Scalar and Tensor Sea Contributions to Nucleons

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Scalar and Tensor Sea Contributions to Nucleons. M. Batra ... coupling of spin 1/2 state coupled to scalar sea ... spin projection in z-direction operator of .
Scalar and Tensor Sea Contributions to Nucleons M. Batra1* and A. Upadhayay2 SPMS , Thapar University Patiala-147004. *email:[email protected]

Introduction: The low energy parameters of baryons like spin distribution, magnetic moment, weak decay, coupling constant ratios are still a challenge for particle physicist. In past various models and theories have been suggested to describe the structure of baryons in these context. The experimental investigation for the structure of baryons came into existence from DeepInelastic experiments. Later on EMC [1] at CERN concluded that spin of proton is carried by quark -anti quark pair in addition to valence part. SMC collaboration [2] measured the spin structure g1(x) of proton and neutron at Q2=10 GeV2. Also different models are put forward with perturbative and non perturbative approach to define the quark interactions. For example, Isgur and Karl [3].defined the confining interaction potentials among quarks. Some models explain the low energy properties through presence of �𝛷𝛷1↑ βŒͺ = 2

constituent quarks inside the baryon. In order to have reliable information about the various properties of hadrons at low energy, selfconsistent approach should be adopted. Low energy parameters for baryons are computed here by assuming baryons as a composite system of three quarks known as valence part and sea-part as made up of quark-antiquark pair and gluons. The wave-function [4] for valence part of the baryon is denoted by 𝚿𝚿 = 𝜱𝜱[|πœ™πœ™βŒͺ|πœ’πœ’βŒͺ|πœ“πœ“βŒͺ]|πœ‰πœ‰βŒͺwhere each part contributes in such a way so as to make the total wave-function anti-symmetric in nature. Here sea is assumed to be flavorless and in Swave, spin of sea part are chosen to be in 0, 1, 2 �����𝑐𝑐 ). The total flavorand color can be (1𝑐𝑐 , 8𝑐𝑐 , 10 spin-color wave function of a spin up baryon which consist of three-valence quarks and sea components can be written as

↑ ↑ ↑ 1 1 οΏ½12�↑ οΏ½1 οΏ½ οΏ½1 οΏ½ 2 [𝛷𝛷1 𝐻𝐻0 𝐺𝐺1 + π‘Žπ‘Ž8 𝛷𝛷8 2 𝐻𝐻0 𝐺𝐺8 + π‘Žπ‘Ž10 𝛷𝛷102 𝐻𝐻0 𝐺𝐺10 οΏ½οΏ½οΏ½οΏ½ + 𝑏𝑏1 �𝛷𝛷1 ⨂𝐻𝐻1 οΏ½ 𝐺𝐺1 + 𝑁𝑁 1

↑

↑

1

3

3

2 2 2 ↑ ↑ + 𝑏𝑏8 �𝛷𝛷82 ⨂𝐻𝐻1 οΏ½ 𝐺𝐺8 + 𝑏𝑏10 �𝛷𝛷10 ⨂𝐻𝐻1 οΏ½ 𝐺𝐺10 οΏ½οΏ½οΏ½οΏ½ + 𝑐𝑐8 (𝛷𝛷8 ⨂𝐻𝐻1 ) 𝐺𝐺8 +𝑑𝑑8 (𝛷𝛷8 ⨂𝐻𝐻2 ) 𝐺𝐺8 ]

2 2 + 𝑏𝑏12 + 𝑏𝑏82 + 𝑏𝑏10 + 𝑐𝑐82 + 𝑑𝑑82 𝑁𝑁 2 = 1 + π‘Žπ‘Ž82 + π‘Žπ‘Ž10

Where

Each term in above defined wave-function consists of two parts where 𝛷𝛷 represents contribution from valence part and other contributes to sea. Various combinations contributing to sea take into account antisymmetry of each term in the wave-function. The first three terms in wave-function come from

coupling of spin 1/2 state coupled to scalar sea and other three b1, b8, b10 represents coupling between spin 1/2 and vector Sea. Finally terms with c8 and d8 are due to coupling of spin 3/2 with vector and tensor sea respectively. Various low energy parameters can be calculated by defining a suitable operator for each property of the system.

1 (↑) οΏ½ (↑) οΏ½ if >πœ†πœ†πœ†πœ† < Οƒ οΏ½ if >𝜌𝜌𝜌𝜌 < Οƒ οΏ½ if >πœ†πœ†πœ†πœ† < Οƒ οΏ½Ξ¦1/2 οΏ½ = 2 [π‘Žπ‘Ž βˆ‘π‘–π‘– < Ο οΏ½iz >πœ†πœ†β†‘πœ†πœ†β†‘ +< Ο οΏ½iz >πœŒπœŒβ†‘πœŒπœŒβ†‘ + 2 < Ο οΏ½iz >πœ†πœ†β†‘πœŒπœŒβ†‘ οΏ½Ξ¦1/2 �Ο 𝑁𝑁

� if Ο

𝜌𝜌𝜌𝜌

Οƒ οΏ½iz

πœ†πœ†β†‘πœ†πœ†β†‘

Οƒ οΏ½iz

� if Ο

πœŒπœŒβ†‘πœŒπœŒβ†‘

πœ†πœ†πœ†πœ†

Οƒ οΏ½iz

� if Ο

πœŒπœŒβ†‘πœŒπœŒβ†‘

𝜌𝜌𝜌𝜌

Οƒ οΏ½iz

πœ†πœ†β†‘πœ†πœ†β†‘

+b

> )οΏ½ < > +< > οΏ½ + 𝑐𝑐 βˆ‘π‘–π‘– [< > < > + < > < > βˆ’2 < 3 3 οΏ½ if >𝜌𝜌𝜌𝜌 ] + π‘’π‘’οΏ½βˆ‘π‘–π‘– ( < Ο οΏ½ if >𝜌𝜌𝜌𝜌 βˆ’< Ο οΏ½ if >πœ†πœ†πœ†πœ† οΏ½ < Οƒ οΏ½ if >πœ†πœ†πœ†πœ† < Οƒ οΏ½ if >πœ†πœ†πœ†πœ† +βˆ‘π‘–π‘– < Ο οΏ½iz >πœ†πœ†β†‘2↑ + 2 βˆ‘π‘–π‘– < Ο οΏ½iz >πœŒπœŒβ†‘2↑ ] 𝑑𝑑[βˆ‘π‘–π‘– < Ο Here

i

� if Ο

οΏ½ if >πœ†πœ†πœ†πœ† + < βˆ‘π‘–π‘– (< Ο

>πœ†πœ†πœ†πœ† < Οƒ οΏ½iz >πœŒπœŒβ†‘πœŒπœŒβ†‘ ] +

i

οΏ½ f >πœ†πœ†πœ†πœ† =οΏ½πœ™πœ™ πœ†πœ† οΏ½ΞŸπ‘“π‘“π‘–π‘– οΏ½πœ™πœ™ πœ†πœ† οΏ½ , < ΟƒοΏ½ iz >πœ†πœ†β†‘πœ†πœ†β†‘ = οΏ½πœ’πœ’ πœ†πœ†β†‘ οΏ½πœŽπœŽπ‘§π‘§π‘–π‘– οΏ½πœ’πœ’ πœ†πœ†β†‘ οΏ½, < Ο οΏ½ f >𝜌𝜌𝜌𝜌 = οΏ½πœ™πœ™πœŒπœŒ οΏ½ΞŸπ‘“π‘“π‘–π‘– οΏ½πœ™πœ™πœŒπœŒ οΏ½and < ΟƒοΏ½ iz >πœŒπœŒβ†‘πœŒπœŒβ†‘ = οΏ½πœ’πœ’ πœŒπœŒβ†‘ οΏ½πœŽπœŽπ‘§π‘§π‘–π‘– οΏ½πœ’πœ’ πœŒπœŒβ†‘ οΏ½