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Quantum Chromodynamics

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Quantum Chromodynamics Quantum Chromo dynamics QCD Contents Evidence for quarks R DIS Colour Gauge Symmetry Gluons Freedom Asymptotic Interquark Forces Mesons and Baryons Flavour SU Learning Outcomes Be able to describ e the evidence for quarks Be able to write down the QCD covariant derivative and explain the origin of gluons in terms of a gauge symmetry Be able to compute the interaction strength b etween quarks at tree level in p erturbation theory Be able to show how comp osite quark states break down into dieren t SU colour and a vour representations Be able to describ e the p ossible colour singlet quark states Bo oks Reference Intro duction to High Energy Physics Perkins yon Elementary Particle Physics Ken QCD We now turn our attention to the description of the strong nuclear force and hadrons in terms of quarks and the gauge theory describing their interactions Evidence For Quarks Let us b egin our study of the strong nuclear force by considering the evidence for the + existence of quarks The simplest studies are through e e collisions at accelerators where jets of hadrons were discovered jet e+ γ _ e jet 0 In the s many such hadrons were catalogued including b easts such as p n Understanding these jets to ok a long time but the settled picture is that a quark anti quark pair are pro duced in the collision Their strong interactions then pro duce more quarks with which they are b ound into the hadrons jet e+ γ q _ q e_ jet We will need to build up this picture slowly to understand how quarks are conned in hadrons The RRatio If we neglect the jet formation which is inevitable once the quarks are formed we can make QED predictions for the jet pro duction cross sections Rememb er 2 + + e e s so 2 2 + e e qq Q multiplicity q s where the multiplicity is the numb er of identical copies of a particular quark Its useful to dene the Rratio as + X e e hadrons 2 multiplicity Q R f + + e e f where Q are charges of the quarks which can b e pro duced in the exp eriment Q f u Q etc d The multiplicity factor surprisingly turns out to b e the rst evidence that quarks come in three colours p Thus for example in a machine with centre of mass energy s GeV one can only create u d s quarks so 2 2 R Whereas in a GeV machine we can also create charm quarks so 2 R c Thus at GeV we observe R ou can see that ab ove GeV there Heres a plot of real exp erimental data for R Y quark is an extra step corresp onding to the discovery of the b Quark Spin + In e e collisions the spin nature of the photon created inuences the spin structure of the nal state particles pro duced The preferable conguration for the incoming electron and p ositron to create the photon is _ e+ e J Z = −1 helicity −1/2 +1/2 For fermionic nal states there is a preference for the particle pair to b e pro duced back to back along the b eam line mimicing the electron p ositron initial state This shows up in the cross section which go es as d 2 cos d This is the distribution we nd for jets which tells us the quarks themselves are spin fermions Deep Inelastic Scattering A high energy virtual photon can b e used to prob e structure within the proton eg as was done at SLAC in the s __ e e γ p p When a lot of energy is exchanged the photon has a short wavelength E h hc The proton app ears as a large environment to such photons and the scattering cross section is well describ ed by it hitting an isolated quark or parton as they were initially called γ −i 2/3 e γ µ u u d A very long wavelength photon on the other hand just sees the whole proton with charge Q p γ µ −i e γ p This transition in the form of the crosssection with scattering energy is observed Colour Multiplicity We have seen that there are identical copies colours of each quark We can call them red green and blue but it is imp ortant to rememb er that they are indistinguishable Any theory we write down must b e invariant to interchanging the lab els In a quantum theory this is even more subtle We can imagine a state with a wave function p p R G which is in a red state of the time and a green state of the time yet we cant distinguish them so we could equally well describ e this as one of our basic quark colour states In general there are an innite set of transformations on the RGB wave function 1 vector which are describ ed by matrices acting on the vector These matrices are called elements of a group called SU these are words b orrowed from mathematical group theory but the details of group theory are not essential to us here 1 A red quark is represented by the state a green quark by etc These matrices must b e unitary as we can see from the following Both b efore and after the transformation on our three basis wave functions we require that Z y 3 d x So when we transform U Z Z 0 y 3 y y 3 d x U U d x From which we see that y U U It turns out that these innite set of matrices may b e written compactly as a a U exp i T and we are using the index summation convention where the index a The are numb ers that you can cho ose any values for dierent values give a the dierent U matrices a The T are matrices called the generators of SU Explicitly they can 1 a b ab so that T r T T b e written as normalized 2 i B C C C B B 1 1 1 2 1 3 i T T T A A A 2 2 2 i C C C B B B 1 1 1 4 5 6 i T T T A A A 2 2 2 i i C B C B 1 1 7 8 p T T A A 2 12 We should also pause to make clear what we mean by the exp onential of a matrix in ab ove We dene it as the sum of the innite series 2 M M e M w the transformation No a a i T e iq (x) lo oks like a generalization of the gauge transform of QED e a set of transformations incidentally called the group U It is therefore a natural step to promote the symmetry to a lo cal level to obtain gauge b osons for the strong nuclear force QCD The Dirac equation for the three colour wave functions is i mI where the I is the identity matrix in colour space We will require that this equation is invariant under the transformation a a (x)T ig s e where g will b e the strong forces coupling and note the lo cal form of the transfor s mation a For ease lets just think ab out a transformation where the are innitesimal so a a ig xT s a As usual the op erators will not let the x pass by unhindered as in QED into the derivative whose shifts under an enlarged we must intro duce vector elds A gauge transformation will undo the damage Clearly we will now need such vector a a a elds to cancel the degrees of freedom x Further each comes with a T so a so must each of the A to o Thus we intro duce the covariant derivative a a ig G T D s a The G are the gluon elds of QCD Their Feynman rules to quarks are _ q gluon a γµ q ig s T Note that the presence of the generators in the vertices means that the gluons interchange quark colours 2 2 eg G has a coupling including T which acts as R G R C B C B C B B C 2 G T R G A A A A B B In other words the vertex lo oks like _ _ R R G G In a sense the gluon here is carrying away G antiR colour the gluon is coloured Note also this guy do es not couple to blue quarks The mathematics of the generalized Maxwell equations is a quagmire that we shall avoid here Not surprisingly though having learnt that the gluons are coloured it turns out that they interact directly with each other unlike photons 2 gs gs Collider Evidence For Gluons The simplest way to lo ok for gluons is to directly make them For example _ q e γ gluon _ e+ q 2 We exp ect to see three jet events suppressed by g rememb er to square the s diagram to get the probability Such events are seen and from them we can extract the strong coupling constant 2 M GeV g s Z s Asymptotic Freedom The quark and gluon mo del of QCD is highly successful at predicting results in high energy collider exp eriments but we need to understand why we never see free quarks or gluons In fact we never see any colour charged ob jects in normal life Formally this is still an op en question within QCD but there is enough understanding of the theory to see that it will answer the question The main concrete understanding for which the Nob el Prize was awarded in is asymptotic freedom We saw in QED that the running coupling runs with energy scale due to lo op corrections In QCD the same happ ens but there are additional contributions since the gluons self interact QED like new The result of the computation is again that the coupling runs logarithmically 2 M s 2 Q s 0 2 2 2 M ln Q M s 4 where is the numb er one extracts from the diagrams ab ove 0 In QED just including a single electron and the running lo oks like 0 α 1/137 Q 10277 GeV In QCD though the extra contributions enter with the opp osite sign 0 N N f c 0 where N the numb er of colours and N counts the numb er of quark avours c f accessible at a particular energy scale If N then and the coupling runs as shown f 0 αs ΛQCD Q At large energies or small quark separation the theory is weakly coupled asymp totically free At low energies or big quark separation the coupling grows At the scale QC D the equations we have give an innite value of the coupling Of course our calculation has b een done in p erturbation theory which assumes that g so to extrap olate to s this p oint is b eyond the validity of the metho d however that
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