Complex mass renormalization in EFT D. Djukanovic InstitutfürKernphysik,JohannesGutenberg-Universität,J.J.Becherweg45, 0 D-55099Mainz,Germany 1 E-mail: [email protected] 0 2 J. Gegelia∗ n a InstitutfürKernphysik,JohannesGutenberg-Universität,J.J.Becherweg45, J D-55099Mainz,Germany 2 and 1 HighEnergyPhysicsInstituteofTSU,Tbilisi,Georgia ] E-mail: [email protected] h p A. Keller - p InstitutfürKernphysik,JohannesGutenberg-Universität,J.J.Becherweg45, e D-55099Mainz,Germany h E-mail: [email protected] [ 1 S. Scherer v InstitutfürKernphysik,JohannesGutenberg-Universität,J.J.Becherweg45, 2 D-55099Mainz,Germany 7 7 E-mail: [email protected] 1 . 1 Weconsideraneffectivefieldtheoryofunstableparticles(resonances)usingthecomplex-mass 0 0 renormalization. As an application we calculate the masses and the widths of the ρ meson and 1 theRoperresonance. : v i X r a 6thInternationalWorkshoponChiralDynamics,CD09 July6-10,2009 Bern,Switzerland ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ Complexmassrenormalization J.Gegelia 1. Introduction Theconstructionofconsistentchiraleffectivefieldtheorieswithheavydegreesoffreedomis anon-trivialproblem. Forexample,inbaryonchiralperturbationtheorytheusualpowercounting isviolatedifoneusesthedimensionalregularizationandtheminimalsubtractionscheme[1]. The current solutions to this problem either involve the heavy-baryon approach [2] or use a suitably chosenrenormalizationcondition[3,4,5,6]. Duetothesmallmassdifferencebetweenthenucleon andthe∆(1232)incomparisonwiththenucleonmass,the∆resonancecanbeconsistentlyincluded intheframeworkofeffectivefieldtheory[7,8,9,10,11]. Ontheotherhand,thetreatmentoftheρ mesonortheinclusionofheavierbaryonresonances suchastheRoperresonanceismorecomplicated. Weaddresstheissueofpowercountinginsuch effective theories by using the complex-mass renormalization scheme [12, 13, 14], which can be understoodasanextensionoftheon-mass-shellrenormalizationschemetounstableparticles. Asanapplicationweconsiderthemassesandthewidthsoftheρ mesonandtheRoperreso- nance. MoredetailscanbefoundinRefs.[15,16]. 2. Rhomeson We start with the most general effective Lagrangian for ρ and ω mesons and pions in the parametrizationofthemodelIIIofRef.[17]: L =L(2)+L +L +L +···. π ρπ ω ωρπ Theindividualexpressionsrelevantforthecalculationsofthisworkread L(2) = F2Tr(cid:104)∂ U(∂µU)†(cid:105)+F2M2Tr(cid:0)U†+U(cid:1), π µ 4 4 L = −1Tr(cid:0)ρ ρµν(cid:1)+(cid:34)M2+cxM2Tr(cid:0)U†+U(cid:1)(cid:35)Tr(cid:20)(cid:18)ρµ−iΓµ(cid:19)(cid:18)ρ −iΓµ(cid:19)(cid:21), ρπ 2 µν ρ 4 g µ g L = −1(cid:0)∂ ω −∂ ω (cid:1)(∂µων−∂νωµ)+Mω2 ωµωµ, ω µ ν ν µ 4 2 1 (cid:16) (cid:17) L = g ε ωνTr ραβuµ , (2.1) ωρπ 2 ωρπ µναβ where (cid:18)i(cid:126)τ·(cid:126)π(cid:19) (cid:126)τ·(cid:126)ρµ U = u2=exp , ρµ = , F 2 ρµν = ∂µρν−∂νρµ−ig[ρµ,ρν], Γ = 1(cid:2)u†∂ u+u∂ u†u =i(cid:2)u†∂ u−u∂ u†(cid:3). (2.2) µ µ µ µ µ µ 2 All the fields and parameters in Eqs. (2.1) are bare quantities. In order to increase the readability of the expressions we have omitted the usual subscript 0. In Eqs. (2.1), F denotes the pion-decay constant in the chiral limit, M2 is the lowest-order expression for the squared pion mass, M and ρ 2 Complexmassrenormalization J.Gegelia M refer to the bare ρ and ω masses, g, c , and g are coupling constants. We use the KSFR ω x ωρπ relation[18,19] M2 = 2g2F2. (2.3) ρ Toperformtherenormalizationweexpressthebarequantitiesintermsofrenormalizedones: M = M +δM , ρ,0 R R c = c +δc , x,0 x x ··· (2.4) Weapplythecomplex-massrenormalizationscheme[12,13,14]andchooseM2=(M −iΓ /2)2 R χ χ as the pole of the ρ-meson propagator in the chiral limit. M and Γ are the pole mass and the χ χ width of the ρ meson in the chiral limit, respectively. Both are input parameters in our approach. Inthecomplex-massrenormalizationscheme,thecountertermsareingeneralcomplexquantities. Thepresenceoflargeexternalmomentaoftheρ mesonleadstoaconsiderablecomplication in the power counting for loop diagrams. It is necessary to investigate all possible flows of the external momenta through the internal lines of a loop diagram. Next, one needs to determine the chiral orders for all flows of external momenta. Finally, the smallest of these orders is defined as thechiralorderofthegivendiagram. Thepowercountingrulesareasfollows. Letqcollectivelystandforasmallquantitysuchas thepionmass. ApionpropagatorcountsasO(q−2)ifitdoesnotcarrylargeexternalmomentaand asO(q0)if itdoes. A vector-mesonpropagator countsasO(q0)if itdoes notcarry large external momenta and as O(q−1) if it does. The pion mass counts as O(q1), the vector-meson mass as O(q0),andthewidthasO(q1). VerticesgeneratedbytheeffectiveLagrangianofGoldstonebosons L(n)countasO(qn). DerivativesactingonheavyvectormesonscountasO(q0). Thecontributions π ofvectormesonloopscanbeabsorbedsystematicallyintheparametersoftheeffectiveLagrangian. Thedressedpropagator,expressedintermsoftheselfenergy iΠab(p)=iδab(cid:2)g Π (p2)+p p Π (p2)(cid:3) (2.5) µν µν 1 µ ν 2 hastheform g −p p 1+Π2(p2) iSab(p)=−iδab µν µ νMR2+Π1(p2)+p2Π2(p2) . (2.6) µν p2−M2−Π (p2)+i0+ R 1 Thepoleofthepropagatorisfoundasthe(complex)solutiontothefollowingequation: z−M2−Π (z)=0. (2.7) R 1 Wedefinethepolemassandthewidthoftheρ mesonbyparameterizing z=(M −iΓ/2)2. (2.8) ρ ThesolutiontoEq.(2.7)canbefoundperturbativelyasaloopexpansion z=z(0)+z(1)+z(2)+···. (2.9) 3 Complexmassrenormalization J.Gegelia (a) (b) (c) Figure1: One-loopcontributionstotheρ-mesonself-energyatO(q3). Thedashed,solid,andwigglylines correspondtothepion,theω meson,andtheρ meson,respectively. Eachofthesetermshasitsownchiralexpansion. Uptothirdchiralorderthepolereads z=z(0)=M2+c M2. (2.10) R x The one-loop contributions to the vector self-energy up to O(q3) are shown in Fig. 1. The contributionsofdiagrams(a)and(b)toΠ aregivenby 1 g2(cid:2)2A (M2)−(cid:0)p2−4M2(cid:1)B (p2,M2,M2)(cid:3) 0 0 Π = , 1(a) 16π2(n−1) (n−2)g2 (cid:26) Π = − ωρπ M4B (p2,M2,M2)−(cid:2)2B (p2,M2,M2)M2 +A (M2)−A (M2) 1(b) 64π2(n−1) 0 ω 0 ω ω 0 0 ω +2B (p2,M2,M2)p2(cid:3)M2+B (p2,M2,M2)p2+M2 (cid:2)B (p2,M2,M2)M2 0 ω 0 ω ω 0 ω ω (cid:27) +A (M2)−A (M2)(cid:3)−(cid:2)2B (p2,M2,M2)M2 +A (M2)+A (M2)(cid:3)p2 . (2.11) 0 0 ω 0 ω ω 0 0 ω Usingdimensionalregularizationwithnspace-timedimensions,theloopfunctionsread A (cid:0)m2(cid:1) = −32π2λm2−2m2lnm, 0 µ B (cid:0)p2,m2,m2(cid:1) = −32π2λ+2ln µ −1 0 1 2 m 2 1(cid:18) m2 (cid:19) (cid:18) m2 (cid:19) ω − 1+ 2 F 1,2;3;1+ 2 − F (1,2;3;ω), 2 m2(ω−1) 2 1 m2(ω−1) 2 2 1 1 1 (cid:113) m2−m2+p2+ (cid:0)m2−m2+p2(cid:1)2−4m2p2 1 2 1 2 1 ω = , (2.12) 2m2 1 where F (a,b;c;z)isthestandardhypergeometricfunction,µ isthescaleparameterand 2 1 (cid:26) (cid:27) λ = 1 1 −1 (cid:2)ln(4π)+Γ(cid:48)(1)+1(cid:3) . (2.13) 16π2 n−4 2 Theρππ vertexindiagram(a)shouldcountasO(q0). However,itslargecomponentdoesnot contributetoΠ . Therefore,theΠ partofdiagram(a)hasorderO(q4). Diagram(c)containsthe 1 1 contributionsofthecounterterms. Indiagram(b)wetakeM =M . ω R 4 Complexmassrenormalization J.Gegelia WefixthecountertermssuchthatthepoleinthechirallimitstaysatM2. Thecontributionsof R diagrams(a),(b),and(c)tothepole,expandeduptoO(q4),read (cid:16) (cid:17) z(1)= g2M4 (cid:32)3−2lnM2 −2iπ(cid:33)−g2ωρπM3Mχ −g2ωρπM4 lnMMχ22 −1 +ig2ωρπM3Γχ . 16π2M2 M2 24π 32π2 48π R χ (2.14) 3. RoperResonance The most general effective Lagrangian, relevant for the subsequent calculation of the pole of theRoperpropagatoratorderO(q3)reads: L =L +L(2)+L +L +L , (3.1) 0 π R NR ∆R whereL isgivenby 0 L = N¯ (i∂/−m )N+R¯(i∂/−m )R 0 N0 R0 (cid:20) (cid:21) −Ψ¯µξ23 (i∂/−m∆0)gµν−i(γµ∂ν+γν∂µ)+iγµ∂/γν+m∆0γµγν ξ32Ψν. (3.2) Here,N andRdenotenucleonandRoperisospindoubletswithbaremassesm andm ,respec- N0 R0 tively. Ψ arethevector-spinorisovector-isospinorRarita-Schwingerfieldsofthe∆resonance[20] ν withbaremassm∆0 andξ23 istheisospin-3/2projector(seeRef.[10]formoredetails). TheinteractiontermsL ,L ,andL areconstructedfollowingRef.[21]. Atleadingorder R NR ∆R g L(1)= RR¯γµγ u R. (3.3) R 2 5 µ Thenext-to-leading-orderRoperLagrangianisgivenby L(2)=c∗ (cid:104)χ (cid:105)R¯R, (3.4) R 1,0 + wherec∗ isacouplingconstantandχ =M2(U+U†). Thenucleon-Roperinteractionreads 1,0 + g L(1)= NRR¯γµγ u N+h.c.. (3.5) NR 2 5 µ Finally,theleading-orderinteractionbetweenthedeltaandtheRoperisgivenby L∆(R1)=−g∆RΨ¯µξ23(gµν+z˜γµγν)uνR+h.c., (3.6) wherewetakethe”off-mass-shellparameter”z˜=−1. Torenormalizetheloopdiagrams,weapplythecomplex-massrenormalizationandwrite: m = z +δz , R0 χ χ m = m+δm, N0 m = z +δz , ∆0 ∆χ ∆χ c∗ = c∗+δc∗, 1,0 1 1 ··· , (3.7) 5 Complexmassrenormalization J.Gegelia (a) (b) (c) Figure2: One-loopself-energydiagramsoftheRoper. Thedashed,solid,double-dashed,anddouble-solid linescorrespondtothepion,nucleon,Roper,anddelta,respectively. where z is the complex pole of the Roper propagator in the chiral limit, m is the mass of the χ nucleoninthechirallimit,andz isthepoleofthedeltapropagatorinthechirallimit. ∆χ WeorganizeourperturbativecalculationbyapplyingthestandardpowercountingofRefs.[22], i.e., an interaction vertex obtained from an O(qn) Lagrangian counts as order qn, a pion propaga- tor as order q−2, a nucleon propagator as order q−1, and the integration of a loop as order q4. In addition, we assign the order q−1 to the ∆ propagator and to the Roper propagator (carrying loop momenta). Withinthecomplex-massrenormalizationscheme,suchapowercountingisrespected intherangeofenergiesclosetotheRopermass. ThedressedpropagatoroftheRoper i iS (p)= , (3.8) R p/−z −Σ (p/) χ R whereΣ (p/)denotestheself-energy,hasacomplexpolewhichisobtainedfromtheequation R z−z −Σ (z)=0. (3.9) χ R ToorderO(q3)theRoperself-energyconsistsofatree-ordercontribution Σ =4c∗M2, (3.10) tree 1 andtheloopdiagramsshowninFig.2. Forthediagrams(a),(b),and(c)ofFig.2weobtain Σ = 3g2NR (cid:2)Oˆ (m)A (cid:0)m2(cid:1)+Oˆ (m)A (cid:0)M2(cid:1)+Oˆ (m)B (cid:0)p2,m2,M2(cid:1)(cid:3), (3.11) (a) 128π2F2 1 0 2 0 3 0 Σ = 3g2R (cid:104)Oˆ (z )A (cid:16)z2(cid:17)+Oˆ (z )A (cid:0)M2(cid:1)+Oˆ (z )B (cid:16)p2,z2,M2(cid:17)(cid:105), (3.12) (b) 128π2F2 1 χ 0 χ 2 χ 0 3 χ 0 χ g2 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) Σ = ∆R Oˆ +Oˆ A z2 +Oˆ A (cid:0)M2(cid:1)+Oˆ B p2,z2 ,M2 , (3.13) (c) 48π2F2 4 5 0 ∆χ 6 0 7 0 ∆χ where (cid:18) x2(cid:19) Oˆ (x) = p/ 1+ +2x, 1 p2 (cid:18) x2(cid:19) Oˆ (x) = p/ 1− , 2 p2 (cid:34) (cid:35) (cid:18) x2(cid:19)2 (cid:18) x2(cid:19) Oˆ (x) = p/ −p2 1− +M2 1+ +2M2x. 3 p2 p2 6 Complexmassrenormalization J.Gegelia (cid:34) (cid:35) 1 p2−3M2 2(p2)2−3M4−8p2M2 Oˆ = 3p/z2 −12p2z −4p/p2+4p2 +p/ , 4 6 ∆χ ∆χ z z2 ∆χ ∆χ Oˆ = 1 (cid:34)p/z2 +2p2z −p/(cid:0)2M2+p2(cid:1)+2p2p2−M2 +p/(cid:0)M2−p2(cid:1)2(cid:35), 5 p2 ∆χ ∆χ z z2 ∆χ ∆χ (cid:34) (cid:35) 1 M2+p2 M4−3p2M2−(p2)2 Oˆ = − p/z2 +2p2z −2M2p/−2p2 +p/ , 6 p2 ∆χ ∆χ z z2 ∆χ ∆χ Oˆ = − 1 (cid:104)p/z2 +2p2z +p/(cid:0)p2−M2(cid:1)(cid:105)(cid:34)z2 −2(cid:0)M2+p2(cid:1)+(cid:0)M2−p2(cid:1)2(cid:35). 7 p2 ∆χ ∆χ ∆χ z2 ∆χ To implement the complex-mass renormalization scheme, in analogy to Ref. [6], we expand the self-energy loop diagrams in powers of M, p/−z , and p2−z2, which all count as O(q). We χ χ subtractthosetermswhichviolatethepowercounting.Thesubtractiontermsat p/ =z read χ 3g2 (m+z )2(cid:20) (cid:16) (cid:17) (cid:21) ΣST = − NR χ (m−z )2B z2,0,m2 −A (cid:0)m2(cid:1) (a) 128π2F2z χ 0 χ 0 χ 3g2 (m+z )M2(cid:20) m + NR χ −2m3ln −iπm3+z2m−32π2z3λ 64π2F2z3 µ χ χ χ (cid:35) (cid:16) (cid:17) z2 −m2 + m3−z3 ln χ +iπz3 , χ µ2 χ 3g2z (cid:16) (cid:17) 3g2z M2(cid:20) z (cid:21) ΣST = R χ A z2 − R χ 32π2λ+2ln χ −1 , (b) 32π2F2 0 χ 32π2F2 µ g2 (cid:104) (cid:16) (cid:17) ΣST = − ∆R 6(z −z )2(z +z )4B z2,0,z2 (c) 288F2π2z2 z ∆χ χ ∆χ χ 0 χ ∆χ ∆χ χ +z2(cid:0)−3z4 +12z z3 +4z2z2 −4z3z −2z4(cid:1) χ ∆χ χ ∆χ χ ∆χ χ ∆χ χ (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) −6 z4 +2z z3 −z2z2 +2z3z +z4 A z2 ∆χ χ ∆χ χ ∆χ χ ∆χ χ 0 ∆χ g2 M2 (cid:20) z + ∆R −6iπz6 −6(2z +3z )z5 ln ∆χ −9iπz z5 +6z2z4 72π2F2z2 z3 ∆χ ∆χ χ ∆χ µ χ ∆χ χ ∆χ ∆χ χ +9z3z3 +3z4z2 −288π2λz5z +9iπz5z +z6 −192π2λz6 χ ∆χ χ ∆χ χ ∆χ χ ∆χ χ χ (cid:35) (cid:16) (cid:17) z2 −z2 + 6z6 +9z z5 −9z5z −6z6 ln χ ∆χ +6iπz6 . (3.14) ∆χ χ ∆χ χ ∆χ χ µ2 χ The above expressions of Eq. (3.14) are exactly canceled by contributions of δz and δc∗. The χ 1 poleoftheRoperpropagatortothirdorderisgivenbytheexpression z=z −4c∗M2+(cid:2)Σ +Σ +Σ (cid:3) −ΣST−ΣST−ΣST. (3.15) χ 1 (a) (b) (c) p/=z (a) (b) (c) χ ItiseasilyshownthattheexpansionofEq.(3.15)satisfiesthepowercounting,i.e.isofO(q3). 4. Summary WehaveconsideredaneffectivefieldtheoryofresonancesinteractingwithGoldstonebosons using the complex-mass renormalization scheme. A systematic power counting emerging within 7 Complexmassrenormalization J.Gegelia this scheme allows one to calculate the physical quantities in powers of small parameters. As an application we have calculated the pole masses and the widths of the ρ meson and the Roper resonance which are of particular interest in the context of lattice extrapolations. The masses and thewidthsinthechirallimitareconsideredasinputparameterswithinthisapproach. 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