5 The Renormalization Group - DAMTP

1 5 The Renormalization GroupEven a humble glass of pure water consists of countless H2O molecules, which are madefrom atoms that involve many electrons perpetually executing complicated orbits around adense nucleus, the nucleus itself is a seething mass of protons and neutrons glued togetherby pion exchange, these hadrons are made from the complicated and still poorly understoodquarks and gluons which themselves maybe all we can make out of tiny vibrations of somestring, or modes of a theory yet undreamed of. How then is it possible to understandanything about water without first solving all the deep mysteries of Quantum Gravity?In classical physics the explanation is really an aspect of the Principle of Least Action:if it costs a great deal of energy to excite a degree of freedom of some system, either byraising it up its potential or by allowing it to whizz around rapidly in space time, then theleast action configuration will be when that degree of freedom is in its ground state.

2 Thecorresponding field will be constant and at a minimum of the potential. This constant is thezero mode of the field, and plays the role of a Lagrange multiplier for the remaining low energy degrees of freedom. You used Lagrange multipliers in mechanics to confine woodenbeads to steel hoops. This is a good description at low energies, but my sledgehammer canexcite degrees of freedom in the hoop that your Lagrange multiplier doesn t must re-examine this question in QFT because we re no longer constrained to sitat an extremum of the action. The danger is already apparent in perturbation theory ,for even in a process where all external momenta are small, momentum conservation ateach vertex still allows for very high momenta to circulate around the loop and the value ofthese loop integrals would seem to depend on all the details of the high energy theory . TheRenormalization Group (RG), via the concept ofuniversality, will emerge as our quantumunderstanding of why it is possible to understand physics at Integrating out degrees of freedomSuppose our QFT is governed by the actionS 0[ ]= ddx[12 + i d di0gi0Oi(x)].

3 ( )Here we ve allowed arbitrary local operatorsOi(x) of dimensiondi>0 to appear in theaction; eachOican be a Lorentz invariant monomial involving some numbernipowers offields and their derivatives, ( )ri siwithri+si=ni. For later convenience,I ve included explicit factors of some energy scale 0in the couplings, chosen so as to ensurethat the coupling constantsgi0themselves are dimensionless, but of course the action isat this point totally general. We ve simply allowed all possible terms we can include this action, we can define a regularized partition function byZ 0(gi0)= C (M) 0D e S 0[ ]/!( )where the integral is taken over the spaceC (M) 0of smooth functions onMwhoseenergy is at most 0. The first thing to note about this integral is that it makes sense: 45 we ve explicitly regularized the theory by declaring that we are only allowing momentummodes up to the cut off19 0. For example, there can be no UV divergences20in anyperturbative loop integral following from ( ), because the UV region is simply let s think what happens as we try to perform the path integral by first integratingthose modes with energy between 0and < 0.

4 The spaceC (M) 0is naturally avector space with addition just being pointwise addition onM. Therefore we can split ageneral field (x) as (x)= |p| 0ddp(2 )deip x (p)= |p| ddp(2 )deip x (p)+ <|p| 0ddp(2 )deip x (p)=: (x)+ (x),( )where C (M) is the low energy part of the field, while C (M)( , 0]has highenergy. The path integral measure onC (M) 0likewise factorizes asD =D D into a product of measures over the low and high energy modes. Performing the integralover the high energy modes provides us with aneffective action at scale Seff [ ] := !log[ C (M)( , 0]D exp ( S 0[ + ]/!)]( )involving the low energy modes only. We call the process of integrating out modeschangingthe scaleof the theory . We can iterate this process, integrating out further modes andobtaining a new effective actionSeff [ ] := !log[ C (M)( , ]D exp( Seff [ + ]/!)]( )at a still lower scale < . For this reason, equation ( ) is known as therenormalizationgroup equationfor the effective out the kinetic part, we write the original action asS 0[ + ]=S0[ ]+S0[ ]+Sint 0[ , ]( )whereS0[ ] is the kinetic termS0[ ]= ddx[12( )2+12m2 2]( )19In writingS 0in terms of dimensionless couplings, we used the same energy scale 0as we chose forthe cut-off.)))

5 This was purely for a non compact space time manifoldMthere can be IR divergences. This is a separate issue,unrelated to Renormalization , that we ll handle later if I get time. If you re worried, think of the theory asliving in a large box of sideLwith either periodic or reflecting boundary conditions on all fields. Momentumis then quantized in units of 2 /L, so the spaceC (M) 0is finite dimensional. 46 for andS0[ ] is similar. Notice that the quadratic terms can contain no cross terms , because these modes have different support in momentum space. For the samereason, the terms in the effective interactionSint 0[ , ] must be at least cubic in the is non dynamical as far as the path integral goes, we can bringS0[ ] out oftherhsof ( ). Observing that the same kinetic action already appears on thelhs, weobtain (!= 1)Sint [ ]= log[ C (M)( , 0]D exp( S0[ ] Sint 0[ , ])]( )which is the Renormalization Group equation for the Running couplings and their -functionsIt should be clear that the partition functionZ (gi( )) = C (M) D e Seff [ ]/!)

6 ( )obtained from the effective action scale (or at any lower scale) is exactly the same as thepartition function we started with:Z (gi( )) =Z 0(gi0; 0)( )because we re just performing the remaining integrals over the low energy modes. Inparticular, as the scale is lowered infinitesimally ( ) becomes the differential equation dZ (g)d =( gi+ gi( ) gi )Z (g)=0.( )Equation ( ) is known as the Renormalization Group equation for the partition function,and is our first example of aCallan Symanzik equation. It just says that as change thescale by integrating out modes, the couplings in the effective actionSeff vary to accountfor the change in the degrees of freedom over which we take the path integral, so thatthe partition function is in fact independent of the scale at which we define our theory ,provided this scale is below our initial cut off fact that the couplings themselves vary, or run , as we change the scale is animportant notion.

7 As we saw in zero and one dimensions, it s quite natural to expect thecouplings to change as we integrate out modes, changing the degrees of freedom we canaccess at low scales. However, it seems strange: you ve learned that the electromagneticcoupling =e24 (0!c can it mean for the fine structure constant to depend on the scale? We ll understandthe answer to such questions later. 47 With a generic initial action, the effective action will also take the general formSeff [ ]= ddx[Z 2 + i d diZni/2 gi( )Oi(x)],( )where thewavefunction Renormalization factorZ accounts for the fact that it s perfectlypossible for the coefficient of the kinetic term itself to receive quantum corrections as weintegrate out modes. (Z is not to be confused with the partition functionZ !) At anygiven scale, we can of course define arenormalized field :=Z1/2 ( )in terms of which the kinetic term will be canonically normalized. We ve also included apower ofZ1/2 in the definition of our scale couplings so that these powers are removedonce one writes the action in terms of the renormalized the running of couplings is so important, we give it a special name and definethebeta function iof the couplinggito be its derivative with respect to the logarithm ofthe scale: i:= gi.)

8 ( )The -functions for dimensionless couplings take the form i(gj( )) = (di d)gi( ) + quanti(gj)( )where the first term just compensates the variation of the explicit power of in front ofthe coupling in ( ). The second term quantirepresents the quantum effect of integratingout the high energy modes. To actually compute this term requires us to perform thepath integral and so will generically introduce dependence on all the other couplings in theoriginal action ( ), so that the -function forgiis a function of all the couplings i(gj).Similarly, although at any given scale we can remove the wavefunction renormalizationfactor, moving to a different scale will generically cause it to re-emerge. We define theanomalous dimension of the field by := 12 lnZ ( )Except for the fact that we re taken the derivative of the logarithm ofZ1/2 , this is just the -function for the coupling in front of the kinetic term. Like any -function, dependson the values of all the couplings in the theory .

9 It gets it s name for reasons that will beapparent momentarily. If our theory contained more than one type of field, then we d havea wavefunction Renormalization factor and anomalous dimension for each fact, in general we d have amatrixof wavefunction Renormalization factors, allowing different fields(of the same quantum numbers such as spin, chargeetc.) to mix their identities as modes are integratedout. 48 Anomalous dimensionsWavefunction Renormalization plays an important role in correlation functions. Supposewe wish to compute then point correlator (x1) (xn) :=1Z C (M) D e Seff [Z1/2 ;gi( 0)] (x1) n(xn)( )of fields inserted at pointsx1, .. , xn Musing the scale theory , allowing for thepossibility that we hadn t canonically normalized the field in the action. In terms of thecanonically normalized field :=Z1/2 this is (x1) (xn) =Z n/2 (x1) (xn) ( )since the change in the measureD D cancels as we ve normalized by the partitionfunction.

10 Upon performing the path integral we will (in principle!) evaluate the re-maining correlator as some function (n) (x1, .. , xn;gi( )) that depends on the scale couplings and on the fixed points{xi}.Now, if the field insertions just involve modes with energies( then we should be ableto compute the same correlator using just a lower scale theory the operator insertionswill be unaffected as we integrate out modes in the range (s , ] for some fractions< for wavefunction Renormalization givesZ n/2s (n)s (x1, .. , xn;gi(s )) =Z n/2 (n) (x1, .. , xn;gi( )),( )or equivalently dd (n) (x1, .. , xn;gi( )) =( + i gi+n ) (n) (x1, .. , xn;gi( )) = 0 ( )infinitesimally. Equation ( ) is the generalized Callan Symanzik equation appropriatefor correlation functions. Once again, it simply says that the couplings and wavefunctionrenormalization factors change as we lower the scale in such a way that correlation functionsremain a Poincar e invariant theory , correlation functions depend the distances betweenpairs of insertion points, as we saw in section ( ).))