Download New Frontiers in Nuclear Physics Kai Hebeler (OSU)

New Frontiers in Nuclear Physics
Kai Hebeler (OSU)
Darmstadt, March 24, 2011
Outline
Nuclear equation
of state
Neutron-rich nuclei
UNEDF
Superfluidity,
Neutron star cooling
Correlations in
nuclear systems
Goal
Superfluidity,
Neutron star cooling
Nuclear equation
of state
Unified description!
Key step:
Choose convenient resolution scale.
Neutron-rich nuclei
UNEDF
Correlations in
nuclear systems
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Wavelength and resolution
size of resolvable structures depends on the wavelength
Question: Which resolution should we choose?
Wavelength and resolution
size of resolvable structures depends on the wavelength
Question: Which resolution should we choose?
Depends on the system and phenomena we are interested in!
Resolution: The higher the better?
in the nuclear physics here we are interested in low-energy observables
(long-wavelength information!)
Resolution: The higher the better?
in the nuclear physics here we are interested in low-energy observables
(long-wavelength information!)
Resolution: The higher the better?
in the nuclear physics here we are interested in low-energy observables
(long-wavelength information!)
Resolution: The higher the better?
in the nuclear physics here we are interested in low-energy observables
(long-wavelength information!)
• resolution of very small (irrelevant) structures can obscure this information
• small details have nothing to do with long-wavelength information!
Strategy: Use a low-resolution version
• long-wavelength information is preserved
• distortion at small distance significantly reduced
• much less information necessary
In nuclear physics:
Use renormalization group (RG) to change resolution!
Overview
Overview RG
RG Summary
Summary Extras
Extras
Physics
Physics Resolution
Resolution Forces
Forces Filter
Filter Coupling
Coupling
Problem:
Traditional
“hard”
NN
interactions
Why is textbook nuclear physics so hard?
Why is textbook nuclear physics so hard?
−k !
k!
VV3N
k
−k
!k ! |V |k"
V
VL=0
(k, kk!!)) ∝
∝
L=0(k,
!!
rr22 dr
dr jj00(kr
(kr)) V
V(r
(r)) jj00(k
(k!!rr)) =
= "k|V
"k|VL=0
|k!!## =⇒
=⇒ V
Vkk
matrix
L=0|k
kk!! matrix
• constructed to fit scattering data (long-wavelength information!)
Momentum
Momentum units
units (!
(! =
= cc =
= 1):
1): typical
typical relative
relative momentum
momentum
interactions
contain
repulsive
core at small relative distance
• “hard”ininNN
−1
large
≈
large nucleus
nucleus ≈
≈ 11 fm
fm−1
≈ 200
200 MeV
MeV but
but .. .. ..
−1
coupling
and(!
components, hard to solve!
Repulsive
core
=⇒
high-k
22 fm
)) components
• strong
Repulsive
corebetween
=⇒ large
largelow
high-k
(!high-momentum
fm−1
components
Dick
Dick Furnstahl
Furnstahl
RG
RG in
in Nuclear
Nuclear Physics
Physics
Claim:
Problems due to high resolution from interaction.
These interactions correspond to using beer coasters!
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
Changing the resolution:
The (Similarity) Renormalization Group
• goal: generate unitary transformation of “hard” Hamiltonian
†
Hλ = Uλ HUλ with the resolution parameter λ
•
dHλ
basic idea: change resolution in small steps:
= [ηλ , Hλ ]
dλ
• SRG only one possibility, also:Vlow k, UCOM, Lee-Suzuki...
Changing the resolution:
The (Similarity) Renormalization Group
• elimination of coupling between low- and high momentum components,
calculations much easier!
• observables unaffected by resolution change (for exact calculations)
• residual resolution dependences can be used as tool to test calculations
Not the full story:
RG transformation also changes three-body (and higher-body) interactions!
excited to resonances
areMeV)
there 3N forces?
nt contribution fromWhy
!(1232
Classical analog
Tidal effects lead to 3N forces in earth-sun-moon system:
(3N) forces?
es,
er-range parts
fects leads to 3-body forces in earth-sun-moon system
are there three-n
of sun
• force between earth and moon depends on the position Why
Nucleons are finite-mass composite
• tidal deformations are internal excitations can be excited to resonances
dominant contribution from !(1232
• nucleons are composite particles, can also be excited
• change of resolution changes the excitations that can be
sun-moon system
described explicitly
change of 3N force
• three-nucleon forces are crucial at low resolutions!
+ shorter-range parts
Nuclear effective degrees of freedom
• if a nucleus is probed at high energies,
nucleon substructure is resolved
• at low energies, details are not resolved
Nuclear effective degrees of freedom
Resolution
• if a nucleus is probed at high energies,
nucleon substructure is resolved
• at low energies, details are not resolved
• replace fine structure by something
simpler (like multipole expansion),
low-energy observables unchanged
Nuclear effective degrees of freedom
Resolution
• if a nucleus is probed at high energies,
nucleon substructure is resolved
• at low energies, details are not resolved
• replace fine structure by something
simpler (like multipole expansion),
low-energy observables unchanged
In essence, effective field theory (EFT) is the method that
describes how to do this replacement in a systematic way!
Basics concepts of chiral effective field theory
NN
• choose effective degrees of
freedom: here nucleons and pions
• short-range physics captured in
few short-range couplings
• separation of scales: Q << Λb,
breakdown scale Λb~500 MeV
• power-counting:
expand in powers Q/Λb
• systematic: work to desired
accuracy, obtain error estimates
Plan: Use EFT interactions
as input to RG evolution.
3N
4N
dominant contribution from !(1232 M
Leading order chiral 3N forces
NN
3N
∆
4N
+ shorter-range parts
long (2π)
intermediate (π)
short-range
tidal effects leads to 3-body forces in e
cE term
c1 , c3 , c4 terms cD term
• large uncertainties in 2π coupling
constants at present:
1.5
leads to theoretical uncertainties in
many-body observables
• cD and cE have to be determined
in A ≥ 3 systems
Outline
Nuclear equation
of state
Neutron-rich nuclei
UNEDF
Superfluidity,
Neutron star cooling
Correlations in
nuclear systems
Equation of state: Many-body perturbation theory
central quantity of interest: energy per particle E/N
H(λ) = T + VNN (λ) + V3N (λ) + ...
kinetic energy
E=
+
VNN
+
VNN
+
+
...
VNN
V3N
+
VNN
Hartree-Fock
V3N
+
VNN
V3N
+
V3N
V3N
2nd-order
+
V3N
V3N
3rd-order
and beyond
• “hard” interactions require non-perturbative summation of diagrams
• with low-momentum interactions much more perturbative
• inclusion of 3N interaction contributions!
Equation of state of pure neutron matter
20
Hartree-Fock
2.0 <
15
2nd-order
E NN+3N,eff
3N
< 2.5 fm -1
10
=
=
=
=
5
0
1.8
2.0
2.4
2.8
fm -1
fm -1
fm -1
fm -1
E NN+3N,eff +c 3 uncertainty
(2)
E(1)
NN + E NN
15
0.05
0.10
[fm -3 ]
0.15 0
0.05
0.10
[fm -3 ]
0.15
3N
10
5
0
0
E NN+3N,eff +c 3 +c 1 uncertainties
20
Energy/nucleon [MeV]
Energy/nucleon [MeV]
E(1)
NN+3N,eff
0
0.05
0.10
0.15
[fm -3 ]
KH and Schwenk PRC 82, 014314 (2010)
• significantly reduced cutoff dependence at 2nd order perturbation theory
• small resolution dependence indicates converged calculation
• energy sensitive to uncertainties in 3N interaction
• variation due to 3N input uncertainty much larger than resolution dependence
Equation of state of pure neutron matter
20
Hartree-Fock
2.0 <
15
2nd-order
E NN+3N,eff
3N
< 2.5 fm -1
10
=
=
=
=
5
0
1.8
2.0
2.4
2.8
fm -1
fm -1
fm -1
fm -1
15
10
5
0
0
0.05
0.10
[fm -3 ]
0.15 0
0.05
0.10
[fm -3 ]
0.15
E NN+3N,eff +c 3 +c 1 uncertainties
Schwenk+Pethick (2005)
Akmal et al. (1998)
QMC s-wave
GFMC v6
GFMC v8’
20
Energy/nucleon [MeV]
Energy/nucleon [MeV]
E(1)
NN+3N,eff
0
0.05
0.10
0.15
[fm -3 ]
KH and Schwenk PRC 82, 014314 (2010)
• significantly reduced cutoff dependence at 2nd order perturbation theory
• small resolution dependence indicates converged calculation
• energy sensitive to uncertainties in 3N interaction
• variation due to 3N input uncertainty much larger than resolution dependence
• good agreement with other approaches (different NN interactions)
The diagrams inset in this panel show top-down
schematics of the binary system at orbital phases of
0.25, 0.5 and 0.75 turns (from left to right). The
–40
neutron star is shown in red, the white dwarf
Figure 1 | Shapiro delay measurement for PSR
30
ab 30
companion
in blue
and the emitted
radio beam,
J1614-2230.
Timing
residual—the
excess delay
pointing
towards
Earth,
in
yellow.
At
20
20
not accounted for by the timing model—as a orbital phase
of 0.25
theorbital
Earth–pulsar
function
of theturns,
pulsar’s
phase. a,line
Fullof sight passes
10
10
doi:10.1038/nature09466
nearest
to the
companion
(,240,000
magnitude
of the
Shapiro
delay when
all otherkm),
producing
the
sharp
peak
in
pulse
We found
0
model parameters are fixed at their best-fitdelay.
values.
0
no evidence
kind of form
pulseofintensity
The solid
line showsfor
theany
functional
the
–10
variations,
as
from
an
eclipse,
near
conjunction.
Shapiro
delay,
and
the
red
points
are
the
1,752
–10
Best-fit residuals
obtained
using an
orbital
timingb,measurements
in our
GBT–GUPPI
data
set. model
–20
RESEARCH
LETTER
The diagrams
this panel
show top-down effects.
–20
that doesinset
not in
account
for general-relativistic
–30
schematics
of case,
the binary
orbital phases
of
In this
somesystem
of theatShapiro
delay signal
is
P.
B. Demorest1, T. Pennucci2, S. M. Ransom1, M. S. E. Roberts3 & J. W. T. Hessels4,5
–30
0.25,
0.5
and
0.75
turns
(from
left
to
right).
The
absorbed
by
covariant
non-relativistic
model
–40
neutron
star is shown
in these
red,delay
the
white
dwarf
Figure
1 | Shapiro
measurement
parameters.
That
residuals
deviate for PSR
30
a
30
b –40
companion
in
blue
and
the
emitted
radio
beam,
Neutron stars are composed of the densest form of matter known long-term data set, parameter covariance and dispersion measure varisignificantly
from
a
random,
Gaussian
distribution
J1614-2230.
Timing
residual—the
excess
delay
30
c to20
20
pointing
towards
Earth,
in
yellow.
At
orbital
phase
exist in our Universe, the composition and properties of which ation can be found in Supplementary Information.
ofnot
zero
mean shows
that
thetiming
Shapiro
delay must
accounted
for by
the
model—as
a be
are20still theoretically uncertain. Measurements of the masses or
As shown in Fig. 1, the Shapiro delay was detected in our
with
of data
0.25included
turns,
the
Earth–pulsar
line
of
sight
passes
to
model
the
pulse
arrival
times
properly,
function of the pulsar’s orbital phase. a, Full
radii
10 of these objects can strongly constrain the neutron star matter extremely high significance, and must be included to model
the arrival
10
nearest
to the companion (,240,000 km),
especially
at of
conjunction.
addition
theother
red
magnitude
the ShapiroIndelay
whentoall
equation
of
state
and
rule
out
theoretical
models
of
their
compositimes
of
the
radio
pulses
correctly.
However,
estimating
parameter
values
101,2
producing
the sharp peak in pulse delay. We found
tion
.
The
observed
range
of
neutron
star
masses,
however,
has
and
uncertainties
can
be
difficult
owing
to
the
high
covariance
between
0
GBT–GUPPI
points,
the
454
grey
points
show
the
model parameters
are fixed at their best-fit values.
0
no evidence
any kind of pulse intensity
hitherto
forprevious
the for ‘long-term’
0 been too narrow to rule out many predictions of ‘exotic’ many orbital timing model terms14. Furthermore, the x2 surfaces
The drastic
The solid line showsdata
the set.
functional
form of the
non-nucleonic
components3–6. The Shapiro delay is a general-relat- Shapiro-derived companion mass (M2) and inclination anglevariations,
(i) are often as from an eclipse, near conjunction.
–10
–10
15
improvement
inand
datathe
quality
isorbital
apparent.
c, 1,752
Post-fit
Shapiro
delay,
redanpoints
are
the
ivistic
–10 increase in light travel time through the curved space-time significantly curved or otherwise non-Gaussian . To obtain b,
robust
error
Best-fit
residuals
obtained
using
model
7
near
a
massive
body
.
For
highly
inclined
(nearly
edge-on)
binary
estimates,
we
used
a
Markov
chain
Monte
Carlo
(MCMC)
approach
to
residuals
for
the
fully
relativistic
timing
model
timing
measurements
in
our
GBT–GUPPI
–20
–20
that does not account for general-relativistic effects. data set.
–20
millisecond
radio pulsar systems, this effect allows us to infer the explore the post-fit x2 space and derive posterior probability distributions
(including
delay),
which
have
mean
The
inset
in this
panel
show
top-down
Inthe
this
case, diagrams
someShapiro
of the
Shapiro
delay
signal
isa root
masses of both the neutron star and its binary companion to high for all timing model parameters (Fig. 2). Our final results for
model
2
–30
–30
8,9
value
ofof
squared
residual
of
1.1
ms
and
a
reduced
x
–30
schematics
of
the
binary
system
at
orbital
phases
absorbed by covariant non-relativistic model
precision . Here we present radio timing observations of the binary
10,11
Table
1
|
Physical
parameters
for
PSR
J1614-2230
1.4
with
degrees
of (from
freedom.
bars,The
1s.
millisecond pulsar J1614-2230
that show a strong Shapiro delay
Credit:
NASA/Dana
Berry
0.25,
0.52,165
and
0.75
turns
leftError
to right).
parameters.
That
these
residuals
deviate
–40
–40
–40
Parameter
Value
signature.
We
calculate
the
pulsar
mass
to
be
(1.976
0.04)
M
,
which
[
fromstar
a random,
Gaussian
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 significantly
neutron
is shown
in red, distribution
the white dwarf
30
c
30 out almost all currently proposed2–5 hyperon or boson con- Ecliptic longitude (l)
b rules
245.78827556(5)u
of
zero
mean
shows
that
the
Shapiro
delay
must
be beam,
companion
in
blue
and
the
emitted
radio
Ecliptic latitude (b)
21.256744(2)u
densate equations of state (M[, solar mass). Quark matter
canphase
sup- (turns)
Orbital
20
Proper
motion
in
l
9.79(7)
mas
yr
included
to
model
the
pulse
arrival
times
properly,
port
quarks are strongly interacting and
pointing towards Earth, in yellow. At orbital phase
20a star this massive only if the
Proper motion in b
230(3) mas yr
are
therefore not ‘free’ quarks12.
especially
at conjunction.
In
addition to
red
system
to
be
remarkably
edge-on,
with an
ofthe
89.17u
6 0.02u.
Demorest
et
al.,
Nature
467,
1081
(2010)
of 0.25
turns,
theinclination
Earth–pulsar
line
of sight
passes
10
Parallax
0.5(6)
mas
parameters,
with MCMC error estimates, are given in Table 1. Owing to
In March 2010, we performed a dense set of observations of J1614- Pulsar spin period
3.1508076534271(6)
ms
GBT–GUPPI
points,
the
454
grey
points
show
the
10
This is the9.6216(9)
most 3inclined
pulsar
system known
at present.
nearest binary
to the companion
(,240,000
km), The
the2230
high
of Astronomy
this detection,
our
MCMC
procedure
derivative and a
10
ss
with significance
the National Radio
Observatory
Green
Bank Period
0
previous
‘long-term’
data
set.
The
drastic
2
Reference
epoch
(MJD)
53,600
producing
the
sharp
peak
in
pulse
delay.
Wewith
found
Telescope x
(GBT),
timed to follow
the system
through one complete
amplitude and
sharpness of the Shapiro delay increase rapidly
fit produce
similar
uncertainties.
standard
Dispersion measure*
34.4865 pc cm
improvement
in
data
quality
is
apparent.
c,
Post-fit
–100 orbit with special
8.7-d
attention
paid
to
the
orbital
conjunction,
where
maxShapiro
! a companion
!binary
any kind
of pulse
Orbital period
8.6866194196(2)
d no evidence
inclination
thefor
overall
scaling
of intensity
the signal is
From
the
detected
measure
mass of increasing
residuals
for and
the fully
relativistic
timing model
the
Shapiro
delay
signal is strongest.
Thesedelay,
data werewe
taken
with the newly
Projected semimajor axis
11.2911975(2)
light s
variations,
as
from
an
eclipse,
near
conjunction.
–20
–10
to the Shapiro
mass ofdelay),
the which
companion
star.mean
Thus, the
(esin v) linearly proportional
1.1(3) 3 10
built Green
Bank Ultimate
Pulsar Processing Instrument (GUPPI). First Laplace
(including
have a root
is aparameter
helium–
(0.500
60.006)M
[, which implies that the companion
Second Laplace parameter (ecos v)
21.29(3) 3 10
b,
Best-fit
residuals
obtained
using
an
orbital
model
2
GUPPI coherently removes interstellar
dispersive
smearing
from
the
16
unique
combination
of
the
high
orbital
inclination
and
massive
squared residual of 1.1 ms and a reduced x value of white
–30
Companion
massbinary
0.500(6)M
.
The
Shapiro
delay
also
shows
the
carbon–oxygen
white
dwarf
pulsar
signal
and
integrates
the
data
modulo
the
current
apparent
pulse
–20
that
does
not account
for general-relativistic
effects.
Sine of inclination angle
0.999894(5)
1.4 with
2,165
degrees
of freedom.
Errordelay
bars, 1s.
dwarf companion
in
J1614-2230
cause
a Shapiro
amplitude
period,
52,331.1701098(3) In this case, some of the Shapiro delay signal is
–40 producing a set of average pulse profiles, or flux-versus-rota- Epoch of ascending node (MJD)
89.24
a tional-phase
b0.4 pulse 0.5
data (MJD) 0.8 orders
52,469–55,330
0.0
0.1 curves. 0.2
0.6 of timing0.7
0.9 of magnitude
1.0
light
From these,0.3
we determined
times of Span
larger
than for most other millisecond pulsars.
–30
Number of TOAs{
2,206 (454, 1,752)
absorbed by covariant non-relativistic model
arrival using standard procedures, with a typical uncertainty of ,1 ms.
Orbital phase (turns)
Root mean squared TOA residual In addition, the 1.1
ms
excellent
timing precision
achievable
the pulsar
89.22
We used the measured arrival times to determine key physical paraparameters.
That these
residualsfrom
deviate
–40
Right
ascension
(J2000)
16
h
14
min
36.5051(5)
s
meters of the neutron star and its binary system by fitting them to a
with
theremarkably
GBT
andedge-on,
GUPPI
provide
afrom
verya random,
high
signal-to-noise
ratio
significantly
Gaussian
distribution
system
to be
with
an inclination
of 89.17u
6 0.02u.
Declination
(J2000)to
222u 309 31.081(7)99
parameters,
MCMC
error
estimates,
arerotation
givenofintheTable
1.
Owing
3089.2 with
timing
model that
accounts
for every
c comprehensive
Orbital eccentricity (e)
3 10 Shapiro
measurement1.30(4)
of both
delay
parameters
a single
of zero
mean
shows that within
the Shapiro
delayorbit.
must be
star
over the timeof
spanned
by the fit. Theour
modelMCMC
predicts atprocedure
theneutron
high
significance
this detection,
and a This is the most inclined
Inclination angle
89.17(2)upulsar binary system known at present. The
89.18
20
1.97(4)M
what times2 pulses should arrive at Earth, taking into account pulsar Pulsar mass
included
to
model
the
pulse
arrival
times
properly,
The
standard
Keplerian
orbital
parameters,
combined
with
the
known
amplitude and sharpness
of the Shapiro delay increase rapidly with
standard
x fit produce similar uncertainties.
1.2 kpc
rotation and spin-down, astrometric terms (sky position and proper Dispersion-derived distance{
especially
at conjunction.
addition
to
companion
mass
and
orbital
inclination,
fully
describe
the
dynamics
of a
89.16
Parallax
distance
.0.9
kpc
increasing
binary
inclination
and
the overall
scaling
ofInthe
signal
is the red
10 the
From
detected
Shapiro time-variable
delay, we measure
a companion
mass of
motion),
binary
orbital parameters,
interstellar disperSurface magnetic field
1.8 3 10 G −1
GBT–GUPPI
points,
the
454
grey
points
show
3
‘clean’
binary system—one
two stable
objects—the
linearly
proportional
of the companion
star.compact
Thus, the
sion and
general-relativistic
effects such as the Shapiro delay (Table 1).
age
5.2to
Gyrthe masscomprising
is a helium–
(0.500
60.006)M
[, which implies that the companionCharacteristic
89.14
0
‘long-term’
data massive
set. the
Thepulsar’s
drastic mass.
1.2 3 10
erg s highprevious
We compared the observed arrival16times with the model predictions, Spin-down luminosity
unique
combination
of the
inclination
and
white
under
general
relativity
andorbital
therefore
also determine
delay
also
the
binary
carbon–oxygen
white
dwarf .byThe
and obtained best-fit
parameters
x2 Shapiro
minimization,
using
theshows
Average
flux
density*
at
1.4
GHz
1.2
mJy
improvement
in
data
quality
is
apparent.
Post-fit
–10
89.12
in21.9(1)
J1614-2230
cause a0.04)M
Shapiro
delay amplitude
13
2 results Spectral index, 1.1–1.92GHzdwarfWecompanion
measure a 2
pulsar
mass of (1.976
is by far thec, high[, which
TEMPO2 software package2
. We also obtained consistent
residuals
for other
the fully
relativisticpulsars.
timing model
89.24the original TEMPO package. The post-fit
a using
b
228.0(3)
rad m than
residuals, that is, Rotation measure
orders
magnitude
larger
forstar
most
millisecond
est of
precisely
measured
neutron
mass determined
to date. In contrast
–2089.1
(including
Shapiro
delay),
which
a root mean
derived from timing model parameter values (middle) and
the differences between the observed and the model-predicted pulse Timing model parameters (top), quantities
17 from the have
Inproperties
addition,
the inexcellent
timing
precision
achievable
pulsar
89.22
radio spectral and interstellar medium
(bottom).
Values
parenthesesmass/radius
represent
the 1s
with
X-ray-based
measurements
,
the
Shapiro
delay
pro-of
arrival
effectively measure how well the timing model describes uncertainty in the final digit, as determined by MCMC error analysis. The fit included both ‘long-term’ squared
residual of 1.1 ms and a reduced x2 value
–30 times,
data
0.48
0.5Fig.0.51
1.9 1.95 spanning
2 seven
2.05years2.1
2.15withdata
the
GBT
and
GUPPI
provide
a
very
high
signal-to-noise
ratio
the89.2
data, and
are 0.49
shown in
1. We0.52
included1.8
both1.85
a previously
and new GBT–GUPPI vides
spanning
three
months.
The
new
data
were
observed
no information about1.4
thewith
neutron
radius.
However,
unlike
2,165star’s
degrees
of freedom.
Error
bars,the
1s.
using an(M
800-MHz-wide
band centred at a radio frequency of 1.5 GHz. The raw profiles were polarizationCompanion
(MGUPPI
)
Pulsar
)
recorded long-term
data setmass,
and ourMnew
data in a single
fit. mass
RESEARCH
LETTER
–30
LETTER
A two-solar-mass neutron star measured using
Shapiro delay
Timing residual (μs)
Timing residual (μs)
Constraints on the nuclear equation of state (EOS)
21
Timing residual (μs)
21
M
221
= 1.65M → 1.97 ± 0.04 M
21
23
27
26
[
Probability density
Inclination angle, i (°)
Structure of a neutron star is determined by
Tolman-Oppenheimer-Volkov (TOV) equation:
26
[
!
"!
"!
"
dP
GM !
P
4πr P
2GM
=−
1+
1+
1−
dr
r
!c
Mc
c r
8
34
21
–40
ability density
ation angle, i (°)
22
measurement
of bothour
Shapiro
parameters
a single orbit.
X-ray methods,
resultdelay
is nearly
modelwithin
independent,
as it depends
crucial ingredient: energy density ! = !(P )
2
(
(
and flux-calibrated and averaged into 100-MHz, 7.5-min intervals using the PSRCHIVE software
0.0
0.1 determine
0.2model parameters
0.3
0.4
0.5 package0.6
0.7
0.9 MJD, modified
1.0Julian date.
The
long-term data
(for
example spin89.18
, from which pulse
times of arrival 0.8
(TOAs) were determined.
The
standard
Keplerian
orbital parameters, combined with the known
quantities vary stochastically on >1-d
timescales.
Values
presented here
are the averages forbeing an adequate description of gravity.
only
on
general
relativity
down rate
and astrometry)
characteristic
longer
than * Theseimage
Figure
2 | Results
of thewith
MCMC
error timescales
analysis.
a,
Grey-scale
shows
the
our GUPPI data set.
Orbital
phase
(turns)
companion
mass and
orbital
inclination,
fully describe
the dynamics ofbased
a
a89.16
few weeks, whereas the new data best constrain parameters on { Shown in parentheses are separate
for theaddition,
long-term (first) andunlike
new (second) data
sets.
In
statistical
pulsar
mass determinations
on
two-dimensional
posterior probability density function (PDF) in the M2–i pulsarvalues
.
distance
model
timescales of the orbital period or less. Additional discussion of the { Calculated using the NE2001
‘clean’ binary system—one comprising two stable
18–20compact objects—
25
26
Neutron star radius constraints
Problem: Solution of TOV equation requires EOS up to very high densities.
Radius of a typical NS (M~1.4 M!) theoretically not well constrained.
But: Radius of NS is relatively insensitive to high density region.
incorporation of beta-equilibrium: neutron matter
parametrize piecewise
high-density extensions of EOS:
p∼ρ
Γ
• range of parameters
Γ1 , ρ12 , Γ2
limited by physics!
log 10 P [dyne / cm 2 ]
• use polytropic ansatz
37
36
neutron star matter
crust EOS
neutron star matter
with c i uncertainties
2
35
1
34
33
32
31
13.0
13.5
log 10
14.0
[g / cm 3 ]
1
12
KH et al., PRL 05, 161102 (2010)
3.0
Neutron star radius
constraints
2.5
use the constraints:
2.0
Mmax > 1.97 M!
M [M ]
recent NS observation
causality
vs (ρ) =
!
Mmax
= 1.5
1 = 2.5
1 = 3.5
1 = 4.5
2 = 1.5
2 = 2.5
2 = 3.5
2 = 4.5
= 1
= 12 = 1.5
= 12 = 2.5
= 12 = 3.5
= 12 = 4.5
1.5
1
1.0
dP/dε < c
KH et al., PRL 05, 161102 (2010)
0.5
0
6
7
8
0
0
0
0
9
10
11
12
13
14
R [km]
• low-density part of EOS sets scale for allowed high-density extensions
• radius constraint after incorporating crust corrections: 10.5 − 13.5 km
15
Overview RG Summary Extras
Physics Resolution Forces Filter Coupling
Equation ofnuclear
state of symmetric
nuclear
matter
Why is textbook
physics so
hard?
0
3
Vlow k NN from N LO (500 MeV)
¯
lS
Energy/nucleon [MeV]
!5
!1
3NF fit to E3H and r4He "3NF = 2.0 fm
NN + 3N
!10
!15
!20
!25
3rd order pp+hh
!1
" = 1.8 fm
!1
" = 2.8 fm
!1
" = 1.8 fm NN only
!1
" = 2.8 fm
!30
VL=0 (k, k ! ) ∝
!
0.8
NN only
NN only
1.0
1.2
1.4
1.6
!1
kF [fm ]
r 2 dr j0 (kr ) V (r ) j0 (k ! r ) = "k|VL=0 |k ! # =⇒
VPRC(R)
kk ! matrix
KH et al.,
83, 031301 (2011)
saturation delicate due to cancellations of large kinetic and
• nuclear
Momentum units (! = c = 1): typical relative momentum
potential energy contributions
in large nucleus ≈ 1 fm−1 ≈ 200−3
MeV but . . .
saturation at nS ∼ 0.16 fm and−1Ebinding /N ∼ −16 MeV
• empirical
Repulsive core =⇒ large high-k (! 2 fm ) components
Here: fitRG3NF
couplings to few-body systems:
• 3N forces are essential!
Dick Furnstahl
in Nuclear Physics
E3H = −8.482 MeV and r4He = 1.95 − 1.96 fm
Equation of state of symmetric nuclear matter
5
Energy/nucleon [MeV]
3
!1
" = 1.8 fm
!1
" = 2.0 fm
!1
" = 2.2 fm
!1
" = 2.8 fm
Vlow k NN from N LO (500 MeV)
0
3NF fit to E3H and r4He
!5
2.0 < "3NF < 2.5 fm
!1
!10
Hartree-Fock
Empirical
saturation
point
0.8
1.4
!15
!20
1.0
1.2
!1
kF [fm ]
2nd order
1.6 0.8
1.0
3rd order pp+hh
1.2
1.4
1.6 0.8
!1
kF [fm ]
1.0
1.2
1.4
1.6
!1
kF [fm ]
KH et al., PRC(R) 83, 031301 (2011)
• saturation point consistent with experiment, without new free parameters
• cutoff dependence at 2nd order significantly reduced
• 3rd order contributions small
• cutoff dependence consistent with expected size of 4N force contributions
Summary: Bulk nuclear properties
efficiently described at low resolution!
Outline
Nuclear equation
of state
Neutron-rich nuclei
UNEDF
Superfluidity,
Neutron star cooling
Correlations in
nuclear systems
and (6), are both s ¼ %!=6 # %1=12. As long as t is only
from the strong suppression of LMU by extensive proton
slightly larger than tC , the transit slope is larger than those
superconductivity. Figure 4 demonstrates that the result
of the asymptotic trajectories by a factor #f. The observed
TC ’ 0:5 ' 109 K does not depend on the star’s mass,
slope over a 10 yr interval is sobs ’ %1:4. Note, however,
but that the slope during the transit is very sensitive to
that the
model
‘‘0.5’’
of
Fig.
1
does
not
exhibit
such
a
large
neutron star transparent to neutrinos the extentneutrino
emission dominates
of proton superconductivity.
Models successful
slope. We are thus led to investigate the origin of the
in reproducing the observed slope require superconducting
5
rapidity
of
Cas
A’s
cooling.
cooling process for about 10 years after protons
formation
in the entire core. Although spectral fits [5] seem
Several factors influence the rapidity of the transit phase.
to indicate
thatin
Cas
A has neutron
a larger thanstars,
canonical mass
pair
formation
process
young
First, Cooper
LPBF depends
on the
shape of thedominant
Tcn ð"Þ curve.cooling
A
(1:4M( ), a recent analysis [6] indicates compatibility, to
weak " dependence, i.e., a wide Tcn ð"Þ curve, results in a
within +
3#,νwith
a smaller mass, 1:25M( . The need for
+
n
→
[nn]
+
ν̄
allows
thickersuperfluidity
PBF neutrino emitting
shell process
and a larger Ln
than
a
PBF
extensive proton superconductivity to reproduce the large
strong " dependence. Second, the T dependence of Te , i.e.,
observed slope favors moderate masses unless superconthe parameter ! in Eq. (7), also affects the slope in Eq. (8).
ductivity extends to much higher densities than current
Third, protons in the core will likely exhibit superconducmodels
predict (see,
Fig 9 inonly
[14] for
large sample
4% Most
temperature
in young
neutron
stare.g.,
within
10ayears!
tivitySpectacular
in the 1 S0 channel.
calculations ofdrop
the proton
of current
models).
Star Provides Direct Evidence for Bizarre Type of Nuclear Matter - ScienceNOW
critical temperature, Tcp ð"Þ, are larger than Tcn ð"Þ at low Neutron
The inferred TC ’ 0:5 ' 109 K, either from Figs. 1, 3,
gap size
andextraction
4 or from of
Eq.approximate
(3), appears pairing
quite robust
and stems from
%1 Þ1=6 .
t
the small exponent in the relation TC / ðC9 L%1
9 C
TC = 1 9
Selected for a Viewpoint in Physics
0 K
Assuming L9 P HisY S Inot
affected by proton
C A L Rvery
E V I E W strongly
LETTERS
PRL 106, 081101 (2011)
Cooling of neutron stars
•
•
TC = 5
.5x10 8
K
week ending
25 FEBRUARY 2011
TC = 0
AAAS.ORG
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LIBRARIANS
Rapid Cooling of the Neutron Star in Cassiopeia A Triggered
by Neutron Superfluidity in Dense Matter
Enter
ALERTS
M/M TC [10 9 K]
Instituto de Astronomı́a, Universidad Nacional Autónoma de México, Mexico D.F. 04510, Mexico
1.9 USA 0.51
Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701-2979,
Home > News > ScienceNOW > February 2011 > Neutron Star Provides Direct Evidence for Bizarre Type of Nuclear Matter
Department of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, New York 11794-3800, USA
Joint Institute for Nuclear Astrophysics, National Superconducting Cyclotron Laboratory and,
of Physics
1.6Department0.52
and Astronomy,
Michigan
State
University,2011
East Lansing, Michigan 48824, USA
Adrian
Cho, 25
February
(Received 29 November 2010; published 22 February 2011) 1.3
0.57
Dany Page,1 Madappa Prakash,2 James M. Lattimer,3 and Andrew W. Steiner4
News Home
1
ScienceNOW
ScienceInsider
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About Science News
2
3
4
We propose that the observed cooling of the neutron star in Cassiopeia A is due to enhanced neutrino
emission Star
from the recent
onset of the breaking
and formation
of neutron Cooper
in the P channel.
Neutron
Provides
Direct
Evidence
forpairs
Bizarre
Type of
We find that the critical temperature for this superfluid transition is ’ 0:5 ! 10 K. The observed rapidity
Nuclear
Matter
of the cooling
implies that protons were already in a superconducting state with a larger critical
3
9
2
This is2011,
the first
direct
thatLink
superfluidity
and superconductivity occur at supranuclear
by Adrian temperature.
Cho on 25 February
2:42
PM evidence
| Permanent
| 6 Comments
densities within neutron stars. Our prediction that this cooling will continue for several decades at the
present rate can be tested by continuous monitoring of this neutron star.
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DOI: 10.1103/PhysRevLett.106.081101
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PACS numbers: 97.60.Jd, 95.30.Cq, 26.60."c
For more than 50 years, astrophysicists have speculated that inside a
ENLARGE IMAGE
superdense
neutron
star, nuclear
matter
might flow without
any
The essence of the minimal cooling paradigm is the
The
neutron star
in Cassiopeia
A (Cas
A), discovered
in
priori exclusion of all fast !-emission mechanisms,
1999resistance
in the Chandra
first light observation
[1] targeting
theearthya materials
whatsoever—much
like electricity
does in
thus restricting ! emission to the ‘‘standard’’ MU process
supernova
remnant, is the youngest known in the Milky
known as superconductors. Now, two teams say they have direct
and nucleon bremsstrahlung processes [11]. However, efWay. An association with the historical supernova SN 1680
evidence of such bizarre "superfluidity" in a neutron star, and
fectsother
of pairing, i.e., neutron superfluidity and/or proton
[2] gives
Cas A an age of 330 yr, in agreement with its
researchers
"I think
it's aisdefensible
says
superconductivity,
are included. At temperatures just bekinematic
age [3].seem
The convinced.
distance to the
remnant
estimated claim,"
Page et al., PRL 106, 081101 (2011)
FIG. 3.
A typical good fit to Cas A’s rapid cooling for a 1:4M
FIG. 4.
Cooling curves with different masses and values of
Superfluidity in neutron and cooling of neutron stars
spin-triplet (3P2-3F2):
spin-singlet (1S0):
3.0
2.0
!(kF) [MeV]
( F ) [MeV]
2.5
1.5
1.0
TC ! 0.5 × 109 K
0.4
0.2
0
0.5
1.0
F
[fm -1 ]
-1
" = 1.8 fm
-1
" = 2.0 fm
-1
" = 2.8 fm
free, NN only
free, NN+3N
HF, NN only
HF, NN+3N
0.5
0
PRELIMINARY
0.6
0
nS
1.5
2
KH and Schwenk PRC 82, 014314 (2010)
1
1.2
1.4
1.6 nS 1.8
-1
kF [fm ]
3 nS 2
2
TC extracted in Page et al., PRL 106, 081101 (2011)
• pairing gap rather well constrained
• active at low densities
• 3N force contributions moderate
• only weakly affects cooling
• only loosely constrained so far
• active at higher densities
• 3N force contributions important
• crucial for cooling
(crustal cooling)
(core cooling)
Summary: Low resolution methods give insight into nuclear superfluidity.
Outline
Nuclear equation
of state
Neutron-rich nuclei
UNEDF
Superfluidity,
Neutron star cooling
Correlations in
nuclear systems
Correlations in nuclear systems
b)
What is this vertex?
e’
k
k!
e
q
k!
N
N
A
A!2
Higinbotham,
arXiv:1010.4433
correlation studies is to cleanly isolate diagram
b) from
a).
h final-state interactions, that can produce the same final state as
matics that were dominated by diagram b) it would finally allow
of the nucleon-nucleon potential.
q
Short-range-correlation
interpretation:
energy and moment transferred, but also the energy and
ween the transferred
k ! and detected energy and momentum
entum, pmiss , respectively. From the theoretical works on
k
momentum distribution of nucleons in the nucleus [6], it
verage potential motion of the nucleon in the nucleus of
correlations.
Electron
Beam
(CEBAF) [7], it was possible to
rel.Facility
momentum
k : low
ic kinematics
into the pmiss > 250 MeV/c region. In the
!
rel. momentum
k : high
3
E89-044
He(e,e’p)pn
and 3 He(e,e’p)d, these kinematics
short-range correlations. And while final results of the
• detection of knocked out pairs
with large relative momenta
• excess of np pairs over pp pairs
Subedi et al., Science 320, 1476 (2008)
Explanation in terms
of low-momentum interactions?
Correlations in nuclear systems
b)
What is this vertex?
e’
k
k!
e
q
k!
N
N
A
A!2
Higinbotham,
arXiv:1010.4433
correlation studies is to cleanly isolate diagram
b) from
a).
h final-state interactions, that can produce the same final state as
matics that were dominated by diagram b) it would finally allow
of the nucleon-nucleon potential.
q
Short-range-correlation
interpretation:
energy and moment transferred, but also the energy and
ween the transferred
k ! and detected energy and momentum
entum, pmiss , respectively. From the theoretical works on
k
momentum distribution of nucleons in the nucleus [6], it
verage potential motion of the nucleon in the nucleus of
correlations.
Electron
Beam
(CEBAF) [7], it was possible to
rel.Facility
momentum
k : low
ic kinematics
into the pmiss > 250 MeV/c region. In the
!
rel. momentum
k : high
3
E89-044
He(e,e’p)pn
and 3 He(e,e’p)d, these kinematics
short-range correlations. And while final results of the
• detection of knocked out pairs
with large relative momenta
• excess of np pairs over pp pairs
Subedi et al., Science 320, 1476 (2008)
Explanation in terms
of low-momentum interactions?
Vertex depends on the resolution!
RG provides systematic
way to calculate such
processes at low resolution.
med for each of the configurations sampled in
In He and He with uncorrelated wave functions, 3/4 of
om walk. The large range of values of x and X
the np pairs are in deuteron-like T, S=0,1 states, while
to obtain converged results, especially for 3 He,
airly large numbers of points; we used grids of
and 80 points for x and X, respectively. We
6
p! + p = Q = 0
10
3
He
over all pairs instead of just pair 12.
!
4
5
p
−
p
=
2
q
3
4
10
He
and pp momentum distributions in He, He,
6
Li
4
Be nuclei are shown in Fig. 1 as functions of the
10
8
Be
momentum q at fixed total pair momentum Q=0,
3
10
nding to nucleons moving back to back. The
2
l errors due to the Monte Carlo integration are
10
np pairs
only for the pp pairs; they are negligibly small
1
10
p pairs. The striking features seen in all cases
he momentum distribution of np pairs is much
0
10
an that of pp pairs for relative momenta in the
-1
pp pairs
10
–3.0 fm−1 , and ii) for the helium and lithium
the node in the pp momentum distribution is
0
1
2
3
4
5
-1
the np one, which instead exhibits a change of
q (fm )
characteristic value of p # 1.5 fm−1 . The nodal
Schiavilla et al., PRL 98, 132501 (2007)
8
FIG. 1: (Color online) The np (lines) and pp (symbols) mois much less prominent in Be. At small valscaling
of momentum
function:
mentum distributions
in various nuclei as functions of the
the ratios
of np tobehavior
pp momentum
distributions distribution
relative momentum q at vanishing total pair momentum Q.
r to those of ρnp
in
(q,pp
Q pair
= 0)numbers,
≈ CA ×which
ρNN,Deuteron
(q, Q = 0) at large q
NNto
6
!NN(q,Q=0) (fm )
Correlations in nuclear systems
•
• dominance of np pairs over pp pairs
• “hard” interaction used (high resolution), calculations hard!
• dominance explained by short-range tensor forces
Nuclear scaling at low resolution
!
"
⇒ ψλ |Oλ |ψλ factorizes into a low-momentum structure and a
universal high momentum part if the initial operator only
weakly couples low and high momenta
explains scaling!
RG transformation of
pair density operator
(induced many-body
terms neglected):
Uλ
simple calculation of pair density at low resolution in nuclear matter:
Vλ
!
"
ρ(P, q) =
+
Vλ
Vλ
+
+
Vλ
Vλ
Nuclear scaling at low resolution
<!(P=0,q)>np/<!(P=0,q)>nn
10
9
PRELIMINARY with tensor
8
interaction
7
6
5
no tensor
interaction
4
-1
" = 1.8 fm
-1
" = 2.0 fm
-1
" = 2.5 fm
-1
" = 3.0 fm
3
2
1
0
1.5
2
3
2.5
-1
q [fm ]
3.5
4
KH et al., in preparation
• pair-densities approximately resolution independent
• significant enhancement of np pairs over nn pairs due to tensor force
• reproduction of previous results using a “simple” calculation at low resolution!
Nuclear scaling at low resolution
<!(P=0,q)>np/<!(P=0,q)>nn
10
9
PRELIMINARY with tensor
8
interaction
7
6
5
no tensor
interaction
4
-1
" = 1.8 fm
-1
" = 2.0 fm
-1
" = 2.5 fm
-1
" = 3.0 fm
3
2
1
0
1.5
2
3
2.5
-1
q [fm ]
3.5
4
KH et al., in preparation
• pair-densities approximately resolution independent
• significant enhancement of np pairs over nn pairs due to tensor force
• reproduction of previous results using a “simple” calculation at low resolution!
Summary:
High-resolution experiments can be explained by low-resolution methods!
Opens door to study other electro-weak processes and higher-body correlations.
Outline
Nuclear equation
of state
Neutron-rich nuclei
UNEDF
Superfluidity,
Neutron star cooling
Correlations in
nuclear systems
Density Functional Theory for nuclei
Introduction
Formalism
Introduction
Introduction
Introduction
Introduction
Results
Formalism
Formalism
Formalism
Formalism
scale
ResultsResults
Results
Results
Ultimate goal Ultimate
Ultimate
goal
goal
Ultimate
Ultimate
goalgoal
Ground state
Spectroscopy
Ground
state
Ground
state
Ground
state
Ground
state
Mass, deformation
Mass,
deformation
Mass,
deformation
Mass,
deformation
Mass, deformation
Spectroscopy
scale
S
Sum
Summar
Spectroscopy
Spectroscopy
Spectroscopy
Spectroscopy
Spectroscopy
Spectroscopy
SpectroscopySpectroscopy
Reaction properties
Reaction
properties
Reaction
properties
Reaction
properties
Reaction
properties
Collective modes
Collective
modes
Collective
modes
Collective
modes
Collective
modes
RPA, QRPA, GCM
RPA,
QRPA,
GCM
RPA,
QRPA,
GCM
RPA, QRPA,
GCM
RPA,
QRPA,
GCM
Introduction
Summary
scale
scale
scale
Fusion, transfer...
Fusion,
transfer...
Fusion,
transfer...
Fusion, transfer...
Fusion, transfer...
Formalism
Results
scale
Summary
Heavy elements
Exotic behaviors
Heavy
elements
Exotic
behaviors
Ultimate goal
Heavy
elements
Exotic
behaviors
Heavy elements
Exotic
behaviors
Heavy elements
Exotic behaviors
Fission, fusion, SHE
Drip-lines, halos
Formalism
Fission,
fusion,
SHE Introduction
Drip-lines,
halos
Fission,
fusion,
SHE
Drip-lines,
halos
Fission, fusion,
SHE
Drip-lines, halos
Fission, fusion, SHE
Drip-lines, halos
Ground state
Astrophysics
Result
Ultimate goal Spectroscopy
Astrophysics
Astrophysics
Astrophysics
Astrophysics
Mass, deformation NS, SN, r-process
NS,
SN,
r-process
NS,
SN,
r-process
NS, SN, r-process
Spectroscopy
NS, SN, r-process
Ground state
Reaction properties
Collective modes
Mass, deformation
Fusion, transfer...
Non-Empirical Pairing Functional for nuclei
Non-Empirical
Pairing
Functional
nuclei
Non-Empirical
Pairing
Functional
forfor
nuclei
Non-Empirical Pairing Functional for nuclei
RPA, QRPA, GCM
T. Duguet
T. T
D
T. Dugue
Non-Empirical Pairing Functional for nuclei
Collective
Exoticmodes
behaviors
Heavy elements
• Density Functional Theory is the tool of choice to describe low-energy
halos
RPA,Drip-lines,
QRPA, GCM
Fission, fusion, SHE
properties of known medium-mass and heavy nucleiAstrophysics
• current functionals are empirical
unknown regions of mass table
NS, SN, r-process
Heavy elements
E
Fission, fusion, SHE
D
lack of predictive power to
Non-Empirical Pairing Functional for nuclei
Astrophysics
T. Duguet
NS, SN, r-process
Goal: Constrain/construct microscopically energy-density functionals.
Non-Empirical Pairing Functional for nuclei
Microscopic Kohn-Sham
Density Functional Theory
Idea behind Kohn-Sham DFT
Lu Jen Sham
Walter Kohn,
Nobel Prize 1998
Turn off the interaction, but change
same density
external potential such that the
Ground state density of
interacting fermions in external
harmonic trap.
density remains that of the
interacting system. The additional
potential is the Kohn-Sham potential.
• use one-body density instead of many-body wavefunction as basic variable
• basic idea: describe an interacting QM system by a non-interacting
Picture from R. Furnstahl: http://trshare.triumf.ca/~schwenk/ECT/furnstahl1.pdf
system in an effective (“Kohn-Sham”) potential, in principle exact!
Basic problem: Determination of the effective potential.
Microscopic Kohn-Sham
Density Functional Theory
Idea behind Kohn-Sham DFT
Lu Jen Sham
Walter Kohn,
Nobel Prize 1998
Turn off the interaction, but change
same density
external potential such that the
Ground state density of
interacting fermions in external
harmonic trap.
density remains that of the
interacting system. The additional
potential is the Kohn-Sham potential.
• use one-body density instead of many-body wavefunction as basic variable
• basic idea: describe an interacting QM system by a non-interacting
Picture from R. Furnstahl: http://trshare.triumf.ca/~schwenk/ECT/furnstahl1.pdf
system in an effective (“Kohn-Sham”) potential, in principle exact!
Basic problem: Determination of the effective potential.
Strategy:
Nuclear interactions at low resolution act more like coulomb in chemistry.
Adapt successful quantum chemistry methods to nuclear physics!
Microscopic Kohn-Sham
Density Functional Theory
Idea behind Kohn-Sham DFT
Walter Kohn,
Nobel Prize 1998
Lu Jen Sham
50
Turn off the interaction, but change
external potential such that the
45
density remains that of the
interacting system. The additional
potential is the Kohn-Sham potential.
First implementation of microscopic
(orbital-based) DFT for nuclear systems:
Ground state density of
interacting fermions in external
harmonic trap.
NCFC
HF or OEP
DME/PSA HF
DME/PSA BHF
20 MeV
• neutrons in an external harmonic
Picture from R. Furnstahl: http://trshare.triumf.ca/~schwenk/ECT/furnstahl1.pdf
potential (“neutron drop”)
same functional as for nuclei
• first step: check DFT approximations
against exact results
• to first order (“exact exchange”) almost
perfect agreement with Hartree-Fock
Etot/N [MeV]
40
N = 20
35
hΩ of
harmonic trap
30
N=8
25
20
15
10 MeV
NN-only
Minnesota potential
2.0
2.5
radius [fm]
3.0
Drut, Platter, KH
Microscopic Kohn-Sham
Density Functional Theory
Idea behind Kohn-Sham DFT
Walter Kohn,
Nobel Prize 1998
Lu Jen Sham
50
Turn off the interaction, but change
external potential such that the
45
density remains that of the
interacting system. The additional
potential is the Kohn-Sham potential.
First implementation of microscopic
(orbital-based) DFT for nuclear systems:
Ground state density of
interacting fermions in external
harmonic trap.
NCFC
HF or OEP
DME/PSA HF
DME/PSA BHF
20 MeV
• neutrons in an external harmonic
Picture from R. Furnstahl: http://trshare.triumf.ca/~schwenk/ECT/furnstahl1.pdf
potential (“neutron drop”)
same functional as for nuclei
• first step: check DFT approximations
against exact results
Etot/N [MeV]
40
N = 20
35
hΩ of
harmonic trap
30
N=8
25
20
NN-only
Minnesota potential
15 enables
Summary: Low resolution
2.0
2.5
exchange”) almost
• to first order (“exact
radius [fm]
microscopic DFT methods for nuclear systems.
perfect agreement with Hartree-Fock
10 MeV
3.0
Drut, Platter, KH
Summary
• low resolution in nuclear many-body systems simplifies calculations dramatically
• change in resolution scale accomplished by RG transformations
• observables invariant under changes in resolution scale
• chiral EFT provides systematic framework for constructing nuclear Hamiltonians
• simplified calculations for various observables at low resolution:
nuclear equation of state
★ superfluidity in nucleonic matter
★ correlations in nuclear systems
★ orbital-based density functional theory
★
Superfluidity,
Neutron star cooling
Nuclear equation
of state
Outlook
Neutron-rich nuclei
UNEDF
Correlations in
nuclear systems
Nuclear equation
of state
Superfluidity,
Neutron star cooling
• constrain chiral EOS extensions by high-density EOSs (quark matter, etc..)
• extend equation of state to finite temperature
• incorporate constraints on nuclear EOS in stellar evolution models
Neutron-rich nuclei
UNEDF
Correlations in
nuclear systems
Nuclear equation
of state
Superfluidity,
Neutron star cooling
• investigate effect of pairing on cooling of neutron stars
• impact of chiral 3N interactions on pairing in finite nuclei
• non-perturbative treatment of unitary Fermi gases using Gorkov formalism
Neutron-rich nuclei
UNEDF
Correlations in
nuclear systems
Nuclear equation
of state
Superfluidity,
Neutron star cooling
• investigation of many-body terms in density operator evolution
scattering experiments
• many-body correlations
• study other operators relevant for electro-weak processes
Neutron-rich nuclei
UNEDF
Correlations in
nuclear systems
Nuclear equation
of state
Superfluidity,
Neutron star cooling
• generalize orbital-based DFT beyond exact exchange, include correlations
• include pairing and 3N forces, compare with explicit functionals
• extend DFT methods to self-bound systems
Neutron-rich nuclei
UNEDF
Correlations in
nuclear systems
In collaboration with:
E. Anderson
R. Furnstahl
J.-P. Blaizot
J. Lattimer
S. Bogner
T. Lesinski
J. Drut
A. Nogga
T. Duguet
C. Pethick
B. Friman
A. Schwenk
Backup slides
Microscopic
Kohn-Sham
Idea behind Kohn-Sham
DFT
Density Functional Theory
Walter Kohn,
Nobel Prize 1998
Ground state density of
interacting fermions in external
harmonic trap.
explicit functionals
Turn off the interaction, but change
external potential such that the
density remains that of the
interacting system. The additional
potential is the Kohn-Sham potential.
• local-density approximation,
orbital-based functionals
• functional depends only implicitly on
Picture from R. Furnstahl: http://trshare.triumf.ca/~schwenk/ECT/furnstahl1.pdf
gradient-corrections...
• most of currently used functionals
are of this type
the density
• cure deficiencies of explicit functionals
• more microscopic but computationally
more expensive
• current efforts in quantum chemistry
towards such functionals
Equation of state of symmetric nuclear matter
0
0
3
EM Vlow k + ci’s
EGM Vlow k + ci’s
EM Vlow k + PWA ci’s
EM SRG + ci’s
EGM SRG + ci’s
EM SRG + PWA ci’s
Vlow k NN from N LO (500 MeV)
!1
3NF fit to E3H and r4He "3NF = 2.0 fm
Energy/nucleon [MeV]
Energy/nucleon [MeV]
!5
NN + 3N
!10
!15
!20
!25
3rd order pp+hh
!1
" = 1.8 fm
!1
" = 2.8 fm
!1
" = 1.8 fm NN only
!1
" = 2.8 fm
!30
0.8
1.0
NN only
!5
!10
!15
!1
kF [fm ]
-1
-1
" = # = 2.0 fm
3NF fit to E3H and r4He
3rd order pp+hh
NN only
1.2
"3NF = 2.0 fm
1.4
1.6
!20
0.8
1.0
1.2
1.4
1.6
!1
kF [fm ]
KH, Bogner, Furnstahl, Schwenk, Nogga, PRC(R) 83, 031301 (2011)
uncertainty due to ci coupling uncertainties, interaction dependence and
RG scheme dependence comparable to cutoff dependence
Pairing gap in semi-magic nuclei including 3N forces
Three-body mass difference:
∆
(3)
(−1)N
(N ) =
[E(N + 1) − 2E(N ) + E(N − 1)]
2
repulsive 3N contributions
lead to suppression
of the pairing gap
Lesinski, KH, Duguet, Schwenk in preparation
Hierarchy of many-body contributions
neutron matter
nuclear matter
40
Energy/nucleon [MeV]
Energy/nucleon [MeV]
40
20
0
-20
Ekinetic
20
0
-20
-40
Ekinetic
-60
-40
0
0.05
0.1
-3
! [fm ]
0.15
0.05
0.1
0.2
0.15
-3
! [fm ]
0.25
0.3
• binding energy results from cancellations of much larger kinetic and potential
energy contributions
• chiral hierarchy of many-body terms preserved for considered density range
• cutoff dependence of natural size, consistent with chiral exp. parameter ∼ 1/3
Hierarchy of many-body contributions
neutron matter
nuclear matter
40
Energy/nucleon [MeV]
Energy/nucleon [MeV]
40
20
0
-20
-40
0
Ekinetic
ENN
Ekinetic + ENN
0.05
20
0
-20
-40
-60
0.1
-3
! [fm ]
0.15
0.05
Ekinetic
ENN
Ekinetic + ENN
0.1
0.2
0.15
-3
! [fm ]
0.25
0.3
• binding energy results from cancellations of much larger kinetic and potential
energy contributions
• chiral hierarchy of many-body terms preserved for considered density range
• cutoff dependence of natural size, consistent with chiral exp. parameter ∼ 1/3
Hierarchy of many-body contributions
neutron matter
nuclear matter
40
Energy/nucleon [MeV]
Energy/nucleon [MeV]
40
20
0
-20
-40
0
Ekinetic
ENN
E3N + 3N-NN
Etotal
0.05
20
0
-20
-40
-60
0.1
-3
! [fm ]
0.15
0.05
Ekinetic
ENN
E3N + 3N-NN
Etotal
0.1
0.2
0.15
-3
! [fm ]
0.25
0.3
• binding energy results from cancellations of much larger kinetic and potential
energy contributions
• chiral hierarchy of many-body terms preserved for considered density range
• cutoff dependence of natural size, consistent with chiral exp. parameter ∼ 1/3
SRG evolution of operators: Deuteron
Instructive test case: density operator in the deuteron
ψDλ |Uλ a†q aq Uλ† |ψDλ
"
2
10
(q = 4.5 fm−1 )
Any λ with no truncation
λ=6.0 fm−1 truncated
AV18
0
10
−1
λ=3.0 fm
truncated
λ=2.0 fm−1 truncated
(a+qaq)deuteron
integrand of
!
−2
10
−4
10
−6
10
Truncation at Λ=2.5 fm−1
after evolution
−8
10
0
1
2
−1
3
4
5
q [fm ]
Anderson, Bogner, Furnstahl, Perry, PRC 82, 054001 (2010)
• perfect invariance of momentum distribution function
• strong/slight evolution of short/long-distance operators
q) factorizes for k < λ and q ! λ : Uλ (k, q) ≈ Kλ (k)Qλ (q)
• Uλ (k,
!
"
FIG. 5. Decoupling in operator matrix elements is tested by calculating the momentum distribut
in the deuteron after evolving the AV18 potential to several different λ and then truncating t
Hamiltonian and evolved occupation operators (i.e., set them to zero above Λ = 2.5 fm−1 ).
⇒ ψλ |Oλ |ψλ
both operators. The integrand of the operator at q = 3.02 fm−1 begins as a sharp spi
corresponding to the original operator, but then flows out along the momentum axes to low
momentum. By the time the integrand reaches lower values of λ in the evolution nearly
of the strength in the expectation value is in the low-momentum region. The original sp
disappears as the wave function dependence at high momentum falls off.
As for the operator at q = 0.34 fm−1 , the strength does begin to flow out to some exte
but remains almost entirely in the low-momentum region. Once again, the display sc
has overemphasized the extent of the evolution in the values of the integrand. The sp
which remains at λ = 1.5 fm−1 actually contains ≈ 96% of the full expectation value. D
to the possibility of misinterpreting these plots on a linear display scale, we also include t
same plots with logarithmic display scale. These pictures show only the magnitude of t
integrands, but display nearly the full range of their values. Now it is conclusive that t
strength of the high momentum operator flows to low momentum, and the strength of t
low-momentum operator remains at low momentum for a low-energy state. We will see t
pattern repeat in the calculation of other operators in the next section.
factorizes into a low-momentum structure
O
and a universal high momentum part if the initial operator
only weakly couples low and high momenta
Alternative explanation of nuclear scaling!
Chiral 3N interaction as density-dependent two-body interaction
(1) calculate antisymmetrized 3N interaction
V3N
=
π
π
-
π
π
-
π
π
-
π
π
+
π
+
π
π
π
(2) construct effective density-dependent NN interaction
Basic idea:
Sum one particle over occupied
states in the Fermi sea
V3N
V3N
k3 σ3
(3) combine with free-space NN interaction
combinatorial factor c depends
on type of diagram!
V
=
V NN
+ 1/c
V3N
Properties of the effective interaction V 3N
General momentum dependence:
V 3N = V 3N (k, k! , P)
P/2 + k!
P/2 − k!
!
k,
k
P
-dependence
much
weaker
than
-dependence!
•
V3N
• neglect P-dependence, set P = 0
• matrix elements have the same form like free-space
P/2 + k
NN interaction matrix elements
• straightforward to include in existing many-body schemes
(1)
Efull
(1)
Eeff
=
=
+
+
VNN
V
+
V3N
Energy/nucleon [MeV]
20
P/2 − k
neutron matter
15
10
E(1)
NN
(1)
E NN+3N,full
E(1)
NN+3N,eff
5
0
0
0.05
0.10
[fm -3 ]
0.15
Properties of the effective interaction V 3N
(Λ3N = 2.0 fm−1 )
F
0.6
F
1.0
1.2
1.4
1.6
fm -1
fm -1
fm -1
fm -1
PNM
0.4
0.4
3
0.2
P0
0.15
0.1
3N (
3N (
, ) [fm]
F
=
=
=
=
0.2
0.05
0
, ) [fm]
F
0.3
3N (
S0
, ) [fm]
1
0.8
0.1
3N (
1
S0
F
F
F
F
1.0
1.2
1.4
1.6
fm -1
fm -1
fm -1
fm -1
1.5
SNM
1.0
0.5
P1
0.05
3
P2
SNM
0.2
0
0.0
-0.05
3
0.2
P2
F
0.15
F
F
0.1
3N (
3N (
, ) [fm]
2.0
=
=
=
=
3
0.4
0.1
, ) [fm]
2.5
PNM
P1
0.15
, ) [fm]
[fm -1 ]
3
0.2
2.5
, ) [fm]
2.0
3N (
1.5
F
=
=
=
=
1.0
1.2
1.4
1.6
fm -1
fm -1
fm -1
fm -1
0.2
, ) [fm]
1.0
0.2
0.05
F
F
F
F
0.1
=
=
=
=
1.0
1.2
1.4
1.6
fm -1
fm -1
fm -1
fm -1
3N (
0.5
0
P0
0.0
-0.05
0
3
0
0.0
-0.05
0.0
0.5
1.0
1.5
[fm -1 ]
2.0
2.5
0
0.5
1.0
1.5
[fm -1 ]
2.0
2.5
0.5
1.0
1.5
[fm -1 ]
2.0
2.5
Λ / Resolution dependence of nuclear forces
with high-energy probes:
quarks+gluons
cf. scale/scheme dependence
of parton distribution functions
Lattice QCD
Effective theory for NN, 3N, many-N interactions and
electroweak operators: resolution scale/Λ-dependent
Λchiral
momenta Q ~ λ-1 ~ mπ: chiral EFT - typical momenta in nuclei
nucleons interacting via pion exchanges and
shorter-range contact interactions
Λpionless
Q << mπ=140 MeV: pionless EFT
large scattering length physics and non-universal corrections
n
9Li
n
Neutron matter:
EOS (first order), Test of fixed-P approximation
(1)
Eeff
=
=
+
+
V NN
+
V
V̄ 3N
20
V
relative difference of
3N contributions only ~3%
Energy/nucleon [MeV]
(1)
Efull
15
10
E(1)
NN
(1)
E NN+3N,full
E(1)
NN+3N,eff
5
0
0
0.05
0.10
0.15
[fm -3 ]
KH and A.Schwenk arXiv:0911.0483, to appear in PRC
P-independent effective NN interaction is a very good approximation
tensive, because complete Gaussian integrations have to
be performed for each of the configurations sampled in
the random walk. The large range of values of x and X
required to obtain converged results, especially for 3 He,
require fairly large numbers of points; we used grids of
up to 96 and 80 points for x and X, respectively. We
also sum over all pairs instead of just pair 12.
The np and pp momentum distributions in 3 He, 4 He,
6
Li, and 8 Be nuclei are shown in Fig. 1 as functions of the
relative momentum q at fixed total pair momentum Q=0,
corresponding to nucleons moving back to back. The
statistical errors due to the Monte Carlo integration are
displayed only for the pp pairs; they are negligibly small
for the np pairs. The striking features seen in all cases
are: i) the momentum distribution of np pairs is much
larger than that of pp pairs for relative momenta in the
range 1.5–3.0 fm−1 , and ii) for the helium and lithium
isotopes the node in the pp momentum distribution is
absent in the np one, which instead exhibits a change of
slope at a characteristic value of p # 1.5 fm−1 . The nodal
structure is much less prominent in 8 Be. At small values of q the ratios of np to pp momentum distributions
taken from
Ciofi of
degli
53, 1689which
(1996)in
are closer
to those
npAtti,
to Simula
pp pairPRC
numbers,
pair and residual (A–2) system are in a relative S-wave.
In 3 He and 4 He with uncorrelated wave functions, 3/4 of
the np pairs are in deuteron-like T, S=0,1 states, while
Short-Range Correlations in nuclear systems
10
p! + p = Q = 0
10
p − p = 2q
6
6
He
He
6
Li
8
Be
!
5
!NN(q,Q=0) (fm )
3
4
4
10
3
10
2
np pairs
10
1
10
0
10
-1
pp pairs
10
0
1
2
3
4
q (fm )
Schiavilla et al. PRL 98, 132501 (2007)
FIG. 1: (Color online) The np (lines) and pp (symbols) momentum distributions in various nuclei as functions of the
relative momentum q at vanishing total pair momentum Q.
• scaling behavior of momentum distribution functions:
nA (p) ≈ CA nD (p)
5
-1
at large p
• explained by invoking dominance of two-body interactions and short-range
correlations in the wave function
• dominance of np pairs over pp pairs at large relative momenta and small
C.M momenta explained by short-range tensor forces
Unitary transformations from the SRG
• unitary transformation of initial Hamiltonian
Hs =
†
Us HUs
≡ T + Vs
with the flow parameter s
• differentiating wrt s
dHs
= [ηs , Hs ]
ds
with
• specifying ηs by a generator Gs
dUs †
ηs ≡
Us = −ηs†
ds
choose Gs = T : AV18 3 S1
(s = λ−1/2 )
ηs = [Gs , Hs ]
choice of Gs determines flow:
decoupling of low and high-momenta
• transformed wave functions and operators
|ψs ! = Us |ψ!
⇒ "ψ|O|ψ# = "ψs |Os |ψs #
Os = Us OUs†
• Us reduces dramatically coupling between low- and high momentum
components (SRC), calculation becomes (more) perturbative
• Us induces many-body terms and non-localities
SRG evolution of operators in nuclear matter
10
PRELIMINARY
<!(P=0,q)>np/<!(P=0,q)>nn
with operator
evolution
0.0001
<!(P=0,q)>np
PRELIMINARY
1e-06
no operator
evolution
-1
1e-08
1e-10
1.5
" = 1.8 fm
-1
" = 2.0 fm
-1
" = 2.5 fm
-1
" = 3.0 fm
2
3
2.5
-1
q [fm ]
3.5
4
with operator
evolution
8
6
no operator
evolution
4
-1
2
0
1.5
" = 1.8 fm
-1
" = 2.0 fm
-1
" = 2.5 fm
-1
" = 3.0 fm
2
3
2.5
-1
q [fm ]
3.5
• approximate invariance of distribution functions with evolved operator
• 3N operator contributions seem small (further investigations necessary)
• significant enhancement of np pairs over nn pairs due to tensor force
4
SRG evolution of operators in nuclear matter
10
PRELIMINARY
<!(P=0,q)>np/<!(P=0,q)>nn
with operator
evolution
0.0001
<!(P=0,q)>np
9
1e-06
no operator
evolution
-1
1e-08
1e-10
1.5
" = 1.8 fm
-1
" = 2.0 fm
-1
" = 2.5 fm
-1
" = 3.0 fm
2
3
2.5
-1
q [fm ]
PRELIMINARY
with tensor
interaction
8
7
6
5
no tensor
interaction
4
-1
" = 1.8 fm
-1
" = 2.0 fm
-1
" = 2.5 fm
-1
" = 3.0 fm
3
2
1
3.5
4
0
1.5
2
3
2.5
-1
q [fm ]
3.5
• approximate invariance of distribution functions with evolved operator
• 3N operator contributions seem small (further investigations necessary)
• significant enhancement of np pairs over nn pairs due to tensor force
4
the
7,8].
n of
ns at
a)
atios
4<
e we
stics
robe
RC,
NN
ratio
the
1999
elecThe
on a
onA
a
and 4 He and 1:18 & 0:02 for 56 Fe. Note that the 3 He yield
in Eq. (2) is also corrected for the beam energy difference
by the difference in the Mott cross sections. The corrected
3 He cross sections at the two energies agree within $ 3:5%
[8].
We calculated the radiative correction factors for the
xB < 2 using Sargsian’s upgraded
reaction A!e; e0 " at e’
code of Ref. [13] and the formalism of Mo and Tsai [14].
These factors change 10%–15% with xB for 1 < xB < 2.
However, their ratios, RArad , for 3 He to the other nuclei are
almost constant (within 2%–3%) for xB > 1:4. We applied
RArad in Eq. (2) event by event for 0:8 < xB < 2. Since there
q calculations at xB > 2, we
are no theoretical cross section
applied the value of RArad averaged over 1:4 < xB < 2 to the
entire 2 < xB < 3 range. Since the xB dependence of RArad
N the
for 4 He and 12 C are very small, this should not affect
ratio r of Eq. (2). For 56 Fe, due to the observed small slope
of RArad with xB , r!A; 3 He" can increase up to 4% at xB #
2:55. This was included in the systematic errors.
Figure 1 shows the resulting ratios integrated over 1:4 <
2
A!1
Q < 2:6 GeV2 . These cross section ratios (a) scale
iniindicates320,
that1476
NN SRCs
tially for 1:5 Subedi
< xB < 2,etwhich
al., Science
(2008)
Correlations in nuclear systems
b)
What is this vertex?
e’
k
e
k!
q
q = k − k!
p!1
p!2
A
N
N
A!2
ν = Ek − E k !
Q2 = −q 2
Q2
xB =
2mN ν
r( He/ He)
Higinbotham,
arXiv:1010.4433
pecFIGURE
1. The simple goal of short-range nucleon-nucleon correlation studies is to cleanly isolate diagram
b) from a).
aloUnfortunately,
there
a) are many other diagrams, including those with final-state interactions, that can produce the same final state as
3
dhetodiagram scientists
would like to isolate. If one could find kinematics that were dominated by diagram b) it would finally allow
2.5
used
electron scattering to provide new insights into the short-range part of the nucleon-nucleon potential.
SRC interpretation:
NN interaction can scatter
states with p1 , p2 ! kF
to intermediate states with
p , p ! k which are
knocked out by the photon
q
3
r( Fe/ He)
12
3
r( C/ He)
4
3
ning
2
ows
1.5
d to
1
ineFor A(e,e’p)
reactions, one can determine not only the energy and moment transferred, but also the energy and
4
b)
drift
momentum
of the knocked-out nucleon. The difference between the transferred
ies.
p!1 and detected energy and momentum
3
referred to as the missing energy, Emiss and missing momentum, pmiss
M,s to
1 , respectively. From the theoretical works on
tak- short-range
2
how
nucleon-nucleon correlations effects the momentum distribution
nucleus [6], it
p!2 of nucleons in the
!
!
ss in
clear one1must probe beyond the simple particle in an average potential motion of the nucleon in the nucleus of
F
1 2
ncep2
approximately
MeV/c in order to observe the effects of correlations.
3 He,
6 250
c)
With the construction of the Jefferson Lab Continuous Electron Beam Facility (CEBAF) [7], it was possible to
an’s
datahigh-luminosity
4
do
knock-out reactions in ideal quasi-elastic kinematics into the pmiss > 250 MeV/c region. In the
all C
3
3
56
early Jefferson Lab knock-out reaction proposals, such as E89-044 He(e,e’p)pn and He(e,e’p)d, these kinematics
2
were
as a argued as the key to cleanly observe the effects of short-range correlations. And while final results of the
LAS
experiments
were1 clearly
effected by
the
presence of correlations, the magnitude of the cross sections in the high
1.25 1.5 1.75
2
2.25 2.5 2.75
ross
missing
momentum
region was
dominated
by final-state interaction effects [8, 9]. Equally striking was the D(e,e’p)n
xB
2
2
Q <taken
2.6 GeV
data1.4
from<CLAS
at Q2 > 5 [GeV/c]2 in xB < 1 kinematics [10]. Here it was shown that meson-exchange currents,
FIG.interaction,
1. Weighted cross
section ratios [seeconfigurations
Eq. (2)] of (a) 4 He, mask cleanly probing nucleon-nucleons even at extremely high
final-state
12
56 and delta-isobar
3
Fe toet He
as a function
of xB for
Q2 >
(b) C, and (c)
Egiyan
al.
PRL
96,
1082501
(2006)
2
2 . The horizontal dashed lines indicate the NN (1:5 <
Q(2)in xB1:4<GeV
1 kinematics.
How to explain cross sections in terms of
low-momentum interactions?
the
7,8].
n of
ns at
a)
atios
4<
e we
stics
robe
RC,
NN
ratio
the
1999
elecThe
on a
onA
a
and 4 He and 1:18 & 0:02 for 56 Fe. Note that the 3 He yield
in Eq. (2) is also corrected for the beam energy difference
by the difference in the Mott cross sections. The corrected
3 He cross sections at the two energies agree within $ 3:5%
[8].
We calculated the radiative correction factors for the
xB < 2 using Sargsian’s upgraded
reaction A!e; e0 " at e’
code of Ref. [13] and the formalism of Mo and Tsai [14].
These factors change 10%–15% with xB for 1 < xB < 2.
However, their ratios, RArad , for 3 He to the other nuclei are
almost constant (within 2%–3%) for xB > 1:4. We applied
RArad in Eq. (2) event by event for 0:8 < xB < 2. Since there
q calculations at xB > 2, we
are no theoretical cross section
applied the value of RArad averaged over 1:4 < xB < 2 to the
entire 2 < xB < 3 range. Since the xB dependence of RArad
N the
for 4 He and 12 C are very small, this should not affect
ratio r of Eq. (2). For 56 Fe, due to the observed small slope
of RArad with xB , r!A; 3 He" can increase up to 4% at xB #
2:55. This was included in the systematic errors.
Figure 1 shows the resulting ratios integrated over 1:4 <
2
A!1
Q < 2:6 GeV2 . These cross section ratios (a) scale
iniindicates320,
that1476
NN SRCs
tially for 1:5 Subedi
< xB < 2,etwhich
al., Science
(2008)
Correlations in nuclear systems
b)
What is this vertex?
e’
k
e
k!
q
q = k − k!
p!1
p!2
A
N
N
A!2
ν = Ek − E k !
Q2 = −q 2
Q2
xB =
2mN ν
r( He/ He)
Higinbotham,
arXiv:1010.4433
pecFIGURE
1. The simple goal of short-range nucleon-nucleon correlation studies is to cleanly isolate diagram
b) from a).
aloUnfortunately,
there
a) are many other diagrams, including those with final-state interactions, that can produce the same final state as
3
dhetodiagram scientists
would like to isolate. If one could find kinematics that were dominated by diagram b) it would finally allow
2.5
used
electron scattering to provide new insights into the short-range part of the nucleon-nucleon potential.
SRC interpretation:
NN interaction can scatter
states with p1 , p2 ! kF
to intermediate states with
p , p ! k which are
knocked out by the photon
q
3
r( Fe/ He)
12
3
r( C/ He)
4
3
ning
2
ows
1.5
d to
1
ineFor A(e,e’p)
reactions, one can determine not only the energy and moment transferred, but also the energy and
4
b)
drift
momentum
of the knocked-out nucleon. The difference between the transferred
ies.
p!1 and detected energy and momentum
3
referred to as the missing energy, Emiss and missing momentum, pmiss
M,s to
1 , respectively. From the theoretical works on
tak- short-range
2
how
nucleon-nucleon correlations effects the momentum distribution
nucleus [6], it
p!2 of nucleons in the
!
!
ss in
clear one1must probe beyond the simple particle in an average potential motion of the nucleon in the nucleus of
F
1 2
ncep2
approximately
MeV/c in order to observe the effects of correlations.
3 He,
6 250
c)
With the construction of the Jefferson Lab Continuous Electron Beam Facility (CEBAF) [7], it was possible to
an’s
datahigh-luminosity
4
do
knock-out reactions in ideal quasi-elastic kinematics into the pmiss > 250 MeV/c region. In the
all C
3
3
56
early Jefferson Lab knock-out reaction proposals, such as E89-044 He(e,e’p)pn and He(e,e’p)d, these kinematics
2
were
as a argued as the key to cleanly observe the effects of short-range correlations. And while final results of the
LAS
experiments
were1 clearly
effected by
the
presence of correlations, the magnitude of the cross sections in the high
1.25 1.5 1.75
2
2.25 2.5 2.75
ross
missing
momentum
region was
dominated
by final-state interaction effects [8, 9]. Equally striking was the D(e,e’p)n
xB
2
2
Q <taken
2.6 GeV
data1.4
from<CLAS
at Q2 > 5 [GeV/c]2 in xB < 1 kinematics [10]. Here it was shown that meson-exchange currents,
FIG.interaction,
1. Weighted cross
section ratios [seeconfigurations
Eq. (2)] of (a) 4 He, mask cleanly probing nucleon-nucleons even at extremely high
final-state
12
56 and delta-isobar
3
Fe toet He
as a function
of xB for
Q2 >
(b) C, and (c)
Egiyan
al.
PRL
96,
1082501
(2006)
2
2 . The horizontal dashed lines indicate the NN (1:5 <
Q(2)in xB1:4<GeV
1 kinematics.
How to explain cross sections in terms of
low-momentum interactions?
Vertex depends on the resolution!
Neutron star radii
Mmax > 1.97 M!
use the constraints:
vs (ρ) =
3.0
36
log 10 P [dyne / cm 2 ]
2.5
M [M ]
2.0
Mmax
= 1.5
1 = 2.5
1 = 3.5
1 = 4.5
2 = 1.5
2 = 2.5
2 = 3.5
2 = 4.5
= 1
= 12 = 1.5
= 12 = 2.5
= 12 = 3.5
= 12 = 4.5
1.5
1
1.0
0.5
0
6
7
8
35
WFF1
WFF2
WFF3
AP4
AP3
MS1
MS3
GM3
ENG
!
dP/dε < c
PAL
GS1
GS2
PCL2
SQM1
SQM2
SQM3
PS
34
0
0
0
33
0
9
10
11
R [km]
12
13
14
15
14.2
14.4
14.6
log 10
14.8
15.0
15.2
[g / cm 3 ]
KH, Lattimer, Pethick, Schwenk, PRL 05, 161102 (2010)
• low-density part of EOS sets scale for allowed high-density extensions
• radius constraint after incorporating crust corrections: 10.5 − 13.5 km