[PPT] - Few-Body in EFT Gautam Rupak Mississippi State University EMMI: PowerPoint Presentation

SLIDE 1

Few-Body in EFT

Gautam Rupak Mississippi State University

EMMI: The Systematic Treatment of the Coulomb Interaction in Few-Body Systems Darmstadt, May 30 - June 3, 2016

SLIDE 2

Outline

N-d scattering
Field-redefinition
p-p fusion

SLIDE 3

A little detour on the lattice

Consider: ;
Need effective “cluster” Hamiltonian -- acts in

cluster coordinates, spins,etc.

Calculate reaction with cluster Hamiltonian. Many

possibilities --- traditional methods, continuum EFT, lattice method

a(b, γ)c a(b, c)d

SLIDE 4

Adiabatic Projection Method

Microscopic Hamiltonian Cluster Hamiltonian

- acts on the cluster CM and spins

L3(A−1) L3

Initial state |~ Ri Evolved state |~ Riτ = e−τH|~ Ri

τh ~

R0|H|~ Riτ

Energy measurements in cluster basis. Divide by the norm matrix as these are not orthogonal basis smaller matrices in practice!!

[N⌧] ~

R, ~ R0 =⌧ h~

R|~ R0i⌧

SLIDE 5

0.1 0.2 0.3 0.4 0.5 0.6 5 10 15 20 25 τ (MeV−1) E (MeV) Convergence: L=7, b=1/100 MeV−1

Pine, Lee, Rupak, EPJA 2013

Neutron-Deuteron System

grouping R found efficient

∼ 30 × 30

SLIDE 6

Something still missing ... long range Coulomb

O ✓ α p2 ◆ O ✓ α p2 αµ p ◆

· · ·

SLIDE 7

Spherical-wall method

Rw L/2 −L/2

ψshort(r) ∝ j0(kr) cot δs − n0(kr), ψCoulomb(r) ∝ F0(kr) cot δsc + G0(kr)

Hard spherical wall boundary conditions, Borasoy et al. 2007 Carlson et al. 1984 Even older ? Adjust from free theory: j0(k0Rw) = 0

IR scale setting

SLIDE 8

p-p Coulomb Subtracted Phase Shift

5 10 15 20 25 30 10 20 30 40 50 60 p (MeV) δsc (degree) Analytic b=1/100 MeV−1 b=1/200 MeV−1

Rupak, Ravi PLB 2014

T =Tc + Tsc Tc ≈2π µ e2iσ − 1 2ip T ≈2π µ e2i(σ+δsc) − 1 2ip

3% error in fits

SLIDE 9

Improvement

−90 −75 −60 −45 −30 −15 20 40 60 80 100 120 140

Quartet−S

Breakup −90 −75 −60 −45 −30 −15 20 40 60 80 100 120 140

Quartet−S

Breakup

p−d (EFT)

−90 −75 −60 −45 −30 −15 20 40 60 80 100 120 140

Quartet−S

Breakup

n−d (EFT)

−90 −75 −60 −45 −30 −15 20 40 60 80 100 120 140

Quartet−S

Breakup

n−d

−90 −75 −60 −45 −30 −15 20 40 60 80 100 120 140

Quartet−S

Breakup

p−d

p (MeV) δ0 (degrees)

50 100 p (MeV)

80
60
40
20

δ0(degree) breakup b=1/100 MeV

1

b=1/150 MeV

1

b=1/200 MeV

1

STM

Pine, Lee, Rupak (2013)

Elhatisari, Lee, Meißner, Rupak (2016)

SLIDE 10

n-d doublet channel

/

(/)

p cot δ = −1/a + rp2/2 1 + p2/p2

a ∼0.65 fm, r ∼ − 150 fm, p0 ∼13 MeV

ERE form van Oers & Seagrave (1967)

what EFT for modified ERE

Virtual state at 0.5 MeV Girard & Fuda (1979)

Efimov physics

SLIDE 11

Efimov plot

Shallow virtual to bound state

lattice QCD with B field, even with heavy pions?

Higa, Rupak, Vaghani, van Kolck Preliminary

SLIDE 12

Phillips-Girard-Fuda

()

()

3-body correlation

Preliminary

SLIDE 13

Adhikari-Torreao

()

()

Adhikari-Torreao

Virtual

Preliminary

SLIDE 14

Field Redefinition

L = N †(i

→

∂ 0 +

→

r

2

2M )N + d†

t(i →

Dt)dt + d†

s(i →

Dt)ds + t†(i

→

DΩ)t gt(di

t †NPiN + h.c) gs(da s †N ¯

PaN + h.c) wt(t†σiNdi

t + h.c.) ws(t†τ aNda s + h.c.)

Start here and write in terms of only nucleon fields

i

→

Dt = (i

→

∂ 0

→

r

2

4M + ∆t), i

→

Ds = (i

→

∂ 0

→

r

2

4M + ∆s), i

→

DΩ = (i

→

∂ 0 +

→

r

2

6M + ∆Ω) .

alternatives possible

SLIDE 15

Integrate out fields

∂L ∂t† = 0 ⇒ t = (i

→

DΩ)−1(wtσiNdi

t + wsτ aNda s)

You see that it generates dimer-nucleon interaction ... more to come

t†(i

→

DΩ)t

∂L ∂di

t † = 0

⇒ dj

t = (−i →

D

j i t )−1[gtNPiN + wtwsN †σiτ a(i →

DΩ)−1Nda

s]

− i

→

D

i j t = −i →

Dt δij − w2

t N †σiσj(i →

DΩ)−1N

Now things get interesting

SLIDE 16

Finally integrate the last dimer field and write

L = N †(i

→

∂ 0 +

→

r

2

2M )N g2

t

2 [(NPiN)†(i

→

D

i j t )−1NPjN + h.c.]

1 2[gs(N ¯ PaN)† + gtwtws(NPkN)†(i

→

D

k l t )−1N †(i →

DΩ)−1σlτ aN](i

→

D

a b

)−1 ⇥ [gsN ¯ PbN + gtwtwsN †(i

→

DΩ)−1σiτ bN(i

→

D

i j t )−1NPjN] + h.c. ,

- generates higher-body terms
- need to remove time-derivatives

(−i

→

D

a b

) = [−i

→

Ds δa b − w2

sN †τ aτ b(i →

DΩ)−1N − w2

t w2 sN †(i →

DΩ)−1σiτ aN(−i

→

D

i j t )−1N †σjτ b(i →

DΩ)−1N]

SLIDE 17

Keep upto 3-body contact interactions

(i

→

D

i j t )−1 = (i →

Dt)−1δi j + w2

t

∆2

t

N †σiσj(i

→

DΩ)−1N + w2

t

∆3

t∆Ω

(i∂0 +

→

r

2

4M )N †σiσjN , (i

→

D

a b

)−1 = (i

→

Ds)−1δa b + w2

s

∆2

s

N †τ aτ b(i

→

DΩ)−1N + w2

s

∆3

s∆Ω

(i∂0 +

→

r

2

4M )N †τ aτ bN ⌘ (i

→

D

a b s )−1 ,

Almost home, and write

L = N †(i

→

∂ 0 +

→

r

2

2M )N g2

t

2 [(NPiN)†(i

→

D

i j t )−1NPjN + h.c.]

g2

s

2 [(N ¯ PaN)†(i

→

D

a b s )−1N ¯

PbN + h.c.] gtgswtws[(N ¯ PaN)†(i

→

Ds)−1N †(i

→

DΩ)−1σiτ aN(i

→

Dt)−1NPiN + h.c.] ⌘ L1 + L2 + L3 +...

SLIDE 18

Field Redefinition

To remove time-derivative from two-body try

N → N + a1P †

i N †(NPiN) + b1 ¯

P †

aN †(N ¯

PaN)

L2 = g2

t

∆t (NPiN)†NPiN g2

s

∆s (N ¯ PaN)†N ¯ PaN

g2

t

8M∆2

t

[(NPiN)†(N

↔

r

2

PiN) + h.c.] g2

s

8M∆2

s

[(NPaN)†(N

↔

r

2

PaN) + h.c.]

After a good amount of elbow grease using

a1 = g2

t /∆2 t

b1 = g2

s/∆2 s

Bedaque, Grießhammer (2000) Bedaque, Rupak, Grießhammer, Hammer (2003)

SLIDE 19

Removing time-derivatives from three-body contact interaction requires, another field redefinition. However, the leading momentum independent term is simple

−  g2

t w2 t

∆2

t∆Ω

+ g4

t

6∆3

t

(NPiN)†N †σiσjN(NPjN) −

 g2

sw2 s

∆2

s∆Ω

+ g4

s

6∆3

s

(N ¯

PaN)†N †τ aτ bN(N ¯ PbN) − gtgswtws ∆t∆s∆Ω − g2

t g2 s

4∆2

t∆s

− g2

t g2 s

4∆t∆2

s

[(N ¯

PaN)†N †τ aσiN(NPiN) + h.c.]

Need to pull out the SU(4) symmetric piece

Bedaque, Rupak, Grießhammer, Hammer (2003)

SLIDE 20

p-p fusion

|NN(s;~ k, p)i = p p 4⇡ 1 (2⇡)3 Z dΩˆ

p

h N(~ k/2 + ~ p)P(s)N(~ k/2 ~ p) i† |0i

cm, relative momentum Projector with projectors

X

Ave. pol

Tr h P(s)P(s0)†i = 1 2δss0 hNN(s0;~ k0, p0)|NN(s;~ k, p)i = (3)(~ k ~ k0)(p p0)ss0

Chen, Rupak, Savage (1999)

SLIDE 21

project the appropriate p-wave channels Chen, Rupak, Savage (1999) Fleming, Mehen, Stewart (2000) s-wave comparison indices contracted

|hd; x|A−

y |ppi| = gACη

r32π γ3 Λ(p)δxy ∼ p ZdhgA(~ ✏∗

d × ~

✏∗)xˆ py(−2µ)2 Z d3q (2⇡)3 (+)

~ p

(~ q) q2 + 2 hNN(s0;~ k0, p0)|N ⇤

aN ⇤ b Oab;cdNcNd|NN(s;~

k, p)i = 1 2 Z d cos ✓P(s0)

ab Oab;cd

 P(s0)

ab † cd

SLIDE 22

Few-Body in EFT Gautam Rupak Mississippi State University EMMI: - - PowerPoint PPT Presentation

Few-Body in EFT

Outline

A little detour on the lattice

Adiabatic Projection Method

Neutron-Deuteron System

Something still missing ... long range Coulomb

Spherical-wall method

p-p Coulomb Subtracted Phase Shift

Improvement

n-d doublet channel

Efimov plot

Phillips-Girard-Fuda

Adhikari-Torreao

Field Redefinition

Integrate out fields

Field Redefinition

p-p fusion

Thank you