1

EFT Approaches to αα and Nα Systems

Renato Higa

Kernfysisch Versneller InstituutUniversity of Groningen

INT Workshop on Weakly Bound Systems, Mar. 09, 2010

2

EFT Approaches to αα and Nα Systems

Outline

• Motivation

• EFT approach and universality

• halo/cluster EFT

? EM interactions? αα scattering? Nα scattering

• Summary and outlook

3

http://www.int.washington.edu/PROGRAMS/weakly bound wkshp/

http://fy.chalmers.se/subatom/halo/halo.html http://www.ornl.gov/info/ornlreview/v34 2 01/search.htm

• N/Z 1: challenge forshell model

• few nucleon systems:formation of halo systems

? 11Be , 19C, 11Li , 6He , 14Be, 8He, 8B, 17Ne, ...

4

http://www.int.washington.edu/PROGRAMS/weakly bound wkshp/

http://fy.chalmers.se/subatom/halo/halo.html http://www.ornl.gov/info/ornlreview/v34 2 01/search.htm

• N/Z 1: challenge forshell model

• few nucleon systems:formation of halo systems

? 11Be , 19C, 11Li , 6He , 14Be, 8He, 8B, 17Ne, ...

5

p + 7Be→ 8B + γsolar neutrinos

8Be + α→ 12C + γred giants

9Be(12+

) + α→ 12C + n

core-colapse supernovae

6

naturalness: where NDA works

R

2π /k

V(r)

r

R

7

strongly interacting systems: fine-tuning

R

a

2π /k

V(r)

r

R

8

“speakers should assume that nobody in the audience knowsanything about the details and differences of the atom-atom /internuclear / intermolecular forces and that those should berepeatedly explained and emphasized.”

9

“speakers should assume that nobody in the audience knowsanything about the details and differences of the atom-atom /internuclear / intermolecular forces and that those should berepeatedly explained and emphasized.”

at least one speaker knows nothing about the details and differ-ences of the atom-atom / internuclear / intermolecular forces.

10

Effective Field Theory

k, 1/a ∼Mlo, 1/R ∼Mhi

L = φ†

24i∂0+~∇2

2m

35φ−b0 (φφ)†(φφ)+

b2

8

»(φφ)

†φ(↔∇)

2φ+ H.c.

–+· · ·

−c0 (φφφ)†(φφφ)+· · · ,

iT(−1)

=0b

0b 0b

+

0b 0b 0b

+ + ...

= −ib0(Λ)

1 + (Λ + ik)m4πb0(Λ)

iT(0)

= (−1)

T(−1)

T

2b

= −ib2(Λ)ˆ

1 + (Λ + ik)m4πb0(Λ)˜2

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“We would be also happy if you could address the advantages /limitations of the few-body method of your choice”

• symmetries• simplicity• able to handle non-local interactions• W/EM interactions• 3-4B extensions• controlled and systematic low-energy expansion

12

“We would be also happy if you could address the advantages /limitations of the few-body method of your choice”

• symmetries• simplicity• able to handle non-local interactions• W/EM interactions• 3-4B extensions• controlled and systematic low-energy expansion

• controlled and systematic low-energy expansion• complexity for >4B (Kirscher et al.)• ...

13

Universality in two-body systems

a is the only relevant scale at LO

f(θ) =1

−1/a− ik, EB,V =

1ma2

,dσ

dΩ=

4a2

1 + a2k2(1)

• a→∞: scale-invariant system⇒ BS at threshold, dσ/dΩ saturates the UB

• RG analysis: non-trivial IR fixed point(Birse et al., Phys. Lett. B 464, 169)

• close analogy to critical phenomena(liquid-gas phase transition, ferromagnets)

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1

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Universality in three-body systems (Braaten and Hammer, Phys. Rept. 428, 259)

10-4

10-3

10-2

10-1

100

101

102

103

1 10 100

|E3|

Λ

-10

-5

0

5

10

100

102

H(Λ

)

Λ

(Hammer and RH, Eur. J. Phys. A 37, 193)

• renormalization requires c0 at LO⇒ limit cycle

• E(n)/E(n+1) → const.(∼ 515 for bosons)

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halo/cluster EFT: separation of scales

• excitation of each cluster√mcE∗c ∼Mhi (& mπ)

• binding of the valence nucleons (clusters) ∼Mlo Mhi

• extension of the core—treated in perturbation theory

• power-counting: modified to account for other effects (resonance/Coulomb)

• expansion around the resonance: rearrangement of the perturbative series,improved convergence

• Coulomb interactions

16

halo/cluster EFT: k mπ,√mcE∗c ∼ Mhi

Physical quantities: k, 1/a0 ∼Mlo, r0 ∼M−1hi ,P ∼M

−3hi , . . .

Tl = −2π

µ

k2l(2l + 1)

k2l+1(cot δl − i)Pl(cos θ)

k2l+1

cot δl ≈ −1/al +rl

2k

2+Pl4k

4+ · · ·

L = φ†

24i∂0+~∇2

4µ

35φ+σ d†

24i∂0+~∇2

8µ−∆

35d+g

»d†φφ+(φφ)

†d

–+· · · ,

+ += +

∆ ∼Mlo → iD(0)d =

iσ

−∆ + iε∼

1

Mlo

(NN)

∆ ∼M2lo/µ → iD

(0)d =

iσ

q0 − q2/8µ−∆ + iε∼

µ

M2lo

(αα)

17

Coulomb photons dominant at very low energies

kC = Z1Z2αemµ

∼ kCk≡ η , ∼ kCm

[12

lnΛ2k

+ · · ·]

For αα: kC = Z2ααemmα/2 ∼Mhi

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• non-perturbative Coulomb (Kong and Ravndal, NPA 665, 137)

Coulomb wave functions: |k〉 → |χ(±)k 〉

T → TC + TCS

η = Z2αemµ/k = kC/k

σl = arg Γ(1 + l + iη) , C(0) 2η = e−πηΓ(1 + iη)Γ(1− iη)

G(±)C (E) =

1E − H0 − VC ± iε

= 2µ∫

d3q

(2π)3

|χ(±)q 〉〈χ(±)

q |2µE − q2 ± iε

= + +

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TCS = 〈χ−k′| VS |χ+k 〉+ 〈χ−k′| VSG

+C VS |χ

+k 〉+ · · ·

T(0)CS = = C(E)χ(−) ∗

k′ (0)χ(+)k (0) = C(E)C(0) 2

η e2iσ0 ,

T(1)CS = = C(E)C(0) 2

η e2iσ0 C(E) J0(E) ,

TCS = + · · ·+ ...

= C(0) 2η

C(E) e2iσ0

1− C(E) J0(E),

J0(E) = 2µ∫

d3q

(2π)3

χ(+)q (0)χ(+) ∗

q (0)k2 − q2 ± iε

= 2µ∫

d3q

(2π)3

2πηqe2πηq − 1

1k2 − q2 + iε

(ηq = kC/q)

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αα scattering

• 0+ resonance (8Be g.s.):

ELABR = 184.15± 0.07 keV , ΓLAB

R = 11.14± 0.50 eV

Mlo ≈√µELAB

R ∼ 20 MeV, Mhi ∼ mπ ∼ 140 MeV

• power-counting: ELAB ≤ 3.0 MeV

• scattering: Afzal et.al. (1969)

? ELAB ≤ 3.0 MeV : data from Heydenburg and Temmer (1956)? ERE parameters from Russell et.al. (1956), Rasche (1967):a0 = (−1.65± 0.17)× 103 fm,r0 = 1.084± 0.011 fm ∼ 1/Mhi, P0 = −1.76± 0.22 fm3 ∼ 1/M3

hi

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TCS = C(0) 2η

C(E) e2iσ0

1− C(E) J0(E)= −2π

µ

C(0) 2η e2iσ0

− 1a0

+ r02 k

2 − iε+ 2πµ J0(E)

= −2πµ

C(0) 2η e2iσ0

− 1ac0

+ r02 k

2 − 2aBH(η)

,

aB =1

Z2αemµ∼ 1Mhi

H(η) = ψ(iη) +1

2iη− ln(iη)⇒

η1→ aB

2 ikη1→ 1

12 (aB k)2 + 1120 (aB k)4

• without Coulomb: conformal invariance in 8Be , Efimov state in 12C at LO(RH, Hammer, van Kolck, Nucl. Phys. A 809, 171)

• with Coulomb: 8Be and 12C 0+ states remain close to threshold

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(RH, Hammer, van Kolck, 2008)

90

120

150

180

0 1 2 3

δ0 [degre

es]

ELAB [MeV]

LONLO

Afzal et.al.

0

0.05

0.1

0.15

0.2

0 1 2 3

k c

ot

δ0 [fm

-1]

ELAB [MeV]

LONLO

Afzal et.al.

a0 (103 fm) r0 (fm) P0 (fm3)

LO −1.80 1.083 —

NLO −1.92± 0.09 1.098± 0.005 −1.46± 0.08

Rasche −1.65± 0.17 1.084± 0.011 −1.76± 0.22

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fine-tuning puzzle

∆(R)︸︷︷︸M2hiµ

= ∆(κ)︸ ︷︷ ︸M2hiµ

−∆(loops)︸ ︷︷ ︸M2hiµ

(natural)

∆(R)︸︷︷︸MhiMlo

µ

= ∆(κ)︸ ︷︷ ︸M2hiµ

−∆(loops)︸ ︷︷ ︸M2hiµ

(fine-tuned like NN)

∆(R)︸︷︷︸M2loµ

= ∆(κ)︸ ︷︷ ︸M2hiµ

−∆(loops)︸ ︷︷ ︸M2hiµ

(fine-tuned to get ER)

∆(R)︸︷︷︸M3lo

Mhiµ

= ∆(κ)︸ ︷︷ ︸M2hiµ

−∆(loops)︸ ︷︷ ︸M2hiµ

(fine-tuned to get ΓR)

∼ factor of 1000!!!(Oberhummer et al., Science 289, 88; RH, Hammer, van Kolck, 2008)

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pα scattering: S1/2, P3/2, P1/2

LLO = φ†

[i∂0+

~∇2

2mα

]φ+N†

[i∂0+

~∇2

2mN

]N

+η1+ t†

[i∂0+

~∇2

2(mα+mN)−∆1+

]t

+g1+

2

t†~S† ·

[N ~∇φ− (~∇N)φ

]+ H.c.− r

[t†~S† · ~∇(Nφ) + H.c.

]LNLO = η0+ s

†[−∆0+

]s+g0+

[s†Nφ+φ†N†s

]+g′1+ t

†

[i∂0 +

~∇2

2(mα+mN)

]2

t

(Bertulani, Hammer, van Kolck, NPA 712, 37;

Bedaque, Hammer, van Kolck, PLB 569, 159)

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(RH, Bertulani, van Kolck, in preparation)

-60

-40

-20

0

20

40

60

80

100

120

140

0.0 1.0 2.0 3.0 4.0 5.0

δ [

de

gre

es]

Ep [MeV]

P3/2

S1/2

LONLO

Arndt et al.

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6

dσ

/dΩ

[b

arn

/sr]

Ep[MeV]

θ=140 deg

LONLOERE

Nurmela et.al.

ab-initio: Nollett em et al., PRL 99, 022502 (2007),

Quaglioni & Navratil, PRL 101, 092501 (2008)

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c/r2+Coulomb: warm-up for 3α

• 3-body problem with large a ∼ 1D Schrodinger Eq. with V (r) = 1/r2

• limit cycle for c < −1/4 ⇔ Efimov spectrum(Beane et al., Bawin and Coon, Braaten and Phillips, Long and van Kolck...)

• counterterm: log-periodic function of the cutoff

• Counterterm parameter Λ∗: iteration of quantum corrections(dimensional transmutation)

• model to study loss of conformal invariance (Kaplan et al.)

27

1/r2+Coulomb: warm-up for 3α

(Hammer, RH, Eur. J. Phys. A 37, 193)

10-4

10-3

10-2

10-1

100

101

102

103

1 10 100

EB

Λ

Efixed

α=0.0

α=0.1

α=0.3

α=0.5

-10

-5

0

5

10

1 10 100

H(Λ

)

Λ

α=0.0

α=0.1

α=0.3

α=0.5

10-4

10-3

10-2

10-1

100

101

102

103

1 10 100

EB

Λ

α=0.0

α=0.1, exact

α=0.1, pert.

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Summary

• Halo nuclei, cluster systems: opportunities for EFT approach

• universality ⇔ limit cycles

• αα scattering

? Coulomb turned off ⇒ conformal invariance @LO, Efimov spectrum in 12C? incredible amount of fine-tuning? LO (parameter-free) works well at very low energies, NLO improves description

up to ELAB ≈ 3 MeV? extraction of the ERE parameters with improved errorbars

• p-α scattering: good description of the P3/2 resonance

• future: 3α, p-7Be, nradcap (Rupak & RH), Borromean halos, heavier nuclei, ...

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selected questions

• What are the unsolved problems in the theory of scattering both in the nuclearand AMO sector?

• Does P-Wave universality exist?

• How can we decide if a nuclear halo is an Efimov state?

• How can the halo EFT approach be useful to ab-initio calcuations?

• When can we do scattering with halo EFT?