8.324 Relativistic Quantum Field Theory II

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1 Lecture 8.3 Relatvstc Quantum Feld Theory II Fall Relatvstc Quantum Feld Theory II MIT OpenCourseWare Lecture Notes Hon Lu, Fall 00 Lecture 5.: RENORMALIZATION GROUP FLOW Consder the bare acton defned at a scale 0 : 0 [ 0 ] d d x ( ϕ) + O, () { } where O s a complete set of local operators formed from ϕ. The theory s specfed by the set. As explaned n the prevous lecture, we can chane the cutoff scale to some < 0 by nteratn out the derees of freedom n the nterval (, 0 ). Ths ves [] d d x ( ϕ) + ()O, () after redefnn ϕ to absorb the feld renormalzaton factor Z. Ths theory s specfed by the set { ()}. mlarly, at another scale <, we obtan [ ], descrbed by { ( )}. These three actons, 0, and, should all descrbe the same physcs at an enery scale E < < < 0. The relatons between them can be found by nteratn out the derees of freedom explctly n the path nteral, vn () (, 0 ; ), ( ) (, 0 ; ) ( (), ; ). Ths process descrbes the renormalzaton roup transformatons, or the renormalzaton roup flow: transformatons between couplns at dfferent scales to ensure they descrbe the same low enery physcs. If we consder, for ( ) 0 () (') Fure : The renormalzaton flow as the flow n the space of all possble coupln parameterzatons to ensure the same low-enery physcs at dfferent scales. smplcty, the dmensonless couplns {λ ()} defned by λ, dfferentatn ves dλ β ({λ j ()}) (3) d d where β ({λ j ()}) d ln λ ({λ j ()}, Z). It s mportant to note that the β are only functons of the Z dmensonless coupln constants {λ j ()}: they do not depend on explctly, as can be seen by consdern

2 Lecture 8.3 Relatvstc Quantum Feld Theory II Fall 00 nteratn out a fracton of the hhest-enery modes n the path nteral. The β-functons ve the tanent vector of the flow, and depend only on the values of {λ j }. Under a relabeln of couplns, we have that The β functons can be computed explctly from the path nteral: [,ϕ d d x Jϕ Z [J] Dϕ e ] k < k < λ λ ({λ j }), () {λ } β ( ) dλ β j ({λ}). (5) dλ j j Dϕ (k) < k < Dϕ (k) e [ϕ ], k < Dϕ (k) e [ϕ + d d x Jϕ. Now, f we let, [ ] [] + [], we have ϕ,] d d x J(ϕ +ϕ ) d F ( ) (6) d Expandn O λ O, (7) (6) ves us the β-functons for all couplns. As an example, let us consder the case of a free scalar feld n four dmensons, wth a cut-off at a scale. Then, we have d k [ϕ] f(k)ϕ (k)ϕ (k). (8) k< (π) We expand f(k) as a power seres n k: f(k) m 0 + k + r k +... r () λ m () + k + k +..., where the coeffcent of k can be chosen to one wth a sutable normalzaton for ϕ. Here, λ m (), r (),... [ ] [ ( ) ] are dmensonless couplns: ϕ, ϕ 6, and so m, r, for example. We now let ϕ (k) ϕ (k) + ϕ (k) wth ϕ (k) supported for k (, ) and ϕ supported for k (0, ). Then we have that [ ] d k [ϕ ] [ϕ ] + ϕ + f(k)ϕ (k)ϕ (k), (9) (π) where the last term s zero as ϕ and ϕ k have dsjont support. Interatn out ϕ only enerates an overall constant for the path nteral, and so d k [ϕ ] [ϕ ] f(k)ϕ (k)ϕ (k) k< (π) where f(k) has not chaned. That s, f(k) m 0 + k + r k +... λ m ( ) + k + r ( ) k +...,

3 Lecture 8.3 Relatvstc Quantum Feld Theory II Fall 00 and we conclude that ( ) ( ) m λ m ( ) λ m () λ m () s a relevant operator, ( ) ( ) m r ( ) r () r () s an rrelevant operator. mlarly, dλ m ( ) β m λ m m λ m < 0, d dr ( ) β r d r m r > 0. We note that dmensonal quanttes lke m and r do not chane at all n ths nstance, but that the dmensonless couplns flow as they are defned wth respect to the cut-off scale. Ths does reflect the rht physcs: the relatve mportance of each term n f(k) as we o to lower eneres, or smaller k. That s, m 0 becomes larer as k becomes smaller, k r k becomes smaller as k becomes smaller. k We wll now derve the full flow equaton for [ϕ]. For ths purpose, we wrte t as [ϕ, ] d k G (k)ϕ(k)ϕ( k) + I [ϕ, ] + U() () (π) where U() s a cosmolocal constant, and the propaator G (k) satsfes { G (k) k k, 0 k. () We have that Z Dϕ(k) e 0[ϕ,] I [ϕ,], (3) where I I + U. There s now no need to mpose an explct cut-off when nteratn over ϕ(k). It s clearly κ (k) k Fure : The propaator G (k) κ (k) has a cut-off around k. k very complcated to obtan the flow equaton for I [ϕ, ] by evaluatn the path nteral drectly. We wll nstead requre dz [] 0, () d whch s an equvalent statement. From ths, we have d k dg ϕ( k)ϕ(k)e I I e. (5) (π) d 3

4 Lecture 8.3 Relatvstc Quantum Feld Theory II Fall 00 Here,... Dϕ... e 0, wth Z 0 Dϕ e 0 Z 0. We would lke to express the left-hand sde of (5) more drectly n terms of I. For ths purpose, consder ( ) 0 Dϕ ϕ(k)e 0 I. (6) ϕ(k) From ths, we have that I (π) () e I G I + ϕ(k) e 0. (7) ϕ(k) The last term n ths equaton s stll complcated. Consder further ( ) 0 Dϕ e 0 I. (8) From ths, we have () G ( ) I (π) e G I G ϕ(k) e + e I I 0. (9) ϕ(k) If we multply 7 by G and add the result to (9), we obtan () G ( e I G ) I (π) e I + 0. Elmnatn I between (5) and ves d d k dg e I U e I U d (π) d (π) () d k d lo G e I U. (π) d Here, the second term s a constant, and so we have where V (π) (), and so where U 0 s ndependent of, and d d d k U V lo G (k), () d d (π) d k U() U 0 + V lo G (k) () (π) d e I d k dg (k) e I, (3) d (π) d or, equvalently, d d k dg (k) [ I I ] I I d (π) d ϕ(k) ϕ( k). ()

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