrangian to obtain inverse quark and gluon propagators. ❖ Quark propagator in momentum space obtained by setting ∂α = −ipα for an incoming field. Result is in Table 1. The iε prescription for pole of propagator is determined by causality, as in QED. ❖ Gluon propagator impossible to define without a choice of gauge. The choice 2 1 α A Lgauge−fixing = − ∂ A 2 λ α “ ” defines covariant gauges with gauge parameter λ. Inverse gluon propagator is then (2) 2 1 Γ (p) = iδAB p gαβ − (1 − )pαpβ . {AB, αβ} λ » – (Check that without gauge-fixing term this function would have no inverse.) Resulting propagator is in Table 1. λ = 1 (0) is Feynman (Landau) gauge. 5 α β A,αp B,β AB αβ p p i £ £ £ £ £ £ £ £ £ £ £ δ −g + (1 − λ) ¦¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¥ " p2 + iε# p2 + iε AB i A- p B δ p2 + iε ab i a, i- p b, j δ (6p − m + iε)ji § γ ¦§¡ B, β ABC αβ ¦§¡ −gf g (p − q) q¦§¡ βγ α ¦§•¤ +g (q − r) §¢¡ ¢¡£ ¤ γα β §§ ¢¡ ¢¡££¤¤ h p§¢¡ ¢¡ ¢¡¢¡£ ¤r +g (r − p) A,α§¢¡ ¢¡£ ¤ C,γ (all momenta incoming) i A,α¤ § B,β ¢¡£ ¤ §¢¡ ¢¡££¤¤ §§ ¢¡ 2 XAC XBD ¢¡¢¡£ ¤ §¢¡ ¢¡ −ig f f (gαβgγδ − gαδgβγ) ¢¡£§•¤§¤¢¡ 2 XAD XBC §§ ¢¡ ¢¡££¤¤ −ig2f XAB fXCD (gαβgγδ − gαγgβδ) §¢¡ ¢¡ ¢¡¢¡£ ¤ §§ ¢¡ ¢¡££¤¤ −ig f f (gαγgβδ − gαδgβγ) C,γ¢¡ ¢¡ D,δ § ¦§¡ A, α ¦§¡ ¦§¡ ¦• ABC α R gf q q B C § ¦§¡ A, α ¦§¡ ¦§¡ ¦ A α •@ @R −ig t (γ )ji @ cb @ b, i c, j ` ´ Table 1: Feynman rules for QCD in a covariant gauge. 6 ● Gauge fixing explicitly breaks gauge invariance. However, in the end physical results will be independent of gauge. For convenience we usually use Feynman gauge. ● In non-Abelian theories like QCD, covariant gauge-fixing term must be supplemented by a ghost term which we do not discuss here. Ghost field, shown by dashed lines in Table 1, cancels unphysical degrees of freedom of gluon which would otherwise propagate in covariant gauges. 7 Running Coupling ● Consider dimensionless physical observable R which depends on a single large energy scale, Q ≫ m where m is any mass. Then we can set m → 0 (assuming this limit exists), and dimensional analysis suggests that R should be independent of Q. ● This is not true in quantum field theory. Calculation of R as a perturbation series in the 2 coupling αS = g /4π requires renormalization to remove ultraviolet divergences. This introduces a second mass scale µ — point at which subtractions which remove divergences are performed. Then R depends on the ratio Q/µ and is not constant. The renormalized coupling αS also depends on µ. ● But µ is arbitrary! Therefore, if we hold bare coupling fixed, R cannot depend on µ. Since 2 2 R is dimensionless, it can only depend on Q /µ and the renormalized coupling αS. Hence 2 d Q ∂ ∂αS ∂ µ2 R , α ≡ µ2 + µ2 R = 0 . 2 2 S 2 2 dµ µ ! " ∂µ ∂µ ∂αS # 8 ● Introducing 2 Q 2 ∂αS τ = ln , β(αS) = µ , µ2 ! ∂µ2 we have ∂ ∂ − + β(αS) R = 0. " ∂τ ∂αS # This renormalization group equation is solved by defining running coupling αS(Q): αS(Q) dx τ = , αS(µ) ≡ αS . β(x) Z αS Then ∂αS(Q) ∂αS(Q) β(αS(Q)) = β(αS(Q)) , = . ∂τ ∂αS β(αS) 2 2 and hence R(Q /µ , αS) = R(1, αS(Q)). Thus all scale dependence in R comes from running of αS(Q). ● We shall see QCD is asymptotically free: αS(Q) → 0 as Q → ∞. Thus for large Q we can safely use perturbation theory. Then knowledge of R(1, αS) to fixed order allows us to predict variation of R with Q. 9 Beta Function ● Running of of the QCD coupling αS is determined by the β function, which has the expansion 2 ′ 4 β(αS) = −bαS(1 + b αS)+ O(αS) 2 (11CA − 2Nf ) (17C − 5CANf − 3CF Nf ) b = , b′ = A , 12π 2π(11CA − 2Nf ) 5 where Nf is number of “active” light flavours. Terms up to O(αS) are known. ● Roughly speaking, quark loop “vacuum polarisation” diagram (a) contributes negative Nf term in b, while gluon loop (b) gives positive CA contribution, which makes β function negative overall. ● QED β function is 1 2 βQED(α) = α + ... 3π Thus b coefficients in QED and QCD have opposite signs. 10 ● From previous section, ∂αS(Q) 2 ′ 4 = −bα (Q) 1+ b αS(Q) + O(α ). ∂τ S S h i Neglecting b′ and higher coefficients gives 2 αS(µ) Q α (Q) = , τ = ln . S 2 1+ αS(µ)bτ µ ! ● As Q becomes large, αS(Q) decreases to zero: this is asymptotic freedom. Notice that sign of b is crucial. In QED, b < 0 and coupling increases at large Q. ′ Including next coefficient b gives implicit equation for αS(Q): 1 1 αS(Q) αS(µ) bτ = − + b′ ln − b′ ln ′ ′ αS(Q) αS(µ) 1+ b αS(Q) 1+ b αS(µ) “ ” “ ” 11 ● What type of terms does the solution of the renormalization group equation take into 2 2 account in the dimensionless physical quantity R(Q /µ , αS)? Assume that R has perturbative expansion 2 3 R(1, αS) = R1αS + R2αS + O(αS) RGE solution R(1, αS(Q)) can be re-expressed in terms of αS(µ): 2 3 αS(Q) = αS(µ) − bτ[αS(µ)] + O(αS) 2 3 R(1, αS(Q)) = R1αS(µ) + (R2 − bτ)αS(µ) + O(αS) Thus there are powers of τ = log(Q2/µ2) that are automatically resummed by using the running coupling. ● Notice that a leading order (LO) evaluation of R (i.e. the coefficient R1) is not very useful since αS(µ) can be given any value by varying the scale µ. ❖ We need the next-to-leading order (NLO) coefficient (R2 − bτ) to gain some control of scale dependence: the µ dependence of τ starts to compensate that of αS(µ). 12 Charge Screening ● In QED the observed electron charge is distance-dependent (⇒ momentum transfer dependent) due to charge screening by the vacuum polarisation: α eff - - + - - + + - + + 1 - + + - 1 128 - + + 137 + + - - + - - 2 2 mZ q ● At short distances (high momentum scales) we see more of the “bare” charge ⇒ effective charge (coupling) increases. ● In contrast, the vacuum polarisation of a non-Abelian gauge field gives anti-screening.
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