Chiral perturbation theory study of the axial N→Δ(1232) transition
L. S. Geng, J. Martin Camalich, L. Alvarez-Ruso, M. J. Vicente Vacas
aa r X i v : . [ h e p - ph ] F e b October 22, 2018 6:1 WSPC/INSTRUCTION FILE proceeding
Chiral perturbation theory study of the axial N → ∆(1232) transition L. S. GENG, J. MARTIN CAMALICH, L. ALVAREZ-RUSO, and M. J. VICENTE VACAS
Departamento de F´ısica Te´orica and IFIC, Centro Mixto, Institutos de Investigaci´on dePaterna - Universidad de Valencia-CSIC
Received (Day Month Year)Revised (Day Month Year)We have performed a theoretical study of the axial Nucleon to Delta(1232) ( N → ∆)transition form factors up to one-loop order in covariant baryon chiral perturbationtheory within a formalism in which the unphysical spin-1/2 components of the ∆ fieldsare decoupled. Keywords : N → ∆ transition form factors; neutrino-nucleon(nucleus) interaction; Chiralperturbation theory.PACS Nos.: 23.40.Bw,12.39.Fe, 14.20.Gk.
1. Introduction
Nowadays, it is generally accepted that Quantum Chromodynamics (QCD) is thetheory of the strong interaction. It has been very successful and tested to greatprecision at high energies; however, its application in the low energy region of ∼ χ PT) and lattice QCD approach has made possiblea model independent study of the low-energy strong phenomena for the first time.Neutrino physics has made remarkable progress in recent years, as evidenced bythe 2002 Nobel prize in physics (awarded partly to Raymond Davis Jr and MasatoshiKoshiba “for pioneering contributions to astrophysics, in particular for the detectionof cosmic neutrinos”). After many years of experimental (and theoretical) efforts,two facts have been firmly established: (i) neutrino have masses and (ii) differentflavors of neutrino can oscillate into each other. Presently, one of the main goals inthe field is to measure accurately the masses and oscillation parameters. A goodunderstanding of pion production is relevant to reduce systematic uncertainties inoscillation experiments. The axial nucleon to ∆(1232) transition, characterized byfour form factors, plays an important role in this reaction at low Q transfer.1Most of our current (experimental) knowledge of the N → ∆ axial transitionform factors comes from neutrino bubble chamber data.2 The possibility to extractthem using parity-violating electron scattering at Jefferson Lab has been extensivelystudied,3 and could shed new light on the nature of these form factors. Present and ctober 22, 2018 6:1 WSPC/INSTRUCTION FILE proceeding L. S. GENG, J. MARTIN CAMALICH, L. ALVAREZ-RUSO, and M. J. VICENTE VACAS future neutrino experiments (MiniBoone, K2K, Fermilab) could also provide furtherinformation.In the past, the theoretical descriptions have been done using different ap-proaches, mostly quark models (for a review, see Ref. 4). In recent years, therehas been an increasing interest on these form factors. They have been calculated,for instance, using the chiral constituent quark model and light cone QCD sumrules. State of the art calculations within lattice QCD have also become available.5While the axial N → ∆ form factors have been addressed in (tree level)HB χ PT,6 no calculation has been performed up to now within the relativistic frame-work . With lattice QCD results becoming available and in view of the many ongo-ing experimental efforts to extract these form factors from electron- and neutrino-induced reactions, it is timely to study the axial N → ∆ transition form factorswithin covariant baryon χ PT.7
2. Theoretical framework: covariant baryon χ PT with explicit ∆’s
The study of the N → ∆ transition form factors using covariant baryon χ PT ismuch more complicated than it seems to be. To begin with, one has to address thefollowing three questions: power counting, chiral Lagrangians, and the appropriateform of the ∆ propagator.(1) A proper power counting scheme is at the center of effective field theories.To include the ∆(1232) explicitly, one has to count the N -∆ mass difference ∆ ≡ M ∆ − M N ∼ . δ expansion scheme, which counts m π / Λ χ SB as δ to maintain the scale hierarchy m π ≪ ∆ ≪ Λ χ SB .8(2) The pion-nucleon and pion-pion Lagrangians are rather standard. The N ∆ and∆∆ Lagrangians, on the other hand, require more attention. The ∆(1232) is aspin-3/2 resonance and, therefore, its spin content can be described in termsof the Rarita-Schwinger (RS) field, ∆ µ , where µ is the Lorentz index. Thisfield, however, contains unphysical spin-1/2 components. In order to tackle thisproblem, we follow Ref. 9 and adopt the “consistent” couplings, which are gauge-invariant under the transformation∆ µ ( x ) → ∆ µ ( x ) + ∂ µ ǫ ( x ) . (1)(3) Different forms of the spin-3/2 propagator have been used in the literature,some of which may lead to serious theoretical problems.10 Due to the spin-3/2gauge symmetric nature of the consistent couplings, we can use the most generalspin-3/2 free field propagator.11A more detailed discussion of these issues and the relevant N ∆ and ∆∆ Lagrangianscan be found in Ref. 7.ctober 22, 2018 6:1 WSPC/INSTRUCTION FILE proceeding ChPT study of the N → ∆ axial transition FF ( a ) ( b ) ( c ) ( d ) ( f )( e )( g ) ( h ) ( i ) ( j ) ∆ p ′ Np q
Fig. 1. Feynman diagrams contributing to the N → ∆ axial transition form factors up to order δ (3) . The double, solid, and dashed lines correspond to the delta, nucleon, and pion, respectively;while the wiggly line denotes the external pseudovector source.
3. Results and Discussions
The N → ∆ axial transition form factors can be parametrized in terms of theusually called Adler form factors:12 , h ∆ + α ( p ′ ) | − A αµ, | P ( p ) i = ¯∆ + α ( p ′ ) (cid:26) C A ( q ) M N (cid:0) g αµ γ · q − q α γ µ (cid:1) + C A ( q ) M N (cid:0) q · p ′ g αµ − q α p ′ µ (cid:1) + C A ( q ) g αµ + C A ( q ) M N q α q µ (cid:27) N, where A αµ, is the third isospin component of the axial current.In the δ expansion scheme, up to order δ (3) , all diagrams contributing to theaxial N → ∆ transition form factors are displayed in Fig. 1. The correspondingresults in terms of low-energy constants (LEC) and loop functions are summarizedin Table 1.We can easily see that at δ (1) , C A (0) = q h A ≈ .
16, where h A is the πN ∆coupling determined from the ∆ width, which is close to the Kitagaki-Adler value2of 1.2. The Kitagaki-Adlder assumption C A = C A M N m π − q , on the other hand, issatisfied only up to δ (2) , i.e., non pion-pole contributions appear at order δ (3) .7 Table 1. The N → ∆ axial transition form factors in covariant baryon χ PT; d , d , d , d areorder 2 LEC (in GeV − ) while f , f , f , f , f , f , f are order 3 LEC (in GeV − ); g ( q ), g ( q ), g ( q ), and g ( q ) are the one-loop contributions as defined in Eq. (31) of Ref. 7.FF δ (1) δ (2) δ (3) − q C A ( q ) M N − d f ∆ + g ( q ) − q C A ( q ) M N − d /M ∆ ( f + f ) ∆/M ∆ + g ( q ) − q C A ( q ) − h A − ( d + d ) ∆ ( f + f ) ∆ + ( f + f ) q + g ( q ) − q C A ( q ) M N h A / q − m π ( d + d ) ∆q − m π − f + g ( q ) + − ( f + f ) ∆ − f q − ( g ( q )+ g ( q ) q ) q − m π ctober 22, 2018 6:1 WSPC/INSTRUCTION FILE proceeding L. S. GENG, J. MARTIN CAMALICH, L. ALVAREZ-RUSO, and M. J. VICENTE VACAS
The form factors C A and C A both start at chiral order 2 and get their q dependence at order 3 from the loops. For C A , we find a small q dependence, whichis quite sensitive to the π ∆∆ coupling constant. On the other hand, its imaginarypart, coming mainly from the N - N internal diagram, is finite ( ∼ .
03 at q = 0)and has a mild q dependence. This suggest that C A is small (compared to C A , , )but not necessarily zero. The C A dependence on q is also found to be rather mildat order δ (3) .In covariant χ PT up to δ (3) , four δ (2) and seven δ (3) LEC appear in the results.However, some of them appear in particular combinations. Therefore, effectivelywe have only five unknown constants. They can be fixed by fitting either to thephenomenological form factors obtained from neutrino bubble chamber data (withseveral assumptions), to the results of other approaches, such as those of variousquark models, or to the lattice QCD results5. For a more detailed discussion, seeRef. 7.
Acknowledgments
We thank Mauro Napsuciale, Stefan Scherer, Wolfram Weise, and in particularMassimiliano Procura and Vladimir Pascalutsa for useful discussions. L. S. Geng ac-knowledges financial support from the Ministerio de Educacion y Ciencia in the Pro-gram “Estancias de doctores y tecnologos extranjeros”. J. Martin Camalich acknowl-edges the same institution for a FPU fellowship. This work was partially supportedby the MEC contract FIS2006-03438, the Generalitat Valenciana ACOMP07/302,and the EU Integrated Infrastructure Initiative Hadron Physics Project contractRII3-CT-2004-506078.
References
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