Renormalizable noncommutative quantum field theory

Transcription

1 Journl of Physis: Conferene Series Renormlizle nonommuttive quntum field theory To ite this rtile: Hrld Grosse nd Rimr Wulkenhr 22 J Phys: Conf Ser View the rtile online for udtes nd enhnements Relted ontent - Non-trivil Bkgrounds in nonerturtive Yng-Mills Theory y the Slvnov-Tylor Identity A Qudri - Closed equtions of the two-oint funtions for tensoril grou field theory Dine Ousmne Smry - Correltion funtions of just renormlizle tensoril grou field theory: the meloni roximtion Dine Ousmne Smry, Crlos I Pérez- Sánhez, Fien Vignes-Tourneret et l Reent ittions - One loo rditive orretions to the trnsltion-invrint nonommuttive Yukw Theory K Bouhhi et l This ontent ws downloded from IP ddress on 3/7/28 t 7:38

2 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 Renormlizle nonommuttive quntum field theory Hrld Grosse nd Rimr Wulkenhr Fkultät für Physik, Universität Wien Mthemtishes Institut der Westfälishen Wilhelms-Universität E-mil: Astrt We disuss seil Euliden nonommuttive φ 4 -quntum field theory models in two nd four dimensions They re exmles of renormlizle field theories Using Wrd identity, it hs een shown, tht the et funtion for the ouling onstnt vnishes to ll orders in erturtion theory We extend this work nd otin from the Shwinger-Dyson eqution non-liner integrl eqution for the renormlised two-oint funtion lone The non-trivil renormlised four-oint funtion fulfils liner integrl eqution with the inhomogeneity determined y the two-oint funtion We otin suh reltions for the four s well for the two dimensionl sitution We exet to lern out renormlistion from this lmost solvle models Introdution Construtive methods led yers go to mny eutiful ides nd results, ut the min gol to onstrut mthemtil onsistent model of four dimensionl lol quntum field theory hs not een rehed Renormlized ertution exnsions llow to get quntum orretions order y order in ouling onstnt The onvergene of this exnsion, for exmle s Borel summle series, n e questioned In reent yers, modifition of the se-time struture led to new models, whih re nonlol in rtiulr sense But these models, in generl suffer under n dditionl disese, whih is lled the Infrred Ultrviolet mixing [2] Additionl infrred singulrities show u A ossile wy to ure this rolem hs een found y us in revious work [3] It led to seil models, whih needed 4 insted of 3 relevnt/mrginl oertors in the defining Lgrngin In ddition new fix oint ered t seil vlue of the dditionl ouling onstnt Tht this fixed oint exists in ertution theory to ll orders hs een shown in work y Rivsseu nd ollortors Pulished under liene y IOP Pulishing Ltd

3 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 The min oen question onerns the nonertutive onstrution of nontrivil nonomuttive quntum field theory, with whih we re deling here This reort is sed on our reent work [] We relized reviously tht the model defined y the tion S = d 4 x 2 φ Ω2 x 2 µ 2 φ λ 4 φ φ φ φ x is renormlisle to ll orders of erturtion theory Here, refers to the Moyl rodut rmetrised y the ntisymmetri 4 4-mtrix Θ, nd x = 2Θ x The model is ovrint under the Lngmnn-Szo dulity trnsformtion [4] nd eomes self-dul t Ω = Certin vrints hve lso een treted, see [5] for review Evlution of the β-funtions for the ouling onstnts Ω, λ in first order of erturtion theory leds to ouled dynmil system whih indites fixed-oint t Ω =, while λ remins ounded [6, 7] The vnishing of the β-funtion t Ω = ws next roven in [8] t three-loo order nd finlly in [9] to ll orders of erturtion theory It imlies tht there is no infinite renormlistion of λ, nd non-erturtive onstrution seems ossile [] The Lndu ghost rolem is solved The vnishing of the β-funtion to ll orders hs een otined using Wrd identity [9] We extend this work nd derive n integrl eqution for the two-oint funtion lone y using the Wrd identity nd Shwinger-Dyson equtions Usully, Shwinger-Dyson equtions oule the two-oint funtion to the four-oint funtion In our model, we show tht the Wrd identity llows to exress the four-oint funtion in terms of the two-oint funtion, resulting in n eqution for the two-oint funtion lone This is hieved in the first ste for the re twooint funtion We re le to erform the mss nd wvefuntion renormlistion diretly in the integrl eqution, giving self-onsistent non-liner eqution for the renormlised two-oint funtion lone Higher n-oint funtions fulfil liner inhomogeneous Shwinger-Dyson eqution, with the inhomogeneity given y m-oint funtions with m < n This mens tht solving our eqution for the two-oint funtion leds to full non-erturtive onstrution of this interting quntum field theory in four dimensions So fr we treted our eqution erturtively u to third order in λ The solution shows n interesting numer-theoreti struture We hoe tht detiled nlysis of our model will hel for non-erturtive tretment of more relisti Euliden quntum field theories We exet tht we n lern muh out non-erturtive renormliztion of Euliden quntum field theories in four dimensions from this lmost solvle model As first ste to understnd the underlying struture, we formulte finlly in the lst Chter the model in two dimensions Agin losed integrl eqution for the renormlized two oint funtion results 2 Mtrix Model It is onvenient to write the tion in the mtrix se of the Moyl se, see [3, 3] It simlifies enormously t the self-dulity oint Ω = We write down the resulting tion funtionls for the re quntities, whih involves the re mss µ re nd the wve funtion 2

4 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 renormlistion φ Z 2φ For simliity we fix the length sle to θ = 4 This gives S = m,n N 2 Λ H mn = Z µ 2 re m n, 2 φ mnh mn φ nm Vφ, 2 Vφ = Z2 λ 4 m,n,k,l N 2 Λ φ mn φ nk φ kl φ lm, 3 It is lredy used tht this model hs no renormlistion of the ouling onstnt [9] All summtion indies m,n, elong to N 2, with m := m m 2 The symol N 2 Λ refers to ut-off in the mtrix size The slr field is rel, φ mn = φ nm 3 Wrd Identity The key ste in the roof [9] tht the β-funtion vnishes is the disovery of Wrd identity indued y inner utomorhisms φ UφU Inserting into the onneted grhs the seil insertion vertex V ins := n H n H n φ n φ n 4 is the sme s the differene of grhs with externl indies nd, resetively, Z G ins [] = G G : We write Feynmn grhs in the self-dul φ 4 4-model s rion grhs on genus-g Riemnn surfe with B externl fes Adding for eh externl fe n externl vertex to get losed surfe, the mtrix index is onstnt t every fe Inserting the seil vertex V ins leds, however, to n index jum from to in n externl fe whih meets n externl vertex The orresonding externl soures t the jumed fe re thus J n nd J m for some other indies m,n Aording to the Wrd identity, this is the sme s the differene etween the grhs with fe index nd, resetively: Z = Z G ins [] = G G 6 The dots in 6 stnd for the remining fe indies We hve used H n H n = Z 5 3

5 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 4 Shwinger-Dyson eqution The Shwinger-Dyson eqution for the one-rtile irreduile two-oint funtion Γ reds Γ = = 7 The sum of the lst two grhs n e reexressed in terms of the two-oint funtion with insertion vertex, Γ = Z 2 λ G G Gins [] = Z 2 λ G G G G 8 Z = Z 2 λ Γ Γ H Γ H Γ H Γ Z This is losed eqution for the two-oint funtion lone It involves the divergent quntities Γ nd Z,µ re 5 Renormliztion Introduing the renormlised lnr two-oint funtion y Tylor exnsion Γ = Zµ 2 re µ2 Z, with Γren = nd =, we otin ouled system of equtions for, Z nd µ re It leds to losed eqution for the renormlised funtion lone, whih is further nlysed in the integrl reresenttion We rele the indies in,,n y ontinuous vriles in R Eqution 8 deends only on the length = 2 of indies The Tylor exnsion resets this feture, so tht we rele N 2 y Λ α ρ Λ d After onvenient hnge of vriles =: µ2 α, =: µ2 ρ nd =: µ 2, 9 α β G αβ nd using n identity resulting from the symmetry G α = G α, we rrive t []: Theorem The renormlised lnr onneted two-oint funtion G αβ of the self-dul nonommuttive φ 4 4-theory stisfies the integrl eqution α G αβ = λ Mβ L β βy β Mα L α αy β Gαβ Mα L α αn α G α α β α β Lβ N αβ N α G αβ Y, 4

6 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 where α,β [,, L α := dρ G αρ G ρ ρ nd Y = lim α M α L α α, M α := dρ αg αρ αρ, N αβ := dρ G ρβ G αβ ρ α, 6 Perturtion exnsion These integrl equtions re the strting oint for erturtive solution In this wy, the renormlised orreltion funtions re diretly otined, without Feynmn grh omuttion nd further renormlistion stes We otin { } G αβ = λ AI β βbi α α λ 2{ AB Iα αiβ βi α αi β βαβζ2 A βiβ B αiα αi α where A := α β, B := I α := Iα βi β αab Iβ 2 2βI β I β βba Iα 2 2αI α I α } Oλ 3, nd the following iterted integrls er: := α dx = ln α, 2 αx dx αi x αx = Li 2α 2 ln α 2 We onjeture tht G αβ is t ny order olynomil with rtionl oeffiients in α,β,a,b nd iterted integrls lelled y rooted trees 7 Four-oint Shwinger-Dyson eqution The knowledge of the two-oint funtion llows suessive onstrution of the whole theory As n exmle we tret the lnr onneted four-oint funtion G d Following the -fe in diretion of n rrow, there is distinguished vertex t whih the first -line strts For this vertex there re two ossiilities for the mtrix index of the digonlly oosite orner to the -fe: either or summtion vertex : d d = d d d d 3 5

7 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 We write the first ontriution s rodut of the vertex Z 2 λ, the left onneted two-oint funtion, the downwrd two-oint funtion nd n insertion, whih is reexressed y mens of the Wrd-identity After muttion of the externl two-oint funtions we otin the Shwinger- Dyson eqution for the renormlised PI four-oint funtion G d = G G G d G d d s follows: d = Zλ Zλ G d G d In terms of the PI funtion we hve Z d = λ λ Γren d Γren d d µ 2 d µ 2 G Gd d G Γren d 4 d d Γren d d µ 2 d Γren d Γren d λ d µ 2 d µ2 d 5 Pssing to the integrl reresenttion nd the vriles α nd β, we find for Γ αβγδ := d n integrl eqution, whih mniulted roritely llows gin to tke the limit ξ fter insertion of the exression for the wve funtion renormlistion onstnt Theorem 2 The renormlised lnr PI four-oint funtion Γ αβγδ of self-dul nonommuttive φ 4 4-theory with ontinuous indies α,β,γ,δ [, stisfies the integrl eqution Γ αβγδ = λ In lowest order we find α γδg αδ G γδ G γδ δα γ G αδ λ ρdρ β αδg βρg δρ βρ δρ M β L β YG αδ Γ ρβγδ Γ αβγδ ρ α dρ G αδg βρ β δρ βρ ρdρ β αδg βρ G ρδ G αδ βρ δρ ρ α Γ αβγδ = λ λ 2 γi α α αi γ γ α γ δi β β βi δ δ β δ Note tht Γ αβγδ is yli in the four indies, nd tht Γ = λoλ 3 6 Oλ 3 7 6

8 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 8 The Two Dimensionl Model As efore, we rele the indies in N y ontinuous vriles in R nd the sum N Λ y the integrl Λ d From eqution 8 we thus otin = λ Λ d µ 2 µ 2 We introdue hnge of vriles Γren µ 2 Γ ren µ 2 2 µ 2 Γ ren 8 =: µ 2 α α, =: β µ2 β, =: ρ µ2 ρ =: µ 2 Γ αβ α β, Λ =: ξ µ2 ξ dρ, d = µ2 ρ 2 9 nd otin Γ αβ α β = λ µ 2 ξ dρ ρ α αρ Γ αρ α Γ βρ Γ βα βρ Γ βρ ρ α β βρ Γ βρ Γ βα βρ Γ βρ We now exress 2 in terms of the onneted funtion G αβ defined y The result is G αβ = λ α β ξ dρ µ 2G αβ ρ λ ξ dρ µ 2 α β ρ Rtionl frtion exnsion yields G αβ = λ µ 2 α β We hve thus roven: Γ ρ Γ ρ ρ 2 Γ ρ 2 Γ αβ = G αβ 2 αgαρ αρ α G ρβ G αβ βρ ρ α 2G ρ β α βρ G ρ ρ G βρ βρ 22 { Gαρ G ρ dρ G αβ αg αρ ρ αρ G ρ ρ βg ρβ βρ G ρβ G } αβ 23 ρ α 7

9 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 Theorem 3 The renormlised lnr onneted two-oint funtion G αβ of self-dul nonommuttive φ 4 2-theory with ontinuous indies stisfies the integrl eqution where α,β [,[ nd G αβ = λ α β } {G µ 2 αβ L α M α N M β N αβ, 24 L α := dρ G αρ G ρ ρ, M α := dρ αg αρ αρ, N αβ := dρ G ρβ G αβ ρ α 25 The first terms of the erturtive exnsion re in terms of C αβ := α β G αβ = λ µ 2C αβ log βlog α given y λ 2Cαβ { µ 2 α 2 α 2 log αα α β 2 β 2 log ββ C αβ ζ2 β 3 C αβ 2 log β2 Li 2 β 3 } 2 [log α]2 Li 2 αlog βlog α These integrl equtions might e the strting oint of nonerturtive onstrution of Euliden quntum field theory on nonommuttive se Aknowledgements OneofusHGenjoyed thehositlity inprgt theconferenendlike tothnkprof Cestmir Burdik for the kind invittion Referenes [] H Grosse nd R Wulkenhr, Progress in solving nonommuttive quntum field theory in four dimensions, rxiv:99389 [he-th] [2] S Minwll, M Vn Rmsdonk nd N Seierg, Nonommuttive erturtive dynmis, JHEP [rxiv:he-th/99272] [3] H Grosse nd R Wulkenhr, Renormlistion of φ 4 -theory on nonommuttive R 4 in the mtrix se, Commun Mth Phys [rxiv:he-th/428] [4] E Lngmnn nd R J Szo, Dulity in slr field theory on nonommuttive hse ses, Phys Lett B [rxiv:he-th/2239] [5] V Rivsseu, Non-ommuttive renormliztion In: Quntum Ses Séminire Poinré X, eds B Dulntier nd V Rivsseu, Birkhäuser Verlg Bsel [rxiv:7575 [he-th]] [6] H Grosse nd R Wulkenhr, The β-funtion in dulity-ovrint nonommuttive φ 4 -theory, Eur Phys J C [rxiv:he-th/4293] [7] H Grosse nd R Wulkenhr, Renormlistionof φ 4 -theory on non-ommuttive R 4 to ll orders, Lett Mth Phys

10 7th Interntionl Conferene on Quntum Theory nd Symmetries QTS7 IOP Pulishing Journl of Physis: Conferene Series doi:88/ /343//243 [8] M Disertori nd V Rivsseu, Two nd three loos et funtion of non ommuttive φ 4 4 theory, Eur Phys J C [rxiv:he-th/6224] [9] M Disertori, R Guru, J Mgnen nd V Rivsseu, Vnishing of et funtion of non ommuttive φ 4 4 theory to ll orders, Phys Lett B [rxiv:he-th/6225] [] V Rivsseu, Construtive Mtrix Theory, JHEP [rxiv:76224 [he-th]] [] A Connes nd D Kreimer, Hof lgers, renormliztion nd nonommuttive geometry, Commun Mth Phys [rxiv:he-th/98842] [2] M Kontsevih nd D Zgier, Periods In: Mthemtis unlimited 2 nd eyond, Sringer- Verlg Berlin [3] H Grosse nd R Wulkenhr, Renormlistion of φ 4 theory on nonommuttive R 2 in the mtrix se, JHEP 32, 9 23 [rxiv:he-th/377] 9