Nuclear states and spectra in holographic QCD Yoshinori Matsuo - - PowerPoint PPT Presentation

Nuclear states and spectra in holographic QCD

Aug 19, 2019@Strings and Fields 2019

Yoshinori Matsuo Osaka University

Based on arXiv:1807.11352

with Koji Hashimoto (Osaka U.), Takeshi Morita (Shizuoka U.)

Introduction

Baryons in Sakai-Sugimoto model

𝒚𝟏 𝒚𝟐 𝒚𝟑 𝒚𝟒 𝒚𝟓 𝑽 𝜾𝟐 𝜾𝟑 𝜾𝟒 𝜾𝟓 D4 ✓ ✓ ✓ ✓ ✓ D8 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ D4’ ✓ ✓ ✓ ✓ ✓

D-branes (D4’) wrapping color D-branes (D4) 𝑦5 ⋯ 𝑦9 D4 (at 𝑉 = 0) D4’ on 𝑇4 𝑉 (radial direction) and 𝑇4 (𝜄1 ⋯ 𝜄4) D4 Background geometry (holography) 𝑇4 Integrate out Effective theory on D8 Effective theory of mesons D4’ in D8 effective theory Instanton on D8 Skyrmion Effective fields on D4’ ADHM data of instantons

Introduction

Baryons in holographic QCD solitonic D4-brane geometry 𝑒𝑡2 = 𝑉 𝑆

3 2

−𝑒𝑢2 + 𝑒𝑦2 + 𝑔 𝑉 𝑒𝑦4

2 +

𝑆 𝑉

3 2

𝑒𝑉2 𝑔 𝑉 + 𝑉2𝑒Ω4

2

Anti-periodic b.c. for 𝑦4 A similar factor to BH 𝑔(𝑉) Geometry ends at some 𝑉 Baryon is located near the tip of geomtry D8-brane Baryon D4-brane Nuclear matrix model Matrix model of Baryon vertex (D4-brane) with bosonic field of D4-D8 open string near the tip of solitonic (color) D4-brane background

Nuclear matrix model

Action for 𝐵 baryons 𝑇 = 𝑇0 + 𝑂𝑑 න𝑒𝑢 tr𝐵𝑢 𝑇0 = න 𝑒𝑢 tr ቈ1 2 𝐸𝑢𝑌𝐽 2 + 1 2 𝐸𝑢 ഥ 𝑥 ሶ

𝛽𝑗

𝐸𝑢𝑥 ሶ

𝛽𝑗 − 1

2 𝑁2 ഥ 𝑥 ሶ

𝛽𝑗𝑥 ሶ 𝛽𝑗

቉ + 1 4𝜇 𝐸𝐽 2 + 𝐸𝐽 2𝑗𝜗𝐽𝐾𝐿𝑌𝐾𝑌𝐿 + ഥ 𝑥 ሶ

𝛽𝑗 𝜐𝐽 ሶ 𝛽 ሶ𝛾𝑥 ሶ 𝛾𝑗

𝐸𝑢 = 𝜖𝑢 − 𝑗𝐵𝑢: covariant derivative 𝑌𝐽: D4-D4 scalar 𝑥 (ഥ 𝑥): D4-D8 scalar Diagonal comp.: position of D-brane (= baryon) 𝐵 × 𝐵 matrix Off-diagonal: interaction between D-branes carries charges of quarks (spin, flavor, baryon number)

[Hashimoto-Iizuka-Yi,’10]

𝐼 = 𝐼0 + 𝑊 𝑊 = − 1 2 𝑛2 𝑌𝐽 2 − 2𝜇 𝑌𝐽, 𝑌𝐾 2 −4𝑗𝜇𝜗𝐽𝐾𝐿𝑌𝐵

𝐾𝑌𝐶 𝐿𝑔 𝐷 𝐵𝐶 ഥ

𝑥𝑏

ሶ 𝛽𝑗 𝜐𝐽 ሶ 𝛽 ሶ𝛾 𝑢𝐷 𝑐 𝑏 𝑥 ሶ 𝛾𝑗 𝑐 + 𝜇 ഥ

𝑥 ሶ

𝛽𝑗 𝜐𝐽 ሶ 𝛽 ሶ𝛾𝑥 ሶ 𝛾𝑗 2

𝐼0 = 1 2 tr ΠI 2 + 1 2 𝑛2tr 𝑌𝐽 2 + 1 2 ത 𝜌 ሶ

𝛽𝑗 𝑏 𝜌𝑏 ሶ 𝛽𝑗 + 1

2 𝑁2 ഥ 𝑥𝑏

ሶ 𝛽𝑗𝑥 ሶ 𝛽𝑗 𝑏

Perturbation around harmonic potential 𝑊: perturbation 𝐵𝑢: gauge field (baryon 𝑇𝑉(𝐵))

Non-dynamical field EOM gives constraints

0 = 𝜀𝑇 𝜀𝐵𝑢 = 𝜀𝑇0 𝜀𝐵𝑢 − 𝑂𝑑𝕁 = 𝑅𝑉 𝐵 − 𝑂𝑑𝕁 𝑅𝑇𝑉 𝐵 = 0 𝑅𝑉 1 𝐶 = 𝑂𝑑𝐵 Eigenstates of Hamiltonian must be Singlet in baryon 𝑇𝑉(𝐵) symmetry Baryon (quark) number must be 𝑂𝑑𝐵

Ground state and constraint

0-th order Hamiltonian 𝐼0 Harmonic oscillators of 𝑌𝐽, 𝑥 and ഥ 𝑥. Constraint 1: baryon 𝑉(1) charge must be 𝑂𝑑𝐵 Baryon 𝑉 1 charges 𝑌𝐽: 0 𝑥: 1 ഥ 𝑥: −1 Number of 𝑥 − Number of ഥ 𝑥 = 𝑂𝑑𝐵 Constraint for excitation of harmonic oscillators Physical ground state for 𝐵 ≤ 2𝑂

𝑔

𝜔0 = 𝜗𝑏1⋯𝑏𝐵𝑥 ሶ

𝛽1𝑗1 𝑏1 ⋯ 𝑥 ሶ 𝛽𝐵𝑗𝐵 𝑏𝐵

× ⋯ × 𝜗𝑐1⋯𝑐𝐵 𝑥𝑐1 ⋯ 𝑥𝑐𝐵 Lowest energy state Smallest number of excitations Number of ഥ 𝑥 = 0 Number of 𝑥 = 𝑂𝑑𝐵 Constraint 2: physical state must be singlet of baryon 𝑇𝑉(𝐵) 𝑂𝑑 of 𝜗𝑏1⋯𝑏𝐵𝑥𝑏1 ⋯ 𝑥𝑏𝐵

𝑥𝑏: raising operator of oscillator 0 : ground state of harmonic oscillators

Physical ground state for 𝐵 ≤ 2𝑂

𝑔

𝜔0 = 𝜗𝑏1⋯𝑏𝐵𝑥 ሶ

𝛽1𝑗1 𝑏1 ⋯ 𝑥 ሶ 𝛽𝐵𝑗𝐵 𝑏𝐵

× ⋯ × 𝜗𝑐1⋯𝑐𝐵 𝑥𝑐1 ⋯ 𝑥𝑐𝐵 𝑂𝑑 of 𝜗𝑏1⋯𝑏𝐵𝑥𝑏1 ⋯ 𝑥𝑏𝐵 Only 2 × 𝑂

𝑔 of different 𝑥 ሶ 𝛽𝑗: 2 different spins, 𝑂 𝑔 different flavors

𝐵 (> 2𝑂

𝑔) of 𝑥𝑏 cannot form antisymmetric combination

Physical ground state for 𝐵 > 2𝑂

𝑔

construct different operator by using 𝑌𝐽: 𝑌𝐽𝑥 𝑏 = 𝑌𝐽

𝑐 𝑏 𝑥𝑐,

𝑌𝐽𝑌𝐾𝑥 𝑏, 𝑌𝐽𝑌𝐾𝑌𝐿𝑥 𝑏, ⋯ 𝜔0 = 𝜗𝑏1⋯𝑏𝐵𝑥𝑏1 ⋯ 𝑥

𝑏2𝑂𝑔 𝑌𝐽𝑥 ⋯ 𝑌𝐾𝑥 ⋯ 𝑌𝐿 ⋯ 𝑌𝑀𝑥 𝑏𝐵

× ⋯ × [𝜗𝑐1⋯𝑐𝐵𝑥𝑐1 ⋯ 𝑥

𝑐2𝑂𝑔 𝑌𝑥 ⋯ 𝑌 ⋯ 𝑌𝑥 𝑐𝐵 ] 0

𝑂𝑑 of 𝜗𝑏1⋯𝑏𝐵𝑥𝑏1 ⋯ (𝑌𝐽 ⋯ 𝑌𝐾)𝑥𝑏𝐵

Magic numbers

Magic number for 𝑣 (or 𝑒) quarks (𝑂𝑑 = 1 case for simplicity) 𝜔0 = 𝜗𝑣↑|0〉 𝐵 = 1 𝑥 = 𝑣 ሶ

𝛽

ሶ 𝛽 = 1, 2 (𝑗 = 1) 𝜔0 = 𝜗𝑣↑𝑣↓|0〉 𝐵 = 2 2 of 𝑣 (spin ↑ and ↓) 𝜔0 = 𝜗𝑣↑𝑣↓𝑣↑ 0 = 0 𝐵 = 3 𝜔0 = 𝜗𝑣𝑣(𝑣𝑌)(𝑣𝑌)|0〉 𝐵 = 4 𝜔0 = 𝜗𝑣𝑣 𝑌𝑣 |0〉 𝑌𝐽 𝐽 = 1, 2, 3 2 × 3 = 6 of 𝑌𝑣 𝜔0 = 𝜗𝑣𝑣 𝑌𝑣 ⋯ 𝑌𝑣 |0〉 𝐵 = 8 ⋮ ⋮ 𝜔0 = 𝜗𝑣𝑣 𝑌𝑣 ⋯ 𝑌𝑣 (𝑌𝑣) 0 = 0 𝐵 = 9 𝐵 = 9

6 of 𝑌𝑣 Additional energy of 𝑌𝐽 Magic number Magic number 6 of 𝑌𝑣

𝜔0 = 𝜗𝑣𝑣 𝑌𝑣 ⋯ 𝑌𝑣 (𝑌𝑌𝑣) 0

6 of 𝑌𝑣

Nuclei are stable (small energy) for some specific numbers of proton (neutron)

Magic numbers

Magic number for either proton or neutron (𝑂𝑑 = 3) 𝑂

𝑔 = 2

Both quarks and nucleons have isospin 𝐽 = 1/2 𝜔0 = 𝑒𝑒 𝑣𝑣(𝑌𝑣)(𝑌𝑣)(𝑌𝑣)(𝑌𝑣)(𝑌𝑣)(𝑌𝑣) × 𝑒𝑒 𝑣𝑣(𝑌𝑣)(𝑌𝑣)(𝑌𝑣)(𝑌𝑣)(𝑌𝑣)(𝑌𝑣) × 𝑣𝑣 𝑒𝑒 𝑌𝑒 𝑌𝑒 𝑌𝑒 𝑌𝑒 𝑌𝑒 𝑌𝑒 0 Example: (𝑂𝑞 = 8, 𝑂𝑜 = 2)

2 neutrons 8 protons

Proton and neutron configurations Quark configurations

?

Example: 𝑂𝑣 = 6, 𝑂𝑒 = 3; same to 3 protons (3 𝑞’s cannot be in ground state) 𝜔0 = 𝑣𝑣 𝑒 𝑣𝑣 𝑒 𝑣𝑣 𝑒 0 State without 𝑌 excitation is possible 3 sets of anti-sym. combinations 𝜗𝑥 ⋯ 𝑌𝑥 corresponds to nucleus with Δ will be heavier if 1st order perturbation 𝑊 is taken into account

First order perturbation

Energy at first order = expectation value of 𝐼 for 𝜔0 𝐹 = 𝜔0 𝐼 𝜔0 = 𝐹0 + 𝜔0 𝑊|𝜔0〉 No 𝑌𝐽 excitation for 𝐵 ≤ 2𝑂

𝑔

𝑊 = 𝜇 ഥ 𝑥 ሶ

𝛽𝑗 𝜐𝐽 ሶ 𝛽 ሶ𝛾𝑥 ሶ 𝛾𝑗 2

𝐹0 = 𝑂𝑑𝐵𝑁 Energy at 0-th order Linear order correction 𝑊 for 𝑂𝑑𝐵 excitations of 𝑥 No excitations of 𝑌𝐽 or ഥ 𝑥 (only 𝑥 excitations) 𝑊 = 4𝜇 𝑁2 𝐷

𝑔 + 𝜇

𝑁2 2𝐵 − 𝑂

𝑔

𝑂

𝑔𝐵

𝑂𝑥

2

𝐷

𝑔: quadratic Casimir of flavor 𝑇𝑉(𝑂 𝑔)

𝑂𝑥: Number of excitations of 𝑥 Smaller flavor charge more stable

Allowed states (𝑂

𝑔 = 2)

𝐵 = 1 and 𝑂𝑑 = 3 𝜔0 = 𝑥 ሶ

𝛽1𝑗1𝑥 ሶ 𝛽2𝑗2𝑥 ሶ 𝛽3𝑗3 0

(No baryon index) 𝑥 are bosonic (= symmetric) Spin 𝐾 and isospin 𝐽 are in same rep. 𝐾, 𝐽 =

1 2, 1 2

𝐾, 𝐽 =

3 2, 3 2

• r

proton or neutron Δ 𝐹 = 3𝑁 + 4𝜇 𝑁2 𝐽(𝐽 + 1) Mass proton and neutron have smaller mass than Δ

𝑥 ሶ

𝛽𝑗: raising operator of 𝑥 ሶ 𝛽𝑗

Allowed states (𝑂

𝑔 = 2)

𝐵 = 2 and 𝑂𝑑 = 1 𝜔0 = 𝜗𝑏1𝑏2𝑥 ሶ

𝛽1𝑗1 𝑏1 𝑥 ሶ 𝛽2𝑗2 𝑏2

𝑥 are antisymmetric spin isospin is symmetric antisymmetric or antisymmetric symmetric 𝐾, 𝐽 = 1,0 𝐾, 𝐽 = 0,1 stable 𝐵 = 2 and 𝑂𝑑 = 3 Symmetric combination of 3 sets of 𝐵 = 2 and 𝑂𝑑 = 1 𝐾, 𝐽 = 1,0 𝐾, 𝐽 = 3,0 𝐾, 𝐽 = 1,2 𝐾, 𝐽 = 0,1 𝐾, 𝐽 = 0,3 𝐾, 𝐽 = 2,1 Most stable states 𝐾, 𝐽 = 1,0 𝐾, 𝐽 = 3,0 Deuteron Dibaryon 𝐸03

Hyperon

𝑂

𝑔 = 3

𝑣, 𝑒, 𝑡 quarks

𝑡 quark has larger mass 𝑁2 ഥ 𝑥𝑏

ሶ 𝛽𝑗𝑥 ሶ 𝛽𝑗 𝑏

𝑁2 ഥ 𝑥𝑏

ሶ 𝛽𝑗𝑥 ሶ 𝛽𝑗 𝑏 + 𝑁𝑇 2(ഥ

𝑥(𝑡))𝑏

ሶ 𝛽(𝑥(𝑡)) ሶ 𝛽 𝑏

෍

𝑗=1 2

෍

𝑗=1 3

Number of 𝑥𝑡 = 𝑥𝑗=3 in |𝜔0〉 Number of 𝑡 quarks in nucleus

Put larger mass for 𝑥(𝑡) = 𝑥𝑗=3 by hand Assumption: no 𝑇𝑉(3) breaking effect in Effect of mass in raising and lowering operators 𝑥 = 1 𝑁 𝑏† + ത 𝑏 𝑏†: raising (creation) operator of 𝑥 ത 𝑏 : lowering (annihilation) operator of ഥ 𝑥 𝑊 = 𝜇 ഥ 𝑥𝑏

ሶ 𝛽𝑗 𝜐𝐽 ሶ 𝛽 ሶ𝛾𝑥 ሶ 𝛾𝑗 𝑏 2

Hyperon mass

Mass formula 𝑁hyperon = ෩ 𝑁D4 + 4 ሚ 𝜇 1 − 𝜀 𝐷

𝑔

− 𝑁𝑇𝜀 − 2 ሚ 𝜇𝜀 1 − 𝜀 𝑍 + 4 ሚ 𝜇𝜀 𝐽 𝐽 + 1 − 1 4 𝑍2 + ሚ 𝜇𝜀2𝑍2 ሚ 𝜇 = 𝜇 𝑁2 𝜀 = 1 − 𝑁 𝑁𝑇 ෩ 𝑁D4: D-brane tension

𝐎 𝚳 𝚻 𝚶 𝚬 𝚻∗ 𝚶∗ 𝛁 𝐽 1/2 1 1/2 3/2 1 1/2 𝑍 1 －1 1 －1 －2 Exp. 939 1116 1193 1318 1232 1385 1533 1672 GMO 939 1117 1183 1328 1238 1383 1528 1673 Our 941 1115 1182 1327 1240 1380 1525 1676

Global fit with hyperon mass ෩ 𝑁D4 = 933 [MeV] 𝑁S = 603 [MeV] ሚ 𝜇 = 24.9 [MeV] 𝜀 = 0.339

GMO: Gell-Mann-Okubo formula Our: Our mass formula

𝑍: Hypercharge 𝐵: Number of baryons

Dibaryon

Octet 𝐎(𝟘𝟒𝟘) 𝚳(𝟐𝟐𝟐𝟕) 𝚻(𝟐𝟐𝟘𝟒) 𝚶(𝟐𝟒𝟐𝟗) GMO 939 1117 1183 1328 Our 975 1126 1237 1347 Decuplet 𝚬(𝟐𝟑𝟒𝟑) 𝚻∗(𝟐𝟒𝟗𝟔) 𝚶∗(𝟐𝟔𝟒𝟒) 𝛁(𝟐𝟕𝟖𝟑) GMO 1238 1383 1528 1673 Our 1311 1407 1516 1639 Dibaryon 𝑬(𝟐𝟗𝟖𝟕) 𝑬𝟏𝟒(𝟑𝟒𝟖𝟏) 𝑰 𝛁𝛁 Our 1876 2285 2084 3007 Dibaryon? 𝑬𝟐𝟏 𝑬𝟒𝟏 𝑬𝟐𝟑(𝟑𝟐𝟕𝟏? ) 𝑬𝟑𝟐 Our 1855 2157 2100 2058 Threshold 𝐎 + 𝐎 𝚬 + 𝚬 𝐎 + 𝚬 𝚳 + 𝚳 𝛁 + 𝛁 Experiment 1878 2464 2171 2232 3344 Our 1950 2622 2286 2252 3278

2nd order corrections are partially calculated

Baryon resonance

Internal excitations in baryons 𝐵 = 1 𝑌𝐽 is 𝑉(1) No internal excitation from 𝑌𝐽 Only possible source of internal excitation is 𝑥 Number of 𝑥 − Number of ഥ 𝑥 = 𝑂𝑑𝐵 Constraint: ഥ 𝑥𝑥 pair excitation(s) 𝜔0 = 𝑥 ሶ

𝛽1𝑗1𝑥 ሶ 𝛽2𝑗2𝑥 ሶ 𝛽3𝑗3𝑥 ሶ 𝛽4𝑗4 ഥ

𝑥 ሶ

𝛽5𝑗5 0

Wave function

𝑥 ሶ

𝛽𝑗: raising operator of 𝑥 ሶ 𝛽𝑗

ഥ 𝑥 ሶ

𝛽𝑗: raising operator of ഥ

𝑥 ሶ

𝛽𝑗

Baryon resonance

Input 𝑂 938 , Δ(1232) and excited state of 𝑂 at 1440 MeV

𝑶(𝑱 = Τ 𝟐 𝟑) PDG Our 𝐾 = 1/2 938(****) (938) 1440(****) (1440) 1720(***) 1880(***) 1859 2100(*) 𝐾 = 3/2 1720(****) 1821 1900(***) 2040(*) 𝚬(𝑱 = 𝟒/𝟑) PDG Our 𝐾 = 1/2 1750(*) 1910(****) 1821 𝐾 = 3/2 1232(****) (1232) 1600(***) 1478 1920(***) 2409 𝐾 = 5/2 1905(****) 2000(**) 2213 𝑱 = 𝟔/𝟑 Our 𝐾 = 3/2 2213 𝐾 = 5/2 1723

All parity +

Summary

Nuclear matrix model from D-branes in holographic QCD 𝑌𝐽: position of baryons (nucleons) 𝑥 ሶ

𝛽𝑗: carries charges of quarks (spin, flavor)

EOM of gauge field constraints U(1) constraint appropriate quark number as nuclei SU(A) constraint anti-symmetric (fermionic) wave function Interaction terms 𝑌𝐽, 𝑌𝐾 2 𝜗𝐽𝐾𝐿 𝑌𝐽, 𝑌𝐾 ഥ 𝑥𝜐𝐿𝑥 ഥ 𝑥𝜐𝐽𝑥 2 Effective trapping potential (?) Spin-orbit interaction? Flavor dependence of mass Matrix model gives good results for small baryon number We calculated mass of hyperons, dibaryons and baryon resonance For larger baryons number, more studies on 𝑌𝐽 are necessary

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