A hint of renormalization

Transcription

1 A hint of enomalization Betand Delamotte a) Laboatoie de Phyique Théoique et Haute Enegie, Univeité Pai VI, Piee et Maie Cuie, Pai VII, Deni Dideot, 2 Place Juieu, Pai Cedex 05, Fance Received 28 Januay 2003; accepted 15 Septembe 2003 An elementay intoduction to petubative enomalization and enomalization goup i peented. No pio knowledge of field theoy i neceay becaue we do not efe to a paticula phyical theoy. We ae thu able to dientangle what i pecific to field theoy and what i intinic to enomalization. We link the geneal agument and eult to eal phenomena encounteed in paticle phyic and tatitical mechanic Ameican Aociation of Phyic Teache. DOI: / I. INTRODUCTION Han Bethe in a eminal 1947 pape wa the fit to calculate the enegy gap, known a the Lamb hift, between the 2 and 2p level of the hydogen atom. 1 Thee level wee found to be degeneate even in Diac theoy, which include elativitic coection. Seveal autho had uggeted that the oigin of the hift could be the inteaction of the electon with it own adiation field and not only with the Coulomb field. Howeve, to quote Bethe, Thi hift come out infinite in all exiting theoie and ha theefoe alway been ignoed. Bethe calculation wa the fit to lead to a finite, accuate eult. Renomalization in it moden petubative ene wa bon. 2 Since then it ha developed into a geneal algoithm to get id of infinitie that appea at each ode of petubation theoy in almot all quantum field theoie QFT. 3 7 In the meantime, the phyical oigin of thee divegence ha alo been explained ee Ref. 8 fo many inteeting contibution on the hitoy and philoophy of enomalization and enomalization goup. In QFT, a in odinay quantum mechanic, the petubative calculation of any phyical poce involve, at each ode, a ummation ove vitual intemediate tate. Howeve, if the theoy i Loentz invaiant, an infinite numbe of upplementay tate exit compaed with the Galilean cae and thei ummation, being geneically divegent, poduce infinitie. The oigin of thee new tate i deeply ooted in quantum mechanic and pecial elativity. When thee two theoie ae combined, a new length cale appea, built out of the ma m of the paticle: the Compton wavelength /mc. It vanihe in both fomal limit 0 and c, coeponding, epectively, to claical and Galilean theoie. Becaue of Heienbeg inequalitie, pobing ditance malle than thi length cale equie enegie highe than mc 2 and thu imply the ceation of paticle. Thi poibility to ceate and annihilate paticle fobid the localization of the oiginal paticle bette than the Compton wavelength becaue the paticle that have jut been ceated ae tictly identical to the oiginal one. Quantum mechanically, thee multi-paticle tate play a ole even when the enegy involved in the poce unde tudy i lowe than mc 2, becaue they ae ummed ove a vitual tate in petubation theoy. Thu, the divegence of petubation theoy in QFT ae diectly linked to it hot ditance tuctue, which i highly nontivial becaue it deciption involve the infinity of multi-paticle tate. Removing thee divegence ha been the nightmae and the delight of many phyicit woking in paticle phyic. It eemed hopele to the non-pecialit to undetand enomalization becaue it equied pio knowledge of quantum mechanic, elativity, electodynamic, etc. Thi tate of affai contibuted to the nobility of the ubject: tudying the ultimate contituent of matte and being incompehenible fit well togethe. Howeve, tangely at leat at fit ight the theoetical beakthough in the undetanding of enomalization beyond it algoithmic apect came fom Wilon wok on continuou phae tanition. 9 The phenomena that take place at thee tanition ae neithe quantum mechanical 10 no elativitic and ae non-tivial becaue of thei coopeative behavio, that i, thei popetie at lage ditance. 11 Thu neithe no c ae neceay fo enomalization. Something ele i at wok that doe not equie quantum mechanic, elativity, ummation ove vitual tate, Compton wavelength, etc., even if in the context of paticle phyic they ae the ingedient that make enomalization neceay. In fact, even divegence that eemed to be the majo poblem of QFT ae now conideed only a bypoduct of the way we have intepeted quantum field theoie. We know now that the inviible hand that ceate divegence in ome theoie i actually the exitence in thee theoie of a no man land in the enegy o length cale fo which coopeative phenomena can take place, moe peciely, fo which fluctuation can add up coheently. 12 In ome cae, they can detabilize the phyical pictue we wee elying on and thi manifet itelf a divegence. Renomalization, and even moe enomalization goup, i the ight way to deal with thee fluctuation. One of the aim of thi aticle i to dientangle what i pecific to field theoy and what i intinic to the enomalization poce. Theefoe, we hall not look fo a phyical model that how divegence, but we hall athe how the geneal mechanim of petubative enomalization and the enomalization goup without pecifying a phyical model. II. A TOY MODEL FOR RENORMALIZATION In the following, we conide an unpecified theoy that involve, by hypothei, only one fee paamete g 0 in tem of which a function F(x), epeenting a phyical quantity, i calculated petubatively, that i, a a powe eie. An example in QFT would be quantum electodynamic QED, which decibe the inteaction of chaged paticle uch a electon with the electomagnetic field. Fo high enegy pocee, the ma of the electon i negligible and the only paamete of thi theoy in thi enegy egime i it chage, 170 Am. J. Phy. 72 2, Febuay Ameican Aociation of Phyic Teache 170

2 which i theefoe the analog of g 0. F can then epeent the co ection of a catteing poce a, fo intance, the catteing of an electon on a heavy nucleu in which cae x i the enegy momentum fou-vecto of the electon. The coupling contant g 0 i defined by the Hamiltonian of the ytem, and F i calculated petubatively uing the uual à la Feynman appoach. Anothe impotant example i continuou phae tanition. Fo fluid, F could epeent a denity denity coelation function and fo magnetim a pin pin coelation function. 18 Yet anothe example i the olution of a diffeential equation that can aie in ome phyical context and that can how divegence ee the following. It i convenient fo what follow to aume that F(x) ha the fom: F x g 0 g 0 2 F 1 x g 0 3 F 2 x. Up to a edefinition of F, thi fom i geneal and coepond to what i eally encounteed in field theoy. Let u now aume that the petubation expanion of F(x) i illdefined and that the F i (x) ae function involving divegent quantitie. An example of uch a function i F 1 x 0 dt t x, which i logaithmically divegent at the uppe limit. Thi example ha been choen becaue it hae many common featue with divegent integal encounteed in QFT: the integal coepond to the ummation ove vitual tate and (t x) 1 epeent the pobability amplitude aociated with each of thee tate. 19 A imple although cucial obevation i that becaue thee i only one fee paamete in the theoy by hypothei, only one meauement of F(x), ay at the point x, i neceay to fully pecify the theoy we ae tudying. Such a meauement i ued to fix the value of g 0 o a to epoduce the expeimental value of F( ). Fo QED fo intance, thi pocedue would mean that: i ii iii 1 2 We tat by witing a geneal Hamiltonian compatible with baic aumption, fo example, elativity, cauality, locality, and gauge invaiance. We calculate phyical pocee at a given ode of petubation theoy. We fix the fee paamete of the initial Hamiltonian to epoduce at thi ode the expeimental data. Thi lat tep equie a much data a thee ae fee paamete. Once the paamete ae fixed, the theoy i completely detemined and thu pedictive. One could then think that it doe not matte whethe we paametize the theoy in tem of g 0, which i only ueful in intemediate calculation, o with a phyical, that i, a meaued quantity F( ), becaue g 0 will be eplaced by thi quantity anyway. Having thi feedom i indeed the geneic ituation in phyic, but the ubtlety hee i that the petubation expanion of F(x) i ingula, and, thu, o i the elationhip between g 0 and F( ). Thu, it eem cucial to epaametize F in tem of F( ) when the expanion i ill-defined. The enomalizability hypothei i that the epaametization of the theoy in tem of a phyical quantity, intead of g 0, i enough to tun the petubation expanion into a welldefined expanion. The hypothei i theefoe that the poblem doe not come fom the petubation expanion itelf, that i, fom the function F i (x), but fom the choice of paamete ued to pefom it. Thi hypothei mean that the phyical quantity, F(x), initially epeented by it illdefined expanion Eq. 1, hould have a well-defined petubation expanion once it i calculated in tem of the phyical paamete F( ). Thi i the implet hypothei we can make, becaue it amount to peeving the x-dependence of the function F i (x) and only modifying the coupling contant g 0. Thu, we aume that F(x) i known at one point, and we define g R by F g R. In the following, and by analogy with QFT, we call g R the enomalized coupling contant and Eq. 3 a enomalization peciption, a babaian name fo uch a tivial opeation. We ae now in a poition to dicu the enomalization pogam. It conit of epaametizing the petubation expanion of F o that it obey the peciption of Eq. 3. The point hee i that we cannot ue Eq. 3 togethe with Eq. 1 becaue Eq. 1 i ill-defined. We fit need to give a welldefined meaning to the petubation expanion. Thi i the egulaization pocedue which i the fit tep of any enomalization. 20,21 The idea i to define the petubation expanion of F by a limit uch that i the F i (x) ae welldefined befoe the limit i taken, and ii afte the enomalization ha been pefomed, the oiginal fomal expanion i ecoveed when the limit i taken. We thu intoduce a new et of egulaized function F and F i,, involving a new paamete, which we call the egulato, and uch that fo finite all thee function ae finite. We thu define F x F x,g 0, g 0 g 2 0 F 1, x g 3 0 F 2, x. 4 Thee ae infinitely many way of egulaizing the F i and fo the example given in Eq. 2, it can conit fo intance in intoducing a cut-off in the following integal: dt F 1, x 0 t x. 5 Diffeent egulaization cheme can lead to vey diffeent intemediate calculation, but mut all lead to identical eult. 22 Fo intance, dimenional egulaization i widely ued in QFT becaue it peeve Loentz and gauge ymmetie ,23 We do not need hee to pecify a egulaization fo the function F, becaue ou agument will be geneal and the few calculation elementay. Once a egulaization cheme ha been choen, it i poible to ue the enomalization peciption, Eq. 3, togethe with the egulaized expanion, Eq. 4, to obtain a well-defined petubation eie fo F in tem of the phyical coupling g R. If thi expanion make ene thi i the enomalizability hypothei it mut be finite even in the limit, becaue it expee a finite phyical quantity F(x) in tem of a phyical quantity g R. Thu, the enomalization pogam conit fit in changing F(x,g 0 ) to F (x,g 0, ), then in ewiting F in tem of g R and, F x,g 0, F x,g R,, Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 171

3 and only then taking the limit at fixed g R and. If thi limit exit, F (x) i by hypothei the function F(x): F x F x,g R, F x,g R,. 7 Of coue, the divegence mut till be omewhee, and we hall ee that they uvive in the elationhip between g 0 and g R ; at fixed g R, g 0 divege when. In the taditional intepetation of enomalization, thi divegence i uppoed to be hamle becaue g 0 i uppoed to be a nonphyical quantity. We hall come back to thi point late. The enomalization pogam i pefomed ecuively, and we now implement it ode by ode to ee how it wok and the containt on the petubation expanion that it implie. Let u emphaize that the eie expanion we hall ue in intemediate calculation ae highly fomal becaue they ae ill-defined in the limit. They ae jutified only by the eult we finally obtain: a good petubation expanion in tem of g R. 24 Renomalization at ode g 0. At thi ode F(x) i contant and given by F x g 0 O g Thu the ue of Eq. 3 lead to g 0 g R O g 2 R. 9 Renomalization at ode g 2 0. Ou only feedom to eliminate the divegence of F (x) i to edefine g 0. Becaue we ae woking petubatively, we expand g 0 a a powe eie in g R. Thu, we et g 0 g R 2 g 3 g, 10 whee n g O(g n R ). At ode g 2 R we obtain F x g R 2 g g R 2 F 1, x O g R 3, 11 whee we have ued g 0 2 g R 2 O(g R 3 ). If we impoe Eq. 3 at thi ode, we obtain 2 g g 2 R F 1,, 12 which divege when. In ou example, Eq. 5, we find dt 2 g g R 2 0 t g R 2 log. 13 If we ubtitute Eq. 12 into Eq. 11, we obtain F to thi ode: F x g R g 2 R F 1, x F 1, O g 3 R. 14 It i clea that thi expeion fo F (x) i finite fo all x at thi ode if and only if the divegent pat of F 1, (x) the pat that become divegent when ) i exactly canceled by that of F 1, ( ), that i, if and only if F 1, x F 1, i egula in x and fo. 15 Thi condition of coue mean that the divegent pat of F 1, (x) mut be a contant, that i, i x-independent. If thi i o, then we define the function F(x) now called enomalized a the limit of F (x) when. The condition 15 i fulfilled fo the example of Eq. 2, and we tivially find that F(x) ead: dt F x g R x g R 2 0 t x t O g R 3, 16 which i obviouly well defined and uch that the peciption of Eq. 3 i veified. We ay that we have enomalized the theoy to thi ode. Befoe going to the next ode of petubation theoy, let u note two impotant fact. Fit, the enomalization pocedue conit of adding a divegent tem 2 g to F to emove it divegence. The cancellation take place between the econd tem of it expanion and the fit one of ode g 0. Both lead to a tem of ode g R 2, the one coming fom the expanion of g 0 in tem of g R being tuned o a to cancel the divegence of the othe. Thi mechanim of cancellation i a geneal phenomenon: a divegence coming fom the nth tem of the petubation expanion i canceled by the expanion in powe of g R of the n 1 peceding tem. Second, thi cancellation i poible fo all x only if the divegence of F 1, (x) i a numbe, that i, i x-independent. If it i not o, then F 1, (x) F 1, ( ) would till be divegent x. Thi divegence would equie the impoition of at leat one moe enomalization peciption to be emoved and thi econd peciption would define a econd, independent, coupling contant ee Appendix A fo two function, one enomalizable and one that i not. The neceity fo a econd meauement of F(x) would contadict ou aumption that thee i only one fee paamete in the theoy. Thu we conclude that thi aumption datically contain the x-dependence of the divegence at ode g 0 2. We actually how in the following that thi containt popagate to any ode of petubation theoy in a nontivial way. We alo will how that togethe with dimenional analyi and fo a vey wide and impotant cla of theoie, thee containt ae ufficient to detemine the analytical fom of the divegence. Renomalization at ode g 0 3. We uppoe that F can be enomalized at ode g R 2, that i, condition 15 i fulfilled. To undetand the tuctue of the enomalization pocedue, it i neceay to go one tep futhe. At ode g R 3 we obtain F x g R 2 g 3 g g R 2 2g R 2 g F 1, x g R 3 F 2, x O g R 4, 17 whee we have ued g 0 3 g R 3 O(g R 4 ) and g 0 2 g R 2 2g R 2 g O(g R 4 ). We again impoe the peciption Eq. 3 and obtain 3 g 2g R 3 F 1, 2 g R 3 F 2,. If we ubtitute Eq. 18 in Eq. 17, we obtain F x g R g R 2 F 1, x F 1, g R 3 F 2, x F 2, 2F 1, F 1, x F 1, O g R Once again, we equie that the divegence ha been ubtacted fo all x which impoe on the x-dependence of the divegent pat of F 2, (x): F 2, x F 2, 2F 1, F 1, x F 1, i egula in x and a Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 172

4 Note that thi containt involve not only F 2,, but alo F 1,. It i convenient to ewite F 1, (x) and F 2, (x) a the um of a egula and of ingula when ) pat: F i, x F i, x F i, x. 21 Becaue anything finite, thi decompoition i not unique: the F i, (x) ae defined up to a egula pat. It i convenient to chooe F 1, (x) uch that F 1, x F 1, 0, 22 which, of coue, implie condition 15. We how in Appendix B that, ecipocally, thi choice i alway poible if 15 i fulfilled. A aleady tated, Eq. 22 mean that the divegent pat of F 1, i x-independent. We can actually impoe a moe tingent condition on F 1, becaue, by again tuning the egula pat of F 1,, we can chooe F 1, to be completely independent of x, fo any. We thu define F 1, x f In ou example, Eq. 5, we can chooe f 1 log, F 1, x log x x. 24 We now ubtitute Eq. 23 into Eq. 20 and, uing the ame kind of agument a in Appendix B, we obtain a containt on the ingula pat of F 2, (x) imila to the one on F 1, (x), Eq. 22 : F 2, x F 2, 2 f 1 F 1, x F 1, 0. Equation 25 can be ewitten a F 2, x 2 f 1 F 1, x F 2, 25 2 f 1 F 1, Equation 26 ha the ame tuctue a Eq. 22 up to the eplacement: F 1, F 2, 2 f 1 ( )F 1, and theefoe ha the ame kind of olution a Eq. 23 : F 2, x 2 f 1 F 1, x f 2, 27 whee f 2 ( ) i any function of and i independent of x. We ee in Eq. 27 that unlike F 1, (x), the divegent pat of F 2, (x) depend on x. Howeve, thi dependence i entiely detemined by the fit ode of the petubation expanion. The 2 g tem, neceay to emove the O(g 2 0 ) divegence, ha poduced at ode g 3 R an x-dependent divegent tem: 2g R 2 gf 1, (x). Thi kind of x-dependence i alo a geneal phenomenon of enomalization: the counte- tem that emove divegence at a given ode poduce divegence at highe ode. If the theoy i enomalizable, thee divegence contibute to the cancellation of divegence peent in the petubation expanion at highe ode. Thu, petubative enomalizability, that i, the poibility of eliminating ode by ode all divegence by the edefinition of the coupling, implie a pecie tuctue of the divegent pat of the ucceive tem of the petubation eie. At ode n, the ingula pat of F n, involve x-dependent tem entiely detemined by the peceding ode plu one new tem that i x-independent. In ou example of Eq. 2 and 24 we find F 2, x 2 2 log log x x f By expanding log log(x )/ x in powe of 1 and by again edefining the egula pat of F 2,, we obtain a imple fom fo F 2, (x): F 2, x 2 2 log log x f Thi elation will be impotant in the following when we hall dicu the enomalization goup. Let u daw ou fit concluion. Infinitie occu in the petubation expanion of the theoy becaue we have aumed that it wa not egulaized. Actually, thee divegence have foced u to egulaize the expanion and thu to intoduce a new cale. Once egulaization ha been pefomed, enomalization can be achieved by eliminating g 0. The limit can then be taken. The poce i ecuive and can be pefomed only if the divegence poe, ode by ode, a vey pecie tuctue. Thi tuctue ultimately expee that thee i only one coupling contant to be enomalized. Thi mean that impoing only one peciption at x i enough to ubtact the divegence fo all x. In geneal, a theoy i aid to be enomalizable if all divegence can be ecuively ubtacted by impoing a many peciption a thee ae independent paamete in the theoy. In QFT, thee ae mae, coupling contant, and the nomalization of the field. An impotant and non-tivial topic i thu to know which paamete ae independent, becaue ymmetie of the theoy like gauge ymmetie can elate diffeent paamete and Geen function. Let u once again ecall that enomalization i nothing but a epaametization in tem of the phyical quantity g R. 25 The pice to pay fo enomalizing F i that g 0 become infinite in the limit, ee Eq. 12. We again emphaize that if g 0 i believed to be no moe than a non-meauable paamete, ueful only in intemediate calculation, it i indeed of no conequence that thi quantity i infinite in the limit. That g 0 wa a divegent non-phyical quantity ha been common belief fo decade in QFT. The phyical eult given by the enomalized quantitie wee thought to be calculable only in tem of unphyical quantitie like g 0 called bae quantitie that the enomalization algoithm could only eliminate aftewad. It wa a if we had to make two mitake that compenated fo each othe: fit intoduce bae quantitie in tem of which eveything wa infinite, and then eliminate them by adding othe divegent quantitie. Undoubtly, the pocedue woked, but, to ay the leat, the intepetation eemed athe obcue. Befoe tudying the enomalization goup, let u now pecialize to a paticula cla of enomalizable theoie. III. RENORMALIZABLE THEORIES WITH DIMENSIONLESS COUPLINGS A vey impotant cla of field theoie coepond to the ituation whee g 0 i dimenionle, and x, which in QFT epeent coodinate o momenta, ha dimenion o moe geneally when g 0 and x have independent dimenion. In fou-dimenional pace time, quantum electodynamic i in thi cla, becaue the fine tuctue contant i dimenionle; quantum chomodynamic and the Weinbeg Salam 173 Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 173

5 model of electo-weak inteaction ae alo in thi cla. In fou pace dimenion, the 4 model elevant fo the Ginzbug Landau Wilon appoach to citical phenomena i in thi cla too. Thi paticula cla of enomalizable theoie i the conetone of enomalization in field theoie. Ou main goal in thi ection i to how that, independently of the undelying phyical model, dimenional analyi togethe with the enomalizability containt detemine almot entiely the tuctue of the divegence. Thi undelying implicity of the natue of the divegence explain that thee i no combinatoial miacle of Feynman diagam in QFT a it might eem at fit glance. Let u now ee in detail how it wok. Becaue F (x) ha the ame dimenion a g 0, it alo i dimenionle and o ae the F i, (x). The only poibility fo a dimenionle quantity like F to be a function of a dimenional vaiable like x i that thee exit anothe dimenional vaiable uch that F depend on x only though the atio of thee two vaiable. Apat fom x, the only othe quantity on which F depend i, which mut theefoe have the ame dimenion a x. Thi i indeed the cae in ou example, Eq. 5. Thu, the function F i, (x) depend on the atio x/ only. 26 Let u how that thi i enough to pove that the F i, (x) ae um of powe of logaithm with, fo mot of them, pecibed pefacto. Let u tat with F 1, (x). On one hand, we have een that by edefining the egula pat of F 1, (x), we could take it ingula pat F 1, (x) independent of x, Eq. 23. Onthe othe hand, we know that F 1, (x) i a function of x/. Thu, by edefining F 1, (x), it mut be poible to extact an x-dependent egula pat, (x), of thi function o a to build the x/ dependence of F 1, (x): F 1, x f x f 1 x. 30 Hence, F 1, i epaable into function of x only and of only which um up to a function of x/. We how in Appendix C the well-known fact that only the logaithm obey thi popety. We obtain ee Eq. C3 and C4 F 1, x f 1 x f 1 f 1 x log x. 31 Theefoe, fo enomalizable theoie and fo dimenionle function uch a F, only logaithmic divegence ae allowed at ode g 2 0 in QFT, thi i the o-called one-loop tem. Thi i the eaon why logaithm ae encounteed eveywhee in QFT. Note that becaue of dimenional analyi, the finite pat of F 1, (x) i nothing but (x), up to an additive contant, at leat fo. Thi can be checked fo the example given in Eq. 5. Thu, by dimenional analyi, the tuctue of the divegence detemine that of the finite pat up to a contant. Notice that thing would not be that imple if F (x) depended on anothe dimenional paamete, which i the cae of maive field theoie whee mae and momenta have the ame dimenion. In thi cae, the finite pat i only patially detemined by the ingula one. Let u now how that the tuctue of F 2, alo i entiely detemined fo enomalizable theoie with dimenionle coupling both by the enomalizability hypothei and by dimenional analyi. We have aleady patially tudied thi cae with the example given in Eq. 5 whee F 1, (x) i logaithmically divegent, a chaacteitic featue of thee enomalizable theoie. In paticula, we have hown that in thi cae, enomalizability impoe at ode g 3 0 that F 2, i of the fom given in Eq. 29. Let u now ue dimenional analyi that once again impoe that F 2, depend only on x/. The only feedom we have to econtuct a function of x/ fom the fom of F 2, given in Eq. 29 i to add a egula function to it. It i not difficult to find how to poceed becaue the only admiible tem including log log x i log 2 /x: log 2 x log2 2 log log x log 2 x. 32 Thu, to obtain the dimenionally coect extenion of the tem 2 2 log log x in Eq. 29, we extact 2 log 2 fom f 2 ( ) and add the egula tem 2 log 2 x: 2 2 log log x f log log x 2 log 2 f 2 2 log log log x 2 log 2 2 log 2 x f 2 2 log 2 2 log 2 x f 2 2 log 2. Thu, we obtain fo the new function F 2, (x): F 2, x 2 log 2 x f 2 2 log Now, fo f 2 ( ) 2 log 2, we can epeat the ame agument a the one ued peviouly fo F 1, (x) which i equal to f 1 ( ), Eq. 23 : it i a function of that mut become a function of x/ only by adding a function of x. It i thu alo a logaithm, ee Eq. 30 and 31 and Appendix C. Theefoe, we add a log x tem to F 2, (x) and obtain the final eult: F 2, x 2 log 2 x log x, 35 whee i a pue numbe. We emphaize that although it i x-independent, the tem 2 log 2 involved in F 2, (x) aie fom the log log x tem thank to dimenional analyi. It i thu entiely detemined by the tem of ode g 2 0 of petubation theoy. Only the ub-leading logaithm log /x i new. It i not difficult now to gue the tuctue of the next ode of petubation: it involve a log 3 /x with a pefacto 3,a log 2 /x tem with a pefacto which i a function of and and a log /x with a pefacto independent of and. A pecie calculation how that 174 Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 174

6 F x g 0 2 log x 2 g 0 3 log 2 x 3 g 0 4 log 3 x By ubtituting thi expeion in Eq. 36 and 39, we obtain at O(g R 4 ): g 0 3 log x 5 2 g 0 4 log 2 x F x g R g R 2 log x 2 g R 3 log 2 x 3 g R 4 log 3 x g 0 4 log x. 36 g R 3 log x 5 2 g R 4 log 2 x We have witten the eie o a to exhibit it tiangula natue: the fit line coepond to the leading logaithm, the econd to the ub-leading, etc., and the nth column to the (n 1)th ode of petubation. The leading logaithm ae entiely contolled by the g 2 0 tem, the ub-leading logaithm by both the g 2 0 and g 3 0 tem, etc. It i clea that ode by ode fo the divegent tem, only the log tem i new, all the log 2, log 3, etc., tem ae detemined by the peceding ode. Thi tuctue tongly ugget that we can, at leat patially, eum the petubation eie. We notice that although the leading logaithm fom a imple geometic eie, thi i no longe tue fo the ub-leading logaithm whee, fo intance, the facto 5 /2 of Eq. 36 i non-tivial. Thank to the enomalization goup, thee exit a ytematic way to pefom thee eummation 27 ee the following. We again emphaize that fo ou imple toy model the divegence togethe with dimenional analyi detemine almot entiely the entie function F(x) in the limit of lage. To how thi explicitly, we ewite F a F x,g 0, g 0 F x,g 0, F x,g 0, 37 with F (x,g 0, ) given by Eq. 36 at O(g 4 0 ) and F (x,g 0, ) O(g 2 0 ). Fom dimenional analyi, F (x,g 0, ) i alo a function of x/ only which, by definition, i finite when. Thu, fo lage, F x,g 0, F x,g 0 F 0,g F (x) i theefoe almot x-independent fo lage : itia (g 0 -dependent numbe in thi limit. Fo the ake of implicity, let u conide the cae whee it i vanihing: F x,g 0, g 0 F x,g 0, 39 with F (x,g 0, ) a function of x/ only. By uing the enomalization peciption, Eq. 3, we can calculate g R a a function of g 0 and / and by fomally inveting the eie, we obtain at O(g R 4 ): g 0 g R g R 2 log g 3 2 log 2 log g R 4 log 5 2 log2 3 log Fig. 1. An illutation of the enomalization goup: the two equivalent way to compoe change of paametization. g 4 R log x. 41 Thu, we find that the enomalization poce leave unchanged the functional fom of F, Eq. 36, and jut conit in eplacing (g 0, ) by (g R, ). Thi vey impotant fact i elated to a elf-imilaity popety that we tudy in detail fom the enomalization goup viewpoint. Notice that of coue any explicit dependence on and g 0 ha been eliminated in Eq. 41 and that the limit can now be afely taken, if deied. Note that we have obtained logaithmic divegence becaue we have tudied the enomalization of a dimenionle coupling contant. If g 0 wa dimenional, we would have obtained powe law divegence. Thi i fo intance what happen in QFT fo the ma tem ee alo in the following the expanion in Eq. 45. IV. RENORMALIZATION GROUP Although the enomalization goup will allow u to patially eum the petubation expanion, we hall not intoduce it in thi way. Rathe, we want to examine the intenal conitency of the enomalization pocedue. We have choen a enomalization peciption at the point x whee g R i defined. Obviouly, thi point i not pecial, and we could have choen any othe point o to paametize the theoy. Becaue thee i only one independent coupling contant, the diffeent coupling contant g R g R ( ), g R g R ( ), g R g R ( ) hould all be elated in uch a way that F(x) F(x,,g R ) F(x,,g R ) F(x,,g R ), etc. Thi mean that thee hould exit an equivalence cla of paametization of the ame theoy and that it hould not matte in pactice which element in the cla i choen. Thi independence of the phyical quantity with epect to the choice of peciption point alo mean that the change of paametization hould be a enomalization goup law: going fom the paametization given by (,g R ) to that given by (,g R ) and then to that given by (,g R ) o going diectly fom the fit paametization (,g R ) to the lat one (,g R ) hould make no diffeence, ee Fig. 1. Put thi way, thi tatement eem to be void. Actually, it i. Moe peciely, it would be o if we wee pefoming exact calculation: we would gain no new phyical infomation by implementing the enomalization goup law. Thi i becaue thi goup law doe not eflect a ymmety of the phyic, but only of the paametization of ou olution. Thi ituation i completely analogou to what happen fo the olution of a diffeential equation: we can paametize it at time t in tem of the initial condition at time t 0 fo intance, o we can ue the equation itelf to calculate the olution at an intemediate time and then ue thi olution 175 Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 175

7 a a new initial condition to paametize the olution at time t. The change of initial condition that peeve the final olution can be compoed thank to a goup law. Let u conide, fo example, the following tivial, but illuminating, example: ẏ t y t, y t 0 0, 42 the olution of which i y t f 0,t t 0 0 e (t t 0 ). 43 The goup law can be witten a 28 f 0,t t 0 f f 0, t 0,t, 44 which you can veify uing the exact olution, Eq. 43. The non-tivial point with thee goup law i that, in geneal, they ae violated at any finite ode of the petubation expanion. In ou peviou example, we obtain to ode, y t f 1 0,t t t t 0, 45 and f 1 f 1 0, t 0,t 0 1 t t t t The goup law i veified to ode becaue the petubation expanion i exact at thi ode. Howeve, it i violated by a tem of ode 2 that can be abitaily lage even fo mall, povided t t 0 i lage enough. The inteet of the goup law, Eq. 44, i that it i poible to enfoce it and then to impove the petubation eult. Actually, when enomalization i neceay, the goup law let u patially eum the petubation eie of divegent tem. Let u now ee how thi impovement of the petubation eie wok fo the example of the diffeential equation 42. In thi cae, the divegence occu fo t 0. Thu, t 0 play the ole of the cut-off, t t 0 of log /, and t of log /. Once t 0 i finite, no divegence emain, but the elic of the divegence occuing fo t 0 ae the lage violation of the goup law becaue both the divegence and thee violation oiginate in the fact that the petubation expanion i pefomed in powe of (t t 0 ) and not of. To futhe tudy the elevance of the goup law, it i inteeting to foget the highe ode tem of the petubation expanion fo a while and to look fo an impoved appoximation that coincide at ode with the petubation eult and that obey the goup law at ode 2 : f imp 1 0,t t t t 0 2 G t t By impoing the goup law, Eq. 44, to ode 2, we obtain a functional equation fo G: G t t 0 G t 0 G t t 0 t. 48 If we diffeentiate Eq. 48 with epect to t 0 and take t 0, we obtain, etting x t, G x x G Becaue G(0) 0, Eq. 49 implie that G x x2 ax, 50 2 whee a i abitay. Fo a 0, thi eult i actually the petubation eult to ode 2 becaue y t 0 1 t t t t 0 2 O Thu, the fit ode in the petubation expanion, togethe with the goup law, detemine entiely the tem of highet degee in t t 0 at the next ode. Of coue, to veify exactly the goup law, we hould puue the expanion in to all ode. It i eay to how that to ode n, the tem of highet degee in t t 0 i completely detemined by both the fitode eult and the goup law and coincide with the petubation eult: n (t t 0 ) n /n!. Thu, the only infomation given by the petubation expanion i that all ubdominant tem, n (t t 0 ) p with p n, vanih in thi example. We could now how how the implementation of the goup law let u eum the petubation expanion. Unfotunately, thi example i too imple and ome impotant featue of the enomalization goup ae mied in thi cae. See Appendix E fo a complete dicuion of the implementation of the enomalization goup on thi example. We theefoe go back to ou toy model fo which we pecialize to enomalizable theoie with dimenionle coupling. Renomalization goup fo enomalizable theoie with dimenionle coupling. We now econide ou toy model, Eq. 4, 36, and 37, fom the point of view of the enomalization goup. Fo the ake of implicity, we keep only the dominant tem at each ode, that i, apat fom g 0, the divegent one in Eq. 39. Fit, notice that in the ame way g R i clealy aociated with the cale, Eq. 3, oig 0 with the cale becaue fom Eq. 36, wefind 29 F x g Let u define a thid coupling contant aociated with the cale, F g R, 53 and tudy the elationhip between thee diffeent coupling contant at ode g 2 0. Fom F x,g 0, g 0 g 0 2 log x O g 0 3, we obtain g R g 0 g 0 2 log O g 0 3, g R g 0 g 0 2 log O g 0 3. By eliminating g 0 between thee two equation, we find g R g R g R 2 log O g R 3, and thu, a expected, the goup law contolling the change of peciption point i veified petubatively. We note that the eential ingedient fo thi compoition law i that Eq. 57 i independent of. Thi i what enue that the ame fom can be ued to change (g 0, ) into (g R, ) and then (g R, ) into (g R, ). Thi independence, in tun, i nothing but the ignatue of petubative enomalizability which let u completely eliminate at each ode (g 0, ) fo (g R, ). Petubatively, eveything look fine. Howeve, the peviou calculation elie on a fomal tep that i not mathematically 176 Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 176

8 coect, at leat fo lage. Indeed, to go fom Eq. 56 to Eq. 57, the eie g R g R (g 0 ) mut be inveted to find g 0 g 0 (g R ) while, fo, the eie g R g R (g 0 ) i clealy not convegent and thu not invetible. Thu, the neglected tem of ode g 3 R in Eq. 57 involve a tem popotional to log / log / analogou to the tem (t )( t 0 ) of Eq. 46 and 48 which i neglected becaue it i of ode g 3 R, but which i vey lage fo lage ee Appendix D. Fom a pactical point of view, the exitence at any ode of thee lage tem of highe ode poil the goup law o that the independence of the phyical eult with epect to the choice of peciption point i not veified. A in the cae of the diffeential equation 47, wecan look fo an impoved function: F imp, F imp x,g 0, g 0 g 2 0 log x g 3 0 G x O g 4 0, 58 fo which the goup law at ode g 0 3 i obeyed. It i hown in Appendix D that thi containt implie that G x 2 log 2 x log x, whee i abitay. Thu, F imp x,g 0, g 0 g 0 2 log x 2 g 0 3 log 2 x g 0 3 log x O g Once again, we find that the goup law togethe with the ode g 0 2 eult detemine the leading behavio at the next ode, hee the log 2 ( /x) tem. Moeove, we find that the goup law impoe the exitence of the ame log 2 tem a the one found fom the enomalizability containt, Eq. 35 and 36, and allow the exitence of a ub-leading logaithm. Although nontivial, thi hould not be too upiing becaue the enomalizability containt mean that once F i well defined at x, it alo i eveywhee and in paticula at x. The enomalizability containt i theefoe cetainly neceay fo the implementation of the goup law. A in the example of the diffeential equation, Eq. 42, we hould puue the expanion to all ode to obtain an exactly veified goup law. It i clea that by doing o, we would find the ame expanion a the one obtained fom the enomalizability containt. Thu, if we ue petubation theoy to calculate the coefficient in font of the fit leading logaithm of ode g 0 2 ) and impoe the goup law, we hould be able to eum all the leading logaithm. To do the eummation of the ub-leading and ub-ub-leading logaithm, a knowledge of, epectively, the ode g 0 3 and g 0 4 tem i equied. Clealy, we need to undetand how to ytematically contuct the function f giving g R in tem of g R and /, 27 g R f g R,, 61 uch that it expanion at ode n i given by the nth ode of petubation theoy, the goup law i exactly veified: f g R, f f g R,,. 62 The function f i then aid to be the elf-imila appoximation at ode n of the exact elationhip between g R and g R. 30 Fit notice one cucial thing. Ou fit aim wa to tudy the petubation expanion of a function F in a powe eie of a coupling contant g 0. Then we have dicoveed that the logaithmic divegence at ode g 2 0 popagate to all ode o that the expanion i actually pefomed in g 0 log / intead of g 0. Becaue i the egulato, it i uppoed to be vey lage compaed with, o that the lage logaithmic tem invalidate the ue of the petubation expanion. Recipocally, it i clea that petubation theoy i pefectly valid if it i pefomed between two cale 1 and 2 which ae vey cloe. Thu, intead of uing petubation theoy to make a big jump between two vey ditinct cale, ay and, we hould ue it to pefom a eie of vey little tep fo which it i valid at each of them. In geometical tem, the fact that the petubative appoach i valid only between two vey cloe cale mean that we hould not ue petubation theoy to appoximate the equation of the cuve given by the function f, Eq. 61, that join the point (,g R ) and (,g R ), but we hould ue it to calculate the field of tangent vecto to thi cuve, that i, it envelope. The cuve itelf hould then be econtucted by integation, ee Appendix E. By doing o, the goup law will be automatically veified becaue, by contuction, the integation peciely conit in compoing infiniteimal change of epaametization infinitely many time. Let u conide again Eq. 55. We want to calculate the evolution of g R ( ) with fo a given model pecified by (,g 0 ). Thu we define g R g R, 63 g 0, which give the infiniteimal evolution of the coupling contant with the cale fo the model coeponding to (g 0, ). We tivially find to thi ode fom Eq. 55, g R g 2 0 O g 3 0, 64 and thu, by tivially inveting the eie of Eq. 55, we obtain g R g 2 R O g 3 R. 65 Now, if we integate Eq. 63 togethe with Eq. 65, we obtain g R g R 1 g R log. 66 Thi elation ha eveal inteeting popetie. i When expanded to ode g R 2, the petubation eult to thi ode i ecoveed, Eq. 57. Thi i quite nomal becaue (g R ) ha been calculated to thi ode. ii When expanded to all ode, the whole eie of leading logaithm i ecoveed. Thi i moe inteeting becaue (g R ) ha been calculated only to ode g 2, but imply mean that all the leading logaithm ae detemined by the fit one. iii The goup law 62 i obeyed exactly. We have thu found the function f of Eq. 61 to thi ode. It i vey 177 Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 177

9 intuctive to check the goup law diectly fom Eq. 66 and to veify that the function found in Eq. 65 i not modified if we add the leading logaithmic tem of ode g 0 3 to elation 55 : g R g 0 g 0 2 log 2 g 0 3 log 2 O g The independence of the function with epect to the addition of the ucceive leading logaithmic tem mean that thi function i indeed the ight object to build elfimila appoximation out of the petubation expanion. Let u now etun to the function itelf. Fit, we have calculated the logaithmic deivative g R / intead of the odinay deivative with epect to becaue we wanted to have a dimenionle function. Second, even the dimenionle quantity, (g R ), could have depended on /. Howeve, the evolution of g R ( ) between and d cannot depend in petubation theoy on becaue the theoy i petubatively enomalizable: the petubative elation between g R ( ) and g R ( ) depend only on and and not on. Thu, being dimenionle, the function cannot depend on alone and i thu only a function of g R. Thi popety i geneal fo any enomalizable theoy: in the pace of coupling contant, the function i alway a local function. Thid, the function i the function to be expanded in petubation theoy becaue it i given by a tue eie in g R and not in g R log /. Thi i clea fo ou example, Eq. 65, whee thee i no logaithm, and can be poven fomally by the following agument. If we ue Eq. 61 and 63, we find that g R f y g R,y y 1. If f i a double eie in g and in log( / ), f g R, n,p n,p g R n log p, it i clea fom Eq. 68 that only tem with p 1 contibute to (g), with the logaithm eplaced by 1. Thu we immediately deduce fom thi agument and fom Eq. 36 that g R g 2 R g 3 R g 4 R O g 5 R. 70 It i eay to check that the fit two coefficient, and, ae univeal in the ene that fo two diffeent theoie, paametized by (g R, ) and (g R, ), the two function have the ame fit two coefficient in thei expanion. Thi method of computing the function alo let u bypa the tange way to calculate it that we have ued in Eq. 64 and 65, whee we have fit expeed g R in tem of g 0 to calculate (g R ) a a function of g 0 and then, by inveion of the eie, e-obtained a function of g R. Thee two tep ae a pioi dangeou becaue they both involve lage logaithm. Actually, they alway cancel each othe. Thi can be een diectly fo the example of Eq. 67 and the eaon fo thi cancellation come fom Eq. 68 and 69, which how that no inveion of eie i needed to calculate (g R ). Thee i no miacle hee, becaue only the behavio at y / 1, which of coue doe not involve, matte. Finally, we mention that the integation of the function at O(g 3 R ) analogou to a two-loop eult in QFT lead to an implicit equation fo g R that genealize Eq. 66 : 1 g R 1 g R log g R g R g R g R log. 71 Thee i no imple olution of thi tancendental equation. It i howeve poible to obtain an iteative olution that i valid if the O(g R 3 ) tem i mall compaed with the O(g R 2 ) one, that i, if g R / 1. It i obtained by eplacing g R in the thid tem of Eq. 71 by it expeion obtained to ode g R 2, Eq. 66 : g R g R 1 g R log g R log 1 g R log. 72 It i eay to check that Eq. 72 eum exactly all the leading and ub-leading logaithm of the petubation expanion Eq. 41. Note that contay to the one-loop eult, Eq. 66, which eum only the leading logaithm, the exact expeion in Eq. 71 contibute alo to the ub-ub-leading logaithm a well a the ub-ub-ub-leading one and o on and o foth. V. SUMMARY 1 The long way of enomalization tat with a theoy depending on only one paamete g 0, which i the mall paamete in which petubation eie ae expanded. In paticle phyic, thi paamete i in geneal a coupling contant like an electic chage involved in a Hamiltonian moe peciely the fine tuctue contant fo electodynamic. Thi paamete i alo the fit ode contibution of a phyical quantity F. In paticle/tatitical phyic, F i a Geen/ coelation function. The fit ode of petubation theoy neglect fluctuation quantum o tatitical and thu coepond to the claical/mean field appoximation. The paamete g 0 alo i to thi ode a meauable quantity becaue it i given by a Geen function. Thu, it i natual to intepet it a the unique and phyical coupling contant of the poblem. If, a we uppoe in the following, g 0 i dimenionle, o i F. Moeove, if x i dimenional it epeent momenta in QFT it i natual that F doe not depend on it a i found in the claical theoy, that i, at fit ode of the petubation expanion. 2 If F doe depend on x, a we uppoe it doe at econd ode of petubation theoy, it mut depend on anothe dimenional paamete,, though the atio of x and. Ifwe have not included thi paamete fom the beginning in the model, the x-dependent tem ae eithe vanihing, which i what happen at fit ode, o infinite a they ae at econd and highe ode. Thi i the vey oigin of divegence fom the technical point of view. 3 Thee divegence equie that we egulaize F. Thi equiement, in tun, equie the intoduction of the cale that wa miing. In the context of field theoy, the divegence occu in Feynman diagam fo high momenta, that i, at hot ditance. The cut-off uppee the fluctuation at hot ditance compaed with 1. In tatitical phyic, thi cale, although intoduced fo fomal eaon, ha a natual intepetation becaue the theoie ae alway effective theoie built at a given micocopic cale. It coepond in geneal to the ange of inteaction of the contituent of the model, fo example, a lattice pacing fo pin, the aveage intemolecula ditance fo fluid. In paticle 178 Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 178

10 phyic, thing ae le imple. At leat pychologically. It wa indeed natual in the ealy day of quantum electodynamic to think that thi theoy wa fundamental, that i, not deived fom a moe fundamental theoy. Moe peciely, it wa believed that QED had to be mathematically intenally conitent, even if in the eal wold new phyic had to occu at highe enegie. Thu, the egulato cale wa intoduced only a a tick to pefom intemediate calculation. The limit wa uppoed to be the ight way to eliminate thi unwanted cale, which anyway eemed to have no intepetation. We hall ee in the following that the community now intepet the enomalization poce diffeently. 4 Once the theoy i egulaized, F can be a nontivial function of x. The pice i that diffeent value of x now coepond to diffeent value of the coupling contant defined a the value of F fo thee x). Actually, it no longe make ene to peak of a coupling contant in itelf. The only meaningful concept i the pai (,g R ( )) of coupling contant at a given cale. The elevant quetion now i, What ae the phyical eaon in paticle/tatitical phyic that make the coupling contant depend on the cale while they ae contant in the claical/mean field appoximation? A mentioned, fo paticle phyic, the anwe i the exitence of new quantum fluctuation coeponding to the poibility of ceating and annihilating paticle at enegie highe than mc 2. What wa cale independent in the claical theoy become cale dependent in the quantum theoy becaue, a the available enegy inceae, moe and moe paticle can be ceated. The pai of vitual paticle uounding an electon ae polaized by it peence and thu ceen it chage. A a conequence, the chage of an electon depend on the ditance o equivalently the enegy at which it i pobed, at leat fo ditance malle than the Compton wavelength. Note that the enegy cale mc 2 hould not be confued with the cut-off cale. mc 2 i the enegy cale above which quantum fluctuation tat to play a ignificant ole while i the cale whee they ae cut-off. Thu, although the Compton wavelength i a hot ditance cale fo the claical theoy, it i a long ditance cale fo QFT, the hot one being 1. Thee ae thu thee domain of length cale in QFT: above the Compton wavelength whee the theoy behave claically up to mall quantum coection coming fom high enegy vitual pocee, between the Compton wavelength and the cut-off cale 1 whee the elativitic and quantum fluctuation play a geat ole, and below 1 whee a new, moe fundamental theoy ha to be invoked. 12 In tatitical phyic, the analog of the Compton wavelength i the coelation length which i a meaue of the ditance at which two micocopic contituent of the ytem ae able to influence each othe though themal fluctuation. 31 Fo the Iing model, fo intance, the coelation length away fom the citical point i the ode of the lattice pacing and the coection to the mean-field appoximation due to fluctuation ae mall. Unlike paticle phyic whee the mae and theefoe the Compton wavelength ae fixed, the coelation length in tatitical mechanic can be tuned by vaying the tempeatue. Nea the citical tempeatue whee the phae tanition take place, the coelation length become extemely lage and fluctuation on all length cale between the micocopic cale of ode 1, a lattice pacing, and the coelation length add up to modify the mean-field behavio ee Ref. 32, 33 and alo Ref. 34 fo a bibliogaphy on thi ubject. We ee hee a key to the elevance of enomalization: two vey diffeent cale mut exit between which a nontivial dynamic quantum o tatitical in ou example can develop. Thi ituation i a pioi athe unnatual a can be een fo phae tanition, whee a fine tuning of tempeatue mut be implemented to obtain coelation length much lage than the micocopic cale. Mot of the time, phyical ytem have an intinic cale of time, enegy, length, etc. and all the othe elevant cale of the poblem ae of the ame ode. All phenomena occuing at vey diffeent cale ae thu almot completely uppeed. The exitence of a unique elevant cale i one of the eaon why enomalization i not neceay in mot phyical theoie. In QFT it i mandatoy becaue the mae of the known paticle ae much malle than a hypothetical cut-off cale, till to be dicoveed, whee new phyic hould take place. Thi i a athe unnatual ituation, becaue, contay to phae tanition, thee i no analog of a tempeatue that could be finetuned to ceate a lage plitting of enegy, that i, ma, cale. The quetion of natualne of the model we have at peent in paticle phyic i till lagely open, although thee ha been much effot in thi diection uing upeymmety. 5 The claical theoy i valid down to the Compton/ coelation length, but cannot be continued naively beyond thi cale; othewie, when mixed with the quantum fomalim, it poduce divegence. Actually, it i known in QFT that the field hould be conideed a ditibution and not a odinay function. The need fo conideing ditibution come fom the nontivial tuctue of the theoy at vey hot length cale whee fluctuation ae vey impotant. At hot ditance, function ae not ufficient to decibe the field tate, which i not mooth but ough, and ditibution ae neceay. Renomalizing the theoy conit actually in building, ode by ode, the coect ditibutional continuation of the claical theoy. The fluctuation ae then coectly taken into account and depend on the cale at which the theoy i pobed: thi nontivial cale dependence can only be taken into account theoetically though the dependence of the analog of the function F with x and thu of the coupling with the cale. 6 If the theoy i petubatively enomalizable, the pai (,g( )) fom an equivalence cla of paametization of the theoy. The change of paametization fom (,g( )) to (,g( )), called a enomalization goup tanfomation, i then pefomed by a law which i elf-imila, that i, uch that it can be iteated eveal time while being fom-invaiant. 27,30 Thi law i obtained by the integation of g R g R. 73 g 0, Thi function ha a tue petubation expanion in tem of g R unlike the petubative elation between g R ( ) and g R ( ) which involve logaithm of / that can be lage. The integation of Eq. 73 patially eum the petubation eie and i thu emi-nonpetubative even if (g R ) ha been calculated petubatively. The elf-imila natue of the goup law i encoded in the fact that (g R ) i independent of. 5 In paticle phyic, the function give the evolution of the tength of the inteaction a the enegy at which it i pobed vaie and the integation of the function eum patially the petubation expanion. Fit, a the enegy inceae, the coupling contant can deceae and eventually 179 Am. J. Phy., Vol. 72, No. 2, Febuay 2004 Betand Delamotte 179