Electric properties of halo nuclei using EFT

Electric properties of halo nuclei using EFT

Daniel PhillipsOhio University

Work done in collaboration with H.-W. Hammer

Research supported by the US Department of Energy and the Deutsche Forschungsgemeinschaft

see also Rupak & Higa arXiv:1101.0207

arXiv:1001.1511 and “in preparation”

Outline

Outline

• Generalities: halo nuclei, experimental techniques

• Example 1: Halo EFT for Carbon-19

• Dissociation

Outline



• Dissociation

Shallow S-wave state

Outline



• Dissociation

• Example 2: Halo EFT for Beryllium-11

• Form factors

• E1 transition from s-state to p-state

• Dissociation

• Conclusion


Outline



• Dissociation

• Example 2: Halo EFT for Beryllium-11

• Form factors

• E1 transition from s-state to p-state

• Dissociation

• Conclusion


Shallow S-wave state Shallow P-wave state

Halo nuclei

Halo nuclei

• Here I define a halo nucleus as one in which the last nucleon (or nucleons) have a <r2>1/2 that is markedly larger than the range, R, of the interaction it has with the rest of the nucleus–the core.

Halo nuclei


• Typically R≡Rcore∼2-3 fm.

Halo nuclei



• And since <r2> is related to the neutron separation energy we are looking for systems with neutron separation energies appreciably less than 1 MeV.

Halo nuclei




• Define Rhalo=<r2>1/2. Seek EFT expansion in Rcore/Rhalo.

Halo nuclei




• Define Rhalo=<r2>1/2. Seek EFT expansion in Rcore/Rhalo.

• By this definition the deuteron is the lightest halo nucleus, and the pionless EFT for few-nucleon systems is a specific case of halo EFT.

Probing halo nuclei

• Typically produced in unstable beams.

• Neutron pickup reactions, e.g. (p,d), in inverse kinematics are one way to investigate

• Here my concern will be with electromagnetic probes.

Probing halo nuclei




• Coulomb dissociation: collide halo nucleus (we hope peripherally) with a high-Z nucleus

Bertulani, arXiv:0908.4307

Probing halo nuclei




• Coulomb dissociation: collide halo nucleus (we hope peripherally) with a high-Z nucleus

• Do with different Z, different nuclear sizes, different energies to test systematics


From disintegration to E1 strength

dσC

2πbdb=

�

πL

�dEγ

EγnπL(Eγ , b)σπL

γ (Eγ)

• Coulomb excitation dissociation cross section (p.v. b>>Rtarget)


• virtual photon numbers, dependent only on kinematic factors. Number of equivalent (virtual) photons that strike the halo nucleus.

dσC

2πbdb=

�

πL

�dEγ


γ (Eγ)

nπL(Eγ , b)



• virtual photon numbers, dependent only on kinematic factors. Number of equivalent (virtual) photons that strike the halo nucleus.

• Virtual photon numbers computable in terms of relative velocity, equivalent photon frequency, impact parameter

dσC

2πbdb=

�

πL

�dEγ


γ (Eγ)

nπL(Eγ , b)



dσC

2πbdb=

�

πL

�dEγ


γ (Eγ)




dσC

2πbdb=

�

πL

�dEγ


γ (Eγ)

σπLγ (Eγ)


• can then be extracted: it’s the (total) cross section for dissociation of the nucleus due to the impact of photons of multipolarity πL.

Our first halo nucleus: Carbon-19


• 19C neutron separation energy=576 keV. Ground state=1/2+

• First excitation in 18C is 1.62 MeV above ground state

• Treat 1/2+ as s-wave halo state: 18C + n





• Blo/Bhi≈1/3⇒Rcore/Rhalo≈0.5






• Data, including cut on impact parameter

Nakamura et al. (2003)

Our approach

Our approach

• S-wave (and P-wave) states generated by cn contact interactions

Our approach


• No discussion of nodes, details of n-core interaction, spectroscopic factors

u0(r) = A0 exp(−γ0r)

Our approach



• “Halo EFT”, expansion in Rcore/Rhalo.


Our approach




• 19C: input at LO: neutron separation energy of s-wave state. Output at LO: Coulomb dissociation of s-wave state, radius of state.


Our approach





• A0 (“wf renormalization”) can be fit at NLO.


Our approach





• A0 (“wf renormalization”) can be fit at NLO.

• Situation is different for P-wave state in11Be, but that comes later....


Lagrangian I: shallow s-wave state

L = c†�

i∂t +∇2

2M

�c + n†

�i∂t +

∇2

2m

�n

+σ†�η0

�i∂t +

∇2

2Mnc

�+ ∆0

�σ − g0[σn†c† + σ†nc]


• c, n: “core”, “neutron” fields. c: boson, n: fermion.

• σ: s-wave field

L = c†�

i∂t +∇2

2M

�c + n†

�i∂t +

∇2

2m

�n

+σ†�η0

�i∂t +

∇2

2Mnc

�+ ∆0

�σ − g0[σn†c† + σ†nc]




• Minimal substitution→dominant EM interactions; other terms suppressed by additional powers of Rcore/Rhalo

L = c†�

i∂t +∇2

2M

�c + n†

�i∂t +

∇2

2m

�n

+σ†�η0

�i∂t +

∇2

2Mnc

�+ ∆0

�σ − g0[σn†c† + σ†nc]




• Minimal substitution→dominant EM interactions; other terms suppressed by additional powers of Rcore/Rhalo

• ...if coefficients natural. But that’s a testable assumption.

L = c†�

i∂t +∇2

2M

�c + n†

�i∂t +

∇2

2m

�n

+σ†�η0

�i∂t +

∇2

2Mnc

�+ ∆0

�σ − g0[σn†c† + σ†nc]

Dressing the s-wave state

= +

Kaplan, Savage, Wise; van Kolck; Gegelia; Birse, Richardson, McGovern

• σnc coupling g0 of order Rhalo, nc loop of order 1/Rhalo. Therefore need to sum all bubbles:


= +



Dσ(p) =1

∆0 + η0[p0 − p2/(2Mnc)]− Σσ(p)


= +



Dσ(p) =1

∆0 + η0[p0 − p2/(2Mnc)]− Σσ(p)


= +


Σσ(p) = −g20mR

2π

�µ + i

�

2mR

�p0 −

p2

2Mnc+ iη

��(PDS)


Dσ(p) =1

∆0 + η0[p0 − p2/(2Mnc)]− Σσ(p)


= +

t =2π

mR

11a0− 1

2r0k2 + ik


Σσ(p) = −g20mR

2π

�µ + i

�

2mR

�p0 −

p2

2Mnc+ iη

��(PDS)


Dσ(p) =1

∆0 + η0[p0 − p2/(2Mnc)]− Σσ(p)


= +

t =2π

mR

11a0− 1

2r0k2 + ik


Σσ(p) = −g20mR

2π

�µ + i

�

2mR

�p0 −

p2

2Mnc+ iη

��(PDS)

+ regularDσ(p) =2πγ0

m2Rg2

0

11− r0γ0

1p0 − p2

2Mnc+ B0

Predicting dissociationc.f. Chen, Savage (1999)

Predicting dissociation

• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.

c.f. Chen, Savage (1999)



• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1





M =eQcg02mR

γ20 +

�p� − m

Mnck�2





ME1 = A0eZeff

√3

2p�

(p�2 + γ20)2


Zeff=6/19




• Final-state interactions suppressed by (Rcore/Rhalo)3

ME1 = A0eZeff

√3

2p�

(p�2 + γ20)2


Zeff=6/19





• First gauge-invariant contact operator: LE1σ†E · (n

↔∇ c) + h.c.

ME1 = A0eZeff

√3

2p�

(p�2 + γ20)2


Zeff=6/19





• First gauge-invariant contact operator:

• Need modified NDA to account for shallow S-wave state. LE1 enters in corrections suppressed by (Rcore/Rhalo)4

LE1σ†E · (n

↔∇ c) + h.c.

ME1 = A0eZeff

√3

2p�

(p�2 + γ20)2


Beane, Savage (2001)

Zeff=6/19





• First gauge-invariant contact operator:

• Need modified NDA to account for shallow S-wave state. LE1 enters in corrections suppressed by (Rcore/Rhalo)4

• Consistent with short-distance piece of FSI loop due to P-wave interactions

LE1σ†E · (n

↔∇ c) + h.c.

ME1 = A0eZeff

√3

2p�

(p�2 + γ20)2


Beane, Savage (2001)

Zeff=6/19

ResultsData: Nakamura et al., PRL, 1999


• Observable is dB(E1)/dE: E1 strength for transition to a core + neutron state, per unit energy, as function of energy (momentum) of the outgoing nc pair

• Multiply by NE1(Eγ):



∼ A20 =

2γ0

1− r0γ0


• r0 fixed from fitting height of peak at NLO

• γ0 determines peak position



∼ A20 =

2γ0

1− r0γ0


• r0 fixed from fitting height of peak at NLO

• γ0 determines peak position



• Determine S-wave 18C-n scattering parameters from dissociation data.

S-wave form factor

-iΓc(q)

q

S-wave form factor

Gc(|q|) = eQc2γ0

f |q| arctan�

f |q|2γ0

�

f=m/Mnc=mR/M

-iΓc(q)

q

S-wave form factor

Gc=eQc at |q|=0 ✔

Gc(|q|) = eQc2γ0

f |q| arctan�

f |q|2γ0

�

f=m/Mnc=mR/M

-iΓc(q)

q

�r2E� =

f2

2γ20

S-wave form factor

Gc=eQc at |q|=0 ✔

Gc(|q|) = eQc2γ0

f |q| arctan�

f |q|2γ0

�

f=m/Mnc=mR/M

-iΓc(q)

q

�r2E� =

f2

2γ20

S-wave form factor

Gc=eQc at |q|=0 ✔

Gc(|q|) = eQc2γ0

f |q| arctan�

f |q|2γ0

�

f=m/Mnc=mR/M

-iΓc(q)

q

Next correction: A0 > √2γ0, i.e. correction to strength of tail of 1/2+ wave function. Obtained from dissociation cross section

�r2E� =

f2

2γ20

S-wave form factor

Gc=eQc at |q|=0 ✔

Gc(|q|) = eQc2γ0

f |q| arctan�

f |q|2γ0

�

f=m/Mnc=mR/M

-iΓc(q)

q


First purely short-distance effect LC0,2 σ✝ ∇2A0 σ: suppressed by (Rcore/Rhalo)3

�r2E� =

f2

2γ20

S-wave form factor

Gc=eQc at |q|=0 ✔

Gc(|q|) = eQc2γ0

f |q| arctan�

f |q|2γ0

�

f=m/Mnc=mR/M

-iΓc(q)

q


First purely short-distance effect LC0,2 σ✝ ∇2A0 σ: suppressed by (Rcore/Rhalo)3

(<rE2>C19-<rE2>C18)1/2=0.23 + 0.08 fm

LO NLO

Beryllium-11 as a (one-neutron)* halo nucleus

*could also be thought of as a 3n halo


• First excitation in 10Be: 3.4 MeV, 10Be ground state is 0+

• 11Be neutron separation energy=504 keV. Ground state=1/2+

• Excited state 320±100 keV above ground state, 1/2-






• TWO halo states, one s-wave and one p-wave







• n10Be scattering, l=1, j=1/2 channel, resonant scattering apparent

*could also be thought of as a 3n haloTypel & Baur, Phys. Rev. Lett. 93, 142502 (2004); Nucl. Phys. A759, 247 (2005); Eur. Phys. J. A 38, 355 (2008)

u0(r) = A0 exp(−γ0r);u1(r) = A1 exp(−γ1r)�

1 +1

γ1r

�






• n10Be scattering, l=1, j=1/2 channel, resonant scattering apparent


*could also be thought of as a 3n haloTypel & Baur, Phys. Rev. Lett. 93, 142502 (2004); Nucl. Phys. A759, 247 (2005); Eur. Phys. J. A 38, 355 (2008)

u0(r) = A0 exp(−γ0r);u1(r) = A1 exp(−γ1r)�

1 +1

γ1r

�

Electromagnetic properties


• B(E1)(1/2+→1/2-)=0.105(12) e2fm2 from intermediate-energy Coulomb excitation

(Summers et al., 2007)

• B(E1)(1/2+→1/2-)=0.116(12) e2fm2 from lifetime measurements (Millener et al., 1983)





Coulomb-induced breakup of 11Be, Palit et al. (2003), c.f. Fukuda et al. (2004)





B(E1, li) = [Z(1)effe]2

34π

�r2�li

Non-energy-weighted sum rule:






B(E1, li) = [Z(1)effe]2

34π

�r2�li


<r2>1/2=5.7(4) fm

Zeff=4/11






B(E1, li) = [Z(1)effe]2

34π

�r2�li


<r2>1/2=5.7(4) fm

Zeff=4/11

c.f. atomic-physics measurement of radiiNoerterhaueser et al., PRL (2009)


Lagrangian II: shallow S- and P-states


• σ, πj: S-wave and P-wave fields

• Compute power of non-minimal EM couplings by NDA with rescaled fields.

L = c†�

i∂t +∇2

2M

�c + n†

�i∂t +

∇2

2m

�n

+σ†�η0

�i∂t +

∇2

2Mnc

�+ ∆0

�σ + π†

j

�η1

�i∂t +

∇2

2Mnc

�+ ∆1

�πj

−g0

�σn†c† + σ†nc

�− g1

2

�π†

j (n↔

i∇j c) + (c†↔

i∇j n†)πj

�

−g1

2M −m

Mnc

�π†

j

→i∇j (nc)−

↔i∇j (n†c†)πj

�+ . . . ,

Dressing the P-wave state

= +

Bertulani, Hammer, van Kolck (2002); Bedaque, Hammer, van Kolck (2003)

• Proceed similarly for p-wave state:


= +




= +


Dπ(p) =1

∆1 + η1[p0 − p2/(2Mnc)]− Σπ(p)



= +


Σπ(p) = −mRg21k2

6π

�32µ + ik

�

Dπ(p) =1

∆1 + η1[p0 − p2/(2Mnc)]− Σπ(p)


• Here both parameters (Δ1 and g1) are mandatory for renormalization at LO


= +



6π

�32µ + ik

�

Dπ(p) =1

∆1 + η1[p0 − p2/(2Mnc)]− Σπ(p)


• Here both parameters (Δ1 and g1) are mandatory for renormalization at LO


= +



6π

�32µ + ik

�

+ regularDπ(p) = − 3π

m2Rg2

1

2r1 + 3γ1

i

p0 − p2/(2Mnc) + B1

Dπ(p) =1

∆1 + η1[p0 − p2/(2Mnc)]− Σπ(p)

Fixing P-wave parameters


• Input at LO: neutron separation energy of s-wave and p-wave state, B0=504 keV, B1=184 keV

• ⇒γ0=0.15 fm-1; γ1=0.09 fm-1 both ∼1/Rhalo




• Also need to fix r1 at LO, we anticipate r1∼1/Rcore





• k3 cot δ1=-1/2 r1 (k2 + γ12) at LO; (k∼γ1⇒r1k2 >> γ13)






• Note: a1∼Rhalo2 Rcore, c.f. original scenario of Bertulani et al. a1∼Rhalo3







• We are going to use the B(E1:1/2+→1/2-) strength to fix r1.







• We are going to use the B(E1:1/2+→1/2-) strength to fix r1.

• No propagation of experimental errors here, but it’s easy to do

+

Irreducible S-to-P vertex: bound-to-bound transition

-iΓjµ(k) k0=ω

i

jq

k

+


Divergences cancel, as they should

-iΓjµ(k) k0=ω

i

jq

k

+



Γji = δjiΓE + kjqiΓM for k·q=0; k·ε=0

-iΓjµ(k) k0=ω

i

jq

k

+




Exploit current conservation kµΓjµ=0

-iΓjµ(k) k0=ω

i

jq

k

+





⇒ωΓj0=kjΓE

-iΓjµ(k) k0=ω

i

jq

k

+

Γj0(k) ∼�

d3ru1(r)

rY1j(r̂)eik·r u0(r)

r





⇒ωΓj0=kjΓE

-iΓjµ(k) k0=ω

i

jq

k

→ kj

�dru1(r)ru0(r) as |k|→ 0

+

Γj0(k) ∼�

d3ru1(r)

rY1j(r̂)eik·r u0(r)

r





⇒ωΓj0=kjΓE

-iΓjµ(k) k0=ω

i

jq

k

E1 matrix element

Converting to result for S-to-P transition

B(E1) =14π

�Γ̄E

ω

�2

Keeping track of constants, defn of B(E1):


B(E1) =Z2

effe2

4π

γ0γ21

(−r1 − 3γ1)43

��dr

�1 +

1γ1r

�e−γ1rre−γ0r

�2

=Z2

effe2

4π

4γ0

−3r1

�2γ1 + γ0

(γ0 + γ1)2

�2

+ . . .

B(E1) =14π

�Γ̄E

ω

�2



• No cutoff parameter needed: integral finite without regularization

B(E1) =Z2

effe2

4π

γ0γ21

(−r1 − 3γ1)43

��dr

�1 +

1γ1r


�2

=Z2

effe2

4π

4γ0

−3r1

�2γ1 + γ0

(γ0 + γ1)2

�2

+ . . .

B(E1) =14π

�Γ̄E

ω

�2




• Numbers: BLO(E1)=0.106 e2fm2 , yields r1=-0.66 fm-1

B(E1) =Z2

effe2

4π

γ0γ21

(−r1 − 3γ1)43

��dr

�1 +

1γ1r


�2

=Z2

effe2

4π

4γ0

−3r1

�2γ1 + γ0

(γ0 + γ1)2

�2

+ . . .

B(E1) =14π

�Γ̄E

ω

�2





• NLO corrections from A0 and γ1/r1 corrections to A1

B(E1) =Z2

effe2

4π

γ0γ21

(−r1 − 3γ1)43

��dr

�1 +

1γ1r


�2

=Z2

effe2

4π

4γ0

−3r1

�2γ1 + γ0

(γ0 + γ1)2

�2

+ . . .

B(E1) =14π

�Γ̄E

ω

�2





• NLO corrections from A0 and γ1/r1 corrections to A1

• Also first contribution of physics at scale Rcore occurs at NLO

B(E1) =Z2

effe2

4π

γ0γ21

(−r1 − 3γ1)43

��dr

�1 +

1γ1r


�2

=Z2

effe2

4π

4γ0

−3r1

�2γ1 + γ0

(γ0 + γ1)2

�2

+ . . .

B(E1) =14π

�Γ̄E

ω

�2


Computing dissociation: γE1 + 11Be→10Be + n


• Note that final state can be spin-1/2 or spin-3/2: final-state interactions are “nautral” in the spin-3/2 channel, i.e. suppressed by three orders.



• FSI in spin-1/2 channel: stronger, but “kinematic” nature of p-wave state implies interaction still perturbative away from resonance:




k3 cot δ1=-1/2 r1 (k2 + γ12) ⇒ δ1 ∼ Rcore/Rhalo if k ∼ 1/Rhalo ∼ γ1.




!

LO NLO





• Also get corrections to A0 (a.k.a. wf renormalization) at NLO

!

LO NLO


Coulomb dissociation: formulae

dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4

�• Straightforward computation of diagrams yields:

c.f. Rupak & Higa arXiv:1101.0207


dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4


Spin-1/2 channel Spin-3/2 channel



dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4


• Expand in Rcore/Rhalo:Spin-1/2 channel Spin-3/2 channel



dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4

�

dB(E1)dE

LO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4 No FSI

• Straightforward computation of diagrams yields:




dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4

�

dB(E1)dE

LO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4 No FSI

dB(E1)dE

NLO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4

�r0γ0 +

2γ0

3r1

γ20 + 3p�2

p�2 + γ21

�





dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4

�

dB(E1)dE

LO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4 No FSI

dB(E1)dE

NLO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4

�r0γ0 +

2γ0

3r1

γ20 + 3p�2

p�2 + γ21

�

Wf renormalization





dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4

�

dB(E1)dE

LO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4 No FSI

dB(E1)dE

NLO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4

�r0γ0 +

2γ0

3r1

γ20 + 3p�2

p�2 + γ21

�

Wf renormalization 2P1/2-wave FSI





dB(E1)dE

= e2Z2eff

mR

2π2A2

0

�p�3[2p�3 cot(δ(1/2)(p�)) + γ3

0 + 3γ0p�2]2

[p�6 + p�6 cot2(δ(1/2)(p�))](p�2 + γ20)4

+8p�3

(p�2 + γ20)4

�

dB(E1)dE

LO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4 No FSI

dB(E1)dE

NLO

= e2Z2eff

3mR

2π2

8γ0p�3

(p�2 + γ20)4

�r0γ0 +

2γ0

3r1

γ20 + 3p�2

p�2 + γ21

�

Wf renormalization 2P1/2-wave FSI


• Higher-order corrections to phase shift at NNLO. Appearance of S-to-2P1/2 E1 counterterm also at that order.



0 1 2 3 4 5 6 7E* [MeV]

0

0.2

0.4

dB(E

1)/d

E [e

2 fm2 /M

eV] Typel & Baur, PRL 2004

LO, FSI not relevantNLO, r1=-0.66 fm-1, A0/(2 γ0)1/2=1.3

Coulomb dissociation: result

• Reasonable convergence

• Information on value of r0

through fitting of A0:

• Value of r1 used to fit B(E1:1/2+→1/2-) works here too.

NLO: (<rc2>+<rBe2>) 1/2=2.44 fm

r0=2.7 fm

0 1 2 3 4 5 6 7E* [MeV]

0

0.2

0.4

dB(E

1)/d

E [e

2 fm2 /M

eV] Typel & Baur, PRL 2004

LO, FSI not relevantNLO, r1=-0.66 fm-1, A0/(2 γ0)1/2=1.3

Coulomb dissociation: result

• Reasonable convergence

• Information on value of r0

through fitting of A0:

• Value of r1 used to fit B(E1:1/2+→1/2-) works here too.

NLO: (<rc2>+<rBe2>) 1/2=2.44 fm

dB(E1)dE

=48

π2B0

y3

(y2 + 1)4

�e2Q2

c∆�r2E�(σ) − 3π

4B(E1)

(1 + x)4(1 + 3y2)(y2 + x2)(1 + 2x)2

�x = γ1/γ0; y = p�/γ0

r0=2.7 fm

Output

Output

• Value obtained for r0 implies (<rE2>+<rE, Be102>)1/2=2.40 fm at LO, 2.43 fm at NLO

Output


• Experimental result <rBe112>1/2=2.463(16) fm Noerterhaueser et al., PRL (2009)

Output


• Experimental result <rBe112>1/2=2.463(16) fm

• r1=-0.66±0.29 fm-1 implies a1=374±150 fm3

Noerterhaueser et al., PRL (2009)

Output



• r1=-0.66±0.29 fm-1 implies a1=374±150 fm3

• Information on P-wave interactions from dissociation, but only at NLO


Output



• r1=-0.66±0.29 fm-1 implies a1=374±150 fm3


• P-wave state’s radius also calculable


Output



• r1=-0.66±0.29 fm-1 implies a1=374±150 fm3


• P-wave state’s radius also calculable

• Universal correlation:


B(E1) =2e2Q2

c

15π�r2

E�(π)x

�1 + 2x

(1 + x)2

�2

; x = γ1/γ0

Conclusions

Conclusions

• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo

Conclusions


• Carbon-19: one-neutron halo, shallow S-wave state

Conclusions



• Beryllium-11: one-neutron halo, shallow S- and P-wave state

Conclusions




• 11Be has big B(E1) strength, can be hard to calculate in ab initio methods because of extended nature of p-wave state. Here controlled by r1.


Conclusions





• NLO in 11Be: fix A0 from dissociation to continuum, predict radii, extract a1


Conclusions






• Correlations between low-energy observables


Conclusions






• Correlations between low-energy observables

• Other one- (and two-?) neutron (?and proton) halos await: “universality”.


Date post:	17-Feb-2022
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Electric properties of halo nuclei using EFT

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