Eur. Phys. J. C 03 73:54 DOI 0.40/epc/s005-03-54-3 Regular Artcle - Theoretcal Physcs The role of the Seberg Wtten feld redefnton n renormalzaton of noncommutatve chral electrodynamcs Maa Burć a DuškoLatas b Blana Nkolć c Voa Radovanovć d Unversty of Belgrade Faculty of Physcs P. O. Box 44 00 Belgrade Serba Receved: 6 June 03 / Revsed: 5 August 03 Sprnger-Verlag Berln Hedelberg and Socetà Italana d Fsca 03 Abstract It has been conectured n the lterature that renormalzablty of the θ-expanded noncommutatve gauge theores mproves when one takes nto account full nonunqueness of the Seberg Wtten expanson whch relates noncommutatve hgh-energy wth commutatve lowenergy felds. In order to check ths conecture we analyze renormalzablty of the θ-expanded noncommutatve chral electrodynamcs by quantzng the acton whch contans all terms mpled by ths nonunqueness. After renormalzaton we arrve at a dfferent theory characterzed by dfferent relatons between the couplng constants: ths means that the θ-expanded noncommutatve chral electrodynamcs s not renormalzable. t opens a whole range of questons conceptual and concrete. Frst one has to defne operatonally noncommutatve space A whch fulflls preferably wth a noton of dfferentablty and second felds on t. Ths for tself s a dffcult mathematcal problem. But a necessary constrant whch one wshes to mpose s that noncommutatve feld theores have good commutatve lmt the one whch s expermentally well establshed at the present length scale provdng at the same tme resoluton to the sngularty problems at the noncommutatvty scale. The most feasble model whch one usually starts wth s the space wth constant canoncal noncommutatvty: [ ˆx μ ˆx ν] = θ μν = const Introducton Basc noncommutatvty of spacetme coordnates [] sa very plausble dea when one consders two sngularty problems of classcal and quantum feld theory: sngular solutons and renormalzablty. Ths can be easly seen: when coordnates are represented by operators ˆx μ the commutaton relatons [ ˆx μ ˆx ν] = ˆθ μν ˆx μν =...d put a lower bound of order θ on coordnate measurements and an upper bound of order θ on momentum measurements; here θ gves the length scale at whch effects of noncommutatvty become sgnfcant. Ths property s desrable as a cure to both mentoned problems but a e-mal: maab@pb.ac.rs b e-mal: latas@pb.ac.rs c e-mal: blana@pb.ac.rs d e-mal: rvoa@pb.ac.rs because felds ˆφ ˆχ on t can be represented by functons on ordnary R 4. The feld multplcaton s gven by the Moyal Weyl star product: ˆφx ˆχx= e θ μν x μ y ν ˆφx ˆχy y x. 3 Ths representaton s called the Moyal space. The Moyal space s a flat noncommutatve space but clearly n any number of dmensons except n d = constant commutator breaks the Lorentz symmetry. Gauge symmetres on the Moyal space can be n prncple ntroduced n a straghtforward way. For example for spnor feld ˆψ one can defne acton of the noncommutatve U gauge group by ˆψ = Û ˆψ 4 where Û are untary elements of A. As coordnates ˆx μ generate A Û are always expressble as functons Û = Ûˆx μ : we are dealng wth local symmetry. The group acton can also be the adont ˆψ = Û ˆψÛ or the rght acton ˆψ = ˆψÛ. Obvously the noncommutatve U group s not abelan and therefore transformaton propertes of the gauge potental resemble those of the usual nonabelan theores.
Page of 0 Eur. Phys. J. C 03 73:54 In fact the expanson of the potental n terms of the Le algebra generators  μ =  a μ T a s possble only for the UN groups. When one consders general noncommutatve spaces qute often one can only defne nfntesmal symmetry transformatons. We wll n ths paper be manly concerned wth gauge theores whch act on fermonc felds. One of the mportant constrants on noncommutatve theores s ther commutatve lmt θ 0. In ths lmt one naturally expects that noncommutatve felds Â μ ˆψ reduce to commutatve gauge and matter felds A μ and ψ  μ θ=0 = A μ ˆψ θ=0 = ψ 5 as for θ = 0 the star product reduces to the ordnary one. Therefore all noncommutatve theores have the same commutatve lmt. On the other hand startng from a specfc commutatve theory we can get varous deformatons: noncommutatve generalzaton s not unque. But the noncommutatve structure of the space tself can gve restrctons whch reduce the number of possble models. We mentoned that for symmetry transformatons on the Moyal space defned by 4 only the UN groups can be consstently represented. Varous aspects of ths representaton were analyzed n the lterature [ 6] and t was shown that n perturbatve quantzaton noncommutatve models behave worse than commutatve gauge theores: the ultravolet dvergences propagate to the nfrared sector UV/IR mxng. Another wdely explored possblty of representng gauge symmetres s the envelopng -algebra formalsm n whch one enlarges the Le algebra of the group to the envelopng algebra expandng the gauge potental n the symmetrzed products of the group generators T a  μ =  a μ T a +  ab { μ T a T b} +. 6 There s no restrcton on the type of the gauge group n ths approach. Formula 6 nduces at the same tme an expanson of all felds n θ called the Seberg Wtten SW map [7 0]. Ths expanson apart from provdng the effectve low-energy theory and the new nteractons can be used to defne quantzaton procedure. The dea s the followng: one frst quantzes the theory n lnear order then proceeds to the second and hgher orders by usng some knd of teratve procedure. Hgher orders of felds n the expanson are related to lower orders by gauge symmetres [ ] so one expects that renormalzablty could be acheved through the noncommutatve Ward denttes. In addton the SW expanson has an amount of nonunqueness whch ncreases the number of possble counterterms for renormalzaton [3 4]. Therefore to dscuss renormalzablty of the θ-expanded theores we must nvestgate behavor of the lnear order. Results obtaned n the lterature are qute encouragng especally for the pure gauge theores of the U SUN symmetres and the gauge sector of a noncommutatve verson of the Standard Model [4 7]. Incluson of the matter on the other hand presents dffculty. For some tme t was beleved that fermons cannot be successfully ncorporated nto a renormalzable theory because of the 4ψdvergence [8 9]; however ths dvergence s absent n chral theores [0 ]. Good results were obtaned n [ 4] too for the GUT nspred anomaly-safe models wth chral fermons: t was found that all lnear-order dvergences are gven by margnal operators. A counterargument was gven by Armon [5] who comparng the expanded to the unexpanded theores argued that the sum of margnal operators of dfferent orders gves a relevant operator whch prevents renormalzablty. To clarfy the queston of renormalzablty and decde whether a potental renormalzablty of the θ-expanded chral electrodynamcs obtaned n [] s n fact actual we attempt n ths paper to renormalze the model explctly. In order to do that one has to take nto account the full nonunqueness of the SW expanson and that gves sx new nteracton terms n the acton. New couplng constants κ however are constraned by relaton 3. We calculate all one-loop dvergences and perform renormalzaton of κ ncludng noncommutatvty parameter θ μν : we fnd that renormalzaton volates the constrant equaton. The concluson therefore s that the SW expanson cannot hold smultaneously for the bare and the dressed felds whch means that our model s not renormalzable. More generally t mples that the SW-expanded gauge theores whch nclude matter are not renormalzable: nonrenormalzablty s effected through a mechansm dfferent from the UV/IR mxng but remans. Whether noncommutatve feld theores can descrbe the basc quantum felds n a dfferent framework or formalsm or serve only as an effectve descrpton of nature at low scales s a queston unsolved yet. Noncommutatve chral electrodynamcs Let us ntroduce the Lagrangan. We wsh to dscuss mnmal noncommutatve extenson of the commutatve chral electrodynamcs defned by the acton S C = d 4 x ϕ σ μ μ + qa μ ϕ 4 F μνf μν 7 where ϕ s a left chral fermon of charge q A μ s the vector potental and F μν = μ A ν ν A μ s the electromagnetc feld strength. Noncommutatve felds ˆϕ  μ and ˆF μν = μ  ν ν  μ + q[â μ  ν ] are also represented
Eur. Phys. J. C 03 73:54 Page 3 of 0 by functons of commutatve coordnates; the correspondng noncommutatve acton s gven by S NC = d 4 x ˆ ϕ σ μ μ + qâ μ ˆϕ 4 ˆF μν ˆF μν. 8 Two sets of felds are related by the Seberg Wtten map:  ρ = A ρ + n= A n ρ ˆϕ = ϕ + n= ϕ n. 9 Ths map s an expanson n powers of noncommutatvty θ μν whch s desgnated by ndex n: n the commutatve lmt hgher powers vansh and the ntal values of felds are ϕ 0 = ϕ A 0 ρ = A ρ. Seberg Wtten expanson 9 can be seen as soluton to the condton that nfntesmal symmetry transformatons close. The smplest soluton n lnear order s gven by [8 0]:  ρ = A ρ + 4 qθμν {A μ ν A ρ + F νρ } 0 ˆF ρσ = F ρσ qθμν {F μρ F νσ } + 4 qθμν{ A μ ν + D ν F ρσ } ˆϕ = ϕ + qθμν A μ ν ϕ where D μ denotes the commutatve covarant dervatve D μ ϕ = μ + qa μ ϕ. It s possble to generate hgher orders n the expanson from lnear order []. In addton whenever we have a partcular soluton A n ρ ϕ n we can obtan a more general one by addng arbtrary gauge covarant expressons A n ρ Φ n of the same order [3 4]: A n ρ = A n ρ + An ρ ϕ n = ϕ n + Φ n. 3 Ths property s called the Seberg Wtten nonunqueness: t means that the relaton between the physcal hgh-energy felds  ρ ˆϕ and the usual expermentally observed lowenergy felds A ρ ϕ s not unquely defned beyond the zeroth order n θ. One ntutvely expects that such a dfference would be unobservable. But n fact the SW feld redefnton can change not only the dsperson relatons or the cross sectons: t changes even the renormalzaton propertes of the theory and ths happens when fermonc matter s ncluded. Therefore renormalzablty of the theory was proposed n the lterature as a crteron whch fxes the nonunqueness of the Seberg Wtten expanson. We shall dscuss θ-lnear order of the chral electrodynamcs. Usng SW expansons 0 we obtan the acton L NC = L 0A + L 0ϕ + L A + L ϕ 4 wth L 0A = 4 F μνf μν 5 L 0ϕ = ϕ σ μ D μ ϕ 6 L A = qθμν F μρ F νσ F ρσ 4 F μνf ρσ F ρσ 7 L ϕ = L = 6 qθμν Δ αβγ μνρ F αβ ϕ σ ρ D γ ϕ + h.c.; 8 we denote n the followng αβγ δ μνρδ = Δ αβγ μνρ. The acton 4 was analyzed n [] and t was shown that as t stands t s not renormalzable. However all dvergences are of the form mpled by a SW redefnton and therefore we conectured that there exsts another expanson wthn allowed class 3 whch gves a renormalzable theory. In order to check ths conecture we need to expand the Lagrangan usng the most general frst-order SW soluton. Therefore nstead of 0 weuse A ρ = A ρ + A ρ ϕ = ϕ + Φ 9 where A ρ n θ: A ρ and Φ are covarant expressons of frst order = θ μν a μρσ τ ν F στ + a μνρτ σ F τσ + a 3 μντσ ρ F τσ 0 Φ = θ μν b σ μν D ϕ + b qf μ ρ σ νρ ϕ + b 3 qf μν ϕ + b 4 q μνρσ F ρσ ϕ and constants a and b are real. Ths changes the ntal acton to S NC = S NC + ΔS SW wth ΔS SW = d 4 x D ρ F ρμ A μ q ϕ σ μ ϕa μ + ϕ σ μ D μ Φ + h.c.. 3 When we ntroduce 0 and smplfy the acton usng varous denttes we obtan ΔL SW = b θ μν μνρσ ϕ σ σ D ρ D ϕ + qθ [ μν b + b F μρ ϕ σ ν D ρ ϕ + b F μρ ϕ σ ρ D ν ϕ + b 3 F μν ϕ σ ρ D ρ ϕ + a + a b μρσ τ F ρσ ϕ σ τ D ν ϕ 4 + a 3 a + b 4 μνρσ F ρσ ϕ σ τ D τ ϕ ] + h.c.
Page 4 of 0 Eur. Phys. J. C 03 73:54 Ths form s n a way canoncal as t contans mnmal number of terms. The new Lagrangan our startng pont for quantzaton reads L NC = L C + L A + L ϕ + [ κ 3 θ μν μνρσ ϕ σ σ D ρ D ϕ + qθ μν κ 4 F μρ ϕ σ ν D ρ ϕ + κ 5 F μν ϕ σ ρ D ρ ϕ + κ 6 μρσ τ F ρσ ϕ σ τ D ν ϕ + κ 7 μνρσ F ρσ ϕ σ τ D τ ϕ + h.c. ] = L C + L A + + κ L + 7 κ L 4 =3 where κ =...7 are the couplng constants ntroduced as κ = b κ 3 = b κ 4 = b b 5 κ 5 = b 4 +b 3 κ 6 = a +a b 4 κ 7 = a 3 a +b 4 and 6 L 3 = θ μν μνρσ ϕ σ σ D ρ D ϕ + h.c. 7 L 4 = qθ μν F μρ ϕ σ ν D ρ ϕ + h.c. 8 L 5 = qθ μν F μν ϕ σ ρ D ρ ϕ + h.c. 9 L 6 = qθ μν μρσ τ F ρσ ϕ σ τ D ν ϕ + h.c. 30 L 7 = qθ μν μνρσ F ρσ ϕ σ τ D τ ϕ + h.c. 3 Note that not all couplng constants are ndependent: there s a relaton between them κ 4κ 3 κ 4 = 0. 3 We shall see that ths relaton s broken n quantzaton. 3 Quantzaton For quantzaton we use the background feld method. The procedure for ths knd of models was developed n detal n [] so we wll not repeat t here. The man dfference s that now nstead of one we have sx fermon-photon vertces; n addton the fermon propagator has noncommutatve correcton whch we also treat perturbatvely. For the purpose of calculaton of functonal ntegrals we ntroduce the Maorana spnor ψ nstead of the chral ϕ ϕα ψ = ϕ α. Rewrtng the acton n Maorana spnors for the commutatve part of the spnor Lagrangan we obtan L 0ϕ = ψγ μ μ qγ 5 A μ ψ; noncommutatve terms are expressed lkewse. The one-loop effectve acton s gven by expanson Γ = STr log I + N + T 0 + T + T = n+ STr N + T 0 n + T + T n. 33 In ths formula N denotes a matrx whch s obtaned from commutatve 3-vertces after the expanson of quantum felds around the statonary classcal confguraton. It s gven by 0 ψγ 5 γ κ / N = q γ 5 γ λ. 34 ψ γ 5 /A / The T 0 T and T are defned n analogy but related to the terms lnear n θ: T 0 corresponds to the -vertex that s to the noncommutatve correcton of the fermon propagator whle T and T are obtaned from 3- and 4-vertces. The expanson gves T 0 = κ 3 θ μν 0 0 μνρσ 0 γ σ / ρ. 35 The T s the sum of sx terms T = wth 7 = T κ 36 T κ = 8 κ qθ μν Δ αβγ μνρ 0 δα κ β ψγ ρ γ / δα λγ ρ β ψ γ F αβ γ ρ γ /
Eur. Phys. J. C 03 73:54 Page 5 of 0 0 ψm ρκ T κ 3 = κ 3 qθ μν αβ μνρσ α β ψ ρ ψ κ γ 5 γ σ / γ 5 γ σ M ρλ αβ α β ψ ρ λ ψ N ραβ α β ρ A τ τ γ 5 γ σ / T κ 4 = κ 4 qθ μν 0 μg ρκ ρ g μκ ψγ ν ρ / γ ν ρ ψg ρλ μ g μλ ρ F μρ γ ν ρ / T κ 5 = κ 5 qθ μν 0 μg νκ ψ γ ρ ρ ψg νλ μ F μν T κ 6 = κ 6 qθ μν μρσ τ T κ 7 = κ 7 qθ μν μνρσ 0 ρ δκ σ ψγ τ γ 5 ν / δλ σ γ τ γ 5 ν ψ ρ F ρσ γ τ γ 5 ν / 0 ρ δκ σ ψγ 5 δλ σ γ τ γ 5 τ ψ ρ F ρσ γ 5 where M ρλ αβ = gρα g αβ + δ ρ αδ λ β and N ραβ = A ρ g αβ + A β g ρα. From 4-vertces we obtan T T = 5 = T κ 37 wth T κ = 8 κ q θ μν Δ αβγ μνρ T κ 3 = κ 3 q θ μν μνρσ T κ 4 = κ 4 q θ μν T κ 5 = κ 5 q θ μν δβ λδκ γ ψγ ρ γ 5 ψ α δγ κ F αβ + αδβ κa γ ψγ ρ γ 5 / γ ρ γ 5 ψf αβ δγ λ + A γ δβ λ α F αβ A γ γ ρ γ 5 / ρ δ λ ψγ σ κ ψ+ g λκ ψγ σ ρ ψ A α ψ β P ρ αβγ γ σ / γ σ A α β ψp ρ αβλ γ σ A τ A ρ τ + A τ A τ ρ / g ρκ ψγ 5 γ ν ψδ[ρ λ μ] F μκ + [μ g ρ]κ A ρ ψγ 5 γ ν / γ 5 γ ν ψf μλ + A ρ g λ[ρ μ] F μρ A ρ γ 5 γ ν / δμ λ ψγ 5 γ κ ψ ν F μν ψγ 5 γ κ + μδν κaρ ψγ 5 γ ρ / F μν γ 5 γ λ ψ + A ρ γ 5 γ ρ ψδν λ. μ F μν γ 5 /A / Here P ρ αβγ = δρ αg βγ + δ ρ β g αγ + δ ρ γ g αβ. Operators above 34 37 are the basc ngredents for the perturbaton theory. 4 Dvergences and renormalzaton By the power countng one can fx the terms whch gve dvergent contrbutons. They come from Γ dv = STr N N + N N N 3 N T N N N T 0 + N N T N T dv. 38 We calculated dvergences by dmensonal regularzaton. In ths very demandng calculaton our man ad apart from the Mathematca-based package MathTensor was the gauge covarance of the background feld method. Omttng the ntermedate steps we wrte only the fnal result as a sum of counterterms Γ dv = L ct. 39
Page 6 of 0 Eur. Phys. J. C 03 73:54 Addng L ct to the classcal acton we obtan the bare Lagrangan whch gves after quantzaton a theory fnte at one loop: κ 7 + q 8 4π 7 + κ 6κ 3 + κ 4 + 4κ 5 6κ 7. 40 L NC + L ct = 4 F μνf μν 4 q 3 4π + ϕ σ μ D μ ϕ + h.c. q 4π + qμ θ μν F μρ F νσ F ρσ 4 F μνf ρσ F ρσ 4 3 q 4π + κ 8κ 3 4κ 4 + 6 qμ θ μν Δ αβγ μνρ F αβ ϕ σ ρ D γ ϕ + h.c. + κ + q 3 4π 5 + 3κ 0κ 3 0κ 4 8κ 5 + 8κ 7 + qμ θ μν μνρσ ϕ σ σ D ρ D ϕ + h.c. κ 3 q 4π κ + 6κ 3 + 4κ 4 4κ 5 + 6κ 7 + qμ θ μν F μρ ϕ σ ν D ρ ϕ + h.c. κ 4 + q 6 4π + 9κ + 68κ 3 + 6κ 4 4κ 5 8κ 7 + qμ θ μν F μν ϕ σ ρ D ρ ϕ + h.c. κ 5 + q 4 4π κ 0κ 3 6κ 4 4κ 5 + 8κ 7 + qμ θ μν μρσ τ F ρσ ϕ σ τ D ν ϕ + h.c. κ 6 + q 36 4π 6 5κ + 3κ 3 4κ 4 4κ 5 7κ 6 + 4κ 7 + qμ θ μν μνρσ F ρσ ϕ σ τ D τ ϕ + h.c. The s the regularzaton parameter from dmensonal regularzaton. We can read off the values of the bare couplngs and felds. From the commutatve part we obtan the known renormalzatons ϕ 0 = Z ϕ = A μ 0 = Z 3 A μ = q 0 = μ / Z 3 Z q 4π ϕ 4 q 4 3 4π Aμ 4 q q 4π = μ + q 3 4π q. 43 The noncommutatve part of dvergences gves the bare couplngs κ 0. But one can notce that along wth L the 3-photon term L A gets a quantum correcton from the fermon loops although ts coeffcent s n the classcal Lagrangan fxed to. Ths mples that a rescalng of the noncommutatvty parameter θ μν s necessary. The bare θ μν 0 s gven by θ μν 0 = 4 q 3 4π κ 8κ 3 4κ 4 θ μν. 44 Usng 4 44 we obtan for the runnng of κ the followng results: κ 0 = κ + q + 3κ 3 4π + 4κ κ 8κ 3 4κ 4 5κ 3 36κ 4 8κ 5 + 8κ 7 45 κ 3 0 = κ 3 + q κ 4π + 40κ 3 + 6κ 3 κ 8κ 3 4κ 4 4κ 5 + 6κ 7 46 κ 4 0 = κ 4 + q + 9κ 6 4π + 68κ 3 + 38κ 4 + 8κ 4 κ 8κ 3 4κ 4 4κ 5 8κ 7 47 κ 5 0 = κ 5 + q 3 3κ 4π 60κ 3 8κ 4 + κ 5 + 6κ 5 κ 8κ 3 4κ 4 + 4κ 7 48 κ 6 0 = κ 6 + q 6 5κ 36 4π + 3κ 3 4κ 4 4κ 5
Eur. Phys. J. C 03 73:54 Page 7 of 0 + 48κ 6 κ 8κ 3 4κ 4 + 4κ 7 49 κ 7 0 = κ 7 + q + 3κ 4 4π 48κ 3 + 6κ 4 + κ 5 + 3κ 7 κ 8κ 3 4κ 4. 50 A somewhat unusual quadratc runnng follows from the renormalzaton of θ μν. In partcular we obtan κ 0 4κ 3 0 κ 4 0 = + 4 q 3 4π κ 8κ 3 4κ 4 κ 4κ 3 κ 4 q + 3 4π 3 + 5κ 60κ 3 74κ 4 an we see easly that constrant 3 s not preserved for the bare couplngs that s the couplng constants κ κ 3 and κ 4 do not renormalze consstently wth the SW expanson. Runnng of κ 5 κ 6 and κ 7 on the other hand obstructs 5 6 not. 5 Anomaly-safe theores Inevtable concluson s therefore that the SW-expanded chral electrodynamcs 4 s not perturbatvely renormalzable. Of course ths theory s not renormalzable for another stronger reason: the exstence of chral anomaly. It was shown namely that n the θ-expanded gauge models the anomaly-cancellaton condtons are exactly the same as n the correspondng commutatve theores [6 7]; on the other hand we know that the commutatve chral electrodynamcs contans an anomaly. So there s a natural the queston: f we constructed a model consstng of several fermons n dfferent representatons of the noncommutatve U and mposed n addton the anomalycancellaton condtons would renormalzablty mprove? Unfortunately as s shown n the Appendx ths s not the case. Though the anomaly-cancellaton condtons q = 0 q 3 = 0 5 remove the 3-photon vertex from the acton and make room for an arbtrary renormalzaton of θ μν ths addtonal renormalzaton cannot mprove overall renormalzablty of the model. 6 Dscusson Before we dscuss the meanng of our result let us consder brefly some lmtng cases. The easest case s when fermons are absent ϕ = 0. We see that then there are no new noncommutatve dvergences and therefore no need to rescale θ μν whch s n accord wth prevously obtaned behavor of the SUN gauge theores [5]. We dscussed n [] the mnmal or lttle case n whch all κ = 0 =...7 that s n whch complete noncommutatve fermon correcton reduces to L. As can be seen from 45 50 ths one term generates after quantzaton all other L ; even the fermon propagaton changes. Ths s n a way nce result as t shows a specfc relaton between the spatal and the gauge degrees of freedom: fermon propagaton changes because of the photon loops. Quantum correctons generate noncommutatve nteractons even n the case when they are absent from the classcal Lagrangan that s for + κ = 0 κ = 0 = 3...7; ths effect was dscussed before n the case of Drac fermons n [9]. Let us summarze the result. We started classcally wth the most general acton for the noncommutatve θ-expanded chral electrodynamcs whch s permtted by the Seberg Wtten map. As the amount of nonunqueness of the SW expanson s huge the ntal Lagrangan 4 contans essentally all terms allowed by dmenson and gauge covarance. Denote n addton to L ϕ = L + 7 κ L 5 = L A = λ θ μν F μρ F νσ F ρσ + λ θ μν F μν F ρσ F ρσ. 53 There are only two condtons on L NC = L 0A + L 0ϕ + L A + L ϕ 54 whch dstngush the orgn of the separate terms that s whch sgnfy that the acton s derved from 8 through the SW map: they are λ + 4λ = 0 κ 4κ 3 κ 4 = 0. 55 The frst relaton the rato between the two 3-photon terms s stable under quantzaton; the second relaton s broken after renormalzaton. Ths means that the SW map s not compatble wth quantzaton: clearly ths happens only when fermons are present. It further mples that the θ-expanded chral electrodynamcs s not renormalzable and we are forced to conclude that more generally the θ-expanded theores cannot be consdered as fundamental or basc theores whch provde representatons of gauge symmetry on the Moyal space. They can probably gve a good effectve descrpton of the effects of noncommutatvty but one should expect that n a fundamental noncommutatve gauge theory the matter wll be ncluded n a dfferent way. Acknowledgements Ths work was supported by the Serban Mnstry of Educaton Scence and Technologcal Development under Grant No. ON703.
Page 8 of 0 Eur. Phys. J. C 03 73:54 Appendx: One-loop dvergences All calculatons mentoned n the text are done actually usng the Maorana spnors and the algebra of γ -matrces. The drect outcome of calculaton for the Lagrangan contanng only one spnor feld s Γ dv = 4π q d 4 x ψγ μ μ qγ 5 A μ ψ 3 F μνf μν + θ μν μνρσ ψγ σ D ρ D τ D τ ψ + q 3 F μρf νσ F ρσ 6 F μνf ρσ F ρσ 5 6 F μρ ψγ ρ D ν ψ + 6 F μρ ψγ ν D ρ ψ + 3 F μν ψγ ρ D ρ ψ + 6 μρσ τ F ρσ ψγ 5 γ τ D ν ψ 7 8 μνρσ F ρσ ψγ 5 γ τ D τ ψ + κ μνρσ ψγ σ D ρ D τ D τ ψ + q 3 F μρf νσ F ρσ 6 F μνf ρσ F ρσ + F μρ ψγ ρ D ν ψ 3 F μρ ψγ ν D ρ ψ + 5 36 μρσ τ F ρσ ψγ 5 γ τ D ν ψ 8 μνρσ F ρσ ψγ 5 γ τ D τ ψ + κ 3 4 3 μνρσ ψγ σ D ρ D τ D τ ψ + q 6 3 F μρf νσ F ρσ + 4 3 F μνf ρσ F ρσ 0 3 F μρ ψγ ρ D ν ψ 34 3 F μρ ψγ ν D ρ ψ + 0 3 F μν ψγ ρ D ρ ψ 3 μρσ τ F ρσ ψγ 5 γ τ D ν ψ + μνρσ F ρσ ψγ 5 γ τ D τ ψ + κ 4 7 6 μνρσ ψγ σ D ρ D τ D τ ψ + q 8 3 F μρf νσ F ρσ + 3 F μνf ρσ F ρσ 0 3 F μρ ψγ ρ D ν ψ 3 3 F μρ ψγ ν D ρ ψ + 9 6 F μν ψγ ρ D ρ ψ + 3 μρσ τ F ρσ ψγ 5 γ τ D ν ψ 4 μνρσ F ρσ ψγ 5 γ τ D τ ψ + κ 5 3 μνρσ ψγ σ D ρ D τ D τ ψ + q 4 3 F μρ ψγ ρ D ν ψ + 3 F μρ ψγ ν D ρ ψ + 5 3 F μν ψγ ρ D ρ ψ + 3 μρσ τ F ρσ ψγ 5 γ τ D ν ψ μνρσ F ρσ ψγ 5 γ τ D τ ψ + κ 6 qμρσ τ F ρσ ψγ 5 γ τ D ν ψ + κ 7 4 3 μνρσ ψγ σ D ρ D τ D τ ψ 4 + q 3 F μρ ψγ ρ D ν ψ + 4 3 F μρ ψγ ν D ρ ψ 8 3 F μν ψγ ρ D ρ ψ 3 μρσ τ F ρσ ψγ 5 γ τ D ν ψ + μνρσ F ρσ ψγ 5 γ τ D τ ψ. A. Ths result reduces to the result found before n [] for κ = 0. One mmedately notces that apart from the usual commutatve dvergences and the 3-photon term L A all other terms are electron-photon nteractons and they should be converted to the canoncal L. Partal ntegratons and other transformatons based on the spnor denttes succeeded by the change from the Maorana to the chral form gve fnal expresson 40. In order to analyze the case wth several matter felds let us frst shortly dscuss the correspondng noncommutatve classcal acton. We assume that we have a set of fermons ˆϕ =...N wth electrc charges q. It s known [8] that n the θ-expanded theores each noncommutatve feld comes wth ts own noncommutatve potental  μ ;allof them however for θ = 0 reduce to the same A μ.theâ μ are dfferent because ther correspondng SW maps dffer: they depend on charges q. Therefore n order to obtan the acton wth the correct lmt n the sum N L C = ϕ σ μ μ + q A μ ϕ N F μν F μν A. 4 = =
Eur. Phys. J. C 03 73:54 Page 9 of 0 we have to rescale the gauge felds and the charges A μ c A μ = c A μ q c q. We get N L C = ϕ σ μ μ + q A μ ϕ N c F μν F μν. 4 = = A.3 In order to assocate A.3 wth the Lagrangan of the usual commutatve theory we fx the sum of weghts c to N= c =. In the noncommutatve U case we are free to choose c = /N; for other symmetres the analogous relatons are more complcated [7]. The same rescalng appled to the noncommutatve part of the Lagrangan gves for the boson vertex L A = q N θ μν F μρ F νσ F ρσ 4 F μνf ρσ F ρσ A.4 + θ μν μνρσ ϕ σ σ D ρ D ϕ + h.c. q κ 3 + α 3 4π + μ θ μν F μρ q ϕ σ ν D ρ ϕ + h.c. q κ 4 + α 4 4π + μ θ μν F μν q ϕ σ ρ D ρ ϕ + h.c. q κ 5 + α 5 4π q ϕ σ τ D ν ϕ + h.c. and clearly n an anomaly-safe model n whch q = 0 we fnd that ths term vanshes L A = 0. Fermon terms are on the other hand unchanged as they are mutually ndependent for each feld: L ϕ = 6 θ μν Δ αβγ μνρ F αβ q ϕ σ ρ γ + q A γ ϕ + h.c. L 3ϕ = θ μν μνρσ ϕ σ σ D ρ D ϕ + h.c. L 4ϕ = θ μν F μρ q ϕ σ ν D ρ ϕ + h.c. L 5ϕ = θ μν F μν q ϕ σ ρ D ρ ϕ + h.c. L 6ϕ = θ μν μρσ τ F ρσ L 7ϕ = θ μν μνρσ F ρσ q ϕ σ τ D ν ϕ + h.c. q ϕ σ τ D τ ϕ + h.c. We can extract the value of the one-loop dvergences from our prevous result ether usng the same rescalng of charges q by c or straghtforwardly by repeatng the calculaton. We obtan for the renormalzed Lagrangan: L NC + L ct q = 4 F μνf μν 4 3 4π + ϕ σ μ q D μ ϕ + h.c. 4π + 6 μ θ μν Δ αβγ μνρ F αβ q ϕ σ ρ D γ ϕ + h.c. q + κ + α 4π wth + μ θ μν μρσ τ F ρσ q κ 6 + α 6 4π + μ θ μν μνρσ F ρσ q ϕ σ τ D τ ϕ + h.c. κ 7 + α 7 q 4π α = 3 5 + 3κ 0κ 3 0κ 4 8κ 5 + 8κ 7 α 3 = κ + 6κ 3 + 4κ 4 4κ 5 + 6κ 7 α 4 = 6 + 9κ + 68κ 3 + 6κ 4 4κ 5 8κ 7 α 5 = 4 κ 0κ 3 6κ 4 4κ 5 + 8κ 7 α 6 = 36 6 5κ + 3κ 3 4κ 4 4κ 5 7κ 6 + 4κ 7 α 7 = 8 7 + κ 6κ 3 + κ 4 + 4κ 5 6κ 7. Whle renormalzaton of felds and charges s standard ϕ 0 = Z ϕ = 4π ϕ A μ 0 = Z 3 A μ = q 0 = μ / Z 3 Z q q 4 3 4π Aμ q 4π q
Page 0 of 0 Eur. Phys. J. C 03 73:54 q = μ + 3 4π q the noncommutatvty θ μν can renormalze arbtrarly. In order to try to use ths fact we assume that t s of the form θ μν 0 = + α q 4π θ μν wth an arbtrary coeffcent α whch s to be determned from some renormalzablty constrant. Renormalzaton of the κ follows: κ 0 = κ α 4π + κ 0κ 3 0κ 4 8κ 5 + 8κ 7 q q + 3 + 9κ κ 3 0 = κ 3 α 4π κ 3 q + κ + 40κ 3 + 4κ 4 4κ 5 + 6κ 7 q κ 4 0 = κ 4 α 4π κ 4 q + 6 + 9κ + 68κ 3 + 38κ 4 4κ 5 8κ 7 q κ 5 0 = κ 5 α 4π κ 5 q + 4 κ 0κ 3 6κ 4 + 4κ 5 + 8κ 7 q κ 6 0 = κ 6 α 4π κ 6 q + 36 6 5κ + 3κ 3 4κ 4 4κ 5 + 4κ 7 q κ 7 0 = κ 7 α 4π κ 7 q + 8 7 + κ 6κ 3 + κ 4 + 4κ 5 q. However we easly observe that n whatever way we fx α the expressons κ 0 4κ 3 0 κ 4 0 = α 4π q κ 4κ 3 κ 4 α 4π 8κ 3 7κ 4 cannot be zero. References q + 4π q 3 3 + 9κ. H.S. Snyder Phys. Rev. 7 38 947. S. Mnwalla M. Van Raamsdonk N. Seberg J. Hgh Energy Phys. 000 00 000 3. M. Hayakawa Phys. Lett. B 478 394 000 4. A. Matuss L. Sussknd N. Toumbas J. Hgh Energy Phys. 00 00 000 5. M. Van Raamsdonk N. Seberg J. Hgh Energy Phys. 0003 035 000 6. A. Armon Nucl. Phys. B 593 9 00 7. N. Seberg E. Wtten J. Hgh Energy Phys. 09 03 999 8. J. Madore S. Schraml P. Schupp J. Wess Eur. Phys. J. C 6 6 000 9. B. Jurco L. Moller S. Schraml P. Schupp J. Wess Eur. Phys. J. C 383 00 0. P. Schupp J. Trampetc J. Wess G. Raffelt Eur. Phys. J. C 36 405 004. K. Ulker B. Yapskan Phys. Rev. D 77 065006 008. P. Ascher L. Castellan J. Hgh Energy Phys. 07 84 0 3. T. Asakawa I. Kshmoto J. Hgh Energy Phys. 99 04 999 4. A. Bchl J. Grmstrup H. Grosse L. Popp M. Schweda R. Wulkenhaar J. Hgh Energy Phys. 06 03 00 5. M. Burc D. Latas V. Radovanovc J. Hgh Energy Phys. 060 046 006 6. D. Latas V. Radovanovc J. Trampetc Phys. Rev. D 76 085006 007 7. M. Burc V. Radovanovc J. Trampetc J. Hgh Energy Phys. 0703 030 007 8. R. Wulkenhaar J. Hgh Energy Phys. 003 04 00 9. M. Burc V. Radovanovc Class. Quantum Gravty 55 005 0. M. Burc D. Latas V. Radovanovc J. Trampetc Phys. Rev. D 77 04503 008. M. Burc D. Latas V. Radovanovc J. Trampetc Phys. Rev. D 83 04503 0. C.P. Martn C. Tamart Phys. Rev. D 80 06503 009 3. C.P. Martn C. Tamart J. Hgh Energy Phys. 09 04 009 4. C. Tamart Phys. Rev. D 8 05006 00 5. A. Armon Phys. Lett. B 704 67 0 6. R. Baneree S. Ghosh Phys. Lett. B 533 6 00 7. C.P. Martn Nucl. Phys. B 65 7 003 8. P. Ascher B. Jurco P. Schupp J. Wess Nucl. Phys. B 65 45 003 9. J.M. Grmstrup R. Wulkenhaar Eur. Phys. J. C 6 39 00