Electric properties of halo nuclei using EFT
Transcript of 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
• Generalities: halo nuclei, experimental techniques
• Example 1: Halo EFT for Carbon-19
• Dissociation
Outline
• Generalities: halo nuclei, experimental techniques
• Example 1: Halo EFT for Carbon-19
• Dissociation
Shallow S-wave state
Outline
• Generalities: halo nuclei, experimental techniques
• Example 1: Halo EFT for Carbon-19
• Dissociation
• Example 2: Halo EFT for Beryllium-11
• Form factors
• E1 transition from s-state to p-state
• Dissociation
• Conclusion
Shallow S-wave state
Outline
• Generalities: halo nuclei, experimental techniques
• Example 1: Halo EFT for Carbon-19
• 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 S-wave state Shallow P-wave state
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
• 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.
• Typically R≡Rcore∼2-3 fm.
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.
• Typically R≡Rcore∼2-3 fm.
• 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
• 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.
• Typically R≡Rcore∼2-3 fm.
• And since <r2> is related to the neutron separation energy we are looking for systems with neutron separation energies appreciably less than 1 MeV.
• Define Rhalo=<r2>1/2. Seek EFT expansion in Rcore/Rhalo.
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.
• Typically R≡Rcore∼2-3 fm.
• And since <r2> is related to the neutron separation energy we are looking for systems with neutron separation energies appreciably less than 1 MeV.
• 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
• 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.
• Coulomb dissociation: collide halo nucleus (we hope peripherally) with a high-Z nucleus
Bertulani, arXiv:0908.4307
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.
• 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
Bertulani, arXiv:0908.4307
From disintegration to E1 strength
dσC
2πbdb=
�
πL
�dEγ
EγnπL(Eγ , b)σπL
γ (Eγ)
• Coulomb excitation dissociation cross section (p.v. b>>Rtarget)
From disintegration to E1 strength
• virtual photon numbers, dependent only on kinematic factors. Number of equivalent (virtual) photons that strike the halo nucleus.
dσC
2πbdb=
�
πL
�dEγ
EγnπL(Eγ , b)σπL
γ (Eγ)
nπL(Eγ , b)
• Coulomb excitation dissociation cross section (p.v. b>>Rtarget)
From disintegration to E1 strength
• 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
EγnπL(Eγ , b)σπL
γ (Eγ)
nπL(Eγ , b)
• Coulomb excitation dissociation cross section (p.v. b>>Rtarget)
From disintegration to E1 strength
dσC
2πbdb=
�
πL
�dEγ
EγnπL(Eγ , b)σπL
γ (Eγ)
Bertulani, arXiv:0908.4307
• Coulomb excitation dissociation cross section (p.v. b>>Rtarget)
From disintegration to E1 strength
dσC
2πbdb=
�
πL
�dEγ
EγnπL(Eγ , b)σπL
γ (Eγ)
σπLγ (Eγ)
• Coulomb excitation dissociation cross section (p.v. b>>Rtarget)
• 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
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
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
• S-wave (and P-wave) states generated by cn contact interactions
• No discussion of nodes, details of n-core interaction, spectroscopic factors
u0(r) = A0 exp(−γ0r)
Our approach
• S-wave (and P-wave) states generated by cn contact interactions
• No discussion of nodes, details of n-core interaction, spectroscopic factors
• “Halo EFT”, expansion in Rcore/Rhalo.
u0(r) = A0 exp(−γ0r)
Our approach
• S-wave (and P-wave) states generated by cn contact interactions
• No discussion of nodes, details of n-core interaction, spectroscopic factors
• “Halo EFT”, expansion in Rcore/Rhalo.
• 19C: input at LO: neutron separation energy of s-wave state. Output at LO: Coulomb dissociation of s-wave state, radius of state.
u0(r) = A0 exp(−γ0r)
Our approach
• S-wave (and P-wave) states generated by cn contact interactions
• No discussion of nodes, details of n-core interaction, spectroscopic factors
• “Halo EFT”, expansion in Rcore/Rhalo.
• 19C: input at LO: neutron separation energy of s-wave state. Output at LO: Coulomb dissociation of s-wave state, radius of state.
• A0 (“wf renormalization”) can be fit at NLO.
u0(r) = A0 exp(−γ0r)
Our approach
• S-wave (and P-wave) states generated by cn contact interactions
• No discussion of nodes, details of n-core interaction, spectroscopic factors
• “Halo EFT”, expansion in Rcore/Rhalo.
• 19C: input at LO: neutron separation energy of s-wave state. Output at LO: Coulomb dissociation of s-wave state, radius of state.
• A0 (“wf renormalization”) can be fit at NLO.
• Situation is different for P-wave state in11Be, but that comes later....
u0(r) = A0 exp(−γ0r)
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]
Lagrangian I: shallow s-wave state
• 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]
Lagrangian I: shallow s-wave state
• c, n: “core”, “neutron” fields. c: boson, n: fermion.
• σ: s-wave field
• 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]
Lagrangian I: shallow s-wave state
• c, n: “core”, “neutron” fields. c: boson, n: fermion.
• σ: s-wave field
• 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]
• σnc coupling g0 of order Rhalo, nc loop of order 1/Rhalo. Therefore need to sum all bubbles:
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)
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)
Dressing the s-wave state
= +
Kaplan, Savage, Wise; van Kolck; Gegelia; Birse, Richardson, McGovern
Σσ(p) = −g20mR
2π
�µ + i
�
2mR
�p0 −
p2
2Mnc+ iη
��(PDS)
• 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)
Dressing the s-wave state
= +
t =2π
mR
11a0− 1
2r0k2 + ik
Kaplan, Savage, Wise; van Kolck; Gegelia; Birse, Richardson, McGovern
Σσ(p) = −g20mR
2π
�µ + i
�
2mR
�p0 −
p2
2Mnc+ iη
��(PDS)
• 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)
Dressing the s-wave state
= +
t =2π
mR
11a0− 1
2r0k2 + ik
Kaplan, Savage, Wise; van Kolck; Gegelia; Birse, Richardson, McGovern
Σσ(p) = −g20mR
2π
�µ + i
�
2mR
�p0 −
p2
2Mnc+ iη
��(PDS)
+ regularDσ(p) =2πγ0
m2Rg2
0
11− r0γ0
1p0 − p2
2Mnc+ B0
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
c.f. Chen, Savage (1999)
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1
c.f. Chen, Savage (1999)
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1
M =eQcg02mR
γ20 +
�p� − m
Mnck�2
c.f. Chen, Savage (1999)
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1
ME1 = A0eZeff
√3
2p�
(p�2 + γ20)2
c.f. Chen, Savage (1999)
Zeff=6/19
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1
• Final-state interactions suppressed by (Rcore/Rhalo)3
ME1 = A0eZeff
√3
2p�
(p�2 + γ20)2
c.f. Chen, Savage (1999)
Zeff=6/19
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1
• Final-state interactions suppressed by (Rcore/Rhalo)3
• First gauge-invariant contact operator: LE1σ†E · (n
↔∇ c) + h.c.
ME1 = A0eZeff
√3
2p�
(p�2 + γ20)2
c.f. Chen, Savage (1999)
Zeff=6/19
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1
• Final-state interactions suppressed by (Rcore/Rhalo)3
• 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
c.f. Chen, Savage (1999)
Beane, Savage (2001)
Zeff=6/19
Predicting dissociation
• Counting in S waves: a0∼Rhalo∼1/γ0; r0∼Rcore. r0=0 at LO.
• Leading order: no FSI, γ0 is only free parameter=0.16 fm-1
• Final-state interactions suppressed by (Rcore/Rhalo)3
• 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
c.f. Chen, Savage (1999)
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):
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
∼ A20 =
2γ0
1− r0γ0
• Multiply by NE1(Eγ):
• r0 fixed from fitting height of peak at NLO
• γ0 determines peak position
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
∼ A20 =
2γ0
1− r0γ0
• Multiply by NE1(Eγ):
• r0 fixed from fitting height of peak at NLO
• γ0 determines peak position
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
• Determine S-wave 18C-n scattering parameters from dissociation data.
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
Next correction: A0 > √2γ0, i.e. correction to strength of tail of 1/2+ wave function. Obtained from dissociation cross section
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
Next correction: A0 > √2γ0, i.e. correction to strength of tail of 1/2+ wave function. Obtained from dissociation cross section
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
• 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-
*could also be thought of as a 3n halo
Beryllium-11 as a (one-neutron)* halo nucleus
• 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
*could also be thought of as a 3n halo
Beryllium-11 as a (one-neutron)* halo nucleus
• 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
�
Beryllium-11 as a (one-neutron)* halo nucleus
• 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
• Blo/Bhi≈1/6⇒Rcore/Rhalo≈0.4
*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)
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)
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)
B(E1, li) = [Z(1)effe]2
34π
�r2�li
Non-energy-weighted sum rule:
Coulomb-induced breakup of 11Be, Palit et al. (2003), c.f. Fukuda et al. (2004)
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)
B(E1, li) = [Z(1)effe]2
34π
�r2�li
Non-energy-weighted sum rule:
<r2>1/2=5.7(4) fm
Zeff=4/11
Coulomb-induced breakup of 11Be, Palit et al. (2003), c.f. Fukuda et al. (2004)
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)
B(E1, li) = [Z(1)effe]2
34π
�r2�li
Non-energy-weighted sum rule:
<r2>1/2=5.7(4) fm
Zeff=4/11
c.f. atomic-physics measurement of radiiNoerterhaueser et al., PRL (2009)
Coulomb-induced breakup of 11Be, Palit et al. (2003), c.f. Fukuda et al. (2004)
Lagrangian II: shallow S- and P-states
• c, n: “core”, “neutron” fields. c: boson, n: fermion.
• 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:
Dressing the P-wave state
= +
Bertulani, Hammer, van Kolck (2002); Bedaque, Hammer, van Kolck (2003)
• Proceed similarly for p-wave state:
Dressing the P-wave state
= +
Bertulani, Hammer, van Kolck (2002); Bedaque, Hammer, van Kolck (2003)
Dπ(p) =1
∆1 + η1[p0 − p2/(2Mnc)]− Σπ(p)
• Proceed similarly for p-wave state:
Dressing the P-wave state
= +
Bertulani, Hammer, van Kolck (2002); Bedaque, Hammer, van Kolck (2003)
Σπ(p) = −mRg21k2
6π
�32µ + ik
�
Dπ(p) =1
∆1 + η1[p0 − p2/(2Mnc)]− Σπ(p)
• Proceed similarly for p-wave state:
• Here both parameters (Δ1 and g1) are mandatory for renormalization at LO
Dressing the P-wave state
= +
Bertulani, Hammer, van Kolck (2002); Bedaque, Hammer, van Kolck (2003)
Σπ(p) = −mRg21k2
6π
�32µ + ik
�
Dπ(p) =1
∆1 + η1[p0 − p2/(2Mnc)]− Σπ(p)
• Proceed similarly for p-wave state:
• Here both parameters (Δ1 and g1) are mandatory for renormalization at LO
Dressing the P-wave state
= +
Bertulani, Hammer, van Kolck (2002); Bedaque, Hammer, van Kolck (2003)
Σπ(p) = −mRg21k2
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
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
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)
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
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.
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.
• No propagation of experimental errors here, but it’s easy to do
+
Irreducible S-to-P vertex: bound-to-bound transition
Divergences cancel, as they should
-iΓjµ(k) k0=ω
i
jq
k
+
Irreducible S-to-P vertex: bound-to-bound transition
Divergences cancel, as they should
Γji = δjiΓE + kjqiΓM for k·q=0; k·ε=0
-iΓjµ(k) k0=ω
i
jq
k
+
Irreducible S-to-P vertex: bound-to-bound transition
Divergences cancel, as they should
Γji = δjiΓE + kjqiΓM for k·q=0; k·ε=0
Exploit current conservation kµΓjµ=0
-iΓjµ(k) k0=ω
i
jq
k
+
Irreducible S-to-P vertex: bound-to-bound transition
Divergences cancel, as they should
Γji = δjiΓE + kjqiΓM for k·q=0; k·ε=0
Exploit current conservation kµΓjµ=0
⇒ωΓj0=kjΓE
-iΓjµ(k) k0=ω
i
jq
k
+
Γj0(k) ∼�
d3ru1(r)
rY1j(r̂)eik·r u0(r)
r
Irreducible S-to-P vertex: bound-to-bound transition
Divergences cancel, as they should
Γji = δjiΓE + kjqiΓM for k·q=0; k·ε=0
Exploit current conservation kµΓjµ=0
⇒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
Irreducible S-to-P vertex: bound-to-bound transition
Divergences cancel, as they should
Γji = δjiΓE + kjqiΓM for k·q=0; k·ε=0
Exploit current conservation kµΓjµ=0
⇒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):
Converting to result for S-to-P transition
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
Keeping track of constants, defn of B(E1):
Converting to result for S-to-P transition
• No cutoff parameter needed: integral finite without regularization
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
Keeping track of constants, defn of B(E1):
Converting to result for S-to-P transition
• No cutoff parameter needed: integral finite without regularization
• Numbers: BLO(E1)=0.106 e2fm2 , yields r1=-0.66 fm-1
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
Keeping track of constants, defn of B(E1):
Converting to result for S-to-P transition
• No cutoff parameter needed: integral finite without regularization
• Numbers: BLO(E1)=0.106 e2fm2 , yields r1=-0.66 fm-1
• NLO corrections from A0 and γ1/r1 corrections to A1
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
Keeping track of constants, defn of B(E1):
Converting to result for S-to-P transition
• No cutoff parameter needed: integral finite without regularization
• Numbers: BLO(E1)=0.106 e2fm2 , yields r1=-0.66 fm-1
• 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
�e−γ1rre−γ0r
�2
=Z2
effe2
4π
4γ0
−3r1
�2γ1 + γ0
(γ0 + γ1)2
�2
+ . . .
B(E1) =14π
�Γ̄E
ω
�2
Keeping track of constants, defn of B(E1):
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.
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:
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.
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:
!
LO NLO
k3 cot δ1=-1/2 r1 (k2 + γ12) ⇒ δ1 ∼ Rcore/Rhalo if k ∼ 1/Rhalo ∼ γ1.
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:
• Also get corrections to A0 (a.k.a. wf renormalization) at NLO
!
LO NLO
k3 cot δ1=-1/2 r1 (k2 + γ12) ⇒ δ1 ∼ Rcore/Rhalo if k ∼ 1/Rhalo ∼ γ1.
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
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:
Spin-1/2 channel Spin-3/2 channel
c.f. Rupak & Higa arXiv:1101.0207
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:
• Expand in Rcore/Rhalo:Spin-1/2 channel Spin-3/2 channel
c.f. Rupak & Higa arXiv:1101.0207
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
�
dB(E1)dE
LO
= e2Z2eff
3mR
2π2
8γ0p�3
(p�2 + γ20)4 No FSI
• Straightforward computation of diagrams yields:
• Expand in Rcore/Rhalo:Spin-1/2 channel Spin-3/2 channel
c.f. Rupak & Higa arXiv:1101.0207
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
�
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
�
• Straightforward computation of diagrams yields:
• Expand in Rcore/Rhalo:Spin-1/2 channel Spin-3/2 channel
c.f. Rupak & Higa arXiv:1101.0207
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
�
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
• Straightforward computation of diagrams yields:
• Expand in Rcore/Rhalo:Spin-1/2 channel Spin-3/2 channel
c.f. Rupak & Higa arXiv:1101.0207
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
�
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
• Straightforward computation of diagrams yields:
• Expand in Rcore/Rhalo:Spin-1/2 channel Spin-3/2 channel
c.f. Rupak & Higa arXiv:1101.0207
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
�
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
• Straightforward computation of diagrams yields:
• Higher-order corrections to phase shift at NNLO. Appearance of S-to-2P1/2 E1 counterterm also at that order.
• Expand in Rcore/Rhalo:Spin-1/2 channel Spin-3/2 channel
c.f. Rupak & Higa arXiv:1101.0207
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
• Value obtained for r0 implies (<rE2>+<rE, Be102>)1/2=2.40 fm at LO, 2.43 fm at NLO
• Experimental result <rBe112>1/2=2.463(16) fm Noerterhaueser et al., PRL (2009)
Output
• Value obtained for r0 implies (<rE2>+<rE, Be102>)1/2=2.40 fm at LO, 2.43 fm at NLO
• 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
• Value obtained for r0 implies (<rE2>+<rE, Be102>)1/2=2.40 fm at LO, 2.43 fm at NLO
• Experimental result <rBe112>1/2=2.463(16) fm
• r1=-0.66±0.29 fm-1 implies a1=374±150 fm3
• Information on P-wave interactions from dissociation, but only at NLO
Noerterhaueser et al., PRL (2009)
Output
• Value obtained for r0 implies (<rE2>+<rE, Be102>)1/2=2.40 fm at LO, 2.43 fm at NLO
• Experimental result <rBe112>1/2=2.463(16) fm
• r1=-0.66±0.29 fm-1 implies a1=374±150 fm3
• Information on P-wave interactions from dissociation, but only at NLO
• P-wave state’s radius also calculable
Noerterhaueser et al., PRL (2009)
Output
• Value obtained for r0 implies (<rE2>+<rE, Be102>)1/2=2.40 fm at LO, 2.43 fm at NLO
• Experimental result <rBe112>1/2=2.463(16) fm
• r1=-0.66±0.29 fm-1 implies a1=374±150 fm3
• Information on P-wave interactions from dissociation, but only at NLO
• P-wave state’s radius also calculable
• Universal correlation:
Noerterhaueser et al., PRL (2009)
B(E1) =2e2Q2
c
15π�r2
E�(π)x
�1 + 2x
(1 + x)2
�2
; x = γ1/γ0
Conclusions
• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo
Conclusions
• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo
• Carbon-19: one-neutron halo, shallow S-wave state
Conclusions
• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo
• Carbon-19: one-neutron halo, shallow S-wave state
• Beryllium-11: one-neutron halo, shallow S- and P-wave state
Conclusions
• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo
• Carbon-19: one-neutron halo, shallow S-wave state
• Beryllium-11: one-neutron halo, shallow S- and P-wave state
• 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.
c.f. Rupak & Higa arXiv:1101.0207
Conclusions
• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo
• Carbon-19: one-neutron halo, shallow S-wave state
• Beryllium-11: one-neutron halo, shallow S- and P-wave state
• 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.
• NLO in 11Be: fix A0 from dissociation to continuum, predict radii, extract a1
c.f. Rupak & Higa arXiv:1101.0207
Conclusions
• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo
• Carbon-19: one-neutron halo, shallow S-wave state
• Beryllium-11: one-neutron halo, shallow S- and P-wave state
• 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.
• NLO in 11Be: fix A0 from dissociation to continuum, predict radii, extract a1
• Correlations between low-energy observables
c.f. Rupak & Higa arXiv:1101.0207
Conclusions
• Halo EFT provides a systematic way to organize observables in halo nuclei in an expansion in Rcore/Rhalo
• Carbon-19: one-neutron halo, shallow S-wave state
• Beryllium-11: one-neutron halo, shallow S- and P-wave state
• 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.
• NLO in 11Be: fix A0 from dissociation to continuum, predict radii, extract a1
• Correlations between low-energy observables
• Other one- (and two-?) neutron (?and proton) halos await: “universality”.
c.f. Rupak & Higa arXiv:1101.0207