CHARGE RADII AND HIGHER ELECTROMAGNETIC MOMENTS WITH LATTICE QCD IN - - PowerPoint PPT Presentation

CHARGE RADII AND HIGHER ELECTROMAGNETIC MOMENTS WITH LATTICE QCD IN NONUNIFORM BACKGROUND FIELDS

T H E 3 4 T H I N T E R N AT I O N A L S Y M P O S I U M O N L AT T I C E G AU G E T H E O R I E S U N I V E R S I T Y O F S O U T H A M P TO N , J U LY 2 4 - 3 0 , 2 0 1 6

Z O H R E H DAVO U D I M I T

ZD and W. Detmold, Phys. Rev. D 92, 074506 (2015), ZD and W. Detmold, Phys. Rev. D 93, 014509 (2016).

E L E C T R O M A G N E T I C E F F E C T S I N S T R O N G LY I N T E R A C T I N G S Y S T E M S

1 ) D Y N A M I C A L P H O T O N S

d ~ A d ~ A

G A U S S ’ S L A W + P E R I O D I C I T Y ?

Lucini, et, al., JHEP02(2016)076. Endres, et, al., to appear in PRL, arXiv: 1507.08916 [hep-lat]. Borsanyi, et al., Science 347:1452-1455 (2015). ZD, M. J. Savage, Phys. Rev. D 90, 054503 (2014) Hayakawa and Uno, Prog. Theor. Phys. 120, 413 (2008)

E L E C T R O M A G N E T I C E F F E C T S I N S T R O N G LY I N T E R A C T I N G S Y S T E M S

2 ) C L A S S I C A L E L E C T R O M A G N E T I S M

d ~ A d ~ A

B A C K G R O U N D F I E L D S + P E R I O D I C I T Y

• G. ’t Hooft, Nuclear Physics B 153, 141 (1979).
• J. Smit and J. C. Vink, Nucl.Phys. B286, 485 (1987).
• M. Al-Hashimi and U.-J. Wiese, Annals Phys. 324, 343 (2009).

S O M E R E C E N T S TAT E - O F - T H E - A R T A P P L I C AT I O N S

F R O M L AT T I C E Q C D

np → dγ

Images by W. Detmold

Beane, at al. [NPLQCD collaboration], Phys. Rev. Lett. 115, 132001 (2015).

S O M E R E C E N T S TAT E - O F - T H E - A R T A P P L I C AT I O N S

.0 .1 .2

• L

51 π

/

L1

di

• ackgr

Beane and Savage, Nucl.Phys. A694, 511 (2001). A N E F F E C T I V E F I E L D T H E O RY R E S U LT N E E D S L AT T I C E Q C D I N P U T

F R O M L AT T I C E Q C D

np → dγ

Beane, at al. [NPLQCD collaboration], Phys. Rev. Lett. 115, 132001 (2015).

S O M E R E C E N T S TAT E - O F - T H E - A R T A P P L I C AT I O N S

S E T U P A B A C K G R O U N D M A G N E T I C F I E L D

Detmold and Savage, Nucl.Phys. A743, 170 (2004).

F R O M L AT T I C E Q C D

np → dγ

Beane, at al. [NPLQCD collaboration], Phys. Rev. Lett. 115, 132001 (2015).

M O R E P H Y S I C S W I T H B A C K G R O U N D F I E L D S ?

1 ) E M C H A R G E R A D I U S

===================================================

A.

2 ) E L E C T R I C Q U A D R U P O L E M O M E N T 4 ) A X I A L B A C K G R O U N D F I E L D S

p p

n

p

e+

νe

3 ) F O R M FA C T O R S

Detmold, Phys.Rev. D71, 054506 (2005)

I M P L E M E N TAT I O N O F U ( 1 ) B A C K G R O U N D G A U G E F I E L D S O N A P E R I O D I C H Y P E R C U B I C L AT T I C E

ZD and W. Detmold, Phys. Rev. D 92, 074506 (2015)

(0, 0)

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

(T, 0)

x3 t

1

(T, L)

U = eie ˆ

Q R Aµ(z)dzµ

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

(0, 0)

1

(T, 0)

x3 t

(T, L)

U = eie ˆ

Q R Aµ(z)dzµ

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2 P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

(0, 0)

(T, 0)

1

(T, L)

x3 t

U = eie ˆ

Q R Aµ(z)dzµ

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2 P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

(0, 0)

x3 t

1

(T, 0) (T, L)

1 1

U = eie ˆ

Q R Aµ(z)dzµ

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2 P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

(0, 0)

x3 t

1

(T, 0) (T, L)

1 1

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2

= eie ˆ

QE0atas/2

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

(0, 0)

x3 t

1

(T, 0) (T, L)

1 1

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2

= eie ˆ

QE0atas/2

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

(0, 0)

x3 t

1

(T, 0) (T, L)

1 1

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2

= eie ˆ

QE0atas/2

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

P E R I O D I C B C

(0, 0)

1

(T, 0) (T, L)

1 1

x3 t

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2

= eie ˆ

QE0atas/2

M O D I F I E D L I N K

×e−ie ˆ

QE0L2t/2

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

P E R I O D I C B C

(0, 0)

x3 t

1

(T, 0) (T, L)

1 1

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2

= eie ˆ

QE0atas/2

M O D I F I E D L I N K

×e−ie ˆ

QE0L2t/2

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

P E R I O D I C B C

(0, 0)

x3 t

1

(T, 0) (T, L)

1 1

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2

= eie ˆ

QE0atas/2

M O D I F I E D L I N K

×e−ie ˆ

QE0L2t/2

P E R I O D I C B C

eie ˆ

QE0L2T/2 = 1

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

(0, 0)

Q U A N T I Z AT I O N C O N D I T I O N F O R T H E S L O P E O F T H E F I E L D

x3 t

1

(T, 0) (T, L)

1 1

→ E = E0x3 ˆ x3

Aµ = ✓ −E0 2 (x3 − hx3 L i L)2, 0 ◆

e−ie ˆ

QE0at/2

e−(L−as)2ie ˆ

QE0at/2

= eie ˆ

QE0atas/2

M O D I F I E D L I N K

×e−ie ˆ

QE0L2t/2

P E R I O D I C B C

eie ˆ

QE0L2T/2 = 1

E0 = 4πn e ˆ QL2T

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

BOUNDARY

1 2 3 4 5 6 7 8 9 10 11

x3 − x(src)

3

• 2.4
• 2.3
• 2.2
• 2.1
• 2.0

No modified links - Nonquantized No modified links - Quantized ⌘ log C(x3, τ) C(x3, τ + 1),

x(src)

3

= 9

N E U T R A L P I O N C O R R E L AT I O N F U N C T I O N

E = E0x3 ˆ x3

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

BOUNDARY

1 2 3 4 5 6 7 8 9 10 11

x3 − x(src)

3

x(src)

3

= 9

⌘ log C(x3, τ) C(x3, τ + 1),

N E U T R A L P I O N C O R R E L AT I O N F U N C T I O N

E = E0x3 ˆ x3

2 4 6 8 10 12

• 2.4
• 2.3
• 2.2
• 2.1
• 2.0

Modified links - Nonquantized Modified links - Quantized

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

h Aν(xµ = 0, xν + aν) − e Aν(xµ = Lµ, xν + aν) i fµ,ν(xν + aν) = h i h Aν(xµ = 0, xν) − e Aν(xµ = Lµ, xν) i (fµ,ν(xν) + aν) 2 4

Lµaµ

Y

xµ=0

eie ˆ

Q[Aµ(xµ,xν=0) e Aµ(xµ,xν=Lν)]aµ

3 5 "Lνaν Y

xν=0

eie ˆ

Q[Aν(xµ=0,xν) e Aν(xµ=Lµ,xν)]aν

# = 1.

U (QCD)

µ

(x) → U (QCD)

µ

(x) × eie ˆ

QAµ(x)aµ ×

Y

ν6=µ

eie ˆ

Q[Aν(xµ=0,xν) e Aν(xµ=Lµ,xν)]fµ,ν(xν)⇥δxµ,Lµ−aµ

M O D I F I E D L I N K S

W I T H L I N K F U N C T I O N S S AT I S F Y I N G Q U A N T I Z AT I O N C O N D I T I O N S

L I N E A R LY VA RY I N G F I E L D S VA R I O U S S PA C E / T I M E D E P E N D E N C E

C H A R G E R A D I U S - Q U A D R U P O L E M O M E N T S P I N P O L A R I Z A B I L I T I E S O F N U C L E O N S

O S C I L L AT O RY F I E L D S F O R M FA C T O R S

.

• W. Detmold, Phys.Rev. D71, 054506 (2005)

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

P E R I O D I C I M P L E M E N TAT I O N O F N O N U N I F O R M B A C K G R O U N D F I E L D S

O S C I L L AT O RY F I E L D S

Aµ = (A0, −A) = ✓ia q3 eiq3x3, 0, 0, 0 ◆ → E = aeiq3x3 ˆ x3

Q U A N T I Z AT I O N C O N D I T I O N F O R T H E F R E Q U E N C Y O F T H E F I E L D

P E R I O D I C B C

(0, 0)

x3 t

1

(T, 0) (T, L)

1

M O D I F I E D L I N K

e− e ˆ

Qa q3 eiq3at e− e ˆ

Qa(L−as) q3

eiq3at

1 × e− e ˆ

Qa q3 (1−eiq3L)t

e− e ˆ

Qa q3 (1−eiq3L)T = 1

q3 = 2πn L

O R

a(Im) = πq3n0 e ˆ QT , a , a(Re) = − sin(q3L) 1 − cos(q3L)a(Im)

Q U A N T I Z AT I O N C O N D I T I O N F O R T H E A M P L I T U D E O F T H E F I E L D

As in: Bali and Endrodi, PhysRevD.92.054506

T O WA R D S A N E X T R A C T I O N O F E L E C T R I C Q U A D R U P O L E M O M E N T A N D C H A R G E R A D I U S

ZD and W. Detmold, Phys. Rev. D 93, 014509 (2016)

O N E - P H O T O N M AT C H I N G

Jµ

A S P I N - 1 T H E O RY

• f a spin-1 field. The Lagrangian, in terms of

and degrees of freedom, can be written as L = 1 2W †µνWµν + M2V †αVα − 1 2W †µν(DµVν − DνVµ) − 1 2((DµVν)† − DνV †

µ)W µν +

ieC(0) Fµν V †µV ν + ieC(2)

1

M2 ∂µF µν((DνV α)†Vα − V †αDνVα) + ieC(2)

2

M2 ∂αF µν((DαVµ)†Vν − V †

ν DαVµ) + ieC(2) 3

M2 ∂νF µα((DµVα)†Vν − V †

ν DµVα) + O

✓ 1 M4 , F 2 ◆ C(0) = µ1 Q0, C(2)

1

= 1 6e

• M2 hr2iE + eQ2
• ,

C(2)

2

= 1 4(Q0 + Q2 + µ1) + 1 6eM2 hr2iM , C(2)

3

= 1 2(Q0 + Q2 + µ1).

T H E G E N E R A L S T R AT E G Y: A N E F F E C T I V E S I N G L E - PA R T I C L E D E S C R I P T I O N

S P E C I A L C A R E M U S T B E G I V E N T O E O M O P E R AT O R S I N N R T H E O RY W I T H B A C K G R O U N D F I E L D S

Lee and Tiburzi, Phys. Rev. D 89, 054017 (2014), Phys. Rev. D 90, 074036 (2014).

ZD and W. Detmold, Phys. Rev. D 92, 074506 (2015)

A (semi) relativistic effective theory NR Green’s functions

b b H(E)

SR = M3 + eQ0 b

' + (3 + i2) b π2 2M − i2 M (S · b π)2 + e 2M2 (1 + 1) ×  iC(0) h b E · b π − SiSj b Ej b πi i − 2C(2)

1 (r · b

E) + 2C(2)

3

 r · b E − 1 2(SiSj + SjSi)ri b Ej

• −ieC(0)

2M2 (1 − 1) h b π · b E − SiSj b πj b Ei i + O ✓ 1 M4 , F 2 ◆ .(35)

G  i d dt ⌥ b H(±)

NR(b

π, b x3)

• G(±)

λ,λ0(x, t; x0, t0) = iδ3(x x0)δ(t t0)δλ,λ

for ±(t t0) > 0, a

Relativistic Green’s functions

dt

b HSR h i d

dt b

HSR i GSR(x, t; x0, t0) = iδ(x x0)δ(t t0) c Hamiltonian

A NR effective theory

b b H(±)

NR = M I3⇥3 ± eQ0ϕ I3⇥3 + b

π2 2M I3⇥3 ⌥ e(µ1 Q0) 2M2 S · ( b E ⇥ b π) ± ie(µ1 Q0) 4M2 S · (r ⇥ b E) ⌥hr2iE 6 r · b E I3⇥3 ⌥ Q2 4  SiSj + SjSi 2 3S2δij

• ri b

Ej + O ✓ 1 M3 , F 2 ◆ . (84)

Lattice QCD correlation functions

Cαβ(x, τ; x0, τ 0) = h0|[Oψ(x, τ)]α[Oψ†(x0, τ 0)]β|0iAµ

Transformed correlation functions

1 ) M AT C H I N G T O C O R R E L AT I O N F U N C T I O N S

b C(±)

MS,M0

S(x, τ; x0, τ 0) = P(±) ⌦ T(MS) U(x; b

p)C(x, τ; x0, τ 0)U1(x0; b p0) P(±) ⌦ T T

(MS)

2 ) M AT C H I N G T O E N E R G I E S

Lattice QCD correlation functions

Cαβ(x, τ; x0, τ 0) = h0|[Oψ(x, τ)]α[Oψ†(x0, τ 0)]β|0iAµ

Spatially projected Correlation functions at large Euclidean times

CMS,M 0

S(τ, τ 0) → ZMSeE (MS ) n

(ττ 0)

2 ) M AT C H I N G T O E N E R G I E S

I S O L AT I N G C H A R G E R A D I U S C O N T R I B U T I O N

I S O L AT I N G Q U A D R U P O L E M O M E N T C O N T R I B U T I O N

1 3(E(MS=1)

n

+ E(MS=0)

n

+ E(MS=1)

n

) = (n + 1 2)|ωE| E0 hr2iE 6 , E(MS=1)

n

+ E(MS=1)

n

2E(MS=0)

n

= E0Q2,

ω2

E = eQE0

M E0 : slope of the field

Lattice QCD correlation functions

Cαβ(x, τ; x0, τ 0) = h0|[Oψ(x, τ)]α[Oψ†(x0, τ 0)]β|0iAµ

Spatially projected Correlation functions at large Euclidean times

CMS,M 0

S(τ, τ 0) → ZMSeE (MS ) n

(ττ 0) A P O S I T I V E LY C H A R G E D PA R T I C L E I N A L I N E A R LY D E C R E A S I N G E L E C T R I C F I E L D I N T H E X 3 D I R E C T I O N

x3

L 2L −2L −L

E3(x3) x3

L 2L −2L −L

ϕ(x3)

L → ∞

P O T E N T I A L

M O R E P H Y S I C S W I T H B A C K G R O U N D F I E L D S ?

1 ) E M C H A R G E R A D I U S

===================================================

A.

2 ) E L E C T R I C Q U A D R U P O L E M O M E N T 4 ) A X I A L B A C K G R O U N D F I E L D S

p p

n

p

e+

νe

S O M E I M P L E M E N TAT I O N S U N D E R WAY

3 ) F O R M FA C T O R S

Detmold, Phys.Rev. D71, 054506 (2005)

T H A N K Y O U