ZETA REGULARIZATION METHOD APPLIED TO THE CALCULATION OF DIVERGENT DIVERGENT INTEGRALS

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1 ZETA REGULARIZATION METOD APPLIED TO TE CALCULATION OF DIVERGENT DIVERGENT INTEGRALS Jose Jver Grc Moret Grdute studet of Physcs t the UPV/EU (Uversty of Bsque coutry) I Sold Stte Physcs Addres: Prctctes Ad y Grjlb 6 G P.O Portuglete Vzcy (Sp) Phoe: () E-ml: josegrc@yhoo.es MSC: 97M5, 4G99, 4G, 8T55 s ABSTRACT: We study geerlzto of the zet regulrzto method ppled to the s cse of the regulrzto of dverget tegrls for postve s, usg the Euler Mclur summto formul, we mge to epress dverget tegrl term of ler combto of dverget seres, these seres c be regulrzed usg the Rem Zet fucto ζ ( s) s >, the cse of the pole t s= we use property Γ ' of the Fuctol determt to obt the regulrzto = ( ), wth ( ) Γ the d of the Luret seres oe d severl vrbles we c eted zet regulrzto to the cses of tegrls f ( ), we beleve ths method c be of terest the regulrzto of the dverget UV tegrls Qutum Feld theory sce our method would ot hve the problems of the Alytc regulrzto or dmesol regulrzto Keywords: = Rem Zet fucto, Fuctol determt, Zet regulrzto, dverget seres. ZETA REGULARIZATION FOR DIVERGENT INTEGRALS: Sometmes mthemtcs d physcs, we must evlute dverget seres of the form k, of course ths seres s dverget ules Re (k) >, however cses lke k= or k=3 pper severl clcultos of strg theory d Csmr effect, for the cse of 3 Csmr effect [3] the result = ppers to gve the correct result for the

2 Fc cπ Csmr force = h here A s the re d d the seprto betwee the 4 A 4 pltes, c d h re the speed of lgth d the Plck s costt. The de behd the Zet regulrzto method s to tke for grted tht for every s the detty s = ζ ( s), follows lthough ths formul s vld just for Re (s) >, to eted the defto of the Rem Zet fucto to egtve rel umbers, oe eed to use the fuctol equto for the Rem fucto s π s ζ ( s) = ( π ) Γ( s)cos ζ ( s) π Γ( s) Γ( s) = () s( π s) Ths gves the epressos t s=, the rmoc seres fte vlue =, = d = due to the pole s NOT zet regulrzble, lthough t c be gve = γ = , ths vlue c be justfed by usg the theory of Zet-regulrzed fte products (determts), s we shll see lter the pper o Zet regulrzto for dverget tegrls: m s Let be f ( ) = wth Re(m-s) < -, the the Euler-Mclur summto formul (see [] d pped A formul (A.) )for ths fucto reds m s = ζ ( s m) m s m s m s m s = r m r s ( m r s) r= B Γ( m s ) ( r)! Γ( m r s) N () ere formul () ll the seres d tegrls re coverget, formul () s usully k k worthless, sce t s trvl to prove tht = for Re(k) >,d the Rem k zet fucto ζ ( ) = m s =, so othg ew c be obted from (), the de s m s to use the Fuctol equto () for the Rem d Zet fucto to eted the defto of equto () to the whole comple ple ecept s=, cse (m-s) s postve there wll be o pole t =, so we c put = d tke the lmt s m B m!( m r ) ( r)!( m r )! m m r m r = ζ ( m) (3) r= Formul (3) s the Alytc cotuto of formul () wth = d c be used to obt fte defto for otherwse dverget tegrls,ppretly ths recurrece

3 equto hs fte umber of terms but the Gmm fucto hs pole t = d t beg some egtve teger, lso f the fucto s of the form f ( ) = α wth α > we c us the Euler-Mclur formul drectly, sce the sum f ( ) f ( ) d ts dervtves re fte. some emples of formul (3) I = ζ () = = ζ ( ) = I I I B I = ζ ( ) I = (4) 3 B 3 I3 = ( I ζ ( ) ) I ζ ( 3) B3I = So our method c provde fte regulrzto to dverget tegrls, wth the Ad of the zet regulrzto lgorthm. Also our formule () (3) d (4) re cosstet wth the usul summto propertes, fct f Λ m s fte for fte Λ d we use the property of the Rem d urwtz Zet fucto [ ] to get the sum of the k-th powers of o the tervl [, Λ ] Λ = m = ζ ( m) ζ ( m, Λ), ζ ( s, Λ ) = ( Λ) s defed for Re(s) > (of course for postve s s Λ the secod term goes to ) m B m!( m r ) ( r)!( m r )! Λ Λ Λ m m r m r = ζ ( m) ζ ( m, Λ) (5) r= B For teger m (, ) ( ) m ζ m = we fd the Beroull Polyomls, the powers m Λ m m Λ of Λ would ccel the tegrl =, so the ed formul (5) we would m B get the usul defto of Zet regulrzto ( ) () m ζ m = for teger m. Of m course oe could rgue tht smpler regulrzto of the dverget tegrls should s s be I( s) = ( ) = d I( ) = ( ) = log s, ths s just droppg out the term proportol to log s or sde the tegrl to mke t fte, however f w plugged ths result to the Euler-Mclur summto formule () (3) or (5) the terms volvg would ccel d we would flly fd tht ζ ( m) = for every m whch clerly s gst the defto of zet regulrzto of seres, for the cse of the logrhmc dvergece, obted from dfferetto wth respect to the eterl 3

4 prmeter ths s result of tkg the fte prt of the tegrl,whch ppretly m s k works. For the cse of the tegrls log ( ), we c smply dfferette k- tmes wth respect to regultor s sde () order to obt fte vlues terms of ζ ( s) d ζ '( s) for egtve vlues of s uless m=- (for other egtve vlues of m we c mke chge of vrble q = ), ths s treted the et secto o Zet-regulrzed determts d the rmoc seres: λ = Gve opertor A wth fte set of ozero Egevlues { } Zet fucto d Zet-regulrzed determt, Voros [] we c defe s { } Tr A = ζ A( s) = λ s dζ A det( A) ep () = λ = ds (6) The proof of the secod formul sde (6) s pretty esy, the dervtve of the log λ Geerlzed zet fucto wll be ζ A '( s) = s ow let s=, use the property of λ the logrthm log(. b) = log logb d tke the epoetl o both sdes. For the cse of the Egevlues of smple Qutum rmoc osclltor oe dmeso [ ] λ =, the Zet fucto s just the urwtz Zet fucto, so we c defe zet-regulrzed fte product the form dζ ( ) ep (, ) = ds dζ ( ) (, ) log ( ) log π ds = Γ (7) I cse we put = we fd the zet-regulrzed product of ll the turl umbers ( ) = π,see [5] f we tke the dervtve wth respect to, we would fd the sme regulrzed Vlue Rmuj dd [] precsely Γ ' = ( ) > ( ) Γ rmoc seres pper due to logrthmc dvergece of the tegrl ( ), f we put m= - sde formul (), usg regultor s, s we hve the Euler Mclur summto formul (8) r B r = s s r s ( ) ( ) r= ( r)! u ( ) = Sce s > the tegrl d the seres sde (8) wll be coverget, ow we c tegrte s over sde (8) d use the defto of the logrthm lm log =, to s s 4

5 regulrze the tegrl s s terms of the fucto ( )s some fte correctos due to the Euler-Mclur summto formul. Γ ' ( ) Γ plus A fster method s just smple dfferette wth respect to sde the tegrl di ( ) =, ow ths tegrl s coverget for every d equl to d, tegrto over g gves the vlue log c plus costt c tht wll ot deped o the vlue of sde the tegrl questo, the proof tht c s uque o mtter wht s comes from the fct tht the dfferece log b =. b dζ (,) For the cse =, the dervtve of the urwtz Zet s = log ( π ) so f ds we ppromte the dverget tegrl by seres, the we c get the regulrzed result =. Appretly t seems tht usg two dfferet regulrztos we get some dfferet results, the de s tht f we use the Strlg symptotc formul ppromto for the logrthm of the Zet fucto B z log Γ ( z) = z log z z log ( π ) r (9) r= r ( r ) If we tke the dervtve wth respect to z sde (9), s ow more ppret tht for µ the logrthmc dervtve log here c = log µ s costt obted from dfferetto wth respect to to regulrze the dverget tegrl, ths costt c must be relted to some physcl costt or cse the qutty hs dmeso of Eergy the µ must hve lso dmesos of eergy so the logrthm s dmesoless, ths costt c would be the oly free djustble prmeter tht would pper sde our clcultos to regulrze tegrls. If s egtve there s etr term due to the vlue log( ) = π, for more comple logrthmc tegrl oe c use the defto k log ( ) log k µ wth the sme eergy scle c = log µ k So f we tke formul () d the detty sde [] ζ ( s, ) s C \ { } s ( ) = reg Γ '( ) s = Γ( ) the we c regulrze y dverget tegrl ( ) m to get fte result by lytc cotuto. Ths result s justfed by tkg the lmt s sde the formul for the Dgmm fucto ( s) s Ψ ( z) = ( ) Γ ( s ) ζ ( s, z) 5

6 o Regulrzto of dverget tegrls f ( ) : I geerl, the dverget tegrls tht pper Qutum Feld Theory [ ] re vrt 4 uder rottos, for emple 4 d p p m or d p ( p q) m p, f we use 4- ( ) ( ) dmesol polr coordtes we c reduce these tegrls to the cse d / π d drf ( r) r Γ d / the the UV dvergeces pper whe r, here d=4 s the ( ) dmeso of the spcetme, depedg o the vlue of d we c hve severl types of dvergeces Λ d m drf ( r) r Λ blog Λ, f b = for m = the UV dvergeces re qudrtc f m = the dvergeces re ler, cse = d b = the dvergeces re 4 of logrthmc type, for emple d p p m hs oly logrthmc dvergece ( ) dmeso 4, for lower vlue of the dmeso (d=3 ) ths tegrl ests. To study the rte of dvergece, we c epd the fucto to Luret seres vld k for z, f ( ) = c ( ) k s fte umber d mes tht the fucto f ( ) hs power lw dvergece for bg, the the de to compute dverget tegrl would be ths, we dd d substrct Polyoml plus term proportol to to splt the tegrl to fte prt d other dverget tegrl, both cses we must lso troduce regultor ( ) s for turl umber so we mke the tegrls coverget for some Re(s) > b k k s f ( ) b ( ) b ( ) b s s ( ) = ( ) () Also we c use the chge of vrble ( ), so the ew lmts of tegrto would be (, ), sce s turl umber, the the followg detty m s m s m s ( ) = = ζ ( m s) holds for every postve d m the sese of zet regulrzed seres. Of course sde () our substrcto we c clude o-teger powers of sce the recurso formul () s stll vld for them. The umber of terms k s chose so the frst tegrl s FINITE, ths frst tegrl c be computed by Numercl or ect methods d yelds to fte vlue, the rest of the tegrls re just the logrthmc d power-lw dvergeces, they c be regulrzed wth the d of formule () (3) (4) (6) (8) to get fte vlue volvg ler 6

7 ζ (,) combto of ζ ( m) m=,,,...,k d other vlue proportol to s µ or log for emple we c lyze ths smple dverget tegrl > s ζ () = Γ B Γ r ' r ( ) ζ ( ) r r= ( r)! u = s () The frst tegrl () s coverget d hve ect vlue of log, order Γ ' to regulrze the logrthmc tegrl we hve used the result = ( ) plus ( ) Γ the Euler-Mclur summto formul. The mthemtcl justfcto of ths s the followg, gve dverget tegrl f ( ) we troduce regultor F( s) = f ( ) s so the tegrl F(s) ests for some bg s, f we dd d substrct powers of the form k s for teger k d ( ) s, we c splt F(s) to coverget tegrl I (s) vld for s d some dverget tegrls of the form ( ) m s d s, usg formule () (3) (4) d (8) we c epress these tegrls terms of the seres ( ) s d ( ) s, whch wll be coverget for m Re( s m) > d Re( s ) >, ow usg the Fuctol equto for the urwtz d Rem Zet fucto we c mke the lytc cotuto of both seres to s vodg the pole t s= by the use of Rem Zet fucto t egtve Γ ' tegers ζ ( ) plus some correctos volvg ( ) of course the rules for chge Γ of vrble d stll vld so ( ) s s f ( ) = duf ( u) u ths c be used to vod some IR dvergeces t = by splttg the tegrl to IR dverget prt d UV dverget prt du = du du. For other types of dverget tegrls lke β α log ( ) for postve α d β oe could dfferette wth respect to m or s sde formul () order to obt recurrece equto for the tegrls 7

8 β α log ( ), ths recurrece equto s fte (ppromtely) sce for Re ( p ) > β log ( ) p s fte d do ot eed to be regulrzed provded >. Other useful dettes c be ( ) / or the epso of the logrthm vld for y.4 > log = to mke logrthms more trctble, lso we could use Luret epsos to hdle complcte o-polyoml epressos lke ( µ ) k by epdg t for bg to symptotc (verse) power seres. REGULARIZATION FOR IR DIVERGENT INTEGRALS: If the tegrd hs pole t = so s IR dverget tegrl, we c mke the Tylor substrcto λ f ( ) b ( ) λ = λ b ( λ ) ( ) b = ( f ) ( )! () For the cse λ = we hve ( ) = = ( ) ( ) ( ) ( ) ( ) (3) To vod the pole t = we sert smll comple qutty Epslo so ε, the formul () c be grte = ( ) ( ε ) ( ε ) ( ε ) (4) The sde (4) we hve oly UV logrthmc dvergece, for λ > there re oly IR dvergeces, f we use the Boml theorem ( ) λ ( ε ) λ k λ k = λ k k λ = ( ( ε ) ) (5) Itegrls sde (4) d (5) c be evluted wth the epresso 8

9 m m r m r Γ ( ) π ( ) = = I(, m, r) r ( ( ε ) ) mπ π m s ( r )! Γ 3 r ε (6) m o Regulrzto of tegrls m : There re cses, for m= we hve ( ) = = ( u) Where we hve mde the chge of vrble u = sde the lst tegrl to tur the IR dvergece to Logrthmc UV dvergece. For m > we troduce regultor s to mke the IR dverget tegrl to coverge = = m s m s m s m s / duu m s u = (7) The lst tegrl sde (7) s dverget the lmt s d c be regulrzd wth λ λ λ formul (). If my IR dvergeces pper ( ).( )...( ) m m we must perform the Tylor substrcto () o ech pot ( ) REGULARIZATION OF MULTIPLE INTEGRALS: I geerl QFT there wll be lso mult-loop (multple) tegrls so we wll lso hve to regulrze tegrls the form. I( s) = d q d q... d q F( q, q,..., q ) (8) = ( q ) ere we hve troduced regultor depedg o eterl prmeter s order the tegrl (8) to coverge for bg s d the use the lytc regulrzto to tke the lmt s, ths regultor must be chose wth cre order ot to spol y symmetres of the Physcl system. Aother method s to cosder the multple tegrl s terte tegrl d the mke the substrcto for every vrble for emple 9

10 k k (,,..., ) (,..., )( ) (,..., )( ) = = q F q q q q q q q q q q (9) The symbol q mes tht the tegrl s mde over the vrble q keepg the other vrbles costt, the umber k s chose so the frst tegrl s fte, ths tegrl wll deped o I( q,..., q ), the dverget tegrls (eve for the logrthmc cse =-) c be regulrzed. Now we hve regulrzed the frst tegrl, we hve reduced oe vrble the multple tegrl, repetg the tertve process for the fuctos ( q,..., q ) k k q ( q..., q) b j( q,..., q )( q ) q b( q,..., q )( q ) j= j= j () Usg (9) d () o every vrble we c reduce the dmeso of the tegrl utl we rech to the oe dmesol cse, whch s eser to hdle. If the tegrd F( q, q,..., q ) hd o sgulrtes for every q j >, we my epd ths tegrd to multple Luret seres of severl vrbles, d the s, s,..., s m m m m, m,..., m m, m,..., m perform the substrcto C ( q b ) ( q b )...( q b ) order to defe fte prt of the tegrl d q d q d q F C q b q b q b () s, s,..., s m m m... m, m,..., m ( ) ( )...( ) m, m,..., m Plus some correctos due to dverget tegrls ( q ) m b dq m=-,,,.... I my cses lthough the tegrls gve () d () re fte they wll hve o ect epresso or the ect epresso wll be too complcte, ths cse we c use the Guss-Lguerre Qudrture formul ( cse the tervl s [, ) ) to ppromte the tegrl by sum over the zeros of Lguerre Polyomls w f ( q, q,..., q, ) wth the wegth epressed terms of Lguerre Polyomls d ther roots w =, L ( ) = () ( ) L ( ) ( ) uas emple we wll regulrze the smples -loop tegrl l dldk k ( k p) ( k l) s s We troduce the zet regultors k l to get fte results d vod dvergeces, the regultor wll be elmted fter clcultos. Itegrto over l gves (here p s costt) =

11 l l dl l dl dl = ( k) ( k l) ( k l)( l ) l s s s (3) To perform tegrto over k we wll eed to regulrze the tegrls k dk A k k s ( k) dk ( ) p / s dl ( k p) k ( k l)( l ) A = (4) The frst tegrl becomes logrthmc UV dverget tegrl u du s wth the ( up ) chge of vrble u = ( s lwys the regultor s teds to ), ths c be regulrzed by ddto d substrcto to ths tegrl of the fctor, ths logrthmclly dverget tegrl c be regulrzed wth the Euler Mclur formul d the detty = γ. reg To evlute the secod tegrl we wll use the Lguerre qudrture formul order to smplfy clcultos ( k) dk dl w ( k) dk j dk ( k p) k ( k l)( l ) (5) ( k p) k ( k ) ( j j j ) Epresso (5) s legtmte sce the tegrl over l s coverget, wth chge of vrble u = ech tegrl of the sum sde (5) becomes logrthmc dverget wju( u ) du tegrl dk, ths s g logrthmc dverget tegrl d c be ( up) u ( j ) evluted by ddg /substrctg term of the form. Usg the equtos ( -5) we hve show emple of how -loop tegrl c be ζ regulrzd, by pplyg the zet regulrzto method of our pper o ech vrble, frst over l keepg k costt d the over k to get fte results CONCLUSIONS AND FINAL REMARKS: We hve eteded the defto of the zet regulrzto of seres to pply t to the Zet regulrzto of dverget tegrl m > m > by usg the Zet regulrzto techque combed wth the Euler Mclur summto formul. For good troducto to the Zet regulrzto techques, there s the book by Elzlde [4] or the Book by Bredt bsed o the mthemtcl dscoveres of Rmuj d ts

12 method of summto equvlet to the Zet regulrzto lgorthm [], other good referece (but bt more dvced) s Zedler [], for the cse of Zet-regulrzed determts [7] s good ole referece descrbg lso the process of Zet regulrzto v lytc cotuto d how t c be ppled to prove the detty ( ) = log π. Appretly there s cotrdcto, sce the Rem Zet fucto hs pole t s= so the rmoc seres could ot be regulrzed, however E usg the defto of fuctol determt E = oe gets the fte µ Γ '( ) result for the rmoc (geerlzed) seres =, wth the d of the Γ( ) Euler-mclur summto formul ths result for the rmoc seres c be used to gve ppromte regulrzed vlue of the logrthmc tegrl, for the cse of other types of dverget tegrls ( ) m we c use g Euler- Mclur summto formul to epress ths dverget tegrls terms of the egtve vlues of the urwtz or Rem Zet fucto ζ ( s,) = ζ ( s ) ζ ( m,) (UV) m=,,,3,4,...d the vlue of the dervtve of urwtz zet fucto log s = ζ (, ) (logrthmc UV), these vlues ecode the UV dvergeces []. For the s cse of the IR (frred ) dvergeces the form vrble to re-terprette these tegrls s q m s oe could mke chge of m s q dq for the cse m= we hve logrthmc dvergece both t = s so we must splt the tegrl / to IR d UV dverget prt = ths tegrl wll be equl to d IR regultor so ( ) fter few smple clcultos / = log µ, sce we c smply troduce forml UV Λ lm log UV Λ = Λ Λ, UV regultor s troduced to esure tht the tegrl wll be coverget.we lso beleve tht smlr procedure c be ppled to eted our Zet regulrzto lgorthm to multple (mult-loop) tegrls d q d q... d q F( q, q,..., q ), oe of the m dvtges of ths lgorthm s tht the dmeso of the spce does ot pper eplctly so our method does ot hve the sme problems s dmesol regulrzto, d c be used whe the Drc mtrces γ 5 = γ γγ γ 3 pper. The mposto formul () tht must be turl umber s order to vod oddtes the process of Zet regulrzto wth the Zet d urwtz Zet fucto, sce uless s postve teger the equlty ζ (, ) = ( ) = does ot hold.

13 Aother dvtge s tht we get oly fte quttes, wheres dmesol regulrzto you wll lwys fd poles of the Gmm fucto Γ ( z) = Ψ () O( z) z the lmt z ths epresso blows up. APPENDIX A: OW TO OVERCOME TE POLE ζ () I ths pper we hve see how due to the pole of the Rem zet t the pot s = we could ot regulrze the tegrl uless we use the result for the rmoc seres = reg ( ) fro > d fte, the f we troduce ths result sde the Γ ' ( ) Γ Euler-Mclur summto formul we c get fte results for Aother ltertve s to use the detty =. ( ) ( α ) = e = log ( )!! α = (A.) I ths cse we c evlute the tegrls sde (A.) by the Euler Mclur formul f ( ) f ( ) B f f f f ( k)! k ( ) = ( ) ( ) ( ) k = (k ) (k ) ( ) (A.) ere The Beroull umbers o the lst sde of (A.) re the coeffcets of the Tylor log ( ) f ( ) = epso of e f we sert sde (A.) the epresó α log ( ) log ( ) ( ) ( ) r log ( ) B r log ( ) = ζ ( α) α α α r s = r= ( r)! u ( ) = (A.3) ( ere α s smll o teger, so the zet fucto d ts dervtves ζ ) ( α) re FINITE Aother ltertve s to look for Pde or Rtol ppromto for the squre root of for emple. P( ) Q( ) ( ) ( )! = > (A.4) ( )(!) 4 3

14 P( ) I ths cse (A.4) we hve the ppromto 3/ Q( ) formul P( ) c 3/ 3/ c c c 5/ Q( ) (A.5), ow f we pply the Isde (A.5) ow there re o logrthmc-dverget tegrls, so the pole ζ () wll ot ow pper Refereces [] Abrmowtz, M. d Stegu, I. A. (Eds.). "Rem Zet Fucto d Other Sums of Recprocl Powers." 3. dbook of Mthemtcl Fuctos. New York: Dover, pp , 97. [] Berdt B. Rmuj's Notebooks: Prt II Sprger; edto (993) ISBN-: [3] Dowlg Joth P, "The Mthemtcs of the Csmr Effect", Mth. Mg. 6, (989). [4] Elzlde E. ; Zet-fucto regulrzto s well-defed, Jourl of Physcs 7 (994), L [5] Elzlde E. ; Zet regulrzto techques wth pplctos Ed. World Scetfc Press ISBN [6] Mzuo, Y. "Geerlzed Lerch Formuls: Emples of Zet-Regulrzed Products." J. Number Th. 8, 55-7, 6. [7] Muñoz Grcí, E. d Pérez Mrco, R. "The Product Over All Prmes s." Preprt IES/M/3/34. My 3. [8] Pvel E "Note o dmesol regulrzto", Qutum felds d strgs: course for mthemtcs, Vol.,(Prceto, NJ, 996/997), Provdece, R.I.: Amer. Mth. Soc., pp , ISBN (999) [9] Prellberg Thoms, The mthemtcs of the Csmr Effect tlk vlble t e-prt t [] Voros A. Zet fuctos over zeros of zet fuctos Ed. SPRINGER () ISBN:

15 [] Zee, A. (3). Qutum Feld Theory Nutshell. Prceto Uversty ISBN ISBN [] Zedler E. Qutum Feld Theory I : Bss Physcs d Mthemtcs Ed : Sprger Verlg ISBN: