Some theoretical issues in HAL QCD method Sinya Aoki Center for - PowerPoint PPT Presentation

Some theoretical issues in HAL QCD method Sinya Aoki Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University International Molecule-type Workshop Frontiers in Lattice QCD and related topics April 15- April 26, 2019, Yukawa Institute for Theoretical Physics, Kyoto University

0. Introduction Some Issues in the HAL QCD method

HAL QCD method Derivative expansion A powerful method to investigate hadron interactions energy-independent non-local potential NBS wave function Strategy q k t = h 0 | N ( ~ ~ k ( ~ x ) e − W ~ ~ k 2 + m 2 ' r, t ) N ( ~ r + ~ x, t ) | NN, W ~ k i W ~ k = 2 N sin( kx + δ l ( k )) X Y lm ( Ω ~ x ) C lm → kx lm ~ k 2 , H 0 = �r 2 Z ~ ~ � � k ( ~ k ( ~ y ) d 3 y, k � H 0 x ) = U ( ~ y ) ' k = E ~ ' x, ~ E ~ , W ~ k ≤ W th = 2 m N + m ⇡ m N m N x, ~ r ) � (3) ( ~ U ( ~ y ) = V ( ~ y ) x, ~ x � ~ x, ~ � 2 ) + V T ( x ) S 12 + V LS ( x ) ~ L · ~ S + O ( ~ r 2 ) V ( ~ r ) = V 0 ( x ) + V σ ( x )( ~ � 1 · ~

Some issues Q2. Validity of the derivative expansion ? small parameter ? II. Definition of the HAL QCD potential with the derivative expansion Q3. Is the HAL QCD potential Hermite ? III. Hermite potential from non-Hermite potential I. The HAL QCD potential from the moving system Sinya Gongyo’s taik on 4/24. Q5. Quark annihilation processes and resonances ? Yutaro Akahoshi’s talk on 4/16. Q1. The HAL QCD potential in the moving system ? Q4. Partial wave mixings in the cubic box ?

I. The HAL QCD potential from the moving system

Our Motivation no “vacuum” state in the NBS true only in the CM system But no definition of the potential directly from the boosted NBS Moving system the potential describes the vacuum as the “deeply bound state” of two pions ? “vacuum” has the same quantum numbers “vacuum” state appears in the NBS wave function in center-of-mass system σ resonance from the I = 0 ππ scattering in the HAL QCD method sin( kx + δ l ( k )) X k t = h 0 | N ( ~ ~ Y lm ( Ω ~ x ) C lm k ( ~ x ) e − W ~ → r, t ) N ( ~ r + ~ x, t ) | NN, W ~ k i ' kx lm

Generalized definition of the potential in the CM system non-equal time NBS wave function in CM system For simplicity, we consider this problem for scalar field. Can we extract the generalized potential from the boosted NBS function ? also in Akahoshi’s talk � ◆� ⌧ ✓ ~ ◆ ✓ � − ~ x 2 , x 4 2 , − x 4 x ~ k ( ~ � � x, x 4 ) = 0 ' � N N � NN, W ~ � � k 2 2 sin( kx + δ l ( k ) + π l/ 2) X A lm Y lm ( Ω ~ x ) k = | ~ ' k | kx lm ~ ~ k ( ~ k ( ~ ( E k � H 0 ) ' x, x 4 ) = V x 4 ( ~ x, r ) ' x, x 4 ) x 4 = 0: equal time scheme the HAL QCD potential in the x 4 scheme

CM and moving systems boost velocity boost velocity depends on energy boost factor Moving important relation CM p, P p ∗ , P ∗ ~ p 1 ~ ~ V p 2 ~ − ~ p ∗ p ∗ ~ P ∗ = 0 ~ p = ( ~ p 1 − ~ p 2 ) / 2 ~ P = ~ p 1 + ~ p 2 1 P ⇤ ( Λ P ) 0 = � ( P 0 � ~ V ~ P ⇤ where � = V 2 , := P ) , k := ( Λ P ) k = � ( P k � V k P 0 ) = 0 , 0 p 1 � ~ as ~ V = ~ P/P 0 . p 2 � ( ~ 0 � ~ p ) 2 = P 2 P 2 P · ~ ? ) 2 + ( ~ k ) 2 = ~ p ⇤ ) 2 p ⇤ p ⇤ � m 2 . ( ~ := ( ~ P 2 4 0

relation of NBS wave functions between 2 systems CM moving Scalar field NBS where x := x 1 � x 2 , i � m 2 = 0 fy p 2 p,P ( x, X ) := h 0 | T { � ( x 1 ) � ( x 2 ) }| p 1 , p 2 i at the total en q q 1 + m 2 + p 2 p 2 P 0 = 2 + m 2 . ~ ~ x := x 1 − x 2 , X = ( x 1 + x 2 ) / 2 h 0 | e i ˆ P · X T { � ( x/ 2) � ( � x/ 2) } e � i ˆ P · X | p 1 , p 2 i p,P ( x, X ) = e � iP · X ' p,P ( x ) , = e � iP · X ' p,P ( x ) = e � iP ∗ · X ∗ ' p ∗ ,P ∗ ( x ⇤ ) . Since P · X = P ⇤ · X ⇤ ' p ∗ ,P ∗ ( x ⇤ ) , ' p,P ( x ) = p ⇤ = (0 , ~ P ⇤ = ( P ⇤ x ⇤ = ⇣ ⌘ � ( x 0 � ~ x k � ~ p ⇤ ) , 0 , ~ 0) x ) , � ( ~ V x 0 ) , ~ V ~ x ?

Euclidean space-time (Wick rotation) moving Minkowski complex in CM ! CM Euclid x 4 = ix 0 � ( X 0 � ~ V ~ X ⇤ = X ) , 0 X ⇤ ~ ~ X ⇤ ~ k = � ( ~ X k � ~ = V X 0 ) , X ? , X ⇤ � ( X 4 � i ~ V ~ X ⇤ k = � ( X k + i ~ X ⇤ = X ) , V X 4 ) , ? = X ? , ? 4 x ⇤ � ( x 0 � ~ x ⇤ � ( x 4 � i ~ x ⇤ k = � ( x k + i ~ x ⇤ = x ) , V ~ = x ) , V x 4 ) , ? = x ? , V ~ 0 4 x k � ~ x ⇤ x ⇤ = k = � ( ~ V x 0 ) , ~ ~ ~ x ? , ? ' p ∗ ,P ∗ ( x ⇤ ) , ' p,P ( x ) = p ⇤ = (0 , ~ P ⇤ = x ⇤ = ⇣ ⌘ ⇣ ⌘ � ( x 4 � i ~ x k + i ~ p ⇤ ) , P ⇤ 0 , ~ 0 V ~ x ) , � ( ~ V x 4 ) , ~ x ? . 0 � ~ ~ p ⇤ ) 2 = P 2 P 2 0 = P 0 P P ⇤ ~ � P 0 X 4 + i ~ P ~ X = � P ⇤ 0 X ⇤ � m 2 , ( ~ � , V = , � iP · X = 4 . 4 P 0

HAL QCD potential from boosted NBS wave function using Boosted NBS generalized definition CM CM CM Moving Moving Moving ( E p ∗ � H 0 ) ' p ∗ ,P ∗ ( x ⇤ ) = V x ∗ x, r ) ' p ∗ ,P ∗ ( x ⇤ ) 4 ( ~ ' p ∗ ,P ∗ ( x ⇤ ) , ' p,P ( x ) = p ∗ ) 2 + r 2 x ⇤ , r ) ' p,P ( x ) = ( ~ x ∗ 4 ( ~ ' p,P ( x ) V x ∗ m ⌘ 2 + r 2 � 2 ⇣ r 2 = r x k + iV @ x 4 x ? . x ⇤ ? r � k k ⌘ 2 + r 2 p ⇤ ) 2 + � 2 ⇣ ( ~ r x k + iV @ x 4 x ? V � ( x 4 � iV x k ) ( � ( x k + iV x 4 ) , x ? , r x ⇤ ) ' p,P ( x ) = ' p,P ( x ) m ⌘ 2 ⇣ x 4 = 0 (equal-time boosted NBS) is required.

Minkowski time from Euclidean correlates ! x 4 = 0 ⌘ 2 + r 2 p ⇤ ) 2 + � 2 ⇣ ( ~ r x k + iV @ x 4 x ? V � i � V x k ( � x k , x ? , r x ⇤ ) ' p,P ( x ) = ' p,P ( x ) . m potential in the x ⇤ Each x k 0 = − γ x k scheme 0 � ~ ~ p ⇤ ) 2 = P 2 P 2 0 = P 0 P P ⇤ ~ � m 2 , ( ~ � , V = , 4 P 0

Time-dependent HAL QCD method R-correlator sum of NBS h { } �� i e i ~ P · ~ A n e � P n X X 0 X 4 ' p n ,P n ( x ) + · · · , P ( x, X ) e i ~ P · ~ = h T { � ( x 1 ) � ( x 2 ) } J † �� (0 , ~ X := P ) i R ~ ~ ~ 0  W th P n X ⌘ 2 + r 2 p ⇤ ) 2 + � 2 ⇣ ( ~ r x k + iV @ x 4 x ? V � i � V x k ( � x k , x ? , r x ⇤ ) ' p,P ( x ) = ' p,P ( x ) . m 0 � ~ ~ p ⇤ ) 2 = P 2 P 2 0 = P 0 P P ⇤ ~ 0 ) 2 � m 2 , ( P n ( ~ � , V = , � 2 n = 4 P 0 0 ) 2 − ~ ( P n P 2 ⇣ P 2 ⌘ 0 ) 2 � ~ X 4 � ~ @ 2 X A n e � P n P 2 ) ' p n ,P n ( x ) , 0 X 4 (( P n A 1 ( x, X ) := R ~ P ( x, X ) = n P 2 ⌘ 2 R ~ 0 ) 2 � ~ ⇣ X 4 � ~ A n e � P n @ 2 X 0 X 4 (( P n P 2 ) 2 ' p n ,P n ( x ) , A 2 ( x, X ) := P ( x, X ) = n r 2 B ( x, X ) := x ? A 1 ( x, X ) , ⌘ 2 R ~ ⇣ � @ X 4 r x k + i | ~ P | @ 2 C ( x, X ) = P ( x, X ) x 4 ⌘ 2 ' p n ,P n ( x ) , 0 ) 2 ⇣ A n e � P n X 0 X 4 ( P n = r x k + iV n @ x 4 (41) n V n = | ~ P | P n 0

LO potentail A 2 ( x, X ) � m 2 A 1 ( x, X ) + B ( x, X ) + C ( x, X ) 4 ✓ ⇣ X "( 0 ) 2 � ~ ) # ⇣ ( P n P 2 ( P n 0 ) 2 � m 2 + r 2 P 2 ⌘ 0 ) 2 � ~ X A n e − P n 0 X 4 P 2 ( r x k + iV n @ x 4 ) 2 ( P n = + ' p n ,P n ( x ) x ? 0 ) 2 � ~ 4 ( P n n = ( ~ n ) 2 p ∗ = γ 2 n ⌘ 2 + r 2 p ⇤ ) 2 + � 2 ⇣ ( ~ r x k + iV @ x 4 Set x 4 = 0 x ? V � i � V x k ( � x k , x ? , r x ⇤ ) ' p,P ( x ) = ' p,P ( x ) . m 0 ) 2 � ~ X A n e � P n 0 X 4 V � i γ n V n x k ( � n x k , x ? , r x ⇤ )(( P n P 2 ) ' p n ,P n ( x ) = m n 0 ) 2 � ~ X A n e � P n ✓ 0 X 4 (( P n P 2 ) ' p n ,P n ( x ) = mV x ∗ 4 =0 ( x k = 0 , x ? , r x ∗ ) X Set x k = 0 n n = mV x ⇤ 4 =0 ( x k = 0 , x ? , r ) A 1 ( x, X ) ound state saturation. At the LO, we have A 2 ( x, X ) / 4 � m 2 A 1 ( x, X ) + B ( x, X ) + C ( x, X ) V LO 4 =0 (0 , x ? ) = . x ⇤ mA 1 ( x, X ) x 4 = x k = 0

Future investigations 1. Check the formula for he simple system 2. resonance in the HAL QCD potential 3. extension to fermions (Baryons) lower components mix relativistic formulation for the “potential” ? I = 2 ππ moving system σ resonance in I = 0 ππ moving system

II. Definition of the HAL QCD potential with the derivative expansion

expansion. Of course, the scheme is not unique. One may use a different one. We here propose a scheme to fix the potential completely using the derivative momentum dependent part of the potential. For simplicity, let us consider the scalar particles and ignore the angular (scheme) explicitly. Therefore this potential is ambiguous. We have to fix the definition of the potential This equation does not fix the non-local potential due to the restriction of energies. Z ~ ~ k ( ~ k ( ~ y ) d 3 y, ( E ~ k − H 0 ) ' x ) = U ( ~ y ) ' x, ~ W ~ k ≤ W th • We consider the expansion in terms of r 2 (but not L 2 ). • Terms with odd number of r are not included. This is our scheme. • The potential must be non-Hermitian. We can make it Hermitian as seen later.