Scale Invariance, Renormalisation

158 9Renormalization

of renormalizable theories without mass terms. These theories are scale invariant at the classical level because the action does not contain any dimensionful parameter. In this case the running of the coupling constants can be seen as resulting from a quantum breaking of classical scale invariance: different energy scales in the theory 158 are distinguished by different values of the9Renormalization coupling constants. Remembering what we learned in Section 8, we conclude that classical scale invariance is an anomalous of renormalizable theoriessymmetry. without massOne heuristic terms. These way theor to seeies how are scale the conformal invariant at anomaly comes about is to the classical level becausenotice the action that the does regularization not contain any of an dimensionful otherwise parameter.scale invariant field theory requires In this case the running158 the of introduction the coupling of constants an energy can scale be seen (e.g.as a resulting cutoff). This from9Renormalization bre a aking of scale invariance quantum breaking9.3 The ofRenormalization classical scale Group invariance: in Statistical different Mechanicsenergy scales in the theory 163 of renormalizablecannot be restored theories without after renormalization. mass terms. These theories are scale invariant at are distinguished by differentNevertheless, values of the scale coupling invariance constants. is not Remembering lost forever inwhat the quantum theory. It is re- we learnedmanaged in Sectionthe toclassical 8, rewrite we conclude level the partition because that classical the function action scale solely does inv not inariance termscontain is of any anthis anomalous dimensionful new (renormalized) parameter. In thiscovered case1 the at running the fixed of the points coupling of the constants beta function can be1 seen where,as resulting by defin fromition, a the coupling symmetry.spin One variables heuristic ways interacting to see how through the conformal a new Hamiltonian anomaly comesH about is to quantumdoes breaking not run. of classical To understand scale invariance: how this different happensenergy we go scales back in to the a sca theoryle invariant clas- notice that the regularization of an otherwise scale invari1 ant1 field theory requires are distinguishedsical field by theory different whose values field of!!H thex couplingstransforma const underants. coordinate Remembering rescalings what as 9.3 The Renormalization Group in Statistical Mechanicsthe introduction 161 of an energy scale (e.g.Z a cutoff). Thise breaking. of scale invariance (9.43) we learned in Section 8, we conclude that classical scale invariance is an anomalous square lattice as the one depicted in Fig 9.2. In terms of the spin variables s 1 , 1 cannot be restoredi 2 after renormalization. µ s µ # 1 where i labels the lattice site, the Hamiltonian of the system is given by symmetry. One heuristic wayx to see" howx , the conformal! x anomaly" !comes" x about, is to (9.23) Nevertheless, scale invariance is not lostScale forever invariance, in the quantum theory. renormalisation It is re- H J si s j, (9.35) notice that the regularization of an otherwise scale invariant field theory requires i, j covered at theLet fixed us pointsnow think of the about beta the function space of where, all possible by defin Hamiltoniansition, the couplingfor our statistical where i, j indicates that the sum extends over nearest neighbors and J is the cou- the introductionwhere # ofis ancalled energy the scale canonical (e.g. a scaling cutoff). This dimension breaking of of the scale field. invariance An example of such pling constant between neighboring spins (here we consider that there issystem no external including allRenormalisation kinds of possible deals couplings with the between scale thdependenceeindividualspinscom- of the the does not run. To understand how this happens we4 go back to a scale invariant clas- magnetic field). The starting point to study the statistical mechanics of this system cannotatheoryisamassless be restored after renormalization.! theory in four dimensions is the partition function defined as sical field theorypatible whose with the field symmetriesphysics! x transform even of the if the under system. original coordinate If denote theory rescalings by isR scalethe decimationinvariant. as operation, ! H Nevertheless, scale invariance is not lost forever in the quantum theory. It is re- Z e , our(9.36) previous analysis shows that R defines a map in this space of Hamiltonians si coveredµ at theµ fixed points of the beta# function11 where,µ byg defin4 ition, the coupling where the sum is over all possible configurations of the spins and 1 is the Virtual phenomena can get more complicated or simplify as ! T x "x , ! x " L! " x ,µ! ! !(9.23), (9.24) inverse temperature. For J 0theIsingmodelpresentsspontaneousmagnetization does not run. To understand how this happens1 2 we go back4! to a scale invariant clas- below a critical temperature Tc,inanydimensionhigherthanone.Awayfromthis we move to largerR : Hand shorterH . distances (9.44) temperature correlations between spins decay exponentially at large distances sical field theory whose field ! x transform under coordinate rescalings as wherexij # is called the canonical scaling dimension of the field. An example of such s s e " , (9.37) where the scalar field has canonical scaling dimension # 1. The Lagrangian den- i j At the same4 time the operation replaces a lattice with spacing a by another one atheoryisamassless! theory in fourµ dimensionsRµ # 1 with xij the distance between the spins located1 in the i-th and j-th sites of the lat- sity transformsx as "x , ! x " ! " x , (9.23) tice. This expression serves as a definitionλ of− the correlation length " whichwith sets the double spacing 2a.Asaconsequencethecorrelationlengthinthenewlattice characteristic length scale at which spins can influence eachotherbytheirinterac- " tion through their nearest neighbors.× measured in units of the1 latticeµ spacingg is4 divided by two,4R : " . where # is calledL theµ canonical! ! scaling! , dimensionL " of theL field.! (9.24) An2 example of such (9.25) Nowatheoryisamassless we can iterate2 the! operation4 theory4! inR fouran indefinite dimensions number of times. Eventually we might reach a Hamiltonian H that is not further modified by the operation R where the scalar field hasand canonical the classical scaling action dimension remains# invariant1. The3 Lagrangian. den- 1 µ g 4 sity transforms as We look at the freeL theoryµ! g ! 0foramoment.Nowtherearenodivergences! , (9.24) H R H 1 R H2 2 R ... 4!R H . (9.45) and all correlation functions can be exactly computed. In particular we consider the where the scalarL field has" canonical4L ! scaling dimension # 1. The(9.25) Lagrangian den- The fixed pointmomentum Hamiltonian spaceHn-pointis scale correlation invariant functionbecause it does not change as sity transformsIn asrelativistic QFT we seem to get only fixed points, no limit and the classicalR is performed. action remains Noticecycles invariant that nor because strange3. 4 of 4attractors this invariance the correlation length of the 9.4Fig. 9.3TheDecimation Renormalization of the spin lattice. Group Each in block Quantum in the upper Field lattice Theory is replaced by an effective G0 165p1,...,pn 2$ % p1 ... pn spin computed according to the rule (9.39). Notice also that the size of the lattice spacingsystem is doubled at the fixed point do not change under 4R.Thisfactiscompatiblewiththe in the process. We look at the free theory g 0foramoment.NowtherearenodivergencesL " L ! (9.25) ! 2 " and all correlationtransformation functions" can2 beonly exactly if " computed.0or" In pa.Herewewillfocusinthecaseofrticular we consider the 4 4 ip1 x31 ... ipn xn momentumnontrivial spaceandn-point the fixed classical correlation points action with functiond infinite remainsx1 ... correlationd invariantxn e . length. 0 T !0 x1 ...!0 xn 0 , (9.26) The spaceWe look of Hamiltonians at the free theory can beg parametrized0foramoment.Nowtherearenodivergences by specifying the values of the and4 all correlation4 functions can be exactly computed. In particular we consider the G0 p1,...,couplingpn 2$ constants% wherep1 associated by...!0 xpn we with denote all possible the free interacti field operator.on terms between Applying individual the rescaling (9.23) we F momentum space n-point correlation function spins of the lattice.find the If following we denote transformation by Oa si these (possibly for the correlation infinite) interaction function terms, 4 4 ip1 x1 ... ipn xn Luis Alvarez-Gaume Natal Lectures March 23-26 2011 the most general Hamiltonian for the spin system under study can be written as d x1 ...! d xn e 4 4 0 T !0 x1 ...!0 xn 0 , (9.26) O G01 p1,...,pn 2$ % p1 ... pn 4 n 1 n# Fig. 9.4 Example of a renormalization group flow. G0 p1,...,pn " G0 " p1,...," pn . (9.27)

4 4 ip1 x1 ... ipn xn where by !0 x we denote thed freex1 ... fieldd x operator.Hn esi Applying#aOa0s theTi , ! rescaling0 x1 ...!0 (9.xn23)0 we, (9.46)(9.26) perturbations the system flows either to the free theory at theoriginortoatheory 65 with infinite values for the couplings.find the following transformation for the correlationa 1 function Sunday, 27 March, 2011 wherewhere# R byare!0 thex couplingwe denote4 n constants1 then# free fieldfor the operator. corresponding Applying operators the rescaling.Thesecon- (9.23) we G0 ap1,...,3 Inp anD-dimensional" theoryG0 the" canonicalp1,...," scalingpn . dimensions(9.27) of thefieldscoincidewithitsengi- 9.4 The Renormalization Group in Quantum Fieldfind Theory the following transformation for the correlation function stants can beneering thought dimension: of as coordinates# D 2 for in bosonic the space fields of and all# HamDiltonians.1 for fermionic Therefore ones. For a Lagrangian the operation R defines a transformation2 in the set of coupling constants2 Let us see now how these ideas of the renormalization group apply to Fieldwith Theory. no dimensionful,..., parameters4 classicaln 1 n# scale invarianc,...,efollowsthenfromdimensionalanalysis.. Let us begin with a quantum field theory defined by the Lagrangian G0 p1 pn " G0 " p1 " pn (9.27) 1 R : #a # . (9.47) L "a 3LIn0 " a D-dimensionalgiOi "a , theory the(9.49) canonical scaling dimensions of thaefieldscoincidewithitsengi- i D 2 D 1 neering dimension: # 2 for bosonic fields and # 2 for fermionic ones. For a Lagrangian where L0 "a is the kinetic part ofwith the Lagrangian no dimensionful andForgi are example, the parameters coupling constants in our classical case scale we invarianc startedefollowsthenfromdimensionalanalysis. with a Hamiltonian in whichonlyoneofthe associated with the operators Oi "a .Inordertomakesenseofthequantumthe-3 In a D-dimensional theory the canonical scaling dimensions of thefieldscoincidewithitsengi- ory we introduce a cutoff in momenta #.Inprincipleweincludealloperatorscoupling constantsOi is different from zero (say #1 J). As a result of the decima- neering dimension: D 2 for bosonic fields and D 1 for fermionic ones. For a Lagrangian compatible with the symmetries of the theory. tion # J J 1# while2 some of the originally# 2 vanishing coupling constants In section 9.2 we saw how in the cases of QED and QCD,with1 the value no o dimensionfulfthecou- parameters classical scale invariancefollowsthenfromdimensionalanalysis. pling constant changed with the scale from its valuewill at the sctakeale #.Wecanunder- a nonzero value. Of course, for the fixed point Hamiltonian the coupling stand now this behavior along the lines of the analysis presented above for the Ising model. If we would like to compute the effective dynamicsconstants of the theory do not at an en-change under the scale transformation R. ergy scale µ # we only have to integrate out all physical models with energies between the cutoff # and the scale of interest µ.Thisisanalogoustowhatwedid in the Ising model by replacing the original spins by the blockspins.Inthecaseof field theory the effective action S "a, µ at scale µ can be written in the language of functional integration as

iS " ,µ iS "a,# e a D"a e . (9.50) a µ p #

Here S "a,# is the action at the cutoff scale 156 9Renormalization i.e. such that they become weakly coupled at high energies. This is not a purely academic question. In the late 1960s a series of deep-inelastic scattering experiments carried out at SLAC showed that the quarks behave essentiallyasfreeparticles inside hadrons. The apparent problem was that no theory was9.2 k Thenown Beta-Function at thatand Asymptotic time Freedom 153 that would become free at very short distances: the example set by QED seem to be followed by all the theories that were studied. This posed a very serious problem for Quantum Field Theory as a way to describe subnuclear physics,sinceitseemedthat

2 2 its predictive power was restricted to electrodynamics but failed miserably" e when# " e 2 # 9.2 The Beta-Function and Asymptotic Freedom ! ve$ ue v $ u 155! ve$ ue & q v $ u applied to describe strong interactions. "# 4%q2 µ µ "# 4%q2 µ µ 2 2 2 run in the loop (MW MZ). Taking this into account, as well" as thresholde effects,e q # Nevertheless, this critical time for Quantum Field Theory turned!"# ve out$ ue to be1 its log vµ $ uµ . (9.11) the value of the electron charge at the scale MZ is found to be [1] 4%q2 12%2 ' 2 finest9.2 The hour. Beta-Function In 1973 and Asymptotic David Gross Freedom and Frank Wilczek [2] and Davi155dPolitzer[3] 2 Now let us imagine that we are performing a e e µ µ with a center of mass showed that nonabelian gauge theories can actually displaye MZ the1 required behavior. run in the loop (M M ). Taking this into account,! asM wellz as thresholdenergy effects,µ..Fromthepreviousresultwecanidentifytheeffectivechar(9.18) ge of the parti- W Z 4" cles128 at.9 this2 energy scale e µ as Forthe the value QCD of the Lagrangian electron charge in at Eq. the (8.37) scale M theis beta found function to be [1] is givenby9.2 The Beta-Function and Asymptotic Freedom 153 This156 growingZ of the effective fine structure constant with energy can be9Renormalization under- 2 stood heuristically32 by remembering that the effect of the polarization of the vac- " e µ # e M g 111 2 ! ve$ ue v $ u . (9.12) i.e. suchZ that they become weakly coupled at high energies. This is"# not a purely 2 µ µ !! Mguumz shown in the diagramNc. of Eq.N (9.2)f . amounts to(9.18) the creation ofaplethoraof(9.21) 4%q electron-positronacademic416" "2 question. pairs1283 around.9 In the the3 late location 1960s of a series the charge. of deep-inelasThese virtualtic scattering pairs behave experiments as dipolescarried that, out as at in a SLAC dielectric showed medium, that tend the to quarksThis screen charge, behave thieschargeanddecreasingµ ,isthequantitythatisphysicallymeasurableinourexperi essentiallyasfreeparticles ment. This growing of the effective fine structure constant with energy can be under- Fixed2 points beta functions2 its valueinside at long hadrons. distances The (i.e. apparent lower energies). problem wasNow we that can no make3 theory sense was of the known formallye at divergent that time result (9.11) by assuminge that 7g v " u v # u v " u q2 v # u Instood particular, heuristically for real by remembering QCD9.2 ( TheN Beta-Function that3, the andN Asymptotic effect6) Freedom of we the have polarization that g of the vac-153 !"# 0.e$ Thise 2 µ $ µ !"# e$ e 2 & µ $ µ TheCthat variation wouldf ofbecome the coupling free at constant very short with! distances:the energy charge is the appearing16 usuall" example2 yencodedinQuan- in the set classical by4% QEDq Lagrangian seem ofto QED be is just a “bare”4%q value that uum shown in the diagram of Eq. (9.2) amounts to the creation ofaplethoraofdepends on the scale ' at which we cut off the theory, e e ' bare.Inorderto means that for a theory thattum is Fieldfollowed weakly Theory by coupled in all the thebeta theories at function an that energydefined were studied. by scale Thisµ0 posedthe coupling a very serious2 problem2 for 2 reconcile (9.11) with the physical" resultse (9.12) wee must assumeq that the dependence# electron-positron pairs around the locationQuantum of Field the charge. TheoryThese as a way virtual to describe pairs subnuclearbehave !"# physicsve$ u,sinceitseemedthate 1 log vµ $ uµ . (9.11) constant decreases as the energy increases µ .Thisexplaintheapparentfree-dg of the bare (unobservable) charge4%e q'2 bare on12 the%2 cutoff '' 2is determined by the as dipoles that, as in a dielectric medium,its predictive tend to power screen was thischargeanddecreasing restricted# g µ to electrodynamics.identity but failed miserably(9.19) when dom of quarks inside the hadrons: when2 the quarks are2 verydµ close together their its value at long distances (i.e. lowerapplied energies)." toe describe# strong interactions." e 2 # Now let us imagine that we are performing2 a2e e µ µ with a center of mass !"# ve$ ue vµ $ uµ !"# ve$ ue & q vµ $ uµ e ' µ 4%q2 4%q2 2 2 bare effectiveThe variation color charge of the coupling tend to constant zero. This with phenomenon energy is usuall isyencodedinQuan- called asymptoticenergy µe.Fromthepreviousresultwecanidentifytheeffectivechar freedomµ e ' bare. 1 log . (9.13) ge of the parti- In the caseNevertheless, of QED the this beta critical function time can for be Quantum computed Field from Eq. Theory (9.15) turned with theout12 to%2 be its ' 2 e2 e2 q2 cles at this energy scale e µ as tum Field Theory in the beta functionresultfinestdefined" hour. by In 1973 David Gross# and. Frank Wilczek [2] and DavidPolitzer[3] Asymptotic free theories display!"# ve$ ue a behavior2 1 2 log that2 is oppositvµ $ uµ etothatfoundabove(9.11) showed that4%q nonabelian12% ' gauge theoriesIf can we stillactually insist in display removingthe the required cutoff, ' behavior.we have to send the bare charge to in QED. At high energies their coupling constant approaches3zerozeroe ' bare whereas0insuchawaythattheeffectivecouplinghasthefinitevalue at low given Now let us imagine thatdg we are performing a e e µ µ withe a center of mass 2 2 For the QCD Lagrangian# ine Eq.QED (8.37) theby. the beta experiment function at isthe give energynby scale(9.20)µ.Itisnotaproblem,however,thatthebare" e µ # energy#µ.Fromthepreviousresultwecanidentifytheeffectivecharg µ . ge2 of the(9.19) parti- !"# ve$ ue vµ $ uµ . (9.12) energies they become strongly coupleddµ (infrared slavery). 12This"charge features is small for largeare values at the of the cutoff, since the only measurable4%q2 quantity is cles at this energy scale e µ as 3 At one loop heart of the success of QCDThe as fact a theory that the coefficient of strong of interactions the leadingg termthe,sincethisisexactly11 effective in the beta-fcharge2 unction that remains is positive finite. Therefore all observable quantities should ! g be expressedNc inN perturbationf . theory as a power series(9.21) in the physical coupling e µ 2 In the case of QED the beta function1 can be computed from Eq.2 (9.15)2 with the 9.2 The Beta-Function#0 and Asymptotic0givesustheoverallbehaviorofthecouplingaswechangeth Freedom " e µ 16"# 1573 3 escale. the type of behavior found in quarks:6" they are! quasi-freeve$ ue parv $ticlesu and. not inside(9.12) inThis the unphysical charge, the hadronse µ bare,isthequantitythatisphysicallymeasurableinourexperi coupling e ' bare. ment. "# 4%q2 µ µ result Eq. (9.20)!(g ) means that, if we start at an energy where theNow elect weric can coupling make sense is small of the formally divergent result (9.11) by assuming that but the interaction potential potential between them increases at large distances. 3 enoughIn forparticular, our perturbative for real QCD treatment (N to3, beN valid,6) the wethe effec have chargetive that charge appearingg grows in the with7g classical0. LagrangianThis of QED is just a “bare” value that 3 C f IR free (QED)! 16"2 Although asymptotic freeThis charge, theoriese µ ,isthequantitythatisphysicallymeasurableinourexperi cane be handled in the ultraviolet,ment.depends they on become the scale at which we cut off the theory, e e .Inorderto the# energyemeans scale. that This for. a growing theory of that the is effective weakly coupling coupled9.2 The(9.20) con Beta-Function at anstant energy with energy sca andle'µ Asymptoticmeans0 the coupling Freedom ' bare Now we canQED make sense of2 the formally divergent result (9.11) by assumingreconcile that (9.11) with the physical results (9.12) we must assume that the dependence thethat charge QEDconstant appearing is infrared in12 decreases the" classical safe, Lagrangian as since the the energy of perturbative QED is increases just a “bare” approxima value that tion.Thisexplaintheapparentfree- gives better and bet- extremely complicated in the infrared. In the case of QCD it isstilltobeunder-µ dg dependster results on the scale as we' at go which to lower we cut off energies. the theory, Actually,e e ' bare because.Inordertoof the the bareelectron (unobservable) is the lighter charge e ' bare on the cutoff ' is determined by the dom of quarks inside the hadrons:g whenWe thecanβ look( quarksg) atg the> are previous0 very,µ discussion, close together an= inβ particular(g theirg∗)+ Eq.... (9.13), from a different stoodThe fact (at thatleast the analytically) coefficientreconcile of how the (9.11) leadingthe with theory the physical term confines results in the (9.12) beta-f we color mustunction assume charges that the is dependence positive andidentity| generates∗ thedµ − ofelectrically the bareeffective (unobservable) charged color charge particle chargee andtendon hasthe to cutoff zero. a finite Thisis determined nonvanishingpoint phenomenon of by view. the mass In order is callthe to removeedrunningasymptotic the of ambiguities the freedom associated. with infinities we have 1 ' bare ' µ ,g1 spectrum#0 6" of0givesustheoverallbehaviorofthecouplingaswechangeth hadrons, as wellidentityfine as structure the breaking constant stops of the at the chiral scale symme inbeenmetry theescale. forced well-known (8.52). to introduce value a dependence.Would of the coupling consta2 nt on the energy2 scale Asymptotic free theories display a behavior that is opposit↑ etothatfoundabove137 2 ↑ 2 e ' bare µ Eq. (9.20) means that, if we startother at charged an energy fermions where with the masses electric below couplingm be is present small in Nature, thee µ effectivee ' bare 1 log . (9.13) In general, the ultraviolet andin QED. infrared At high properties energiese ' 2 their ofµ coupling2 a theeory constant are controlled approaches zero by whereas at low 12%2 ' 2 e µ 2 e ' 2 1 bare log . (9.13) enough for our perturbative treatmentvalueenergies of the to be fine they valid, structure becomebare the constant effec12 strongly%2 tive in the charge' coupled2 interaction grows (infraredUV betwe with free slavery).en (QCD) these particlesThis features would are at the the fixed points of the betarun function, further to i.e. lower those values values at energies of below the coupling the electronIf we ma constant stillss. insist ing removingfor the cutoff, ' we have to send the bare charge to the energy scale. This growingIf of we still theheart insist effective in of removing the coupling success the cutoff, of' con QCDstantwe have as with a to theory send energy the bare of strongcharge means to interactions,sincethisisexactly g* g* g* zero e 2 0insuchawaythattheeffectivecouplinghasthefinitevaluegiven On the other hand if we increase3 the energy scale e 'growsbare until at some whichthat QED it vanishes is infrared safe, sincezero thee ' perturbativebare1 0insuchawaythattheeffectivecouplinghasthefinitevalue2 approximation gives better andgiven bet-µ dg by the experimentthe type at the of energy behavior scale µ found.Itisnotaproblem,however,thatthebare in quarks: they are quasi-freeby the experiment particles at the inside energy the scale hadronsµ.Itisnotaproblem,however,thatthebare Fig. 9.1 Beta functionscale for the a hypothetical coupling theory is withof order three fixed one points andg1 , g the2 and perturbativeg3 .Apertur- β(g appro) g∗ ximation< 0 ,µ breaks down.= β(g g∗)+... ter results as we go to lowerbative analysis energies.charge would is capturebut small the Actually,only for largethe interaction regions values shown because of the in potential the cutoff, boxes. the sinceelectron potential the only measurable isbetween the quantity lighter themcharge| is incre is smallases for at large largedµ values distances. of the− cutoff, since the only measurable quantity is theIn effective QED charge this is that known remains as finite. the Therefore problem all ofobse thervable Landau quantities pole should but in factitdoesnotpose electrically charged particle and has aAlthough finite! g nonvanishing asymptotic0. freemass theories the running can be handled ofthe the2 effectiveµ in the charge,g ultr(9.22)aviolet, that remains they become finite. Therefore all observable quantities should beany expressed serious in perturbation threat to theory the asreliability a power series of in QED the physical perturbation coupling e µ theory: a simple calculation Luis Alvarez-Gaume Natal Lectures March 23-26 2011 ↑ ↓ 2 and not in the unphysical bare coupling e 180 . 1 be expressed in perturbation theory as a power series10 in The the p Originhysical of coupling Mass e µ fine structure constantwhich stops the at couplingshows theextremely constant scale that the approachesme energy complicatedin the its scale critical well-known at value.' inbare whichThis the is theinfrared. in value fact theory governed137 In would.Would the case become of QCD strongly it i coupledsstilltobeunder- is by the sign of the beta function around the critical coupling. and not in the unphysical bare coupling e ' bare. Usingother charged perturbation fermions theory with masses weThere havestood below10 (at seen277 is leastmGeV. ae thatbe analytically) dynamically However, present for both we in how know Nature, QED generated the that theory the QED and effective confines does Q CD scale not onelivecolor that of charges long! such2 At and much2 generates the We have$ seenLandau above that when the beta function is negative close to the fixed p ! . Λ m ,m (10.37) point (the case ofresponsible QCD)spectrum the coupling of for hadrons, tends most to its criticalof as wellthe value, mass asg the0, of breaking as thethe en- nucleons of the chiral symmetry (8.52).QCD QCD u d fixedvalue points of the fine occurs structure at zero constant9.2 coupling,lower The in Beta-Functionscales the interactiong we expect and0. However, electromagnetismAsymptotic between Freedom these our particles toanalysis be unified would also with showed other interactions, that and ergy is increased. This meansIn general, that the critical the point ultraviolet is ultraviolet and stable infrared,i.e.itisan properties of a theory are controlled by 66 run further to lower valuesattractor at as energies weeven evolve if towards this below is higher not the theenergies. electron case If, we on the mawill contrary,ss. enter the the beta uncharted function territory of quantum gravity the two theories present radically different behavior19 So at far hi wegh have and not low made9.2 energies. The any Beta-Function hypothesis From as to and the Asymptotic mass of the Freedom quarks. Let us now On the other hand ifis positiveSunday, we increaseWe (as 27at it can March, energieshappens lookthe 2011 at the in fixed the QED) of previous energy the thepoints couplingorder discussion, scale of of constant the 10 ane beta in apprµ particularGeV.oaches2 function,grows Eq. the critical(9.13), until i.e. valuefrom those at a different some values of the coupling constant g for the point of view ofas the the energy betapoint decreases. ofSo function, view.which much ThisIn order isitfor the vanishes to QED. case remove the of an The the differenceinfrared ambiguities nextassume stable question associatedfixed thatlies point. thatwith we in one infinities arethe may dealing we energy ask have at with this regime staglighteiswhetherit quarks at .Theyaredefinedasthosewhosemass scale the coupling is of orderThis analysisbeen one forced andthat we to the introduce have perturbative motivated a dependence with the of appro examples thesatisfies couplingximation of QED constam andnt breakson QCD the energy is .Thisisthecaseofthe down.We scale can look at the previousu, discussion,d and s quarks an in particular that make Eq. (9.13), up most from ofa different completely generalis possible and can tobe carried find quantum out for any quantum field theories fieldtheory.InFig.9.1q with! aQCD behavior opposite to that of QED, In QED this is known aswe the have problem represented the of beta the function Landau for a pole hypotheticalthe but matter the inory fac withtitdoesnotpose that! threeg we fixed see0. aroundpoint of view. us. In In this order case, to remove eq. (10.37)the(9.22) ambiguities can be associated recast aswith infinities we have any serious threat to thepoints reliability located at couplingsof QEDg1 , perturbationg2 and g3 .Thearrowsinthelinebelowtheplot theory: a simple calculationbeen forced to introduce a dependence of the coupling constant on the energy scale represent the evolution of the coupling constant as the energy increases. From the 2 shows that the energy scale at whichUsing the theory perturbation would theory become we shavetrongly seen coupled that for2 is both QED2 and QCD one of such The expression of the betaanalysis function presented above of QCD we see that wasg1 also0andg known3 are ultraviolet to ’t stableHooft fixed points, [4]. Therep are evenmq earlier (q u,d,s). (10.38) 277 while the fixed point g is infrared stable. $ 10 GeV. However, we knowfixed2 points that QED occurs does at zero not coupling, live that long!g 0. At However, much our analysis also showed that computationsLandau in the russianIn order literature to understand [5]. the high and low energy behavior of a quantum field the- lower scales we expectory electromagnetism it is then crucialthe to know two the to theories structure be unified of present the beta withThis functi radically otons meansher associated interactions, different that with light its behavior quarks and at inside high and the lowhadrons energies. are relativi From stic. What is more impor- couplings. This canthe be a point very difficult of view task, since of perturbat thetant, betaion eq. theory function, (10.37) only allows implies the difference that the lies typical in the energy energy of regime these at quarks if of order and even if this is not the casethe study we of will the theory enter around the “trivial” uncharted fixed points, territ i.e. thoseory that of occur quantum at zero cou- gravity !QCD 19 at energies of the orderpling of 10 like theGeV. case of g1 in Fig. 9.1. On the other hand,therefore any “nontrivial” we fixed are point in regime where QCD is strongly coupled. occurring in a theory (like g and g )cannotbecapturedinperturbationtheoryand So much for QED. The next question2 that3 one may ask atThere this stag areeiswhetherit two conclusions to be extracted from this discussion. The first one requires a full nonperturbative2 The expression analysis. of the beta function of QCD was also known to ’t Hooft [4]. There are even earlier is possible to find quantumThe moralfield to theories be learned from with our discussion a behavior aboveis is oppo that dealsite weing with have to the that ultra- found of QED, the reason behind the technical problemsincalculatingthe violet divergences incomputations a quantum field intheory the has russian the conseque literaturence, among [5]. others, of introducing an energy dependence in the measured valuemasses of thecouplingconstants of hadrons such as protons or neutrons from first principles: we would have of the theory (for example the electric charge in QED).to This deal happens with even a in theory the case in a regime in which perturbation theory does not work. Hence we have to resort to numerical approaches such as lattice fieldtheory. The second lesson we have learned is that the Higgs mechanism actually contributes very little to explaining the mass we see around us. In fact, most of the mass of the atoms comes from the nucleus (from about 99.95% for hydrogen to 99.9997% for uranium) that is made of protons and neutrons. What we haveseenaboveisthat quark mass parameter mq generated through the Higgs mechanism contributes very little to the mass of these hadrons: most of the mass of protonsandneutrons,and therefore of the world we see, comes from !QCD. That all difficulties in computing hadron masses comes from having light quarks can be seen is a toy model due to Howard Georgi [2]. He imagines aworldessen- tially identical to our own but with a single crucial difference: the masses of the u and d quarks satisfy 1 m m m ! . (10.39) u d 3 proton QCD

2 2 Because of this fact p mq and the quarks can be treated nonrelativistically. Thus, the typical energy of the processes inside the proton is mq and the condition (10.39) implies that theory at this scale is weakly coupled. Tuning mq !QCD we can even make

2 g mq 1 "s mq . (10.40) 4# 137

42 This sets !QCD 10 mq. Given all this, it should be possible to study the bound state of the three quarks in the proton using the techniques of atomic physics. Since the theory is in a coupling regime where perturbation theory can be used, the static potential between the quarks is obtained from the diagram where the two quarks interchange a gluon. If fact we do not even have to compute the diagram. It sufﬁces to compare the corresponding processes in QCD and QED Farewell

‣ QFT is a vast and complex subject

‣ SM is a big achievement

‣ It summarises our knowledge of the fundamental laws of Nature

‣ But also our ignorance

‣ Many puzzles and unanswered questions remain

‣ We may be at the end of a cycle. Perhaps the symmetry paradigm has been exhausted. ‣ Naturalness, a red herring? Higgs or not Higgs Thank you ‣Gravity into the picture ﬁnally?

‣Hopefully we are entering a golden decade

Luis Alvarez-Gaume Natal Lectures March 23-26 2011

Sunday, 27 March, 2011