The neutron-neutron scatteringlength
Anders [email protected]
Relevant publicationsIn collaboration with Daniel Phillips (supported by NSF andDOE):
A.G. and D.R. PhillipsPhys. Rev. C 73, 014002 (2006)arXiv.org/abs/nucl-th/0501049
A.G. and D.R. PhillipsPhys. Rev. Lett. 96, 232301 (2006)arXiv.org/abs/nucl-th/0603045
A.G.Phys. Rev. C 74, 017001 (2006)arXiv.org/abs/nucl-th/0604035
Review on ann:A.G., J. Phys. G: Nucl. Part. Phys. 35, 053001 (2009).
INT, Seattle, WA, 3/25/20 – p.1/27
Effective range expansionAt low energies NN the s-wave phase shift can be written as
p cot δ = −1
a+
1
2r0p
2
where r0 is the effective range parameter
a and r0 are independent of potential shape!precursor to (pionless) effective field theory
Current values:NN a (fm) r0 (fm)nn −18.9 ± 0.4 2.75 ± 0.11
np −23.740 ± 0.020 2.77 ± 0.05
pp −17.3 ± 0.4 2.85 ± 0.04
Similar, but different! Why?
OINT, Seattle, WA, 3/25/20 – p.2/27
Effective range expansionAt low energies NN the s-wave phase shift can be written as
p cot δ = −1
a+
1
2r0p
2
where r0 is the effective range parametera and r0 are independent of potential shape!precursor to (pionless) effective field theory
Current values:NN a (fm) r0 (fm)nn −18.9 ± 0.4 2.75 ± 0.11
np −23.740 ± 0.020 2.77 ± 0.05
pp −17.3 ± 0.4 2.85 ± 0.04
Similar, but different! Why?
OINT, Seattle, WA, 3/25/20 – p.2/27
Effective range expansionAt low energies NN the s-wave phase shift can be written as
p cot δ = −1
a+
1
2r0p
2
where r0 is the effective range parametera and r0 are independent of potential shape!precursor to (pionless) effective field theory
Current values:NN a (fm) r0 (fm)nn −18.9 ± 0.4 2.75 ± 0.11
np −23.740 ± 0.020 2.77 ± 0.05
pp −17.3 ± 0.4 2.85 ± 0.04
Similar, but different! Why?
OINT, Seattle, WA, 3/25/20 – p.2/27
Effective range expansionAt low energies NN the s-wave phase shift can be written as
p cot δ = −1
a+
1
2r0p
2
where r0 is the effective range parametera and r0 are independent of potential shape!precursor to (pionless) effective field theory
Current values:NN a (fm) r0 (fm)nn −18.9 ± 0.4 2.75 ± 0.11
np −23.740 ± 0.020 2.77 ± 0.05
pp −17.3 ± 0.4 2.85 ± 0.04
Similar, but different! Why?INT, Seattle, WA, 3/25/20 – p.2/27
Charge Symmetry BreakingQCD Lagrangian almost symmetric under u ↔ d exchange(Charge Symmetry, CS), PCS = exp(iπτ2/2)broken by mu 6= md (and EM effects)Charge Symmetry Breaking (CSB)
Experimental evidence:
n-p mass differenceρ0-ω mixing (e+e− → π+π−)mirror nuclei (e.g. 3He-3H) binding energy, N-S anomalynp → np: An(θn) 6= Ap(θp) analyzing powersAfb(np → dπ0) (TRIUMF) and dd → απ0 (IUCF, soon COSY)
CSB reviews:[Miller, Nefkens, and Šlaus, PRt194, 1 (1990);Miller and van Oers, nucl-th/9409013;
Miller, Opper, and Stephenson, ARNPS56, 293 (2006), nucl-ex/0602021]
OINT, Seattle, WA, 3/25/20 – p.3/27
Charge Symmetry BreakingQCD Lagrangian almost symmetric under u ↔ d exchange(Charge Symmetry, CS), PCS = exp(iπτ2/2)broken by mu 6= md (and EM effects)Charge Symmetry Breaking (CSB)
Experimental evidence:
n-p mass differenceρ0-ω mixing (e+e− → π+π−)mirror nuclei (e.g. 3He-3H) binding energy, N-S anomalynp → np: An(θn) 6= Ap(θp) analyzing powers
Afb(np → dπ0) (TRIUMF) and dd → απ0 (IUCF, soon COSY)
CSB reviews:[Miller, Nefkens, and Šlaus, PRt194, 1 (1990);Miller and van Oers, nucl-th/9409013;
Miller, Opper, and Stephenson, ARNPS56, 293 (2006), nucl-ex/0602021]
OINT, Seattle, WA, 3/25/20 – p.3/27
Charge Symmetry BreakingQCD Lagrangian almost symmetric under u ↔ d exchange(Charge Symmetry, CS), PCS = exp(iπτ2/2)broken by mu 6= md (and EM effects)Charge Symmetry Breaking (CSB)
Experimental evidence:
n-p mass differenceρ0-ω mixing (e+e− → π+π−)mirror nuclei (e.g. 3He-3H) binding energy, N-S anomalynp → np: An(θn) 6= Ap(θp) analyzing powersAfb(np → dπ0) (TRIUMF) and dd → απ0 (IUCF, soon COSY)
CSB reviews:[Miller, Nefkens, and Šlaus, PRt194, 1 (1990);Miller and van Oers, nucl-th/9409013;
Miller, Opper, and Stephenson, ARNPS56, 293 (2006), nucl-ex/0602021]
OINT, Seattle, WA, 3/25/20 – p.3/27
Charge Symmetry BreakingQCD Lagrangian almost symmetric under u ↔ d exchange(Charge Symmetry, CS), PCS = exp(iπτ2/2)broken by mu 6= md (and EM effects)Charge Symmetry Breaking (CSB)
Experimental evidence:
n-p mass differenceρ0-ω mixing (e+e− → π+π−)mirror nuclei (e.g. 3He-3H) binding energy, N-S anomalynp → np: An(θn) 6= Ap(θp) analyzing powersAfb(np → dπ0) (TRIUMF) and dd → απ0 (IUCF, soon COSY)
CSB reviews:[Miller, Nefkens, and Šlaus, PRt194, 1 (1990);Miller and van Oers, nucl-th/9409013;
Miller, Opper, and Stephenson, ARNPS56, 293 (2006), nucl-ex/0602021]INT, Seattle, WA, 3/25/20 – p.3/27
CSB and scattering lengths
CSB ⇒ astrnn 6= astr
pp
astrnn 6= astr
pp ↔ enhancement factor: ∆aCSB
a = (10 − 15)∆VCSB
V
Calcs of B(3H) − B(3He) rely on |astrpp | < |astr
nn|, fails if|astr
pp | > |astrnn|!
astrnn = −18.9 ± 0.4 fm, astr
pp = −17.3 ± 0.4 fm
Difficulties: EM corrections (astrNN ), no free n target (astr
nn)
WHAT TO DO?
OINT, Seattle, WA, 3/25/20 – p.4/27
CSB and scattering lengths
CSB ⇒ astrnn 6= astr
pp
astrnn 6= astr
pp ↔ enhancement factor: ∆aCSB
a = (10 − 15)∆VCSB
V
Calcs of B(3H) − B(3He) rely on |astrpp | < |astr
nn|, fails if|astr
pp | > |astrnn|!
astrnn = −18.9 ± 0.4 fm, astr
pp = −17.3 ± 0.4 fm
Difficulties: EM corrections (astrNN ), no free n target (astr
nn)
WHAT TO DO?
OINT, Seattle, WA, 3/25/20 – p.4/27
CSB and scattering lengths
CSB ⇒ astrnn 6= astr
pp
astrnn 6= astr
pp ↔ enhancement factor: ∆aCSB
a = (10 − 15)∆VCSB
V
Calcs of B(3H) − B(3He) rely on |astrpp | < |astr
nn|, fails if|astr
pp | > |astrnn|!
astrnn = −18.9 ± 0.4 fm, astr
pp = −17.3 ± 0.4 fm
Difficulties: EM corrections (astrNN ), no free n target (astr
nn)
WHAT TO DO?
OINT, Seattle, WA, 3/25/20 – p.4/27
CSB and scattering lengths
CSB ⇒ astrnn 6= astr
pp
astrnn 6= astr
pp ↔ enhancement factor: ∆aCSB
a = (10 − 15)∆VCSB
V
Calcs of B(3H) − B(3He) rely on |astrpp | < |astr
nn|, fails if|astr
pp | > |astrnn|!
astrnn = −18.9 ± 0.4 fm, astr
pp = −17.3 ± 0.4 fm
Difficulties: EM corrections (astrNN ), no free n target (astr
nn)
WHAT TO DO?
INT, Seattle, WA, 3/25/20 – p.4/27
Solutions(?)Direct measurements:Wild idea #1: Simultaneous underground nuclear explosions
Wild idea #2: Launch a pulsed reactor into orbitRecent idea: Pulsed reactor YAGUAR in Snezhinsk, Russia
Indirect nn experiments:Implemented idea: Reactions giving nn with small rel. energy
nd → nnp: 3-body forces needed, expts differ:ann = −16.1 ± 0.4 fm (n, np) [Huhn et al., PRL85, 1190 (2000)] andann = −16.5 ± 0.9 fm (n, p) [von Witsch et al., PRC74, 014001 (2006)] vsann = −18.7 ± 0.7 fm (n, nnp) [González Trotter et al., PRC73, 034001 (’06)]
π−d → nnγ: −18.59 ± 0.40 fm (π−, nγ) ⇒ standard value(PSI and LAMPF) [Machleidt and Slaus, JPG:NPP27, R69 (2001)]
Need accurate theoretical input for extraction!
OINT, Seattle, WA, 3/25/20 – p.5/27
Solutions(?)Direct measurements:Wild idea #1: Simultaneous underground nuclear explosionsWild idea #2: Launch a pulsed reactor into orbit
Recent idea: Pulsed reactor YAGUAR in Snezhinsk, Russia
Indirect nn experiments:Implemented idea: Reactions giving nn with small rel. energy
nd → nnp: 3-body forces needed, expts differ:ann = −16.1 ± 0.4 fm (n, np) [Huhn et al., PRL85, 1190 (2000)] andann = −16.5 ± 0.9 fm (n, p) [von Witsch et al., PRC74, 014001 (2006)] vsann = −18.7 ± 0.7 fm (n, nnp) [González Trotter et al., PRC73, 034001 (’06)]
π−d → nnγ: −18.59 ± 0.40 fm (π−, nγ) ⇒ standard value(PSI and LAMPF) [Machleidt and Slaus, JPG:NPP27, R69 (2001)]
Need accurate theoretical input for extraction!
OINT, Seattle, WA, 3/25/20 – p.5/27
Solutions(?)Direct measurements:Wild idea #1: Simultaneous underground nuclear explosionsWild idea #2: Launch a pulsed reactor into orbitRecent idea: Pulsed reactor YAGUAR in Snezhinsk, Russia
2630 W I Furman et al
Figure 1. Possible arrangement for the nn-scattering experiment at YAGUAR. See the text for
explanations.
duration another salt, CdSO4, is added. The cadmium concentration is about 5 g per litre. The
effective diameter of the cylindrical active core is 40 cm with an effective inside diameter of
17 cm. The critical height is about 39 cm, depending on the position of the startup rods. The
body of the reactor has a central channel of 15 cm diameter, which in the standard operating
mode has startup rods that leave a cylindrical space of 12 cm diameter for the experimental
channel.
A direct measurement of the nn-scattering length is to be performed by the time-of-flight
method by counting the scattered neutrons arriving after the reactor burst to a detector placed
behind special collimators far away from the reactor. If only the colliding neutrons contribute
to the detector counting rate, and if the parameters of the neutron field are determined and
the geometry is known, then the detector counts, integrated over the thermal neutron part of
the time-of-flight spectrum, measure (see equation (15)) the nn-scattering cross section. One
possible arrangement for the nn-scattering experiment at YAGUAR is shown schematically in
figure 1. The active core 1 is placed on three supports at 2 m above the floor level. The neutron
CH2 moderator 2 is inserted inside the channel. An evacuated tube 3 contains a collimation
system 4, and the neutron detector 5 is placed at the flight path of about 12 m long. The
collimation system is designed to screen the source and to eliminate the background due to
neutrons scattered from the walls. With neutron fluxes of 1018 cm−2 s−1 and a vacuum of
10−6 Torr or better, the scattering on the residual gas is negligible relative to the nn scattering.
The 3He absorber 6 reduces the neutron scattering from the back wall, while the time-of-flight
measurement partially separates (due to the difference in flight paths) the remaining backwall
background from the nn signal. The shields 7 serve to screen the detector from epithermal
and fast neutrons.
Direct measurement of the nn scattering 2631
1
2
3
4
5
Figure 2. Simplified geometry of the reactor active core with the neutron moderator inserted intothe channel: 1—polyethylene moderator, 2—reactor active core, 3—Cu and Au foils covered by
Cd, 4—Cu and Au foils without Cd, 5—Cu wire.
4. Monte Carlo modelling
The key issues for the current nn project at the reactor YAGUAR are the conversion of the fast
neutrons to thermal energy and the characteristics of the neutron field—the angular distribution
of the neutrons and the neutron source intensity distribution along the surface of the nn-cavity.
To address these issues we performed Monte Carlo modelling with the use of the code
MCNP-4b [22]. The problem was set up in the simplified geometry shown in figure 2, but
without the activation foils shown in the figure. These foils are employed in subsequent
measurements described in section 6. The cells, materials, masses and appropriate nuclear
data were specified in the proper input sections to account for the reactor parameters reported
in section 3. To set up a realistic neutron source input we run the MCNP code in the
criticality option KCODE, which uses successive fission cycles to model neutron sources.
This option allowed us to model the fast neutron source distribution over the YAGUAR active
core as it exists during the fission burst. The MCNP code follows the transport of each
fast neutron recording all needed tallies, and in particular PTRAC tables which contain all
relevant information, such as the neutron energy, time, space and track angles on the surfaces
of interest.
According to Amaldi and Fermi [23], the angular distribution of neutrons emerging per
unit solid angle from an infinite slab of a moderator is proportional to n(A) cos δ(1 +A cos δ),
withA =√
3. Here δ is the angle from the normal, n(A) = 2/(1 + 2A/3) is the normalization
factor that sets the integral over δ from 0 to 90 equal to unity. Since the neutron flux is
always defined as the above quantity divided by cos δ, a nonzero A indicates the presence
of anisotropy in the neutron flux. The anisotropy originates from the existence of a neutron
density gradient near the surface of a diffusive medium. This distribution was also shown to
fit the experimental data for finite water slabs (see, e.g., [24]). However, for the ‘re-entrant’
internal surface of the YAGUAR cylindrical moderator our Monte Carlo modelling indicated
the value of A = 0. This implies an isotropy of the surface flux angular distribution, which
seems reasonable because of the thermal neutrons multiple crossing of the internal surface
without leaving the moderator.
The second output of our Monte Carlo modelling was the z distribution of the thermal
neutron source density S along the channel surface. These data were approximated analytically
as S(z1) = S cos(πz1/L′) with L′ = 48.5 cm. In such a description, the number of neutrons
q emitted inside the cavity per second from 1 cm2 of the wall at the cavity centre plane is
q = 2πSv, where v is the average velocity of the Maxwellian neutron density spectrum.
The third output of the Monte Carlo modelling was the result for fast neutron conversion
to thermal energies by hollow cylindrical moderators of different thicknesses. According to
[Furman et al., JPG 28, 2627 (2002);Muzichka et al., NPA 789, 30 (2007)]
Indirect nn experiments:Implemented idea: Reactions giving nn with small rel. energy
nd → nnp: 3-body forces needed, expts differ:ann = −16.1 ± 0.4 fm (n, np) [Huhn et al., PRL85, 1190 (2000)] andann = −16.5 ± 0.9 fm (n, p) [von Witsch et al., PRC74, 014001 (2006)] vsann = −18.7 ± 0.7 fm (n, nnp) [González Trotter et al., PRC73, 034001 (’06)]
π−d → nnγ: −18.59 ± 0.40 fm (π−, nγ) ⇒ standard value(PSI and LAMPF) [Machleidt and Slaus, JPG:NPP27, R69 (2001)]
Need accurate theoretical input for extraction!
OINT, Seattle, WA, 3/25/20 – p.5/27
Solutions(?)Direct measurements:Wild idea #1: Simultaneous underground nuclear explosionsWild idea #2: Launch a pulsed reactor into orbitRecent idea: Pulsed reactor YAGUAR in Snezhinsk, Russia
Indirect nn experiments:Implemented idea: Reactions giving nn with small rel. energy
nd → nnp: 3-body forces needed, expts differ:ann = −16.1 ± 0.4 fm (n, np) [Huhn et al., PRL85, 1190 (2000)] andann = −16.5 ± 0.9 fm (n, p) [von Witsch et al., PRC74, 014001 (2006)] vsann = −18.7 ± 0.7 fm (n, nnp) [González Trotter et al., PRC73, 034001 (’06)]
π−d → nnγ: −18.59 ± 0.40 fm (π−, nγ) ⇒ standard value(PSI and LAMPF) [Machleidt and Slaus, JPG:NPP27, R69 (2001)]
Need accurate theoretical input for extraction!
OINT, Seattle, WA, 3/25/20 – p.5/27
Solutions(?)Direct measurements:Wild idea #1: Simultaneous underground nuclear explosionsWild idea #2: Launch a pulsed reactor into orbitRecent idea: Pulsed reactor YAGUAR in Snezhinsk, Russia
Indirect nn experiments:Implemented idea: Reactions giving nn with small rel. energy
nd → nnp: 3-body forces needed, expts differ:ann = −16.1 ± 0.4 fm (n, np) [Huhn et al., PRL85, 1190 (2000)] andann = −16.5 ± 0.9 fm (n, p) [von Witsch et al., PRC74, 014001 (2006)] vsann = −18.7 ± 0.7 fm (n, nnp) [González Trotter et al., PRC73, 034001 (’06)]
π−d → nnγ: −18.59 ± 0.40 fm (π−, nγ) ⇒ standard value(PSI and LAMPF) [Machleidt and Slaus, JPG:NPP27, R69 (2001)]
Need accurate theoretical input for extraction!INT, Seattle, WA, 3/25/20 – p.5/27
Bonn nd → nnp set-up
INT, Seattle, WA, 3/25/20 – p.6/27
TUNL nd → nnp set-up
INT, Seattle, WA, 3/25/20 – p.7/27
LAMPF set-up.
Stopped pionscaptured on din atomic s-wave orbitals
( )C.R. Howell et al.rPhysics Letters B 444 1998 252–259254
Fig. 1. Schematic of the mid-level cut-away view of the experimental layout.
plastic scintillator paddles were placed in front of theneutron and g-ray detectors to reject events due tocharged particles.
Data were accumulated for both double-coinci-Ž . Ž .dences DC the g ray and one neutron and for
Ž . Žtriple coincidences TC the g ray and both neu-.trons . For both types of data the momenta of all
detected particles were measured, thereby fully de-
termining the reaction kinematics in the case of theDC events and overdetermining the kinematics in theTC measurements. A total of 1.9=106 DC eventsand 5.7=104 TC events were stored on tape. Thevalue of a was determined from the shape of thenn
Ž .measured neutron time-of-flight NTOF spectrum ofthe DC data. The NTOF was the measured timedifference between the detection of the g ray in the
Lab = cm
γ and ndetected incoincidence
INT, Seattle, WA, 3/25/20 – p.8/27
π−d → nnγ data (LAMPF).
θ3
p2
p1k
γ and n1 detected at 0.05 < θ3 < 0.1 (rad) [Howell et al., PLB444, 252 (1998)]( )C.R. Howell et al.rPhysics Letters B 444 1998 252–259256
Ž .Fig. 2. a The top plot is a comparison of the shape of the experimental NTOF spectrum with that of the Monte Carlo simulated TOFŽ . 2spectrum for a u cut between 0.05 and 0.1 radian. The spectra are for the entire neutron-detector array. b The total x versus a .3 n n
stopped pions in the target, e.g., neutrons producednear the edge of the target facing the neutron detec-tors were less attenuated than those on the oppositeside of the target. The NTOF spectra resulting fromthese two extreme locations have different shapes.Therefore, a very precise description of the stopped-pion distribution inside the target and an accurateevaluation of in-scattering were needed for a correct
Ž y .determination of a . We used the H p ,ng two-nnbody reaction to make tomographic measurements ofthe stopped-pion distribution inside the target. Theempirically determined distributions were used in thesimulations.
In addition to the expected QF and FSI peaks, it isimportant to identify any non-statistical structures inthe spectra. No such structures were observed in our
w xNTOF spectra, in contrast to the PSI experiment 9 .Several parameters were regularly checked duringthe data acquisition, e.g., the gain of each neutrondetector and the location of the prompt g-ray peak inthe TOF spectrum. The system was extraordinarilystable and robust.
The deuterium-target data contained some eventsy Ž y .due to p capture on hydrogen. Since the H p ,ng
reaction is a two-body process, the presence of hy-drogen in the deuterium target influences the NTOF
QF FSI
Unnormalized, but shape fitted to give ann! INT, Seattle, WA, 3/25/20 – p.9/27
Old theory for π−d → nnγ
Gibbs, Gibson, and Stephenson (GGS) [PRC11, 90 (1975)]:
π−p → γn, rel corr up to O(p/M)
estimated pion rescattering
tried different wave functions
theoretical error (mainly SD): ∆ann = ±0.3 fm
Only accurate under the FSI peak!
de Téramond et al., [PRC16, 1976 (1977);36, 691 (1987)]
similar error
Can chiral perturbation theory (χPT) do better?
OINT, Seattle, WA, 3/25/20 – p.10/27
Old theory for π−d → nnγ
Gibbs, Gibson, and Stephenson (GGS) [PRC11, 90 (1975)]:
π−p → γn, rel corr up to O(p/M)
estimated pion rescattering
tried different wave functions
theoretical error (mainly SD): ∆ann = ±0.3 fm
Only accurate under the FSI peak!
de Téramond et al., [PRC16, 1976 (1977);36, 691 (1987)]
similar error
Can chiral perturbation theory (χPT) do better?
INT, Seattle, WA, 3/25/20 – p.10/27
EFT credoAdvantages of an effective field theory like χPT:
Consistent amplitudes and wave functions
Recipe to estimate theoretical error
Systematic improvement possible
χPT = low-energy limit of QCD, retains chiral symmetry ofQCD
At low E: expansion in αS ∼ 1 not possible. Instead:Power counting gives hierarchy of amplitudes. Here:
Q ∼ mπ small momentum/energy of problem
Λχ ∼ M ∼ 4πfπ ∼ 1 GeV energy scale where χPT breaksdown
Expand in Q/Λχ
OINT, Seattle, WA, 3/25/20 – p.11/27
EFT credoAdvantages of an effective field theory like χPT:
Consistent amplitudes and wave functions
Recipe to estimate theoretical error
Systematic improvement possible
χPT = low-energy limit of QCD, retains chiral symmetry ofQCD
At low E: expansion in αS ∼ 1 not possible. Instead:Power counting gives hierarchy of amplitudes. Here:
Q ∼ mπ small momentum/energy of problem
Λχ ∼ M ∼ 4πfπ ∼ 1 GeV energy scale where χPT breaksdown
Expand in Q/ΛχINT, Seattle, WA, 3/25/20 – p.11/27
χPT for π−d → nnγ.
For π−d → nnγ we get
O(Q3) = GGS + π loops + 2-body
O(Q3) πN → γN fitted to data ⇒ no free parameters
For capture on d: qπ = 0, only one CGLN amplitude (F1)survives
2B O(Q4): π exchanges and contact term1B O(Q4): under investigation
⇒ High precision possible
OINT, Seattle, WA, 3/25/20 – p.12/27
χPT for π−d → nnγ.
For π−d → nnγ we get
O(Q3) = GGS + π loops + 2-body
O(Q3) πN → γN fitted to data ⇒ no free parameters
For capture on d: qπ = 0, only one CGLN amplitude (F1)survives
2B O(Q4): π exchanges and contact term1B O(Q4): under investigation
⇒ High precision possible
OINT, Seattle, WA, 3/25/20 – p.12/27
χPT for π−d → nnγ.
For π−d → nnγ we get
O(Q3) = GGS + π loops + 2-body
O(Q3) πN → γN fitted to data ⇒ no free parameters
For capture on d: qπ = 0, only one CGLN amplitude (F1)survives
2B O(Q4): π exchanges and contact term1B O(Q4): under investigation
⇒ High precision possible
INT, Seattle, WA, 3/25/20 – p.12/27
Generic diagrams
A1 A1
Quasifree (QF) Final State Interaction (FSI)
A2 A2
Two-body effects (2)
Γ ∝ |MQF + MFSI + M2|2 INT, Seattle, WA, 3/25/20 – p.13/27
Chirally inspired wave functions.Start from asymptotic wave functionsSE integrated in from r = ∞ with chiral OPEP and TPEP[Phillips & Cohen, NPA668, 45 (2000)]:
Coupled integral equations for d (3S1–3D1)
Uncoupled integral equations for nn (1S0, 3PJ , 1D2, no 3F2)
Match with spherical well solution at r = R = 1.4 to 3.0 fm(Regulates unknown short-distance physics)
Calc indep of R?
OINT, Seattle, WA, 3/25/20 – p.14/27
Chirally inspired wave functions.Start from asymptotic wave functionsSE integrated in from r = ∞ with chiral OPEP and TPEP[Phillips & Cohen, NPA668, 45 (2000)]:
Coupled integral equations for d (3S1–3D1)
Uncoupled integral equations for nn (1S0, 3PJ , 1D2, no 3F2)
Match with spherical well solution at r = R = 1.4 to 3.0 fm(Regulates unknown short-distance physics)
Calc indep of R?
INT, Seattle, WA, 3/25/20 – p.14/27
One-body amplitudes.
EFT to O(Q3)[Fearing et al., PRC62, 054006 (2000)]:
+pion loops at O(Q3) ⇒ A1
all parameters fitted to data
m11
(0) ~mp!5
mN
4p~mN1mp!
eGA
2F F ~mp1mn!
6mpmN2
1
12mN2 1
~mp1mn!
6mN2 1
2b10
3GA~4pF !2G , ~24!
m12
(0) ~mp!5
mN
4p~mN1mp!
eGA
2F F2
~mp1mn!
3mpmN1
7
24mN2 2
~mp1mn!
3mN2 1
2b10
3GA~4pF !2G , ~25!
e11
(0) ~mp!5
mN
4p~mN1mp!
eGA
2F1
24mN2 , ~26!
and for the (2) isospin channel
FIG. 1. The cross section for the pion capture reaction, quoted as the reduced center-of-mass cross section for the inverse gn→p2p or@for ~f!# gp→p1n reaction. Experimental data are compared to the HBChPT predictions at O(p) ~dotted line!, O(p2) ~dashed line!, andO(p3) ~solid line!. The O(p3) result corresponds to A~35! and B~35! which are indistinguishable in these plots. ~a! Tp59.88 MeV, ~b!
Tp514.62 MeV, ~c! Tp519.85 MeV, ~d! Tp527.4 MeV, ~e! Tp539.3 MeV, ~f! Tg5153 MeV, Tp53.06 MeV.
RADIATIVE PION CAPTURE BY A NUCLEON PHYSICAL REVIEW C 62 054006
054006-5
INT, Seattle, WA, 3/25/20 – p.15/27
Two-body amplitudes O(Q3)
⇒ A2
First diagram has a Coulomb-like propagator, 1/~q 2
Second diagram has 1/~q 2 and also an off-shell pion propThird diagram (2 off-shell props) vanishes in Coulomb gauge
INT, Seattle, WA, 3/25/20 – p.16/27
R-dep error at O(Q3)
at O(Q4)
200 400 600 800TOF (channels)
0
0.2
0.4
0.6
0.8
Dec
ay r
ate
(arb
r. u
nits
)
R = 1.4 fm, N2LO, OPER = 2.2 fmR = 3.0 fm
QF FSI
∆ann(theory) = ±0.2 fm (FSI only)∆ann(theory) = ±1 fm (full spectrum)
200 400 600 800TOF (channels)
0
0.2
0.4
0.6
0.8
Dec
ay r
ate
(arb
r. u
nits
)
R = 1.4 fm, N3LO, OTPER = 2.2 fmR = 3.0 fm
QF FSI
∆ann(theory) = ±0.05 fm (FSI only)
∆ann(theory) = ±0.3 fm (full spectrum)
INT, Seattle, WA, 3/25/20 – p.17/27
Chiral relationsChiral Lagrangian:
L = N †(iv · D + gAS · u)N
− 2d1N†S · uNN †N + 2d2ε
abcεκλµνvκuλ,aN †Sµτ bNN †Sντ cN . . .
where fπuµ = −τa∂µπa−ε3baVµπbτa+fπAµ + O(π3)
1N (gA): Goldberger-Treiman and Kroll-Ruderman
gA
fπ=
gπNN
M|AKR| =
egA
fπ
relate axial coupling to πN coupling and γπN coupling
2N (di): Axial isovector coupling to NN (3S1 ↔ 1S0)
Connects π (photo)prod to EW reactions and chiral 3NF!
OINT, Seattle, WA, 3/25/20 – p.18/27
Chiral relationsChiral Lagrangian:
L = N †(iv · D + gAS · u)N
− 2d1N†S · uNN †N + 2d2ε
abcεκλµνvκuλ,aN †Sµτ bNN †Sντ cN . . .
where fπuµ = −τa∂µπa−ε3baVµπbτa+fπAµ + O(π3)
2N (di): Axial isovector coupling to NN (3S1 ↔ 1S0)
Connects π (photo)prod to EW reactions
and chiral 3NF!
OINT, Seattle, WA, 3/25/20 – p.18/27
Chiral relationsChiral Lagrangian:
L = N †(iv · D + gAS · u)N
− 2d1N†S · uNN †N + 2d2ε
abcεκλµνvκuλ,aN †Sµτ bNN †Sντ cN . . .
where fπuµ = −τa∂µπa−ε3baVµπbτa+fπAµ + O(π3)
2N (di): Axial isovector coupling to NN (3S1 ↔ 1S0)
Connects π (photo)prod to EW reactions and chiral 3NF!
INT, Seattle, WA, 3/25/20 – p.18/27
O(Q4) axial isovector contact termFor 3S1 ↔ 1S0 one single LEC:
d ≡ d1 + 2d2 +c3
3+
2c4
3+
1
6
Relates SD physics of
pp fusion, 3H → 3He e−νe (not EFT): [Schiavilla et al., PRC58, 1263 (1998)]
p-wave π prod+3NF: [Hanhart, van Kolck, Miller, PRL85, 2905 (2000)]
µ−d → nnνµ: [Ando et al., PLB533, 25 (2002)]
ν(ν)d breakup: [Ando et al., PLB555, 49 (2003)]
pp fusion, hep, 3H → 3He e−νe: [Park et al., PRC67, 055206 (2003)]
pp fusion, π−d → nnγ, γd → nnπ+: [AG+DRP, PRL96, 232301 (2006);
AG, PRC74, 017001 (2006)]
pp fusion, ν(ν)d, µ−d → nnνµ: [Butler et al., PLB520, 97 (2001);
EFT(6π): d ↔ L1,A Chen et al., PRC72, 061001(R) (2005)]
OINT, Seattle, WA, 3/25/20 – p.19/27
O(Q4) axial isovector contact termFor 3S1 ↔ 1S0 one single LEC:
d ≡ d1 + 2d2 +c3
3+
2c4
3+
1
6
Relates SD physics of
pp fusion, 3H → 3He e−νe (not EFT): [Schiavilla et al., PRC58, 1263 (1998)]
p-wave π prod+3NF: [Hanhart, van Kolck, Miller, PRL85, 2905 (2000)]
µ−d → nnνµ: [Ando et al., PLB533, 25 (2002)]
ν(ν)d breakup: [Ando et al., PLB555, 49 (2003)]
pp fusion, hep, 3H → 3He e−νe: [Park et al., PRC67, 055206 (2003)]
pp fusion, π−d → nnγ, γd → nnπ+: [AG+DRP, PRL96, 232301 (2006);
AG, PRC74, 017001 (2006)]
pp fusion, ν(ν)d, µ−d → nnνµ: [Butler et al., PLB520, 97 (2001);
EFT( 6π): d ↔ L1,A Chen et al., PRC72, 061001(R) (2005)]INT, Seattle, WA, 3/25/20 – p.19/27
Two-body amplitudes O(Q4)
(b)
(f) (g)
(j) (k)
(l) (m)
(a)
(e)
(i)
(c) (d)
(h)
INT, Seattle, WA, 3/25/20 – p.20/27
Constraining contact term
For axial isovector 3S1 ↔ 1S0 (Gamow-Teller) transitionscommon in NN systems only one LEC combination:
d ≡ d1 + 2d2 +c3
3+
2c4
3+
1
6
d ↔ “gA” for two-nucleon systems, but pot/wf/reg dep!
Possible constrain candidate: pp → de+νe (solar fusion)
Calculated by [Park et al., PRC67, 055206 (2003)]
constrained by tritium beta decay
Let’s do a numerical experiment!
Remember:Tjon line: B(4He) vs B(3H)
Phillips line: 2and vs B(3H)
OINT, Seattle, WA, 3/25/20 – p.21/27
Constraining contact term
For axial isovector 3S1 ↔ 1S0 (Gamow-Teller) transitionscommon in NN systems only one LEC combination:
d ≡ d1 + 2d2 +c3
3+
2c4
3+
1
6
d ↔ “gA” for two-nucleon systems, but pot/wf/reg dep!
Possible constrain candidate: pp → de+νe (solar fusion)
Calculated by [Park et al., PRC67, 055206 (2003)]
constrained by tritium beta decay
Let’s do a numerical experiment!
Remember:Tjon line: B(4He) vs B(3H)
Phillips line: 2and vs B(3H)
OINT, Seattle, WA, 3/25/20 – p.21/27
Constraining contact term
For axial isovector 3S1 ↔ 1S0 (Gamow-Teller) transitionscommon in NN systems only one LEC combination:
d ≡ d1 + 2d2 +c3
3+
2c4
3+
1
6
d ↔ “gA” for two-nucleon systems, but pot/wf/reg dep!
Possible constrain candidate: pp → de+νe (solar fusion)
Calculated by [Park et al., PRC67, 055206 (2003)]
constrained by tritium beta decay
Let’s do a numerical experiment!
Remember:Tjon line: B(4He) vs B(3H)
Phillips line: 2and vs B(3H)INT, Seattle, WA, 3/25/20 – p.21/27
Tjon line [Nogga, Kamada, Glöckle, PRL85, 944 (2000)]
7.5 7.75 8.0 8.25 8.5
Bt
25.0
27.5
B
E x p
C D BO N N
N ijm 9 3
N ijm II
N ijm I
A V 1 8
INT, Seattle, WA, 3/25/20 – p.22/27
Phillips line [Witała et al., PRC68, 034002 (2003)]
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.42and [fm]
7.4
7.6
7.8
8.0
8.2
8.4
8.6
8.8
-E3 H
[MeV
]
0.60 0.68
8.44
8.50
8.56
FIG. 4. The results for 2and and E3H from Table I: np-nnforces alone ~pluses!, np-pp forces alone ~squares!, and np-nnand np-pp forces plus electromagnetic interactions ~starsand circles, respectively!. The four straight lines ~Phillips lines!are x2 fits (np-nn , solid; np-pp , dashed-dotted; np-nn withEMI’s, dashed; np-pp with EMI’s, dotted!. The lines with EMI’smiss the experimental error bar for 2and @33#. The physicallyinteresting domain around the experimental values is shown in theinset.
INT, Seattle, WA, 3/25/20 – p.23/27
Gamow-Teller vs FSI
1.4 fm1.6 fm1.8 fm
2 fm2.2 fm
2.4 fm2.6 fm
2.8 fm3 fm
1.4 fm
1.6 fm
1.8 fm
2 fm
2.2 fm
2.4 fm
2.6 fm
2.8 fm
3 fm
0.3 0.32 0.34 0.36 0.38 0.4 0.42ΓFSI (arbr. units)
4.8
4.9
5
5.1
5.2
MG
T (fm
)
OPE, N2LOTPE, N3LO, no CT
MGT of pp fusion vs FSI peak heightOINT, Seattle, WA, 3/25/20 – p.24/27
Gårdestig-Phillips line(s)
1.4 fm1.6 fm1.8 fm
2 fm2.2 fm
2.4 fm2.6 fm
2.8 fm3 fm
1.4 fm
1.6 fm
1.8 fm
2 fm
2.2 fm
2.4 fm
2.6 fm
2.8 fm
3 fm
0.3 0.32 0.34 0.36 0.38 0.4 0.42ΓFSI (arbr. units)
4.8
4.9
5
5.1
5.2
MG
T (fm
)
OPE, N2LOTPE, N3LO, no CTTPE, N3LO with CT
MGT of pp fusion vs FSI peak heightINT, Seattle, WA, 3/25/20 – p.24/27
R-dep error at O(Q3)
at O(Q4)
200 400 600 800TOF (channels)
0
0.2
0.4
0.6
0.8
Dec
ay r
ate
(arb
r. u
nits
)
R = 1.4 fm, N2LO, OPER = 2.2 fmR = 3.0 fm
QF FSI
∆ann(theory) = ±0.2 fm (FSI only)∆ann(theory) = ±1 fm (full spectrum)
200 400 600 800TOF (channels)
0
0.2
0.4
0.6
0.8
Dec
ay r
ate
(arb
r. u
nits
)
R = 1.4 fm, N3LO, OTPER = 2.2 fmR = 3.0 fm
QF FSI
∆ann(theory) = ±0.05 fm (FSI only)∆ann(theory) = ±0.3 fm (full spectrum)
OINT, Seattle, WA, 3/25/20 – p.25/27
R-dep error at O(Q4)
200 400 600 800TOF (channels)
0
0.2
0.4
0.6
0.8
Dec
ay r
ate
(arb
r. u
nits
)
R = 1.4 fm, N3LO, OTPER = 2.2 fmR = 3.0 fm
QF FSI
∆ann(theory) = ±0.05 fm (FSI only)∆ann(theory) = ±0.3 fm (full spectrum) INT, Seattle, WA, 3/25/20 – p.25/27
Summary and Conclusions IχS relates SD physics of 2B EW reactions to (γ)πNN !
Chiral 3NF constrained by EW 2B obs!
χPT reduces theory error for ann, ∆ann = ±0.05 fm,a factor >3 better than previous calcs![AG+DRP, PRL 96, 232301 (2006)]
What remains:Higher order EM corrections in wfs?N3LO 1BOrthonormalization
Better input possible from γd → nnπ+ or µ−d → nnνµ?
Complete the circle:µ−d → nnνµ (1%) at PSI; expt and calc under way!ν(ν)d breakup (SNO) with chiral wfs3H → 3Hee−νe with (r-space) chiral wfs?
OINT, Seattle, WA, 3/25/20 – p.26/27
Summary and Conclusions IχS relates SD physics of 2B EW reactions to (γ)πNN !
Chiral 3NF constrained by EW 2B obs!
χPT reduces theory error for ann, ∆ann = ±0.05 fm,a factor >3 better than previous calcs![AG+DRP, PRL 96, 232301 (2006)]
What remains:Higher order EM corrections in wfs?N3LO 1BOrthonormalization
Better input possible from γd → nnπ+ or µ−d → nnνµ?
Complete the circle:µ−d → nnνµ (1%) at PSI; expt and calc under way!ν(ν)d breakup (SNO) with chiral wfs3H → 3Hee−νe with (r-space) chiral wfs?
OINT, Seattle, WA, 3/25/20 – p.26/27
Summary and Conclusions IχS relates SD physics of 2B EW reactions to (γ)πNN !
Chiral 3NF constrained by EW 2B obs!
χPT reduces theory error for ann, ∆ann = ±0.05 fm,a factor >3 better than previous calcs![AG+DRP, PRL 96, 232301 (2006)]
What remains:Higher order EM corrections in wfs?N3LO 1BOrthonormalization
Better input possible from γd → nnπ+ or µ−d → nnνµ?
Complete the circle:µ−d → nnνµ (1%) at PSI; expt and calc under way!ν(ν)d breakup (SNO) with chiral wfs3H → 3Hee−νe with (r-space) chiral wfs?
OINT, Seattle, WA, 3/25/20 – p.26/27
Summary and Conclusions IχS relates SD physics of 2B EW reactions to (γ)πNN !
Chiral 3NF constrained by EW 2B obs!
χPT reduces theory error for ann, ∆ann = ±0.05 fm,a factor >3 better than previous calcs![AG+DRP, PRL 96, 232301 (2006)]
What remains:Higher order EM corrections in wfs?N3LO 1BOrthonormalization
Better input possible from γd → nnπ+ or µ−d → nnνµ?
Complete the circle:µ−d → nnνµ (1%) at PSI; expt and calc under way!ν(ν)d breakup (SNO) with chiral wfs3H → 3Hee−νe with (r-space) chiral wfs?
OINT, Seattle, WA, 3/25/20 – p.26/27
Summary and Conclusions IχS relates SD physics of 2B EW reactions to (γ)πNN !
Chiral 3NF constrained by EW 2B obs!
χPT reduces theory error for ann, ∆ann = ±0.05 fm,a factor >3 better than previous calcs![AG+DRP, PRL 96, 232301 (2006)]
What remains:Higher order EM corrections in wfs?N3LO 1BOrthonormalization
Better input possible from γd → nnπ+ or µ−d → nnνµ?
Complete the circle:µ−d → nnνµ (1%) at PSI; expt and calc under way!ν(ν)d breakup (SNO) with chiral wfs3H → 3Hee−νe with (r-space) chiral wfs?
INT, Seattle, WA, 3/25/20 – p.26/27
Summary and Conclusions II
π−d → nnγ under good experimental and theoreticalcontrol
Bonn-TUNL discrepancy needs to be resolved, workunder way
direct measurement of ann finally possible?
OINT, Seattle, WA, 3/25/20 – p.27/27
Summary and Conclusions II
π−d → nnγ under good experimental and theoreticalcontrol
Bonn-TUNL discrepancy needs to be resolved, workunder way
direct measurement of ann finally possible?
OINT, Seattle, WA, 3/25/20 – p.27/27
Summary and Conclusions II
π−d → nnγ under good experimental and theoreticalcontrol
Bonn-TUNL discrepancy needs to be resolved, workunder way
direct measurement of ann finally possible?
INT, Seattle, WA, 3/25/20 – p.27/27
Intro to Phase ShiftsFree wave (radial part of 3D): ul ∼ sin(pr − lπ
2 )
0 20 40 60 80 100
0
free wave
Interaction V ⇒ phase shift: δ > 0 attractive potentialδ < 0 repulsive potential
OINT, Seattle, WA, 3/25/20
Intro to Phase ShiftsScattered wave: ul ∼ sin(pr − lπ
2 + δl)
0 20 40 60 80 100
0
-V
scattered wavefree wave
Interaction V ⇒ phase shift: δ > 0 attractive potentialδ < 0 repulsive potential
OINT, Seattle, WA, 3/25/20
Intro to Phase ShiftsScattered wave: ul ∼ sin(pr − lπ
2 + δl)
0 20 40 60 80 100
0
-V
scattered wavefree wave
δ
Interaction V ⇒ phase shift: δ > 0 attractive potentialδ < 0 repulsive potential
INT, Seattle, WA, 3/25/20
Definition of Scattering LengthThe low energy cross section
dσ
dΩ=
1
p2sin2 δ0
p→0→ a2
defining the (S-wave) scattering length
a ≡ − limp→0
δ0
p
where the sign is conventional.
For impenetrable sphere: a > 0 and σ = 4πa2
INT, Seattle, WA, 3/25/20
Scattering length and wfsAt zero energy the asymptotic NN wave function behaves as1 − r/a
a) a < 0, attractive potential, scattering state
b) a > 0, repulsive potential
c) a > 0, attractive potential, bound state
INT, Seattle, WA, 3/25/20
CGLN amplitudesSpin decomposition
AI(γN → πN) = F1(Eπ, x)iσ · εγ + F2(Eπ, x)σ · qσ · (k× εγ)
+ F3(Eπ, x)iσ · k q · εγ + F4(Eπ, x)iσ · q q · εγ
Isospin
F ai (Eπ, x) = F
(−)i (Eπ, x)iεa3bτ b + F
(0)i (Eπ, x)τa + F
(+)i (Eπ, x)δa3
and for γn → π−p
Fi(γn → π−p) =√
2[F(0)i − F
(−)i ]
q = 0 ⇒ only F1, dominated by KR for charged pions
INT, Seattle, WA, 3/25/20
Deuteron wave functions (OPE).
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0r (fm)
0
0.1
0.2
0.3
0.4
w(r
) (fm
)
-1/2
-1/2
asymptoticR = 0 fmR = 1.4 fmR = 1.5 fmR = 2.0 fmR = 2.5 fmNijm93
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00
0.5
1
u(r)
(fm
)
asymptoticR = 0 fmR = 1.4 fmR = 1.5 fmR = 2.0 fmR = 2.5 fmNijm93
INT, Seattle, WA, 3/25/20
Deuteron wave functions (TPE).
0 1 2 3 4 5 6 7 8 9 10r (fm)
0
0,1
0,2
0,3
0,4
w(r
) (fm
)
-1/2
-1/2
asymptoticRd = 0 fm, OPERd = 0 fm, TPE
0 1 2 3 4 5 6 7 8 9 100
0,2
0,4
0,6
0,8
1
u(r)
(fm
)
asymptoticRd = 0 fm, OPERd = 0 fm, TPE
INT, Seattle, WA, 3/25/20
nn scattering wfs, GGS
vs GP
OINT, Seattle, WA, 3/25/20
nn scattering wfs, GGS vs GP
R=1.4 fmR=1.5 fmR=2.0 fm
INT, Seattle, WA, 3/25/20
nn scattering wfs, GGS
vs GP
R=1.4 fmR=1.5 fmR=2.0 fm
GGS use P5(r)R = 1.4 fmRSC
GP use sph wellvarying ROPEP+TPEP
OINT, Seattle, WA, 3/25/20
nn scattering wfs, GGS vs GP
R=1.4 fmR=1.5 fmR=2.0 fm
GGS use P5(r)R = 1.4 fmRSC
GP use sph wellvarying ROPEP+TPEP
INT, Seattle, WA, 3/25/20
TPE wfs
0 1 2 3 4 5r (fm)
0
2
4
6
8
10
12
14
16
wf (
fm)
v0R = 2.5 fm, OPER = 2.5 fm, TPER = 2 fm, OPER = 2 fm, TPER = 1.5 fm, OPER = 1.5 fm, TPER = 1.0 fm, OPER = 1.0 fm,TPEAV18
p = 10 MeV/ccomparison of OPE and TPE
INT, Seattle, WA, 3/25/20
Feynman rule for γππNN
d can only be established if new FR derived:
(c4 +
1
4M
)2ie
f2π
[(δabτ3 − δa3τ b
)[S · q1, S · εγ ]
−(δabτ3 − δb3τa
)[S · q2, S · εγ ]
]
Not published before (not in [Bernard, Kaiser, Meißner, IJMPE 4, 193 (1995)])
[AG, PRC 74, 017001 (2006)]
INT, Seattle, WA, 3/25/20
Convergence
200 400 600 800TOF (channels)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Dec
ay r
ate
(arb
r. u
nits
)
QF FSI
LO (KR), R = 2.2 fmNLON2LO 1BN2LO 1B+2BN3LO 2B+CT
ΓQF
ΓFSI= 2.422(1 − 0.0035 + 0.0003 + 0.013 − 0.035)
INT, Seattle, WA, 3/25/20
Role of higher partial waves
200 400 600 8000
0.2
0.4
0.6
0.8 N2LO, 2.2 fmN3LO, 2.2 fm+3Pj from a+r+1D2 from Nijm pwa+3Pj from Nijm pwa+1D2 from Nijm pwa
INT, Seattle, WA, 3/25/20
Sensitivity to ann
200 400 600 800neutron TOF (channels)
0
0.1
0.2
0.3
0.4
deca
y ra
te (a
rbr.
units
)
a = -20 fma = -18 fma = -16 fm
QF FSI
Neutron TOF spectrum at θ3 = 0.075 rad ⇒ ∆ann
ann= 0.83∆Γ
Γ
INT, Seattle, WA, 3/25/20
Boost correctionsCorrections to CGLN
∆F(0)1 (Eπ) =
egA
2fπ
−(Eπpn ·bk + E2
π)
2M2(µp + µn)
∆F(−)1 (Eπ) =
egA
2fπ
Eπpn ·bk + E2
π
M2
New spin-momentum structures
G(0)(Eπ) =egA
2fπ
iEπpn · εγσ ·bk
2M2(µp + µn − 1)
G(−)(Eπ) =egA
2fπ
Eπpn · (bk × εγ)
2M2(µp − µn +
1
2) −
ipn · εγσ · (2pn + Eπbk)
M2
!
µp − µn + 12 = 5.2, but pn · (k × εγ) ≈ E2
π sin θ3 with θ3 = 0.075 radsimilarly pn · εγ ≈ Eπ sin θ3
Thus only CGLN corr’s important, O(µ2/2M2) ∼ 1%
INT, Seattle, WA, 3/25/20
To boost or not to boost?
200 400 600 800neutron TOF (channels)
0
0.1
0.2
0.3
0.4
0.5
deca
y ra
te (a
rbr.
uni
ts)
no boostsboostsno boosts rescaled
Both peaks scale the same way ⇒ 0.10% for ann
INT, Seattle, WA, 3/25/20
‘Off-shellness’
(a)
(c)
=
(b)
(d)
=
+
Off-shell nucleon transformed into 2B and on-shell 1BNew 2B O(Q5) ⇔ p2/M2 ∼ µ2/M2 ∼ 2% of O(Q3) 2B
⇒ ∆ann = 0.02 fmINT, Seattle, WA, 3/25/20
Subthreshold extrapolation
200 400 600 800neutron TOF (channels)
0
0.1
0.2
0.3
0.4
0.5
deca
y ra
te (a
rbr.
uni
ts)
subthresholdno subthreshold
INT, Seattle, WA, 3/25/20
Error from d wfs
200 400 600 800neutron TOF (channels)
0
0.1
0.2
0.3
0.4
deca
y ra
te (a
rbr.
uni
ts)
Rd = 2.0 fmRd = 1.5 fmBonn B
INT, Seattle, WA, 3/25/20
Error from nn wfs
200 400 600 800neutron TOF (channels)
0
0.1
0.2
0.3
0.4
deca
y ra
te (a
rbr.
units
)
R = 2 fm, ann = -17.15 fmR = 1.5 fm, ann = -17.15 fmNijm INijm II
INT, Seattle, WA, 3/25/20
FSI only
500 520 540 560 580neutron TOF (channels)
0.14
0.145
0.15
deca
y ra
te (a
rbr.
units
)
R = 2 fm, ann = -17.15 fmR = 1.5 fm, ann = -17.15 fmNijm INijm II
INT, Seattle, WA, 3/25/20
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