Quark correlations in the ground and excited nucleonstates via the photo-absorption sum rules

Sergo Gerasimov1,∗

1Joint Institute for Nuclear Research, Dubna, Russia

Abstract. The account of the dynamical correlations of quarks composing thenucleons is necessary for relevant description of nucleon and nuclear interac-tions. The present work is devoted to the use of the technique of known integralsum rules for the cross-sections and amplitude of the photo-hadron processes.In particular, a kind of the visual representation and quantification of the va-lence quark correlation characteristics referring to the photo-excitation of thenucleon resonances can be presented.

1 Introduction: the past experience

Instead of well-known non-relativistic dipole sum rules [1] for atomic and nuclear photo-effect,derived in the center-of-mass of the non-relativistic target systems:

σn(E1) =∫∞

thrdω ωn σE1(ω),

where: n = −2 refers to Kramers-Heisenberg sum rule (SR) for static electric-dipole polar-izability of a given quantum system; n = −1 is the bremsstrahlung-weighted SR, dependingof charged “parton” correlation in a given system; n = 0 is the famous Thomas-Reiche-KuhnSR, known as a precursor of the Quantum Mechanics itself, we should accept the pictureof the transverse quark current densities of the fast moving nucleons. Just this frameworkgives fruitful and widely used approach for the description of the electromagnetic structureparameters of the nucleon.

In this paper, we return to the application of the current algebra technique in the infinitemomentum frame used previously in the derivation of the Cabibbo–Radicati [2] sum rule andin the derivation of the earlier known from the dispersion theory [3–5] and current algebra [6]sum rules for the anomalous magnetic moments of the nucleon. In this way, a certain numberof relations will be obtained directly depending on the flavor quark correlation in the groundnucleon state and in the states including an overlap of the ground and excited states of thenucleon.

2 Sum rules with relativistic spin degrees of freedom

In the relativistic theories the operators of electric dipole moments depend on the spin andfollowing formally to the pz → ∞ techniques derivation of the Cabibbo-Radicati [2] or the∗e-mail: [email protected]

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© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the CreativeCommons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

GDH s.r. derivation in the infinite momentum frame [6] we can derive the relations

4π2α(13< ~D2 > −(

κN2mN

)2) =∫

dννσrestot (ν), (1)

4π2α[13< ~D2 >] =

∫∞

thr

dνν

(σresp (ν), (2)

where σp(a) refers to the interaction cross section of the polarized nucleon and polarizedphoton with parallel (or anti-parallel) spins, κn-anomalous magnetic moment of the nucleonin the natural units,

D̂ =∫

~xρ̂(~x)d3 x =3∑

j=1

Qq( j)~d j. (3)

The replacement of the constant κ2 by its integral form illustrates and fulfills the essentialelement of the relationship and interdependence of the description of the static characteristicsof a given system and known properties of its excited states.

The defined operators Qq( j) and ~d j are the electric charges and configuration variables ofpoint-like interacting quarks in the infinite-momentum frame of the bound system. Finally,we relate the electric dipole moment operator correlations, successively for the proton, theneutron and the pure "isovector-nucleon" part equal for both nucleons and the isovector partof the mean-squared radii operators entering the Cabibbo-Radicati sum rule, which all aresandwiched by the nucleon state vectors in the "infinite – momentum frame", with experi-mentally measurable data on the resonance parts of the photoabsorption cross sections onthe proton and neutron presently known below ∼ 2 GeV. The listed operator mean values areparameterized as follows [7]

RV =16

(< r21 >P − < r21 >N ) =

13

(α−12β), (4)

JP =13< D̂2 >P=

827α +

127β +

827γ−

827δ, (5)

JN =13< D̂2 >N =

227α +

427β +

227γ−

827δ, (6)

JV =13< D̂2 >V =

212α +

112β +

212γ−

412δ, (7)

where < ~d12>=< ~d2

2>=α,< ~d3

2>=β, < ~d1 · ~d2 >=γ, < ~d1 · ~d3 >=< ~d2 · ~d3 >=δ and indices

"1" and "2" refer to the like quarks, and "3" to the odd quark.Evaluation of the relativistic electric dipole moment fluctuation was carried out with the

available compilation [8] of the resonance pion-photoproduction data on the proton and neu-tron AP(N)1/2 and A

P(N)3/2 the numerical values we calculated and all integrals over photoexcited

nucleon resonances were taken in the narrow resonance approximation, when

Jresp(a) '4πmn|Ares3/2(1/2)|

2

m2res − m2n, (8)

where mn(res) is the nucleon (or resonance) mass. Solving the system of the linear equationsand evaluating the RV, JP,N,V with the help of experimentally known partial amplitudes ofmain photo-excited resonances, we find our final results for the numerical values α, β and theopening angle θ12 and θ13 between vectors ~d1 and ~d2 and vectors ~d1 and ~d3, lying in the planetransverse to the "infinite momentum"’ pz→∞ vector [9]:

α1/2 = 0.75 ± 0.06 fm, β1/2 = 0.77 ± 0.12 fm, θ12 ' 1200, θ13 ' θ12 ∼ 120

0.

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We note that we could use the isoscalar input

JS =13< D̂2 >S =

1108

(2α + β + 2γ + 4δ) (9)

instead of the isovector one in Eq. (7). The numerical values of the parameters α,β,γ,δ wouldturn out to be the same due to the use of the standard isotopic symmetry relations relating JVand JS with the JP and JN .

Summarizing the results of the global approach via the use of the sum rules, we obtain akind of averaged picture across the spectrum pattern with the presumably dominating influ-ence of the first ∆(1236)-resonance with its magnetic-dipole photo-excitation of the nucleonspin degrees of freedom, which is also hiding explication of the qualitative isospin-dependingfeatures of the (mainly, electric-dipole) 2-nd and (the electric-quadrupole) 3-rd resonancephoto-excitation regions. The value of effective dipole moments, determining the excitationcross section of the "electric-dipole-type" nucleon resonances, includes apparently the pionicdegrees of freedom of the constituent quarks. This is supported by the fundamental principleof the charge symmetry of strong interaction dynamics. The participation of the pion degreesof freedom plays the role in the dynamical formation of the quark-diquark spatially lookingnucleon structure. We suggest to illustrate it numerically choosing the experimental data onthe resonance photo-absorption data on the J P = 3/2−, P13(1500) and J P = 3/2−, D33(1600)-resonances,composing the region of the "second" resonance area. Performing the evaluationsanalogous the earlier presented for all the resonances with the data from the PDG table [8],we obtain [9] the results differing from earlier cited

α1/2 = 0.82 fm, β1/2 = 0.53 fm, θ12 ' 1300, θ13 ' 70

0, θ23 ∼ 1600.

A pronounced asymmetry of the sides of the triangles Oud and Ouu

CminudCuu

'0.8 fm1.5 fm

(10)

can testify to the validity of the quark-diquark model of the considered resonances thoughdiquarks are not apparently strongly bound. On the other hand, the obvious need to fulfillthe charged symmetry requirements implies a significant role of the isospin-dependent quark-quark potentials, that is the role of the interactions caused by the pion exchanges, for instance.Ultimately, however, both factors can be the sides of the same phenomenon.

References

[1] J.S. Levinger, H.A. Bethe, Phys. Rev. 78, 115 (1950)[2] N. Cabibbo, L.A. Radicati, Phys. Lett. 19, 678 (1966)[3] S.B. Gerasimov, Sov. J. Nucl. Phys 2, 430 (1966)[4] S.B. Gerasimov, Sov. J. Nucl. Phys 5, 1263 (1967)[5] S.D. Drell, A.C. Hearn, Phys. Rev. Lett. 16, 908 (1966)[6] M. Hosoda, K. Yamamoto, Prog. Theor. Phys. 36, 425 (1966)[7] S.B. Gerasimov, Czech. J. Phys. 55, A209 (2005)[8] K.A. Olive, et al. Chin. Phys. G 38, 090001 (2014)[9] S. B. Gerasimov, J. Phys.: Conf. Ser. 678 012009 (2016)

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