PoS(Bormio2012)024 http://pos.sissa.it/ and Carla Terschlüsen ∗ [email protected] Speaker. The assumption that vector dominate(vector- the dominance interactions — of VMD) provides withother an hand, important a phenomenological clear microscopic concept. derivation iscally On fails, still the e.g. missing for and the there omega are transitionsystematic cases form expansion where factor. and VMD In power principle, drasti- counting effective field could theories providemore with a generally their tool to to describe assess the the interactions validity ofatic of vector development VMD mesons and is at low still energies. in Thoughand an the vector system- infant mesons stage which we is presentused inspired to here calculate by a electromagnetic ideas meson Lagrangian form from for factors.successes light effective It of turns field pseudoscalar out VMD theories. that concerning one can the The reproducethe Lagrangian both omega form the transition is form factors factor. and the deviations from VMD concerning ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

50th International Winter Meeting on Nuclear23-27 Physics January 2012 Bormio, Italy Stefan Leupold Department of Physics and Astronomy, UppsalaE-mail: University, Sweden Towards an effective field theory for vector mesons PoS(Bormio2012)024 , the 0 ]. c) A π 3 − , ]. In the e 2 ]. Follow- 5 1 → Stefan Leupold γ ]. 6 − where the VMD e 2 . It constitutes one of the − π ]. 4 + π → ]. For the eta transition form factor, 8 − , e shows how well VMD works for this 7 + 1 e [GeV] 2 2 q √ ]. Figure 5 0 , VMD works very well as can be seen on the right-hand γ Barkov 85 present VMD − µ = 4.93 ρ + g g = 6.05 µ 0.68 0.73 0.78 → 50.0 40.0 30.0 20.0 η . This is displayed one the left-hand side of figure 0 2 | π π |F The pion form factor as compared to VMD. Figure taken from [ − µ + µ . Also for the pion transition form factor, extracted, e.g., from → 2 Figure 1: ω So, what do we know about the interactions between hadrons and real or virtual ? The A good understanding of the interactions of hadrons with electromagnetism is a key ingredient The pion form factor is determined by the reaction traditional successes of the VMD scenario [ form factor. In contrast, VMD drastically fails for the omega transition form factor, extracted from neutral vector mesons which have thethe same . quantum numbers Indeed, as these the vector photonand mesons can are decay couple prominently directly rates. seen to in the This corresponding gave cross sections rise to the notion of vector-meson dominance (VMD) [ promising candidate to search for physicsthe beyond . the At standard present model the isis largest the provided uncertainty by gyromagnetic on the ratio the hadronic of theory contribution side to for this gyromagnetic the ratio standard-model [ calculation the reaction Towards an effective field theory for vector mesons 1. Introduction for several areas of hadronthat and the physics. has To an nameup intrinsic a structure experiments few came for examples: from elastic elastic a)structure, and -proton The scattering (deep) first nowadays [ indication quantitatively inelastic encoded scatteringgeneralized in revealed parton structure more distributions and functions, and so on. more electromagnetic b)placed of form The in change the factors, a of strongly proton the interacting properties environment of isstar hadrons an physics. once active they Such field are in-medium of properties research of in hadrons heavy-ion can and be - accessed in dilepton spectra [ following, we will concentrate on electromagnetic mesonhow form well factors. such We form start factors with a are brief described review by VMD. prediction is compared to data fromextracted NA60 from and the reaction G [ side of figure VMD scenario works well. Here, a key quantity is the slope of the form factor at vanishing mass PoS(Bormio2012)024 basebase Entries Entries Mean x Mean x Mean y Mean y 0 0 0 0 RMS x RMS x 0 0 RMS y RMS y 0 0 0 0 (1.1) (1.2) (1.3) -2 M (GeV) Stefan Leupold -2 0.04GeV ± -2 0.17 0.4 GeV ± ± γ - µ =1.8 GeV =1.95 =1.90 + -2 η -2 η -2 η µ Λ Λ Λ → η pion transition form factor is of ] VMD : NA60 : Lepton G: 9 ]. Therefore, it is important to clarify

0 0.1 0.2 0.3 0.4 0.5 0.6

1

4

η 10 |F | .

GeV. 2 0 ) = s 77GeV 05

. . ) 0 0 double-virtual s ( ± ≈ ds 3 dF V 76 . m 0 = : basebase = -meson mass) Entries Entries Mean x Mean x Mean y Mean y 0 0 0 0 RMS x RMS x 0 0 RMS y RMS y 0 0 0 0 pion transition form factor, i.e. the electromagnetic tran- 2 have to be studied. We will come back to this issue below. ρ = ( − 0 Λ π VMD PDG Λ + -2 e Λ − M (GeV) e -2 → denotes the 0.02 GeV single-virtual + V ± -2 e m − e 0.06 0.21 GeV ± ± 0 π - ]. and 8 µ =2.24 =1.68 GeV =2.36 + − -2 ω -2 ω -2 ω µ e Λ Λ Λ 2 → + ω e ]). This issue has not been settled yet from the experimental side. In principle, reactions 2 The transition form factors of omega to pion (left) and of eta to photon (right) as compared to 12 NA60 : VMD : Lepton G: → ,

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1

2 3

In the following we will describe first steps towards an effective field theory for vector mesons.

11 ω π 10

|F | , 10 10 2 10 2. Effective field theories One aim of such an enterprise is the reliable calculation of electromagnetic form factors and other considerable interest. For example, it provides antering important contribution part to of the the hadronic magnetic light-by-lightwhether scat- moment the of double-virtual pion the transition muon form [ [ factor can be reliably described by VMD (see, e.g., like which is in excellent agreement with the experimental result [ This quantity concerns the The VMD prediction is ( Figure 2: VMD. The only theory curves areFigures the taken dashed from VMD [ lines. The full and dash-dotted lines are fits toof the the data. virtual photon, sition between a pion and a real photon. Also the Towards an effective field theory for vector mesons PoS(Bormio2012)024 Then, the 1 Stefan Leupold , if the Taylor expansion in the in- effective 4 quantity the Taylor expansion (power counting) is based on? In the numerator possible terms which are in agreement with the symmetries of the system. all dimensionless For the in the sector of hadrons made out of light this includes , charge conjugation The basic idea of an effective field theory is to perform a Taylor expansion in the involved mo- 1 and approximate chiral symmetry. one has the momenta, theThis so-called “soft “hard scale”. scale” But needs whichthen to quantity the be appears effective significantly in field the larger theoryspecified denominator? than can the work. the respective Lagrangians. considered This momentum discussion region. will be Only continued below when we have Towards an effective field theory for vector mesons quantities where vector mesons constitutepossible important to degrees figure of out where freedom. or under Inbe which that conditions stressed way VMD right works it and away should where that be itare does at encouraging not. present It — this should as developmentlow-energy will is expansion in be is an shown feasible infant below and stage. —important. convergent it Below, Though for a has first Lagrangian the not for results energy lightis been region inspired vector established by and where the yet pseudoscalar ideas vector of mesons that mesons effective will fieldthis a are theories. be Lagrangian systematic presented Without an might which established still systematic power be counting tree-level regarded model. at However, present the as ambitionunder of a way this — to enterprise rather scrutinize is successful the higher validity — and of phenomenological one-loop this approach calculations as are a systematic effectivementa. field theory. (For this general outline— we see do the not more specify detailed whether discussionprocesses we below.) with talk This low enough is about momenta. a four- reasonable ortively. The We approach three-momenta crucial will as question come long is back as what to oneand “low this sticks a enough” question phenomenological to means below. model quantita- The (like, difference e.g.,considers between VMD) is an that effective for field the theory constructionbasic of idea the Lagrangian is one to classifysimply these is possible that terms derivatives translate according to to momenta.has, their Thus, the the number less more of derivatives important a derivatives. it term The is inory reason the for is Lagrangian that low-energy one processes. can The systematically genericexpansion. improve a feature In calculation of by that an working effective sense out field it theone the- next is does order like in not an the expand ordinary Taylor inmomenta. perturbative field powers Typically, theory. of higher The the powers difference coupling incontrast, is constants the in just a Taylor but that phenomenological expansion in model involve powers one higher-loopto of demands tree calculations. from level the or outside In involved to to (small!) one-loop restrictAn level the without effective calculations, a field e.g., reason theory why is thewith neglected defined a contributions by should power the be counting small. most (andclassified general the according Lagrangian quantization to procedure). one the can All power write counting. emergingpansion down Feynman which Therefore, diagrams diagrams together one contribute. can knows In be at contrast, whichLagrangian a together order phenomenological with in model a the is “by-hand” Taylor defined prescription ex- by whichIn Feynman a that diagrams specific should sense be an considered. effective fieldother theory is hand, systematic one while cannot a always phenomenologicalpoint formulate model is is an that not. an effective effective On field field the theory theoryvolved momenta is for converges only a (fast feasible, enough). given i.e. Therefore energy theis region. quantitatively the important The question is: What PoS(Bormio2012)024 , ω (2.3) (2.4) (2.5) (2.6) (2.1) (2.2) ’s, ∗ K ] 20 ’s, , ρ Stefan Leupold the pion mass. 19 , π ) is caused by the m 18 . Such an extension , 0 2.4 η 17 ) in ( and † u η . 2.1 ) ) U U µ + . D † ( , † U µν , ( u µ ! tr 1 2 A ! ] 2 π ω + + = m U 0 ! , + π ρ 2 µ π 1 3 f 0 2 2 Q 0 [ U − ρ − 1 4 √ √ ie − 2 3 5 − to the physical states 0 + ) + + − 0 π ω , µ η U ρ 2 π )

+ µ f U 2 √ 0 ∂ † / µ = ) together with the nonet of vector mesons ( √ ρ

U Φ = η Q ( i

( = tr U ’s, 2 µ = Φ f K ]. exp D 90MeV denotes the pion decay constant and µν = ’s, 16 = V ≈ ]. We shall briefly review the respective leading-order chiral La- π 2 f PT u χ 20 , = L . We have neglected breaking and external fields other than the 19 U µ , A , the charge matrix 18 ] which included three active quark flavors — up, down and strange. There, the e , 12 17 , and its mixing with the octet PT Lagrangian relevant for the present purpose is given by [ χ 15 η , PT) [ 14 χ , 13 In the following we restrict ourselves to mesons built out of the lightest two quark flavors, For the energy region where the light pseudoscalar mesons are the only relevant degrees of In the sector of natural parity — where one only has terms with an even number of pseu- ) has been taken into account. However, a fully consistent framework had to include the full fla- spontaneous breaking of chiral symmetry. One implication is that already the chirally extended The pion fields are encoded in and the photon field electromagnetic one. The non-linear appearance of the pion fields ( with the gauge covariant derivative the charge Here, “tr” is the flavor trace, vor nonets of pseudoscalar and vectorflavor mesons. singlet This implies that one needs to incorporate also the ϕ is presently under construction [ freedom the appropriate effective fieldtheory” theory ( is well established;grangian it for is the called sector with “chiral an perturbation Afterwards even and we with shall an discuss odd the number inclusion of of pseudoscalar vector mesons, mesons. respectively. doscalar mesons and where onleading-order the formal level no Levi-Civita tensor appears — the part of the Strictly speaking, most of the resultswork which [ will be presented below have been obtained in a frame- namely the pseudoscalar , encoded in the flavor matrix and the rho and omega vector mesons, collected in octet of pseudoscalar mesons ( Towards an effective field theory for vector mesons PoS(Bormio2012)024 ) . ] 2.3 Φ (2.7) µ which ∂ ) are all ) , 2 p Φ 2.3 ( ][ O is the leading Stefan Leupold Φ . In the second ) µ p 2 ∂ p , . Thus, there are ∼ ( ) Φ [ 2 O 2 p 1 f ( O ∼ breaks down if the loop diagrams p ) have formally the same order. is limited for, at least, two reasons. p 2.5 contributions. They emerge from one- ) and from tree-level diagrams based on ) 4 2.3 p ( (or smaller). As a consequence also the pion O p 1GeV. 6 to such reactions, then all involved particle energies ≈ p because of the Goldstone- nature of the pions. f ) ]. This restricts the predictive power of more accurate π ) is the most general Lagrangian of order 0 . The terms in the leading-order Lagrangian ( 4 p 22 p . They can be adjusted to data or ideally deduced from the ( because of Lorentz invariance. Thus, 2.3 , π O ) ∼ 1 m 21 π p ( m O . Therefore, a tree-level calculation is sufficient at low enough energies. , contains interactions with a fixed interaction strength given by the pion ) ) 4 µ p ( U † µ O U and the pion mass ( tr f : In the first term the derivatives translate to momenta which are 2 ) f 2 p ( If one assigns a generic momentum O 2 What “small enough” means quantitatively will be specified in a moment. If one wants to describe reactions at somewhat higher energies or the low-energy reactions For reactions of pions with pions and/or photons the tree-level diagrams emerging from the There can be no terms of order A second limitation comes from the fact that only the pions (or, for three flavors, the pions, However, there is an additional problem if one would like to increase the applicability range 2 . In practice, one limitation comes from the plethora of additional parameters which come in . In that way, both terms in the gauge covariant derivative ( p p mass should be of this order, i.e. and three-momenta should roughly be as large as with larger accuracy, then one has to work out the leading-order Lagrangian provide a decentenough. description — as long as the reaction energy is small are at least of order loop diagrams based on the leading-order Lagrangian ( In the following, this scale“intrinsic emerging hard from scale”. a comparison of tree-level and loop diagrams iskaons called and eta) are taken into account. Therefore, the convergence of the series must break down in the next-to-leading-order Lagrangian. The latter contains allmasses. terms with It four derivatives involves and/or additional pion low-energydata constants or which deduced again needed from to QCD. be In fitted to principle, additional onewith the can higher-order continue Lagrangians this [ calculations. procedure to higher powers of to higher energies. The convergence of the power series in is in agreement withcharge the conjugation symmetries and of approximate the chiralpions strong symmetry. as interaction, For active i.e., the degreesonly for given of setup, two the freedom, i.e. parameters case considering there (“low-energy at only decay constants”) are hand, constant governing no parity, the other lowest-order termsmicroscopic Lagrangian: of theory, the QCD. order pion Once thesepower for parameters low-energy are reactions. determined, In addition, the one Lagrangian can show has that predictive loop diagrams emerging from ( One limit emerges from the loop diagrams.become The as power important series as in the treechiral diagrams perturbation based theory on one the finds same that Lagrangian. the scale If is one set works by that out for decay constant. For example, the corresponding four-point interaction is Towards an effective field theory for vector mesons kinetic term, of order order and it is important to note that ( There can be no terms of order term one has the square of the pion mass. It is advantageous to count also the photon field as order PoS(Bormio2012)024 5 denotes V m Stefan Leupold -momenta of the . The mass(es) of p three ]. Obviously the expansion of 24 ] and references therein). If this is , 26 23 , 25 , cf., e.g., the right-hand side of figure ) ) set by the loops. 2 V m 2.7 − 2 ]. This external hard scale seems to set the actual p 9 7 ( / PT. The lowest not considered degree of freedom is χ what the phrase “an effective field theory works at low one can expand this vector-meson propagator in a power V m  p quantitatively meson. It contributes to pion-pion scattering and the corresponding Feynman . For light degrees of freedom all masses, energies and momenta should be ρ p . To turn the argument around, the masses and momenta define the soft scale denotes the typical energy/momentum of the considered reaction, and p . In chiral perturbation theory, where vector mesons are not considered explicitly, p 2 V m / 2 ]. Here it is crucial that the cannot decay into mesons, in particular not into p 20 What was presented so far is the conservative picture. However, there might be some twists We have determined In principle, it is tempting to extend chiral perturbation theory by including heavier states. In For additional heavy degrees of freedom a power series in terms of the . This makes only sense if the soft scale remains significantly smaller than the intrinsic hard p true, then there is no compelling reason to include the sigma as an explicit degree of freedom in an light mesons. Otherwise the scalesproducts are inherit intertwined, the large because scale the of three-momentathe the of inclusion decaying of the heavy state. vector light mesons This decay so is challenging one (see of the below). aspects which makes limit because it is lower than the intrinsic hard scale ( to the previous arguments: Thereantiquark are state but studies rather which a two-pion suggest correlation that (see, e.g., the [ sigma meson is not a quark- enough energies” means for strict mesonic diagrams contain a vector-meson propagator, 1 below. Again, Towards an effective field theory for vector mesons the energy region where alsomight other consider mesons the become active degrees of freedom. For example, one the vector-meson mass. For series of the vector-meson propagator breaks downof if the the vector typical meson. momentum becomes Moreenergetic as generally, limit large the as to masses the the of mass the applicabilitymass gap not range between considered the of degrees heaviest an considered of andtheory. effective freedom the field For set lightest chiral theory. not an considered perturbation state This theory ofall it range the the is effective scales field other very with mesons advantageous because the thatto of the the the pions neglected Goldstone-boson are degrees character much of freedom lighter of is than the called pions. “external hard The scale” scale inthat connected what way follows. one could include moreto processes higher and it energies. might be There possibleto are to include push several these the aspects additional external to degrees hardmomentum of scale consider: freedom scale One as light has or to as decide heavy whether as one compared to wants the considered the heavy states aretheory part [ of the hard scale. This is the framework of chiral perturbation the sigma meson with a mass of about 600 MeV [ such contributions are encoded in the low-energy constants [ of the order of scale introduced above. This providesconsidered a as limit “light”. for In thebetween addition, masses the the of heaviest states convergence of of which the the couldperturbation power included be theory series reasonably states the requires large and gap a the emergesthe significant lightest from rest gap the of of fact the the that not one spectrum includedbe deals the picked states. with identification up Goldstone of below. For . a chiral sizable For gap is not so obvious. Thisheavy issue states will and the four-momenta oflight the states light and states the might involved be three-momenta reasonable. of Here all the states masses of define the the soft scale PoS(Bormio2012)024 ) 2.7 (2.8) µναβ ε PT up to ]. It will χ 20 PT, the sigma Stefan Leupold χ . For the present ) 4 p ( O below. PT. The power counting 4 ] and references therein). χ PT, has been introduced. It 28 χ ) Φ 2 Q ( tr . This parameter-free Lagrangian pro- µ αβ A F ν ∂ µν − F ν A ]. It is fully determined by the chiral anomaly and 8 µ µναβ ∂ 30 ε , 2 = f e PT have produced very promising results in the last few 29 term given by χ 2 µν F π γγ 3 0 32 π = WZW L Born series, i.e. really solve the Lippmann-Schwinger equation instead ). The next external hard scale is provided by the vector mesons. From all 2.7 whole ]. In spite of the mentioned ambiguities the common feature of such approaches Born series applied to the problem of a or a resonance. One rather 26 , 25 , 27 directly applied to the observable quantities like the scattering amplitude. This would be truncated We have seen that it might be possible to circumvent both limitations of strict In the last section the effective field theory for Goldstone bosons, In the sector of anomalous parity — where on the formal level a Levi-Civita tensor not vides an excellent description of the decay of the neutral pion into two real photons [ mass and the scale ( with the electromagnetic field strength once the two-particle reducible diagrams are properly resummed. Therefore, they need towhich be is supposed included to as work inis explicit the a degrees region of of of low-lying freedom mesonic activein resonances. research. in the If next an Further this section. discussions is effective Before, possible andnumber Lagrangian however, at of first the all pions steps leading-order will towards chiral be this Lagrangian briefly discussed. for goal the are sector presented of anis odd involved — the leading-ordergiven contribution by to the the Wess-Zumino-Witten action effective [ fieldfree theory of for any Goldstone undetermined bosons low-energy is purpose parameters. the only relevant It part contributes is at the order constitute one contribution to the pion transition form factor, see section 3. The vector-meson Lagrangian and its parameters has also been used to illustrate the general ideas of an effective field theory. For the formulation of we know about QCD at present,correlations. it is They highly are unlikely that dominantly the quark-antiquark vector states mesons (see, are two- e.g. or [ three-pion is the resummation ofcorrespond specific to rescattering). classes In of suchOne diagrams a needs framework (the the to two-particle intrinsic compare hard reducibleanalysis scale the has diagrams needs size not to which been of be carried re-evaluated. two-particle out irreducible yet. loops However, the and successes tree-level of unitarized terms. versions of Such an Towards an effective field theory for vector mesons effective Lagrangian. An illustrative analogy might befrom the a deuteron calculation which based emerges as on a a boundway, Lagrangian state the which contains sigma only could appear (andnot mesons). “dynamically” contain In in the a the similar sigma pion-pion meson.is phase shift. However, such The a Lagrangianlike formalism might a can only workhas if to the sum power up counting the of a perturbative treatment. This translatesshould to be the applied demand to to the “unitarize” other determination words, of rescattering scattering processes kernelsconcept need and is to not unsatisfying to be because there scattering resummed.On are amplitudes. the ambiguities From other in In hand, the the unitarized unitarization/resummation versions puristic of procedure. point of view this energies of 1 GeV and beyond suggest that the actual intrinsic hard scale is higher up than ( years [ PoS(Bormio2012)024 ]. 34 (3.1) ] and ] it has 13 emerges 13 ) of (strict) , 2.7 Stefan Leupold 33 , 32 , . 31 µν F ) Q the same Lagrangian µν V ( tr V m ). Slightly exaggerating, one might say V The first Lagrangian concerns the decays 2 is already soft. This suggests to discuss e / 3 2.7 8 1 V m − ]) ν 9 U , µ U [ µν V ( tr V m ]. Based on the hadrogenesis conjecture [ , is still heavy while P V 26 h m , i 4 15 − , = 14 1 L More vector-meson fields are not needed at tree level for the processes of interest. The leading-order Lagrangian collecting all terms with one vector-meson field is given by In the following, only preliminary answers can be provided which need to be further scruti- The inclusion of vector mesons as light degrees of freedom has been proposed in [ Whether the intrinsic hard scale is significantly larger than the vector-meson masses needs to The inclusion of vector mesons as strict heavy degrees of freedom has been pioneered in [ PT suggest that the intrinsic hard scale could be significantly larger than 1 GeV. This would 3 χ PT. In that sense they are heavy. On the other hand, if vector mesons decay, the momenta of the χ of vector mesons while theabsorbing second or emitting Lagrangian light concerns . the Whethera propagation this systematic pragmatic of approach point vector for of loop mesons view is calculations, while feasible, needs i.e. to provides be seen in the future. separately the Lagrangian which includes the termswhich with one contains vector-meson the field terms and the with Lagrangian two vector-meson fields. The focus has been set to reactionsthe where whole a constant respective number process. of vector In mesonshas remained that been throughout avoided way, by the hand. problem In howare the neither to present truly deal work heavy, nor with a truly more decaying light. pragmatic vector Their way mesons mass is is suggested: close Vector to mesons the intrinsic hard scale ( decay products are still small compared to the scale ( that the vector-meson mass, further explored in [ Towards an effective field theory for vector mesons an effective field theory forquestions: the energy Should region of the vector vectorlight, mesons mesons which one other been has mesons introduced to limit consider the asconsidered applicability the light and range, following i.e. or not is considered heavy there states? degrees adeal large of If with enough the light, freedom? gap fact where between that is the If vector the mesons intrinsic can decay hard into scale? light If degreesnized of heavy, in freedom how the (pions, to future. photons)? However, one remarkable(at feature the is respective that leading basically order), no matterdegrees whether of one treats freedom! the vector This mesons gives asexplain some light its credit or phenomenological as to heavy success the even proposedtablished if Lagrangian yet a and (or might full-fledged maybe to effective even field some cannot theory extent be has established). not been es- been argued that the other mesonsgenerated in from the the energy coupled-channel range close interactionssuch to states of the need pseudoscalar vector not mesons and be are considered vectora dynamically explicitly mesons. sizable in gap the Therefore, between effective Lagrangian. thethe This energy not implies region considered that of other there (quark-antiquark) the is states. lowest-lying vector and pseudoscalar mesons and be explored in loop calculations.tion It of should be two-particle stressed reducible again diagrams,in that is exact the mandatory unitarity, region for i.e. of an the hadronic resumma- effectiveof resonances. field theory As which already operates mentioned,provide the some successes room of where an unitarized effective versions field theory of light vector mesons can be applicable. PoS(Bormio2012)024 . ]. V 13 . m [ (3.2) ]. ]. As ) . 2 − 28 23 p e  , ( β + e interaction O 19 U A } → h Stefan Leupold . In the latter ]. The phrase λα ρ µ V 13 ], e.g. one would V λ , and 24 D ]. The ] is not considered in , 23 π [ 2 20 µν 35 V µν → ) are the only ones which { ], the counting is the same, ], one counts all derivatives V ) and is of order ρ  3.1 13 13 tr 3.1 ) is the complete leading-order . Note that the process proceeds 3.2 3 ). It is important to stress that this µναβ ω ε A 3.1 ]. h PT Lagrangian [ i 15 8 χ , . Here, it is crucial that the vector mesons ) . . There are several more terms of this order. 13 2 ) + ) ) p 3 2 ( term in ( µν p p ( O ( P V 10 O h O ρ µν V ( tr 2 V ). The terms appearing in ( m 3.1 8 1 term of pion- interaction. ¡ 0 γ A can be determined from the decays can be fitted to the decay of an omega meson into a pion and π g ) + instead of the V A e h ) να 3 V p ν . The complete Lagrangian is given in ( ( and , then one might count all derivatives (and the photon field encoded p D p O P h ) as soft in ( Two-step decay of an omega meson into a pion and a real photon. µα ∼ ∼ µν terms are possible for structures with only one . In any other V ]) F µ ) ν 2 D U p ( , ). The Lagrangian is of order ( tr µ O 4 1 Figure 3: 3.2 U [ − ν V = µ ∂ 2 ( Lagrangian. It resembles the case of the nucleon L If the vector mesons are considered as (semi-)heavy, then ( If the vector mesons are considered as light degrees of freedom [ The single free parameter If one considers the vector mesons as light degrees of freedom [ The two parameters For structures with two vector mesons the pertinent Lagrangian is ) p ( a consequence the masses of rho and omega meson agree in this approach. Both are given by term translates to the well-known appearing in ( a real photon. The corresponding diagram is shown in figure However, only one contributes tothe the pion processes mass of and interest is here. numerically rather This unimportant term [ involves the square of Terms with more than one flavor trace have been neglected on account of the OZI rule [ have one vector-meson field and appear at order have tr statement applies to vector-meson fields whichchiral transform transformations. as The ordinary treatment matter of fields vector with mesons respect as to gauge bosons [ all derivatives are of order The same results, but with higher accuracy, are obtained from a fit to the pion form factor [ O are represented by anti-symmetric tensor fields and not, e.g., by vector fields, representation such structures would be at least Towards an effective field theory for vector mesons The vector mesons are“leading represented order” by has anti-symmetric thein tensor following the fields, meaning: sense that If theiructs one mass are considers is already the large small vector (part of mesons the as hard semi-heavy scale) but the momenta of the decay prod- case one would need more derivatives to construct the corresponding terms [ the present work. Thethat use only of here the anti-symmetric tensor representation is singled out by the fact in the field strength PoS(Bormio2012)024 V V e e (4.1) from | V e · ) provides ]. A direct A 15 h 3.1 | relative to the Stefan Leupold , PT as given in χ A 13 ). This is rather h [ and A 3.2 − h has been determined e is pure convention, but + V ), ( e e P h decay and the constant 3.1 → . . ρ 1 ρπ − , . e 2 − to + π , e + from − | ≈ ω − e | π A e + h V can be used to fix + | e e → e | γ γ and the coupling of the to the , − + → V π e for the → e 0 2 0 + , A 0 , π e π h ∼ to the photon. Since . To pin down the signs of the parameters π − → 30 P l . → ρ h 0 + ρ l ). In the second case the Lagrangian ( 11 ] ω . In the first case, the pions couple via their electric 0 ≈ 4 π 15 2.3 P , h from → 28 p , h ω 13 , 22 ) there are three free parameters. As already discussed they can . ), the process can take place via a direct coupling of the pions to the photon or 0 would involve more than two derivatives, i.e. it is of higher power − ). 3.2 − e γ π + is a physics issue. The same applies to the sign of | ≈ + e 2.8 ), ( + V 0 P e π | h π → 3.1 → ρ → ). − e ω + 3.1 e ) and the sign of the parameter relative to 2.8 ). Their consequences are worked out at tree level or, if needed, including rescattering V e . PT. 2.8 P ) for the direct coupling of the virtual . The numerical results are [ χ h γ one needs more complex reactions. In other words: The sign of 0 3.1 the single-virtual pion transition form factor, the double-virtual pion transition form factor, the pion form factor, contained in the reaction the omega transition form factor, ∼ A π The process Having fixed the parameter values (at least their absolute sizes), no further parameters are To summarize this section: No matter whether one treats vector mesons as semi-heavy or light In the Lagrangians ( In the following, these Lagrangians are used together with the Lagrangians of h ) and ( • • • • → 2.3 Wess-Zumino-Witten term ( the coupling of the virtual photonpions to the rho meson via a vector meson. This is depicted in figure ( It is important to noteresult that of at the no calculations, but stage it any is VMD not assumption put has in by entered. hand. It might come out as a needed to determine charge. This is described by the Lagrangian ( from ( encouraging in view ofscales the of fact that the vector-meson mass is in between the hard and the soft be determined from two-body decays: There is a freedom toof choose vector-mesons the fields and sign the ofpseudoscalar sign one fields. of coupling one constant This coupling which freedom constant multiplies hasWitten which been term an multiplies used odd ( an to number odd specify numberand the of overall sign in thethe Wess-Zumino- sign of degrees of freedom, one ends up with the Lagrangians presented in ( effects. Power counting issues will not be discussed anymore but deferred to4. future work. Electromagnetic form factors Towards an effective field theory for vector mesons in two steps which involves the coupling constants from another reaction ( interaction term for than the terms of ( ω PoS(Bormio2012)024 ) π π 4.1 (4.2) (4.3) . The − π matrix of ρ ) and the + π T ]. ) is given by 2.3 S 28 Stefan Leupold 3.1 ¡ + to 1, i.e. by dropping ) and ( S 2.3 ]. 065. With the values ( ρ to the final state . ). The interaction between 28 0 − ¡ PT Lagrangian ( e 2.3 -wave channel. The χ ≈ + p e ). . Figure taken from [ by putting 2 V = K 4 m 3.1 . / 2 s ), but now including the rescattering of − e f s . 2 V K and the pion form factor requires a resum- s 3.1 16 . The VMD prediction is m + − ] this has been achieved by a Bethe-Salpeter − s ¡ e term of ( e 2 − V = π ¡ 2 V . The scattering kernel consists of two parts: 28 2 P 2 m V P + m 5 h h m π P ) and ( e f V h . The form factor is again normalized to the photon 12 e → 7 16 V 2.3 ) = e s − ( e + π π + 1 e VMD π F ) = s ( π F S ), respectively, conspire such that the final result is close to VMD. to 1). In that way, the pion form factor becomes unity at the photon 3.1 KT . Obviously one obtains an excellent result. Note that isospin breaking S 6 066 which is obviously very close, provided one chooses a positive sign . ¡ 0 + ≈ P h | K V mixing. This has not been included. Therefore, the sharp omega peak seen in the data e | ¡ when putting ω Bethe-Salpeter equation for the rescattering of pions in the The two possible processes which lead from the initial state - 4 = ρ . Then a cancellation takes place: The two terms from the V A reliable description of the reaction The full result based on the Lagrangians ( The next quantity which will be considered is the electromagnetic form factor of the transition Concerning the comparison to VMD it is illuminating to discuss first the tree-level result for The tree-level result is obtained from the diagrams of figure e T in the figure. It is common practice to normalize the pion form factor to the direct term (first term + − ¡ e e mation of the rescattering processes of the pions. In [ in figure S one obtains for pions is shown in figure The two formulae would analytically agree for leads to point. The tree-level pion form factor, as obtained from the Lagrangians ( The of the virtual photon is denoted by cannot be reproduced. between an omega meson and a pion, see figure box denoted by “S” signals the rescattering of the pions. Figure taken from [ The direct contact interaction between the pions is obtained from ( vector-meson Lagrangian ( Figure 5: equation. Schematically this is depicted in figure the pion form factor.compared to Afterwards the the data. full calculation including the rescattering process will be Figure 4: Towards an effective field theory for vector mesons pion-pion scattering is obtained from the scattering kernel denoted by the pions and the rho meson comes again from the ¡ PoS(Bormio2012)024 . 0 − ¯ l K + K ωπ l 0 π → − is given by → e , if not + 0 ω e Stefan Leupold ρωπ ωπ and − . For the form factor, denotes a lepton. π ]. V l e + 28 · π 1 A h ω ω ρ 0.9 data with 0.8 . V e . In principle, the intermediate rho meson 8 0.7 ρ s [GeV] √ 13 0.6 0 0 π π 0.5 is proportional to the product 0.4 8 The pion form factor as compared to data [ + + − − l l 0.3 l l 5 0

45 40 35 30 25 20 15 10 50

π | |F 2 is not sensitive to the pion rescattering effects. Therefore, only the tree-level Figure 6: µ , e Generic picture for the omega-to-pion transition form factor. = l The omega-to-pion transition form factor in the present approach. The coupling Figure 7: The process shown in figure and the direct coupling of the rho meson to the photon by A with the lepton result will be presented in thewhich following. probes The case the would be transition differentwould form for require the a factor process coupled-channel at calculation larger at least invariant with masses. the channels There, a proper description could decay into two pions. Thusmeson the should rescattering be of considered. these pions It which causes has the been width checked, of however, the that rho the decay process point. Thus, it parametrizes theprocess deviation is from described a by structureless the decay. diagram In shown the in present figure approach the Figure 8: h This is beyond the scope of the present work. Towards an effective field theory for vector mesons PoS(Bormio2012)024 (4.4) (4.5) Stefan Leupold ). A follow-up 4.5 ) and one from the 2.8 . The present approach has two . Obviously, the NA60 data are 9 10 0.6 . . s s s ] in figure 2 − V 8 0.4 − + [GeV] 2 m V − ) than by the VMD formula ( l 2 2 V V m + l m m (P1) (P2) 4.4 m 14 NA60 ) = 0.2 s ) = ( s ( stand. VMD ωπ . It is depicted in figure VMD 0 ωπ F − ) provide a parameter independent prediction for the omega F ). Note that the latter contribution vanishes if both photons e 1 + 10 3.2 e 100 3.2

γ , one from the Wess-Zumino-Witten term (

ωπ

| |F ), (

0 ), ( 2 11 → 3.1 0 3.1 π ]. . Thus, the excellent description of the latter process by the Wess-Zumino- γ 15 2 → 0 π The omega to pion transition form factor. The present calculations are depicted by the full (red) Next, the pion transition form factor is considered; at first, the single-virtual form factor which Witten Lagrangian is not spoiled. The pion transition form factor, normalized to the photon point, contributions, shown in figure are real, i.e. for Both form factors are compared to data from NA60 [ line. Figure taken from [ transition form factor: This deviates significantly from VMD: Figure 9: however, this overall factor dropsTherefore, out the because Lagrangians the ( form factor is normalized to the photon point. much better described by the presentexperiment approach using ( dielectrons instead of dimuons is planned by the WASA atcorresponds COSY to collaboration. the process vector-meson Lagrangians ( Towards an effective field theory for vector mesons PoS(Bormio2012)024 (4.8) (4.6) (4.7) 11. Using the Stefan Leupold . 0 ≈ 0 2 } ) ) induce an additional π ) is again numerically 2 π s / denotes a lepton. 3.2 2 l 4.6 e − ), ( 2 V 12 m 4 3.1 V = )( {z 0 m 0 1 2 V s . Another cancellation has taken e ρ π ω A A VMD type − ) h . h 2 2 V s s m ) − − ( | 2 s . s 2 2 V V 2 transition form factor: It is given by s m m + + ) 2 − V } 1 2 2 )( 2 V s s s 1 2 m e V 2 ( s 10. Thus, the formula ( 2 V e A ) m . − 2 V − 2 h m 0 − 2 V s m 2 V 12 2 15 2 V m ≈ m π + ) = m s 2 V 1 ( )( − ( + s e 1 ( | {z 1 1 s 2 V double-virtual A s γ 2 e V 2 0 h − 2 V e VMD | A m − πγ The vector-meson Lagrangians ( h 2 ) = V π m F 2 “half” VMD V s 12 2 m ( m π ( πγ − + + F | ) would agree analytically for Right: + 1 1 1 − l 4.7 l ≈ = ) = ) one obtains 2 s , 4.1 ) and ( 1 s ( 4.6 F Generic picture for the pion-to-photon transition form factor. The Wess-Zumino-Witten action contains a point interaction for one pion and two (real Left: Figure 10: However, the picture changes for the The formulae ( Note that the masseskinematical of situations where rho the and tree-level results omega are notapproach meson enough. to are For the the VMD not comparison result, of distinguished however, the it present here. is most Of transparent to course, look there at the are tree-level formulae. Towards an effective field theory for vector mesons or virtual) photons (wavyinteraction lines). between a pion and two photons. i.e. normalized to the Wess-Zumino-Witten contribution, is given by The corresponding VMD formula is close to the VMD result,place, provided one now between uses the a Wess-Zumino-Witten positive term signmentioned and for in the the vector-meson introduction, contribution. the Asthe VMD already present prediction approach is agrees close also toform with the factor. the experimental result. experimental result Therefore, for the single-virtual pion transition Figure 11: numerical values from ( PoS(Bormio2012)024 , − → e ρ (4.9) + e γ → 0 π Stefan Leupold ]. -wave pion-pion 14 p ) is different from VMD ]. 15 4.8 ]. Like for the pion transition . 36 ) 2 s − 2 V . Obviously, ( 2 m s 4 V )( m ]. 1 s and 13 [ 1 − s ]. γ 2 V 16 0 m 28 π ( → ) = ω 2 s is reproduced very well (not shown here) [ is presently investigated [ , 0 1 0 s π and the corresponding pion form factor is in good agreement ( π − − − π π π + and the corresponding omega transition form factor is much better VMD + + π F π − π lepton) and l → + = ). Also these ingredients have been motivated by effective field theory → l → l 0 ω ( − − π 3.2 e e − l + + e → e + l ω ) and ( → ω 3.1 / 0. Future experimental data for the double-virtual transition form factor are crucial to ρ 6= , 2 π s 2 described than with the vector-meson dominance (VMD) model [ A prediction is provided forwith the the VMD double-virtual prediction. pion transition form factor whichThe disagrees decay rate of The reaction The decay The single-virtual pion transition form factor, as, e.g., measured in the decay The reaction All three parameters of the vector-meson Lagrangian are fitted to the two-body decays with the data. Intimately connected toscattering this phase is also shift a (not good shown description here) of [ the turns out to be close to the VMD result and therefore also close to the data. form factor there is an interplay between thecontribution. Wess-Zumino-Witten term and the vector-meson , It is worth to summarize the comparison to the VMD model: Where the experimental data The respective lowest-order Lagrangians of chiral perturbation theory for the sectors with an The developed formalism can be used to calculate the following quantities: 1 • • • • • • • s are well described by VMDpresent (pion approach contains form two factor terms, and one single-virtual from chiral pion perturbation transition theory form and one factor), including the vector for distinguish between the presentintroduction approach such and data the are also VMDto an scenario. the important gyromagnetic input ratio As for of already the the calculation mentioned muon. of in the the hadronic contribution 5. Further discussion, summary and outlook even and with an oddas number of given pions in have ( been combined with a Lagrangian for vector mesons ideas and different options and openregion problems of concerning the vector proper mesons power have counting beenthat in discussed. the the energy constructed Though vector-meson several Lagrangian issuesvector can are mesons be unsettled, as motivated it light in or is as two appealing (semi-)heavy different degrees way, of treating freedom. the The two virtualities of the photons are denoted by Towards an effective field theory for vector mesons while the VMD formula is simply PoS(Bormio2012)024 , 19 , instead of ) s Stefan Leupold − 2 V m ( / ) s Int. J. Mod. Phys. E , (1985) 301. + 2 V resembling the VMD model, 128 (2009) 260 [arXiv:0902.2547 m ( 677 (2009) 1 [arXiv:0902.3360 [hep-ph]]. 477 , Phys. Rept. Qualitatively Phys. Lett. B , 17 Phys. Rept. , Study of the electromagnetic transition form-factors in Hadrons in strongly interacting matter different from VMD, namely ]. It is not clear whether high-energy constraints should be Effective Lagrangian approach to vector mesons, their structure , University of Chicago Press, Chicago, USA, 1969. decays with NA60 24 0 The muon g-2 π − (1996) 193 [arXiv:hep-ph/9607431]. µ + µ 356 S.L. thanks M.F.M. Lutz for collaboration and discussions. He also ac- (2011) 1. → (1991) 597 (Nobel prize lecture 1990). quantitatively ω 814 Electromagnetic decays of light mesons The CBM physics book: Compressed baryonic matter in laboratory experiments [NA60 Collaboration], 63 Deep inelastic scattering: Experiments on the proton and the observation of scaling . Finally, a better (microscopic?) understanding of the observed cancellations 0 and Currents and mesons η Z. Phys. A γ et al. , − . The former expression provides a much better description of the experimental data. µ ) ]. On the other hand, the other cancellation which takes place for the (single-virtual) + s µ 24 − 2 → V m η Lect. Notes Phys. F. Klingl, N. Kaiser and W.and Weise, decays L. G. Landsberg, R. Arnaldi [hep-ph]]. B. Friman et al., F. Jegerlehner and A. Nyffeler, J. J. Sakurai, H. W. Kendall, S. Leupold, V. Metag and U. Mosel, Rev. Mod. Phys. (2010) 147 [arXiv:0907.2388 [nucl-th]]. In any case it is appealing to have a formalism which is flexible enough to reproduce VMD A challenge for the future is to establish a systematic power counting for the energy region of Acknowledgments: ( / 2 V [6] [7] [8] [3] [4] [5] [1] [2] the present approach provides afactor. parameter However, independent it prediction is form the omega transition form vector mesons. Here calculations beyondare leading an order, important including issue. ininclusion particular A of loop the second calculations, line ofwould be development desirable is to the further appreciate extension the to successes of three VMD flavors and with to see the its limitations. results but can also go beyonddrastically it. deviates Indeed, from for the the data. omegatwo transition Here, which form the could factor present potentially the approach show VMD a prediction provides cancellation only effect. one contribution, not knowledges stimulating discussions with A.Bormio 2012 Kupsc. for their Finally excellent he work. wants to thank theReferences organizers of pion transition form factornot has follow nothing from chiral to symmetry dobehavior alone. of with On the KSFR. way pion In to formapplied derive addition, factor it to the [ is a KSFR to low-energy use relationtheir theory. in microscopic does addition reason Obviously, remains the the an high-energy open observed issue cancellations at present. are very interesting, but Towards an effective field theory for vector mesons mesons. 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