arXiv:1112.5576v1 [hep-ph] 23 Dec 2011 rcse)i suulyngetd hs nti ae ewl ou on focus will we paper this in Thus, neglected. usually is it processes) otenurn rcse ecnie etrwa F swl det well as FFs weak vector consider we of processes the to w axial two ntrso nsfo lsi lcrnncensatrn.Ti rela This scattering. [2]: - (CVC) hypotheses elastic current from vector ones of terms in urn us lsi (CCQE) elastic quasi current rencenpoess ula ffcswr o ae noacco into taken not experim were available effects with nuclear results the processes, compare nucleon and free factors form CCQE of rcs nweg fwa ulo F svtlfrpeetadfutu and present for vital is is experiments. FFs FFs nucleon neutrino nucleon weak axial of on knowledge of data precise structure experimental axial of probe interpretation to retical opportunity unique the give Introduction 1 ofra hswr sjs eeaiaino h hoyof theory the of generalization a just is work this as far So h eerhIsiueo hsc,Suhr eea Univer Federal Southern Physics, of Institute Research The ∗ † G CEpoessaeprmtrzdb orfntosof functions four by parameterized are processes CCQE nti okw ugs hnmnlgclbro-eo ag th gauge - phenomenological suggest we work this In [email protected] [email protected] 3 utpido QDaypoe si odareetwt exp with agreement good a in is asymptotes pQCD on multiplied a on htwa ulo omfco (FF) factor form nucleon weak that o pr consta found hadronization scattering the the was (anti)neutrino-nucleon model of elastic parameters the quasi and of Then, masses framework meson the by within formed meso that vector/pseudovector shown and was It interactions strong of metry otecosscini rprinlt etnms,s ncs fligh of case in so mass, to proportional is section cross the to hnmnlgclmliag oe fnurn-ulo in neutrino-nucleon of model multigauge Phenomenological xa omfcosfrqaielastic quasi for factors form Axial G neatosi h vector/pseudovector the in interactions 1 G , 3 omfcos oee etrFsaentidpnet hyca they independent, not are FFs vector However factors. form F a ν ( .Kanshin K. oiac model. dominance Q − 2 N = ) cteig npriua,w nrdc utpl expansions multipole introduce we particular, In scattering. F a 400 Russia 344090, ep ( Q Abstract ∗ 2 ) n .Vereshkov G. and − 1 F a en G A ( Q ( Q 2 2 ) cse aebe netgtdadit and investigated been have ocesses osdrda utpl expansion multipole as considered ) a , e − oiac oe ssuggested. is model dominance n Q N unt. 1 = eato ae ncia sym- chiral on based teraction 2 lsi cteigpeetdi [1] in presented scattering elastic udmna etrbosons. vector fundamental f ulos hti h la theo- clear why is that , = rmna data. erimental fgetiprac.Moreover, importance. great of to eto eadcyis decay beta of nt rie.Bsds contribution Besides, ermined. , † eaclrtradatmospheric and accelerator re inoiiae rmconserved from originates tion 2 . − oyof eory na aa esuidonly studied We data. ental q only iy Rostov-on-Don, sity, 2 w vector two : etn (non t G β 1 ν dcyadcharged and -decay ( Q − 2 ). eexpressed be n N F 1 τ F , -lepton 2 and (1) 2 Basics and features

The main assumption of the model is the vector and axial dominance. In case of charged current ν N interaction it claims that intermediate vector W ± hadronize to the sets of vector and− axial , so interaction between neutrino and nucleon occurs via meson exchange. In order to account the sets of mesons with different masses we use multigauge approach. Use of multigauge groups allows to introduce a set of gauge fields with their own coupling parameters, for example in Dirac terms of lagrangian it means:

N multiplifying Dirac: gΨ¯ γµΨA g w Ψ¯ γµΨAi , µ −−−−−−−−−→ i µ Xi=1 where g is a coupling constant, Ψ is any fermionic field, i is generation (family) index, N is a i total number of generations (will be discussed later), Aµ are multigauge fields and wi - are actual parameters of the theory. Following relation is required to save gauge invariance:

N wi =1. (2) Xi=1 This is the first sum rule (SR) imposed on parameters. Moreover, to account anomalous magnetic dipole moment of nucleon Pauli terms are necessary:

N multiplifying Pauli: gΨ¯ σµν ΨA g κ Ψ¯ σµν ΨAi , µν −−−−−−−−−→ i µν Xi=1 however there is no similar relation for κi motivated by gauge symmetries. Thus we have two κ types of parameters: Dirac wi and PauliP i. Full theory of both neutral and charged current ν N interactions involves at least four sets of i i −i i mesons which could be classified by UL(1) SUL(2) UR(1) SUR(2) isotopic chiral groups. In case × ×i ×i i i of charged currents, only sets of isovector mesons ρ and a1 corresponding to SUL(2) SUR(2) i i ×i have to be taking into consideration, while isoscalar ω which are formed by UL(1) UR(1) gauge fields act only in elastic ν N and e N scattering. × Finally, chiral symmetry− of strong− interactions requires coupling constants of left and right fields to be equal and this way reduces number of parameters. Moreover, it leads to equality of ρ N and a1 N coupling. Consequently, parameters of vector and axial form factors will be the same.− − By adding nucleon-meson terms to the lepton doublets we receive fermionic sector of lagrangian. In order to construct effective lepton-meson vertexes one has to introduce Higgs fields as a representation of both mesonic and electro-weak gauge groups. To generate meson masses additional Higgs fields are required. Summary of the objects of the theory is given in the table:

2 Field Group Vacuum shift p N = U i (1) SU i (2) U i (1) SU i (2) n L × L × R × R e µ EW EW l = or U (1) SUL (2) νe νµ × SU EW (2) U i (1) SU i (2) m2 L × L × L ∼ ρ U EW (1) U i (1) SU i (2) m2 × R × R ∼ ρ Higgs sector U i (1) SU i (2) U i (1) SU i (2) m2 m2 L × L × R × R ∼ a1 − ρ U i (1) U i (1) m2 m2 L × R ∼ f − ω U EW (1) SU EW (2) m2 × L ∼ W Assuming chiral symmetry of the vacuum we put vacuum shifts of left and right (1st and 2nd in the table) Higgs fields equal. Spontaneous symmetry breaking generates quadratic form of fields. Its diagonalization leads to the small mixing (only charged bosons considered):

± ± i ±i ±i W = W + ε ρ + a1 , (3)  i2 i mρ where ε 2 1. This way lepton-meson vertex is introduced. ∼ mW ≪ The remaining degrees of freedom have to be mixed with the Standard Model and then identified with additional meson sets. However lepton-Higgs vertex is suppressed by enormous (245 GeV) value of vacuum shift, so contribution of these mesons is expected to be small and it has been neglected. After lagrangian had been constructed one could study real physical processes.

3 β-decay

Standard expression for squared β-decay amplitude which originates from Fermi theory of weak interactions is following: 2 2 = 16G cos θ (1+3α). (4) |M| F C 2 gW EW Here GF = 2 is the Fermi constant, where gW is the coupling constant of SUL (2), θC is 4√2mW the Cabibbo mixing angle, α =1.2689 [3] is a dimensionless phenomenological parameter, precisely determined by experiments. Squared amplitude has been calculated within the framework of the theory as well and param- eters of the theory has been identified with ones from the Fermi theory. Thus, GF is formed by 2 mixing parameters, coupling constants of gauge groups and meson propagators at Q = 0; cos θC must be inserted due to hadronization via non-perturbative ud¯ twist, and finally we have got the following expression for α: N m2 α = w ρi (5) i m2 Xi=1 a1i

Taking into account this relation and the fact that parameters wi are the same for both electron and CC neutrino parts of the theory one had to refit data on electron scattering. This has been done and no any significant changes occurred, parameters have not changed a lot.

3 4 ν N scattering and the axial FF − 2 As we have already mentioned the main purpose of studying CCQE scatterng was fitting of G1(Q ). At low Q2 region the conventional dipole form of axial form factor reveals satisfactory fit of the experimental data: dipole 2 α G1 (Q )= 2 (6) Q2 1+ M 2  A  It is parameterized by the β-decay constant α and the phenomenological parameter MA, the so- called axial-vector dipole mass. Within the framework of presented model we have obtained the following multipole expressions of form factors: 2 2 N N κ 2 wi mρi 2 i mρi F1(Q )= 2 2 F2(Q )= 2 2 =1 m + Q =1 m + Q iP ρi iP ρi (7) e N e 2 2 wi ma1i G1(Q )= 2 2 =1 m + Q iP ρi e To reproduce the correct asymptotic behavior of the form factors at infinity the additional sum rules are required:

N 2 2 1 2 F1(Q ), G1(Q ) 4 wi mρi =0 ∼ Q → =1 iP

N κ 2 (8) i mρi =0 2 1 i=1 F2(Q ) P ∼ Q6 → N κ 4 i mρi =0 =1 iP p n Besides, from (1) and F2 (0) = µp 1, F2 (0) = µn, where µp and µn are magnetic moments of and neutron correspondingly we− get:

N κ = µ µ 1 3.70 (9) i p − n − ≃ Xi=1 Now let us discuss the number of meson generations N. We have obtained six sum rules (2), (5), (8), (9) which have to be imposed on parameters. To satisfy them at least three generations must be taken into account. The first reason to use only three families is that there is no precise data on heavy axial mesons. Another one is that contribution of i > 3 mesons is suppressed by increasing mass. And finally, three generations fit of electron data was accurate enough. In the case of N = 3 both sets of parameters wi and κi (i = 1, 2, 3) are determined from six sum rules. Thus, we consider only three generations, however in the theory there is no any restrictions on number N. These mesons are following:

ρ(770) a1(1260) ρ(1450) a1(1640) (10) ρ(1700) a1(1930)

To account pQCD asymptotes in high Q2 region one has to multiply form factors (7) on 2 2 logarithmic functions qa(Q ) of Q calculated for Dirac a = 1 and Pauli a = 2 FFs in [4], [5], [6], [7]. The general expression is following:

2 2 2 2 2 2 −pa/2 qa(Q )= 1+ ha ln(1 + Q /Λ )+ ga ln 1+ Q /Λ , (11)  4 1,3

1,2 ANL 82, D , K.L. Miller et al.

2

1,1 BNL 81, D , N.J. Baker et al.

2

FNAL 83, D , T. Kitagaki et al. 1,0

2

__ Reconstruction from eN data 0,9

0,8 ) 2

0,7 (Q A 0,6 G

0,5

0,4

0,3

0,2

0,1

0,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

2 2

Q , GeV

Figure 1: Reconstruction of axial formfactor

where all the parameters ha ga pa are the same with ones used in electron part of theory. So, they have been already determined from electron data. Thus we obtain final expressions of FFs:

2 2 2 2 2 2 2 2 2 F1(Q )= q1(Q )F1(Q ) F2(Q )= q2(Q )F2(Q ) G1(Q )= q1(Q )G1(Q ) (12) We emphasize thate there is no free fit parameters ein final FFs. So, we present juste a reconstruction of ν N form factors based on e N data but not fit. As we have already mentioned nuclear − − 2 effects was out of consideration. The comparison of G1(Q ) from (12) with available deuterium experiments [8],[9],[10] is presented in the Fig. 1.

5 Perspectives and conclusion

We are planning to apply this multigauge scheme based on chiral symmetry and meson dominance to the neutral current as well and thus to complete the full theory of (quasi)elastic lepton-nucleon interactions which consists of three parts:

1. Elastic electron-nucleon scattering (presented in [1]) 2. Charged current quasi elastic neutrino-nucleon scattering (shortly described here)

5 3. Neutral current elastic neutrino-nucleon scattering (under investigation at the moment) The third part is more complicated then previous ones because of mixing of five boson fields 0 0 (ρ ,a1, ω - mesons, Z-boson and ). Moreover, in order to introduce effective ω-lepton vertex an additional mixing mechanism is required. It is based on the fact that stress tensor of U(1) groups is gauge invariant, so cross term in the kinematic sector of the lagrangian is not forbidden: 1 B ωµν , −4 µν

EW Here Bµ field corresponds to U (1) group, Bµν = ∂µBν ∂ν Bµ. Diagonalization of both kinematic and Higgs sectors leads to the required mixing. The− theory of NC will not contain undetermined parameters. Another perspective application of the approach is a studying of some other processes caused by strong interactions, such as resonance and single production or many-pion production.

References

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